THE PREDICTION AND VALIDATION OF GREENHOUSE TOMATO YIELD USING MATHEMATICAL MODELS AND EXPERT SYSTEMS by WINSTON C. TANG B.A.Sc, The University of British Columbia, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Bio-Resource Engineering We accept this thesis as corrforrning to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1995 © Winston C. Tang, 1995 In presenting degree freely at the available copying of department publication this of in partial fulfilment University of British Columbia, for this or thesis reference thesis by this for his thesis and study. scholarly or for her of I shall of not Bio-Resource Engineering The University of British C o l u m b i a Vancouver, Canada DE-6 (2/88) be It permission. Department that agree purposes may gain requirements agree I further representatives. financial the is be that the for an Library shall permission for granted by understood allowed advanced the make extensive head that without it of copying my my or written 11 Abstract Greenhouse tomato yields were predicted using two mathematical models developed in this study - the empirical and deterministic models. Weekly yield predictions for an entire growing season were compared with actual yields and results from an expert system model developed by the Agassiz Research Station. The deterministic math model involved using first principle equations of photosynthesis and respiration to simulate crop growth. Utilizing a known tomato yield conversion factor, net photosynthesis rates (P ) were converted to weekly yield predictions and compared net with actual recorded yields. A deterministic model using two week cumulations of P t ne converted to yield was used successfully to predict actual tomato yields 6 weeks ahead of time with a root-mean-square-error (RMSE) of 0.38 kg/m . 2 The empirical math model employed regression techniques to fit historical greenhouse climate data to recorded yields. Correlations between light, temperature, and weekly tomato yields were derived into equations to predict yields for future growing seasons. An empirical model cumulating 3, 6, and 9 weeks of light and temperature data was developed to predict yields 4 weeks ahead of time with a R M S E of 0.45 kg/m . 2 When one-week-ahead predictions from the Agassiz expert system model were compared with actual recorded yields a R M S E of 0.401 kg/m was calculated. The expert 2 system model utilizing trend recognition techniques was also used as a comparison with the two math models. When compared and ranked for prediction accuracy, application flexibility, and user-friendliness, the expert system was chosen as the overall best model for tomato yield prediction. TABLE OF CONTENTS Abstract ii Table of Contents iii List of Tables iv List of Figures v Acknowledgements vii INTRODUCTION 1 1.0 .3 2.0 THEORY AND REVIEW 1.1 The Development of Mathematical Yield Models 3 1.2 The Application of Expert System Models for Yield Prediction 6 1.3 Research Objectives 9 MATERIALS AND METHODS 10 2.1 3.0 Data Source 2.2 Data Collection 2.3 Data Analysis 2.4 Selecting the Deterministic Math Model 2.5 Developing the Empirical Math Model 2.6 The Agassiz Expert System Model 2.7 Validation and Comparison of the Yield Predictive Models RESULTS AND DISCUSSION 3.1 3.2 3.3 3.4 3.5 10 12 12 14 20 25 27 29 Analysis of Graphical Plots of Hourly Greenhouse Climate Data 29 Deterministic Math Model Results 31 Empirical Math Model Results 39 Agassiz Expert System Results 54 Validation and Comparison Results of the Yield Predictive Models .... 59 4.0 CONCLUSIONS 70 5.0 RECOMMENDATIONS 72 6.0 BIBLIOGRAPHY 76 TABLE OF CONTENTS APPENDIX A Plots of Inside Temperature, Outside Temperature, & Solar Intensity Plots of Greenhouse Carbon Dioxide Concentrations Plots of Greenhouse Relative Humidity 81 APPENDIX B Deterministic Math Model Spreadsheet Layout Leaf Area Index vs. Time Graph 91 92 96 APPENDIX C Summary of Total PPF vs. Crop Yield Results Yearly Plots of Total PPF vs. Crop Yield 97 98 99 82 85 88 APPENDED D 106 Empirical Math Model Regression Results for Equation #1 (Light Alone) .... 107 Plots of 1994 Predicted Yields - Cumulative Light Based 108 Empirical Math Model Regression Results for Equation #2 (Light & Heat) .110 Plots of 1994 Predicted Yields - Light and Temperature Based Ill Stepwise Regression Results for Light and Temperature Based Equations ... 113 Empirical Math Model Regression Results for Equations #3.1 & 4.1 127 Empirical Model Regression Results for Alternative Regression Method 131 APPENDTXE Empirical Model Humidity Trial Step-Wise Regression Results 1994 Predicted Yield Plot - Humidity, Light, and Temperature Based 134 135 140 LIST OF TABLES Table 1a lb 2 Empirical Math Model Results for 1994 Yield Predictions (Eqn 1 & 2) .... 43 Empirical Math Model Results for 1994 Yield Predictions (Eqn 3.1/4.1)... 49 Agassiz Expert System Results 56 3 Comparison of Yield Predictive Models 66 Appendix Tables A-1 A-2 A-3 Summary of Total PPF vs. Crop Yield Results Empirical Model Regression Results - Light Only - (Equation #1) Empirical Model Regression Results - Light & Heat - (Equation #2) 98 107 110 V LIST OF FIGURES Figure 1 2 3 4 5 6a 6b 6c 7 8 9 10 11 12 13 14 15 16 17 18 Greenhouse Tomato Yield Prediction Modeling Approaches 4 Location of Measurement Transducers in a Greenhouse 7 Research Objectives 11 Data Collection and Analysis Layout 13 Development of the Deterministic Model 15 Gross Photosynthesis and Maintenance Respiration Rate Equations 17 Crop Growth Rate, Net Photosynthesis (P t), and Coefficient Values 18 Enoch's Generalized Multiplication Factor for Tomato Yield 19 Development of the Empirical Math Model 21 Development of the Agassiz Expert System Yield Prediction Model 26 Root Mean Square Error (RMSE) or Standard Error of Estimate 28 Cumulative P t Based Yield - Immediate Yield Predictions 32 Cumulative P t Based Yield - 4 Week Ahead Predictions 35 Cumulative P t Based Yield - 6 Week Ahead Predictions 37 Trend Identification of Light (PPF) vs. Crop Yield 41 1994 Predicted Yield - Cumulative Light Based (CL ) 44 1994 Predicted Yield - Light (CL ) and Temperature (CH ) 47 1994 Predictions - 3 Light & 3 Heat Variables (CL , e , 9 & C H 3 , 6 , 9 ) 51 1994 Predictions (4 weeks ahead) - 3 Light & 3 Heat Variables 53 Agassiz Expert System Predictions (1 week ahead) 57 ne ne ne ne 9 9 9 3 Appendix Figures A-l A-2 A-3 A-4 A-5 A-6 A-7 A-8 A-9 A-10 Inside & Outside Temperature & Solar Intensity Plots - January 1992.... 82 Inside & Outside Temperature & Solar Intensity Plots - February 1992 .83 Inside & Outside Temperature & Solar Intensity Plots - March 1992 84 Greenhouse Carbon Dioxide Concentration Plots - January 1992 85 Greenhouse Carbon Dioxide Concentration Plots - February 1992 86 Greenhouse Carbon Dioxide Concentration Plots - March 1992 87 Greenhouse Relative Humidity Plots - January 1992 88 Greenhouse Relative Humidity Plots - February 1992 89 Greenhouse Relative Humidity Plots - March 1992 90 Leaf Area Index (LAI) vs. Time 96 vi LIST OF FIGURES (cont'd) Figure A-11 A-12 A-13 A-14 A-15 A-16 A-17 A-18 A-19 A-20 A-21 A-22 1987: Total PPF vs. Crop Yield..., 99 1989: Total PPF vs. Crop Yield 100 1990: Total PPF vs. Crop Yield 101 1991: Total PPF vs. Crop Yield 102 1992: Total PPF vs. Crop Yield 103 1993: Total PPF vs. Crop Yield 104 1987-1993: Total PPF vs. Crop Yield 105 1994 Predicted Yields - Cumulative Light Based (1987 to 1992 Eqns.) 108 1994 Predicted Yields - Cumulative Light (1992-93 & 1987-1993 Eqns.).. 109 1994 Predicted Yields - Light & Temperature (1987 to 1992 Eqns.) Ill 1994 Predicted Yields - Light & Temperature (1993 & 1987-1993 Eqns.). 112 1994 Predicted Yields - Humidity, Light, & Temperature (1992 Eqn.) 140 Vll Acknowledgements I would like to express my sincerest thanks and gratitude to my thesis advisor, Dr. Anthony K. Lau, for his thoughtful advice and assistance throughout this project. The kindness, encouragement, and friendship that has been shared will not be forgotten. Thank you Anthony, for all your help throughout my university experience. A special thank you to Dr. Wei C. Lin, for providing resultsfromthe Agassiz Expert System for comparison with the mathematical models developed in this study. Thank you Wei, your assistance, advice, and explanations were always helpful and will always be remembered. To Dr. Sie-Tan Chieng, thank you for your insightful ideas and suggestions throughout this research project. Your many helpful suggestions and recommendations were always welcome and greatly appreciated. Thank you Sie-Tan! A special thank you to my wonderful Princess who helped me endure all the long hours of writing and analysis. encouragement, You were always there for me when I needed compassion, or ranch-dipped pretzels. Thank you so much for all the love, laughter, and the most unforgettable moments of my life! I.L.U.F.B. U.R.E.2.Me.W.B. I.W.2.M.U. W,W.U.M.M.? What do you say? O "If you believe, it does exist. If you are determined, it can be yours." - 7V&7 W 2 & P e k k l e (TLF) 1 Introduction Greenhouse produced crops have become an important part of the agricultural industry (BCFA, 1994) and are well-known in the marketplace for their high quality and consistency. However, while striving to provide a top quality product, greenhouse producers have had to incur heavy capital costs, and these costs are then inevitably reflected to the consumer. Consequently, any advances that may improve the yield and lower the cost of producing greenhouse crops will benefit both growers and consumers. Current decision-making for greenhouse climate control and production relies heavily on the past experience of the individual grower and general guidelines published in production manuals. As a result, grower yields tend to vary greatly on an individual and yearly basis. In an attempt to solve these problems, computers have been implemented in climate control to regulate the aerial and root-zone environmental conditions for crops and thereby improve yields. However, computer climate control still remains largely dependent on settings prescribed by individual growers. Although computers have been useful in providing automated control of environmental parameters (Hashimoto et al., 1993), the actual dynamic relationships that exist between climatic conditions and crop yields still remain unclear. Models of the greenhouse environment and simulations of crop growth are needed to enhance environmental control strategies and to encourage growth of the greenhouse industry (Jones et al., 1991). entire 2 One approach to improving crop yields and identifying plant response relationships lies in developing mathematical growth models that can predict future yields given a set of climatic conditions. Successful growth models can then help to identify the environmental factors with the greatest influence on crop yield and aid the grower in making essential management decisions to improve productivity. Affected decisions may rangefromsimple adjustments of lighting or temperature setpoints to justifying the cost of investing in new equipment. A valid growth model can also be a useful tool for the greenhouse grower by providing a quick estimate of future yields based on current practises. Another approach for yield prediction involves expert system modeling which is an advanced information processing method that is able to recognize patterns and compensate for incomplete or ambiguous data (Rehbein, et al., 1992). Provided with historical data, a properly trained expert system will be able to analyse trends in production as they relate to input data (ie. light, temperature, and other greenhouse climatic variables) and predict future yields. By providing reliable future yield and growth forecasts, both the mathematical modeling and expert system approaches aim to improve the overall productivity of greenhouse crops and enhance their marketability to the consumer. Successful yield prediction is a valuable tool for detennining better greenhouse management practises, reducing resource wastage, and identifying the relationships between plant responses, crop yield, and climate. 3 1.0 Theory and Review As world populations continue their steady growth, greenhouse production systems must become more efficient to meet the increasing demands for food. Much of the recent advances in greenhouse crop production have been linked to the application of computers in climate control. Computer automation has already allowed greenhouse environmental conditions to be properly monitored and maintained to increase crop productivity while reducing energy requirements. However, in order to make further advances in greenhouse productivity a better understanding of plant responses to climatic conditions is required (Tantau, 1980). The emphasis on developing innovative methods for optimizing greenhouse production has intensified globally over the past decade (Jones, 1991). Devising models or methods for predicting future crop yields is an effective way to investigate plant growth and climate relationships. 1.1 The Development of Mathematical Yield Models Developing mathematical models to simulate crop growth is one method that can be used to predictfinalyield. Math models are generally divided into two main classes - those that employ the deterministic method and those that apply the empirical approach. {Figure 1). The deterministic approach relies on general laws or basicfirstprinciples to derive a scientifically-based model. Researchers have already used this approach to produce models to make predictions of canopy photosynthesis (Acock et al, transpiration (Stanghellini and van Meurs, 1992;Yang et al., 1990). 1978), and plant Greenhouse Tomato Yield Prediction Modeling Expert System Modeling (Agassiz Research Station) A/ fro • Empirical Approach Mathematical H I Modeling Deterministic Approach (General - based on First Principles) Figure 1. Shown above are the approaches that will be used to predict Greenhouse Tomato Crop Yield. (Illustration by W.C.Tang, 1995) 5 The deterministic approach could also be useful in predicting yield. By applying the first principles that relate plant photosynthesis, respiration, and plant growth, estimates of yield could be made. Deterministic equations involve systems of differential equations to describe dynamic processes such as plant growth. If plant models are developed using this approach, each reaction between the seedling to harvest stage is described by a differential equation (ie. change in dry matter production). However, as plant growth typically involves long time spans, most studies limit their scope for testing plant models to only a few months (Takakura, 1993). The ultimate goal of the deterministic model is to reach the best possible agreement between the model predictions and the experimental data over a set time period. Empirical models examine historical data and use mathematical regression techniques to find correlations between independent and dependent variables. In this case, the independent variables would be the environmental parameters such as greenhouse temperature, light, and humidity and the dependent variable would be the actual yield recorded for that period. Empirical models search for basic cause and effect relationships to simplify a complicated process into a functional equation which can be applied to the general case. The problem that exists for equations derived purely from an empirical approach is that they are generally linear-based (straight line equations) while plant growth is an example of a non-linear, dynamic system (Takakura, 1993). As a result, empirical models are restricted to providing good estimates within the range of data they were derived from. 6 Beyond the range of values found in the experimental case, empirical models can only provide extrapolations as a basis for predictions. 