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The development of a stochastic model for predicting championship squash performance McGarry, James T. 1993

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THE DEVELOPMENT OF A STOCHASTIC MODEL FOR PREDICTING CHAMPIONSHIP SQUASHPERFORMANCEbyJAMES TIMOTHY McGARRYB.Sc.(Hons), Liverpool John Moores University, 1988M.Sc., The University of Bradford, 1989A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTSFOR THE DEGREE OFMASTER OF PHYSICAL EDUCATIONinTHE FACULTY OF GRADUATE STUDIESSchool of Human KineticsWe accept this thesis as conforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1993© James Timothy McGarry, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my written(Signature) Department of^kflitA4n. ek.rkThe University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)ABSTRACTSport analysis provides objective data regarding individual or teamperformance in the widespread belief that the information available isapplicable to a future setting. Recent matches, often against differentopponents, of two competitors scheduled to meet next are frequently analyzedin some form for a priori match information. An underlying assumption isthat consistency in athletic performance exists, to the extent that theseanalyses duly inform the coaching process in preparation for the forthcomingcontest. The hypothesis of this study was that an elite player's match"profile" characterizes individual playing habits in squash and can be usedto accurately predict future sport performance. The data comprised thequarter-finals through final of the men's Canadian Open Championship, 1988.The hypothesis was tested through the development and application of astochastic Markov model for predicting future competition squash match playfrom past performance.The results, using two way chi-square analysis, revealed consistency inindividual athletic behavior between games (and also between matches) againstthe same opponent (p>.25), but not between matches against differentopponents (p<.25). It seems that a player's profile is dependent on theparticular interaction between two individuals, and that this pattern isreproducible within this interaction. Importantly, however, the interactionis unique to the two players and not generalizable to other opponents. Thissuggests that additional information from the opponent is required if themodel is to successfully predict future sport performance.The present model possesses no predictive validity beyond the scope of itsprevious observation(s), a restriction which unfortunately consigns the pastto forever repeat itself. The application of the model for theidentification of future optimal strategy has therefore little presentutility. The attempt to model competitive sport performance is nonetheless aworthy pursuit, not least because it directs attention to the criticalaspects of data which determine sport success. Further research isrecommended to investigate whether an underlying order which characterizesplayer performance can be established. If no such individual signature canbe identified in sport, the utility of commonly applying tactical strategyfrom previous observation(s) against different opponents must be questioned.iiiTABLE OF CONTENTSABSTRACT^  iiList of Tables^  viList of FiguresAcknowledgements^  xi1^INTRODUCTION ^11.1^Assumptions of the Study ^21.2^Research Hypothesis ^31.3^Limitations of the Study ^41.4^Delimitations of the Study ^51.5^Definition of Terms ^62^LITERATURE REVIEW^  102.1^Introduction to Stochastic Processes^  102.2^Analytic and Simulation Models^  132.3^Stochastic Models in Sport  142.3.1 Sport Scoring Structures  162.3.2 Optimal Strategies^  212.4^Sport Analysis  292.5^Decision-Making in Anticipatory Shot Response in Squash and Tennis ^ 312.6. Predicting Sport Performance^  333^METHOD^  353.1^Introduction^  353.2^Data Collection Procedure^  363.3^System Software  383.3.1 Data Collection Software  383.3.2 Data Analysis Software^  403.3.3 Stochastic Model of Competition Squash Play^  413.3.4 Operation of the Stochastic Model^  483.4^Validation and Reliability of the Instrument and Observer^ 503.4.1 Introduction^  503.4.2 Procedure  513.4.3 Measures of Reliability^  533.4.4 Intra-Rater and Inter-Rater Reliability^  553.5^Methodology^  594^RESULTS AND DISCUSSION^  604.1^Overview^  604.2^Two Way Chi-Square Analysis  624.3^Validity of the Playing Profiles^  644.4^Validity of the Stochastic Model  734.5^Violation of the Assumption of Stationarity^  804.6^Interaction of the Winner-Error Profiles  864.7^Identification of Optimal Strategies^  93ivTABLE OF CONTENTS cont.5^CONCLUSIONS^  102REFERENCES  104APPENDIX A^  158APPENDIX B  166APPENDIX C  167APPENDIX D^  168APPENDIX E  184APPENDIX F  198APPENDIX G^  218APPENDIX H  223VLIST OF TABLES3.1^Table of Shot and Outcome Event Codes for the Systematic Recordingof Squash Match Play^  1103.2^Table of Example Outcome Profiles for Four Players^  1103.3^Table of an Example Cumulative Probability Matrix Denoting PlayerX's Match Play Profile Against Player Y^  1113.4^Table of Example Rally Simulation from Empirical Data^ 1123.5^Table of Example Coded Rally Simulation from Empirical Data^ 1133.6 Table of Difference in Simulation Data When Running One Hundred andOne Thousand Simulations for Prediction of Norman versus Nancarrow 1133.7^Table of Randomly Selected Matches and Match Rallies forConstruction of the Training Video Excerpts^  1143.8 Table of Example Independent Observations Between Two Observers forFour Dependent Variables A Through D^  1143.9^Table of Kappa Coefficient and Strength of Agreement (Altman, 1991) 1143.10 Table of Grouping Classifications for Position and Shot^ 1153.11 Table of Total Agreement and Kappa Coefficients of Intra-RaterReliability for Grouped and Non-Grouped Dependent Variables^ 1153.12 Table of Total Agreement and Kappa Coefficients of Intra-RaterReliability for Non-Grouped Dependent Variables^  1153.13 Table of Total Agreement and Kappa Coefficients of Inter-RaterReliability for Grouped Dependent Variables^  1164.1^Table of Shot Conditions Collapsed for Two Way Chi-Square Analysis ^ 1174.2^Table of Frequency (Percent) Shot Responses and Two Way Chi-SquareAnalysis of Jahangir's Shot Response to a Drive Across Opponent ^ 1174.3^Table of Two Way Chi-Square Analysis for Martin and Jahangir's ShotResponse Profiles Between Games Against Each Other^ 1184.4^Table of Martin and Jahangir's Shot Frequencies for CollapsedLevels Between Games Against Each Other^  1194.5 Table of Two Way Chi-Square Analysis for Jansher and Dittmar's ShotResponse Profiles Between Games Against Each Other^ 1204.6^Table of Jansher and Dittmar's Shot Frequencies for CollapsedLevels Between Games Against Each Other^  121viLIST OF TABLES cont^4.7^Table of Two Way Chi-Square Analysis for Dittmar and Jahangir'sShot Response Profiles Between Games Against Each Other^ 122^4.8^Table of Dittmar and Jahangir's Shot Frequencies for CollapsedLevels Between Games Against Each Other^  1234.9^Table of Two Way Chi-Square Analysis for Shot Response and Winner-Error Profiles for Two Matches Between Jansher and Dittmar^ 1244.10 Table of Shot Frequencies for Collapsed Levels for Two MatchesBetween Dittmar and Jansher (1987 Final, 1988 Semi-Final) ^ 1254.11 Table of Two Way Chi-Square Analyses for Martin, Jansher, Jahangirand Dittmar Between Matches Against Different Opponents^ 1264.12 Table of Two Way Chi-Square Analyses for Different Players AgainstDifferent Opponents^  1274.13 Table of Validation: Replication of Empirical Data (Final) forRallies, Shots and Outcome Frequencies By Match and By Game forDittmar and Jahangir^  1294.14 Table of Validation: Comparison of Empirical (Semi-Final) andPredicted (Quarter-Final) Data for Rallies, Shots and OutcomeFrequencies By Match and By Game for Martin and Jahangir  1304.15 Table of Validation: Comparison of Empirical (Semi-Final) andPredicted (Quarter-Final) Data for Rallies, Shots and OutcomeFrequencies By Match and By Game for Jansher and Dittmar  1314.16 Table of Validation: Comparison of Empirical (Final) and Predicted(Quarter-Final) Data for Rallies, Shots and Outcome Frequencies ByMatch and By Game for Dittmar and Jahangir  1324.17 Table of Validation: Comparison of Empirical (Final) and Predicted(Semi-Final) Data for Rallies, Shots and Outcome Frequencies ByMatch and By Game for Dittmar and Jahangir^  1334.18 Table of Validation Data: Replication of the Empirical (Semi-Final)Playing Profiles of Dittmar and Jahangir in Forecasting (Final)Outcome^  1344.19 Table of Jahangir and Dittmar's Shot Frequencies from Empirical(Semi-Final) and Forecasted (Final) Data^  1354.20 Table of Validation Data: Comparison of Forecasted (Final) andEmpirical (Final) Playing^  1364.21 Table of Jahangir and Dittmar's Shot Frequencies from Forecasted(Final) and Empirical (Final) Data^  137vi iviiiLIST OF TABLES cont4.22 Table of Predicted Match Outcomes from Various Superimpositions ofthe Shot Response Profiles from the Semi-Finals and Final on theCorresponding Quarter-Final Playing Profiles  1384.23 Table of Predicted Match Outcomes from Various Superimpositions ofthe Winner-Error Profiles from the Quarter-Final and Semi-Finalson the Corresponding Semi-Finals and Final Playing Profiles  1394.24 Table of 1988 World Ranking and Simulated Player Rank Ordering forthe Previous (McGarry and Franks, 1993) and Reported StochasticModels from the Quarter-Final Data^  1404.25 Table of Predicted Match Wins for the Previous Stochastic Model(McGarry and Franks, 1993) from the Quarter-Final Data^ 1404.26 Table of Predicted Match Wins for the Reported Stochastic Modelfrom the Quarter-Final Data^  1404.27 Table of Discernible Difference Between the Previous and ReportedModel in Predicting Match Outcome Between Players from theQuarter-Final Data^  1414.28 Table Summary of Player's Winners and Errors and Computed Ratiosfrom Empirical Data^  1424.29 Table Summary of Player's Winners and Errors and Computed RatiosProjected from Empirical Data to Predicted Future Performance inthe Final  1424.30 Table of Jahangir's Quarter-Final Playing Profile^  1434.31 Table of Thorne's Quarter-Final Playing Profile  1444.32 Table of Effect of Optimal Strategy on Quarter-Final BetweenJahangir and Thorne^  1454.33 Table of Jahangir's Semi-Final Playing Profile^  1464.34 Table of Martin's Semi-Final Playing Profile  1474.35 Table of Effect of Optimal Strategy on Semi-Final Between Martinand Jahangir^  1484.36 Table of Jahangir's Final Playing Profile^  1494.37 Table of Dittmar's Final Playing Profile  1504.38 Table of Effect of Optimal Strategy on Final Between Dittmar andJahangir^  151LIST OF TABLES cont4.39 Table of Simulated (N=5) Effects of Optimal Came Strategy Used ByDittmar and Jahangir^  152ixLIST OF FIGURES3.1 Schematic Representation of a Squash Court^ 1533.2 Schematic Representation of Application Program Hierarchy^ 1543.3 File Representation of a Player's Match Profile^ 1553.4 Stochastic Model for Predicting Squash Match Play 1564.1 Observed Tournament Progression from the Quarter-Final ThroughFinal of the Men's Canadian Open Squash Championship,^1988^ 157XACKNOWLEDGEMENTSI would like to express my gratitude to all those who have assisted in thisinvestigation. Special thanks are extended to each of the committee membersDr. Ian Franks, (Human Kinetics), Dr. Robert Schutz, (Human Kinetics) and Dr.Hugh Dempster (Computer Science), for their generous expertise andconsiderable effort expended during the course of this research. Myappreciation is also offered to Mr. Jon Huberman, for his instructivecomments and careful analysis of the training video excerpts, and to allothers who have shown, and continue to show, an interest in my work.Finally, thanks to all those who offer genuine and lasting friendship.xi1 IntroductionA principal aim of the coaching process is to enhance athletic performance.The coach attempts to effect behavioral change in the athlete(s) by providingrelevant instruction in a conducive learning environment. Accurate andappropriate information is therefore essential if coaching effectiveness isto be maximized and optimal sport behavior attained.Sport analysis is beneficial to the coaching process: It promotes theobjective assessment of sport behavior and directs attention to the keyelements of performance for future practice. It is essentially, however, adescriptive analytic tool of past athletic behavior which is unfortunatebecause it is the next contest, often against a different opponent, which thecoach and athlete must prepare for. It is the preparation for a forthcomingcontest, rather than the post mortem analysis per se of a past conflict,which provides the focus for this investigation.Sport analysis typically allows the association of a particular event toearlier occurrences, or alternately, the retrospective prediction of suchevents from their antecedent(s). A stochastic analysis is an appropriatemathematical technique for modeling championship performance in squash andforecasting results from past athletic behavior. It lends itself to theprediction of future events from prior circumstance through inferring likelyresponse to a particular occurrence, or set of occurrences. These responsesare known to be likely from previous empirical data. An appealingapplication of a successful stochastic model is the objective identificationof prescriptive match strategies for use against a particular opponent.1Further, the efficacy of these adopted sport tactics can be subsequentlyevaluated.Not only will a stochastic model provide valuable information preceding aparticular sport contest, it will also offer important insight into the keydeterminants which affect the outcome of contests in general. This isimportant because it will focus attention to the critical aspects of datawhich discriminate sport success. An immediate application is the modelingof practice to these critical data to more fully fit a winning profile. Thepursuit, and subsequent release, of a successful stochastic model ofcompetition squash match play, is thus an important step towards theobjective of understanding and achieving optimal sport performance at thehighest level of competition.1.1 Assumptions of the StudySport analysis traditionally derives playing patterns from a match's outcomefor the explanation of observed athletic performance. Remedial action isprescribed on the basis of these findings, in the belief that a stableunderlying order exists at the elite level of sport performance, and,moreover, that this order is assumed to influence subsequent match behavior.An assumption of this study is that the playing "profiles" of elitecompetition squash match players do not significantly change over arelatively short period of time. If this is true, the empirical playingprofiles are generalizable to future opponents. Elite players are thereforeassumed to reliably produce an expected shot response to a variety of2conditions and opponents, and an observation of past performance ishypothesized to provide a signature to likely future athletic behavior.Each sport situation is complex and unique. What is the cause and extent ofthis observed uniqueness? The causal root may lie in sampling error, theremay be more chance in elite sport than we generally credit, or theinteraction between individuals may be so unique that we simply cannotextrapolate beyond the competing players we analyze. A test of this latterassertion would require that there be profile stability between two competingplayers over some time interval, in addition to simultaneously observingprofile instability against different opponents.1.2 Research HypothesisSimulation models are frequently used to derive approximate solutions toproblems which are often too complex to be solved by analytic means. Thesesimulations are modeled on well known systems, the entities of which can berigorously defined and the relationships between these entities explicitlystated. The behavior of the system can be investigated through simulationprocedure and the various effects of the complex interactions of the modelentities inspected.Simulation models can also be used to test theory (Lehman, 1977). In theseinstances, the system is represented by a theoretical model which purports toaccurately reflect the system under observation. If the model can predictfuture system behavior better than chance, the model is interpreted to3capture some fundamental characteristic of the system, which advances presentunderstanding. The focus of this research investigates whether a consistentunderlying order can be identified at the elite level of competition squashmatch play.The research hypothesis is that consistent patterns of play exist inchampionship performance, and that the individual match profiles are adeterminant feature of future athletic competition. The hypothesis is testedthrough observing the playing profiles from championship knockout tournamentdata and contesting the profiles from earlier rounds to forecast subsequentmatch outcome using a stochastic Markov model.1.3 Limitations of the StudyThe data source comprised seven knockout matches from the quarter-finalthrough final of the men's Canadian Open Squash Championship, 1988, all butone match being won by a game score 3-0. These game scores impact the datafor two reasons: A shorter game score generally lessens the amount of matchinformation which is available, and the data tend to overly bias the playingprofiles in favor of the match victor. More data, with respect to theinformation from a match, and more matches are therefore preferable forpredicting to a future contest, although the data must be contemporary.Preceding stages of a knockout tournament are ideal sources of recent apriori match information, especially if this data can be demonstrated to besuccessfully utilized in predicting successive match behavior.4A limitation of the study is imposed in the derivation of the playingprofiles. A shot response is assigned only from the preceding shot, and notfrom additional information which may be pertinent to the subsequentlyobserved behavioral event. Further, the findings of this study are notgeneralizable to players of a different playing standard or gender, nor arethey generalizable to different sports.1.4 Delimitations of the StudyThis study investigated whether patterns of future athletic behavior could bepredicted from a priori observation. A necessary delimitation of the studywas in the selection of the sport and the subsequent sample chosen from thisparticular sport. Squash was selected because this sport seemingly exhibitscertain playing patterns, typically repetitious long drives (especially tothe rear left side of the squash court), drop shots from a preceding boast,drop and lob shots from a preceding drop, and volleys to increase pressurefrom preceding drives and lobs. Elite male squash players were chosenbecause, if established behavioral responses exist in individual athleticperformance, they were considered most likely to be prevalent at this levelof play.Further delimitations of the study were the range of defined shot types andthe use of a first order Markov model for predicting competition squashperformance. (First order reflects that the transition between event statesin the Markov model is dependent upon the preceding state only.) The choicesof shot type and the model structure naturally determine the playing5profiles, and consequently the future behavior and predictions of thestochastic model.1.5 Definition of TermsGame - a sub-set of a sport contest, usually comprising a number of points.Match - a sport contest, usually comprising a number of games.Playing profile - a pattern of play which characterizes an individual'sathletic behavior. Playing profiles in this study comprise a player'sobserved shot response to the opponent's preceding shot, and the observedoutcome (winner or error type, let) associated with a player's shot.Stochastic - A stochastic model is conjectural, or random (i.e.,it is non-deterministic). A prescribed condition is thus likely to elicit a responsefrom a range of probable responses.Markov - the name given to a process which transits to the next event statein the model with dependence only upon its present state and not on paststates.Simulation - the operation of a theoretical model which purports to representsome observed phenomenon.6Theory - A theory possesses a dependency on empirical data, an explanation ofobserved behavior, a predictive capacity and, often, the use of hypothesizedconstructs which relate to the phenomenon under investigation (Lehman, 1977).Winner - A legal squash shot which is awarded a point since it fails toelicit a shot response from the opponent.Unconditional Winner - A type of winner deemed not to be a direct result ofthe opponent's last play. Unconditional winners are good shots, generallycharacterized by "nicks" or extremely tight shots which tend to "die"unexpectedly.Conditional Winner - A type of winner deemed to arise as a direct result ofthe opponent's last play. Conditional winners are generally considered to beany winner which is "up" and in "open play".Error - An illegal squash shot which, under American scoring, awards a pointto the opponent. Strokes awarded against a player are also considered to bean error on their part.Forced Error - A type of error deemed to arise as a direct result of theopponent's play. Forced errors are not considered to be primarily the faultof the player, rather to the credit of the opponent.Unforced Error - A type of error, usually technical, deemed to be unnecessaryand not as a direct result of the opponent's play. The player is expected tobe capable of returning a legitimate shot in such a situation.7Let - A neutral rally outcome assigned to an unresolved stoppage in play.Drive - A paced shot following a floor bounce which strikes the front walland finishes in the rear court without crossing the mid-court line. Thisdefinition is extended to accommodate "high" drives.XC-Drive - A rapid shot following a floor bounce which strikes the front walland finishes in the rear court crossing the mid-court line. This definitionis extended to accommodate "high" cross-court drives.Boast - A shot following a floor bounce which strikes another (rear and/orside wall) before striking the front wall and finishes in the front courtwithout crossing the mid-court line.XC-Boast - A shot following a floor bounce which strikes another (rear and/orside wall) before striking the front wall and finishes in the front courtcrossing the mid-court line.Drop - A shot following a floor bounce which strikes the front wall andfinishes in the front court without crossing the mid-court line.XC-Drop - A shot following a floor bounce which strikes the front wall andfinishes in the front court crossing the mid-court line.Volley(S) - A shot pre-empting a floor bounce which strikes the front walland finishes in the front court without crossing the mid-court line.8XC-Volley(S) - A shot pre-empting a floor bounce which strikes the front walland finishes in the front court crossing the mid-court line.Volley(L) - A shot pre-empting a floor bounce which strikes the front walland finishes in the rear court without crossing the mid-court line.XC-Volley(L) - A shot pre-empting a floor bounce which strikes the front walland finishes in the rear court crossing the mid-court line.Volley-Boast - A shot pre-empting a floor bounce which strikes another (rearand/or side wall) before striking the front wall and finishes in the frontcourt without crossing the mid-court line.XC-Volley-Boast - A shot pre-empting a floor bounce which strikes another(rear and/or side wall) before striking the front wall and finishes in thefront court crossing the mid-court line.Lob - A shot following a floor bounce which strikes the front wall, usuallyfinishing in the rear court, without crossing the mid-court line. Lobs areusually employed to buy time and are generally played from the front of thecourt.XC-Lob - A shot following a floor bounce which strikes the front wall,usually finishing in the rear court, crossing the mid-court line. Lobs areusually employed to buy time and are generally played from the front of thecourt.92 LITERATURE REVIEW2.1 Introduction to Stochastic Processes"A system S moves in a random way. At every instant, it is in one stateEi among a set of states E and we are interested in the sequence ofstates the system is in during its evolution."Girault (1966,p51)A random model, or function, is used by mathematicians to denote an abstractscheme of a probable nature which represents real behavior (Girault, 1966).If the model is an adequate representation of the phenomenon underobservation, the outcome of certain situations can be forecast. A stochasticprocess is a random function whose argument is time, which enables the studyof the relationship between the past, present and future. It is preciselythis relationship which is of interest to stochastic analysis.A stochastic process can be formally defined as any family of randomvariables {xt,tET}, where x t is the observation at time t, and T is the timerange involved (Doob, 1953). A stochastic model is thus a mathematicalrepresentation of a system which specifies the probable occurrence of aseries, or set, of events. The likelihood of the next event at any giveninstance can therefore be predicted. Stochastic processes may be of either adiscrete or a continuous nature. A discrete process is such that a discreteevent(s) determines the subsequent outcome, whereas a continuous stochasticprocess is concerned with the rate of change of the dependent variable. Themajority of applications of stochastic processes in sport are discrete.1 0A stochastic process may be independent or dependent upon past events. Anindependent process is such that the event state is not reliant on any of itspreceding events, whereas a dependent process is contingent on one or more ofits antecedents. The latter possess Markov properties (Kenyon & Schutz,1970), although the usual interpretation is to consider the next state in theMarkov chain rather than the previous one(s). The usual interpretation of aMarkov process is retained here.A Markov process formally considers the future to be dependent upon thepresent state only and to be independent of the past. Thus, if the historyof the system to the present is known, the present state summarizes theinformation necessary to know its future behavior (Girault, 1966). Astochastic process X1,X2, ... XN is a Markov process if for any a1,a2 ... aNPr{Xn+ 1^I(Xn=an),(Xn_i=an+1^=an-1)---(Xl=a1)/ = Pr {Xn+1 =an+1I(Xn=an)}^(2.1)(modified from Kenyon & Schutz, 1970)where n and N are integers and 1 n < N.The probability of transition to the next state (Xn+ 1) is contingent onlyupon the present state (X n ) and not upon its preceding states (Xn _l, Xn _2 , -..x1)-The probability of traversing event states within a stochastic process isreferred to as the transition probability. A stochastic process isconsidered stationary if the transition probabilities remain stable over11time. Consider a stochastic process X1,X2, ... X N, then the process isstationary if for any a1,a2, time t and integer hPr{Xn+1,t+l'an+11(Xn,t'an)} = Pr {Xn+1,t+1+h'an+1 1( Xn,t+h=an )}^(2.2)(modified from Kenyon & Schutz, 1970)where n and N are integers and 15 n < N.The probability of transition to the next state (Xn+l, t+1) is contingent onlyupon the present state (Xn, t) and independent of the value t and h.Girault (1966) credited the recent emergence of operational research as ascientific discipline with renewing interest in Markov models throughrevealing new applications. Markov models are general enough to representnumerous situations, while simple enough to allow a detailed analysis. Theiradvantage is their inferential capacity to forecast events from a certain setof circumstances. Their limitation lies in the necessary constraints imposedon the model by assumptions, accepted either for model simplicity or as abest guess to unknown quantities. Notwithstanding these restrictions, aMarkov analysis affords useful, and often otherwise unattainable, insight ofmodel behavior. Indeed, Markov models have subsequently proved a profitableanalytic tool in the study of sport, and frequently yield results which areclose approximations to empirical findings (Schutz, 1980).122.2 Analytic and Simulation ModelsThe derivation of a model to represent theory, or an abstraction of the keyelements of theory, is a useful research exercise. It promotes unambiguoustheory because a model requires the identification and quantification of asystem's entities, and explicit definitions of any relationships which existbetween them (Schutz, 1970a). The consequences of theory can be exploredthrough investigating model behavior in carefully controlled settings, andthe theory can be tested through comparing the forecasted results toempirical data. The model, however, does not explain behavior.The behavior of a system can be modeled through analytic or simulationprocedures. The term simulation in this context refers only to computersimulation although other types of simulation are possible. An analyticmodel provides a lasting solution to a particular set of initial conditions,whereas a simulation model does not. A simulation model is characterized byrepetitious analyses, which subsequently yield approximate mean and variancemeasures of expected model behavior. Analytic models are preferable, butsimulation models are particularly useful when investigating the behavior ofa complex system where the derivation of a mathematical model is daunting, oreven impossible. Future reference to models in this study are synonymouswith simulation models rather than analytic models unless otherwise stated.The technique of simulation affords investigation and exploration of aprocess through the examination of model behavior. A computer is a usefulinstrument for