1.2 The Application of Expert Systems for Yield Prediction An innovative approach to modeling involves the application of expert system models in greenhouse crop production (Figure 1). An expert system model is simply a method of processing information that involves a self-learning model that adapts to changing data inputs (ie. environmental parameters). Expert systems are based on the most up-to-date 'expert' information in afieldof study and incorporate this knowledge in decision-making processes and weighting factors (Levine et al, 1990). For predicting tomato crop yield, expert information would include up-to-date researchfindingsof plantclimate relationships from greenhouse growers and plant physiologists. This expert knowledge would then be incorporated into a computer program by applying weighting factors to measured environmental parameters such as light, temperature, and humidity (Figure 2). By relating the weighted environmental parameters to recorded actual yields, an expert system model can provide yield predictions. The advantage of using expert systems for yield prediction is their potential ability to provide nonlinear systems control for cases such as plant growth and their ability to forecast yields even in cases where data sets are incomplete or ambiguous. This is accomplished by assigning weights {W ) to all selected t data inputs ( X,) (ie. light, temperature, humidity values) based on expert (greenhouse growers, plant physiologists) knowledge and then applying these weighted data as input for U u 0 0 c S3 o 0 o o u 1 's I £ 1— c 0 1 3 ON GO U, 11 i + cu H H s CU O 3 -a CO C 3 c <u B g 3 3 CD CU to 3 O •C C CD <U o c o '£ o o s Si I 8 a processing element (internal algorithm) which in turn produces one output value. In reaction to the data input, the model will then gradually adjust the weighting of each factor to produce more accurate and reliable predictions and minimize the error for the next iteration. By repeating this process, expert systems learn to recognize trends in the data inputs and are able to make predictions based oh historical cases. Ideally, a properly trained expert system would then be interfaced with an on-line greenhouse climate control system to determine the best environmental setpoints to maximize future crop yields and minimize operating costs. Expert system models have recently been tested in determining ideal temperature setpoints in greenhouses (Lacroix, et al., 1993). In this case, an artificial expert system temperature controller was created to predict crop responses to changes in greenhouse setpoint temperatures. By examining and testing various temperature setpoints with this technique, researchers were able to reduce energy consumption by 10% while maintaining productivity. In another study, expert system methods were combined with standard data analysis techniques to predict apple quality (Boucherau et al., 1992). Measurements of apple sugar content obtained from infra-red spectrometry were used as a basis for detenmning quality. By employing standard regression techniques, a simple linear model was produced. An expert system was then employed to handle the non-linear data components and this improved the accuracy of apple quality predictions byfiveper cent. 9 Expert systems have also helped determine suitable environmental settings for greenhouse lettuce crops (Seginer et al., 1992). Historical data was used to train an expert system to aid in management decisions and determine environmental setpoints (ie. temperature settings). When compared, mathematical models provide reasonable linear projections of future values but are less flexible than expert system models in terms of overall application. Mathematical models require many parameters for the submodels that account for various physiological growth and development processes (Figures 6a & 6b). These parameters need to be determined by scientific experiments or at the very least approximated. Uncertainties are therefore bound to be associated with mathematical models. Conversely, expert systems focus on trend and pattern analysis eliminating the need to approximate environmental coefficients to provide predictions. Expert systems are able to adapt and learn by continually analysing trends to minimize the error with each subsequent prediction. 1.3 Research Objectives The main objective of this study is to adapt (deterministic model) and develop a model (empirical model) that can successfully predict greenhouse tomato yield, of the tomatofruititself, in terms of quantity (kg/m ). 2 10 Steps (Figure 3) required to achieve the research objectives include: 1. To collect and analyse historical environmental and crop yield data from greenhouse growers via graphical hourly plots of greenhouse data to identify any obvious trends, patterns, or relationships. 2. To develop and test a deteraiinistic yield prediction model (one based on first principles and known yield conversion factors). 3. To develop an empirical yield prediction modelfrommultiple regression analysis of greenhouse data. The best preliminary model will then be selected for further development to determine the best overall empirical model. 4. To analyse yield predictionsfroman alternative, expert system model adapted by the Agassiz Research Station (Lin, 1994). 5. To compare and validate the empirical, deterministic, and Agassiz expert system yield prediction models. The best overall prediction model will then be identified. 2.0 Materials and Methods 2.1 Data Source All of the data used in this study will be collectedfromgrowth records of greenhouse tomatoes provided by independent growers and the B.C. Greenhouse Growers Co-operative. Research Objectives Main Objective IDENTIFY BEST YIELD PREDICTION MODEL •/ Main Objective Comparison of Predictive Models Deterministic Prediction Model Collect and fft Analyse Greenhouse Climate Data Figure 3. S h o w n a b o v e a r e the r e s e a r c h o b j e c t i v e s in this study. 12 2.2 Data Collection Temperature, carbon dioxide, humidity, and solar radiation data used in this study were collected by a greenhouse climate computer with measurement transducers. Typical locations for these transducers within a greenhouse environment are shown on Figure 2. 2.3 Data Analysis The environmental data required for developing the various mathematical and expert system models will be obtained from two independent tomato greenhouse growers. In order to maintain the confidentiality of their information, data shall be identified as either from Greenhouse Grower 'A' or Greenhouse Grower 'B' (Figure 4). Greenhouse Grower A's data set includes hourly environmental values (values for every hour of the day) for light intensity, greenhouse temperature, inside relative humidity, carbon dioxide concentrations, and actual recorded yield for a number of growing seasons during 1992. Hourly data is necessary for developing a deterministic mathematical growth model based on first principles (ie. photosynthesis and respiration equations) as the model attempts to simulate actual crop growth in order to provide yield predictions. Data Collection & Analysis Tomato Crop Climate & Production Data Figure 4. Shown above are the data sets used in this study 14 One complete set of greenhouse tomato hourly climate data will be plotted for a 60day period (1440 hours) during the 1992 growing season. Graphical plots of: inside temperature, outside temperature, solar intensity (light), relative humidity, and carbon dioxide concentrations, will be analysed for any obvious trends or patterns in greenhouse conditions or management practises. Greenhouse Grower B's data set includes weekly values for: light intensity (weekly cumulative total in W/m ), average temperatures (a weekly average based on every 2 hour temperature readings), day and night temperatures (weekly averages of hourly daytime and night-time temperatures, respectively), day and night humidity (weekly averages of hourly day and night humidity readings, respectively) and weekly recorded actual yield values (kg/m ) for the years between 1987 to 1993 (excluding 1988 due to missing data). 2 Weekly datafromgrower B will be used in this study to prepare an empirical math model using mathematical regression methods. The same weekly datafromgrower B was also used in testing the expert system model developed by the Agassiz Research Station (Agassiz, B.C.), a Research Branch of Agriculture Canada. The weekly predictions from the Agassiz expert system will then be used for comparison purposes. 2.4 Selecting the Deterministic Math Model Using the hourly data provided by Greenhouse Grower A, a deterministic crop growth model will be developed to predict yield (Figure 5). Thefirstprinciple equations that will be used for this approach are mathematical relationships between environmental 15 Development of the Deterministic Model ± Validated Deterministic Deterministic Math Model Validation & Analysis Fully Developed of Deterministic Predictions Math Model Yield Prediction using known Yield Conversion Factor First Principle Validation Equations of (Photosynthesis, First Principle Respiration) Equation Results Import, Format, and Manipulate Start Climate Data Greenhouse Grower A Hourly Climate Data (Illustration by W.C.Tang, 1995) Figure 5. Shown above is the general process train for the development of the Deterministic Model. 16 conditions, dry weight growth, respiration rate, and photosynthesis as detailed in literature (Jones, 1991) and shown on Figures 6a & 6b. An alternative method to compute crop yield is by applying a multiplication factor that relates net photosynthesis to crop yield. assumptions: For tomatoes, this was found to be 7 (Enoch, 1978) with the following greenhouse light transmission of 80%, plant production based on C O 2 uptake at optimal leaf temperature, and that one absorbed C0 molecule is used to create 2 one molecule of dry matter (CH2O), with 50% of this dry matter being fruit yield, representing between 5% - 7% of the total fruit weight including both wet and dry portions (Ho and Hewitt, 1983) (Figure 6c). In order to narrow the scope of this study, only hourly datafromthe 1992 growing season for Grower A will be used. The 1992 environmental data contained the most complete set of readings for greenhouse climate: light intensity, temperature, and carbon dioxide concentrations. (The deterministic math model will not utilize humidity data because transpiration is not being modeled.) Using a computer spreadsheet program, the appropriate column headings for the data will be configured and formula cells based on first equations found in literature (Jones, 1991) will be applied. Predictions for respiration, gross photosynthesis, net photosynthesis, and crop dry weight are then obtained. The results of net photosynthesis can now be converted to crop yield (Figure 6c) using the generalized multiplication factor of 7 (Enoch, 1978). Weekly cumulations of net photosynthesis based yield predictions will then be plotted and compared with actual recorded tomato yields for 1992. Once these immediate yield results are validated and corrected for missing data, the model will be used to predict yield 4 and 6 weeks ahead. 17 P - Gross Photosynthesis Equation g p - D x* C* K g where p(9) * In f q* K *L + ( 1 - m )* x * C * p(9) L a * K * I * exp (-KL) + (1 - m)* x * C * 0 J p(8) J Pg = gross photosynthesis measured in units of g (CH 0) / m -h 2 2 = coefficient to convert photosynthesis calculations from umol (CO2) / m -s to g (CH 0) / m -h x = leaf conductance to C 0 , pmol (C0 )V m -s C = C 0 concentration of the air, (umol (C0 ) / mol (air)) = ppm p(6) = dirhensionless function of temperature, the effect of temperature on the maximum rate of photosynthesis for a single leaf, as a quadratic equation: D 2 2 2 2 2 2 2 2 P(6)= { l - ((9h-e)/(0 -e,)) } 2 h where: 9h is the temperature at which leaf photosynthesis is maximum, °C 9 is the measured temperature in the greenhouse, °C 9i is the temperature at which leaf photosynthesis is zero, °C leaf light utilization efficiency, pmol (C0 ) / umol (photon) canopy light extinction coefficient lightfluxdensity at the top of the canopy, umol (photon) / m -s light transmission coefficient of leaves canopy leaf area index, m (leaf) / m (ground) a K 2 2 Io and m L 2 R m - Rm where m 2 - Maintenance Respiration Rate k * exp (0.0693 [ 9 - 25 ] ) m maintenance respiration rate, g (CH 0) /.g tissue - h (maintenance respiration is the loss of C 0 due to breakdown and re-synthesis of existing tissue and depends on temperature) respiration rate at 25°C, g (CH 0) / g tissue - h measured greenhouse/inside temperature, °C 2 2 2 9 Figure 6a. Shown above are the P and R„ equations used in developing the deterministic math model (Jones, 1991). g 18 dW/dt - Crop Dry Weight Growth Rate dW/dt = E ( P - R * W ) g where dW/dt = rate of dry weight of the crop, g (tissue) - h W = total plant dry weight, g / m E = conversion efficiency of CH 0 to plant tissue, g(tissue) / g(CH 0) 2 2 R and m 2 = maintenance respiration rate, g (CH 0) / g(tissue) - h m 2 Pg = canopy gross photosynthesis rate, S (CH 0) / m - h 2 2 Pnet - Net Photosynthesis Pn where and P =Pg - Rm*W net photosynthesis rate, n g (CH 0) / m - h 2 2 Pg canopy gross photosynthesis rate, g (CH 0) / m - h Rm maintenance respiration rate, W total plant dry weight, g/rh 2 2 g (CH 0) / g(tissue) - h 2 2 Coefficient Values used in these Equations a =0.056 E=0.70 0 =3O°C, K = 0.58 h 9i=5°C x = 0.0664 k =0.0006 m = 0.10 D = 0.108 L = variesfrom0.6 to 3.31 (deKoning, 1993) ra Figure 6b. Shown above are the dW/dt and P equations, and coefficient values used in developing the deterministic math model (Jones, 1991). n 19 Photosynthesis: 6C0 + 6H 0 - C H 0 + 60 2 2 6 12 6 2 Simplified Carbon Conversion for determining Yield Conversion Factor: 1 C 0 - 1 CH 0 molecule 2 2 (MW) = 44g (MW) = 30g (MW) = Molecular Weight • For 1 molecule of C0 used, (ie. every 44 g of C0 used) we get 30 g of CH 0. • 1/2 of this 30 g of CH 0 is partitioned to become fruit yield (dry portion) = 15 g. • This 15g is 5% of the total weight of thefruit(wet and dry portions) • Therefore, the total fruit weight is actually: • For 44 g of C 0 used (absorbed) we have 300 g offruitweight. 2 2 2 ( 300 g / 44g) = 7 goffruitweight (wet & dry). Enoch's Generalized Multiplication Factor for Tomato Yield (Enoch, 1978) 30 g 44g • • • 15 g * (100 / 5) =300 grams 2 .'. 1 gof CO2 used leads to _ 2 * I * 100 2 5 s 7 = Yield Multiplication Factor for Tomatoes Molecular Weight for C 0 (44 g) and CH 0 (30 g) Half the dry matter is yield (1/2). This dry matter represents 5% of totalfruitweight (dry & wet) (100/5). 2 2 Figure 6c. Shown above is Enoch's Yield Conversion Factor (1978) for tomatoes that will be used in developing the deterministic math model. 20 Predicted and actual yield values will then be compared with the standard error of estimate (S ) or root-mean-square-error (RMSE) method (Figure 9) for evaluating the e accuracy of results using the deterministic modeling approach. 2.5 Developing the Empirical Math Model Weekly light and temperature datafromGreenhouse Grower B will be used in this portion of the study to develop an empirical prediction model using multiple regression techniques (Figure 7). Regression techniques are useful for deriving linear relationships between input and output variables. Such analysis will be carried out in the advanced math tools available in spreadsheet programs such as Quattro Pro (for simple regression) as well as with more advanced statistical analysis packages such as Systat (for multiple and stepwise regressions). The first step in developing the empirical model involves identifying the primary environmental condition that is most closely linked to tomato crop production. From literature review McAvoy et al. (1989), have found strong relationships between photosynthetic photo flux (light intensity) and total yield. In this study they performed comparisons and regressions of the total photosynthetic photonflux(PPF) received during a 60-day production cycle. A close linear trend (r = 0.947) between recorded tomato 2 yields and total PPF was observed. Following a similar approach, the weekly data for 1987 to 1993 from Greenhouse Grower B will be analysed for a light/yield linear relationship. Nine week (63 day) cumulative light intensity (PPF) values will be plotted against recorded yield. This method Development of the Empirical Math Model Validated Empirical Math Model X I Selection of Best Validated Empirical Model 1 Validation of Models by RMSE Approach 3,6,9 Week Light & 3,6,9 Week Heat Equation for Immediate & Future Yield Predictions Predict 1994 Yields using Light Alone and Light & Heat Equations Select Data Set with most correlated yearly climate data I Preliminary Light & Heat Regressions (Immediate Predictions) Start Figure 7. mom Preliminary Light Only Regressions Basic Weekly Climate Data from Greenhouse Grower B Shown above is the general process train for developing the empirical math model. 22 will identify the yearly data that has the closest relationship between 9-week cumulated PPF and yield (in g/plant), as indicated by the statistical r , and derive a set of linear 2 mathematical expressions for each year of data as well as for 6- year cumulative data (1987, 1989, 1990, 1991, 1992, 1993, excluding 1988 data). The r , or correlation 2 coefficient, measures the linear association or clustering around a line with higher values indicating a closer fit to actual yields (Freedman et al, 1991). Extending the light and yield relationship further, a second set of regression results will be performed by cumulating light data for 9 weeks (CL ) in units of W/m and 2 9 regressing these totals against recorded yield in units of (kg/m ). These linear equations 2 based on 9-week summations of light will then be used to predict 1994 crop yields based on light readings recorded for 1994. The actual 1994 yields and immediate weekly predicted yields from these equations will then be plotted on the same graph and compared. A standard error of estimate (S ) or root-mean-square-error analysis (RMSE) will also be e performed on each set of predictions versus actual yield values (Figure 9) to determine the equation that most closely approximates the actual yield. The next step in developing the empirical model will be to investigate the relationships between yield and other recorded environmental parameters. Tomato plants have been found to be very responsive to changes in temperature and light levels and the status of these conditions strongly affects carbonfixationand dry matter partitioning into fruit (Jones et al, 1991). 23 Humidity also affects tomato growth but to a lesser degree, and a single trial run will be included development of the empirical model to examine its effectiveness as a factor in yield predictions, research time-permitting. As a further step in the empirical analysis, the effect of temperature will be entered into the regression model. The general rule for tomato production is that the 24-hour greenhouse temperature is generally responsible for the rate of growth, with higher average temperatures increasing growth rates. The target setpoint temperature for tomatoes reported in the 1993 Greenhouse Vegetable Production Guide is 19 °C (B.C. Ministry of Agriculture, 1993). Below 10°C, little tomato growth / photosynthetic activity occurs. Provided with this information, a cumulation of temperature in terms of heat units (Wolf et al., 1986) will be performed. Each heat unit (HU) is defined as the difference between the daily 24-hour average temperature and a baseline temperature of 10 °C. The first trial run using this approach will involve cumulating 9 weeks of recorded temperatures and subtracting the baseline 10°Cfromeach value ( 9 weeks of values x 10°C = 90°C subtractedfrom9 week cumulative total of temperature readings). These adjusted values will then be referred to as 9-week cumulative heat units (CH ). 9 By performing a multiple step-wise regression analysis on both the 9 week heat units (CH ) and 9-week cumulative light values (CLg), 9 linear relationships involving heat, light, and yield will result. This approach will provide immediate yield predictions and provide an indication as to which factor (light or temperature) plays a bigger role in tomato yield prediction. 24 Continuing on with the study, the yearly data set that contains the best linear relationship with light (highest r - value from 9-week light (PPF) regression) will be further 2 analysed. Combinations of the environmental factors; 3, 6, and 9 week cumulative heat units (CH , CH6, CH ) and 3, 6, and 9 week cumulative light values (CL , CL , CL9) will 3 9 3 6 be regressed to determine the best prediction equation (Y ) for yield (Y = f { CH , CH*, p p 3 CH , CL , CL , CL9} ) However, at this point of the analysis process instead of using the 9 3 6 model to produce immediate weekly yield values, the model as a true predict-ahead model. The initial regression results of empirical model identified the most correlated data, and whether light or temperature was the greatest influencing factor on yield. However, an empirical model that uses historical data to predict crop yield a few weeks ahead is necessary if the model is to be more flexible for growers and the Greenhouse Growers Cooperative Association. By using the cumulated values of light and temperature and regressing them with actual yield results recorded four weeks ahead, a four-week ahead predictive model can be obtained. As afinalstep of the empirical approach to yield prediction, the 1994 yield will be predicted 4 weeks ahead by an multi-variable linear equation that combines 3,6, and 9 week cumulative light and 3, 6, and 9 week cumulative heat data. This equation will be evaluated by standard error analysis (S or RMSE) for usefulness and flexibility of the model to the e greenhouse grower to identify the best overall empirical model. 25 2.6 The Agassiz Expert System Model The resultsfroman expert system, tomato yield prediction model, developed by the Agassiz Research station (Lin, 1994) will be analysed and used solely as a comparison model for the two mathematical models developed in this study. The specific methodology used to develop the Agassiz expert system is currently being licensed and must remain confidential as part of a collaborative agreement to protect intellectual rights between the Agassiz Research Station, the University of British Columbia, and the participating members of the greenhouse industry. However, an overview of the general methodology used to develop the Agassiz expert system is presented on Figure 8. First, the same weekly environmental data (actual yield, light intensity, temperature, and humidity readings from 1987 to 1993) from Greenhouse Grower B used in developing the empirical math model, was imported into the expert system. Then by using up-to-date expert knowledge from research findings, greenhouse growers, and plant physiologists (Lin, 1994), weighting of environmental variables as they related to yield were applied in training the expert model. The advantage to the Agassiz expert system is that as the database of expert knowledge grows, the model can be modified to reflect these newfindingsby adjusting the environmental weighting factors. The expert system is also a self-learning system, in that it recognizes trends between environmental variables and actual yield, and adjusts the weighting factors to reduce the error with each subsequent yield prediction. 26 Development of the Agassiz Expert System Yield Prediction Model Validated Expert System Model I Tomatoes Expert System Comparison of Yield Predictions Expert System Predictions 1,2,3,4 W e e k s ahead with Empirical & Deterministic Models Training of Expert System Import & Configuration of Climatic Data & Variables Historical Weekly Climatic Data from Greenhouse Grower B (Illustration by W.C.Tang, 1995) Figure 8. Shown above is the general process scheme for developing the Agassiz Expert System yield prediction model. 27 The researchers at Agassiz have configured their expert system to provide 1 week, 2 week, 3 week, and 4 week ahead predictions of 1994 yield and these results will then be validated by comparison with actual recorded yields by the RMSE approach. The results from the Agassiz expert system will also be compared with the predictions from the mathematical models developed in this study. 2.7 Validation and Comparison of the Yield Predictive Models In order to be a useful management tool for greenhouse growers, crop growth and yield prediction models must be validated before they are used (de Koning, 1993a). The validation of the deterministic, empirical, and Agassiz expert system model can be performed by analysing the magnitude of the error in their predictions. Calculation of the standard error of estimate (S ) (Alder and Roessler, 1960) or e root-mean-square-error (RMSE) for prediction results measures the overall size of differences between predicted and actual values for yield (Freedman et al, 1991 and Kozak, 1995). The S e or simplified RMSE equation used in this study is shown on Figure 9. The magnitude of these error values depends on the range of values being compared and the number of predicted values. With a larger range and greater number of values, generally larger error values are expected. In this study, typical tomato crop yield results range from 0 kg/m to 4 kg/m , with a minimum of 30 predicted yield values provided by each model. 2 2 To provide a basis for validation and comparison, models providing error values of less than 0.8 (kg/m ) for a year of predictions (approximately 50% error for an average 1.6 kg/m 2 yield throughout 2 the year) will be considered valid models in this study. 28 Root-Mean-Square-Error Analysis (RMSE) or Standard Error of Estimate ( S ) e m n RMSE or S = e •S X (Actual - Predicted) : 1 i= 1 N = Standard Error of Estimate, kg/m 2 e • RMSE = Root Mean Square Error, kg/m 2 • Predicted = Predicted Yield Value, kg/m 2 • Actual = Actual Yield Value, kg/m •N = Total Number of Predictions 2 Figure 9. Shown above is the S or RMSE method used to compare the predictive models in this study. Lower values indicate less error and better predictions. e (Freedman et al, 1991. Alder & Roessler, 1960. and Kozak, 1995) 29 The lowest RMSE or S values indicate the least error between predicted and actual e values (ideally an error value of 0 kg/m ), and best predictive performance. 2 The best overall model will be determined by the accuracy of predictions (lowest RMSE or S values) for an entire growing season and during periods of peak productivity 6 (RMSE for a specific peak interval), flexibility of the model for application in site-specific and general cases, and overall user-friendliness. 3.0 Results and Discussion 3.1 Analysis of Graphical Plots of Hourly Greenhouse Climate Data Shown in Appendix A are graphical plots of inside greenhouse temperature, outside temperature, solar intensity, carbon dioxide concentrations, and relative humidity for a 60day growth period (between January 1, 1992 to March 5, 1992). Trends observed during the 60-day growth period included the maintenance of a fairly consistent inside greenhouse temperature between the 18 to 23°C range, with peaks in daily inside temperature corresponding directly to peaks in solar intensity (Figures A-l, A-2, A-3 in Appendix A). Solar intensity peaks are recorded for noon periods each day. Guidelinesfromthe 1993 Greenhouse Vegetable Production Guide indicate average 24 hour temperatures as being responsible for the rate of growth, with higher averages leading to faster growth rates. The target setpoint temperature for tomatoes is currently 19°C (B.C. Ministry of Agriculture, Fisheries, and Foods, 1993). Successful environmental 30 temperature control for this plotted growth period is evidenced by the minimal fluctuations even during periods of outside temperature extremes including sub-zero periods. Plots of carbon dioxide concentrations for the same period (Figures A-4, AS, A-6 in Appendix A) revealed a range between 200 to 2000 parts per million (ppm), which is typical enrichment for a tomato crop (Gould, 1983). Concentrations frequently reached their peaks during the noon hour of each day, and showed a daily maximum fluctuation of about 1800 ppm. Inside venting for temperature control can be held accountable for part of the variability in carbon dioxide concentrations. Relative humidity readings plotted for the same growth period showed peaks and valleys approximately every ten hours, with peak humidity during late afternoon and evening hours. Humidity was found to vary greatly between 60 to 95 percent for the early part of the growth period (January 1 to January 24, 1992) with less variation occurring during the following two months (January 25 to March 5, 1992). Low humidities for a few hours a day will not hinder the overall climate balance within the greenhouse but levels should typically be maintained above 50% to prevent extreme moisture losses. Proper maintenance of high humidity levels (80% and above) will encourage stomatal opening and increased carbon dioxide uptake, leading to increased water evaporation, and cooling of the plant canopy (B.C. Ministry of Agriculture, Fisheries, and Foods, 1993). The patterns discovered from these plots illustrate both the importance of climate control and the possibility of applying trend recognition techniques to predict these parameters. Trends in greenhouse climate can then be related to actual recorded yields for particular growth periods for the development of a yield prediction model. 31 3.2 Results of the Deterministic Math Model Applying thefirstprinciple equations of photosynthesis, respiration, and dry weight growth rate detailed in literature (Jones, 1991) a deterministic math model was developed. Using the yield conversion factor of 7 described in literature (Enoch, 1978) hourly net photosynthesis calculations were converted into hourly yield predictions (ie. hourly P values multiplied by 7). net Monthly variations in leaf area index (LAI) throughout the year (Appendix B) affected the photosynthetic values and were corrected for accordingly. Highest LAI values were reportedfromliterature (de Koning, 1993a) for the peak growth period between March to June. A sample of the spreadsheet used in developing the deterministic math model is shown in Appendix B. For graphical plotting of the results, the hourly yield predictions were summed into weekly cumulative totals before plotting versus week number. Week numbers correspond to an annual chronological scheme where week 1 represents the first week in January and where week 52 represents the last week of December. The initial plot of immediate (same week) yield predictions compared with actual recorded yield (Figure 10) produced an good range of results. By cumulating net photosynthetic (P t) rates for periods of 1 week (158 hours) and converting these totals to ne yields (1 Week C. P net Yield) a low range of yield predictions resulted. The magnitude of the predictions were approximately half of the actual recorded yield for the 1 week cumulations, indicating that a week of net photosynthetic activity may not be sufficient for predicting actual yields. 32 Cumulative Pnet based Yield Immediate Yield Predictions ~~ 3.5-r Week # -A- Actual Yield Figure 10 - X - 1 Week C.Pnet-Yield - B - 2 Week C.Pnet-Yield -+— 3 Week C.Pnet-Yield Shown above are the immediate yield predictions using'the yield conversion factor applied to Pnet. 33 The idea of cumulating net photosynthetic activity and converting to yield was carried on further by cumulating 2 and 3 weeks of P t in order to obtain a better fit of the actual ne yield curve. The plotted results (Figure 10) revealed the closest correlation between a two week cumulation of P net and actual yield. For the first half of the year (up to week 23) predictions showed a good range of peaks simulating actual recorded yield. Predicted yield peaks were similar especially during the 10 to 20 week period after a short initial period of over-prediction (weeks 1 to 10). The over-prediction during the initial few weeks was expected as the deterministic model assumes that all net photosynthetic activity is being directly converted to yield. In actual greenhouse practise however, tomato crops are not being harvested for thefirst8 to 9 weeks at the start of the growth season. The general shapes of the curves for each of the weekly cumulation trials were similar and encountered a shared problem. In cases of missing or incomplete data sets, yield predictions cannot be made accurately. Yield predictions for weeks 23 to 25 and 32 to 35 show evidence of the problem that exists when required values for the climatic variables used in predictive equations are missing. The successful prediction and cumulation of P t activity relies on complete data sets. ne Calculated cumulative yields following periods of missing data lead to inaccurate and incomplete predictions due to zero value predictions being included in the weekly totals. These zero values are then carried forward to the next week of predictions and then the effect is magnified as these zero values dampen the results leading to under-predictions. 34 In order to compensate for those weeks of missing or ambiguous data, manual extrapolation of the predictions between those weeks with complete data had to be performed. The two intervals of weeks with incomplete or zero data (weeks 23 to 25 and 32 to 35) were corrected by extrapolating prediction lines between the two weeks with reliable data (weeks 22 to 26 and 32 to 36 respectively). The yield prediction corrections for missing data are shown as intervals connected with dotted lines on the future yield prediction plots that followed. In order for the deterministic prediction model to be useful for a grower, it must be able to forecast yields ahead of time. Predictions of yield were made 4 weeks ahead of time using the same 1, 2, and 3 week cumulations of net photosynthetic converted activity (Figure 11). Better results were found for the 4 week ahead set of data than those reported for the immediate, same week predictions. In fact, by predicting 4 weeks ahead, the problem of over-prediction during the first nine weeks of the growing season was alleviated. The 4 week ahead shift in yield prediction also led to closer alignment of the predicted and actual yield peaks and valleys. In particular, the 2 week cumulation of P net showed the closest relationship to actual yield and was calculated to have a RMSE of 0.464 kg/m for the full year of predictions. The 2 dotted intervals between weeks 25 to 29 and 36 to 38 show the correction by extrapolation for missing data during these periods (Figure 11). However, correction for yield predictions in this manner was still not sufficient to maintain the close prediction trend experienced in thefirst25 weeks plotted. 35 Cumulative Pnet based Yield Predictions 4 weeks ahead 3.5-1 : Week # T A T Actual Yield - X - 1 Week C.Pn-Y - B - 2 Weeks C.Pn-Y —t— 3 Weeks C.Pn-Y Figure 11 Shown above are the 4 week ahead yield predictions using the yield conversion factor applied to Pnet 36 The reason for the loss in accuracy and precision after the periods of missing data is that the deterministic model attempted to simulate actual, continuous growth of a tomato crop. Any discontinuities, or periods of missing data would not only affect that particular period, but also the summations leading to yield predictions for subsequent time periods. The effect of the missing data periods was even magnified further for the last quarter of predictions (weeks 35 to 52). In a further attempt to provide predictions further ahead, yields were predicted 6 weeks ahead of time, again by shifting immediate yields forward and comparing by the RMSE method. Even better results were found for predicting 6 weeks ahead as shown on the plotted graph (Figure 12). The major peaks in yield became even more closely aligned with actual yields and the 6 week shifting reduced the prediction error of the extrapolated intervals with missing data. Root-mean-square analysis of the 2 week cumulative yield plot for 6 weeks ahead produced a value of 0.3818 kg/m , even closer than the 4 week ahead 2 plot. The use of two week cumulative P ne t values converted to yield for 6 week ahead predictions actually becomes a 8 week ahead forecast considering 2 weeks of historical data are required to make these predictions. Basis for cumulating two week P net activity as a yield predictor was found from the rate of starch accumulation of a tomato plant found from literature. Daily starch accumulation for tomato crops was found to reach its peak during the 15 to 25 day period after pollination (Ho and Hewitt, 1986). 