simulation, and its technological advance is a primary causeof the recent proliferation of computer simulation as a means of13investigative research. Paul (1991), for instance, reported computersimulation to be an increasingly widespread and popular technique inoperations research.A computer simulation can be either discrete or continuous, and event or timeoriented (i.e.,the transition step between states in the process is triggeredby the next event or time increment respectively). Whatever itscharacteristic, a computer simulation permits the consequential analysis of amodel's behavior without disturbance to the real process which the model issupposed to represent. A useful function of simulation is the compression orexpansion of time to provide a macroscopic or microscopic view of thesystem's behavior. Simulation techniques possess a number of otheradvantages, and are generally cost efficient, low risk solutions whichpredict the dynamic behavior of complex systems.2.3 Stochastic Models in SportStochastic sport models require the a priori probabilities (p, q=1-p) of aplayer, or team's, response to a usually finite number of events. Theresponse might be a point win at a specific stage of the game, or a decisionstrategy with regard to a particular situation. Values of p and q are oftentheoretically assigned, but can equally be determined from the ratio ofobserved occurrences (sample space) to the number of possible occurrences(finite, or discrete, space). The stronger player, or team, are thought topossess the higher assigned probability of winning any particular pointwithin the scoring structure (p>.5), although Cover (1989) maintained that14the stronger player is ill defined unless the duration of the particularsport contest is known.The assumptions of stationarity and independence are generally accepted andmay not be unreasonable. Pollard (1987), cited previous research in tennisto conclude that, since these assumptions cannot be rejected on statisticalgrounds, they are appropriate for modeling the practical situation. It isplausible that the transition probabilities do change over time, as a resultof factors such as physical and mental fatigue, and perhaps psychologicalvariables which arise from previous play, despite the assumption ofindependence between points. With the exception of Pollard, the existence,or otherwise, of transient probabilities, and their degree of divergence fromthe assumed conditions of stationarity and independence remain largelyunexplored.Carter and Crews (1974) reported the averaging of probabilities over anentire tennis match agree quite well with the results of actual matches, afinding replicated by McGarry and Franks (1993) in squash. Pollard (1987)contended that, for tennis, the averaging assumption of Carter and Crews(1974) is restrictive, and produces inaccurate results for the cases of twoplayers each with strong serves. The inaccuracy is likely to be a functionof the sport and its adopted scoring structure. A break in a service game inelite tennis is likely to result in a favorable set outcome, particularly ina contest involving two strong servers. In elite squash, however, serviceassumes less importance in a game, especially in American scoring, and abreak in service is common and frequently countered with a subsequent breakof service by the opponent. The reduction in probability of winning from15service in squash, coupled with the lengthening of points awarded beforeassigning game outcome, probably offsets the restrictions cited by Pollard(1987) for tennis.Despite the constraints necessarily imposed on any model by its assumptions,Markov models have successfully been used in the theoretical investigation ofsport. Primary applications lie in the analysis of sport scoring systems andin the identification of optimal decision making strategies.2.3.1 Sport Scoring StructuresThe investigation of sport scoring mechanisms evaluate both the efficacy andefficiency of the scoring system. While scoring systems evolved with thenatural development of the individual sport, it is nonetheless reasonable toexpect a scoring system to maximize the likelihood of success for thestronger player or team, and so lessen the role of chance in influencing thefinal result. Moreover, this expectation is tethered by the realisticconstraint of match duration, which reflects such factors as competitorfatigue and spectator appeal, when considering a scoring system's merit.The analysis of scoring systems has, for the most part, been applied toracket sports. A typical example is the examination of the theoreticalconsequences of rule scoring changes. Croucher (1982), reported the effectsof the introduction of the "tie-break" in tennis, rather than the previous"win by two games" rule, to decrease the expected match duration, but to alsoreward the weaker player. Increasing the efficiency of the scoring system in16this instance compromises its efficacy, and so increases the role of chancein determining the outcome of a set.The merits of various tennis scoring methods have been compared and moreeffective scoring systems proposed (Schutz, 1970b; Pollard, 1987). Schutz(1970b) found the discarding of score information in tennis to be detrimentalto maximizing the better player's chance of match success. Point scoreinformation is discarded after a game is assigned, and game score informationsimilarly renounced when a set is awarded. It was reasonably concluded thatthe retention of all tennis score information provides a fairer assessment ofthe player's true ability. Cover (1989) suggested that increasing theduration, or information, of a match does not necessarily favor the strongerplayer (where p>.5), and cited mathematical instances where monotonic growthof match success is not achieved as duration increases. These findings,however, generally apply to contests which use an irregular scoringstructure, rather than those sports (such as tennis) which accumulate scoresof even size.Pollard (1987) proposed a new tennis scoring system which, while similar tothe present system of the best of three, or five, tie-breaker sets, typicallypossesses a considerably smaller variance of match duration, as measured bythe number of points played. Further, the reported existence of a negativedistribution, which particularly arises in those cases in which both playershave strong serve-volley characteristics, was considered a desirable propertysince it is these conditions which classically cause match delay problems.The results of the proposed scoring structure are matches of more predictable17duration, which would notably ease the present scheduling problems of tennistournaments.The relative merits of the International and American scoring systems insquash have also been analyzed (Schutz & Kinsey, 1977; Clarke & Norman,1979). International rules contest the best of five games where each gameconsists of the first to nine points, or ten if selected by the non-servingplayer from a score of eight-eight. A point can only be won by the servingplayer; if the non-serving player wins the rally a "hand out" is awarded andthis player now serves. The American rules also contest the best of fivegames, but each game comprises the first to fifteen points and a point isawarded irrespective of service. A hand out still occurs if the non-servingplayer wins the rally. A game score of first to fifteen can be selected bythe non-server from a tied score of thirteen, or fifteen, seventeen oreighteen from a tied score of fourteen, providing the score has not alreadypassed "thirteen all".Schutz and Kinsey (1977) and Clarke and Norman (1979) both assumedindependence between squash points and stationarity of the theoreticallyassigned probabilities. The analyses showed the International scoring systembetter promotes the winning chances of the stronger player (p>.5), but thatgame duration generally increases, as measured by the expected number ofrallies. The increase in expected game duration is true except under theextreme condition where one player is much stronger than the opponent. Thisobservation is not surprising, given that a minimum of nine rallies areneeded to win a game to International scoring but at least fifteen ralliesare required in American squash. Moreover, an International game requires a18theoretically infinite number of rallies to determine the game victor(because of the win on service rule), whereas the American scoring structureimposes a maximum of thirty five rallies. The larger extremes which arepossible in International squash explain the greater expected variance ingame duration reported by Schutz and Kinsey (1977) and Clarke and Norman(1979). The implication is that very short or, more likely, very long gamesarise in International squash.An interesting consequence is that, since the stronger player is more likelyto win a game in International squash, the matches are generally shorter interms of the expected number of games (Clarke & Norman, 1979). It may thenbe that International squash is a more efficient scoring system, if the matchoutcome favours the stronger player in a shorter contest period. WhileInternational squash scoring generally increases the duration of the game(Schutz & Kinsey, 1977), it remains unclear at present whether this alsotranslates to an increase in match duration because of the expected fewergames in a match (Clarke & Norman, 1979). Assessing match duration by theexpected number of rallies, rather than the expected number of games, wouldresolve this present uncertainty.Pfeifer and Deutsch (1981) similarly used a Markov model, on assumptions ofindependence and stationarity, to investigate the effects of a proposed newscoring system in volleyball (FIN-30). The scoring procedure in atraditional volleyball game requires that the winning team reach a minimum offifteen points, or thereafter a two point advantage. A point is won onlyfrom service, and service is relinquished when the non-serving team wins therally. The proposed scoring structure would require the winning team to19reach a minimum of thirty points, or thereafter a two point advantage. Inthe proposed scoring system a point is awarded irrespective of servicealthough service protocol is retained. A match usually comprises the best offive games in either scoring system.The model showed the expected duration of a contest to be dependent on boththe scoring structure and the preassigned probabilities of winning a rallyfrom service. If the probability of winning from service is low (p<0.45),the FIN-30 shortens the expected duration of the game and increases theweaker team's chance of success. Conversely, as the probability of winningfrom service increases (p>0.45), the FIN-30 lengthens the expected durationof the contest and enhances the stronger team's chance of success. As papproaches 0.45 few differences in expected game duration and winning chancesexist between the two scoring structures. An interesting property of theFIN-30 is that it is more likely to change the results of a particularcontest, although this change is unbiased and equally likely to favor eitherteam. A general finding was that the FIN-30 tends to reduce the variance ofthe expected game duration through lessening the possibility of very shortand very long volleyball games. The effect is a more consistent matchduration for the same reason as that cited earlier in squash.A noteworthy feature of Pfeifer and Deutsch's (1981) study is that the meritsof the different volleyball scoring systems depend on the level of play.Elite volleyball tends to reduce the probability of winning from servicebecause of the receiving team's increased ability to execute a predeterminedoffensive play from a defensive setting. Service, however, assumesincreasing importance at lower standards of play as the probability of20winning increases. The present scoring system is therefore favored for elitecompetition, while the FIN-30 is better suited to lower playing standards.This suggests that a volleyball tournament should perhaps account for thelevel of play when selecting a particular scoring structure. A similarfinding has been found in squash, where service assumes greater importance,as measured by winning probability, at lower levels of squash competition(Hughes, 1985; McGarry, 1988).A related example of the use of mathematical models in sport is Clarke's(1988) identification of optimal batting rates in one day test cricket. Animportant application of Clarke's (1988) work is the equitable assessment ofa batting team's target score if play should be interrupted. The need for afair assessment policy is evident from the recent World Cup Cricket semi-final between England and South Africa, March 1992. South Africa (batting)required 22 runs off the remaining 13 balls for victory. The imminent onsetof rain forced an extrapolation of the target score, leaving South Africarequiring 21 runs off the final, single ball ("Sport in Brief", 1992);clearly a nonsense figure. This recent example demonstrably portrays thedilemma which some sport scoring systems face, despite the existence of aviable remedy through appropriate mathematical procedure.2.3.2 Optimal StrategiesBaseball has proved a popular sport for the statistical analysis andevaluation of offensive athletic performance, in part due to the culturalinterest and wealth of easily available data generated from America's21national sport. Present standardized baseball statistics comprise frequencycounts, or percentages, of observed discrete behaviors and purport to providea measure of offensive baseball performance. Since these statistics provedistinctly inferior estimators of team run production and weak evaluators ofindividual performance, a number of statistical models have been proposed tomore equitably assess the offensive performance of individual players (seeBennett & Flueck, 1983). In an attempt to evaluate twelve of the great homerun hitters of all time, Albert (1992), using Bayesian analysis, proposed aquadratic log-linear model to accommodate the traditional rise and fall in aplayer's performance over a career. An advantage of this approach overprevious models is that it assumed "true" home run ability to peak, positedto occur in midst career, rather than to remain constant over a performancelifetime. Further, the model was not found to be unduly influenced byaberrant observations (outliers) which arise from sampling error. It isinteresting to note the results of Albert's model provided a differentranking order from the one derived through more traditional means.The discrete nature of baseball is ideally suited to Markovian analysis(Bellman, 1977). Trueman (1977) reported a simplified Markov model for therepresentation of a baseball game which, despite a tendency to overestimate ateam's run score over nine innings by ten to fifteen percent, is useful forcomparative analysis and investigating game strategy. The results from themodel suggest the offensive tactic of the "steal" to be generally favorable,although its usefulness is dependent upon both the player and the tacticalobjective of whether to attempt to increase the probability of a single runscore, or to try to increase the expected number of runs before the thirdout. The offensive tactic of the "sacrifice" and the defensive tactic of22"walking" a batter, however, prove inadvisable. The sacrifice is a generallypoor strategy which, under certain conditions, is always unsuccessful, andthe walking of a batter has little recommendation, except perhaps for walkingthe eighth batter to expose the weakest hitter (Trueman, 1976). A futureextension of this model could be its utility for player evaluation throughanalysis of individual contribution to team performance.The advantage of using empirical data in the model is the practical, ratherthan theoretical, application of the model's findings to the real situation.Identifying an optimal strategy from empirical data is preferential toidentifying from theoretical data which may not adequately describe the realsystem. The model's results must be interpreted, however, in light of itsassumptions and limitations which plainly affect system behavior. Afundamental criticism of Trueman's (1977) model is its failure to replicatethe observed data which drives it.The analysis of optimal batting order has been tackled using both Monte Carlosimulation (Freeze, 1974) and Markov modeling properties (Peterson, Jr.,1977; Trueman, 1977). These researchers reached similar conclusions andshowed the present policy of placing hitters with a high frequency ofreaching the bases at the top of the batting order, followed by the powerhitters and lastly the weak hitters to produce near optimum results. Thedifference in optimizing the line ups in accordance to individual specificswas generally considered insignificant (Freeze, 1974; Peterson, Jr., 1977;Trueman, 1977).23Racket sports have similarly been subject to considerable Markovian analysis.Ap Simon (1951, 1957) was amongst the first to investigate the advantage ofbeing awarded service at the start of the game in squash, and also amongstthe first to attempt to derive a mathematical solution for evaluating theprobability of winning a game at any stage from the probability of winning agiven rally. Watson (1970) also investigated the effect of the sport scoringstructure on the probability of winning the game, and the advantage of firstservice for a range of probabilities of winning a given rally. One scoringstructure (table tennis) consisted of the first player to nine points with apoint awarded independent of service, the other structure (International, orEnglish, squash) consisted of the first player to nine points with a pointawarded only if won from service. It has been shown by Ap Simon (1951) andWatson (1970), amongst others, that the probability of winning a game fromfirst service is .53, if equal players are matched. This translates to a .51probability of winning the match. The players in this instance are deemed tobe equal if each has a .50 probability of winning any particular rallyirrespective of service. The effects of winning from first service differ ofcourse according to the preassigned probabilities of winning the rally(Schutz & Kinsey, 1977; Clarke & Norman, 1979).Renick (1976) investigated the optimal decision making strategy from a tiedscore option of thirteen or fourteen in (North) American squash. This wasachieved through extending the Markov model used by Watson to account for theprobabilities of winning a game from the probabilities of winning a rallywhen either serving or receiving, which may reasonably be expected to differ.Renick (1977) developed a subsequent model to identify the optimal decisionmaking strategy in International squash, which uses a different scoring24system, and erroneously concluded that no generality between the two scoringsystems exists if a small number of points (one to five) are involved.Clarke (1979) showed that generality between the two scoring systems doesexist, and that the optimal tie point strategy in one system is directlytransferrable to the other.Interest in the probabilistic modeling of squash has continued, as evidencedin more recent papers by Alexander, McClements and Simmons (1988) and Wright(1988). Wright (1988) extended the present modeling approach yet further toinclude the service court (right or left), as well as the probability ofwinning a rally from service and receipt of service. The rationale for theinclusion of the service court was that many players are "better" (ineffecting a winning rally outcome) at returning from the forehand court thanthe backhand court. This may, or may not, be true. Wright (1988) concludedthat it is not inconceivable that a player's sub-optimal selection of theirbetter court, which may be dependent more on the opponent's capacities thantheir own, could reduce the player's probability of winning by a substantialamount. This conclusion is theoretically sound but remains unsubstantiatedin practice and warrants further investigation from empirical data. King andBaker's (1979) reference to the "principles of optimal adaptation toindividual strengths" would suggest that a player's better court may alreadybe used, and, if so, need not be accounted for in the modeling of competitivesquash.Hannan (1976) used a Markov modeling approach to analyze the differentserving strategies used in tennis (strong-strong, strong-weak, weak-weak) inan attempt to identify the optimal mix for maximizing performance. The25optimal strategy is dependent upon the assigned probabilities of a successfulservice and return of service to both type of serve, and thus dependent uponthe server and opponent. A parallel study (King & Baker, 1979) found littleevidence to support the traditional strong serve-weak serve strategy, ratherthat the optimal strategy rests on the particular opponent. Norman (1985)used dynamic programming techniques to derive optimal serving strategies foreach of the three serve combinations listed above, and found the favorablestrategy to be dependent upon the ratio of the probability of placing astrong serve in bounds and subsequently winning the point to the probabilityof placing a weak serve in bounds and subsequently winning the point. Theoptimization of strategy is important to match success. Hannan (1976) notedthat a minor advantage to winning each point leads to an escalated likelihoodof winning a game, or match, and that this probability amplifies wheneverp>.5. This observation, which results from combinatorial probability,clearly demonstrates the importance of effecting an increase in theprobability of winning each point.The often necessary assumption of stationarity is limiting since transitionprobabilities are likely to change with match duration; a limitation which isespecially applicable to sports which discard information by virtue of theirscoring system. This is best understood by an example: A player can win asquash match under American scoring, say, 15-12; 15-9; 3-13; 4-15; 15-11.The match winner thus wins fifty two points and the loser sixty points which,assuming stationarity, means the loser is more likely to win any particularpoint. In a model assuming stationarity, the losing player, in thisinstance, is more likely to win the match. This begs the question of apoint's importance at a particular stage of the game. Does a point increase26in worth as the player's chance of winning, or losing, the game increasesand, if so, should a player attach competitive effort to the relativeimportance of a particular point? This question is contentious, althoughsubjective observation suggests the successful players often win the criticalpoints. This view is supported by Pollard (1985) who cited evidence fromsquash of successful players being capable of winning the most importantrallies.The concept of "point importance" in tennis, and an attempt at itsquantification, was first raised by Morris (1977). A point's importance isdefined to be the difference between two conditional probabilities: Theprobability that the server wins the game given that he/she wins the point,minus the probability that the server wins the game given that he/she losesthe point. Since the receiver's probabilities are the complements of theserver's, a point is equally important to either player. Pollard (1985) andCroucher (1986) extended Morris' idea to determine the importance of a pointat a particular stage of the game in squash and tennis respectively.Present studies have served to focus attention on particular aspects of asport contest. Tennis point information, for instance, is not consideredequivalent because of the scoring nature of tennis. Information from a gameis discarded when a game is won and game information is discarded when a setis awarded. A parallel situation exists for squash and badminton. The ideaof point importance is pertinent to the sport participant. A tacticalimplication is to play most aggressively on the "important" points whiletrying to tax the opponent on the less important points. This would suggesta need for intensity training, a view offered for squash by Pollard (1985).27Notwithstanding their academic appeal, these studies seem to have had littleimpact in practice, perhaps through failing to relay their findings to awider audience. This is unfortunate because some studies have forciblydemonstrated how sports performers fail to use a particular situation totheir maximum advantage. Morrison (1976) used a Poisson process to determinethe optimal time to pull the goalie in ice hockey, a common tactic which,from Morisson's findings, is usually employed too late in the contest tomaximize potential gain. Janssen and Schutz (1985) similarly cited a sportobservation from American football of a team who failed to maximize theirchance of success through employing a sub-optimal decision making strategy.These sport examples exist, despite the fact that such optimal decisions havebeen identified and could be taken a priori to the situation.A modeling approach is an informative analytic tool which can benefit thesport situation. Present models have generally been developed away from thesport setting using theoretical data to draw conclusions to a genericsituation, for instance, to identify the optimal decision making strategy ata particular stage of a sport contest. If sport performance, however, is tobe optimized, there is a further requirement to develop models which areapplicable to a particular sport situation. Sport data are thereforenecessary, which requires the systematic observation of sport contests andfrom which representative models can be developed. Sport analysis willtherefore likely adopt an increasingly important role in the development offuture sports models. An important caveat is that the theoretical modeldictates the data to be collected, rather than the data collection processitself determine the model's future development.282.4 Sport Analysis"Sport by its very nature is not entirely predictable. There is,however, considerable structure to most sports such that a carefulacquisition and evaluation of the observable details associated with thegame can lead to meaningful trends analysis."Franks, Goodman and Miller (1983, p383)Sport notation analysis is used to reinforce desired athletic behavior and,where necessary, prescribe appropriate remedial action. This requires theobjective collection of data for the examination and inference of trends orpatterns of play. Sport analysis has received increased attention in recentyears, partly because computer analytic procedures now afford the easycollection, storage and analysis of large amounts of sequential data fromvarious sport contests (Franks, Wilson and Goodman, 1987). While matchanalysis has been applied to various sports, this literature scope isrestricted to racket sports only.The first, comprehensive racket sport analysis system was produced in tennisin 1970, and recently updated in a presentation by Downey (1992). The firstanalysis system for squash was subsequently developed by Sanderson and Way(1977), later to be successfully computerized by Hughes (1984) withmodifications to permit the collection of match play data at court side.Sharp in 1986 developed a squash analysis system using a touch sensitivekeypad for ease of data entry. McGarry (1988) further advanced this workthrough the development of three dimensional graphics for representation ofselected patterns of play. The graphics could be interactively rotated for29the best view, and the software later extended to incorporate two squashprofiles in a single screen image (Hughes & McGarry, 1991).An objective analysis of sport generally increases both the accuracy andaptness of the feedback relayed to the athlete(s). Nonetheless, sportanalysis is presently constrained in its informational use because itdescribes and explains past observations but does not infer likely futureoutcome. The analysis therefore offers hindsight but not foresight to thecoaching process. This is unfortunate since it is invariably the nextcontest, often against a different opponent, which the coach and athlete mustprepare for. The focus of this study is to determine whether sport analysisprovides an appropriate signature of athletic performance for application toa future setting.It is often the case that certain events lead to meaningful behavioraloccurrences, such as a strike on goal in soccer, or a winner in tennis orsquash. While it is useful to identify such predictors of sport performance,present analysis frequently fails to isolate these important event sequences.Eom and Schutz (1992) investigated the sequential dependencies which arise involleyball and, using log-linear analysis, found statistical dependencybetween successive behavioral events. Recent developments in statisticaltheory and commercial computer applications afford easier analysis of suchevent sequences, and provide an analytic tool for the investigation ofunderlying probability sport structures. The identification of stableunderlying structures which exist over a period of time would imply thereliable projection of future sport performance. The result would be to30advance sport notation from a descriptive methodology to a prescriptiveapplication.2.5 Decision -Making in Anticipatory Shot Response in Squash and TennisThe anticipatory priming by the defending player of the next shot to beplayed by the opponent is an integral feature of competitive racket play.While the strategy is risky because of the negative consequences which followan incorrect prediction, the strategy is often necessary because of timepressure. The player therefore seeks to identify advance cues from which toprime a suitable response and so reduce the subsequent reaction time from thestimulus of shot execution. Further, accomplished players predict theintentions of their opponents at a level better than chance (Buckolz,Prapavesis & Fairs, 1988).Buckolz et al (1988) reported that a priori information, used to successfullyforecast the intentions of the opponent, is derived from two primary sourceswhich they referred to as "context" and "body language". The context cuesinclude situational variables such as knowledge of the opponent's strength,weaknesses and preferences, position of the players on court andenvironmental conditions. Body language cues refer to the "propheticpotential incorporated in the movements of one's adversary", and includestance, visual focus and particular racket preparation. The contextual cuesare reported to be normally available first and may, or may not, be latercombined with the following body language cues. Buckolz et al (1988) foundevidence for the existence of body language cues to accurately predict the31passing shot type in tennis and, furthermore, that advanced players aregenerally better able to predict these outcomes than intermediate players.Sarrazin and Alain, through modeling procedures, have pioneered investigativeresearch regarding the decision making process for the selection of the shotresponse of the defending player in squash (Sarrazin, Lacombe, Alain andJoly, 1983; Sarrazin, Alain and Lacombe, 1986; Alain and Sarrazin, 1990).The defending player reputedly uses a variety of information sources forformulating subjective probabilities of the opponent's expected shot. Theseare: (1)the respective court positions of the players, (2)play habits of theattacking player, (3)the attacking player's ability to accurately place theball in the intended location, (4)the angle of the attacking player relativeto the front wall, (5)the player's motion when striking the ball and (6)theball's flight trajectory. Sarrazin et al (1983) report that while all theinformation need not necessarily be utilized for a single prediction, therespective court positions of the players is used most of the time, or inconjunction with the other sources. Their model further suggested thecognitive strategy of a player's performance is influenced by a variety offactors, such as the experience of the player, the time pressure, theexpected shot responses of the opponent, the performer's confidence in theirprediction and the mistakes caused by incorrect predictions. These findingssuggest that a player's shot response is thus contingent on tangible criteriawhich generically describe performance in squash.322.6 Predicting Sport PerformanceIt is the scientist's task to describe and explain observed phenomena, topredict future observation and to relay their findings back to the scientificand social community. Sport analysis affords description and explanation ofobserved sport behavior, but does not readily allow the prediction of futureevents. If, however, the prediction of sport performance is attainable to adegree beyond that of chance, then scientific knowledge is extended becausethe governing behavioral patterns which are important determinants of futureperformance have been identified. A successful mathematical model wouldindicate that the selected model entities and their correspondingrelationships accurately model reality.Naumienko and Naumienko (1986) are conspicuous in their attempt to modelcompetitive performance (of judo) from past data. While their work fallsshort of predicting future performance from past analysis (a past performanceis re-simulated from the respective past data), they reported a simulationsystem capable of achieving this objective. They also indicated the possibleapplication of the model for optimal decision making strategy to minimizefailure, or maximize desired outcome, and to introduce user interactionthrough gaming simulation. In this fashion the simulation model could beextended for further exploration of the underlying characteristics of judocompetition.It is logical for sport analysis to provide authentic data for theconstruction of a mathematical model. The model may then be rigorouslytested for predictive capability to ascertain whether the underlying33assumption(s) and/or theory(ies) are tenable. This is important: If themodel can be verified to that of the real system then our understanding ofthe nature of competitive sport is advanced; if it cannot, the descriptiveinferences derived from sport analysis are still true, but are notgeneralizable to future sport behavior. A recent study by McGarry and Franks(1993) found evidence of stability in playing patterns in elite squashplayers when competing against the same opponent, but no evidence ofstability in playing profiles when competing against different opponents.Consistency in competition squash play therefore exists, but this isseemingly dependent upon the player-player interaction. The implication isthat the inferences of sport analysis can be extended to future sportbehavior only if the conditions which relate to the analysis remain constant.A computer simulation of a stochastic model of sport, based on genuinecompetition data, may prove invaluable in furthering insight into underlyingtheory, identifying optimal strategy and assisting athletic performance inspecific competition. Surprisingly little research has been performed tocombine sport analysis and mathematical modeling although a successfulimplementation of the two could prove an important advance on our currentunderstanding of, and preparation for, competitive sport.343 METHOD3.1 IntroductionSquash is a competitive game contested between two opponents. A match isplayed to the best of five games, to either the International (English) orAmerican scoring system, the latter having recently gained favour inInternational competition. The 1988 Men's Canadian Open Squash Championshipanalyzed in this study was contested to American scoring. A game, underAmerican scoring, consists of the first player to reach fifteen points wherea point is awarded to the player who wins the rally irrespective of serviceService at the start of the match is decided by the spin of a racket, andfrom there is granted to the winner of the previous rally. The winner of agame begins service at the start of the next. Tie-break options exist butare not described since they are not accounted for in the stochastic model.A basic tactical objective of squash is to control the T-position (Fig.3.1)since this affords dominance of the court (Sports Illustrated Squash, 1971).The stratagem is to push one's opponent out of position, force weak balls andinduce error (McKenzie, 1986). It is characteristic of elite squash matchplay to find much of the play away from the T-position, often to the sidewalls and to the rear of the court (McGarry, 1988).INSERT FIG. 3.1 HERE353.2 Data Collection ProcedureSquash match behavior was recorded for the objective quantification ofathletic performance. This was achieved by the division of the squash courtinto a number of position cells and the identification of a range of definedshot types and outcomes which describe observed behavior. These discretebehaviours were recorded in sequence of occurrence according to the listedrally structure:Server, Position, Shot(Serve), Position, Shot, ^ , Position, OutcomeThe process was repeated rally by rally for the entire match.PlayerBoth players were uniquely identified and coded by the observer at thebeginning of the data collection process, together with other fileinformation. The server was then only identified at the beginning of eachrally according to the above rally structure.PositionThe squash court surface was divided into a four by four position matrixcoded alphabetically "a" through "p" (Fig.3.1). These positions were, ifnecessary, collapsed in subsequent analysis to a two by two representation ofthe squash court, coded P1 through P4 (where P1=i,j,m,n; P2=a,b,e,f;P3=c,d,g,h; and P4=k,l,o,p). The four by four division of the squash court,and the further collapsing to a two by two matrix, was retained from anearlier study (McGarry, 1988).36ShotThe range of shots observed in a squash match were classified to six distinctcategories: serve, drive, boast, drop, volley and lob. A seventh shotcategory, volley-boast, was also included to retain consistency with earlierdata (McGarry, 1988). Shot condition with respect to straight or cross-court, and/or long or short characteristics, was automatically assignedthrough software, yielding a range of fifteen possible shot behaviours (Table3.1). The shot condition was assigned through examination of positional data(i.e.,from where the shot was played to where the shot falls). The shotcondition does not apply to the serve.INSERT TABLE 3.1 HEREOutcomeA rally outcome is generally classified as a winner, error or let. Aprevious study (McGarry & Franks, 1993) attempted to model squash match playusing these classifications but recommended defining the winner and erroroutcomes further, in an attempt to better discriminate between playingpatterns. Winners were thus classified in this study as unconditional orconditional, and errors as unforced or forced (Table 3.1). An unconditionalwinner is awarded as a result of an outstanding shot action with noreasonable expectation of a shot response by any opponent, whereas aconditional winner is deemed to be contingent on the particular opponent'splay. Similarly, an unforced error is considered to be committed without duecause, but a forced error to be a direct result of the opponent's preceding37athletic behavior. A let is a neutral outcome and is an unambiguousobservation on the part of the investigator because it is awarded by thematch referee. Strokes awarded against a player were considered to be anerror on the player's part, usually unforced.3.3 System SoftwareThe system software was developed in the programming language Turbo-Pascalusing an IBM compatible personal computer. The software is hierarchical instructure and all applications - data collection, analysis, presentation andsimulation software - are invoked from a parent program to which allapplications return after program execution (Fig. 3.2). A genericdescription of the application software is detailed below.INSERT FIG. 3.2 HERE3.3.1 Data Collection SoftwareThe data collection program allows the user to systematically record athleticbehavior of competition squash match play. The software prompts the userthrough menu selection to enter data according to the rally structure listedearlier. The data entry process is repeated rally by rally until the end of38data collection is reached. This is denoted by the user through entering the<Esc> (Escape) key.The initial intention was to write the coded data to disc as it is collected,and so negate the adverse effects of a possible system failure, due to say aloss of power. This notion, however, is contrary to the requirement for anon-line edit facility, which is necessary if erroneous data entries are to becorrected during the data collection process. This is because the datashould be accepted by the user before being committed to a permanent record.A compromise was to write the rally data to file following a suitable(optional) lag: The first rally was committed to disc as the end of thethird rally was recorded, the second was written at the end of the fourthrally and so on.This solution failed to prevent data loss in the event of unexpected systemfailure because an unclosed file in Turbo-Pascal will not permit completerecovery of the data. For this reason a temporary file write, where the usermay elect to save the data collected thus far, and an automatic backupfacility were also included. The temporary file write can be invoked by theuser at any time, and saves all the rallies collected so far to that instantin time. The automatic backup writes the collected data to file in blocks offive rallies (optional). Thus the data is automatically saved after everyfifth rally. In the worst case of data collection using the automatic backupfacility, all complete rallies except for the last four entries arerecoverable; in the best case all recorded rallies can be retrieved. Thisfunctionality insures against the onerous task of repeating data collectionin the unfortunate circumstance of system failure.393.3.2 Data Analysis SoftwareThe analysis software produces probability distributions (playing profiles)from data collection for execution of the mathematical model. The data fileis analyzed and shot distribution counts with respect to position andpreceding shot are established for each player. Shot distributionfrequencies with respect to position are produced for possible match analysisand presentation purposes only. Shot distribution frequencies with respectto the opponent's preceding shot are produced for use by the stochasticmodel. The resultant shot distribution matrix (Fig.3.3) is written to ananalysis file. The analysis files are additive and can be collated oninstruction which permits an average playing profile to be established.INSERT FIG. 3.3 HEREThe data analysis files comprise four profiles within a single structure(Fig.3.3): Firstly, player shot and outcome distribution with respect toposition; secondly, player shot response to the opponent's preceding shot;thirdly, player outcome with respect to the player's shot; and fourthly,opponent outcome with respect to the opponent's shot.403.3.3 Stochastic Model of Competition Squash Match PlayA Markov model considers the likelihood of transition to a subsequent stateto be dependent only upon the present state in which the process resides andindependent of past states. The stochastic model used in this investigation(Fig.3.4) exhibits Markov properties and is a refinement of a previous model(McGarry & Franks, 1993). An assumption of the model is that a player's shotresponse is dependent only upon the opponent's preceding shot.INSERT FIG. 3.4 HEREThe process initially resides in the state Shot=S since the first shot of acontest must be a serve. The process then advances to the next state whichdetermines whether the ball is in or out of play. (The transition betweenstates within the model is based on a simulated random call and thecontesting playing profiles.) If the ball is in play ("Ball-In"), theprocess continues to the next state and the receiver's shot response isassigned from A (drive) through N (cross-court lob). The process repeatsrecursively until a shot is assigned out of play whereupon the processtransits to either the "Ball-In/Out" or "Ball-Out" state. The Ball-Out stateis terminal and automatically awards the outcome of unconditional winner,unforced error or let. This outcome state mimics the behavior of theprevious model reported by McGarry and Franks (1993). In the Ball-In/Outstate the outcome is contingent, in part, upon the opponent's profile. Thus,the process either confirms the outcome of conditional winner or unforced41error, or returns to the Ball-In state for continuation of the rally. Theoutcome of a rally denotes the start of the next until the contest ends whena winning score is reached.The model differs from the previous model through obliging play to continuein certain instances, if conflict regarding the number of conditionalwinners, or forced errors, exists between competing players. The concept isbest illustrated with an example. Consider two matches, one consisting ofplayer A versus player B and the other consisting of player X versus playerY. These matches yield four match play profiles, the outcomes of which arelisted in Table 3.2. For simplification purposes, the outcomes onlyrepresent a particular shot type or, alternately, a summation of all shottypes. Example observations associated with these outcomes are listed inparentheses.INSERT TABLE 3.2 HEREThe initial transit to a conditional winner or forced error from a precedingshot state is schematically described above (Fig.3.4). (An example of howthe model functions using the playing profiles is given in Section 3.3.4,Operation of the Stochastic Model.) What follows next is a description ofhow the simulation determines whether to award or revoke these conditionalassignments.42Consider contesting profiles A versus X from the empirical data in Table 3.2.A is simulated to strike conditional winners against X because of his/herobserved profile against B. The simulation must somehow account for whetherX permits many conditional winners against him/her. The profile whichcontains this is that of Y, in other words, As opponent's previous opponent.A similar situation arises for the assignment, or otherwise, of forcederrors. The requirement to inspect X's history of allowing conditionalwinners, or forcing error, necessitates that X carries this informationwithin his/her own playing profile. The number of let outcomes which X isawarded from Y are not included in this information since they do notdirectly affect the result of the contest.The relationship, if any, which exists with respect to the number ofconditional winners awarded to A against X is ambiguous. A ratio isdesirable because absolutes depend upon the size of the data source. Moreconditional winners, for instance, are likely if a player wins with a gamescore 3-2 than if the same player wins 3-0. Ratios also allow for thepossibility of adding playing profiles to represent a competitor's "average"playing profile. An additional requirement of any relationship for modelingconditional winners and forced errors is that, if two profiles are to besimulated from their empirical data, the behavior of the model shouldreplicate the behavior of the previous model. In other words, anyconditional winners or forced errors awarded to A against B in the simulationshould be granted because these observations actually occurred. It isimperative then, that the ratio for modeling the interaction between outcomeprofiles of two competing players, does not adversely affect the modeled43behavior when contesting players against whom the two profiles were initiallyobserved.It seems logical that the ratio which determines whether or not a conditionalwinner is awarded must be predominantly influenced by the opponent's profile.Does the opponent usually allow many conditional winners against him/her?The number of conditional winners previously observed against the opponent,is therefore included in the ratio's numerator. The denominator term is moreproblematic. The number of shots could be used but is not recommended sincethe initial assignment of conditional winner in the simulation is indeed afunction of this observation. Another possible term for the denominator,which is used in the simulation, is the total number of winners awardedagainst the opponent. The ratio is thus simply the proportion of observedwinners previously struck against the opponent which were conditional, and,likewise, for the forced error ratio the proportion of previously observederrors which were forced. This is considered to provide some measure of theopponent's strength or weakness. It is assumed that few conditional winnerswould be observed against a strong player because of a high capacity forretrieval, while many conditional winners would be expected against a weakplayer because of a low retrieval capacity. The frequency of observedconditional winners and forced errors, however, is contingent on the abilityof the opponent also.Let Rw (p) = observed conditional winner ratio for any particular playerand Rq (p ) = observed forced error ratio for any particular player44then Rw (p) = W(p)/(W(p)-1-14(p)) and Rq( p) = q(p)/(Q ( p)+q ( p))where P denotes the player.Similarly, let Rw 'm = observed conditional winner ratio against anyparticular opponentand Rg '( 0 ) = observed forced error ratio against anyparticular opponentthen Rw ( 0 ) 1 = w( 0 )'/(W( 0 )'+w( 0 )') and Rq ( 0 )' = q( 0 )'/(Q( 0 ).+q(0)')where 0 denotes the opponent.Consider player A versus opponent B and A is assigned a conditional winner.We must inspect B's profile to determine whether he/she allows manyconditional winners against him/her? (This is equivalent to inspecting B'sprevious opponent's profile, in this case A).From Table 3.2. the conditional winner ratio is:Rw(B)' = wW/ (WB'+wW ) = 8 /16 = .50Thus, .50 of all winners against B are conditional. However, since A versusB is a simulation from the original data, all assigned conditional winnersshould be awarded. One solution is to multiply the fraction by itscomplement which assures a ratio of 1.00, which denotes certainty, whenevertwo players are simulated to compete from their original data. Thelikelihood of awarding a conditional winner is thus the product of theproportion of conditional winners against the opponent (Rw ( 0 )') and the45inverse proportion of conditional winners awarded to the player (1/R w (p)). Asimilar relationship exists for the assignment of forced errors. Theproducts of these ratios are termed the conditional winner coefficient andthe forced error coefficient respectively.Let w* = conditional winner coefficientand q* = forced error coefficientthen wpo * = Rw (0)'/Rw (p) = [w0 1 /(W0'+wo')]/[wp/(Wp+wp)]= [wo'/(Wo'+wo')]*[(Wp+wp)/wp]   (3.1)and^ciP0 * = Rq(0)'/Rq(P) = [(40'/ ( Q0 .1- c10' )] / [ qP/ ( QPI-c1P )]= No'/(Q0 . +clo')]*[(Q0- qp)/qPi (3.2)where PO denotes player versus opponent.Consider player A versus opponent B and A is assigned a conditional winner.Does B historically allow many conditional winners against him/her? We mustinspect B's history, against A, to determine whether B allowed manyconditional winners. We therefore compute the conditional winner coefficientfor A against B.From (3.1) wAB* = [ww/(ww+ww)]*[(wA+wA )/wA] = [8/(8+8)]*[(8+8)/8] = 1.0046As expected, every conditional winner by A is awarded against B.47Now consider player A versus opponent X and A is assigned a conditionalwinner. Does X historically allow many conditional winners against him/her?We must inspect X's history, against Y, to determine whether X allowed manyconditional winners.From (3.1) wAX* = [wx'/(Wx'+wx . )]*[(WA+wA)/wA] = [7/(11+7)]*[(8+8)/8] = .78The ratio denotes that A will be granted a conditional assignment against Xwith a probability .78. (The ratio is considered a probability since itsrange lies from zero through one. This is because ratios greater than oneare pared to one.) This implies that X will successfully revoke .22 of Asconditional winners first awarded in the simulation.Similarly, consider player X versus opponent Y and X commits a forced error.Does Y historically force many errors from his/her opponent? We must inspectY's history, against X, to determine whether Y forces many errors in his/heropponent.From (3.2)^qxy * = [qy'/(Qy'+qy')]*[(Qx+qx)/qx] = [4/(6+4)]*[(6+4)/4] = 1.00As expected, every forced error committed by X against Y is awarded.Now consider B versus Y and B commits a forced error. Does Y historicallyforce many errors from his/her opponent? We must inspect Y's history againstX to determine whether B should commit many forced errors against Y, or, moreappropriately, whether Y forces many errors.48From (3.2)^qBy * = [qy '/(Qy '+qy ')]*[(QB+qB)/qB ] = [4/(6+4)]*[(10+3)/3] = 1.73The forced error coefficient in this example is 1.73 and, because aprobability of 1.00 denotes certainty, it is evident that any forced error ofB from Y will be awarded. However, because the forced errors in thesimulation stem from B and not Y, B is not expected to incur more forcederrors than in the initial observation. In other words, Y is not allowed toinduce proportionally more forced errors in B than initially observed againstA. Similarly, nor is a player awarded more conditional winners than observedin their initial profile either. These restrictions, which necessarilyrequire the initial observation before a subsequent outcome can be simulated,are considered a limitation of the present model.3.3.4 Operation of the Stochastic ModelThe playing profile is converted, in Random Access Memory (RAM), to acumulative probability matrix for use by the stochastic model. Thecumulation of probabilities allow the assignment of the generated randomnumber, range r (05.r<l), to a particular shot response within the playingprofile. The cumulative probability matrix therefore depicts the likelihoodof a particular shot response or outcome to any preceding condition. Anexample of the cumulative probabilities for shot response selection followinga serve (shot=S), drive (shot=A) and cross-court drop (shot=F) are enclosedin parentheses (Table 3.3). Contesting the matrices of two competing playersusing random number sequences generated by the stochastic simulation allowsfor the prediction of a rally. An example of a rally simulation is providedin Table 3.4. If the service box is known (right or left), the position data(Pi through P4) can be automatically assigned and a rally reconstructed inits entirety. If the right service box is used in the above example, thenthe rally is coded as detailed in Table 3.5. The prediction of rally outcomeis extended to predict match outcome by repeating the process according tothe scoring rules of squash.INSERT TABLE 3.3, 3.4, AND 3.5 HEREThe playing profiles were contested in a single simulation where a simulationconsisted of one thousand runs, each of which began with a different startingseed within the random number stream. A random number stream is a sequenceof quasi-random numbers produced according to a predefined computeralgorithm, and the starting seed refers to the starting place within thisrandom number sequence. (The numbers are quasi-random because a number isdeterministically produced from its predecessor.) One thousand runs waschosen because this provides an acceptable degree of stability in the finalmatch prediction (standard error5.020 for p=q=.5). This was confirmed onfinding closer agreement between predictions from repetitious runs of onethousand using different starting seeds than between corresponding runs ofone hundred (Table 3.6). The player initially identified as server wasalways awarded service at the onset of each simulation run and care was takento match the server to the observed data.49INSERT TABLE 3.6 HERE3.4 Validation and Reliability of the Instrument and Observer3.4.1 IntroductionThe rationale for the development and use of systematic observationinstruments for the objective evaluation of sport performance is theprovision of accurate data to assist future decision making. This isdesirable because an increase in the information accuracy enhances thecoaching process through more aptly directing future athletic practice. Thedevelopment of a measuring instrument for the collection of sport data raisespsychometric issues, particularly those of reliability and validity. Datareliability considers whether the data is reproducible and data validityaddresses whether the data is an accurate representation of the dependentvariable(s) the instrument is being used to measure. Intra- and inter-raterreliability relate to the coherence of the data within and between observersrespectively. Intra-rater reliability measures the degree of dataconsistency over time, and inter-rater reliability provides a measure of dataobjectivity.It is important for sport analysis, and the derivation of a stochastic model,that the observed data are both reliable and objective, if appropriate andaccurate information regarding sport performance is to be attained.50Reliability is necessary if the instrument and observer are to be used in asystematic manner and credible comparisons between the data are to be drawn.Objectivity is necessary if the instrument is to provide accurateinformation, independent of observer bias. Unreliable or invalid data wouldabrogate subsequent model predictions because the results could not berepeated from the same observation source, on a separate occasion by eitherthe same or an independent observer.3.4.2 ProcedureSixteen position cells, fifteen shot types and five outcome conditions wereidentified for recording competitive athletic behavior in squash. Theseentities and their corresponding definitions were initially agreed upon withan expert and a preliminary study undertaken to assess intra- and inter-reliability coefficients of both the observer and the instrument.The original data source consisted of video recordings of four quarter-finalmatches, two semi-final matches and a final match from the men's CanadianOpen Squash Championship, 1988. The score progression of each match wasrecorded for reference purposes, which allowed the number of rallies within acontest to be identified. Three matches from seven were randomly chosen andtwenty rallies from each of these three matches randomly selected forconstruction of the "training" excerpts. The randomly selected matches, andthe subsequently randomly selected rallies from these matches which formedthe training excerpts, are detailed in Table 3.7. These excerpts comprised51the sample data used in the analysis of intra- and inter-rater reliabilitymeasures.INSERT TABLE 3.7 HEREThe random selections were made without replacement using the computer randomnumber generator and a recursive algorithm. A random selection procedure wasadopted since this was considered a fair representation of the typical dataavailable from the complete data source. The data were checked to ensure theexcerpts contained the full range of possible position, shot and outcomedata.The experimental design comprised a principal observer and an expertobserver, the latter being sophisticated in provincial competition squashmatch play and refereeing. The definitions of position, shot and outcomewere verbally agreed upon between the two observers during a visual exampleof the original data source (not the training excerpts). The subsequentwritten definitions were also agreed upon and were considered to be thearbitrary guide to which both observers adhered during the data collectionprocess.The principal observer analyzed the three video excerpts in their entiretyand repeated the process after six weeks. The expert observer analyzed thefirst video excerpt in its entirety and the rally outcomes only from theother two training excerpts. Intra-rater reliability measures for the52principal