37 Cumulative Pnet based Yield Predictions 6 weeks ahead Week # Actual Yield Figure'12 - X - 1 Week C.Pn-Y - S - 2 Weeks C.Pn-Y - 4 - 3 Weeks C.Pn-Y S h o w n above are the 6 week ahead yield predictions using the yield conversion factor applied to Pnet. 38 As the rate of starch accumulation may vary slightly for different tomato varieties, it is likely that the peak starch accumulation period for the tomatoes used in this study were near the two week (14 day) interval. From literature, the peak of starch accumulation accounts for approximately 30% of the daily accumulated dry matter by day 20 after fruit pollination (Ho and Hewitt, 1983). Use of the two week P net results and converting with the assumptions (Figure 6c) built into the yield conversion factor (Enoch, 1978) for yield predictions may be justified. As much of the starch and dry weight growth of the tomato may have occurred by the two week rapid growth period, the conversion factor already has an assumption that only 50% of the dry matter is being converted to yield. With the 50% assumption of dry matter to yield conversion the magnitude of the prediction error is already being reduced as this ratio may vary under changing climatic conditions. For example, under conditions of very low light flux densities (below 40 W/m ) no net photosynthesis was recorded for tomatoes of any 2 size (Tanaka et al., 1974), however based on the spreadsheet structure of the deterministic model and first principle equations, a value greater than zero may be calculated. One possible reason for the discrepancy between results in actual practise and in theory is that emphasis of the effects of light in the generalized photosynthetic equation may not be great enough. The effect of temperature, carbon dioxide concentrations, leaf area index, and light are all incorporated into the photosynthetic equation and each additional parameter may dampen the significant effect of light levels on P t activity. Consequently, ne when P net values are directly converted into yield predictions using the conversion factor, 39 the deterministic model may over-predict during periods of low light levels and underpredict under high light conditions. Other possible sources of error for the predictions is the effect of respiration upon the magnitude of P„ . It is known that respiration rates in mature fruits and young fruits et vary greatly. Respiration rates in two-week-old tomatoes were reported to change from 0.4-0.6 mg C O 2 per gram offreshweight per hour to 0.05 to 0.07 mg C O 2 per gram of fresh weight per hour as they reached maturity (Tanaka et al., 1974). The deterministic model developed here does not account for the variability of respiration rates due to fruit age and this may be a source of error for the predictions. 3.3 Results of the Empirical Math Model Following studies that found a close correlation between photosynthetic photon flux (PPF) received during a 60-day production cycle and tomato yield (McAvoy et al, 1989), an empirical model was developed. Weekly solar radiation datafromGreenhouse Grower B in units of moles/m were plotted versus crop yield (g/plant) assuming a tomato plant 2 density of 2.5 tomato plants / m . 2 Results from these plots of total PPF for a 9-week (approximately 60-days) production period versus crop yield were plotted and linear regressions were performed to derive equations for each year of data. A summary of the regression equations derived from the total PPF plots can be found in the appendix {Table A-l in Appendix Q. For each plot, nine weeks of PPF were cumulated and then plotted against yield (g/plant) to find the year with the closest correlation (Figures A-ll, A-12, A-13, A-14, A-15, A-16 and A-l 7 in Appendix C). As 40 well, for one plot, data from all 6 years (1987 to 1993 inclusive, excluding 1988 due to missing data) was plotted as a 1987-1993 graph to identify a general trend relationship between light and yield (Figure 13). The 1987-1993 plot had a calculated r value of 0.668 2 which indicates a fairly consistent and direct relationship between light and yield for 6 years of historical data. From r analysis, or comparison by the correlation coefficient, the 1992 data set 2 showed the closest fit with the highest r value of 0.746. 2 The higher the r value (with 2 maximum value being 1) the better the linear fit between the independent variable of light (PPF) and dependent variable (yield). The yearly scatter plots showed a fairly good linear trend between light and yield with r results ranging from 0.61 to 0.746. 2 The r correlation results calculated from 2 Grower B's weekly data were generally lower than those reported in literature for tomato crops (McAvoy, 1989) where an r value of 0.896 (or r = 0.947) was obtained. 2 From the results of these preliminary plots, a second set of regression equations was obtained by plotting light and yield data in different units (Table A-2 in Appendix D). Light data was cumulated for 9 weeks (assuming a 60-day production period as before) in units of W/m and regressed with yield measured in units of kg/m to develop an empirical 2 2 model which would produce more meaningful results (in more useful and common units) for greenhouse growers. The whole range of regression equations derived from each year of data were used to predict 1994 tomato yields and compared with actual yields reported for the same period (Figures A-18 & A-19 in Appendix D). 41 1987-1993: Total PPF vs. Crop Yield 900 800 700 600 =H" c CO a. 500 3 T3 400 t" CD >- K i-fr = F + + 4 ft-+ + + ^ _±—_—±i_ + 300 + "F + -I- ++ + 200 100 0 Figure 13 i mil ii iiirtrihnHr—H- 100 200 300 400 500 600 Total PPF (moles/m ~ 2) 700 800 900 Trend identification of light (PPF) vs. Crop Yield 42 Analysis by the root-mean-square-error analysis revealed the best results (RMSE = 0.382 kg/m , r = 0.625) when the equation derivedfromthe 1990 set of data based on 2 2 light alone (CL9) was used to predict yields for 1994. The coefficients for 1990 set of data were then identified as predictive equation 1.1 (Table 1, and Appendix C - Table A-l). Using a similar approach, the regression equation derivedfromthe 1992 set of data was identified as predictive equation 1.2 to provide an upper-range value for prediction (Table la). When equation 1.2 (1992 equation) was used to predict 1994 tomato yields an RMSE value of 0.4574 kg/m (r = 0.465) was calculated. 2 2 Plots of both sets of same-week yield predictionsfromequation 1.1 and 1.2 against actual recorded yields is shown on Figure 14. From an examination of the plotted predicted yields, the regression equations were found to produce fairly good results for thefirst20 weeks of the growing season when yields rose steadily. However, for certain intervals of rapid growth (between weeks 20 to 30 and weeks 40 to 46), peaks in actual recorded yields could not be adequately predicted by the regression models. In fact from week 20 onward, the light-only based models (CL9) were unable to predict the yield extremes found in actual greenhouse production practises. The greatest discrepancy in actual recorded yields and predicted yields was found during weeks 25 and 32 where the regression models consistently under-predicted actual yields. Although an empirical model developed solely on cumulative light had its basis, adequate correction for predictions during the peak periods of actual yield could not be obtained using a light-only model. 43 Table la EMPIRICAL MATH MODEL Results for 1994 Yield Predictions 1.0 Light Alone (CL ) 9 • Y i = ao + ai(CL ) p 0) 9 r RMSE ao ai Predictive equation (1.1) (1990 Data) -0.07057 0.0000165 0.625 0.382832 Predictive equation (1.2) (1992 Data) -0.08383 0.0000145 0.465 0.457382 2 The predictive equation (1.1) is based on fitting 1990 data to regression equation (1) and produced a RMSE of 0.34827 and r = 0.625 for 1994 yield predictions. Predictive equation (1.2) is based'dnfitting1992 data to regression equation (1) and produced a RMSE of 0.457382 and r = 0.465 when used to predict 1994 yield. 2 2 2.0 Light and Heat (C L and CH ) 9 Y p 2 9 = ao + ai (CL ) + a (CH ) 9 2 ao (2) 9 ai a r 2 2 RMSE Predictive.equation(2.1) 2.39584 (1992 Data) 0.00001513 -0.03003 0.68 0.353099 Predictive equation (2.2) 2.38054 (1990 Data) 0.00002444 -0.033809 0.247 0.542205 The predictive equation (2.1) is based on fitting 1992 data to regression equation (2) and produced a RMSE of 0.353 and r = 0.68 for 1994 yield predictions. Predictive equation (2.2) is based on 1990 data and produced a RMSE of 0.542 and r =0.247 when used to predict 1994 yield. 2 2 44 1994 P r e d i c t e d Y i e l d s Cumulative Light Based (CL J 2.5CM < RMSE = 0.3828 2- E 1.5"D O >- 10.50- Week Number Actual Yield Figure 14 - e - Predictive Eqn# 1.2 Predictive Eqn# 1.1 P r e d i c t e d Y i e l d s for 1994 p r o d u c t i o n b a s e d o n cumulative light d a t a 45 Greenhouse growers require the best yield forecasts during peak growth periods to ensure that availability of adequate harvesting supplies and workers to capitalize on the crop productivity. In an attempt to provide closer predictions during these peak yield periods, another climatic variable had to be incorporated into the regression equation. From literature review, temperature was also found to strongly affect the yield of tomato crops (Jones et al, 1991). As well, in another study, the development time between flowering and harvest for tomato crops was found to be mainly dependent on temperature (de Koning, 1993b). Consequently, the next climatic variable entered into the empirical model for predicting tomato yield was temperature. Using a cumulative heat unit approach described in literature (Wolf et al., 1986) 9-week cumulations of corrected temperatures (CH ) were regressed with 9-week cumulations of 9 light data (CL9) to create a light and heat based empirical model. Multiple step-wise regressions were performed for each year of data to provide detailed analysis of the relationships between corrected temperatures (CH ), light (CL9), and yield 9 (Appendix D). The resulting regression equations from this approach were then summarized (Appendix D, Table A-2) and used to predict 1994 tomato yields. Prediction plots versus actual recorded yields for 1994 were performed (Appendix D, Figures A-20, A-21). Analysing the plotted results it was found that the incorporation of the temperature variable provided better predictions during the peak production periods. Where the lightonly plots (CL9) were unable to predict for these extreme intervals, the 9 week cumulative 46 temperature (CH ) and light (CL9) based equations were able to correct for the peak 9 periods. Isolating the equations from the same two years (1990 and 1992) of predictions (Table 1), a good range of predictions was obtained. The best predictions were found using the heat and light method with the 1992 set of data (RMSE = 0.353 kg/m ) and was 2 identified as predictive equation 2.1 (Table la). The 1990 equation was identified as predictive equation 2.2 (RMSE = 0.542 kg/m ) 2 both for comparison with the previous light-only model (CL9) and also to illustrate the improved predictions during the peak yield periods using this method. From the plot of predicted yields and actual yield (Figure 15) a better range of predictions for high and low values were found. Predictive equation 2.1 (1992 data) was found to produce yield predictions very close to the actual yields for thefirst25 weeks of the production period. Problems with predicting yields during peak periods still existed using equation 2.1 but predictions were generally higher during these intervals producing an overall improved and lower RMSE of 0.353 kg/m . 2 Using predictive equation 2.2, yields were generally over-predicted for the entire production periods (RMSE = 0.542 kg/m ) but the possibility for adequate correction using 2 this method (CH & CL<> regressions) was shown. If RMSE analysis was performed for the 9 peak periods alone, equation 2.2 with its over-predictions may actually be proven to provide the most useful results to a greenhouse grower. During peak periods it may be preferable for a grower to be prepared to handle higher yields than those actually experienced to maximize productivity. 47 1994 P r e d i c t e d Y i e l d s Light(CL ) and Temperature(CH ) Based Week Number Actual Yield Figure 15 - e - Predictive Eqn# 2.1 Predictive Eqn# 2.2 Predicted yields for 1994 production based on cumulative light and heat data. 48 The encouraging results from these light and heat multiple regressions led to experimentation with other combinations of cumulative light and heat data. In particular, it was postulated that by cumulating heat and light units for different time intervals (using a similar approach as used for the 9-week period) 3 and 6 week cumulations would be made. For this experimental process, only one yearly characteristic set of data was used for developing the empirical model. As detailed in the materials and methods section the year of data with the best linear relationship with light was further analysed. The 1992 data set was chosen for further analysis because it had the closest relationship between total PPF and crop yield (g/plant) from thefirstset of regression results (r = 0.746). As well, the 1992 equation was selected for a better comparison of results between the deterministic and empirical models. A comparison of prediction results from the deterministic and empirical models developed from the same year of data (the former being hourly and the latter being weekly 1992 data) would ideally identify the better model rather than variations in predictions based on climatic differences. By incorporating the 3, 6 and 9-week cumulative totals for light and heat into an allinclusive equation, Equation 3.1 (Table lb) (where Yp = f{CH , CHg, CH , CL , CL , 3 9 3 6 CL9}) it was hoped that this method would correct for yield peaks and provide an overall improvement in prediction accuracy. Using three light and heat variables in the yield prediction was based on the fact that each interval (3, 6, and 9 weeks) may have varying effects on the development of the tomato crop related to yield. 49 Table lb EMPIRICAL M A T H MODEL 3.0 Light and Heat (C L Y p3 3 . .9 6 and CH . .9) - Immediate Yield Prediction 3 6 = ao + a,(CL ) + a (CL ) + a (CL ) + a4(CH ) + 3 2 6 3 9 3 a (CH«) +a6(CH ) 5 ao 0.34917 Equation (3.1) (1992 Data) (3) 9 ai 0.0000355 a 0.00000548 2 a a -0.001127 -0.03108 4 a -0.000667 &6 -0.02212 5 3 r 0.527 RMSE 0.429766 2 The predictive equation (3.1) is based on fitting 1992 data to regression equation (3) with a RMSE of 0.429766 and r = 0.527 for 1994 predictions. 2 4.0 Light and Heat (C L , , and CH 3,6.9) - Future Yield (4 Weeks ahead) 3 Y p4 6 9 = ao + a!(CL ) + a (CL ) + a (CL ) + 3 4 ( C H ) + a (CH5) +a6(CH ) 3 2 6 5 Equation (4.1) (1992Data) 3 9 3 (4) 9 ao 0.34917 ai 0.0000355 a -0.001127 a 0.00000548 2 a -0.03108 4 ae -0.02212 5 a -0.000667 3 r 0.491 RMSE 0.44579 2 The predictive equation (4.1) is based onfitting1992 data to regression equation (4) and produced a RMSE of 0.44579 and r = 0.491 for 1994 predictions when using 3,6, and 9 week cumulated data (4 weeks ahead of actual yield) to make future yield predictions. 2 Yp4 = 0.35 + 0.0000355(CL ) + 0.0000055(CL ) 0.031(CH6)-0.022(CH ) 3 6 9 RMSE =0.4458 kg/m 2 0.00067(01^) - 0.0011(CH ) 3 50 By incorporating each cumulative interval into the yield predictive equation an attempt was made to have a model that would predict for peak periods and more closely follow the trends of the actual yields. Regressing (Appendix D) and plotting the results of this approach (Figure 16) a better pattern of predictions was found even though prediction error increased (RMSE = 0.43 kg/m ). 2 Predicted yields showed more high and low variation and demonstrated a greater degree of association with the pattern of tomato crop productivity. From these regression results it became more evident that light or solar intensity was the primary influencing factor affecting yield with temperature effects being the secondary factor mainly influencing peak periods of productivity. From these stages in developing the empirical model, an attempt was made to predict yield weeks ahead of time to be a useful management tool for growers. From the results of the three approaches used to develop the empirical model (light-only, light and heat, and 3 variables of light and heat) it was decided that the method using 3 variables of light and heat be pursued for creating a 4-week predict ahead model. Two approaches were used for developing this 4-week predict ahead model. The first approach involved cumulating 3, 6, and 9-week data for light and heat and regressing these results with actual yield values recorded 4 weeks ahead of time. Previously, predictions were only made for the same week by relating 3, 6, and 9-week data leading up to that week. With the 4 week ahead approach, the actual yields from 1992 were related to data cumulated 4 weeks prior to the actual reported yield. 51 1994 P r e d i c t e d Y i e l d Using 3 Light and 3 Heat Variables Week Number Actual Yield Figure 16 - e - Predictive Eqn# 3.1 Predicted Immediate yield using 3 variables of light and 3 variables of heat. (CL3, CL6, CL9, and CH3, CH6, CH9) 52 The regression equation derived using this approach (Appendix D, Alternative Regression Method) produced predictions with fairly high error and low correlation with environmental variables (RMSE value of 0.615 kg/m and r = 0.321) 2 2 The second approach that was employed to predict 4 weeks ahead, was to simply use the same prediction equation developed for immediate, same-week predictions for the 3 variables of light and heat (Equation 4.1, Table lb) and use weekly light and temperature values 4 weeks ahead in the equation. Use of this method produced a much lower RMSE value (0.45 kg/m ) and showed better correlation (r = 0.491) with environmental variables 2 2 than predictions made using the regression equation derived from 4 week ahead data and yield (Appendix D, Alternative Regression Method). The predictions using the same week equation (4.1) with 4 week ahead data are shown on Figure 17. Analysing and comparing with the actual yield plot, prediction error for thefirst25 weeks is minimal. For the remaining 25 weeks of the growing season, the highest prediction error interval still remains between week 25 to 30 (June, July) when crop productivity is typically highest. A reason for the greater error during this interval include the strong influence of warmer temperatures on crop development time during this period which cannot be easily accommodated by a regression model that tries to predict weekly yields for an entire growing season. In fact, researchers are still having difficulty estimating the amount of plant assimilates (nutrients from photosynthesis and root absorption) being formed and translated at different temperatures. 