observer were computed from analysis of the three trainingexcerpts, and inter-rater reliability measures computed from analysis of thefirst video excerpt only and the combined rally outcome data of the threetraining excerpts.3.4.3 Measures of ReliabilityReliability measures attempt to quantify the amount of coherence between twoobservations. A common reliability measure is the total percent agreement(House, House & Campbell, 1981) which simply computes the ratio of observeragreement to the number of observations from which these agreements arise. Amajor criticism of this elementary computation is that it does not accountfor chance observation, and so over-estimates the true degree of coherencebetween two observations. The kappa coefficient (k) is an alternativereliability measure which does account for chance agreement, and consequentlyis the much preferred statistic for measures of reliability (Altman, 1991).An example of the total percent agreement and the Kappa computation is givenbelow (Table 3.8).INSERT TABLE 3.8 HEREThe total percent agreement is 62% ([18+10+15+191/100). There are twoinherent weaknesses to this approach: it does not account for where in thetable the agreement occurs, nor does it account for agreement which occurs as53a result of chance. The expected frequency of a cell in a frequency table,assuming no association, is the product of the corresponding column total andthe corresponding row total, divided by the grand total. The expectedfrequencies along the diagonal in Table 3.8 are thus:A: 35*22/100 = 7.70B: 16*27/100 = 4.32C: 22*27/100 = 5.94D: 27*24/100 = 6.48Total:^24.44Hence 24.44 observations, or 24.44% agreement, is expected from chance alone.How much better than chance is the observed agreement of 62%? This can beexpressed as a proportion of the possible scope for doing better than chance,thus:Kappa k = Observation - Chance =^62 - 24.44 = .497 Total Agreement - Chance^100 - 24.44A kappa value of 1.00 indicates perfect agreement, .00 shows no agreementbetter than chance and a negative kappa, which is unlikely, shows worse thanchance agreement. Generally accepted kappa values for behavioral observationstudies are listed in Table 3.9. This study considered k.80 to be anacceptable level of reliability.54INSERT TABLE 3.9 HEREWhile the kappa statistic accounts for chance agreement, it does not accountfor the degree of disagreement since it treats all observationaldisagreements equally. In instances where the categories are ordered, aweighted kappa, which treats disagreements further from the diagonal moreseriously than those closer to the diagonal, is applicable. In the exampleabove (Table 3.8), frequency disagreements in cell AC would thus beproportionately weighted heavier than those in cell AB, if computing aweighted kappa.Limitations of the kappa and weighted kappa coefficients are that thecomputation is dependent upon both the proportion of subjects in eachcategory (prevalence), and also on the number of categories (Altman, 1991).This leads to difficulty in comparing kappa values between studies where theprevalence of categories differ. Notwithstanding these limitations, theweighted kappa was used in this investigation as the most appropriatestatistic for intra- and inter-reliability measures.3.4.4 Intra-Rater and Inter-Rater ReliabilityThe nature of squash determines that the shot data are disproportionatelydistributed across position and preceding shot. This, coupled with thenumber of position (sixteen) and shot (fifteen) variables necessitated the55collapsing of position and shot data into appropriate classification for thecomputation of the kappa coefficient (Table 3.10). This is because zerocolumn and/or zero row totals are not permitted in the computation of thekappa statistic.INSERT TABLE 3.10 HEREIt is expected that most of the discrepancies between observations in theanalysis of the position data will likely have arisen between contiguouspositions on court (Fig.3.1). The grouping of position data, according tothe above classification, ensures that such discrepancies are carried intothe kappa computation and not lost as a function of the initial datatreatment. The grouping of shot type coincides with the data recorded by theobserver, with the exception of the volley-boast which is coded as a separateentity. Since the shot condition of straight/cross-court and short/long isautomatically assigned by software as a function of its associated position,any such discrepancies which may exist within the shot groupings are directlyattributable to their corresponding position data, and so have already beenaccounted for. Rally outcome are not grouped since the conditionalassignments of winners and errors are an important discriminator of the data.The number of observations of position and shot behavior in the abovegroupings is sufficient to compute kappa from a single training excerpt, butthis is not so for the outcome conditions because there are more cells in theobservation matrix than there are observations. The data from the three56training excerpts were therefore collated. Intra-reliability measures werecomputed for the position, shot and outcome observations under both groupedand non-grouped classification for the three separate and combined trainingexcerpts (Tables 3.11. and 3.12).INSERT TABLE 3.11 AND 3.12 HEREThe results (Tables 3.11 and 3.12) show an acceptable intra-rater reliabilityfor all position, shot and outcome conditions. (The non-grouped datacoefficients support the findings of high intra-reliability for the groupeddata and indicate these results are not unduly influenced as a function ofthe data treatment.) The principal observer was therefore consistent inusing the data collection instrument for objectively recording competitionsquash match play. Inter-rater reliability measures were obtained bycomparing the principal and expert observations (Table 3.13). The expertobserver recorded data in its entirety for the first video excerpt only whichrevealed high inter-reliability for position and shot observations. The datapertaining to just the rally outcomes was recorded thereafter because of thedisproportionate number of position, shot and outcome data associated with asquash rally. The expert data in each instance were contrasted to bothobservations recorded by the principal observer.INSERT TABLE 3.13 HERE57Inter-reliability analysis revealed that the requirement for k>.80 wassatisfactorily met for the position and shot conditions, but that thiscriterion was not satisfied for outcome condition (Table 3.13). Thediscrepancy between the observer's assessment of outcome was in part a resultof difference in interpretation between unconditional and conditionalwinners, and unforced and forced errors, but also in part due to confusionbetween some winner and error outcomes (Appendix B). The occasionalconfusion arose from limited video information which created difficulty ininterpreting the actual outcome condition.The observation process was therefore repeated with both observers recordingtogether the rally outcome data for the second training excerpt in itsentirety. Differences in viewpoint were reconciled through verbalnegotiation. This process was repeated for the rally outcome data from thethird training excerpt, and the subsequent written outcome definitionstightened and agreed upon. The principal observer and the expert observerthen independently recorded the rally outcome data only from the first semi-final (Appendix C), arbitrarily selected from the remaining data source, andthe inter-rater reliability reassessed. The subsequent results indicatedhigh agreement in observation of rally outcome between observers (h=.865;k=.817), thereby implying consistency and objectivity in data collection.These findings signaled the process of collecting the research data couldbegin.583.5 MethodologyVideotaped match data from a raised standpoint behind the glass back wall wasobtained from the quarter-final matches through final of the men's CanadianOpen Squash Championship, Toronto, 1988. The tournament knockout matcheswere contested to American scoring and all the matches were recorded in theirentirety, except for the second semi-final between Jansher and Dittmar. Thevideo recording of this match unfortunately omitted the rallies in the secondgame between the observed let scores 8-3 and 12-10 (i.e.,four points awardedto Jansher and seven points awarded to Dittmar were lost, together with theassociated rally information). The match data were normalized by removingthree contiguous rallies, all of which were point wins for Jansher, whichimmediately preceded the data omission. Seven points awarded to both Jansherand Dittmar were therefore omitted from the analysis.The data were collected from videotape and coded through computer software inorder of tournament match play. The quarter-final matches were thus analyzedfirst, followed by the semi-finals and latterly the final. Individualplaying profiles were established from each match for use in the simulationmodel.The playing profiles were analyzed for data integrity. American scoring issuch that a player's winner or an opponent's error results in a point awardedto that player, thus a summation of winners (unconditional and conditional)and the opponent's errors (unforced and forced) tallies to the eventualsummarized match score (i.e.,to the number of points awarded to the player).For example, if player A beats B with a match score 15-4; 15-10; 15-6 then A59scores forty-five points and B scores twenty points throughout the match.A's winners and B's errors are expected to summate to forty-five, and B'swinners and A's errors to twenty. Using this logic check revealedinconsistencies in the data (approximately one rally per match) which weresubsequently rectified. In each instance, these inconsistencies arose fromidentifying the wrong server, which of course had a concomitant effect onassigning position, shot and outcome data from that rally to the otherplayer. These findings suggest the need for a comprehensive software dataintegrity and logic check to precede the production of the playing profiles.4 RESULTS AND DISCUSSION4.1 OverviewThe eight contestants and their observed progression through the tournamentstructure are detailed in Fig.4.1. Seven matches were analyzed whichresulted in fourteen playing profiles for use by the stochastic model(Appendix E). As previously stated, these playing profiles comprise fourindividual profiles, the profiles which pertain to this discussion are theshot response and the winner-error profiles.60INSERT FIC.4.1. HEREThe shot response profiles were analyzed for consistency in playing patternbetween games against the same opponent, and between matches againstdifferent opponents, using two way chi-square analysis. The playing profilesfrom the quarter-finals were contested for projection of likely matchoutcome, and likewise for the semi-final profiles. The model predictionswere compared to the observed results for an indication of model validity andutility. The effects of the modeled interaction of the winner-error profileswere investigated, together with the relative importance of the shot responseprofile in determining subsequent match predictions.An appealing application of sport analysis and the stochastic modelingapproach is the possible identification of optimal strategy. Common playingtraits in the observed profiles were sought and possible optimal tacticsidentified. These tactics were tested through the reasonable manipulation ofthe shot response profiles and the consequential outcomes inspected.Analyses of adjacent game profiles were also conducted and the resultingdifferences used to amend the former game response profile in an attempt toevaluate the efficacy of the observed strategy shift. In other words, wouldthe empirical change in shot response profile between two adjacent games beexpected to benefit performance, given the information available from theformer game? These analyses offered valuable insight regarding the behaviorand utility of the stochastic model through comparison to the empiricalresults.614.2 Two Way Chi-Square AnalysisThe assumption of stability of a player's shot response profile against thesame opponent and against different opponents over a short period of time wasformally tested using two way chi-square analysis. This is an appropriatestatistical procedure which tests for independence between two independentvariables. The null hypothesis of chi-square is that the variables areindependent in the population of interest, the alternate hypothesis is thatthe variables are related. The null hypothesis for this analysis is that theopponent and the player's shot response to a preceding shot are independent.That is, a player's shot response to a drive, say, is consistent acrossopponent. A value p>.25 is taken to indicate that the opponent and shotresponse are not related, and a value p<.25 to indicate that the opponent andshot response are related. This study interprets a value p>.25 to suggestthe existence of an underlying consistent shot response, irrespective of theopponent. The criterion of p=.25 is selected for practical rather thaninflexible statistical reasoning, and p values around this criterion areliberally interpreted.Chi-square analysis requires that there are at least five expectedindependent observations (shots) associated with each cell, and at least tenobservations if the degrees of freedom equal one, correcting for continuity.To meet these requirements, the levels of the independent variable shotresponse were grouped according to similar function (i.e.,the number oflevels of the independent variable were collapsed to the original dataentries of serve, drive, boast, drop, volley and lob. See Table 4.1). Ifappropriate, boasts were further grouped with drops. No further collapsing62of shot responses were allowed and, if necessary, deficient categories wereomitted from the analysis. The number of actual categories used in any chi-square analysis were therefore dependent upon the empirical data. An exampleof the chi-square analysis is detailed in Table 4.2. This analysis revealsthat Jahangir's shot response to a drive condition was related to theopponent (X2 obs^x2.25,4; p<.25). Jahangir's response was thereforeinconsistent across these two opponents.INSERT TABLE 4.1 AND 4.2 ABOUT HEREThe shot conditions (i.e.,the shot which elicits a response) were similarlycollapsed to the original data entries listed above, except for the volleywhich retained its short or long identity. This is because the shotresponses to these conditions are expected to differ: More lob shots, forexample, are expected in response to short volleys, and more volley responsesare expected in response to long volleys. This exception is necessarybecause a different observed response profile to the volley may otherwisearise as a function of the type of volley condition (short or long), ratherthan a variable response to the volley itself.The players from the semi-final and final were tested for evidence ofstability of shot response between games within these matches. A matchcomprising three games allows a three way pairing of game data, while a matchcomprising four games allows a six way pairing. These pairings provide moreinformation than adjacent game pairings only. A significant difference in a63shot response profile between Game-1 and Game-2, and between Game-2 and Game-3, for instance, may be interpreted as a changing situation across all threegames. However, it may be that only the response profile of Game-2 issignificantly different. A test of difference between Game-1 and Game-3offers more information in this respect. Similarly, no significantdifference between Game-1 and Game-2, and between Game-2 and Game-3, may beinterpreted as a stable situation across all three games. A test ofdifference between Game-1 and Game-3 would reveal whether this interpretationis reasonable, or whether the transition in playing pattern, which perhapsexists between Game-1 and Game-3, is dissipated by Game-2. For thesereasons, all possible game pairings were subject to chi-square analysis.4.3 Validity of the Playing ProfilesThe relatively few discrete behaviors associated with any particular gamecreated difficulty in applying chi-square analysis, such that only two shotconditions could be identified to elicit an "acceptable" number ofobservations. (Formal violations of the assumption of chi-square withrespect to the expected frequencies associated with any particular cell wereidentified in the results.) These shot conditions were the drive and thoseshots - boast, drop and short volley - which force the opponent into a "shortgame" at the front of the court. The other shot conditions could not belogically grouped and were omitted from the statistical analysis. These datawere subjectively inspected ("eyeballed") for similarity in playing patternof shot response.64Table 4.3 reveals that Martin's general game profile was consistent acrossthe semi-final match against Jahangir, although the response profile to thedrive condition was different between game-2 and the other two games. Visualinspection of the data (Table 4.4) shows further consistency in playingpattern in Martin's response to the serve, the long volley and the lob(p>.25). The exception was the response to the lob condition in game-2 whichis likely a result of few available data.Jahangir's profile on the other hand was found to be more variable. Theresponse to the drive was inconsistent in game-1 when compared to the othergames, and likewise the response to the short game in game-3 was different.Further inspection of the data (Table 4.4) reveals variability in Jahangir'sresponse to the serve and lob, while retaining consistency in response to thelong volley. These findings suggest the possibility that each player triedto maximize performance through changing their playing pattern to thecompetitive environment. If this is true, the results indicate Jahangir tohave been more versatile than Martin in adapting his playing pattern.INSERT TABLE 4.3 AND 4.4 ABOUT HERETables 4.5 and 4.6 contain data from the other semi-final between Jansher andDittmar. Table 4.5 provides evidence that Jansher's response to the drivechanged across games as he perhaps tried to adjust his game to find asuccessful response. A closer inspection of the chi-square values, however,reveal that Game-2 is primarily where the change in response to the drive is65located (supported from data in Table 4.6), while the differences betweenGame-3 and Game-4, and between Game-1 and Game-4, approximate p..25.Jansher's response profile to the drive is probably not as marked thereforeas one may initially interpret. Jansher's short game response also differedbetween the earlier and later games, although evidence of a shift betweenadjacent games is not established. Table 4.6 indicates Jansher's response tothe serve and long volley to be consistent, with the exception of serviceresponse in Game-2, but this observation is not repeated for the response tothe lob condition. Dittmar's profile was slightly more stable.Inconsistency in response profile to the drive occurred in Game-3, andevidence of a change in response to the short game is found between Game-1,Game-3 and Game-4. Inspection of Table 4.6 shows consistency in Dittmar'sresponse to the serve and long volley (except Game-4) and the lob condition.INSERT TABLE 4.5 AND 4.6 ABOUT HEREThe profiles from the final between Dittmar and Jahangir reveal similarfindings to the semi-final profiles (Tables 4.7 and 4.8). Dittmar's profileshows a difference in response to the drive between Game-1 and Game-3 and acontinuously changing response to Jahangir's short game, evidenced by changebetween adjacent games and also between the first and last game (p<.25).This implies that Dittmar consistently adapted his response profile to theshort game in the search for a favorable outcome. Dittmar's responses to theserve, long volley and lob were consistent, despite the few observationsassociated with the latter condition.66INSERT TABLE 4.7 AND 4.8 ABOUT HEREJahangir in contrast was mostly consistent in his shot response profile. Theresponse to the drive is different in Game 3 while the difference inassigning response to the short game between Game-1 and Game-2 is notrepeated when compared to Game-3. Jahangir's responses to the serve, longvolley and lob conditions were also consistent, except for the lob responseprofile which differs in Game-2.It is possible that the type and frequency of the preceding shots by theopponent affects a player's shot response. If a player favors a short game,for instance, and often, say, responds to an opponent's boast and drop shotswith a drop shot, a theoretical contingency between the player's drop shotand the opponent's preceding boast and drop shots exist. If the opponentwere to reduce the frequency of the boast and drop shots, a correspondingreduction in the player's use of the drop shot may follow. It is notinconceivable that the player would alter his/her shot response profile toaccommodate any such strategy shift by the opponent. If the player adapts assuggested, and the adaptation is consistent across all preceding shotconditions, the expectation would be a reasonably consistent player shotresponse profile across all grouped shot conditions (i.e.,a player's summaryshot response profile would be stable). This would be revealed in the chi-square statistics for the shot responses to all the preceding shot conditionsby the opponent between games. Tables 4.3, 4.5 and 4.7 offer no statisticalevidence of such stability. The reported inconsistencies in shot response67profiles observed with respect to the preceding (grouped) shot condition donot provide evidence of a systematic adaptation in shot response to maintaina particular shot distribution..Tables 4.3 through 4.8 provide evidence of general stability in a player'sresponse profile between games against the same opponent, although somevariability within these profiles is observed. That some profile changeexists between games against the same opponent is not necessarily surprisinggiven that the human is an intelligent and adaptive organism who, in acompetitive environment, presumably tries to respond in such a manner as tomaximize their outcome, while at the same time minimizing the potential gainfor the opponent. A game theoretic approach is a particularly useful andappropriate technique for analysis of this type of player-player interaction.It seems that while some of this adaptation in performance is reflected inthe shot response profiles between games, these profiles do nonethelessgenerally capture an underlying characteristic which signifies a particularplayer's pattern when competing between games against the same opponent.The assertion that elite players, for the most part, tend to exhibitstability in their playing pattern when contested against the same opponentwas further tested form the available data. Squash data from an earlierstudy (McGarry & Franks, 1993) included the 1987 final of the men's WorldOpen Championship contested between Jansher and Dittmar. (Jansher won the1987 final and lost the 1988 semi-final.) Data from the 1987 final wascompared to the 1988 semi-final data, thereby allowing a more completestatistical analysis for evidence of consistency in the shot responseprofiles. The expectation, when comparing matches rather than games against68the same opponent, is for greater stability in the playing patterns. This isbecause the observed adaptation of the players is expected to dissipatesomewhat over a number of games.Table 4.9 shows both Jansher and Dittmar were consistent in shot responseacross matches when competing against the same opponent. Jansher, however,changed his response profile to the boast and drop conditions while Dittmarresponded in a different way to the drive and short volley. It isinstructive to investigate these data further since these results may assistin explaining the difference in match outcome (Table 4.10).INSERT TABLE 4.9 AND 4.10 ABOUT HEREJansher's boast response and drop response profile show a consistent patternof change which may suggest the deliberate use of tactical strategy. Table4.10 shows that Jansher increased the likelihood of responding to Dittmar'sboast and drop shot with a drive, at the expense of both the drop and lobshot. It is interesting to note, however, that Dittmar's response to thedrive changed also, perhaps to accommodate Jansher's game. The most notablechange in Dittmar's drive response is the increase in the use of volley, sopressurizing Jansher by reducing his time to respond. Dittmar's response tothe short volley also altered between matches, primarily through using thelob more frequently in the later match. Analysis of the player's winner-error profiles offer further match information (Table 4.9) and inconsistencyis expected given the change in match outcome. While Jansher's winner-error69profile was consistent between matches (p>.25), Dittmar's profile is found tobe different for both the winner and error distribution. An interpretationis that Dittmar won the later contest, rather than Jansher losing it, if thechange in match outcome is associated to the change in the winner-errorprofile.The evidence of stability in a player's profile between games and betweenmatches when competing against the same opponent is encouraging for thedevelopment of a model to describe and predict future match play. Is thisobserved stability consistent across opponent, such that an underlyingpattern exists which provides a signature to a player's likely futurebehavior? Further chi-square analyses were conducted to determine whetherplayer profile stability exists between matches when competing againstdifferent opponents.Table 4.11 provides compelling evidence to suggest that profile stability isnon-existent when competing against different opponents. This findingsupports the assertion that the interaction between players is unique to thatsituation, and refutes the assumption that a player's shot response isindependent of the opponent in elite squash match competition. While aplayer's shot response profile is seemingly dependent upon the particularinteraction between two players, the profiles from these interactions havebeen shown to be stable, and thus replicable, over a short period of time.The data does not support the assumption that a player's match play profileis generalizable to other opponents.70INSERT TABLE 4.11 ABOUT HEREAn interesting comparison is to test whether these observed dissimilaritiesin shot response profiles against different opponents is better than chanceobservation. That is, is there evidence for some underlying patternassociated with a player's profile, even if this evidence is notstatistically valid? A test of this assertion is to examine chi-squareresults for evidence of stability in shot response profiles between differentplayers when matched against different opponents. No interaction betweenthese profiles exist and so there is no reason to expect a result other thanchance. Contrasting these results (Table 4.12) to those obtained for playerscompeting against different opponents (Table 4.11) would indicate whether ornot a player tends to exert some identity in his/her playing profile.INSERT TABLE 4.12 ABOUT HERETable 4.12 reveals, as expected, differences in playing profiles betweendifferent players when competing against different opponents. Thesedifferences, however, are conspicuously not unlike, either in frequency ormagnitude of the observed chi-squares, the differences identified in aplayer's response patterns to different opponents. The differences observedin a player's shot response profile between matches across opponent are nottherefore unexpected from chance alone. In other words, the players appear71to be unable to stamp a common identity of response between differentmatches. This may be a failing of the individual player to exert authorityon the opponent, a task which should increase in difficulty as the playerprogresses through each stage of the seeded tournament and playing abilitypresumably converges. More reasonably, it may be desirable for the player tobe adaptive and therefore responsive to the changing competitive environment.The conclusion from these data is that a player's response profile, asdefined by the likelihood of a response to a preceding shot condition, isunique to the individual player-player interaction and inferences drawn fromsuch interactions cannot be reliably extended to a future setting against adifferent opponent. There is need to investigate this situation further,since these findings may be a reflection of the specific definition of aplaying profile. The present definition awards a player's shot response fromonly the discrete observation of the opponent's previous shot, with noadditional information regarding any further actions of the opponent. Theinteraction which exists between the players in assigning shot responses isnot therefore captured within the present definition. Additional informationof the opponent's play is also required, if an appropriate shot response isto be elicited for suitable prediction to future behavior.It has been established that the shot response profiles change over timeagainst different opponents, thereby violating the model's assumption ofstationarity in the playing matrix. The effect of this violation on themodel's prediction of future match outcome is unclear, because theseforecasts seem to be largely dependent upon the winner-error profiles awardedfrom the empirical data. The shot response profiles are the vehicle through72which these winner-error profiles are accessed, and so influence outcome tosome degree through providing a "channeling" or "weighting" effect. Theextent of this influence provides the focus for further investigation of themodel's behavior.4.4 Validity of the Stochastic ModelModel validity attempts to statistically determine the degree of congruencebetween the model and the real system which it purports to represent. Thevalidity of the reported model addresses two primary concerns: Does themodel correctly predict match outcome and does the model correctly predictmatch behavior? The first concern was referred to by Stanislaw (1986) as the"function" and the second concern as the "form" of the model. Match outcomerefers to the match victor and match behavior refers to the manner in whichthe model arrives at this outcome.An immediate evaluation of the model's function is easily attained throughcomparing the discrete prediction to the discrete observation. An assessmentof the model's form, however, is more problematic. While the simulatedresults yield mean and variance scores of such simple dependent measures asfrequency of rallies, shots and outcomes, the empirical data provides onlydiscrete values and a reasonable approximation of their respective populationdistributions is difficult. This study proposes that a general indication ofmodel validity can be attained through inspection of whether the discretevalues lie within a standard deviation of the simulated mean. 68% rather73than 95% confidence intervals were selected since this offers a morestringent assessment of agreement between observed and simulated data.Table 4.13 is selected as exemplar evidence from the 1988 final that,according to the above validation criteria, the stochastic model accuratelyreflects match behavior and outcome from the empirical data. The simulationdata are replicated from the empirical data on both a match and game basis.Analysis of these data at the level of an individual player further explainsmodel behavior in the production of these data. (For validation of thequarter-finals and semi-finals see Appendix G.) The mean values of thedependent measures for the entire match approximate closely to the empiricaldata and easily lie within a standard deviation (Table 4.13). This findingis expected for multiple simulation runs given the stochastic nature of themodel. However, this variation in stochastic prediction generally causes aslight overestimation of these simulated means from the empirical data. Thereason is that, since the empirical data comprises an observed game score 3-0, simulated scores of 3-1 and 3-2 (awarded to either player) willoverestimate the frequencies of rallies and, necessarily therefore, thefrequencies of shots and outcomes also. The result is a slight inflation ofthe simulated means in comparison to the empirical data.INSERT TABLE 4.13 ABOUT HEREA solution is to collect match data for game scores of 3-0 or 0-3 only. Apreferred alternative, however, is to analyze by game rather than by match,74which allows for truer approximation without determining the simulationresult a priori. The simulations were re-run one thousand times with thevictor being the first to a game, rather than the first to three (best offive) games. These subsequent game data provided closer approximations tothe empirical data (Table 4.13). A useful further function of these gamedata is that they offer a logic check on the model's game scoring structurebecause, if the probability of winning a game is known, the probability ofwinning a match and by what game score can be approximated throughcombinatorial probability. The probability of winning the match from winningthe game is approximate because it is assumed that the winning gameprobability is independent of the initial server. This is not strictly trueand the scoring nature of squash is such that the winner of the previous gameserves at the start of the next. Nonetheless, since the game winningprobabilities usually differ little between starting servers in Americansquash, the following computation is considered a close enough approximationto offer a reasonable check of model function. For an exact mathematicalsolution see Appendix H.Consider player X versus Y.Let the probability that X wins a game = pand the probability that Y wins a game = q = 1-pAssuming p and q are independent of the starting server, a random walkthrough the possible routes that allows X and Y to win the match yields:75P{X wins 3-0} = p 3 [ppp]P{X wins 3-1} = 3qp 3 [qppp + pqpp + ppqp]P{X wins 3-2} = 6q2p3 [qqppp + qpqpp + qppqp + pqqpp + pqpqp + ppqqp]P{Y wins 3-0} = q3 [qqq]P{Y wins 3-1} = 3pq3 [pqqq + qpqq + qqpq]NY wins 3-2} = 6p2q3 [ppqqq + pqpqq + pqqpq + qppqq + qpqpq + qqppq]Thus P{X wins match} = p 3 + 3qp3 + 6q2p3and^P{Y wins match} = q 3 + 3pq3 + 6p2 q3Subsequent checks of the data indicated the model behaves in the expectedfashion in assigning match score from game score, allowing for approximationof computation and stochastic variation. Similar checks of course could beperformed further for the probability of winning a game from the probabilityof winning a point.That the stochastic model accurately replicates the empirical data whichdrives it is not surprising. What is required is an indication as to whetheror not the model accurately predicts future match behavior and outcome frompast data. The dependent measures from the model's predictions of the semi-7 6finals and final from earlier profiles were therefore compared to theirrespective observed data.INSERT TABLES 4.14, 4.15, 4.16 AND 4.17 ABOUT HEREThe data from tables 4.14, 4.15, 4.16 and 4.17 provide reasonableapproximation to the observed game data, suggesting that the model tends tocapture some of the general characteristics of competitive squash, at leastin so far as the dependent measures inform. They also offer an overview ofmodel behavior. Table 4.14, which predicts one of the semi-finals from thequarter-final data, both exaggerates the expected winners and underrates thepredicted errors for Martin. Jahangir's winner and error predictions,however, fit the empirical data. The other semi-final prediction (Table4.15) overestimates the length of the contest, as inferred from the number ofshots per game. The prediction approximates Dittmar's winners and errors butoverestimates the winners and underestimates the errors awarded to Jansher,the combined effect being an erroneous prediction of match outcome.Tables 4.16 and 4.17 represent the predictions of the final from the quarter-final and semi-final data respectively. Table 4.16 shows that the quarter-final prediction, in addition to underestimating the length of the contest,underestimates the error count for Dittmar but overestimates the predictedwinners of Jahangir. Table 4.17 on the other hand shows a good estimation ofthe empirical data, although the predicted winners for Jahangir are inflated.The model's behavior therefore notably differs in its projections from the77quarter-final and semi-final profiles for the same two players. This findingis disconcerting, and concurs with the preceding chi-square analysis whichinforms that the form (behavior) of the predictive model is invalid, becauseconsistency of shot response between matches against different opponents isnot observed.A more suitable criterion of model validation is possible, which, inaddition, accounts for validity of the "software model" itself (i.e.,theveracity of the application software). The validation technique consists ofgenerating a player's profile through simulation from its empirical profileand then contrasting the simulated profile to the empirical profile whichproduced it using two way chi-square analysis. If the software model isvalid, a high degree of consistency in shot response and winner-errorprofiles is expected. The concept can be extended to inspect the predictivevalidity of the model also, through comparing the simulated (predicted)profile to the observed profile.Dittmar and Jahangir's observed semi-final profiles were contested for fiveruns of the simulation. Five runs were selected because this offered amanageable level of information with which to provide a reasonable indicationof model behavior. Each run produced an historical data file of simulatedmatch play (described in 3.3.4. Operation of the Stochastic Model) throughwriting the results, as the simulation progressed, to magnetic disc. Asimilar technique is used in producing the dynamic graphics display of thesimulation through writing the results to screen instead. These generateddata were analyzed, and the shot response and winner-error profiles summatedfor both players before averaging over the five runs and rounding decimal7879places to the nearest integer. These generated profiles constitute thesimulated profiles. Two way chi-square analyses of the simulated data to theobserved semi-final (Table 4.18) and final profiles (Table 4.20) thenprovided a statistical test of model behavior. Tables 4.19 and 4.21 describethe empirical and the simulated playing profiles of Dittmar and Jahangir forthe semi-final and final respectively, from which the respective chi-squareanalyses were produced. The statistic for Dittmar's shot response to thedrive, for instance, is x 2 (3, N=1084)=.772, p>.25 (second row, Tables 4.18 and4.19).INSERT TABLES 4.18, 4.19, 4.20 AND 4.21 ABOUT HEREThe conclusion from these validation data is that the software model isstatistically valid (Table 4.18), but that the model has no predictivevalidity beyond the scope of its empirical data (Table 4.20), a restrictionwhich unfortunately consigns the past to forever repeat itself. Thisapplication has little practical utility and, perhaps also, face validity.Using the model, for instance, to forecast from the 1987 final to the 1988semi-final of Jansher versus Dittmar, a repeat of the 1987 observation isprojected (McGarry & Franks, 1993). While the chi-square results (Table 4.9)indicate that the behavior of the model is statistically valid, the predictedmatch outcome is incorrect. What is required is additional information fromearlier match(es), information which may, or may not, be contained inpreceding tournament data. While the model cannot be statistically verifiedon grounds of profile stability of shot response, or of winner-error outcome,it may nevertheless prove instructive to investigate further the nature ofthe data which governs match outcome.4.5 Violation of the Assumption of StationarityThe first task in exploring the predictive characteristics of the model, andthe interaction of the data which determine these properties, is to ascertainwhether violating the assumption of stationarity of a player's shot responsebetween opponents has a marked effect on the simulated match outcome. Non-identical shot response profiles will produce different outcome results dueto alternate channeling paths through which the winner-error outcomes areaccessed. While inconsistency in some of the player's shot responses betweenmatches against different opponents has been established from empirical data(p<.25), it is not clear whether these findings translate to a meaningfuldifference in predicted match outcome. The uncertainty arises from thedependency of the simulated match outcome on the assigned winner-errorprofiles of both players, and their modeled interaction. The magnitude ofobserved change in predicted match outcome is therefore important. Apredicted swing in match outcome of .015 is negligible and can likely beattributed to the stochastic variation from a single simulation comprising athousand runs (Table 3.6). A predicted swing in match outcome of greaterthan .015 is taken to indicate a more credible shift, and attributable toviolating the assumption of stationarity. It, however, carries littlepractical effect and a predicted change in outcome of the order of 0.100, orgreater, is considered to be meaningful for practical purposes. Predictedchanges within these bounds (.015>p>.100) are interpreted as an acceptableviolation of the assumption of stationarity.80A test of consequence of violating the assumption of stationarity can beachieved by superimposing a different shot response, or winner-error, profileon a preceding, or subsequent, playing matrix. An attractive feature of thistest is that it provides some indication of the relative importance of boththe shot response and the winner-error profile in the prediction of simulatedmatch performance. The benefit is that the findings should guide the modelerin directing future attention to the predominating feature which determinesmatch outcome, in successive attempts to replicate and accurately extend theempirical data to future performance in squash competition.The technique of superimposing profiles was used to identify the resultingeffect of assuming stationarity in shot response profiles when predictingfuture performance from past data. The shot response profiles from the semi-final and final were superimposed on the corresponding quarter-final andsemi-final data as appropriate (while retaining the quarter-final or semi-final winner-error profile), and the projections from the "new" quarter-finalplaying profiles compared to the initial predictions (Table 4.22). Thedifference in prediction is due to the difference between shot responseprofiles and their individual effect on the simulated match outcome. Thesame process was repeated for the final shot-response profiles, which weresuperimposed on both the corresponding semi-final and quarter-final profiles.Various paired combinations of the new profiles were contested in simulationfor inspection of model behavior.INSERT TABLE 4.22 ABOUT HERE81Forecasting the semi-final (Martin versus Jahangir) from the quarter-finaldata forecasted a win for Jahangir (p=.599; see Table 4.22). Superimposingthe semi-final shot response patterns in the playing profile increasedJahangir's predicted chance of victory in each instance: SuperimposingMartin's profile (p=.799), Jahangir's profile (p..696), and both profilessimultaneously (p=.756). In each instance the winner-error profiles remainedfixed. These results suggest that violating the assumption of stationarityof shot response impacts on subsequent model prediction. The results of thissimulation suggest an error of approximately .157 (.756-.599) from failing toaccount for the difference in the player's shot response profiles between thequarter-final and semi-final. The behavior (form) and outcome (function) ofthe model are therefore dependent on the shot response profiles.The observed shifts in model prediction as a function of shot response arenot repeated to the same extent when predicting to the other semi-final(Jansher versus Dittmar) or final (Dittmar versus Jahangir) from the quarter-final data. Errors in prediction from failing to account for shot responsedifferences approximate to .032 and .000 respectively. This finding isalmost certainly due to the assigned differences in "strengths" of thewinner-error profiles of the two contesting players, the strengths of whichbeing contingent, in part, on their previous opponent. The effect of thisdifference in winner-error profile strength between two competing playerswithin the simulation is to desensitize the model to changes in the shotresponse profiles. The model becomes more sensitive to these changes as theplayers approach equality in winner-error profile (i.e.,as p approaches .5),as observed in the Martin-Jahangir prediction. This assertion is confirmedon inspecting the predicted error from projection of the final from the semi-8 2final (.110), where Jahangir is simulated to win the contest from the semi-final data (p=.754; Table 4.22). An alternative explanation is that thepredicted changes are primarily influenced by the size of change in the shotresponse profile, indicated by the frequency and magnitude of the differencesin shot responses. The chi-square results from table 4.11 do not tend tosupport the latter interpretation.The same technique was used to superimpose different winner-error profiles onthe same shot response matrix, in order to confirm the expected dependency ofthe simulation outcome on the winner-error profiles. The winner-errorprofiles from the quarter-final and semi-final were superimposed on the semi-final and final playing profiles as appropriate. The advantage withsuperimposing preceding winner-error profiles on successive playing profiles,rather than subsequent winner-error profiles on earlier playing profiles, isthat the winner-error profiles can be tested for predictive capability. Ineach instance the shot response profiles remained fixed. Variouscombinations of these profiles were contested and the simulation projectionscompared to the initial predictions (Table 4.23). Discrepancies betweencorresponding predictions were attributed to the effect of the winner-errorprofiles on the simulation outcome.INSERT TABLE 4.23 ABOUT HERETable 4.23 contains interesting but disconcerting data. SuperimposingMartin's quarter-final winner-error profile in the semi-final produced a83dramatic shift in the prediction of Jahangir's probability of match success(.999-.133..866), while superimposing Jahangir's winner-error profilereinforced Jahangir's dominance, as evidenced from the game scorefrequencies. The effect of superimposing both winner-error profiles in thesimulation is for each to negate the effect of the other to some degree,although Jahangir's profile proved the stronger (p=.756).The other semi-final simulation data contain the same pattern, such that thesuperimposed quarter-final profile outstrips the opposing semi-final profilein each instance. The inference is that the winner-error profiles from theearlier round contain stronger winning characteristics than their successivecounterparts, presumably because they were achieved against easier opponents,therefore resulting in a wider victory margin. While the simulationpredictions from contesting the quarter-final profiles (p=.037; Table 4.22)and the corresponding semi-final shot response and quarter-final winner-errorprofiles closely agree (p=.069; Table 4.23), they both forecast the wrongplayer (Jansher) to win the contest. This prediction results from theirpreceding winner-error profiles, which were obtained against differentopponents. A confounding issue is that while the prediction does not matchthe empirical result, it agrees with the players' respective world rankingsand so raises the spectre of an unexpected observation.The predictions from the quarter-finals and semi-finals to the final showthat superimposing successively earlier winner-error profiles favors thatparticular player, for the reason identified earlier. It is worthy of notethough that Dittmar's semi-final winner-error profile exerts strongerinfluence in Dittmar's direction than its quarter-final counterpart. This is84perhaps surprising given that the semi-final was contested against a strongeropponent, and so would perhaps be expected to weaken Dittmar's assignedwinner-error profile. Inspection of the respective playing profiles(Appendix E) reveals that Dittmar's quarter-final opponent (Harris) commitsmany errors and is consequently the primary agent in influencing matchoutcome. This is at the expense of Dittmar's playing profile. This problemwas identified in an earlier study (McGarry & Franks, 1993) and was amotivating reason to introduce the notion of conditionality in winner anderror assignment, in an attempt to model some interaction between competingplayers in assigning outcome. The success, or otherwise, of this modeledinteraction is addressed shortly.It is evident that both the shot response profile and the winner-errorprofile exert influence in determining match outcome. The predominatingprofile is the winner-error profile, which, if reasonably equal betweencompeting players, is sensitive to the individual shot responsecharacteristics of each player. Unequal winner-error profiles cause a strongbias to the preferred profile, and the subsequent predicted match outcomeremains largely unaffected by the individual shot response profiles. Thedependency of prediction upon the winner-error and the shot responseprofiles, coupled with the prior observation that these profiles exhibit noestablished stability, dictate that the present model has no authority toproceed with the valid prediction of future sport performance. Caution musttherefore be exercised in subsequent predictions for further comprehension ofpresent model behavior.854.6 Interaction of the Winner-Error ProfilesA study by McGarry and Franks (1993) reported an underlying problem of aprevious stochastic model for the prediction of future squash performance tolie in the independent assignment of winner and error outcomes from previousobservation, with no reflection of how these outcomes interrelate to theprevious and future opponent. That is, there seemed to be an overdependenceon the winner-error ratio which, although conditional to the particularopponent, was attributed solely to the individual player. A resultingconstraint imposed by this independent assignment of outcome is that aparticularly strong, or weak, profile is often awarded at the expense of, orbenefit to, the opponent respectively. A strong player, for instance, mayforce their opponent to be assigned a weaker profile than is merited while,conversely, a weak player may exaggerate a moderate player's profile to thedetriment of the simulation model. Other possible combinations also exist.A recommendation of McGarry and Franks (1993) was to attempt to model theinteraction in winner-error profile which plainly exists between competingplayers. The modeled interaction of these profiles, and their subsequenteffect on the simulated match outcome, provides the present focus for thisdiscussion.The effects of the interactions of the profiles within the simulation can betested through rank ordering the players according to their assessed playingstrength. Contesting all players to each other in Round Robin fashionenables their rank order to be evaluated from the data. Each simulation wasrepeated after reversing the starting server and receiver. This procedurewas repeated using the previous stochastic model (McGarry & Franks, 1993) for86comparison purposes. If the above strong-weak relationship between profilesexists, the quarter-final pairs would be expected to associate strongest rankto weakest rank, next strongest to next weakest and so on. The results(Table 4.24) show that the stronger profiles are not accompanied by anassociated weaker profile of the opponent. Indeed the reverse is true forboth models, and the possibility that a strong profile somehow supplementsthe opponent's profile raises itself. An explanation of this suggestion forthe reported model may lie in the introduction of a complementary effect bythe modeled winner-error profile interaction. This explanation, however,does not extend itself to the perhaps unexpected findings from the previousmodel.INSERT TABLES 4.24 ABOUT HERENo difference in player ranking from the data was found between the twomodels, a result which is probably due to the large margins of successobserved in each quarter-final contest (3-0). A comparatively "easily" woncontest unduly biases the simulation in favor of the stronger player (i.e.,ptends to 1.00) and the model desensitizes to other external effects such asthe modeled interaction of the winner-error profiles. A more sensitivemodel, which occurs as p tends to 0.5 would be expected to be moresusceptible to the effects of these interactions.87INSERT TABLES 4.25 AND 4.26 ABOUT HERETable 4.25 contains the simulated match outcomes from the previous model(McGarry & Franks, 1993), where winner, error or let are independentlyassigned to rally outcome, and Table 4.26 contains the results of the modelused in this study. The tables read row by column for a win probability.For example, Table 4.26 shows the predicted probability that Norman wouldbeat Jahangir from initial service is .007, or, conversely, Jahangir wouldbeat Norman from initial receipt of service with probability .993. Thepredicted probability that Jahangir would beat Norman from initial service,however, is .990, which results in a difference in match prediction as aresult of first service of -.03. This finding is almost certainlyattributable to stochastic variation within the result. It is interesting tonote that both models suggest the effect of first service to have very littleimpact on the eventual outcome at this level of squash competition. Thesefindings, while lower, concur with the earlier theoretical reports of Schutzand Kinsey (1977) and Wright (1988).A comparison of the results from both models (Tables 4.25 and 4.26) suggestthat the modeled interaction of the winner-error profiles exert considerableinfluence in the predicted match outcome, and that, from these data (Table4.25), the model begins to become sensitive to this influence in the region.080 p.5. .920. The insensitivity of the model was observed at the extremerange of p as expected (p5_.002; 1:).998).88The results show the modeled interaction of the winner-error profilesdecreased Martin's predicted chance of a successful outcome when contestedagainst Jansher (p=.753-.913=-.160). Their respective quarter-finalopponents (Norman and Nancarrow) exhibited a similar tendency (p=.567-.679=-.112). This finding infers the possibility of a systematic relationshipwhich results from the modeled interaction, such that a player's winner-errorprofile proportionately complements the opponent's profile. If this is so,the expectation is a repeat observation amongst other profile interactions.Martin's predicted winning chance when matched against Jahangir increased(p=.221-.401=-.180), but in contrast to this trend, their respective quarter-final opponents (Norman and Thorne) revealed a reverse tendency: Norman'spredicted chance of victory dramatically decreased when the winner-profilemodeled interactions were introduced (p=.450-.087=.363). This implies thatthe effect of the modeled winner-error profile does not necessarily introducea complementary relationship between the player and the opponent.The relationship, if any, between the quarter-final pairings and thepredicted match outcomes, as a result of the modeled winner-error interactionwas further investigated. Table 4.27 contains data which representdirectional differences between predictions from the two models:represents an increase in a predicted win for the reported model from theprevious model; "-" represents a decrease in a predicted win, and "*"represents a "don't care" value. The don't care values were assigned todifferences of less than .015, which may have arisen through stochasticvariation, and so cannot be reliably credited as a result of difference inmodel behavior. The frequency of increase and decrease in prediction foreach player were established and evidence was sought for a complementary89pattern, as indicated from the data in Table 4.24. Table 4.27 provides noevidence of a complementary relationship, or indeed any other discernibleassociation, between quarter-final pairs within the simulation as a result ofthe modeled interaction.INSERT TABLES 4.27 ABOUT HEREA general investigation of the winner-error profiles through inspecting thewinner-error ratios lends useful insight into the empirical data andsubsequent model behavior. The observed winner-error ratios of the quarter-finals, the semi-finals and the final are detailed in Table 4.28 which reads,left to right, tournament stage by player by opponent. That is, the firstquarter-final of Norman versus Martin resulted in winners and errors, and thesubsequent derived ratios, for both players. Norman, for instance, scoredfour unconditional winners (W), thirteen conditional winners (w), seventeenunforced errors (Q) and two forced errors (q) against Martin. Martin, on theother hand, in the same contest against Norman, scored ten unconditionalwinners (W), sixteen conditional winners (w), eleven unforced errors (Q) andone forced error (q). The unconditional winner-unforced error (W/Q), theconditional winner-forced error (w/q) and the absolute winner-absolute error(W+w/Q+q) ratios for each player are easily computed from these data. Avivid demonstration of the importance of these ratios in determining thepredicted outcome is found through rank ordering, in descending order, theabsolute winner-absolute error ratio for each player from the empiricalquarter-final data. The subsequent order is in close agreement with the rank90order produced by both models after contesting each player with each otherusing the Round Robin procedure. The corresponding order of unconditionalwinner-unforced error and conditional winner-forced error ratios by contrastprove much more complex.INSERT TABLES 4.28 ABOUT HEREA cursory glance at the winner-error profiles provides little informationwith regard to model behavior, but it does allow projection of theconditional winner and forced error ratios for the final from the respectivequarter-final and semi-final data. These ratios were used to predict theexpected number of conditional winners and forced errors respectively (Table4.29). Dittmar's conditional winner coefficient from the quarter-final data,predicted for the final against Jahangir, was computed as follows:Let Dittmar be X and Jahangir be Y; from (3.1)w^[wY'/(WY'+wY')]*[(WX+wX)/wX]Does Jahangir allow many conditional winners against him? We must thereforeinspect Jahangir's previous opponent's (Thorne) profile to see.91Jahangir allows wy '/(Wy '+wy ')^4/5, or .8, conditional winners against him.Substituting into (3.1) gives a conditional winner coefficient ofw * = (4/5)*[(3+9)/9]^1.07Thus, all of Dittmar's conditional winners against Jahangir will be granted.The product of the conditional winner and forced error coefficients withtheir respective initial observation yields their predicted frequency in thesimulated contest. It is important to appreciate that, in this example, thecoefficients, and their subsequent projection of conditional winners andforced errors, apply for all the shots grouped together. The simulationoperates in the same way but according to individual shot type, which allowsthe shot