53 1994 Predicted Yield - 4 W e e k s A h e a d Using 3 Light and 3 Heat Variables 3 2.5 CM < 2- E 1.5•a CD 10.50- 10 15 20 25 30 Week Number Actual Yield F i g u r e 17 - s - 35 40 Predictive E q n # 45 4.1 Predicted yield for 4 weeks ahead using 3 light and 3 heat variables. (CL3, C L 6 , C L 9 , and C H 3 , C H 6 , CH9) 50 54 Nutrient translocation rates have been reported to be about four times higher at 27°C than at 15°C (Barendse, 1993) and this may provide a clue to sharp increases in productivity during high temperature intervals. Similarly, tomato development times between flowering to harvesting of the tomato fruit have been found to vary from 70 days at 17°C to 44 days at 25°C, demonstrating strong a strong relationship to temperature (de Koning, 1993b). In order to improve the accuracy of predictions during the peak temperature and solar intensity periods (weeks 25 to 30), it may be necessary to develop regression equations for those specific intervals. Actual yield and environmental data for those peak periods may need to be isolated and regressed separately to derive separate equations for peak productivity intervals throughout the year. The grower may then use these 'peak period equations' to provide an upper- end estimate of yields during high productivity periods. When a trial test run was performed to incorporate the effects of humidity as well as light and heat (Appendix E, Humidity Trial Regression Results & Figure A-22), prediction accuracy was found to be worse than with only light and heat, (RMSE = 0.472 kg/m ). 2 Overall, the empirical model with 4-week ahead predictions using equation (4.1) was found to be the mostflexibleand useful for actual application by greenhouse growers. 3.4 Results of the Agassiz Expert System Using weekly data from Greenhouse Grower B, the Agassiz Expert System model was trained to provide tomato yield predictions 1, 2, 3, and 4 weeks ahead of time. The specific weighting and configuration of environmental variables was decided upon by the plant physiologist researchers at Agassiz (Lin, 1994) and the detailed methodology used in the development of the model remained confidential for licensing purposes. However, 55 prediction results using this model were shared for academic comparison with the empirical and deterministic models developed in this study. The tabulated results from the Agassiz Expert System are shown on Table 2. For comparison with the two mathematical models in the study, RMSE analysis was performed on the four sets of predictions. Weekly predictions were identified as PRE-1, PRE-2, PRE-3, and PRE-4 results, representing predictions made 1, 2, 3, and 4 weeks before actual yields are recorded, respectively. Error analysis by RMSE revealed a range from a high 0.722 kg/m for the PRE-2 results (2 weeks before actual yield is 2 recorded) to a low value of 0.401 kg/m for the PRE-1 results. 2 The best predictions in terms of lowest error when compared with actual yield were found for the PRE-1 (1 week before actual yield is recorded) set of results with an RMSE of 0.401 kg/m . 2 The predictionsfromthe Agassiz model were then compared graphically by plotting the PRE-1 results with actual recorded yields for better analysis of prediction performance throughout the growth season (Figure 18). Comparing the actual and predicted yields (PRE-1) the advantages of the trend recognition techniques employed by the Agassiz expert system become more evident. The expert system is capable of predicting more variability in yield trends throughout the growing season than the mathematical models. During the first 20 weeks, although predictions have less accuracy than those of the mathematical models, the general trend of actual yields is well predicted. Between weeks 20 to 30 there is evidence of the improved ability to predict for peaks and valleys in crop productivity during intervals of high temperature and solar intensity. 1994 TOMATO YIELD PREDICTION pRE-4 PRE-3 KG/M PRE-2 PRE-1 ~2 PREDICTED YIELDS ACTUAL YIELD KG/M ~ 2 1994 DATE (Mon --> Sun) ACTUAL YIELD NA NA NA NA 0.85 1.16 1.12 0.36 1.28 1.45 1.63 1.26 NA NA NA 0.77 0.82 0.54 0.77 1.12 1.58 1.25 0.96 0.87 NA NA 0.66 0.77 0.24 0.39 1.11 1.14 1.23 0.85 0.87 1.45 NA 0.47 0.66 0.47 0.66 0.95 0.95 1.26 0.82 1.23 1.56 1.38 0.01 0.53 0.54 0.85 0.95 0.92 1.12 1.19 1.31 1.45 1.81 1.6 FEB28- MAR6 MAR07 -MAR13 MAR14 -MAR20 MAR21 -MAR27 MAR28 APR03 APR04 •APR10 APR11 APR17 APR18 APR24 APR25 MAY01 MAY02 -MAY08 MAY09 MAY15 MAY16 MAY22 1.47 2.1 1.93 2.08 2.31 2.05 1.77 1.33 1.74 1.62 1.42 1.51 1.53 1.97 1.83 2.94 1.19 1.84 1.62 1.46 1.39 1.59 1.83 1.54 3.34 4.34 2.12 1.87 1.92 1.75 1.71 1.99 2.18 1.83 1.74 2.41 1.48 1.65 1.92 1.96 1.83 1.56 1.37 1.31 1.49 0.78 0.74 0.54 1.06 1.44 1.26 1.72 1.88 2.14 1.94 1.7 1.3 1.31 1.81 0.99 0.78 0.7 0.96 1.48 1.61 2 1.63 1.4 2.92 1.75 2.11 2.06 2.65 2.41 1.99 2.99 2.1 2.16 2.12 1.52 2.06 2 1.88 1.21 1.28 1.8 1.49 1.12 1.37 0.82 0.95 1.53 1.75 1.75 0.71 MAY23-MAY29 MAY30-JUN05 JUN06-JUN12 JUN13-JUN19 IJUN20-JUN26 |JUN27-JUL03 (JUL04-JUL10 JUL11-JUL1 7 JUL1 8-JUL24 IJUL25-JUL31 AUG01-AUG07 AUG08-AUG14 |AUG15-AUG21 AUG22-AUG28 AUG29-SEP04 SEP05-SEP11 SEP12-SEP18 SEP19-SEP25 SEP26-OCT02 OCT03-OCT09 OCT10-OCT16 OCT17-OCT23 OCT24-OCT30 OCT31-NOV06 NOV07-NOV13 NOV14-NOV20 INOV21-NOV27 2.09 1.88 2.52 2.35 2.63 3.63 3.94 1.36 1.48 1.49 1.05 1.31 1.32 1.29 1.04 0.93 1.78 1.48 1.59 1.58 1.41 1.32 1.24 1.22 1.1 1.1 1.06 1.62 1.27 1.21 0.96 1.12 1.12 0.7 1.23 0.99 0.77 1.08 1.93 1.67 RMSE || 0.63 10.603 10.722 NA = NOT AVAILABLE BECAUSE OF NO PREVIOUS ACTUAL YIELD RECORD Table 2 Tabulated results for the Agassiz Expert System 5 7 A g a s s i z Expert S y s t e m Predictions 1994 0 3 6 Y i e l d s (1-Week A h e a d Predictions) 9 12 15 18 21 24 27 30 33 36 39 42 Week # Actual Yield Figure 18 1-Week A h e a d Shown above are the 1 -Week Ahead Yield Predictions for 1994 using the Agassiz Expert System. 45 58 For the last interval,fromweeks 30 to 50, the expert system continues to demonstrate its general ability to provide close predictions for peak productivity periods and an excellent forecast of the yield trend expected. For the purpose as a greenhouse management tool, yield predictions 1-week ahead (PRE-1) may be sufficient for preparing for weekly harvests depending on the size of the greenhouse operation. Larger operations may require an earlier forecast of yields for decision-making regarding capital cost investments such as the purchase of higher capacity harvesting equipment. However, the advantage of the predictions provided by the Agassiz Expert System is its ease of application, and flexibility. With the expert system designed to provide a range of predictions (1, 2, 3, and 4 week ahead), the same model could be applied to another set of data from another greenhouse site quite easily as only the weighting factors and input data may change. In fact, in terms of flexibility and user-friendliness, the expert system has the greatest potential and advantage. As the database of actual yield and environmental parameters grows, the predictions by the trend recognition expert system will improve. The expert system will be able to account forfluctuationsin yield and better accommodate for yearly trends than the mathematical models, without tedious modifications. As an added advantage, the expert system approach begins as a general model for yield prediction, but then as it is trained with grower and site specific data it learns yield trends. The expert system model becomes particular to trends of a specific grower and gradually becomes a customized model. 59 The results from the Agassiz expert system demonstrate the dynamic and flexible nature of this approach and encourages further research and experimentation with this method of yield prediction. 3.5 Validation and Comparison of Results from the Yield Predictive Models The guideline for validation of the models in this study was established to be an RMSE error value of less than 0.8 kg/m as mentioned earlier (Section 2.7). 2 For the deterministic model based on the first-principle equations of net photosynthesis and respiration, the best model developed was a 2-week cumulative P t ne based yield model predicting 6 weeks ahead with an RMSE value of 0.3618 kg/m . 2 The error value obtained within the set guideline required for validation so the model was considered valid. The advantages of the deterministic modeling method was that it was flexible, and based on well-acceptedfirst-principleequations. The model was considered flexible because the equations applied relied purely on environmental parameters and did not attempt to measure trends or simulate patterns of yield productivity. Consequently, by theory, the model should be applicable to any greenhouse site without bias, and provide the same predictions for any tomato crop under the same environmental conditions. The deterministic model is not season-dependent nor grower-specific because the structure of this approach relies only on the magnitude of environmental parameters such as solar intensity, temperature, and carbon dioxide concentrations. Disadvantages to the deterministic approach include the strict environmental monitoring required, the use of many growth parameters, assumptions in the yield conversion factor, and the tedious nature of application. In order to provide accurate and 60 useful predictions, the development of the model required a full year of hourly environmental data (8760 hours / year). Missing or ambiguous data for any extended periods of time led would result in 'zero values' for predictions during that interval because the equation would not have any input values. The problem of missing data becomes greater once the effect of cumulating weekly net photosynthetic rates (P cumulative totals of P t ne ne t) is considered. If are being converted and used to predict yield, low totals due to missing data for normally productive yield periods would lead to inaccurate predictions. In this study, zero yield periods were solved by extrapolation between known prediction points but without the far prediction point (weeks ahead of time) no yield value could be reported. Asidefromthe problem of missing data, it may not be time or cost-effective to require such a strict environmental monitoring scheme to ensure proper predictions by this method. The use of a great number of coefficients and growth parameters is also a weakness of the deterministic approach (Figures 6a & 6b). Small errors in assumptions or parameter values become magnified with subsequent predictions andfindingthe source of the errorfromthe many assumptions becomes a difficult task. Another uncertainty in developing the deterministic model was the application of the yield conversion factor found in literature (Enoch, 1978). Examining the assumptions involved in applying the conversion factor more carefully (Figure 6c) two of the main parameters on which the factor is based are quite variable. First, the assumption that half the dry matter of a tomato is partitioned to yield will strongly affect the magnitude of the conversion factor. For example, if the dry matter partitioning for the tomato is actually 61 closer to 60% in practise, rather than the assumed 50%, a 17% increase in the yield conversion factor results (factor changes value from 7 to 8.18). Similarly,fromliterature the dry matter content for tomatoes was found to vary between 5 to 10% at different stages offruitdevelopment (Ho and Hewitt, 1986). If the assumed value for dry matter content changed from 5% to 10%, the yield conversion factor would change from a value of 7 to 3.4. Further, if both assumptions are slightly inaccurate then the changed yield conversion factor would produce significantly different results. Changes in temperature are known to strongly affect the partitioning of assimilates in plants (Barendse, 1993) as well as development time of tomatoes (de Koning, 1993b). Assuming a constant or averaged dry matter content for the entire growth season in order to apply the yield conversion factor may only lead to predictions that apply to certain periods offruitdevelopment. The yield conversion factor assumes that by averaging the high and low percentage periods of dry matter partitioning throughout a growing season a generalized yield prediction can be made. As a result of the many assumptions, sources of error, and possibilities for variations in the yield conversion factor, the low error value (RMSE = 0.3618 kg/m ) of the deterministic model may not prove overall prediction 2 accuracy. The empirical model based on multiple regressions of historical data produced reliable and useful predictions for the 1994 growing season. Beginning with a light-only based model and expanding to a model which incorporated temperature in terms of heat units, the empirical model gradually reduced its prediction error. The best, most useful empirical model developed, involved using cumulations of 3, 6, and 9-weeks of light and 62 temperature data to predict yields 4-weeks ahead (Equation 4.1, Figure 17). Error analysis of this approach produced an RMSE value of 0.45 kg/m , well within the required 2 validation guideline of 0.8 kg/m established for this study. 2 The advantages of the empirical approach include the straight-forward method of development and reliable prediction results for general yield trends. Using regression equations derivedfromhistorical data to relate environmental variables to actual yield had the advantage of not requiring plant physiology assumptions or conversion factors. Based on previous trends of production for a specific grower and site, regression equations were developed and analysed for their prediction accuracy. the The prediction equation required only two parameters, temperature and light on a weekly basis to produce results. The empirical approach was much simpler to apply since less stringent environmental monitoring would be required for obtaining weekly averaged values for only light and temperature data. From the graphical plots, the empirical approach was able to adequately predict tomato yield trends for the first 25 weeks with fairly good accuracy. Difficulty in providing close predictions was encountered during peak yield periods (JuneJuly) most likely due to changes in plant partitioning and shortened development times induced by warmer temperatures (Barendse, 1993) mentioned earlier. Disadvantages to the empirical method would be that the fact that equations derived for the data sets would only be site or grower-specific, and the inability to predict for periods. Since the empirical method employs regression techniques for a specific set of data, the equations derived for one grower could not be applied to other sites or greenhouses. Equations or methods that produced the best-fitting predictions for a 63 particular grower may not necessarily produce good forecasts for other growers. For each grower, a new set of regression equations would have to be produced, and new analysis of results would have be performed to identify the best equations. As a result, the methodology of developing the model actual makes the empirical model the least flexible of the three models compared in this study. However, comparatively, the predictions from the empirical model are more reliable than thosefromthe deterministic model because the method used its development requires fewer assumptions. Thefinalmodel examined in this study was the Agassiz Expert System model which employed trend recognition techniques to make yield predictions. The one-week ahead prediction model (PRE-1) produced an RMSE value of 0.401 kg/m and was considered a 2 valid model by the validation guideline of 0.8 kg/m . 2 The plotted results of the one-week ahead model versus actual yield (Figure 18) demonstrated the ability of the expert approach to provide the good variability to predict high and low peaks and during high productivity periods (weeks 20 to 35). As a comparison model, the Agassiz Expert System model showed the most flexibility. The expert system could be applied to new sites or growers without making major changes to the model itself, other than adjusting weighting factors applied and training the model with new sets of data. Further, the expert system would be the most user-friendly model to apply for yield prediction as it quickly provides an understandable range of predictions (PRE-1, PRE-2, PRE-3, PRE-4) without requiring the user to have detailed knowledge of plant physiology. 64 In order to summarize the weaknesses and strengths of each approach, a three-point ranking system was devised to determine the best overall model. The best model in each category is ranked with a value of 1 (for first / best) and the worst in each category is ranked with a value of 3 (for last / worst). out of a maximum of The model with the overall lowest total score 15 points for 5 categories will be identified as the best overall prediction model. The categories used for comparison included: RMSE error for the entire annual growing period, RMSE error for the most productive period (Weeks 20 to 35 of the annual growing season), flexibility of application for individual grower sites, application flexibility for the general case, and user-friendliness of the model {Table 3). When the root-mean-square-error was calculated and compared for each model the deterministic model reported the lowest error (0.3818 kg/m ) and was ranked with a value 2 of 1. The Agassiz expert system was ranked a close second for overall RMSE (0.401 kg/m ), while the empirical model reported the greatest error of the comparison group 2 (0.45 kg/m ) and was ranked with a value of 3. 