response profile to exert some influence on the simulatedprediction. This explains the difference in projection from the semi-finalto the final between Table 4.29 and the simulation data.INSERT TABLES 4.29 ABOUT HEREThe above example illustrates a constraint of the present model. Since themodel is dependent upon the observation to determine whether the outcome isgranted or rescinded, it in effect imposes an upper limit on the conditionaland forced error coefficient of 1.00 (i.e.,more conditional winners or forcederrors cannot be awarded than occurred in the initial observation). Thisposes at least a philosophical limitation: If a player competes against aweaker opponent in a successive contest, then it is reasonable to expect an92increase in their winner count; similarly, if a player competes against astronger opponent then it is reasonable to expect that the player will incurmore forced errors. This limitation results as a function of the presentmodel, which invokes a player's outcome from their empirical data only, andis not conditional upon the opponent's past. The expectation is thataccounting for the present opponent's history in generating an outcome eventwill notably affect the predicted match outcome.4.7 Identification of Optimal StrategiesAn appealing application of sport analysis, and the development of astochastic model for the prediction of future sport performance, is theidentification of optimal strategies. A fundamental component of any suchstrategy is that it possesses consistency across some particular condition,even if this condition should be restricted to a specific opponent orindividual shot response. This basic requirement is necessary if the strategyis to be successfully applied to a future setting. It is reasonable toinvestigate championship sport performance for evidence of consistent playcharacteristics and the subsequent identification of optimal strategy. Aninspection of the data from this study may provide useful insight regardingoptimal strategy, if idiosyncratic performance can be identified within aplayer's shot response profiles. If an optimal strategy can be established,its efficacy can be tested through the stochastic model.93Table 4.30 contains Jahangir's playing profile contested against Thorne inthe quarter-final. Inspection of the winner-error profile suggests that themost profitable shot for Jahangir against Thorne is the (straight) drop shot(E), since this shot produced most of Jahangir's winners and relatively fewerrors. It is also an efficient shot for Jahangir in the sense that manyoutcomes are associated with relatively few shots. Indeed, the ratio ofoutcome to drop shot for Jahangir is .42 (i.e.,approximately four from tendrop shots resulted in an outcome, mostly winners) which suggests a verystrong short game. A realistic, yet retrospective, tactic for Thorne wouldbe to somehow discourage Jahangir from playing that many drop shots in thematch. A closer inspection of Jahangir's data (Table 4.30) reveals that mostof his drop responses (row E) are played in response to Thorne's cross-courtboast and drop shot (columns D and E). What if Thorne should reduce thefrequency of his boast and drop shots, and so lessen the supply of the shotconditions to which Jahangir mostly responds with a drop?INSERT TABLES 4.30 AND 4.31 ABOUT HERETable 4.31 contains Thorne's corresponding playing profile and informs thatmost of Thorne's boasts (row D) were struck in response to Jahangir'sstraight or cross-court drive (columns A and B) and, to a lesser extent, thecross-court long volley (column J). Most of Thorne's drop shots on the otherhand arise in response to Jahangir's boast and drop shot (columns D and E).A reasonable manipulation of Thorne's boast and drop shot profile in responseto the identified shot conditions, and a subsequent simulation of the playing94profiles would indicate whether such a tactic is favorable to Thorne or not(Table 4.32). Thorne's boast response, a shot which is usually employed as adefensive strategy, was therefore switched to a straight drive, and Thorne'sdrop shot response changed to a cross-court lob. This was achieved in thefollowing manner: Ten shots from cells DA and DB were taken and primed tocells AA and AB respectively, and likewise five shots from each of the cellsDJ, ED and EE taken and primed to cells AJ, ND and NE. This is considered tobe a reasonable manipulation of the playing profile because each of theprimed cells were already well established responses to these particular shotconditions, as evidenced from the playing matrix (Table 4.31). In this waythe status of the playing matrix was preserved.INSERT TABLE 4.32 ABOUT HEREThe simulation was repeated using Jahangir's empirical quarter-final profileand Thorne's optimized profile and the predicted match outcome compared tothe original forecast for assessment of the suggested strategy shift (Table4.32). The results provide good evidence to suggest that, all other thingsequal, reducing Thorne's boast and drop shot response would marginally favorhis chances of a successful outcome. It is evident that Jahangir's winner-error profile is far too strong for this tactic to be visible, in so far aswinning the match is concerned, but the tactic nonetheless demonstrated adirectional shift in favor of the identified optimal strategy. Moreover,this shift is conspicuous given the insensitivity of the model as a result ofthe heavy bias previously identified in performance. However, does evidence95for restricting Jahangir's drop game to optimize competitive matchperformance against him extend beyond this particular interaction?It is interesting to note from Jahangir's semi-final profile (Table 4.33)against Martin that the drop remains a highly efficient shot in eliciting anoutcome (ratio .37), although less profitable to Jahangir since it incurredproportionately fewer winners and more errors, perhaps as a result of toughercompetition. Once more Jahangir's data suggests a high drop response to theopponent's boast, although drop shots are now introduced in response to drivecondition also. Martin's profile (Table 4.34) is similar to Thorne's in thatmost boasts are played in response to the drive. Repeating the manipulationof Martin's profile in a similar fashion to that described for Thorne, andcontesting the subsequent profiles, once again indicates the strategy to bemarginally beneficial to Jahangir's opponent (Table 4.35).INSERT TABLES 4.33, 4.34 AND 4.35 ABOUT HEREJahangir's drop shot in the final against Dittmar (Table 4.36) is lessefficient than the preceding two contests, and less favorable in that nowmost of the associated outcomes were lets. (This finding is extended acrossshot types and across player also, albeit to a lesser extent.) Theobservation of an increase in let outcomes may be circumstantial - a moreprevalent outcome as the competitors converge in ability - or evidence of atactical manoeuvre on Dittmar's part. Inspecting Jahangir's profile (Table4.36) reveals that most drops were once again in response to Dittmar's boast96and drop shot. Dittmar's opposing profile (Table 4.37) shows most boastswere produced in response to the drive conditions (straight and cross-court),and most drop shots in response to the drop and boast respectively.Manipulating Dittmar's profile to minimize Jahangir's frequency of drop shotshad little favorable effect on match outcome (Table 4.38). This finding,however, is not interpreted as being detrimental to the general tactic ofrestricting Jahangir's short game for optimizing the opponent's performance.In essence, this tactic had already been invoked, as evidenced by theinordinate number of lets associated with Jahangir's drop shot.INSERT TABLES 4.36, 4.37 AND 4.38 ABOUT HEREThe preceding example suggests that the search for optimal strategy withinindividual sport performance has promising utility. Its application in apractical setting is to direct future practice to shape athletic behavior tothe desired profile, and for the athlete to subsequently adopt this patternin match competition. It is acknowledged, however, that the use of tacticscannot be assumed to be static because players have been found to adapt theirgame pattern, to some extent, to suit the changing competitive environment.A corresponding application would be for the model to provide currenttactical information of performance between games. This approach was used toinvestigate whether some of the observed behavioral shifts in gameperformance would be expected to benefit performance, given the availablematch information at the time.97The game profiles of Dittmar and Jahangir were obtained (Appendix F) andamended to accommodate the differences in shot response observed betweenadjacent games. Table 4.7 shows Dittmar's short game profile was differentbetween adjacent games, and further inspection reveals the difference in shotresponse between Game-1 and Game-2 to have been primarily due to aconspicuous increase in drop response to the drop and short volley condition,at the expense of the drive. An "optimized" profile of Dittmar's Game-1profile, which reflects this behavioral change, is to take drop responsesaway from the drive condition and prime the drop and short volley conditionsaccordingly. A similar procedure was followed for the change in Dittmar'sprofile between Game-2 and Game-3, and also for those adaptations inJahangir's adjacent game profiles. The subsequent results from variouscombinations of contrasting these profiles are detailed in Table 4.39.INSERT TABLE 4.39 ABOUT HEREAn assumption of this analysis is that the changes in shot response behaviorbetween games is a reflection of prior experience (i.e.,the data from theprevious game). In other words, the profile from the preceding game isresponsible for the observed behavioral shift in the present game. Analysisof difference between adjacent games therefore is considered sufficient toprovide the necessary information from which the strategy relating to theformer game can be inferred. A consequence of adjacent paired analyses isthat for an n-game situation (where n is an integer 3 through 5), n-1strategies can be identified. Thus, for a 3-0 game score, only two98strategies can be analyzed. The reference to Game-1 (Table 4.39) relates tothe strategy used from Game-1 in Game-2, and, likewise, the reference toGame-2 relates to the strategy used from Game-2 in Game-3. The observedbehavioral changes in shot response for both players between Game-1 and Game-2, and between Game-2 and Game-3, were identified and accommodated in newlyassigned playing profiles, hereafter referred to as the player's "optimal"profiles, for Game-1 and Game-2 respectively. The actual Game-1 and Game-2profiles are subsequently referred to as the "observed" profiles.Table 4.39 contains data averaged over five simulations (N=5), eachsimulation comprising one thousand runs (n=1000) of a single game only. Thedata therefore relates to the probability of winning a game rather than amatch, the mathematical relationship between the two having been identifiedearlier. Dittmar was awarded game service at the onset of each run of thesimulation. The data are read row by column for winning probability, forinstance, the probability that Jahangir will win the game from empirical dataagainst Dittmar is .871. Alternatively, the probability that Dittmar willbeat Jahangir is .129. Five simulations were selected to provide a morestable indicator than a single simulation which is more sensitive tostochastic variation.Assuming that a behavioral shift in shot response is attributable to theexperiences of the preceding game, a reasonable expectation is that theobserved change between adjacent games should effect a predicted improvementin performance in the earlier game, if all other conditions remain constant.This is easily tested through matching the observed profile to the optimalprofile for the same player for the earlier game of the adjacent game pair.99A value, reading row by column, of p<.5 would indicate an advantageousoptimal strategy, since the observed profile is credited with winning lessthan half of the simulated contests. The corresponding data (Table 4.39)suggest Dittmar's strategy from Game-1 should be effective in Game-2(p=.408), while the strategy used from Game-2 data will likely beinconsequential in Game-3 (p=.502). Jahangir's strategy conversely is likelyto be of no consequence from Game-1 data (p=.504), but detrimental tosubsequent performance from Game-2 data (p=.618). Tables 4.7 and 4.8 showDittmar's Game-1/Game-2 strategy comprised a change in short game response,due to a proportionate increase in the number of drop responses to the dropand short volley condition. Dittmar's Game-2/Game-3 strategy is also due toa change in short game response, but, in this instance in a reverse directionto the earlier change. A reason for this apparent contradiction will likelybe a change in winner-error profile which has not been accounted for in thisanalysis.Jahangir's Game-1/Game-2 strategy consisted of a change in short gameresponse also, resulting in a decrease in drop response to the short gameconditions. The inconsequence of this strategy change may be a result of themany lets observed throughout the contest in association with this particularresponse. Jahangir's Game-2/Game-3 strategy, which projected a deteriorationin game performance, was to change his response to the drive condition,through reducing his drive frequency and producing relatively more boasts,drops and volleys (including long volleys). It seems then that the shift inbehavior from previous data would be expected to favor Dittmar and notJahangir during the later games of the contest. However, these presentinterpretations exist largely in isolation to the opponent's coincident100change in shot response profile. Contrasting the results of the observedgame outcomes to the optimal strategies will further help in assessing theseempirical shifts in playing patterns.Further analyses concur with the above suggestion that Dittmar succeeds andJahangir fails to enhance their own individual chance of success. Jahangiris simulated to win Game-1 from empirical data (p=.871), whereas thislikelihood decreases when contrasting the corresponding "optimal" profiles(p=.831). (It is the direction of change within the simulation which is ofprimary note, rather than the magnitude which is dependent on the particularcell priming of the optimized profiles.) This situation is repeated forGame-2 with values of p=.878 and p=.837 respectively. These findings are notsupported empirically, and contrast starkly with the point scores for eachgame 15-10, 15-10, 15-4, which indicate little effective change in successbetween players for the first two games and growing ease of victory forJahangir in the last game. The identification of optimal strategy from thisanalysis, between games at least, therefore has little face validity.The inability to account for the change in the winner-error game profile is aprimary reason for these analytic results which contradict the empiricaldata, although it is unclear whether such change can be accounted for apriori. It is clear, however, that while the change in shot response profilein the subsequently observed direction may optimize performance based uponthe present winner-error game profile, this finding cannot be reliablyextended to the next game which naturally produces a different winner-errorprofile. The limitation of few data from a single game undeniably furtherexacerbates this situation.1 01A limitation in identifying optimal strategy on a game by game basis isnecessarily encountered in assuming the observed behavioral shift to bedependent upon previous experience from the preceding game. It is notunreasonable to suggest that the observed adjustment is in response topresent events, and the subsequent observed data reflect these events and notthe preceding events from which the predictions are modeled. The consequentconclusion would be that the preceding data have little bearing on futurebehavior in the next game. The averaging of data over a match, or matches,may help to reduce the variation between games to some extent.5 CONCLUSIONSThe research hypothesis that consistent patterns of play exist at the elitelevel of squash match play, and that these patterns provide a signature tofuture athletic performance is not supported from this study. If suchpatterns exist, they were not captured by eliciting shot response from thepreceding shot by the opponent, or by assigning outcome from shot response.A player's shot response profile and winner-error profile were found tochange when competing across different opponents. This finding suggeststhat, either the model failed to capture the critical data which contain theappropriate information from which future behavior can be accuratelyforecast, or that more variation in competitive sport performance exists thancan be accounted for. This latter suggestion is analogous somewhat toattempting to predict from a single run of a stochastic simulation.102An encouraging result of the study is that, from the available data,consistency in sport performance between two competing players wasestablished, both between games in a match and also between matches. Thestability in shot response pattern therefore seems to be contingent on theparticular interaction which exists between players. This finding suggeststhat there may be additional information in an opponent's performance, notidentified from this investigation, which may reliably produce the particularobserved shot response of the individual player. Further research isrequired to determine whether such additional information can be identified.It is evident from the data that the model is dependent upon both the shotresponse and the winner-error profile in the prediction of the simulatedresult. The winner-error profile is the dominating profile and the modeledinteraction between the players was found to have a marked effect on thepredicted result. The results of this study suggest a similar interactionbetween the shot response profiles is also necessary for assigning playershot response. Unless a heavy bias in winner-error profile existed betweenthe two competing players, the model was sensitive to the shot responseprofile. The requirement to accurately model both the winner-error and theshot response profile is therefore important, if a subsequent model is toreliably project future athletic performance in squash.The application of the model for the identification of optimal strategy haslittle present utility, because consistency in playing performance acrossdifferent competitors has not been established. This finding could havetheoretical impact on the benefit of scouting prospective opponents insquash. While these findings are not generalizable to other sports, the same103deduction holds if consistency in athletic performance in these sports cannotbe identified across opponents either. Nonetheless, an inspection ofJahangir's playing profile indicated a general playing pattern with respectto his use, and derived gain, of the straight drop shot, and the shots towhich this response was most likely to arise. This was considered to beimportant information regarding Jahangir's match play and from which anappropriate optimal strategy was identified. The application of optimalstrategy, as identified from the empirical shift in athletic performance, foruse between successive games could not, however, be demonstrated to be ofbenefit.While the reported stochastic model failed to accurately predict futureperformance from past athletic behavior, further research is recommended toinvestigate whether an underlying order which characterizes individualperformance can be established. The attempt to model competitive sportperformance is a worthy pursuit, not least because it directs attention tothe critical aspects of data that discriminate sport success. Further, if nosuch individual signature in playing profile can be identified, the utilityof commonly applying tactical strategy derived from previous observationsagainst different opponents must be questioned.REFERENCESAlain, C. & Sarrazin, C. (1990). Study of Decision-Making in SquashCompetition: A Computer Simulation Approach. Canadian Journal of Sport Science. 15(3), 193-200.104Albert, J. (1992). A Bayesian Analysis of a Poisson Random Effects Model forHome Run Hitters. The American Statistician. 46(4), 246-253.Alexander, D., McClements, K., & Simmons, J. (1988). Calculating to Win. NewScientist. 120(1642), 30-33.Ap Simon, H.G. (1951). The Luck of the Toss in Squash Rackets. Mathematical Gazette. 35, 193-194.Ap Simon, H.G. (1957). Squash Chances. Mathematical Gazette. 41(336), 136-137.Altman, D.C. (1991). Practical Statistics for Medical Research. London:Chapman and Hall.Bellman, R. (1977). Dynamic Programming and Markovian Decision Processes,with Application to Baseball. In Ladany, S.P. and Machol, R.E. (Eds.),Optimal Strategies in Sports (pp 77-85). Amsterdam: North Holland.Bennett, J.M., & Flueck, J.A. (1983). An Evaluation of Major League BaseballOffensive Performance Models. The American Statistician. 37(1), 76-82.Buckolz, E., Prapavesis, H. & Fairs, J. (1988). Advance Cues and Their Use inPredicting Tennis Passing Shots. Canadian Journal of Sport Science. 13(1).20-30.Carter, W.H.Jr., & Crews, S.L. (1974). An Analysis of the Game of Tennis. TheAmerican Statistician. 28(4), 130-134.Clarke, S.R. (1979). Tie Point Strategy in American and International Squashand Badminton. Research Quarterly. 50(4), 729-734.Clarke, S.R. (1988). Dynamic Programming in One Day Cricket - Optimal ScoringRates. Journal of the Operational Research Society. 39(4), 331-337.Clarke, S.R., & Norman, J.M. (1979). Comparison of North American andInternational Squash Scoring System - Analytic Results. Research Quarterly.50, 723-728.Cover, T.M. (1989). Do Longer Games Favor the Stronger Player? The AmericanStatistician. 12(4), 277-278.105Croucher, J.S. (1982). The Effect of the Tennis Tie-Breaker. ResearchQuarterly for Exercise and Sport. 53(4), 336-339.Croucher, J.S. (1986). The Conditional Probability of Winning Games inTennis. Research Quarterly for Exercise and Sport. 57(1), 23-26.Doob J.L.(1953). Stochastic Processes. New York: John Wiley & Sons.Downey, J. (1992, November). Keynote Address. Paper presented at First WorldCongress of Notational Analysis in Sport. Wirral. UK.Eom, H.J., & Schutz, R.W. (1992). Transition Play in Team Performance ofVolleyball: A Log-Linear Analysis. Research Quarterly for Exercise and Sport.63(3), 261-269.Franks, I.M., Goodman, D., & Miller, G. (1983). Human Factors in SportsSystems. An Empirical Investigation of Events in Team Games. Human FactorsSociety, 27th Annual Meeting. Virginia.USA. 383-386.Franks, I.M., Wilson. G.E. & Goodman, D. (1987). Analyzing a team sport withthe aid of computers. Canadian Journal of Sport Sciences. 12(2). 120-125.Freeze, R.A. (1974). An Analysis of Baseball Batting Order by Monte CarloSimulation. Operations Research. 22(4).Girault M.(1966). Stochastic Processes. New York: Springer-Verlag.Hannan, E.L. (1976). An Analysis of Different Serving Strategies in Tennis.In Machol, R.E., Ladany, S.P., and Morrison, D.G. (Eds.), Management Sciencein Sports 4 (pp 125-135). New York: North Holland.House, A.E., House, B.J., and Campbell, M.B. (1981). Measures ofInterobserver Agreement: Calculation Formulas and Distribution Effects.Journal of Behavioral Assessment. 3(1). 37-57.Hughes, M.D. (1984). Using a Microcomputer for Notational Analysis in Squash.Proceedings of Sport Science Conference. Bedford. UK.Hughes, M.D. (1985). Patterns of Squash at Higher Competitive Levels.Proceedings of Sport and Science Conference. Chichester. UK.106Hughes, M.D., & McGarry, T. (1991,April). The Development of Three Dimensional Graphics to Illustrate Data Presentation from a ComputerizedNotational Analysis of Squash. Paper presented at 9th Eurographics UKConference. Sheffield. UK.Janssen, C.T.L., & Schutz, R.W. (1985). Optimal Convert Strategies inFootball. Research Quarterly for Exercise and Sport. 56(4), 339-344.Kenyon, G.S., & Schutz, R.W. (1970). Patterns of Involvement in Sport: AStochastic View. In G.S.Kenyon (Ed.), Contemporary Psychology of Sport. (pp781-797). Chicago: Athletic InstituteKing, H.A. & Baker, J.A.W. (1979). Statistical Analysis of Service and Match-Play Strategies in Tennis. Canadian Journal of Applied Sport Science. 4(4).298-301.Ladany, S.P. & Machol, R.E. (1977). Optimal Strategies in Sports. Amsterdam:North Holland.Lehman, R.S. (1977). Computer Simulation and Modeling: An Introduction. NewYork: Lawrence Erlbaum Associates.McGarry, T. (1988). Computer-Assisted Notational Analysis of Squash: TheDevelopment of a Computerized Squash Notation System With Three DimensionalGraphics for Representation of Patterns of Play. Unpublished Undergraduate Thesis. Liverpool Polytechnic. UK.McGarry, T., & Franks, I.M. (1993). Manuscript under review. Journal of Sport Sciences.McKenzie, I. (1986). Squash. The Skills of the Game. The Crowood Press.Ramsbury.Morris, C. (1977). The Most Important Points in Tennis. In Ladany, S.P. andMachol, R.E. (Eds.), Optimal Strategies in Sports (pp 131-140). Amsterdam:North Holland.Morrison, D.G. (1976). On the Optimal Time to Pull the Goalie: A PoissonModel Applied to a Common Strategy Used in Ice Hockey. In Machol, R.E.,Ladany, S.P., and Morrison, D.G. (Eds.), Management Science in Sports 4 (pp137-144). New York: North Holland.107Naumienko, E.Z., & Naumienko, B.J. (1986). Simulations of Behavior inCompetitive Situations. Simulations and Games. 17(3), 301-319.Norman, J.M. (1985). Dynamic Programming in Tennis - When to Use a FastServe. Journal of the Operational Research Society. 36(1). 75-77.Paul, R.J. (1991). Recent Developments in Simulation Modelling. Journal of the Royal Operational Research Society. 42.(3) 217-226.Peterson, A.V. (Jr). (1977). Comparing the Run-Scoring Abilities of TwoDifferent Batting Orders: Results of a Simulation. In Ladany, S.P. andMachol, R.E. (Eds.), Optimal Strategies in Sports (pp 86-88). Amsterdam:North Holland.Pfeifer, P.E., & Deutsch, S.J. (1981). A Probabilistic Model for Evaluationof Volleyball Scoring Systems. Research Ouarterly for Exercise and Sport.52(3), 330-338.Pollard, G.H. (1985). A Statistical Investigation of Squash. ResearchQuarterly for Exercise and Sport. 56(2), 144-150.Pollard, G.H. (1987). A New Tennis Scoring System. Research Quarterly for Exercise and Sport. 58(3), 229-233.Porter, R.C. (1977). Extra-Point Strategy in Football. In Ladany, S.P. andMachol, R.E. (Eds.), Optimal Strategies in Sports (pp 109-111). Amsterdam:North Holland.Renick, J. (1976). Optimal Strategy at Decision Points in Singles Squash.Research Quarterly. 47(3), 562-568.Renick, J. (1977). Tie Point Strategy in Badminton and International Squash.Research Ouarterly. 48(2), 492-498.Sanderson, F.H., & Way, K.I.M. (1977). The Development of an Objective Methodof Game Analysis in Squash Rackets. British Journal of Sports Medicine. 11,188.Sarrazin, C., Lacombe, D., Alain, C. & Joly, J. (1983). Simulation Study of aDecision-Making Model of Squash Competition, Phase One: The Analysis of theProtocol. Human Movement Science. 2. 279-306.108Sarrazin, C., Alain, C. & Lacombe, D. (1986). Simulation Study of a Decision-Making Model of Squash Competition, Phase Two: Testing the Model Through theUse of Computer Simulation. Human Movement Science. 5. 373-391.Schutz, R.W. (1970a). Stochastic Processes: Their Nature and Use in the Studyof Sport and Physical Activity. Research Quarterly. 41(2), 205-213.Schutz, R.W. (1970b). A Mathematical Model for Evaluating Scoring Systemswith Specific Reference to Tennis. Research Quarterly. 41(4), 552-561.Schutz, R.W. (1980). Sport and Mathematics: A Definition and Delineation.Research Quarterly for Exercise and Sport. 51(1), 37-49.Schutz, R.W., & Kinsey, W.J. (1977). Comparison of North American andInternational Squash Scoring Systems - A Computer Simulation. ResearchQuarterly. 48(1), 248-251.Sport in Brief. (1992, March). The Globe and Mail. 44398. All.Sports Illustrated Squash. (1971). Philadelphia: J.B.Lippincott.Trueman, R.E. (1976). A Computer Simulation Model of Baseball: WithParticular Application to Strategy Analysis. In Machol, R.E., Ladany, S.P.,and Morrison, D.G. (Eds.), Management Science in Sports. 4 (pp 1-14). NewYork: North Holland.Trueman, R.E. (1977). Analysis of Baseball as a Markov Process. In Ladany,S.P. and Machol, R.E. (Eds.), Optimal Strategies in Sports (pp 68-76).Amsterdam: North Holland.Watson, D.W. (1970). On Scoring in Games. Mathematical Gazette. 54(388), 110-113.Wright, M.B. (1988). Probabilities and Decision Rules for the Game of SquashRackets. Journal of the Operational Research Society. 39(1), 91-99.109Table 3.1 Table of Shotof Squash Match Playand Outcome Event Codes for the Systematic RecordingCode Shot Code OutcomeS Serve W Unconditional WinnerA Drive w Conditional WinnerB XC-Drive Q Unforced ErrorC Boast q Forced ErrorD XC-Boast R LetE Drop W' Unconditional Winner(Opponent)F XC-Drop w Conditional Winner(Opponent)G Volley Q' Unforced Error(Opponent)H XC-Vol(Short) q Forced Error(Opponent)I Volley(Short)J XC-Vol(Long)K Volley(Long)L XC-Vol-BoastM LobN XC-LobTable 3.2 Table of Example Outcome Profiles for Four Players.APlayerB X YPlayerUnconditional Winner(W) WA (8) WB(4) WX(5) Wy(7)Conditional Winner (w) WA(8) wB(7) wx(10) wy(11)Unforced Error (Q) QA(11) QB(10) Qx(6) Qy(2)Forced Error (q) qA(4) qB(3) qX(4) qy(5)Let (R) RA (3) RB(0) Rx(2) RY (4)OpponentUnconditional Winner(W') WA' (4) WB'(8) WX'(7) WY'(5)Conditional Winner (w') wA'(7) wB'(8) wx'(11) wy'(10)Unforced Error (Q') QA'(10) QB'(11) Qx '(2) Qy'(6)Forced Error (W) clA .(3) qB'(4) cIX'(5) clY'(4)110Table 3.3 Table of an Example Cumulative Probability Matrix Denoting PlayerX's Match Play Profile Against Player Y.Player:XOpponent:YPreceding ShotShot^S^A^BCDE^FGHI JKLMNS Serve 0 (0.00) 0 (0.00) 0 (0.00)A Drive 8 (0.40) 23 (0.23) 9 (0.23)B XC-Drive 2 (0.50) 15 (0.38) 2 (0.28)C Boast 0 (0.50) 4 (0.42) 0 (0.28)D XC-Boast 4 (0.70) 9 (0.51) 6 (0.43)E Drop 0 (0.70) 17 (0.68) 11 (0.70)F XC-Drop 0 (0.70) 3 (0.71) 4 (0.80)G Volley(S) 2 (0.80) 5 (0.76) 2 (0.85)H XC-Volley(S) 0 (0.80) 1 (0.77) 0 (0.85)I Volley(L) 2 (0.90) 9 (0.86) 0 (0.85)J XC-Volley(L) 2 (1.00) 3 (0.89) 0 (0.85)K Vol-Boast 0 (1.00) 2 (0.91) 0 (0.85)L XC-Vol-Boast 0 (1.00) 0 (0.91) 0 (0.85)M Lob 0 (1.00) 9 (1.00) 5 (0.98)N XC-Lob 0 (1.00) 0 (1.00) 1 (1.00)TOTAL 20 100 40W WINNER 0 (0.00) 5 (0.33) 4 (0.40)w winner 0 (0.00) 3 (0.83) 1 (0.50)Q ERROR 0 (0.00) 4 (0.80) 2 (0.70)q error 0 (0.00) 1 (0.87) 2 (0.90)R LET 0 (0.00) 2 (1.00) 1 (1.00)TOTAL 0 (0.00) 15 (0.15) 10 (0.25)W' WINNERy 0 (0.00) 3 (0.33) 1 (0.13)w' winnery 0 (0.00) 1 (0.44) 3 (0.50)Q° ERRORy 0 (0.00) 5 (1.00) 3 (0.88)q' errory 0 (0.00) 0 (1.00) 1 (1.00)111Table 3.4 Table of Example Rally Simulation from Empirical DataConsider Y to servePlyr Rand-No-Call Process Result^CodeY Shot = 0 SERVE^S0.63^(no data) Ball-InX 0.81^(Table 3.3) Prev-Shot = 0; Shot = 9; VOLLEY(L)^I0.44^(Table 3.3) Ball-InY 0.21^(no data) Prev-Shot = 9; Shot = 1; DRIVE^A0.79^(no data) Ball-InX 0.16^(Table 3.3) Prev-Shot = 1; Shot = 1; DRIVE^A0.04^(Table 3.3) Assign Outcome0.52^(Table 3.3) Conditional Winner: Confirm?Revoke?0.82^(w * =0.66) Revoke Conditional WinnerY 0.20^(no data) Prev-Shot = 1; Shot = 1; DRIVE^A0.79^(no data) Ball-InX 0.59^(Table 3.3) Prev-Shot = 1; Shot = 5; DROP^E0.17^(Table 3.3) Assign Outcome0.73^(Table 3.3) Forced Error: Confirm?Revoke?0.26^(q * =0.50) Confirm Forced Error FORCED ERROR^qRally: Player-Y; Serve; Volley(L); XC-Drop; XC-Drop; Error112Table 3.5 Table of Example Coded Rally Simulation from Empirical DataPlyr Posn Shot Posn Shot Posn Shot Posn Shot Posn Shot Posn Shot Posn OutcomeY^RR ServeY^4^SRL^Volley(L)1^IRL Drive1^AFR Drive1^AFL Drive1^AFL Drop2^EFL error2^qRally: Y4S1I1A1A1A2E2qTable 3.6 Table of Difference in Simulation Data When Running One Hundred andOne Thousand Simulations for Prediction of Norman versus Nancarrow fromQuarter-Final DataSimulation Runs Mean Runs Mean1 100 .590 1000 .5672 100 .670 1000 .5823 100 .640 1000 .5734 100 .600 1000 .5655 100 .570 1000 .5886 100 .560 1000 .5767 100 .480 1000 .5538 100 .570 1000 .5599 100 .630 1000 .55510 100 .610 1000 .608Grand Mean .592 .573SEM .052 .017113Table 3.7 Table of Randomly Selected Matches and Match Rallies forConstruction of the Training Video Excerpts.Match Match^Match RallyNumber Rallies 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 206 SF2^115 48 9 66 79 85 13 46 87 52 65 15 39 61 23 33 89 43 11 28 174 QF4 87 82 9 62 22 2 21 23 42 30 35 38 29 79 24 84 67 16 51 32 561 QF1^97 64 44 13 7 25 53 75 46 35 59 33 73 21 51 56 85 71 72 17 87Table 3.8 Table of Example Independent Observations Between Two Observers forFour Dependent Variables A Through DObserver 1Observer 2^A^B^C^D TotalA^18^3^1^0^22B 11^10^4^2 27C^5^1^15^6^27D 1^2^2^19 24Total^35^16^22^27^100Table 3.9 Table of Kappa Coefficient and Strength of Agreement (Altman, 1991)Kappa^Strength of Agreement<.20^Poor^.21- .40^Fair.41- .60^Moderate.61- .80^Good.81-1.00^Excellent114Table 3.10 Table of Grouping Classifications for Position and Shot forComputation of the Kappa StatisticPosition Groupp osn Shot Group shotA 1 Serve ServeB 2 Drive DriveC 3 XC-DriveD 4 Boast BoastE 5 XC-BoastF 6 Drop DropG 7 XC-DropH 8 Volley(S) VolleyI 1 XC-Volley(S)J 2 Volley(L)K 3 XC-Volley(L)L 4 Volley-BoastM 5 XC-Volley-BoastN 6 Lob Lob0 7 XC-LobP 8Table 3.11 Table of Total Agreement and Kappa Coefficients of Intra-RaterReliability for Grouped and Non-Grouped Dependent VariablesExcerpt'GroupExcerpt2^Excerpt3 Excerpt1 , 2 , 3Non-GroupExcerpt1 , 2 , 3Position h=.934 h=.842 h=.914 h=.892 h=.892k=.918 k=.808 k=.894 k=.867 k=.875Shot h=.971 h=.959 h=.977 h=.967 h=.928k=.958 k=.934 k=.968 k=.952 k=.914Table 3.12 Table of Total Agreement and Kappa Coefficients of Intra-RaterReliability for Non-Grouped Dependent VariablesNon-GroupExcerpt' Excerpt2 Excerpt3 Excerpt1 , 2 , 3Outcome^h=.800^h=.850^h=.900^h=.850k=.800115Table 3.13 Table of Total Agreement and Kappa Coefficients of Inter-RaterReliability for Grouped Dependent VariablesFirst Principal ObservationExcerpt' Excerpt2 Excerpt3^Excerpt1,2,3Position h=.833k=.789Shot h=.873k=.808Outcome h=.800 h=.750 h=.650 h=.733k=.658Second Principal ObservationExcerpt' Excerpt2 Excerpt3^Excerpt1,2,3Position h=.866k=.831Shot h=.863k=.797Outcome h=.850 h=.750 h=.650 h=.750k=.673116Table 4.1 Table of Shot Conditions Collapsed for Two Way Chi-Square AnalysisIndependent Variable^Level^Collapsed LevelOpponentShot ResponseFirst OpponentSecond OpponentServe^ServeDrive DriveXC-DriveBoast^BoastXC-BoastDrop DropXC-DropVolley(S)^VolleyXC-Volley(S)Volley(L)XC-Volley(L)Volley-BoastXC-Volley-BoastLob^ LobXC-LobTable 4.2 Table of Frequency (Percent) Shot Responses and Two Way Chi-SquareAnalysis of Jahangir's Shot Response to a Drive Across Opponent1988 Semi-Final (Jahangir v Martin)1988 Final^(Jahangir v Dittmar)OPPONENT SHOT RESPONSE^ CHI-SQUAREDrive^Boast^Drop Volley^chiobs chi .25,3Martin^225(79%)^18( 6%)^18(6%)^25( 9%)Dittmar^252(68%)^40(11%)^18(5%)^60(16%)13.754^4.108117Table 4.3 Table of Two Way Chi-Square Analysis for Martin and Jahangir's ShotResponse Profiles Between Games Against Each Other1988 Semi-Final^(Martin v Jahangir)GAMEMARTINchiobs chi 25 p VJAHANGIRchiobs chi 25 p V1v2 Drive 7.452 2.773 * 9.229 2.773 * #Boast/Drop/Vol(S) 1.002 2.773 # .134 2.773 #Total Shots 9.415 4.108 * 5.296 4.108 *2v3 Drive 4.978 2.773 * .482 2.773Boast/Drop/Vol(S) 1.375 2.773 # 3.232 2.773 * #Total Shots 2.288 4.108 1.586 4.1081v3 Drive .027 2.773 7.249 2.773 *Boast/Drop/Vol(S) .955 2.773 # 5.887 2.773 * #Total Shots 2.783 4.108 5.917 4.108 ** p<.25 significance# chi-square violation: less than five/ten expected observations in a cell118Table 4.4 Table of Martin and Jahangir's Shot Frequencies for CollapsedLevels Between Games Against Each Other1988 Semi-Final (Martin v Jahangir)GAME-1^MARTIN SHOT RESPONSE^JAHANGIR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^4^2^0^10^0^11^3^0^4^0Drive 132^11^9^21^1 135^5^8 9^0Boast^4^1^4 0^1^2^1^7^0^1Drop 7^0^4^0^3 5^1^4 0^4Volley(S)^1^0^0 0^0^3^0^4^0^1Volley(L)^11^0^2^1^0 23^1^1 2^1Lob^1^0^0 6^0^1^1^0^3^0Total Shots 160^14^19^38^5^180^12^24^18^7GAME-2^MARTIN SHOT RESPONSE^JAHANGIR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^3^0^0^15^0^2^0^0^3^0Drive 32^6^8 6^0 29^8^1 6^0Boast^2^0^2^0^2^1^0^4^0^1Drop 2^0^1 1^2 4^1^2 0^2Volley(S)^1^0^2^0^0^1^0^1^0^0Volley(L) 5^0^0 0^0 14^0^1 1^0Lob^1^0^0^2^0^4^0^0^0^0Total Shots^46^6^13^24^4^55^9^9^10^3GAME-3^MARTIN SHOT RESPONSE^JAHANGIR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^4^0^0^15^0^2^3^0^8^0Drive 68^6^5^11^0 61^5^9^10^1Boast^3^1^3 0^1^4^0^2 0^0Drop 4^0^6^0^0 7^0^1^0^3Volley(S)^1^0^1 0^2^1^0^1 0^0Volley(L)^11^2^1^1^1 18^1^2^0^1Lob^0^0^0 5^0^1^0^0 3^0Total Shots^91^9^16^32^4^94^9^15^21^5119Table 4.5 Table of Two Way Chi-Square Analysis for Jansher and Dittmar's ShotResponse Profiles Between Games Against Each Other1988 Semi-Final +^(Jansher v Dittmar)GAMEJANSHERchiobs chi 25 p VDITTMARchi obs chi 25 p V1v2 Drive 6.129 2.773 * # .205 2.773Boast/Drop/Vol(S) .751 2.773 .281 2.773 #Total Shots 5.687 5.385 * .419 4.1082v3 Drive 6.157 2.773 * # 5.227 2.773 *Boast/Drop/Vol(S) 2.200 2.773 # 2.275 2.773 #Total Shots 7.364 5.385 * 4.018 4.1083v4 Drive 2.776 2.773 * # 3.815 2.773 *Boast/Drop/Vol(S) 1.926 2.773 5.447 2.773 *Total Shots 6.765 5.385 * 7.160 4.108 *1v3 Drive .869 2.773 5.408 2.773 *Boast/Drop/Vol(S) .752 2.773 6.174 2.773 *Total Shots 2.971 5.385 9.467 4.108 *1v4 Drive 2.880 2.773 * # 1.639 2.773Boast/Drop/Vol(S) 3.651 2.773 * 6.174 2.773 *Total Shots 13.807 5.385 * 14.401 4.108 *2v4 Drive 12.343 2.773 * # 1.455 2.773Boast/Drop/Vol(S) 4.460 2.773 * 2.115 2.773Total Shots 18.607 5.385 * 9.537 4.108 ** p<.25 significance# chi-square violation: less than five/ten expected observations in a cell+ Incomplete data in Game 2120Table 4.6 Table of Jansher and Dittmar's Shot Frequencies for CollapsedLevels Between Games Against Each Other1988 Semi-Final (Jansher v Dittmar)GAME-1^JANSHER SHOT RESPONSE^DITTMAR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^4^1^0^11^0^5^1^0^13^0Drive 130^15^8^12^0 126^23^7^38^0Boast^16^2^7 0^2^10^3^5 1^2Drop 9^1^4^0^6 9^0^5^0^5Volley(S)^6^0^7 0^4^6^1^1 0^5Volley(L)^34^5^1^3^2 14^1^1^5^0Lob^2^0^0^10^0^2^0^1^11^0Total Shots 201^24^27^36^16 172^29^20^68 12GAME-2 +^JANSHER SHOT RESPONSE^DITTMAR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^4^0^0^5^0^2^0^0^11^0Drive 61^1^1 5^0 61^10^5^16^0Boast^6^0^2^0^1^3^0^0 0^1Drop 3^0^2 0^3 2^0^3^0^0Volley(S)^4^0^2^0^2^0^0^1 0^1Volley(L)^16^1^0 2^1 5^0^0^2^1Lob^0^2^0^0^0^1^1^0 4^0Total Shots^94^4^7^12^7 74^11^9^33^3GAME-3^JANSHER SHOT RESPONSE^DITTMAR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^4^1^0^14^0^1^0^0^11^0Drive 75^8^5 4^0 77^11^5^41^0Boast^7^0^5^0^0^2^0^5 0^2Drop 8^0^7 0^3 4^0^7^0^5Volley(S)^8^1^8^3^5^1^0^4 1^2Volley(L)^29^2^0 3^0 11^2^2^4^0Lob^4^0^0^5^0^1^1^0 6^0Total Shots 135^12^25^29^8 97^14^23^63^9GAME-4^JANSHER SHOT RESPONSE^DITTMAR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^6^1^0^12^0^0^0^0^5^0Drive 31^5^6 3^0 34^10^3^13^0Boast^3^0^7^0^2^3^0^4 0^0Drop 6^0^6 0^3 7^0^10^0^1Volley(S)^1^0^4^0^2^1^1^3 0^1Volley(L)^11^1^1 3^0 2^2^0^3^0Lob^2^0^0^0^0^1^1^0 4^0Total Shots^60^7^24^18^7 48^14^20^25^2121+ Incomplete dataTable 4.7 Table of Two Way Chi-Square Analysis for Dittmar and Jahangir'sShot Response Profiles Between Games Against Each Other1988 Final^(Jahangir v Dittmar)GAMEDITTMARchiobs chi 25 p VJAHANGIRchiobs chi 25 p V1v2 Drive 2.134 2.773 .684 2.773Boast/Drop/Vol(S) 4.326 2.773 * 4.997 2.773 *Total Shots 6.270 5.385 * .735 4.1082v3 Drive 1.825 2.773 3.073 2.773 *Boast/Drop/Vol(S) 5.847 2.773 * # 1.942 2.773Total Shots 10.642 5.385 * 1.360 4.1081v3 Drive 6.901 2.773 * 5.898 2.773 *Boast/Drop/Vol(S) 4.958 2.773 * # 1.009 2.773Total Shots 15.791 5.385 * 1.868 4.108* p<.25 significance# chi-square violation: less than five/ten expected observations in a cell122Table 4.8 Table of Dittmar and Jahangir's Shot Frequencies for CollapsedLevels Between Games Against Each Other1988 Final (Jansher v Dittmar)GAME-1^DITTMAR SHOT RESPONSE^JAHANGIR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^1^0^0^26^0^4^1^0^13^0Drive 136^25^5^17^0 132^15^6^29^1Boast^5^0^9 0^7^10^2^16 0^1Drop 14^1^12^0^5 15^1^10^0^3Volley(S)^10^1^4 0^1^3^0^7 1^0Volley(L)^25^3^4^1^0 24^3^2^4^0Lob^1^1^0 3^0^1^1^0^10^0Total Shots 192^31^34^47^13^189^23^41^57^5GAME-2^DITTMAR SHOT RESPONSE^JAHANGIR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe^0^0^0^25^0^1^1^0^9^0Drive 96^17^7^16^0 85^15^7^20^0Boast^6^0^11 0^3^5^0^8 0^2Drop 4^0^14^0^4 21^0^8^0^2Volley(S)^2^0^4 0^4^7^0^4 0^1Volley(L)^21^0^3^1^0 23^3^1^2^0Lob^1^1^0 3^0^3^1^0 7^0Total Shots 130^18^39^45^11^145^20^28^38^5GAME-3^DITTMAR SHOT RESPONSE^JAHANGIR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe 0 0 0 23 0 0 0 0 6 0Drive 45 8 1 14 0 35 10 5 11 0Boast 5 1 4 1 2 3 2 4 0 0Drop 4 1 5 0 3 2 0 3 0 2Volley(S) 1 0 1 0 2 4 1 4 0 4Volley(L) 10 2 0 2 0 25 1 0 1 0Lob 1 0 0 2 0 1 0 0 5 0Total Shots 66 12 11 42 7 70 14 16 23 6123Table 4.9 Table of Two Way Chi-Square Analysis for Shot Response and Winner-Error Profiles for Two Matches Between Jansher and Dittmar1987 Final^: Jansher v Dittmar1988 Semi-Final: Jansher v DittmarOPPONENTSHOTJANSHER SHOT RESPONSEDittmar-87/Dittmar-88chiobs hic25^p^VDITTMAR SHOT RESPONSEJansher-87/Jansher-88chiobs chi 25 p VServe 2.105 2.773 # 1.564 2.773 #Drive 2.444 4.108 24.098 4.108 *Boast 4.362 2.773 * .219 2.773Drop 3.157 2.773 * .540 2.773Volley(S) .321 2.773 9.805 2.773 *Volley(L) 2.696 2.773 1.064 2.773Lob .132 2.773 # 2.277 2.773 #Winner .329 2.773 2.916 2.773 * #Error 2.427 2.773 3.392 2.773 * #* p<.25 significance# chi-square violation: less than five/ten expected observations in a cell124Table 4.10 Table of Shot Frequencies for Collapsed Levels for Two MatchesBetween Jansher and Dittmar1987 Final^Jansher v Dittmar1988 Semi-Final: Jansher v DittmarDITTMARJANSHER 1987 SHOT RESPONSEDrive Boast Drop Volley LobJANSHER 1988 SHOT RESPONSEDrive Boast Drop Volley LobServe 18^2^2 73^0 18^3^0 42^0Drive 391^35^38 43^22 297^29^20 24^0Boast 34^2^42 3^14 32^2^21 0^5Drop 31^1^44 1^30 26^1^19 0^15Volley(S) 30^4^24 1^20 19^1^21 3^13Volley(L) 79^12^7 16^11 90^9^2 11^3Lob 14^4^1 36^2 8^2^0 15^0Winner 6^2^11 3^0 5^0^7 2^0Error 10^6^15 9^2 3^6^13 7^2Let 4^1^8 3^2 2^0^5 2^0DITTMAR 1987 SHOT RESPONSE DITTMAR 1988 SHOT RESPONSEJANSHER Drive Boast Drop Volley Lob Drive Boast Drop Volley LobServe 22^4^1 73^0 8^1^0 40^0Drive 369^86^37 68^17 298^54^20 108^0Boast 23^3^17 0^8 18^3^14 1^5Drop 41^1^55 0^27 22^0^24 0^11Volley(S) 18^3^20 3^2 8^2^9 1^9Volley(L) 72^8^4 24^4 32^5^3 14^1Lob 25^3^8 58^1 5^3^1 25^0Winner 7^1^5 4^0 3^5^4 7^0Error 5^7^8 3^2 7^3^4 5^0Let 9^1^11 8^0 11^0^3 5^1125Table 4.11 Table of Two Way Chi-Square Analyses for Martin, Jansher, Jahangirand Dittmar Between Matches Against Different Opponents1988 Quarter-Final(Martin v Norman)^1988 Quarter-Final(Jansher v N'carrow)1988 Semi-Final(Martin v Jahangir)^1988 Semi-Final(Jansher v Dittmar)MARTIN SHOT RESPONSE^JANSHER SHOT RESPONSEOPPONENT^Norman/Jahangir Nancarrow/DittmarSHOT^chiobs chi 25 p V^chiobs chi 25 p VServe^.009^1.323^4^.010^1.323Drive 30.139^4.108 * 6.726^4.108 *Boast^8.269^2.773 * 4^.609^2.773^4Drop 2.773^2.773^4 2.625^2.773Volley(S)^12.240^2.773 * #^--Not Enough Data--Volley(L)^1.304^2.773 * 4 2.750^2.773^4Lob^1.373^1.323 * 4^.552^2.773^#1988 Quarter-Final (Jahangir v Thorne)1988 Semi-Final^(Jahangir v Martin)1988 Final^(Jahangir v Dittmar)JAHANGIR SHOT RESPONSEOPPONENT^Thorne/Martin^Martin/Dittmar^Thorne/DittmarSHOT^chiobs chi 25 p V chiobs chi 25 p V chiobs chi 25 p VServe^13.385^2.773 * 4^10.918^2.773 * 4^.631^1.323^4Drive 29.635^4.108 *^13.754^4.108 *^5.412^4.108 *Boast^.049^1.323^4^.003^1.323^#^.017^1.323Drop 6.719^2.773 *^4.346^2.773 *^8.903^2.773 * #Volley(S)^.196^1.323^4^.081^1.323^4^.576^1.323^4Volley(L)^9.434^2.773 *^.812^2.773^#^6.418^2.773 *Lob^.530^1.323^4^2.660^1.323^* 4^2.761^2.773^41988 Quarter-Final (Dittmar v Harris)1988 Semi-Final^(Dittmar v Jansher)1988 Final^(Dittmar v Jahangir)DITTMAR SHOT RESPONSEOPPONENT^Harris/Jansher^Jansher/Jahangir^Harris/JahangirSHOT^chiobs chi 25 p V chiobs chi 25 p V chiobs chi 25 p VServe^.002^1.323^4^8.012^1.323 * 4^8.097^1.323 * 4Drive 19.591^4.108 *^16.628^4.108 *^2.981^4.108Boast^.039^2.773 2.761^2.773 3.900^2.773 *Drop 4.068^2.773 *^.502^2.773^6.820^2.773 *Volley(S)^4.056^2.773 * 1.421^2.773 .967^2.773^4Volley(L)^8.175^2.773 *^10.545^2.773 *^.261^2.773^4Lob^3.818^1.323 * .099^1.323 .397^1.323^4126* p<.25 significance# chi-square violation: less than five/ten expected observations in a cellTable 4.12 Table of Two Way Chi-Square Analyses for Different Players AgainstDifferent Opponents1988 Quarter-Final(Norman v Martin)^1988 Quarter-Final(Jansher v Nancarrow)1988 Quarter-Final(Jahangir v Thorne) 1988 Quarter-Final(Dittmar v Harris)MARTIN/JAHANGIR SHOT RESPONSE^JANSHER/DITTMAR SHOT RESPONSESHOT^chiobs^ chiobschi ^p V^SHOT chi 25 p VServe^1.547^1.323 * #^Serve^.837^1.323^#Drive 2.092^4.108^Drive 20.893^4.108 *Boast^14.774^1.323 * Boast^1.945^2.773^#Drop 7.126^2.773 * #^Drop 5.622^2.773 *Volley(S)^--Not Enough Data--- Volley(S)^4.852^2.773 * #Volley(L)^.549^2.773^Volley(L)^.329^2.773^#Lob^.311^2.773^#^Lob^.097^1.323^#MARTIN/DITTMAR SHOT RESPONSE^JANSHER/JAHANGIR SHOT RESPONSESHOT^chiobs chi 25 p V^SHOT^chiobs chi 25 p VServe^.063^1.323^#^Serve^4.836^1.323 * #Drive 14.093^4.108 * Drive 23.899^4.108 *Boast^7.529^2.773 * #^Boast^4.722^1.323 * #Drop 3.091^2.773 * # Drop .363^2.773Volley(S)^--Not Enough Data---^Volley(S)^6.342^1.323 * #Volley(L)^3.261^2.773 *^Volley(L)^1.964^2.773Lob^.094^1.323^#^Lob^.060^2.773 * #MARTIN/JANSHER SHOT RESPONSE^JAHANGIR/DITTMAR SHOT RESPONSESHOT^chiobs chi 25 p V^SHOT^chiobs chi 25 p VServe^.942^1.323^#^Serve^1.199^1.323^#Drive 33.017^4.108 Drive 14.347^4.108 *Boast^1.218^1.323^#^Boast^6.763^2.773 * #Drop 5.375^2.773 * # Drop 8.890^2.773 *Volley(S)^--Not Enough Data---^Volley(S)^3.447^1.323^#Volley(L)^1.269^2.773^Volley(L)^4.531^2.773^#Lob^.072^1.323^#^Lob^1.764^2.773^#127Table 4.12 Continued1988 Semi-Final (Martin v Jahangir)^1988 Semi-Final (Jansher v Dittmar)MARTIN/JAHANGIR SHOT RESPONSE JANSHER/DITTMAR SHOT RESPONSESHOT^chiobs chi 25 p V^SHOT^chiobs chi 25 p VServe 5.762 1.323^*^# Serve 1.915 1.323 *Drive 2.406 4.108 Drive 47.547 4.108 *Boast 1.256 2.773^# Boast .854 2.773Drop 1.264 2.773 Drop 1.430 2.773Volley(S) --Not Enough Data--- Volley(S) .702 2.773Volley(L) .234 1.323^# Volley(L) 9.275 2.773 *Lob 2.720 1.323^*^# Lob 1.433 1.323 * #1988 Final (Jahangir v Dittmar)JAHANGIR/DITTMAR SHOTchiobs^chi 25RESPONSEp^VServe 5.914 1.323 * #Drive 4.299 4.108 *Boast 6.377 2.773 *Drop 7.782 2.773 *Volley(S) 4.139 2.773 * #Volley(L) 1.219 2.773Lob .026 1.323* p<.25 significance# chi-square violation: less than five/ten expected observations in a cell128Table 4.13 Table of Validation: Replication of Empirical Data (Final) forRallies, Shots and Outcome Frequencies By Match and By Game for Dittmar andJahangir1988 Final (Dittmar v Jahangir)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation:^ WinRevised Model:^0^1^1^14 127 857MATCH GAME^OBSERVED SIMULATION^OBSERVED^SIMULATIONDependent Variable^Mean^SD^Mean^SD Mean^SDRallies^ 110^114.8 21.2^36.7^6.5^36.0^6.9Shots 1485^1550.9 305.8^495.0 192.8 488.2 117.0Unconditional Winner^10^10.5^3.5 3.3^0.6^3.3^1.7Conditional Winner 24^25.1^6.3^8.0^3.0^7.9^2.6Unforced Error^32^33.1^6.8^10.7^1.2^10.5^2.8Forced Error 3^3.1^1.8 1.0^0.0^1.0^1.0Let^ 41^42.9^10.9^13.7^5.5^13.3^5.1WINNER Jahangir P{Win}=.998 P{Win}=.942DITTMAR(MATCH)^DITTMAR(GAME)OBSERVED SIMULATION^OBSERVED^SIMULATIONDependent Variable^Mean^SD^Mean^SD Mean^SDShots^ 733^768.0 153.8^244.3 95.9 242.1 59.0Unconditional Winner^3^3.2^2.0 1.0 1.0^1.0^1.0Conditional Winner 11^11.5^4.4^3.7 1.5^3.5^2.0Unforced Error^22^22.8^4.1 7.3 1.2^7.2^2.0Forced Error 3^3.1^1.8^1.0 0.0^1.0^1.0Let^ 17^17.6^5.7 5.7^1.2^5.4^2.8JAHANGIR(MATCH) JAHANGIR(GAME)OBSERVED SIMULATION^OBSERVED^SIMULATIONDependent Variable^Mean^SD^Mean^SD Mean^SDShots^ 752^782.9 152.1^250.7 96.9 246.1 58.1Unconditional Winner^7^7.4^2.6 2.3 0.6^2.3^1.3Conditional Winner 13^13.6^3.4^4.3 1.5^4.4^1.8Unforced Error^10^10.4^4.2 3.3 0.9^3.3^2.1Forced Error 0^0.0^0.0^0.0 0.0^0.0^0.0Let^ 24^25.2^7.0 8.0^6.2^7.9^3.4129Table 4.14 Table of Validation: Comparison of Empirical (Semi-Final) andPredicted (Quarter-Final) Data for Rallies, Shots and Outcome Frequencies ByMatch and By Game for Martin and Jahangir1988 Quarter-Final (Martin v Jahangir)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation: WinRevised Model: 102 136 163 194 223 182MATCH^ GAMEDependent VariableOBSERVED^SIMULATIONMean^SDOBSERVEDMean^SDSIMULATIONMean^SDRallies 89^127.8 26.5 29.7 5.9 31.4^4.2Shots 1041^1228.3 268.1 347.0 155.4 304.2^56.7Unconditional Winner 9^29.7 7.7 3.0 1.7 7.2^2.4Conditional Winner 25^45.2 10.3 8.3 4.2 11.1^2.7Unforced 33^28.3 7.4 11.0 1.7 7.1^2.4Forced 4^1.4 1.3 1.3 0.6 0.3^0.6Let 18^23.1 7.1 6.0 3.0 5.7^2.6WINNER Jahangir P{Win}=0.599 P{Win}=.554MARTIN(MATCH)OBSERVED^SIMULATIONDependent Variable^Mean^SDMARTIN(GAME)OBSERVED^SIMULATIONMean^SD^Mean^SDShots 517 611.5 134.4 172.3 78.3 151.7 28.6Unconditional Winner 5 13.9 4.7 1.7 1.2 3.4 1.7Conditional Winner 8 21.0 6.3 2.7 2.1 5.1 2.1Unforced 21 13.5 4.1 7.0 2.0 3.4 1.7Forced 3 0.0 0.0 1.0 0.0 0.0 0.0Let 9 9.1 3.4 3.0 2.6 2.3 1.6JAHANGIR(MATCH)OBSERVED^SIMULATIONDependent Variable^Mean^SDJAHANGIR(GAME)OBSERVED^SIMULATIONMean^SD^Mean^SDShots 524 616.8 133.9 174.7 77.2 152.5 28.3Unconditional Winner 4 15.8 4.8 1.3 0.6 3.8 1.9Conditional Winner 17 24.2 6.0 5.7 2.5 6.0 2.2Unforced 12 14.9 4.9 4.0 2.6 3.6 1.8Forced 1 1.4 1.3 0.3 0.6 0.3 0.6Let 9 14.0 5.0 3.0 2.0 3.4 1.9130Table 4.15 Table of Validation: Comparison of Empirical (Semi-Final + ) andPredicted (Quarter-Final) Data for Rallies, Shots and Outcome Frequencies ByMatch and By Game for Jansher and DittmarQuarter-Final (Jansher v Dittmar)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation:^ WinRevised Model: 570 281 112^16^16^5MATCH GAMEDependent VariableOBSERVED SIMULATIONMean^SDOBSERVEDMean^SDSIMULATIONMean^SDRallies 115 123.1 28.5 28.0 6.1 34.8^5.5Shots 1649 2621.7 649.2 402.3 179.7 740.9^168.4Unconditional Winner 12 23.1 6.4 3.0 0.8 6.4^2.2Conditional Winner 22 31.9 8.1 5.3 3.7 9.0^2.5Unforced 48 32.4 9.0 11.5 3.3 9.2^2.8Forced 4 0.0 0.0 1.0 0.8 0.0^0.0Let 29 35.6 10.8 7.3 2.4 10.1^4.0WINNER Dittmar P{Win}=.037 P{Win}=.184JANSHER(MATCH)OBSERVED^SIMULATIONDependent Variable^Mean^SDJANSHER(GAME)OBSERVED^SIMULATIONMean^SD^Mean^SDShots 799 1321.2 324.2 200.0 91.8 373.6 83.9Unconditional Winner 8 16.4 4.4 1.8 0.5 4.6 1.9Conditional Winner 6 20.3 4.7 1.8 2.2 5.7 2.0Unforced 28 17.8 6.6 7.0 2.7 5.1 2.4Forced 3 0.0 0.0 0.8 1.0 0.0 0.0Let 9 16.1 5.7 2.3 1.5 4.6 2.5DITTMAR(MATCH)OBSERVED^SIMULATIONDependent Variable^Mean^SDDITTMAR(GAME)OBSERVED^SIMULATIONMean^SD^Mean^SDShots 810 1300.4 325.0 202.3 87.9 367.4 84.5Unconditional Winner 4 6.8 3.3 1.3 0.5 1.9 1.4Conditional Winner 15 11.6 4.8 3.5 2.4 3.3 1.8Unforced 18 14.5 3.9 4.5 1.7 4.1 1.8Forced 1 0.0 0.0 0.3 0.5 0.0 0.0Let 20 19.5 6.6 5.0 2.2 5.6 2.6131+ Incomplete data in Game 2Table 4.16 Table of Validation: Comparison of Empirical (Final) and Predicted(Quarter-Final) Data for Rallies, Shots and Outcome Frequencies By Match andBy Game for Dittmar and JahangirQuarter-Final (Dittmar v Jahangir)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation:^ WinRevised Model:^0^1^1^29 185 784MATCH GAMEDependent VariableOBSERVED SIMULATIONMean^SDOBSERVEDMean^SDSIMULATIONMean^SDRallies 110 99.6 18.9 36.7 6.5 30.9^5.0Shots 1485 1226.2 256.2 495.0 192.8 379.9^85.3Unconditional Winner 10 19.7 4.6 3.3 0.6 6.1^2.0Conditional Winner 24 32.9 7.0 8.0 3.0 10.2^2.6Unforced 32 23.5 6.8 10.7 1.2 7.3^2.8Forced 3 0.0 0.0 1.0 0.0 0.0^0.0Let 41 23.5 6.9 13.7 5.5 7.4^3.1WINNER Jahangir P{Win}=.998 P{Win}=.928DITTMAR(MATCH)OBSERVED^SIMULATIONDependent Variable^Mean^SDDITTMAR(GAME)OBSERVED^SIMULATIONMean^SD^Mean^SDShots 733 596.9 127.2 244.3 95.9 185.3 43.0Unconditional Winner 3 2.8 1.8 1.0 1.0 0.8 0.9Conditional Winner 11 9.4 3.7 3.7 1.5 2.9 1.7Unforced 22 7.5 2.6 7.3 1.2 2.3 1.4Forced 3 0.0 0.0 1.0 0.0 0.0 0.0Let 17 8.9 3.4 5.7 1.2 2.8 1.8JAHANGIR(MATCH)OBSERVED^SIMULATIONDependent Variable^Mean^SDJAHANGIR(GAME)OBSERVED^SIMULATIONMean^SD^Mean^SDShots 752 629.3 129.1 250.7 96.9 194.6 42.4Unconditional Winner 7 16.9 3.9 2.3 0.6 5.2 1.9Conditional Winner 13 23.6 4.6 4.3 1.5 7.3 2.0Unforced 10 16.0 5.7 3.3 0.9 5.0 2.5Forced 0 0.0 0.0 0.0 0.0 0.0 0.0Let 24 14.5 5.0 8.0 6.2 4.6 2.4132Table 4.17 Table of Validation: Comparison of Empirical (Final) and Predicted(Semi-Final) Data for Rallies, Shots and Outcome Frequencies By Match and ByGame for Dittmar and Jahangir1988 Semi-Final (Dittmar v Jahangir)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation:^ WinRevised Model: 59^77 110 211 300 243MATCH^ GAMEDependent VariableOBSERVED SIMULATIONMean^SDOBSERVEDMean^SDSIMULATIONMean^SDRallies 110 148.3 31.2 36.7 6.5 37.2^5.5Shots 1485 2044.7 450.4 495.0 192.8 516.4^103.4Unconditional Winner 10 15.0 4.6 3.3 0.6 3.8^1.8Conditional Winner 24 36.6 8.9 8.0 3.0 9.2^2.6Unforced 32 51.1 11.7 10.7 1.2 12.7^2.9Forced 3 0.0 0.0 1.0 0.0 0.0^0.0Let 41 45.6 12.4 13.7 5.5 11.5^4.2WINNER Jahangir P{Win}=.754 P{win}=.622DITTMAR(MATCH)OBSERVED^SIMULATIONDependent Variable^Mean^SDDITTMAR(GAME)OBSERVED^SIMULATIONMean^SD^Mean^SDShots 733 1017.7 225.5 244.3 95.9 257.4 52.0Unconditional Winner 3 6.2 2.9 1.0 1.0 1.6 1.3Conditional Winner 11 13.4 4.8 3.7 1.5 3.3 1.8Unforced 22 22.8 5.5 7.3 1.2 5.6 2.1Forced 3 0.0 0.0 1.0 0.0 0.0 0.0Let 17 26.5 7.9 5.7 1.2 6.7 2.9JAHANGIR(MATCH)OBSERVED^SIMULATIONDependent Variable^Mean^SDJAHANGIR(GAME)OBSERVED^SIMULATIONMean^SD^Mean^SDShots 752 1027.0 225.0 250.7 96.9 259.0 51.5Unconditional Winner 7 8.7 3.1 2.3 0.6 2.1 1.4Conditional Winner 13 23.2 5.8 4.3 1.5 5.8 2.1Unforced 10 28.4 8.1 3.3 0.9 7.1 2.5Forced 0 0.0 0.0 0.0 0.0 0.0 0.0Let 24 19.1 6.2 8.0 6.2 4.9 2.5133Table 4.18 Table of Validation Data: Replication of the Empirical (Semi-Final) Playing Profiles of Dittmar and Jahangir in Forecasting (Final)Outcome.1988 Semi-Final (Jansher v Dittmar)^1988 Semi-Final (Martin v Jahangir)1988 Model Simulation Data (n=5)^1988 Model Simulation Data (n=5)DITTMAR/SIMULATED SHOT RESPONSEchiobs^chi 25^p^VJAHANGIR/SIMULATED SHOT RESPONSEchiobs^chi 25^p^VServe .024 1.323 # Serve .038 2.773Drive .772 4.108 Drive .443 4.108Boast .837 2.773 Boast .042 2.773 #Drop .566 2.773 Drop .093 2.773Volley(S) 1.184 2.773 Volley(S) .070 2.773 #Volley(L) .706 2.773 Volley(L) .016 4.108 4Lob .003 2.773 # Lob .005 1.323 #Winner .368 2.773 Winner .351 2.773Error 1.246 2.773 Error .111 1.323134Table 4.19 Table of Jahangir and Dittmar's Shot Frequencies from Empirical(Semi-Final) and Forecasted (Final) Data1988 Semi-Final(Jansher v Dittmar)^1988 Semi-Final(Martin v Jahangir)1988 Model Simulation Data (n=5)^1988 Model Simulation Data (n=5)DITTMAR SHOT RESPONSE JAHANGIR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe 8 1 0 40 0 15 6 0 15 0Drive 298 54 20 108 0 225 18 18 25 1Boast 18 3 14 1 5 7 1 13 0 2Drop 22 0 25 0 11 16 2 7 0 9Volley(S) 8 2 9 1 9 5 0 6 0 1Volley(L) 32 5 3 14 1 55 2 4 3 2Lob 5 3 1 25 0 6 1 0 6 0Winner 3 5 4 7 0 7 1 11 2 0Error 7 3 4 5 0 1 4 6 2 0SIMULATION DATA (FINAL)Drive Boast Drop Volley LobSIMULATION DATA (FINAL)Drive Boast Drop Volley LobServe 13 3 0 68 0 29 13 0 30 0Drive 368 68 32 136 0 363 31 26 46 2Boast 17 6 18 1 7 22 4 44 0 6Drop 40 0 39 0 14 37 5 16 0 24Volley(S) 6 1 3 1 7 33 0 35 0 5Volley(L) 44 5 2 16 2 130 7 8 7 5Lob 6 5 0 31 0 12 1 0 16 0Winner 2 8 3 8 0 13 1 17 2 0Error 8 2 4 10 0 1 7 16 4 0135Table 4.20 Table of Validation Data: Comparison of Forecasted (Final) andEmpirical (Final) Playing1988 Final (Dittmar v Jahangir)1988 Model Simulation Data (n=5)DITTMAR/SIMULATED SHOT RESPONSE^JAHANGIR/SIMULATED SHOT RESPONSEchiobs chi 25 p V chiobs chi 25 p VServe 8.601 1.323 * # Serve 13.951 2.773 * #Drive 20.350 4.108 * Drive 13.364 4.108 *Boast 1.122 2.773 Boast .516 2.773 #Drop 1.700 2.773 Drop 7.929 2.773 *Volley(S) 1.683 2.773 Volley(S) .042 2.773 #Volley(L) 9.789 2.773 Volley(L) 1.271 2.773Lob .774 2.773 # Lob 2.758 1.323 * #Winner 4.005 2.773 Winner 5.133 2.773 * #Error 11.287 2.773 Error 6.675 1.323 * #136Table 4.21 Table of Jahangir and Dittmar's Shot Frequencies from Forecasted(Final) and Empirical (Final) Data1988 Final (Dittmar v Jahangir)1988 Model Simulation Data (n=5)DITTMAR SHOT RESPONSE^JAHANGIR SHOT RESPONSEDrive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe 1 0 0 74 0 5 2 0 28 0Drive 277 50 13 47 0 252 40 18 60 1Boast 16 1 24 1 12 18 4 28 0 3Drop 22 2 31 0 12 38 1 21 0 7Volley(S) 13 1 9 0 7 14 1 15 1 2Volley(L) 56 5 7 4 0 72 7 3 7 0Lob 3 2 0 8 0 5 2 0 22 0Winner 5 0 4 5 0 9 1 5 5 0Error 3 8 9 3 2 2 1 3 4 0SIMULATION DATA (FINAL)^SIMULATION DATA (FINAL)Drive Boast Drop Volley Lob^Drive Boast Drop Volley LobServe 13 3 0 68 0 29 13 0 30 0Drive 368 68 32 136 0 363 31 26 46 2Boast 17 6 18 1 7 22 4 44 0 6Drop 40 0 39 0 14 37 5 16 0 24Volley(S) 6 1 3 1 7 33 0 35 0 5Volley(L) 44 5 2 16 2 130 7 8 7 5Lob 6 5 0 31 0 12 1 0 16 0Winner 2 8 3 8 0 13 1 17 2 0Error 8 2 4 10 0 1 7 16 4 0137Table 4.22 Table of Predicted Match Outcomes from Various Superimpositions ofthe Shot Response Profiles from the Semi-Finals and Final on theCorresponding Quarter-Final Playing ProfilesSR : Shot Response ProfileW-E: Winner-Error ProfileF : FinalSF : Semi-FinalQF : Quarter-FinalMartinSR^W-EJahangirSR^W-E3-0 3-1 3-2 2-3 1-3 0-3WinWin Pred-ErrorQF^QF QF^QF .102 .136 .163 .194 .223 .182 .599SF^QF QF^QF .024 .082 .095 .186 .295 .318 .799QF^QF SF^QF .046 .101 .157 .197 .261 .238 .696SF^QF SF^QF .044 .092 .108 .190 .295 .271 .756-.157Jansher Dittmar 3-0 3-1 3-2 2-3 1-3 0-3 WinSR^W-E SR^W-E WinQF^QF QF^QF .570 .281 .112 .016 016 .005 .037SF^QF QF^QF .508 .308 .121 .034 023 .006 .063QF^QF SF^QF .653 .258 .075 .008 004 .002 .014SF^QF SF^QF .510 .300 .121 .031 023 .015 .069-.032Dittmar Jahangir 3-0 3-1 3-2 2-3 1-3 0-3 WinSR^W-E SR^W-E WinQF^QF QF^QF .000 .001 .001 .029 .185 .784 .998F^QF QF^QF .000 .000 .001 .009 .103 .887 .999QF^QF F^QF .000 .005 .009 .057 .238 .691 .986F^QF F^QF .001 .000 .001 .028 .158 .812 .998.000SF^SF SF^SF .059 .077 .110 .211 .300 .243 .754F^SF SF^SF .030 .067 .099 .196 .319 .289 .804SF^SF F^SF .026 .060 .094 .183 .294 .343 .820F^SF F^SF .019 .051 .066 .169 .315 .380 .864-.110138Table 4.23 Table of Predicted Match Outcomes from Various Superimpositions ofthe Winner-Error Profiles from the Quarter-Final and Semi-Finals on theCorresponding Semi-Finals and Final Playing Profiles.SR : Shot Response ProfileW-E: Winner-Error ProfileF : FinalSF : Semi-FinalQF : Quarter-FinalMartinSR^W-EJahangirSR^W-E3-0 3-1 3-2 2-3 1-3 0-3WinWin Pred-ErrorSF^SF SF^SF .000 .001 .000 .022 .139 .844 .999SF^QF SF^SF .360 .318 .189 .062 .044 .027 .133SF^SF SF^QF .000 .000 .000 .000 .018 .982 1.000SF^QF SF^QF .044 .092 .108 .190 .295 .271 .756.243Jansher Dittmar 3-0 3-1 3-2 2-3 1-3 0-3 WinSR^W-E SR^W-E WinSF^SF SF^SF .002 .005 .008 .070 .265 .650 .985SF^QF SF^SF .508 .308 .121 .034 .023 .006 .063SF^SF SF^QF .000 .000 .001 .006 .084 .909 .999SF^QF SF^QF .510 .300 .121 .031 .023 .015 .069.916Dittmar Jahangir 3-0 3-1 3-2 2-3 1-3 0-3 WinSR^W-E SR^W-E WinF^F F^F .000 .001 001 014 127 857 .998F^SF F^F .032 .044 078 198 325 323 .846F^F F^SF .000 .000 0^0 007 072 921 1.000F^SF F^SF .019 .051 066 169 315 380 .864.134F^F F^F .000 .001 001 014 127 857 .998F^QF F^F .018 .037 046 158 324 417 .899F^F F^QF .000 .000 000 000 035 965 1.000F^QF F^QF .001 .000 001 028 158 812 .998.000139Table 4.24 Table of 1988 World Ranking and Simulated Player Rank Ordering forthe Previous (McGarry and Franks, 1993) and Reported Stochastic Models fromthe Quarter-Final DataWorldRankingRank OrderPrevious Model^Reported ModelNorman - 6 6Martin 5 2 2Jahangir 1 1 1Thorne - 5 5Jansher 2 3 3Nancarrow - 7 7Dittmar 3 4 4Harris - 8 8Table 4.25 