2 Although the deterministic model reported the lowest overall RMSE for the year of predictions it must also be noted that during part of the growing season, incomplete data sets led to zero predictions for certain intervals. As a result, the deterministic model required extrapolations between known prediction points to provide a full year of predictions. Further, the known variability in fruit water content and dry matter partitioning throughout fruit development may require the application of a variable instead of a constant yield conversion factor. The advantage of the 65 expert system approach was that even with missing or ambiguous data sets it had the ability to make predictions based on analysed trends of historical data. Finally, the empirical model (Figure 17) produced the best predictions for the first 25 weeks of the growing season but had difficulty predicting the high productivity periods. Based on numerical analysis alone, the deterministic model had the best overall RMSE results but in terms of consistency, the empirical and Agassiz expert system may be more reliable (Table 3). For a better comparison of the results during the peak productivity period, a separate root-mean-square-error analysis was performed between weeks 20 to 35 of the growing season. The results are shown in Table 3 with the best results being reported for the deterministic model, followed by the empirical, and Agassiz expert system (ranked 1, 2, and 3, respectively). Although numerically the error results were the best for the deterministic method, the expert system actually had the best prediction of variability during this period, in terms of high and low peaks. The deterministic produced good results (compared with actual recorded yields) during this period as well but required a complete set of hourly data to produce these predictions. The feasibility of such strict monitoring of environmental parameters and extrapolation of results in cases of missing data is questioned if this method were to be actually applied in-situ. The empirical model had fairly good results as well, but had less ability to predict for the general trends of yield productivity when results were analysed graphically (Figure 17). 66 Table 3: Comparison of Yield Predictive Models OVERALL RMSE RANK Wk 20 to 35 RMSE RANK Application Flexibility User Friendly OVERALL SCORE (kg/m ) (1-3) (kg/m 2) (1 - 3) Rank: (1 - 3) Site / General (1-3) Score/Rank 0.3818 1 0.170 1 2 3 10 / 2 EMPIRICAL 0.45 3 0.233 2 2 3 2 12 / 3 AGASSIZ EXPERT SYSTEM 0.401 2 0.247 3 1 / 1 9 / 1 PREDICTIVE MODEL 2 DETERMINISTIC A 3 / / 1 * Deterministic Model: 2 Week Cumulative P„ based model - 6 weeks ahead * Empirical Model : 3 Light & 3 Heat Variables - 4 weeks ahead * Agassiz Expert Sys. : Trend Recognition Based -1 week ahead ot In terms of flexibility of application on a site-specific basis, the Agassiz expert system was ranked highest because of its well-defined model structure (Table 3). The Agassiz model could be easily applied to new sites without any major modification to the model itself. New historical data for each site would be required to train the model, but the model itself could be easily applied to new sites with minor difficulties. The empirical approach was ranked second because it is based purely on site-specific data and the model 67 is customized for particular growers. The deterministic model using first principle equations of net photosynthesis was ranked third because it applied generalized equations regardless of a specific site. Based on environmental variables alone, the deterministic model would make predictions without considering any yield trendsfromprevious years. When theflexibilityof applying the various models to general cases was analysed, the Agassiz expert system was rankedfirstbecause it would be able to accommodate the greatest variations in available data. While the deterministic and empirical models require complete data sets and a more rigid environmental monitoring scheme, the expert model would be reliable even in cases of missing data sets. For actual field application, it is believed that providing reasonable predictions regardless of missing data sets would be extremely beneficial to the greenhouse grower and management staff. Comparing user-friendliness of the three models, a number of factors were considered. First, the prediction results would be considered user-friendly if they were based on very few assumptions, and could be easily interpreted without extensive knowledge or thought of plant physiology. It was also decided that yield predictions one-week ahead of time were sufficient for most greenhouse management decisions. The Agassiz expert system was the only model that had the advantage of providing an immediate 1, 2, 3, and 4 week ahead range of yield predictions and produced'some of the best results from its 1week ahead forecasts. Generally, larger management decisions for capital cost expenditures would be dependent on analysing annual productivity trends rather than weekly yield patterns. Consequently, the best use for the yield prediction models would be 68 to prepare adequate equipment and manpower for harvests and to maximize overall crop productivity. As the goal of the prediction models was to purely to provide good tomato crop yield predictions, the model requiring the least manipulation or customization by the user was considered the best model. The empirical model based on multiple trials of linear regression would require an experienced technician or researcher to properlyfitthe data for each individual grower site. To properly apply the deterministic model approach a strict environmental monitoring scheme would be required, which may not be a cost-effective option for many small-scale growers. However, the deterministic model would have the advantage of being applied effectively in general cases if predictions were properly adjusted with variable yield conversion factors throughout different stages of fruit development. As well, in situations where predictions are required fairly far in advance, the deterministic model may be favored for its 6-week-ahead predictions. Similarly, the empirical model has the advantage of predictions made 4 weeks ahead but requires many regression trials to properly fit each growers' historical yield data. Comparatively, the Agassiz expert system would require the least customization of all three models to provide reliable and cost-effective yield predictions. Based on the overall scores reported for each model, the Agassiz Expert system was rankedfirst,with the deterministic model a close second. The empirical model was ranked lower than the other two models due to its lack of flexibility in actual application and the need for customization by skilled technicians or researchers. For cases when predictions are required far in advance (i.e. 6 weeks) then the deterministic model may be applied 69 effectively. However, for the general case, it is believed that the range of 1,2, 3, and 4 week predictions offered by the Agassiz expert system would be sufficient for most endusers. Overall, the best prediction approach was determined to be the Agassiz expert system model, favoured for its simplicity, reliable and accurate predictions,flexibilityin application, and user-friendliness. 70 4.0 Conclusions In this study, two mathematical models were successfully adapted (deterministic model), developed (empirical model) and validated to predict greenhouse tomato yield in terms of quantity (kg/m ). 2 The final empirical math model developed was based on 3, 6, and 9 week cumulations of light and temperature (in heat units) to predict tomato yields 4 weeks ahead of time. Error analysis of the empirical method produced the largest root-mean-square-error value of 0.45 kg/m for the entire growing season. 2 Predictions during the peak productivity periods (weeks 20 to 35) were fairly good using this approach (RMSE = 0.233 kg/m ). 2 Weaknesses of the empirical model included its site-specific nature, and difficulty in applying to general grower cases. The final detenriinistic math model based on first principle equations of net photosynthesis (P ) predicted yields 6 weeks ahead using a yield conversion factor and two Mt week cumulations of P . net From error analysis, the most accurate predictions were achieved with the deterministic model, with an overall RMSE value of 0.381 kg/m for the 2 entire growing season. The deterministic approach was also fairly flexible for general applications but there were a number of weaknesses identified with this approach. The application of a constant conversion factor was based on assumptions that fruit moisture contents and dry matter partitioning remained constant throughout the tomato development period. Ideally, a variable conversion factor reflecting changes in crop growth could be developed and applied to this model to improve prediction accuracy. 71 Another weakness identified with the deterministic model was the need for an extensive understanding of plant physiological processes to successfully apply the equations used to make the predictions. Further, the requirement for a continuous and complete set of hourly data to make predictions may not be time or cost-effective for many greenhouse growers. In terms of user-friendliness, the deterministic model was ranked last for its large number of plant parameters and assumptions and its inability to provide predictions without a comprehensive set of hourly environmental data. The predictions from the Agassiz expert system model were used for comparison with the two mathematical models developed. With an overall RMSE value of 0.401 kg/m 2 for the entire growing season, the expert system was considered a valid model. When an overall comparison was made with the two math models developed in this study, the expert model was top-ranked in terms of total score (Table 3). Although ranked third for prediction accuracy during the peak productivity periods it became evident that the expert system had the greatest potential for predicting yield variations during these intervals (Figure 18). The expert system would be capable of improving its predictive ability with further training with site-specific, historical data sets, to become a fully customized management tool. Another advantage of the Agassiz model was its ability to provide predictions even for cases of missing or ambiguous data. When the three tomato yield prediction models were compared, the Agassiz expert system was identified as the best overall model for its prediction accuracy, ease of application to both site-specific and general conditions, and overall user-friendliness. 72 5.0 Recommendations From the results of this study the inherent weaknesses and strengths of each modeling approach became clearly evident. The empirical modeling approach using mathematical regression techniques was effective for providing accurate yield predictions for a site-specific case, but these sitespecific equations could not be readily applied to other greenhouses. In order to successfully apply the approach developed in this study to other sites, the same methodology would have to be repeated to identify the best empirical model. Regression equations with new coefficient values would need to be derived for each individual site and compared with a similar error analysis method. However, an advantage of the empirical approach was that it attempted to predict trends in yield productivity using weekly environmental data. By simply applying the methodology outlined in this study, a grower with a full year of weekly historical data and a spreadsheet program could establish yield predictive equations for their current growing season. Greenhouse growers may favor the empirical modeling approach for its consistent predictability, straight-forward methodology, and low cost for implementation. The deterministic model applyingfirstprinciple equations of photosynthesis had the advantage of being applicable to all general cases of tomato crop production. Using a generalized yield conversion factor (Enoch, 1978) and cumulations of net photosynthesis activity, the deterministic model attempted to simulate actual tomato crop productivity. However, in order to improve the accuracy of predictions during different stages of crop growth, the development of a variable yield conversion factor may be necessary. The 73 current yield conversion factor is based onfixedassumptions of dry matter partitioning and moisture content. It is suggested that a yield conversion factor should be developed to account for plant physiological changes in partitioning ratios throughout the various growth phases of the tomato Suit. Modifying the method found in literature (Enoch, 1978) average values for tomato moisture content and partitioning on a weekly or bi-weekly basis could be used to calculate a new series of yield conversion factors for weekly intervals to convert P t values. ne Further research into the plant physiology of the tomato crop at different developmental stages may be required to properly apply the deterministic model. The Agassiz expert system used for comparison in this study was identified as a successful method for yield prediction. The structure of the model would allow its application to both site-specific and generalized cases with minimal modification to the model itself. In actual practise, the end-user would only be required establish a historical database and input new environmental data weekly, to predict yields 1, 2, 3, and 4 weeks ahead of time. The success of the expert system is dependent on the data with which it is trained. As a result, it is recommended that for the initial training of the expert system, historical data sets with the most complete sets of parameters and characteristic yields be used as input. Ideally, the more sets of data the model is trained with, the greater its capacity to predict for trends during peak productivity periods. A recommended improvement to the Agassiz expert model would be to include an algorithm within the model itself, for an on-going error analysis with each week of predictions. Further, when training the system for predictions, error analysis for each predict ahead model should be performed (PRE-1, PRE-2, PRE-3, PRE-4) for each 74 additional year of historical data. The model could be further integrated to provide graphical analysis of actual and predictive yields to allow end-users to monitor predictive performance visually. A report generation program within the model including yield forecasts, weekly error analyses, and graphical plots would be a useful management tool for greenhouse growers. Ideally, a generalized expert system model for tomato crops could be developed based on generalized sets of data from a variety of greenhouse growers. With a generalized yield predictive model, the end-user may only need to customize the expert model by entering one or two years of historical data to establish good predictions for their site-specific case. Once the Agassiz expert system isfine-tunedto provide accurate predictions for a site-specific case, implementation of yield predictions to an on-line computer climate control system may be feasible. Using computer algorithms related to expected yield values, adjustments may be made to environmental settings to improve tomato crop productivity via the climate control system. Another possibility for the expert model would be to present the end-user with a software driven option to view the yield forecast if modifications are made to environmental settings. Ideally, the greenhouse grower would be allowed to experimentally change the weekly environmental settings within the software program and view the change in productivity if those changes were actually made. Further, once the expert system software is advanced enough to allow experimental changes to parameters, additional menu options could be added to the model itself. If the user desired to increase or decrease yield productivity, the model could be programmed to provide suggestions to achieve this effect. Eventually, the expert system could be user- 75 customized with default settings to maximize yield during selected intervals and become a companion program to the main climate control algorithm. Great potential exists for the expert system approach for yield prediction. A similar method could be applied to other greenhouse crops to develop crop and site-specific models for each individual grower. With the encouraging results found in this study, further research and development of expert system modeling is recommended for greenhouse tomato yield prediction. 76 6.0 Bibliography Acta Horticulturae. 1992. First International Workshop on Sensors in Horticulture, No.304 - March 1992. 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New York. Jones, J.W. 1991. Crop growth, development, and production modelling. In: Proceedings of the symposium on automated agriculture for the 21st Century. December 16-17. Chicago, IL. pp. 447-457 Jones, J.W. etal. 1991. A Dynamic Tomato Growth and Yield Model (TOMGRO). Trans. ASAE 34(2): 663-672 Jones, P., J.W. Jones, and Y. Hwang. 1990. Simulation for determining greenhouse temperature setpoints. Trans. ASAE 33(5): 1722-1728 Kozak, Antal. 1995. U B C . Department of Forestry. Personal communication with Dr. Kozak validating the RMSE method as analysis tool and basis for comparing yield predictive models. Lacroix, R., R. Kok, and L. Gauthier. 1993. Simulation-based determination of greenhouse temperature setpoints. Paper no. 934043. ASAE/CSAE Summer Meeting, Spokane, June 20-23, 1993. 15 pp. Lau, A K . and Staley, L.M. "Solar radiation transmission and capture in Greenhouses" Canadian Agricultural Engineering, p.205-213. March 1989. Lau, Anthony K. 1993 Notes/File on greenhouse management, BIOE 456 course work, Fall 1993. 79 Levine, RI,Drang,DE, andB. Edelson. 1990. AI and Expert Systems - A Comprehensive Guide. Mc-Graw Hill Publishing Company. New York. Lin, Wei C. (Ph.D.) 1994. Post Harvest Physiologist - Agassiz Research Station. Agriculture Canada Research Branch. P.O. Box 1000, Agassiz, B.C. VOM 1A0 Lin, W.C. and J.C.W. Keng. 1989. "Use of Digital Image Processing for NonDestructive Measurement of Cucumber Fruit Color" CSAE. Paper No.PNR89-304. Sept 24-25/89. McAvoy, R. J. et al. 1989. The Effect of Total Available Photosynthetic Photon Flux on Single Truss Tomato Growth and Production. J. of Hort. Science 64(3): 331-338. Ministry of Agriculture, Fisheries, and Food (Province of B.C.). 1993, Greenhouse Vegetable Production Guide - 1993/94. Ohta, H. et al. "Measurement of Photosynthetic and Transpiration Rates under low total pressures" ASAE, June 20-23, 1993. Paper No. 934009. Quattro-Pro Spreadsheet Program Version 4. 1992. Borland International Inc. Rehbein, D A , S.M. Maze, and JP. Havener. 1992. The Application of Neural Networks in the Process Industry. ISA Transactions 31(4): 7-13 Seginer, I. and R.W. McClendon. 1992. Methods for optimal control of the greenhouse environment. Trans ASAE 35 (4): 1299-1307 Systat Version 5.03 - Statistical Package. 1991. Systat, Inc. Stangehellini, C. 1985. Transpiration and temperature of greenhouse crops in relation to internal and external resistances. Acta Hort No. 174: 87-95 Stangehellini, C. and W. Th.M. van Meurs. 1992. Environmental Control of Greenhouse crop transpiration. J. Agric. Eng. Research 51(4): 297 - 311 Takakura, T. 1993. Climate under Cover - Digital Dynamic Simulation in Plant Bio-Engineering. Kluwer Academic Publishers. Netherlands. 80 Takakura, Tadashi. ASAE. Dec 16-17, 1991. Proceedings of the 1991 Symposium on "Automated Agriculture for the 21st Century." Chicago, IL. Tanaka, A. Fujita, K, and K. Kikuchi. 1974. Nutri-physiological studies on the tomato plant. III. Photosynthetic rate of individual leaves in relation to the dry matter production of plants. Soil Sci., PI. Nutri., 20, 173-83. Tang, Winston C. 1995. The Prediction and Validation of Greenhouse Tomato Yield using Mathematical Models and Expert Systems. Contact Address: 5051 Quebec Street, Vancouver, B.C., V5W 2N3 Contact Ph. Number: (604) 325-3847 Willits, D.H. 1993. "The Effect of Evaporative Cooling on the Efficiency of External Greenhouse Shade Cloths". ASAE. Paper No. 93-4042. Wolf, S., J.Rudich, A. Marani and Y. Rekah. 1986. Predicting harvesting date of processing tomatoes by a simulation model. J. Amer. Soc. Hort. Sci. 111(1): 11-16 Yang, X. et al. "Dynamic Modelling of the Microclimate of a Greenhouse Cucumber Row-Crop Part I. Theoretical Model" ASAE. Vol. 33(5) Sept-Oct/1990. p. 1701-1709 Yang, X. et al. "Dynamic Modelling of the Microclimate of a Greenhouse Cucumber Row-Crop Part II. Validation and Simulation." ASAE. Vol. 33(5) Sept-Oct/1990 p. 1710-1716. 81 APPENDIX A 82 83 84 85 86 87 91 APPENDIX B 92 C R O P YIELD M A T H E M A T I C A L M O D E L G R O S S P H O T O S Y N T H E S I S (Pg),''-'': %*W?fr&f<#&43$ 1 B a s e d on Eqn (7) in "Crop Growth, Development, and Production Modelling" by J.W. J o n e s Pg = D ((t*C*p($)) / K ) * { In [ ( & * K * l o + (1 - m)* t * C *p($)) / ( & * K * l o * e x p (-KL) + (1 - m)*t*C*p($)) ] } For Tomato C r o p / Variables & Unit Definitions: F O R : G R O S S P H O T O S Y N T H E S I S (Pg) Term Abbrev. Units Gross Photo. Pg D g ( C H 2 0 ) / m ~ 2*h Conv. Factor Leaf Conduct.to C 0 2 t umol (C02) / m ~ 2 * s C 0 2 cone, of air C u m o l ( C 0 2 ) / mol (air) = p p m Function of Temp. p($) = p(theta) Dimensionless Note: " $ " = Theta (for notation) Canopy L.E. Coeff. K Dimensionless Leaf L.U. Eff. & = alpha umol ( C 0 2 ) / umol (photon) Ligh Flux Density lo umol (photon)/m ~ 2*s * * Conversion from W / m " 2 Respiration rate @ 2 5 * C km Light T. Coeff. Leaves m 0.0006 Dimensionless Leaf Area Index (LAI) FOR: m " 2 ( l e a f ) / m~2(ground) p($) p($) = [1 - ( 1 -{ ( $ h - $ ) / ( $ h - $ i ) }~2] p($) = expresses the effect of temperature on the max. rate of photo, for a single leaf $h = temp at which leaf photosynthesis is maximum, 3 0 * C $i = is temp below which leaf photosynthesis is zero, 5 * C $ = measured temp = Tin = temperature inside RESPIRATION RATE,-;' B a s e d on E q n (5) in J . W . J o n e s ' paper Rm = km * exp (0.0693*[$ - 25)) Rm = maintenance respiration rate, g C H 2 0 / g tissue * h km = respiration rate at 2 5 * C , g C H 2 0 / g tissue * h $ = measured temp. 93 DRY WEIGHT••' - / ^ l Based on Eqn (6) in J.W. Jones' paper 1 fdW / dt = E* ( Pg • Rm'W) dW / dt = rate of dry wt. growth of crop, g tissue / m ~ 2*h W = total plant dry weight, g/m ~ 2 E = conversion efficiency of CH20 to plant tissue, g (tissue) / g(CH20) Pg = canopy gross photosnthesis rate, g(CH20) / m ^2*h Rm = maintenance respiration rate, g CH20 / g tissue *h Then, once dW/dt is known the new weight W(new) can be found: fw(new) = W(i) + dW / dt ~| W(new) = new weight, g/m ~ 2 for that HOUR W(i) = total initial plant dry weight, g/m ~ 2 dW/dt = rate of dry weight growth of the crop ( g tissue / m "2*h) V',''^M^^^Mj''^^^^ M U L T | P. LIG ATI C N . F A C T O R ' ^ Y Based on paper by H.Z. Enoch (Acta Hort 76. 1978 p. 48) Y = [ (30 * 100) / (44* 5) ] * X or: 30/44 * X = Y * (5/100) Y = multiplication factor to relate photosynthesis to yield X = percentage of dry matter that is yield (ie . 50% = 0.50 ) 30, 44 = Mol. Weights of CH20 and C02 respectively 5/100 = represents 5% dry matter in yield. Based on Paper: Y = 7 (approx., based on X= 50% ) YIELD. V A L U E - ; y.V^;.: -.. V ^c^o Based on Pg and multiplication factor "Y" Yield Value #1 = Pg * Y N E T P > i O T O S Y N T H ESI S''/ f if £ Pnet = Pg - Rm*W Pnet = net photosynthesis, g (CH20) / m ~2*h Rm = maintenance resp. rate, g (CH20) / g tissue*h W(new) = total plant dry weight, g/m " 2 =W Pg = gross photosynthesis, g (CH20) / m ~2*h •YIELDJVALUE #2 i ~ > t t > Based on Net Photosynthesis (Pnet) and Multiplication Factor "Y" Yield Value #2 = Pnet * Y = W(i) for the first assumed weight hour 94 YIELD CONVERSION FROM GROWER'S DATA: Weight ( kg / m ~ 2 ) = [ Total number of 20 pound cases * 20 / (2.2*25400) ] * * Greenhouse Area = 25,400 m " 2 LIGHT INTENSITY CONVERSION F R O M W/'m ~ 2 to umol / m ~ 2*s lo = u m o l / m ^ s = (I, W / m ~ 2 / 0.22 ) * 0.80 * 0.45 lo = converted units from I, raw data entered as W/m ~ 2 I = raw data from grower in units of W/m ~ 2 LEAF AREA INDEX (LAI) , L Separate Column for varying LAI, L Shaded Areas are = Formula Blocks To Follow is the Spreadsheet Layout for the Deterministic Math Model $i = 5.0 and $h = 30.0 95 Sample Spreadsheet Column Headings for the Deterministic Model To LAJ P~ 24 Hour Yield #2 Y Factor I = Light Flux Density (umol (photon) / m *s ) 2 0 LAI - Leaf Area Index P = Gross Photosynthesis (g(CH 0)/m *h) 2 g 2 Rm = Maintenance Respiration Rate (g(CH 0) / g tissue* h) 2 Pnet = Net Photosynthesis (g (CH 0) / m * h) 2 2 24 Hour Yield #2 = Yield for a 24 hour period (kg / m ) based on Net Photosynthesis 2 Y Factor = Enoch's Yield factor of 7 Sample Spreadsheet Layout for the DETERMINISTIC MATH MODEL 96 LAI v s . T i m e (month) Month M " \ \ / m l ^\ i \ •. \ 0.6 February March April 274"4"" 3.13 3.31 ...... May June 2.51 July 1.93 1.31 August September 1.18 October 1.00 November 0.62 December 0.00 \ 1 1 i LAI Value January \ / / / / i i 2 4 i — 6 Month # Figure A-IO - — i 8 1 10 Changes in Leaf Area Index (LAI) throughout the year Reference: de Koning. 1993 ii 12 97 APPENDIX C 98 Summary of: Total PPF vs. Crop Yield Results Year r 1987 0.722917 y = 0.95 x- 35.28 1989 0.711196 y = 0.99 x-29.43 1990 0.618388 y= 1.12x-28.23 1991 0.613489 y= 1.07x-41.15 1992 0.746478 y = 0.99 x - 33.53 1993 0.672635 y= 1.02 x- 68.31 1987 - 1993 0.668588 y = 0.00025 x- 0.07858 Equation 2 Where: x = 9 week cumulated photosynthetic photonflux(PPF) y = crop yield in grams/plant Given yield in kg/m . and plant density of 2.5 plants/m : 2 2 Crop Yield (grams/plant) = Yield (kg/m ) *(Plants / m )* (1000 g / kg) 2 2 The best correlated data for light (9 week PPF) and crop yield (g/plant) as analysed by the r correlation coefficient was found to be the 1992 data with a value of 0.746478. 2 Table A - l (D W h OJ iri • r~ 00 T O) O) W OJ O £ TO CM CD O CO in o d 9 o E o M c O > 2 w LU >- TI "5 T> CD o I] C- o CD nb « UJ o co rr z o CO UJ -o CO + ++-ti + + + + + + + Ico s o o o o o o co r— o o CD O O in o o o o co (;uB|d/6) p|9|A o o CM o o T- CO CO CO CO 05 CO CM •3; i O) ai <M CM . T~ T- r 1 ' o cn a O) o d E o x> CD LU O O >- LU cr co CO o o ++ Q. O u_ Q_ o C/) > LL CL CL o o CD o M— ° LO O CO c o -4—' CM < r£ a> o E CO E c o O fc " CO CL CL _ CO o y - O h- o o CD CM *8 oo CM cu s- • CD CT o o CJ) o o 00 o o o o CD o o LO o o o o CO (;uB|d/6) p|8|A o o CM o o i - -r- co i - 00 CO T — CO CM IO CO OJ (M CO OJ r-. co CM CO cn O c o E o TS D) CD CD O UJ >- rr « o o < o CD CD in co cc D) CD Q 9> o .9. 5= CD O O X o o CO T3 o o CD > O o o CD LL. + o^ . CO > UL OL CL "crj -»—> O CL + CM < x+t- O E toc To (D o +-» cd o o o E V E E CL O ± O CO CO - 13 W A : o o C D +t o CM O O CD CD I O o CO o o 00 o o o o CD o o o m o o o o o o co o CM (lUB|d/6) p|8|A 102 LO CD ID O) O CO CNJ •cr CD d cn o Cvi fo O E o •a rr w 3= LU § 2 O CO Ui CD OC o O o 0) O O a o o oo o o >- + •ti 4 o o Q_ O LL CD Q. Q_ O m CO > O O O CO io E CO a) o E c _o '-«-• L L CO CL E CL E CC CM < \_ o CO O 05 o o H—' o CL (0 o CM O O I CU s- o o CD O O 00 o o o O r-- CD O O LO o o o o o o CO CM (}UB|d/6) p|9|A o o -i- 3 CJ 103 T- CO CO O) CO OJ in co co q co O) co co £ o XI o o o cr 2 co o w o o Q X o -o o -o CO o -o 4 + cvj o< -o \_ co E lo o -o LO o -o + o -o CO + + CD o E, u_ 0_ Q. "CO •I—• o It- + o o o o o o o o o o CD CO IO CD o -o C\J o o o o o o o o CO CM (}UB|d/6) piajA T- o -o 104 in CT) cn O to CO CO CM CO CO CO CO E o T3 CD £ LL LU >CD t m LU CT c CO o U co rr z "S 3, ~ O o Q ° 2 X CO o o oo o o T3 C D + •H-H+ O o o CD +: LL. CM . Cf) LO O to E CO CD >S o ca CL EE CL =3 CO IB •+-• > O O CO LL CL CL O o o £i C\J LO| O CO O) CD o I < CU s- o o CO o o o o CD o o LO o o o o o o CO CM (;uB|d/6) pistA o o i— D" 105 CO LO CO rO o CO V LO CM OJ OJ co to CD CO CO co d CO CO CM cn T— o LO OJ o o d o o o d E o T3 CD CD LL w UJ >- 3 « o CJ CO "o3 O CD O o i: i_ CD UJ o Ti CD CD O) CD CO UJ TI c "o "O CD c 0) o X Q CO o o o o CO + 1+ + o o ! 4>f o ^ + 1, + 4 + +; O CJ CD , o o LO +4- 4 o o +\ ++ + ++ " 0_ 0_ 8 o + CO ^ 4-- CM o o o o o o o o CO O) o o o o CO LO o o o o o o CO (lUB|d/6) p|G|A CM o o i- cr •I - Ll_ 106 APPENDIX D 107 EMPIRICAL MATH MODEL REGRESSION RESULTS (Equation 1) REGRESSION RESULTS FOR LIGHT ALONE (CL ) (Cumulative Light for 9 weeks, W/m ) 9 2 Y = a + a (CL,) p2 0 (1) x RMSE Results for 1994 Predictions ao ai 1987 Equation -0.08819 0.000014 0.489258 1989 Equation -0.0736 0.0000147 0.441116 -0.07057 0.0000165 0.382832 (Eqn. 1.1) 1991 Equation -0.10287 0.0000158 0.408559 1992 Equation 1993 Equation -0.08383 -0.17077 0.0000145 0.0000150 0.457382 0.482722 1987-1993 Eqn -0.07858 0.00001479 0.439355 1990 Equation Where: .. (Eqn. 1.2) a is the constant for predictive equation 1 ai is the coefficient for predictive equation 1 (CL ) 0 9 "1992 Equation" means the regression equation based on fitting 1992 data to regression equation 1. RMSE for 1994 Predictions: is the calculated root-mean-square-error of tomato yield predictions for 1994 using that particular year's regression equation. For example, using the 1992 Equation of: Y = -0.08383 + 0.0000145 (CL ) to predict 1994 yield (with 1994 data), the root-mean-square-error of the predictions was found to be: 0.457382 9 Table A-2 108 1994 Predicted Yields Cumulative Light Based (CL J 3-i 2.5< 21.5- CD > 10.5- o- 20 25 30 35 1994 Week Number Actual Yield - E3- 1987 E q n . -A- 1990 E q n . Figure A-16 40 45 -+ - 1989 E q n . - §5- 1991 E q n . Predicted yields for 1994 production based on regression equations derived from light data from 1987 to 1991. 50 109 1994 Predicted Yields Cumulative Light Based (CL ) 5 10 15 Actual Yield Figure A-19 20 25 30 35 1994 Week Number - B - 1993 Eqn. 40 - 4 - 1987 to 1993 Eqn. 45 1992 Eqn. Predicted yields for 1994 production based on regression equations derived from light data (1992,1993, and '87 - '93) 50 110 EMPIRICAL MATH MODEL REGRESSION RESULTS (Equation 2) REGRESSION RESULTS FOR LIGHT and HEAT (CL and CH ) (Cumulative Light for 9 weeks, W/m and Cumulative Heat Units for 9 weeks) 9 9 2 Y p2 = a + a (CL ) + a (CH )..... 0 x 9 2 9 (2) RMSE Results for 1994 Predictions a 0 a! a 2 1987 Equation 0.52199 0.00001207 -0.0043454 0.440142 1989 Equation 4.25232 0.00002708 -0.057994 0.692027 1990 Equation.,,,.... 2.38054 . 0.00002444 -0.033809 1991 Equation 0.542205 (Eqn. 2.2) 3.76864 0.00002193 -0.0480305 0.527883 1992 Equation , 2.39584 0.00001513 -0.0300318 0.353099 (Eqn. 2.1) 1993 Equation 0.77455 0.00001349 -0.0092041 1987-1993 Eqn 0.770725 0.00001364 -0.0080933 0.379579 0.425785 Where: ao is the constant for predictive equation 2 ai is the coefficient for predictive equation 2 (CL ) a2 is the coefficient for predictive equation 2 (CH ) 9 9 "1992 Equation" means the regression equation based on fitting 1992 data to regression equation 2. RMSE for 1994 Predictions: is the calculated root-mean-square-error of tomato yield predictions for 1994 using that particular year's regression equation. For example, using the 1992 Equation of: Y = 2.39584 +0.00001513 (CL*) -0.030018(CH ) to predict 1994 yields (with 1994 data), the root-mean-square-error of the predictions was found to be: 0.353099 9 Table A-3 Ill 1994 Predicted Yields Light(CL ) and Temperature(CH ^ Based g 3.5n 1994 Week Number Actual Yield - B - 1987 Eqn. - B - 1990 Eqn. - X - 1 9 9 1 Eqn. Figure A-20 -*K~ 1989 Eqn. - A r 1992 Eqn. Predicted yields for 1994 production based on regression equations derived from light & heat data from 1987 to 1992. 112 1994 Predicted Yields Light(CL ) and Temperature(CH ^ Based g 1994 Week Number Actual Yield Figure /\-2/ - B - 1993 E q a - * r - 1987 to 1993 E q n . Predicted yields for 1994 production based on regression equations derived from light & heat data (1993 and '87 - 93) 113 1987 Year STEPWISE REGRESSION RESULTS 1987 D-RYALL (MR87.SYS from Systat) Data Set. Variables: VAR(3) = Yield in kg/m VAR(4) = 9 week cumulative light readings in W / m VAR(5) = 9 week cumulative heat units, degrees Celsius (each 9 wk. period = sum of 9 wk. avg temp *C - 90) 2 2 Statistical Method: Stepwise Regression for light VAR(4) and heat units VAR(5). All results based on dependent variable VAR(3), yield in kg/m -> D E P E N D E N T VARIABLE VAR(3), Yield in kg/m MINIMUM T O L E R A N C E F O R ENTRY INTO M O D E L = 2 2 <- .010000 F O R W A R D S T E P W I S E WITH FIRST 4 VARIABLES F O R C E D IN M O D E L A L P H A - T O - E N T E R = .150 AND A L P H A - T O - R E M O V E = .150 MAX # STEPS= STEP # 0 R= .000 VARIABLE COEFFICIENT RSQUARE= STD E R R O R STD C O E F T O L E R A N C E F .000 'P' IN 1 CONSTANT OUT PART. C O R R 2VAR(4) 3VAR(5) STEP # 0.75591487 0.59594473 .1E+01 .45E+02 0.0000 .1E+01 .19E+02 0.0001 1 R= .756 RSQUARE= .571 T E R M E N T E R E D : VAR(4) VARIABLE COEFFICIENT STD ERROR STD C O E F T O L E R A N C E F IN 1 CONSTANT 2VAR(4) OUT 3VAR(5) 0.00001021 0.00000152 .076E+01 .1E+01 .45E+02 0.0000 PART. C O R R -0.13153800 0.27982 0.58103 0.4513 'P' 114 1987 STEP# STEPWISE REGRESSION RESULTS 2 R= .761 RSQUARE= .579 T E R M E N T E R E D : VAR(5) VARIABLE COEFFICIENT STD ERROR STD C O E F TOLERANCE F 'P' IN 1 CONSTANT 2VAR(4) 0.00001207 0.00000288 .089E+01 0.27982 .18E+02 0.0002 3VAR(5) -0.00434545 -0.00570082 -.016E+01 0.27982 0.58103 0.4513 OUT PART. CORR none T H E S U B S E T M O D E L INCLUDES T H E FOLLOWING PREDICTORS: CONSTANT VAR(4) VAR(5) DEP VAR: VAR(3) N: 36 MULTIPLE R: 0.761 S Q U A R E D MULTIPLE R: 0.579 A D J U S T E D S Q U A R E D MULTIPLE R: .553 S T A N D A R D E R R O R O F ESTIMATE: 0.28657972 VARIABLE CONSTANT VAR(4) VAR(5) COEFFICIENT STD ERROR STD C O E F TOLERANCE 0.52199056^0.30523030 0.00000000 0.00001207 '0.00000288 0.89406748 -0.00434545'0.00570082 -0.16279360 0.2798160 0.2798160 P(2 TAIL) T 1.71015 0.09663 4.18631 0.00020 -0.76225 0.45132 ANALYSIS O F VARIANCE SOURCE REGRESSION RESIDUAL SUM-OF-SQUARES 3.72465308 2.71022192 D E P E N D E N T VARIABLE DF M E A N - S Q U A R E 2 33 1.86232654 0.08212794 F-RATIO P 22.67592009 0.00000064 VAR(3) MINIMUM T O L E R A N C E F O R E N T R Y INTO M O D E L = .010000 F O R W A R D S T E P W I S E WITH FIRST 4 VARIABLES F O R C E D IN M O D E L A L P H A - T O - E N T E R = .150 A N D A L P H A - T O - R E M O V E = .150 MAX # S T E P S = 6 115 STEPWISE REGRESSION RESULTS 1 9 8 9 Year. 1989 Data Set. D-RYALL (MR89.SYS from Systat) Variables: VAR(3) = Yield i n k g / m VAR(4) = 9 week cumulative light readings in W/m VAR(5) = 9 week cumulative heat units, degrees celsius (each 9 wk. period = sum of 9 wk. avg temp *C - 90) 2 2 Statistical Method: Stepwise Regression for light VAR(4) and heat units VAR(5). All results based on dependent variable VAR(3), yield in k g / m -> D E P E N D E N T VARIABLE D E P E N D E N T VARIABLE VAR(3), Yield in kg/m 2 2 <- VAR(3) MINIMUM T O L E R A N C E F O R ENTRY INTO MODEL = .010000 F O R W A R D STEPWISE WITH FIRST 4 VARIABLES F O R C E D IN M O D E L ALPHA-TO-ENTER= .150 AND ALPHA-TO-REMOVE= .150 MAX # STEPS= STEP# R= .000 0 VARIABLE COEFFICIENT RSQUARE= 6 .000 STD E R R O R STD C O E F T O L E R A N C E F V IN 1 CONSTANT OUT PART. C O R R 2VAR(4) 3VAR(5) STEP* 0.76284622 0.43172526 .1E+01 .52E+02 0.0000 .1E+01 8.47615 0.0061 R= .763 1 RSQUARE= .582 T E R M E N T E R E D : VAR(4) VARIABLE COEFFICIENT STD E R R O R STD C O E F T O L E R A N C E F IN 1 CONSTANT 2VAR(4) 0.00001285 OUT 3VAR(5) PART. C O R R -0.73518613 0.00000179 .076E+01 .1E+01 .52E+02 0.0000 . 