Table of Predicted Match Wins for the Previous Stochastic Model(McGarry and Franks, 1993) from the Quarter-Final DataSERVERRECEIVERNorman Martin Jahangir Thorne Jansher Nancarrow Dittmar HarrisNorman .017 .001 .450 .038 .679 .122 .922Martin .982  .221 .999 .913 .998 .988 1.000Jahangir .998 .786 ^ 1.000 .985 .999 .999 1.000Thorne .547 .001 .000 ^ .009 .848 .080 .992Jansher .960 .093 .015 .991 ^ .990 .886 1.000Nancarrow .296 .001 .001 .162 .011 ^ .024 .815Dittmar .869 .011 .000 .927 .116 .976 ^ .999Harris .073 .000 .000 .008 .000 .176 .001 ^Table 4.26 Table of Predicted Match Winsfrom the Quarter-Final Datafor the Reported Stochastic ModelSERVERRECEIVERNorman Martin Jahangir Thorne Jansher Nancarrow Dittmar HarrisNorman .007 .002 .087 .001 .567 .078 .933Martin .990  .401 .998 .753 .998 .992 .999Jahangir .996 .597 ^ 1.000 .936 1.000 .998 1.000Thorne .906 .001 .000 ^ .003 .908 .081 1.000Jansher .998 .236 .062 .997  .987 .963 1.000Nancarrow .440 .002 .000 .090 .011 ^ .080 .864Dittmar .921 .007 .002 .911 .038 .914 ^ .998Harris .068 .000 .000 .000 .000 .123 .002 ^140Table 4.27 Table of Discernible Difference Between the Previous and ReportedModel in Predicting Match Outcome Between Players from the Quarter-Final DataPlayerNorman Martin Jahangir Thorne Jansher Nancarrow Dittmar Harris + -Norman ^MartinJahangir* ^** *+-**----**-*****0^41^10 2Thorne + * * ^ * + * * 2^0Jansher + + + * ^ * + * 4 0Nancarrow + * * - * ^ + + 3^1Dittmar + * * - - * 1 2Harris * * * * * - * ^ 0^1Loss 4 1 2 0 0 1 2 1+ positive increase in model prediction- negative increase in model prediction* don't care value141Table 4.28 Table Summary of Player's Winners and Errors and Computed Ratiosfrom Empirical DataPlayerRound^Player^Opponent W w Q q W/Q^w/q 14-1-wN -1-qQuarter Norman^Martin^4 13 17^2^.24^6.50^.89Quarter Martin^Norman^10 16 11^1^.91 16.00^2.17Quarter Jansher^Nancarrow 6 11^6^3 1.00^3.67^1.89Quarter Nancarrow Jansher^11^8 24^3^.46^2.67 .70Quarter Jahangir Thorne^13 26 12^3 1.08^8.67^2.60Quarter Thorne^Jahangir^1^4^5^1^.20^4.00 .83Quarter Dittmar^Harris^3^9^8^0^.38^I^1.50Quarter Harris^Dittmar^8^9 33^2^.24^4.50 .49Semi^Martin^Jahangir^5^8 21^3^.24^2.67^.54Semi^Jahangir Martin^4 17 12^1^.33 17.00^1.62Semi^Jansher^Dittmar^7^7 28^3^.25^2.33^.45Semi^Dittmar^Jansher^5 14 18^1^.28 14.00^1.00Final^Dittmar^Jahangir^3 11 22^3^.14^3.67^.56Final^Jahangir Dittmar^7 13 10^0^.70 I^2.00I indeterminate numberTable 4.29 Table Summary of Player's Winners and Errors and Computed RatiosProjected from Empirical Data to Predicted Future Performance in the FinalPlayerRound Player Opponent^w* q* W^w Q^q W/Q^w/q W+w/Q+qQuarter Dittmar Jahangir 1.07^I^3^9.6^8^0^.38^I^1.58Quarter Jahangir Dittmar^.79 .29 13 20.5 12 .87 1.08 23.56^2.60Semi^Dittmar Jahangir^.68 1.84^5^9.5 18 1.84^.28 13.97^.78Semi^Jahangir Dittmar^.76 1.63^4 12.9 12 1.63^.33 16.97^1.32Final^Dittmar Jahangir 1.00 1.00^3^11 22^3^.14^3.67^.56Final^Jahangir Dittmar^1.00 1.00^7^13 10^0^.70 I^2.00w and q * are the conditional winner and forced error coefficients and arecomputed from equations (3.1) and (3.2) respectively.I indeterminate number142Table 4.30 Table of Jahangir's Quarter-Final Playing Profile1988 Quarter-Final (Jahangir v Thorne)Shot Condition*S^A^B^C D E F G H I J K L M N TotalS^0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0A 1 56 24 1 6 2 3 1 1 17 7 0 0 1 6 126B 0 34 11 2 5 0 4 0 0 13 4 1 0 0 3 77C 0 0 0 0 2 1 0 0 0 0 2 0 0 0 1 6D 2 19 9 0 4 0 0 1 0 4 4 0 0 1 2 36E 0 6 2 0 18 11 6 3 2 0 0 0 0 0 2 50F 0 2 0 1 4 1 1 1 1 1 1 0 0 0 0 13G 0 14 7 0 0 0 0 0 0 1 4 0 0 0 5 31H 0 6 0 0 0 0 0 0 0 1 1 0 0 0 0 8I 9 1 2 0 0 0 0 0 0 0 0 0 0 2 12 26J 12 10 2 0 0 0 0 0 0 2 3 0 0 0 2 31K 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1L 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 5M 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1N 0 0 0 1 1 2 3 1 0 0 0 1 0 0 0 9W 0 1 1 0 0 4 4 3 0 0 0 0 0 0 0 13w 0 4 0 3 1 10 1 2 2 2 1 0 0 0 0 26Q 0 2 1 0 2 4 0 2 0 0 0 0 1 0 0 12q 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 3R 0 3 1 0 0 2 0 1 0 4 0 0 0 0 0 11* See Table 3.1. for shot and outcome codes143Table 4.31 Table of Thorne's Quarter-Final Playing Profile1988 Quarter-Final (Jahangir v Thorne)Shot Condition*S^A^B^C D E F G H I J K L M N TotalS^0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0A 6 62 41 0 10 0 1 6 4 10 13 0 0 1 1 155B 0 21 10 0 8 4 2 2 1 5 5 0 1 0 0 59C 0 0 3 0 1 0 0 0 0 0 1 0 0 0 0 5D 3 14 14 0 2 1 0 1 0 2 7 0 0 0 0 44E 0 4 0 1 6 6 1 3 1 1 0 0 1 0 0 24F 0 0 0 0 4 6 1 4 0 0 0 1 1 0 0 17G 1 4 1 0 0 0 0 0 0 0 1 0 0 0 1 8H 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 5I 23 6 2 0 1 0 0 0 0 1 1 0 0 0 6 40J 21 1 2 0 0 0 0 0 0 0 1 0 0 0 1 26K 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 2L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0M 0 1 0 0 0 1 0 2 0 0 0 0 0 0 0 4N 0 1 0 1 10 11 3 5 0 0 1 0 1 0 0 33W 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1w 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 4Q 0 0 1 0 1 2 0 0 1 0 0 0 0 0 0 5q 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1R 0 2 0 0 1 0 0 1 0 1 0 0 0 0 0 5* See Table 3.1. for shot and outcome codes144Table 4.32 Table of Effect of Optimal Strategy on Quarter-Final BetweenJahangir and Thorne1988 Quarter-Final (Jahangir v Thorne)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 959 38 3 0 0 0Optimization: 933 62 4 1 0 0145Table 4.33 Table of Jahangir's Semi-Final Playing Profile1988 Semi-Final (Martin v Jahangir)Shot Condition*S^A^B^C D E F G H I J K L M N TotalS^0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0A 12 113 36 0 3 5 3 0 0 10 20 0 0 1 2 205B 3 55 21 1 3 8 0 4 1 15 10 0 0 1 2 124C 0 2 1 0 0 2 0 0 0 0 1 0 0 0 0 6D 6 11 4 0 1 0 0 0 0 1 0 0 0 1 0 24E 0 14 2 1 9 3 0 3 2 0 3 0 1 0 0 38F 0 2 0 0 3 2 2 0 0 1 0 0 0 0 0 10G 1 4 4 0 0 0 0 0 0 0 1 0 0 0 0 10H 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 2I 7 3 2 0 0 0 0 0 0 1 0 0 0 0 1 14J 7 8 3 0 0 0 0 0 0 0 1 0 0 1 3 23K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0M 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1N 0 0 0 0 2 8 1 1 0 2 0 0 0 0 0 14W 0 1 0 0 0 0 3 0 0 0 0 0 0 0 0 4w 0 3 3 0 1 7 1 1 0 0 1 0 0 0 0 17Q 0 1 0 2 1 6 0 2 0 0 0 0 0 0 0 12q 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1R 0 5 0 1 0 1 0 0 1 1 0 0 0 0 0 9* See Table 3.1. for shot and outcome codes146Table 4.34 Table of Martin's Semi-Final Playing Profile1988 Semi-Final (Martin v Jahangir)Shot Condition*S^A^B^C D E F G H I J K L M N TotalS^0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0A 10 113 62 0 4 4 3 1 1 9 11 0 0 0 2 220B 1 32 25 0 5 5 1 1 0 1 6 0 0 0 0 77C 0 2 2 0 1 0 0 0 0 0 0 0 0 0 0 5D 2 8 11 0 1 0 0 0 0 1 1 0 0 0 0 24E 0 16 5 1 6 8 1 1 0 1 2 0 0 0 0 41F 0 1 0 0 2 2 0 2 0 0 0 0 0 0 0 7G 1 8 4 0 0 0 0 0 0 0 0 0 0 1 0 13H 3 3 0 0 0 0 1 0 0 0 0 0 0 0 1 8I 16 3 9 0 0 0 0 0 0 0 2 0 0 0 4 34J 20 7 3 0 0 0 0 0 0 0 0 0 0 0 7 30K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0L 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1M 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 4N 0 0 0 1 3 4 0 1 0 0 0 0 0 0 0 9W 0 0 1 0 0 2 0 1 1 0 0 0 0 0 0 5w 0 0 1 1 0 3 0 0 0 2 1 0 0 0 0 8Q 0 2 1 1 2 6 1 4 4 0 0 0 0 0 0 21q 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 3R 0 4 0 0 0 2 0 1 0 2 0 0 0 0 0 9* See Table 3.1. for shot and outcome codes147Table 4.35 Table of Effect of Optimal Strategy on Semi-Final Between Martinand Jahangir1988 Quarter-Final (Martin v Jahangir)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 0 1 0 22 156 821Optimization: 0 0 3 17 162 818148Table 4.36 Table of Jahangir's Final Playing Profile1988 Semi-Final (Dittmar v Jahangir)Shot Condition*S^A^B^C D E F G H I J K L M N TotalS^0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0A 3 97 61 3 4 7 2 1 2 19 26 0 0 0 1 226B 2 62 32 0 11 26 3 8 2 17 10 0 1 0 4 178C 0 2 0 1 2 1 0 0 1 0 0 0 0 0 0 7D 2 24 14 0 1 0 0 0 0 2 5 0 0 0 2 50E 0 11 5 1 25 15 5 5 8 3 0 0 0 0 0 78F 0 2 0 1 1 1 0 0 1 0 0 1 0 0 0 7G 1 14 11 0 0 0 0 0 1 1 3 0 0 0 1 32H 1 1 3 0 0 0 0 0 0 0 0 0 0 0 0 5I 11 0 8 0 0 0 0 0 0 0 3 0 0 0 17 39J 15 14 6 0 0 0 0 0 0 0 0 0 0 0 4 39K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0L 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 3M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0N 0 1 0 0 3 7 0 1 1 0 0 0 0 0 0 13W 0 1 2 0 0 1 0 2 1 0 0 0 0 0 0 7w 0 1 5 1 0 4 0 1 0 0 1 0 0 0 0 13Q 0 2 0 1 0 3 0 2 1 1 0 0 0 0 0 10q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0R 0 5 1 1 0 10 0 3 0 4 0 0 0 0 0 24* See Table 3.1. for shot and outcome codes149Table 4.37 Table of Dittmar's Final Playing Profile1988 Semi-Final (Dittmar v Jahangir)Shot Condition*S^A^B^C D E F G H I J K L M N TotalS^0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0A 1 96 93 1 6 6 0 5 0 15 17 0 0 0 2 242B 0 54 34 2 7 12 4 8 0 10 14 0 0 0 1 146C 0 2 3 0 1 2 0 0 0 2 0 0 0 0 0 10D 0 26 19 0 0 0 0 1 0 1 2 0 0 0 2 51E 0 8 4 0 18 28 1 5 0 2 4 0 2 0 0 72F 0 1 0 0 6 1 1 1 1 1 0 0 0 0 0 12G 0 12 1 0 0 0 0 0 0 3 0 0 0 0 1 17H 10 8 2 0 0 0 0 0 0 0 0 0 0 0 0 20I 34 5 1 0 0 0 0 0 0 0 0 0 0 0 4 44J 30 4 13 0 0 0 0 0 0 0 1 0 0 0 2 50K 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 2L 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0N 0 0 0 0 12 11 1 4 2 0 0 0 1 0 0 31W 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 3w 0 0 3 0 0 3 0 1 2 0 2 0 0 0 0 11Q 0 1 2 3 2 7 2 0 2 0 1 0 0 0 2 22q 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 3R 0 8 1 0 0 4 0 1 0 2 0 1 0 0 0 17* See Table 3.1. for shot and outcome codes150Table 4.38 Table of Effect of Optimal Strategy on Final Between Dittmar andJahangir1988 Quarter-Final (Dittmar v Jahangir)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 0 1 1 14 127 857Optimization: 0 2 0 8 137 853151Table 4.39 Table of Simulated (N=5) Effects of Optimal Strategy Used BetweenGames by Dittmar and Jahangir1988 Final (Dittmar v Jahangir)Observed ProfileDittmar^JahangirMean^SD^Mean^SDOptimized ProfileDittmar^JahangirMean^SD^Mean^SDGAME-1:Observed ProfileDittmar .129 .007 .408 .005 .149 .004Jahangir .871 .007 .849 .009 .504 .011Optimized ProfileDittmar .592 .005 .151 .009 .169 .012Jahangir .851 .004 .496 .011 .831 .012GAME-2:Observed ProfileDittmar .122 .009 .502 .012 .139 .011Jahangir .878 .009 .866 .015 .618 .022Optimized ProfileDittmar .498 .012 .134 .015 .163 .012Jahangir .861 .011 .382 .022 .837 .012152Code Shot Code Shot Code OutcomeS Serve G Volley(S) W Unconditional WinnerA Drive H XC-Volley(S) w Conditional WinnerB XC-Drive I Volley(L) Q Unforced ErrorC Boast J XC-Volley(L) q Forced ErrorD XC-Boast K Volley-Boast R LetE Drop L XC-Volley-BoastF XC-Drop M LobN XC-Lob153Fig.3.1 Schematic Representation of a Squash CourtMAIN MENU154DATAPRESENTATIONSIMULATIONDATACOLLECTIONDATAANALYSIS^/filename.DAT filename.PRO filename.STSFig.3.2 Schematic Representation of Application Program HierarchyResponse(Shot)PositionP1^P2^P3^P4Condition (preceding shot)S^A^B^C^D^E^F^G^H^I^J^K^L^M^NS ServeA DriveB XC-DriveC BoastD XC-BoastE DropF XC-DropG Vol(S)H XC-Vol(S)I Vol(L)J XC-Vol(L)K Vol-BoastL XC-Vol-BtM LobN XC-LobPlayer's ShotDistributionWith Respect toPositionPlayer's Shot Distribution WithRespect to the Opponent's ShotW^U-Winnerw^C-WinnerQ^U-Errorq^F-ErrorR^LetPlayer'sOutcomeDistributionWith Respectto PositionPlayer's Outcome DistributionWith Respect to His/Her ShotW . U-Winnerw^C-WinnerQ ' U-Errorq ' F-ErrorNO DATA CARRIED Opponent's Outcome^DistributionWith Respect to His/Her Shot155Fig.3.3 File Representation of a Player's Match Profile.SHOT=A to N^/  BALL-IN1 ^H BALL-IN/OUT\ BALL-OUTI■yla etcN BALL-INSHOT=SBALL-OUTBALL-IN/OUTUnconditionalConditional7 WINNEREND-I ERRORUnforcedForced LET/  BALL-INSHOT=A to N -H BALL-IN/OUTBALL-OUT156Fig.3.4 Stochastic Model for Predicting Squash Match PlayEntrants:Ross NormanRodney MartinJahangir KhanRoss ThorneJansher KhanTristan NancarrowChris DittmarDel HarrisNorman(0)^—1--^Martin(0)Martin(3)^----^Jahangir(3)^--Jahangir(3) ----^Jahangir(3)^--Thorne(0)-- JahangirJansher(3)^--^Jansher(1)^—Nancarrow--^Dittmar(0)^--Dittmar(3)^--^Dittmar(3)^--Harris(0)^--Fig.4.1 Observed Tournament Progression from the Quarter-Final Through Finalof the Men's Canadian Open Squash Championship, 1988.157APPENDIX AAppendix A comprises data from three disparate video training excerptsrecorded by the principal observer on two independent occassions and a singlerecording of the first video training excerpt by the expert observer.Analysis of these data produced two dimensional matrices of position, shotand outcome data for any two independent observations. The columns denotethe observations from the first observer, the rows the observations from thesecond observer. The superdiagonal (top left through bottom right) denotethe level of agreement between the two observations.The asterisks pertain to the outcome data and denote "unpaired" values whichwere recorded during the analysis of the observed data. These values ariseif the analysis was unable to pair observer outcomes because of a differencein the number of shots recorded by the two observers. Such differences arelisted and outcome pairings were resolved before proceeding with thecomputation of the total percent agreement and kappa coefficients.158intra-l.relReliability rating between:tape#1-1.sqdtape#1-2.sqdError: Unable to match 5 rallies from 20POSITION BY POSITION MATRIXa^b^c^d^e^f^g h^i^jklmnopa^3^0^0^0^1^0^0^0^0^0^0^0^0^0^0^0b^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0c^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0d 1^0^0^7^0^0^0^0^0^0^0^0^0^0^0^0e 0^0^0^0 13^0^0^0^0^0^0^0^0^0^0^0f^0^1^0^0^0^1^0^0^0^0^0^0^0^0^0^0g 0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0h 0^0^0^0^0^0^0 17^0^0^0^0^0^0^0^0i^0^0^0^0^0^0^0^0 38^0^0^0^2^0^0^0j^0^0^0^0^0^0^0^0^0^2^0^0^0^0^0^0k^0^0^0^0^0^0^0^0^0^2^4^1^0^0^0^00^0^0^0^0^0^0^0^0^0^0^1 50^0^0^0^2m^0^0^0^0^0^0^0^0^2^0^0^0 41^1^0^0n 0^0^0^0^0^0^0^0^0^0^0^0^0 15^0^0o 0^0^0^0^0^0^0^0^0^0^0^0^0^0^7^0p^0^0^0^0^0^0^0^0^1^0^0^2^0^0^1 55SHOT(OUTCOME) BY SHOT(OUTCOME) MATRIXS A B C D E F G H I J K L M N W w Q q R *S 20^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0A^0^79^4^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0B 0^0^47^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0C^0^2^0^5^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0D 0^0^0^0 15^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0E 0^0^0^1^0 18^2^0^0^0^0^0^0^0^0^0^0^0^0^0^0F^0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0G 0^0^0^0^0^0^0^8^1^1^0^0^0^0^0^0^0^0^0^0^0H 0^0^0^0^0^0^0^1^0^0^0^0^1^0^0^0^0^0^0^0^0I^0^1^0^0^0^0^0^0^0 16^2^0^0^0^0^0^0^0^0^0^0J 0^0^1^0^0^0^0^0^0^0 22^0^0^0^0^0^0^0^0^0^0K 0^0^0^0^0^0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0L 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0M^0^1^0^0^0^0^0^0^0^1^0^0^0^8^1^0^0^0^0^0^0N 0^0^0^0^0^0^0^0^0^0^0^0^0^0 13^0^0^0^0^0^0W 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^0^0^0^0^0w 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^1^0^0^0^0Q 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^4^1^0^1q 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1R^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^7^0• 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^2^0^0159intra-2.relReliability rating between:tape#2-1.sqdtape#2-2.sqdError: Unable to match 3 rallies from 20POSITION BY POSITION MATRIXa b^c d e^f g h i^j^k 1 m n o pa^5^0^0^0^0^1^0^0^0^0^0^0^0^0^0^0b^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0c^0^0^4^0^0^0^1^0^0^0^0^0^0^0^0^0d^0^0^1^6^0^0^0^0^0^0^0^0^0^0^0^0e 0^0^0^0 13^0^0^0^0^0^0^0^0^0^0^0f^1^1^0^0^0^3^0^0^0^0^0^0^0^0^0^1g 0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0h 0^0^0^2^0^1^4 15^0^1^0^0^0^0^0^00^0^0^0^1^0^0^0 42^2^0^1^4^0^0^0j^0^0^0^0^0^0^0^0^0^6^1^0^1^0^0^0k^0^0^0^0^0^0^0^0^0^0^3^1^0^0^0^01^0^0^0^0^0^0^0^1^2^0^4 38^0^0^0^3m^0^0^0^0^0^0^0^1^0^0^0^0 51^2^0^1n 0^0^0^0^0^0^0^0^0^0^0^0^0 20^1^0o 0^0^0^0^0^0^0^0^0^0^0^0^0^0^6^0p^0^0^0^0^0^0^0^0^0^0^0^1^0^0^9 54SHOT(OUTCOME) BY SHOT(OUTCOME) MATRIXS A B C D E F G H I J K L M N W w Q q RS 20^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0A^0^100^0^0^1^0^1^0^0^1^0^0^0^0^0^0^0^0^0^0^0B 0^2^78^0^0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0^0C^0^0^0^3^0^2^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0D 0^0^2^0 29^0^0^0^0^0^0^0^0^0^1^0^0^0^0^0^0E 0^1^1^0^0 13^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0F^0^0^0^0^0^0^4^0^0^0^0^0^0^0^0^0^0^0^0^0^0G 0^0^0^0^0^0^0 10^1^0^0^0^0^0^0^0^0^0^0^0^0H 0^0^0^0^0^0^0^0^4^0^0^0^0^0^0^0^0^0^0^0^0I^0^0^0^0^0^0^0^1^0^8^0^0^0^0^0^0^0^0^0^0^0J 0^0^0^0^0^0^0^0^1^3 13^0^0^0^0^0^0^0^0^0^0K 0^0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0L 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0M^0^1^0^0^0^0^0^0^0^0^0^0^0^2^1^0^0^0^0^0^0N 0^0^0^0^0^0^0^0^0^0^1^0^0^0 10^0^0^0^0^0^0W 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^3^1^0^0^0^1w 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^0^0^0^0^0Q 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^8^0^0^0q 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0R^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^4^0• 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^0^1^0^0^0160intra-3.relReliability rating between:tape#3-1.sqdtape#3-2.sqdError: Unable to match 3 rallies from 20POSITION BY POSITION MATRIXa b^c^d e^f^g h^i^j^k^1^m n o pa^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0b^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0c^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0d 0^0^0^2^0^0^0^1^0^0^0^0^0^0^0^0e 0^0^0^0^8^0^0^0^0^0^0^0^0^0^0^0f^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0^0g 0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0h 0^0^0^0^0^0^0^8^0^0^0^0^0^0^0^0i^0^0^0^0^1^1^0^0 26^0^0^0^4^0^0^0j^0^0^0^0^0^0^0^0^2^7^0^0^0^0^0^0k^0^0^0^0^0^0^0^0^1^0^2^0^0^0^0^01^0^0^0^0^0^0^0^0^1^0^1 33^0^0^0^0m^0^0^0^0^0^0^0^0^0^0^0^0 37^0^0^0n 0^0^0^0^0^0^0^0^0^0^0^0^0^9^0^0O 0^0^0^0^0^0^0^0^0^0^0^0^0^1^4^0p^0^0^0^0^0^0^0^0^0^0^0^0^0^1^1 19SHOT(OUTCOME) BY SHOT(OUTCOME) MATRIXS A B C D E F G H I J K L M N W w Q q RS 20^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0A^0^46^0^1^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0B 0^4^27^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0C^0^0^0^2^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0D 0^0^0^0 15^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0E 0^1^0^0^0^4^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0F^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0G 0^0^0^0^0^0^0^6^0^0^0^0^0^0^0^0^0^0^0^0^0H 0^0^0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0I^0^0^0^0^0^0^0^1^0^7^0^0^0^0^0^0^0^0^0^0^0J 0^0^0^0^0^0^0^0^1^0 20^0^0^0^0^0^0^0^0^0^0K 0^0^0^0^0^0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0L 0^0^0^0^0^0^0^0^0^0^0^0^2^0^0^0^0^0^0^0^0M^0^0^0^0^0^0^0^0^0^0^0^0^0^3^0^0^0^0^0^0^0N 0^0^0^0^0^0^0^0^0^0^0^0^0^0^9^0^0^0^0^0^0W 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^2^0^0^0^0^0w^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^3^0^0^0^0Q 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^6^0^0^0q 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^2^0^0R^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^3^1• 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^1^0^0^0161inter-l.relReliability rating between:tape#1-1.sqdexpert-l.sqdError: Unable to match 4 rallies from 20POSITION BY POSITION MATRIXa b c d e f g h^k 1 m n o pa^2^0^0^0^1^0^0^0^0^0^0^0^0^0^0^0b^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0c^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0d^1^0^0^6^0^0^0^0^0^0^0^0^0^0^0^0e 2^0^0^0 13^0^0^0^1^0^0^0^0^0^0^0f^0^1^0^0^0^1^0^0^0^0^0^0^0^0^0^0g 0^0^0^0^0^0^1^1^0^0^1^0^0^0^0^0h 0^0^0^1^0^0^0 16^0^0^1^1^0^0^0^0i^0^0^0^0^0^0^0^0 34^2^0^0^0^0^0^0j^0^0^0^0^0^0^0^0^0^2^0^0^0^0^0^0k^0^0^0^0^0^0^0^0^0^0^1^1^0^0^0^01^0^0^0^0^0^0^0^0^0^0^2 40^0^0^0^3m^0^0^0^0^0^0^0^0^6^0^0^0 43^7^0^0n 0^0^0^0^0^0^0^0^0^0^0^0^0^9^0^0o 0^0^0^0^0^0^0^0^0^0^0^0^0^0^6^1p^0^0^0^0^0^0^0^0^0^0^0 11^0^0^2 53SHOT(OUTCOME) BY SHOT(OUTCOME) MATRIXS A B C D E F G H I J K L M N W w Q q R *S 20^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0A^0^82^0^1^0^0^0^0^0^8^1^0^0^4^0^0^0^0^0^0^0B 0^0^52^0^0^0^0^0^0^0 10^0^0^0^5^0^0^0^0^0^0C^0^0^0^5^2^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0D 0^0^0^0 13^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0E 0^1^0^1^0 19^2^2^0^0^0^0^0^2^0^0^0^0^0^0^0F^0^0^0^0^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0G 0^0^0^0^0^0^0^7^1^2^0^1^0^0^0^0^0^0^0^0^0H 0^0^0^0^0^0^0^0^0^0^0^0^1^0^0^0^0^0^0^0^0I^0^0^0^0^0^0^0^0^0^8^1^0^0^0^0^0^0^0^0^0^0J 0^0^0^0^0^0^0^0^0^0 12^0^0^0^0^0^0^0^0^0^0K 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0L 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0M^0^0^0^0^0^0^0^0^0^0^0^0^0^2^1^0^0^0^0^0^0N 0^0^0^0^0^0^0^0^0^0^0^0^0^0^8^0^0^0^0^0^0W 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^0^0^0^0^1w 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^2^1^0^0^0^0Q 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^4^1^0^0q 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0R^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^7^0• 0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^1^2^0^0162inter-2.relReliability rating between:tape#1-2.sqdexpert-l.sqdError: Unable to match 1 rallies from 20POSITION BY POSITION MATRIXa b c d e^f g h i^j^k 1 m n o pa^4^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0b^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0^0c^0^0^1^0^0^0^0^0^0^0^0^0^0^0^0^0d^0^0^0^7^0^0^0^0^0^0^0^0^0^0^0^0e 1^0^0^0 13^0^0^0^1^0^0^0^0^0^0^0f^0^0^0^0^0^2^0^0^0^0^0^0^0^0^0^0g 0^0^0^0^0^0^1^1^0^0^1^0^0^0^0^0h 0^0^0^1^0^0^0 16^0^0^1^1^0^0^0^00^0^0^0^0^0^0^0 35^0^2^0^0^0^0^0j^0^0^0^0^0^0^0^0^0^2^0^0^0^0^0^0k^0^0^0^0^0^0^0^0^0^0^2^0^0^0^0^01^0^0^0^0^0^0^0^0^0^0^1 42^0^0^0^2m^0^0^0^0^0^0^0^0^6^0^0^0 44^6^0^1n 0^0^0^0^0^0^0^0^0^0^0^0^0^9^0^0o 0^0^0^0^0^0^0^0^0^0^0^0^0^0^6^1p^0^0^0^0^0^0^0^0^0^0^0 10^0^0^1 55SHOT(OUTCOME) BY SHOT(OUTCOME) MATRIXS A B C D E F G H I J K L M N W w Q q RS 20^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0A^0^80^0^1^0^0^0^0^0^9^0^0^0^6^0^0^0^0^0^0^0B 0^4^47^0^0^0^0^0^0^1 10^0^0^0^5^0^0^0^0^0^0C^0^0^0^5^3^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0^0D 0^0^0^0 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0 0 0 0 0 0 0 0 0M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0N 0 0 3 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0W 1 1 0 1 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0w 0 2 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0Q 2 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0R 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0w' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0w' 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0Q' 0 0 0 0 0 0 0 2 0 2 1 0 2 0 0 0 0 0 1q' 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0217APPENDIX GTable G.1. Table of Validation: Replication of Empirical Quarter-Final Datafor Rallies, Shots and Outcome Frequencies By Match, By Game and By Player1988 Quarter-Final (Norman v Martin)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 0 2 5 58 235 700MATCHOBSERVED^SIMULATIONMean^SDOBSERVEDGAMESIMULATIONMean^SDRallies 94 103.1^22.4 31.3 30.5 4.7Shots 885 969.6^222.1 295.0 288.0 62.4Unconditional Winner 14 15.2^4.4 4.7 4.5 1.9Conditional Winner 29 31.7^8.4 9.7 9.4 2.8Unforced Error 28 30.8^7.6 9.3 9.1 2.4Forced Error 3 3.3^1.9 1.0 1.0 1.0Let 20 22.1^7.2 6.7 6.5 3.0WINNER Martin P{Win}=.993 P{Win}=.890Table G.2. Table of Validation: Replication of Empirical Quarter-Final Datafor Rallies, Shots and Outcome Frequencies By Match, By Game and By Player1988 Quarter-Final^(Jahangir v Thorne)Score Result:^3-0^3-1^3-2Observation : WinPrediction^:^959^38^32-3^1-300-30^0MATCH GAMEOBSERVED SIMULATION OBSERVED^SIMULATIONMean SD Mean SDRallies 81 82.1 11.2 27.0^26.9 4.6Shots 933 944.6 158.4 311.0^310.4 73.2Unconditional Winner 14 14.2 3.3 4.7^4.6 1.8Conditional Winner 30 30.2 4.6 10.0^10.0 2.2Unforced Error 17 17.2 4.8 5.7^5.7 2.6Forced Error 4 4.1 2.1 1.3^1.3 1.2Let 16 16.3 5.1 5.3^5.3 2.7WINNER Jahangir P{Win}=1.00 P{Win}=.986218Table G.3. Table of Validation: Replication of Empirical Quarter-Final Datafor Rallies, Shots and Outcome Frequencies By Match, By Game and By Player1988 Quarter-Final (Jansher v Nancarrow)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 692 251 44 6 7 0MATCHOBSERVED^SIMULATIONMean^SDOBSERVEDGAMESIMULATIONMean^SDRallies 87 97.3 20.0 29.0 29.2 4.4Shots 1004 1122.2 253.6 334.7 337.3 73.0Unconditional Winner 17 19.1 6.0 5.7 5.7 2.3Conditional Winner 19 21.0 5.9 6.3 6.3 2.3Unforced Error 30 33.5 7.0 10.0 10.0 2.5Forced Error 6 6.9 3.0 2.0 2.1 1.4Let 15 16.8 5.7 5.0 5.1 2.6WINNER Jansher P{Win}=.987 P{Win}..883Table G.4. Table of Validation: Replication of Empirical Quarter-Final Datafor Rallies, Shots and Outcome Frequencies By Match, By Game and By Player1988 Quarter-Final (Dittmar v Harris)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 864 123 11 0 2 0MATCHOBSERVED^SIMULATIONMean^SDOBSERVEDGAMESIMULATIONMean^SDRallies 87 86.8 15.3 29.0 27.5^4.6Shots 1178 1173.2 222.7 392.7 371.7^86.5Unconditional Winner 11 10.9 3.9 3.7 3.4^1.9Conditional Winner 18 18.1 5.3 6.0 5.7^2.3Unforced Error 41 40.8 6.6 13.7 13.0^2.4Forced Error 2 2.0 1.4 0.7 0.6^0.8Let 20 15.0 5.1 6.7 4.7^2.6WINNER Dittmar P{Win}=.998 P{Win}=.955219Table G.5. Table of Validation: Replication of Empirical Semi-Final Data forRallies, Shots and Outcome Frequencies By Match, By Game and By Player1988 Semi-Final (Martin v Jahangir)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 0 1 0 22 156 821MATCHOBSERVED^SIMULATIONMean^SDOBSERVEDGAMESIMULATIONMean^SDRalliesShotsUnconditional WinnerConditional WinnerUnforced ErrorForced ErrorLet89104192533418^93.8^17.41094.8^227.19.4^3.426.3^5.934.7^7.74.3^2.219.1^6.029.7347.03.08.311.01.36.029.3343.83.08.210.71.46.04.778.21.72.52.81.22.8WINNER Jahangir P{Win}=.999 P{Win}=945MARTIN OBSERVED SIMULATION OBSERVED SIMULATIONMean^SD Mean SDShots 517 542.7^114.4 172.33 170.9 39.7Unconditional Winner 5 5.2^2.6 1.67 1.7 1.3Conditional Winner 8 8.4^3.4 2.67 2.6 1.7Unforced Error 21 22.1^4.4 7.00 6.9 2.0Forced Error 3 3.2^1.8 1.00 1.0 1.0Let 9 9.7^3.7 3.00 3.0 1.9JAHANGIR OBSERVED SIMULATION OBSERVED SIMULATIONMean^SD Mean SDShots 524 552.0^112.8 174.67 173.0 38.5Unconditional Winner 4 4.2^2.0 1.33 1.4 1.1Conditional Winner 17 17.9^4.0 5.67 5.6 1.9Unforced Error 12 12.6^4.8 4.00 3.9 2.2Forced Error 1 1.1^1.0 0.33 0.4 0.6Let 9 9.4^3.7 3.00 2.9 1.8220Table G.6. Table of Validation: Replication of Empirical Semi-Final Data forRallies, Shots and Outcome Frequencies By Match, By Game and By Player1988 Semi-Final (Jansher v Dittmar)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 2 5 8 70 265 650MATCHOBSERVED^SIMULATIONMean^SDOBSERVEDGAMESIMULATIONMean^SDRalliesShotsUnconditional WinnerConditional WinnerUnforced ErrorForced ErrorLet1151649122248429^112.2^24.71612.6^384.112.1^4.420.9^5.746.1^10.94.0^2.129.1^8.832.9471.13.46.313.71.18.332.8473.83.56.213.41.28.65.3106.11.82.32.81.13.4WINNER Dittmar P{Win}=.985 P{Win}=.861JANSHER OBSERVED SIMULATION OBSERVED SIMULATIONMean^SD Mean SDShots 799 800.7^192.4 228.3 235.7 53.5Unconditional Winner 8 7.0^3.3 2.3 2.0 1.4Conditional Winner 6 7.1^3.2 1.7 2.1 1.5Unforced Error 28 28.0^5.6 8.0 8.2 2.0Forced Error 3 3.0^1.7 0.9 0.9 0.9Let 9 9.0^3.7 2.6 2.6 1.7DITTMAR OBSERVED SIMULATION OBSERVED SIMULATIONMean^SD Mean SDShots 810 811.9^191.8 231.4 238.2 52.7Unconditional Winner 4 5.1^2.3 1.1 1.4 1.1Conditional Winner 15 13.8^3.7 4.3 4.0 1.7Unforced Error 18 18.0^6.7 5.1 5.2 2.5Forced Error 1 1.0^1.1 0.3 0.3 0.6Let 20 20.1^6.5 5.7 6.0 2.8221Table G.7. Table of Validation: Replication of Empirical Final Data forRallies, Shots and Outcome Frequencies By Match, By Game and By Player1988 Final (Dittmar v Jahangir)Score Result: 3-0 3-1 3-2 2-3 1-3 0-3Observation : WinPrediction^: 0 1 1 14 127 857MATCHOBSERVED^SIMULATIONMean^SDOBSERVEDGAMESIMULATIONMean^SDRalliesShotsUnconditional WinnerConditional WinnerUnforced ErrorForced ErrorLet1101485102432341^114.8^21.21550.9^305.810.5^3.525.1^6.333.1^6.83.1^1.842.9^10.936.7495.03.38.010.71.013.736.0488.23.37.910.51.013.36.9117.01.72.62.81.05.1WINNER Jahangir P(Win)=.998 P{Win}=.942DITTMAR OBSERVED SIMULATION OBSERVED SIMULATIONMean^SD Mean SDShots 733 768.0^153.8 244.3 242.1 59.0Unconditional Winner 3 3.2^2.0 1.0 1.0 1.0Conditional Winner 11 11.5^4.4 3.7 3.5 2.0Unforced Error 22 22.8^4.1 7.3 7.2 2.0Forced Error 3 3.1^1.8 1.0 1.0 1.0Let 17 17.6^5.7 5.7 5.4 2.8JAHANGIR OBSERVED SIMULATION OBSERVED SIMULATIONMean^SD Mean SDShots 752 782.9^152.1 250.7 246.1 58.1Unconditional Winner 7 7.4^2.6 2.3 2.3 1.3Conditional Winner 13 13.6^3.4 4.3 4.4 1.8Unforced Error 10 10.4^4.2 3.3 3.3 2.1Forced Error 0 0.0^0.0 0.0 0.0 0.0Let 24 25.2^7.0 8.0 7.9 3.4222APPENDIX HConsider player X versus Y.Let the probability that X wins a game from initial service be Pand the probability that X wins a game from initial receipt of service be p.Similarly, let the probability that Y wins a game from initial service be Qand the probability that Y wins a game from initial receipt of service be q.Thus, P+q=p+Q= 1.The scoring structure of squash is such that the player who wins the gamebegins service for the next. A random walk through the possible game scores,given X begins service of the first game, yields:3-0: PPP3-1: qpPP + PqpP + PPqp3-2: qQpPP + qpqpP + qpPqp + PqQpP + Pqpqp + PPqQp2-3: PPqQQ + PqpqQ + PqQpq + qpPqQ + qpqpq + qQpPq1-3: qpqQ + qQpq + PqQQ0-3: qQQThe respective probabilities therefore are:Pr{X wins 3-0IXserves first} = P 3Pr{X wins 3-1IXserves first} = 3P 2pqPr{X wins 3-2IXserves first} = 3P2Qpq + 3p p2 (12Pr{Y wins 2-3IXserves first) = P 2 Q2 q + 4PQpq2 + p2 q3Pr{Y wins 1-3IXserves first} = PQ 2 q + 2Qpq2Pr{Y wins 0-3IXserves first} = qQ 2Pr{X wins matchIX serves first} = P3 + 3P2pq + 3P2Qpq + 3pp2 q2Pr{Y wins matchIX serves first} = qQ 2 + PQ2q + 2Qpq2 + P2 Q2q + 4PQpq2 + p2 q3223

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