0.24047 .42E+02 0.0000 'P' 116 1989 STEPWISE REGRESSION RESULTS STEP* 2 TERM ENTERED: VAR(5) VARIABLE R= .899 COEFFICIENT RSQUARE= STD ERROR STD COEF TOLERANCE .808 F "P" IN 1 CONSTANT 2 VAR(4) 0.00002708 0.00000251 .161E+01 0.24047 .12E+03 0.0000 3VAR(5) -0.05799486 -0.00891219-.097E+01 0.24047 .42E+02 0.0000 OUT PART. CORR none THE SUBSET MODEL INCLUDES THE FOLLOWING PREDICTORS: CONSTANT VAR(4) VAR(5) DEPVAR: VAR(3) N: 39 MULTIPLE R: 0.899 SQUARED MULTIPLE R: 0.808 ADJUSTED SQUARED MULTIPLE R: .797 STANDARD ERROR OF ESTIMATE: 0.22600488 VARIABLE CONSTANT VAR(4) VAR(5) COEFFICIENT STD ERROR STD COEF TOLERANCE 4.25232655' 0.64683641 0.00000000 0.00002708 ,0.00000251 1.60766401 -0.05799486^0.00891219 -0.96937144 0.2404687 0.2404687 ANALYSIS OF VARIANCE SOURCE REGRESSION RESIDUAL SUM-OF-SQUARES DF MEAN-SQUARE F-RATIO P 7.73327686 2 3.86663843 75.70035554 0.00000000 1.83881545 36 0.05107821 T P(2 TAIL) 6.57404 0.00000 .11E+02 0.00000 -6.50736 0.00000 117 1990 Year Data Set. STEPWISE REGRESSION RESULTS 1990 D-RYALL (MR90.SYS from Systat) Variables: VAR(3) = Yield in kg/m VAR(4) = 9 week cumulative light readings in W/m VAR(5) = 9 week cumulative heat units, degrees Celsius (each 9 wk. period = sum of 9 wk. avg temp *C - 90) 2 2 Statistical Method: Stepwise Regression for light VAR(4) and heat units VAR(5). All results based on dependent variable VAR(3), yield in kg/m 2 -> DEPENDENT VARIABLE DEPENDENT VARIABLE VAR(3), Yield in kg/m <2 VAR(3) MINIMUM TOLERANCE FOR ENTRY INTO MODEL = .010000 FORWARD STEPWISE WITH FIRST 4 VARIABLES FORCED IN MODEL ALPHA-TO-ENTER= .150 AND ALPHA-TO-REMOVE= .150 MAX # STEPS= STEP* R= .000 0 VARIABLE COEFFICIENT 6 RSQUARE= .000 STD ERROR STD COEF TOLERANCE F 'P' IN 1 CONSTANT OUT PART. CORR 2VAR(4) 3VAR(5) 0.67698642 0.53005288 STEP* 1 TERM ENTERED: VAR(4) VARIABLE .1E+01 .31E+02 0.0000 .1E+01 .14E+02 0.0005 R= .677 COEFFICIENT RSQUARE= .458 STD ERROR STD COEF TOLERANCE F 'P' IN 1 CONSTANT 2VAR(4) OUT 3VAR(5) 0.00001443 0.00000258 .068E+01 .1E+01 .31E+02 0.0000 PART. CORR -0.28777015 0.16902 3.25039 0.0798 118 STEPWISE REGRESSION RESULTS 1990 STEP# 2 TERM ENTERED: VAR(5) VARIABLE R= .709 COEFFICIENT RSQUARE= STD ERROR STD COEF TOLERANCE .503 F 'P' IN 1 CONSTANT 2 VAR(4) 0.00002444 0.00000609 .115E+01 0.16902 .16E+02 0.0003 3 VAR(5) -0.03380935 -0.01875293 -.052E+01 0.16902 3.25039 0.0798 OUT PART. CORR none THE SUBSET MODEL INCLUDES THE FOLLOWING PREDICTORS: CONSTANT VAR(4) VAR(5) DEP VAR: VAR(3) N: 39 MULTIPLE R: 0.709 SQUARED MULTIPLE R: 0.503 ADJUSTED SQUARED MULTIPLE R: .476 STANDARD ERROR OF ESTIMATE: 0.43162224 VARIABLE CONSTANT VAR(4) VAR(5) COEFFICIENT STD ERROR STD COEF TOLERANCE 2.38053976' 1.27331469 0.00002444 ' 0.00000609 -0.03380935' 0.01875293 0.00000000 1.14660486 0.1690213 -0.51516956 0.1690213 T P(2 TAIL) 1.86956 0.06970 4.01265 0.00029 -1.80288 0.07978 ANALYSIS OF VARIANCE SOURCE SUM-OF-SQUARES DF MEAN-SQUARE F-RATIO REGRESSION RESIDUAL 6.79227052 6.70671922 18.22960904 0.00000340 2 3.39613526 36 0.18629776 P 119 STEPWISE REGRESSION RESULTS 1991 Year: 1991 Data Set. D-RYALL (MR91.SYS from Systat) Variables: VAR(3) = Yield in k g / m VAR(4) = 9 week cumulative light readings in W/m VAR(5) = 9 week cumulative heat units, degrees Celsius (each 9 wk. period = sum of 9 wk. avg temp "C - 90) 2 2 Statistical Method: Stepwise Regression for light VAR(4) and heat units VAR(5). All results based on dependent variable VAR(3), yield in k g / m -> D E P E N D E N T VARIABLE DEPENDENT VARIABLE VAR(3), Yield in kg/m 2 <- VAR(3) MINIMUM T O L E R A N C E F O R E N T R Y INTO M O D E L = F O R W A R D S T E P W I S E WITH FIRST .010000 4 V A R I A B L E S F O R C E D IN M O D E L A L P H A - T O - E N T E R = .150 A N D A L P H A - T O - R E M O V E = .150 M A X # S T E P S = STEP #0 R= VARIABLE 2 COEFFICIENT .000 RSQUARE= 6 .000 STD E R R O R STD C O E F TOLERANCE F 'P' F 'P' IN 1 CONSTANT OUT PART. CORR 2VAR(4) 3VAR(5) STEP # 0.65059951 0.26286050 .1E+01 .26E+02 0.0000 .1E+01 2.67207 0.1108 R= 1 .651 RSQUARE= .423 T E R M E N T E R E D : VAR(4) VARIABLE COEFFICIENT STD ERROR STD C O E F TOLERANCE IN 1 CONSTANT 2VAR(4) OUT 3VAR(5) 0.00001201 0.00000234 .065E+01 .1E+01 .26E+02 PART. CORR -0.54492557 0.37242 .15E+02 0.0005 0.0000 120 STEPWISE REGRESSION 1991 STEP # 2 T E R M E N T E R E D : VAR(5) VARIABLE R= .771 RSQUARE= COEFFICIENT STD ERROR 0.00002193 -0.04803047 0.00000326 -0.01249225 RESULTS .595 STD C O E F TOLERANCE F 'P' IN 1 CONSTANT 2VAR(4) 3VAR(5) OUT .119E+01 -.068E+01 0.37242 0.37242 .45E+02 0.0000 .15E+02 0.0005 PART. C O R R none T H E S U B S E T M O D E L INCLUDES T H E FOLLOWING P R E D I C T O R S : CONSTANT VAR(4) VAR(5) DEP VAR: VAR(3) N: 38 MULTIPLE R: 0.771 S Q U A R E D MULTIPLE R: 0.595 A D J U S T E D S Q U A R E D MULTIPLE R: .571 STANDARD E R R O R O F ESTIMATE: 0.31662425 VARIABLE CONSTANT VAR(4) VAR(5) COEFFICIENT 3.76864580 ' 0.00002193' -0.04803047' STD ERROR 0.92989756 0.00000326 0.01249225 STD C O E F TOLERANCE 0.00000000 1.18780098 -0.67811428 0.3724207 0.3724207 4.05275 0.00027 6.73468 0.00000 -3.84482 0.00049 ANALYSIS O F VARIANCE SOURCE REGRESSION RESIDUAL SUM-OF-SQUARES DF M E A N - S Q U A R E P(2 TAIL) F-RATIO 5.14490480 2 2.57245240 25.66013879 0.00000014 3.50878204 35 0.10025092 121 1992 Yean STEPWISE REGRESSION RESULTS 1992 Data Set. D-RYAJJL (MR92.SYS from Systat) Variables: VAR(3) = Yield in kg/m VAR(4) = 9 week cumulative light readings in W/m VAR(5) = 9 week cumulative heat units, degrees Celsius (each 9 wk. period = sum of 9 wk. avg temp *C - 90) 2 2 Statistical Method: Stepwise Regression for light VAR(4) and heat units VAR(5). All results based on dependent variable VAR(3), yield in kg/m 2 -> D E P E N D E N T VARIABLE DEPENDENT VARIABLE VAR(3), Yield in kg/m 2 <- VAR(3) MINIMUM T O L E R A N C E F O R E N T R Y INTO M O D E L = .010000 F O R W A R D S T E P W I S E W I T H F I R S T 4 V A R I A B L E S F O R C E D IN M O D E L A L P H A - T O - E N T E R = .150 A N D A L P H A - T O - R E M O V E = .150 M A X # S T E P S = STEP# R= 0 VARIABLE COEFFICIENT .000 RSQUARE= STD ERROR STD COEF TOLERANCE 6 .000 F 'P' IN 1 CONSTANT OUT PART. CORR 2VAR(4) 3VAR(5) 0.78102804 0.24309995 STEP* 1 T E R M E N T E R E D : VAR(4) VARIABLE .1E+01 .56E+02 0.0000 .1E+01 2.26114 0.1414 R= COEFFICIENT .781 RSQUARE= STD ERROR STD COEF TOLERANCE .610 F IN 1 CONSTANT 2VAR(4) OUT 3VAR(5) 0.00001186 0.00000158 .078E+01 .1E+01 .56E+02 PART. CORR -0.45593461 . 0.63726 9.18502 0.0046 0.0000 'P 1 122 1992 STEP* 2 T E R M E N T E R E D : VAR(5) VARIABLE STEPWISE REGRESSION RESULTS R= COEFFICIENT .831 RSQUARE= .691 STD ERROR STD COEF TOLERANCE F "P" IN 1 CONSTANT 2VAR(4) 3VAR(5) OUT 0.00001513 0.00000179 .100E+01 0.63726 .72E+02 0.0000 - 0 . 0 3 0 0 3 1 8 3 -0.00990927 -.036E+01 0.63726 9.18502 0.0046 PART. CORR none T H E S U B S E T M O D E L INCLUDES T H E FOLLOWING P R E D I C T O R S : CONSTANT VAR(4) VAR(5) DEP VAR: VAR(3) N: 38 M U L T I P L E R: 0.831 S Q U A R E D M U L T I P L E R: 0.691 A D J U S T E D S Q U A R E D M U L T I P L E R: .673 STANDARD E R R O R OF ESTIMATE: 0.29923584 VARIABLE COEFFICIENT STD ERROR STD COEF TOLERANCE T P ( 2 TAIL) CONSTANT 2 . 3 9 5 8 4 0 1 5 ' 0.73325947 0.00000000 . 3.26738 0.00244 VAR(4) 0 . 0 0 0 0 1 5 1 3 ' 0.00000179 0.99584564 0.6372618 8.46172 0.00000 VAR(5) - 0 . 0 3 0 0 3 1 8 3 / 0.00990927 -0.35667547 0 . 6 3 7 2 6 1 8 - 3 . 0 3 0 6 8 . 0 . 0 0 4 5 7 ANALYSIS OF VARIANCE SOURCE REGRESSION RESIDUAL SUM-OF-SQUARES DF M E A N - S Q U A R E F-RATIO 7.01081632 2 3.50540816 39.14816096 3.13397316 35 0.08954209 0.00000000 P 123 STEPWISE REGRESSION RESULTS 1 9 9 3 Year 1993 Data Set. D-RYALL (MR93.SYS from Systat) Variables: VAR(3) = Yield in k g / m VAR(4) = 9 week cumulative light readings in W / m VAR(5) = 9 week cumulative heat units, degrees C e l s i u s (each 9 wk. period = sum of 9 wk. avg temp *C - 90) 2 2 Statistical Method: Stepwise Regression for light VAR(4) and heat units VAR(5). All results based on dependent variable VAR(3), yield in k g / m -> D E P E N D E N T VARIABLE DEPENDENT VARIABLE VAR(3), Yield in kg/m 2 <- VAR(3) MINIMUM T O L E R A N C E F O R E N T R Y INTO M O D E L = .010000 F O R W A R D S T E P W I S E W I T H F I R S T 4 V A R I A B L E S F O R C E D IN M O D E L A L P H A - T O - E N T E R = .150 A N D A L P H A - T O - R E M O V E = .150 M A X # S T E P S = STEP* R= 0 VARIABLE 2 COEFFICIENT .000 RSQUARE= STD ERROR 6 .000 STD COEF TOLERANCE F 'P' IN 1 CONSTANT OUT PART. CORR 2VAR(4) 3VAR(5) 0.67378618 0.31215946 STEP* 1 T E R M E N T E R E D : VAR(4) VARIABLE .1E+01 .29E+02 0.0000 .1E+01 3.77874 0.0600 R= COEFFICIENT .674 RSQUARE= STD ERROR .454 STD COEF TOLERANCE F IN 1 CONSTANT 2VAR(4) OUT 3VAR(5) 0.00001149 0.00000213 .067E+01 .1E+01 .29E+02 PART. CORR -0.19585125 0.60300 1.35618 0.2523 0.0000 'P' 124 1993 STEP* 2 T E R M E N T E R E D : VAR(5) VARIABLE .STEPWISE REGRESSION RESULTS R= COEFFICIENT .689 RSQUARE= .475 STD ERROR STD COEF TOLERANCE F 'P* IN 1 CONSTANT 2VAR(4) 0.00001349 3VAR(5) -0.00920407 OUT 0.00000273 -0.00790353 .079E+01 -.019E+01 0.60300 0.60300 .24E+02 0.0000 1.35618 0.2523 PART. CORR none THE SUBSET MODEL INCLUDES THE FOLLOWING PREDICTORS: CONSTANT VAR(4) VAR(5) DEP VAR: V A R ( 3 ) N: 37 M U L T I P L E R: 0.689 S Q U A R E D M U L T I P L E R: 0.475 A D J U S T E D S Q U A R E D M U L T I P L E R: .444 STANDARD E R R O R OF ESTIMATE: 0.32356666 VARIABLE CONSTANT VAR(4) VAR(5) COEFFICIENT 0.77455001' 0.00001349 ' -0.00920407' STD ERROR 0.54026852 0.00000273 0.00790353 STD COEF 0.00000000 0.79121212 -0.18636692 TOLERANCE T 1.43364 0.16081 4.94405 0.00002 -1.16455 0.25231 0.6030001 0.6030001 ANALYSIS OF VARIANCE SOURCE REGRESSION RESIDUAL SUM-OF-SQUARES 3.21974609 3.55964310 DF M E A N - S Q U A R E F-RATIO P 2 34 15.37673359 0.00001753 1.60987305 0.10469539 P(2 TAIL) 125 S T E P W I S E 1 9 8 7 - 1 9 9 3 Year. R E G R E S S I O N R E S U L T S 1987-1993 Data Set. D-RYALL (MR8793.SYS from Systat) Variables: VAR(3) = Yield in k g / m VAR(4) = 9 week ciimulative light readings in W / m VAR(5) = 9 week cumulative heat units, degrees Celsius (each 9 wk. period = sum of 9 wk. avg temp "C - 90) 2 2 Statistical Method: Stepwise Regression for light VAR(4) and heat units VAR(5). All results based on dependent variable VAR(3), yield in k g / m -> DEPENDENT VARIABLE DEPENDENT VARIABLE VAR(3), Yield in kg/m <2 VAR(3) MINIMUM T O L E R A N C E F O R E N T R Y INTO M O D E L = .010000 F O R W A R D S T E P W I S E W I T H F I R S T 4 V A R I A B L E S F O R C E D IN M O D E L A L P H A - T O - E N T E R = .150 A N D A L P H A - T O - R E M O V E = .150 M A X # S T E P S = STEP # 0 R= .000 VARIABLE 2 COEFFICIENT RSQUARE= 6 .000 STD ERROR STD COEF TOLERANCE F 'P' F 'P' IN 1 CONSTANT OUT PART. CORR 2VAR(4) 3VAR(5) STEP# 0.71031954 0.34628936 . .1E+01 .1E+01 R= .710 1 .23E+03 .31E+02 0.0000 0.0000 RSQUARE= .505 T E R M E N T E R E D : VAR(4) VARIABLE COEFFICIENT STD ERROR STD COEF TOLERANCE IN 1 CONSTANT 2VAR(4) OUT 3VAR(5) 0.00001190 0.00000079 .071E+01 .1E+01 .23E+03 PART. CORR -0.18303353 0.60489 7.76440 0.0058 0.0000 126 STEPWISE REGRESSION RESULTS 1987 - 1993 STEP # 2 R= .722 RSQUARE= .521 T E R M E N T E R E D : VAR(5) VARIABLE COEFFICIENT STD ERROR STD COEF TOLERANCE F 'P' IN 1 CONSTANT 2VAR(4) 3VAR(5) OUT 0.00001364 -0.00809330 0.00000100 -0.00290450 .081E+01 -.017E+01 0.60489 0.60489 .19E+03 0.0000 7.76440 0.0058 PART. CORR none T H E S U B S E T M O D E L INCLUDES T H E FOLLOWING P R E D I C T O R S : CONSTANT VAR(4) VAR(5) DEP VAR: VAR(3) N: 2 2 7 M U L T I P L E R: 0.722 S Q U A R E D M U L T I P L E R: 0.521 A D J U S T E D S Q U A R E D M U L T I P L E R: .517 STANDARD E R R O R OF ESTIMATE: 0.34617793 VARIABLE CONSTANT VAR(4) VAR(5) COEFFICIENT 0.77072466' 0.00001364/ -0.00809330/ STD ERROR 0.21153971 0.00000100 0.00290450 STD COEF 0.00000000 0.81444264 -0.16564933 TOLERANCE 0.6048929 0.6048929 T 3.64340 0.00033 .14E+02 0.00000 -2.78647 0.00579 ANALYSIS OF VARIANCE SOURCE REGRESSION RESIDUAL SUM-OF-SQUARES 29.21550328 26.84397160 DF 2 224 MEAN-SQUARE 14.60775164 0.11983916 F-RATIO .121895E+03 P(2 TAIL) P 0.00000000 127 EMPIRICAL MATH MODEL STATISTICAL RESULTS FOR EQUATIONS 3.1 and 4.1 Light & Heat (3,6,9 week) Combined Step-Wise Regression 1992 Equation for Yield Prediction Definition of Variables: VAR(10) VAR( 5) VAR( 6) VAR( 4) VAR(16) VAR(15) VAR(14) = Yield (kg/m 2) = Cumulative Light = Cumulative Light = Cumulative Light = Cumulative Heat = Cumulative Heat = Cumulative Heat A for for for for for for 3 weeks 6 weeks 9 weeks 3 weeks 6 weeks 9 weeks (W/m2) (W/m2) (W/m2) (*C) (*C) (*C) A A A EQUATION LAYOUT: VAR(10) = CONSTANT + a,VAR(5) + a VAR(6) + a VAR(4) + a VAR(16) + a VAR(15) + a VAR(14) 2 3 4 5 Where 'CONSTANT' and a, to a are to be determined. 6 Step-Wise Regression R e s u l t s DEPENDENT VARIABLE (Based on 1992 Data) VAR(IO) = Four Week Ahead A c t u a l Y i e l d MINIMUM TOLERANCE FOR ENTRY INTO MODEL = .010000 FORWARD STEPWISE WITH FIRST 8 VARIABLES FORCED IN MODEL ALPHA-TO-ENTER= .150 AND ALPHA-TO-REMOVE= .150 MAX # STEPS= STEP # 0 R= VARIABLE .000 RSQUARE= COEFFICIENT 14 .000 STD ERROR STD COEF TOLERANCE IN 1 CONSTANT OUT 2 VAR (5) 3 VAR(6) 4 VAR (4) 5 VAR(16) 6 VAR(15) 7 VAR (14) PART. CORR 0. 83258920 78185025 68540368 0.60663037 56819254 53326428 STEP # 1 R= .833 RSQUAR& TERM ENTERED: VAR(5) VARIABLE .1E+01 .1E+01 .1E+01 .1E+01 .1E+01 .1E+01 .88E+02 .61E+02 .35E+02 .23E+02 .19E+02 .15E+02 0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 .88E+02 0.0000 .693 COEFFICIENT STD ERROR 0.00003932 0.00000419 STD COEF TOLERANCE IN 1 CONSTANT 2 VAR(5) OUT PART. CORR .083E+01 .1E+01 6 3 4 5 6 7 VAR(6) VAR(4) VAR(16) VAR(15) VAR(14) -0.07246533 -0,11529719 0.11202455 0.10239134 0.06869768 STEP # 2 R= .834 RSQUARE= TERM ENTERED: VAR(6) VARIABLE 0.09069 0.25680 0.54637 0.60371 0.63510 0.20060 0.51196 0. 48294 0.40261 0.18019 0.6568 0.4787 0.4913 0.5295 0.6736 . 695 COEFFICIENT STD ERROR 0.00004532 -0.00000332 0.00001405 -0.00000742 STD COEF TOLERANCE F IN 1 CONSTANT 2 VAR(5) 3 VAR(6) OUT 4 5 6 7 .096E+01 •.013E+01 0.09069 0.09069 .10E+02 0.20060 0.0026 0.6568 0.03675 0.53004 0.57265.0.59089 0.61435 0.60454 0.56211 0.31090 0.4381 0.4418 0.4582 0.5805 0.04630 0.01298 0.03675 3.04195 0.31025 0.61435 0.0894 0.5809 0.4381 0.52853 0.56987 0.59087 0.53617 0.47950 0.30193 0.4688 0.4931 0.5861 0.00001991 .072E+01 0.04622 0.00001998 .037E+01 0.01281 -0.00141062 -.035E+01 0.03664 0.01135264 .09129784 0.52853 2.89251 0.21668 0.54570 0.53617 0.0976 0.6444 0.4649 0.4688 0.05454 0.09570 0.00011 0.08532 0.9916 0.7719 PART. CORR VAR(4) VAR(16) VAR(15) VAR (14) -0.12780013 0.12679220 0.12233083 0.09128331 STEP # 3 R= .837 RSQUARE= TERM ENTERED: VAR(4) VARIABLE .700 COEFFICIENT STD ERROR 0.00003448 0.00001098 -0.00109713 0.00001977 0.00001972 -0.00139975 STD COEF TOLERANCE IN 1 2 3 4 CONSTANT VAR(5) VAR{6) VAR (4) OUT .073E+01 .044E+01 •.037E+01 PART. CORR 5 VAR(16) 6 VAR115) 7 VAR(14) 0.12114063 0.11464885 0.09119794 STEP # 4 R= .839 RSQUARE= TERM ENTERED: VAR(16) VARIABLE COEFFICIENT .704 STD ERROR STD COEF TOLERANCE IN 1 2 3 4 5 CONSTANT VAR (5) VAR (6) VAR(4) VAR(16) OUT 0.00003387 0.00000930 -0.00104205 0.00831282 PART. CORR 6 VAR(15) 7 VAR(14) -0.00178971 -0.04931388 STEP # 5 R=" .839 RSQUARE= TERM ENTERED: VAR(IS) VARIABLE .704 COEFFICIENT STD ERROR 0.00003382 0.00000933 -0.00104297 0.00868756 -0.00022289 0.00002069 0.00002045 -0.00143331 0.03721875 -0.02105054 STD COEF TOLERANCE 'P' IN CONSTANT VAR(5) VAR(6) VAR(4) VAR(16) VARU5) OUT PART. CORR .072E+01 04405 .037E+01 01257 -.035E+01 03650 09541359 05058 00416809 0.05454 67310 20809 52950 05448 00011 1110 6511 4717 8168 9916 3 4 5 6 7 VAR(6) VAR (4) VAR(16) VAR(15) VAR(14) -0.07246533 -0.^11529719 0.11202455 0. 10239134 0.06869768 STEP # 2 R= . 8 3 4 RSQUARE= TERM E N T E R E D : V A R ( 6 ) VARIABLE 0.09069 0.25680 0.54637 0.60371 0.63510 0.20060 0.51196 0.48294 0.40261 0.18019 0.6568 0.4787 0.4913 0.5295 0.6736 .096E+01 0.09069 -.013E+01 0.09069 .10E+02 0.20060 0.0026 0.6568 0.03675 0.53004 0.57265 0.59089 0.61435 0.60454 0.56211 0.31090 0.4381 0.4418 0.4582 0.5805 0.00001977 .073E+01 0.04630 0.00001972 .044E+01 0.01298 -0.00139975 -.037E+01 0.03675 3.04195 0.31025 0.61435 0.0894 0.5809 0.4381 0.52853 0.56987 0.59087 0.53617 0.47950 0.30193 0.4688 0.4931 0.5861 0.04622 0.01281 0.03664 0.52853 2.89251 0.21668 0.54570 0.53617 0.0976 0.6444 0.4649 0.4688 0.05454 0.09570 0.00011 0.08532 0.9916 0.7719 2.67310 0.20809 0.52950 0.05448 0.00011 0.1110 0.6511 0.4717 0.8168 0.9916 . 695 COEFFICIENT STD ERROR 0.00004532 -0.00000332 0.00001405 -0.00000742 STD COEF TOLERANCE IN 1 CONSTANT 2 VAR(5) 3 VAR(6) OUT 4 5 6 7 PART. CORR VAR (4) VAR(16) VAR(15) VAR(14) -0.12780013 0.12679220 0.12233083 0.09128331 STEP # 3 Br . 8 3 7 RSQUARE= TERM E N T E R E D : V A R ( 4 ) VARIABLE COEFFICIENT .700 STD ERROR STD COEF TOLERANCE IN 1 CONSTANT VAR(5) VAR(6) 4 VAR (4) OUT 0.00003448 0.00001098 -0.00109713 PART.' CORR 5 VAR(16) 6 VAR(15) 7 VAR(14) 0.12114063 0.11464885 0.09119794 STEP # 4 R= . 8 3 9 RSQUARE= TERM E N T E R E D : V A R ( 1 6 ) VARIABLE COEFFICIENT .704 STD ERROR STD COEF TOLERANCE IN 1 2 3 4 5 CONSTANT VAR(5) VAR(6) VAR (4) VAR(16) OUT 0.00003387 0.00001991 .072E+01 0.00000930 • 0.00001998 .037E+01 -0.00104205 -0.00141062 -.035E+01 0.00831282 0.01135264 ,09129784 PART. CORR 6 VAR(15) 7 VAR(14) -0.00178971 -0.04931388 STEP # 5 R= . 8 3 9 RSQUARE= TERM E N T E R E D : V A R ( 1 5 ) VARIABLE .704 COEFFICIENT STD ERROR 0.00003382 0.00000933 -0.00104297 0.00868756 -0.00022289 0.00002069 0.00002045 -0.00143331 0.03721875 -0.02105054 STD COEF TOLERANCE IN 1 2 3 4 5 6 OUT CONSTANT VAR (5) VAR(6) VAR (4) VAR (16) VAR (15) PART. CORR -.072E+01 .037E+01 -.035E+01 .09541359 .00416809 0.04405 0.01257 0.03650 0.05058 0.05454 130 7 VAR(14) -0.11478835 . STEP # 6 R= .841 RSQUARE= TERM ENTERED: VAR(14) VARIABLE . 0.01651 0.45398 0.5050 2 0 0 0 0 0 0 0 0 0 0 0 .708 COEFFICIENT STD ERROR STD COEF TOLERANCE IN CONSTANT VAR (5) VAR(6) VAR (4) VAR(16) VAR (15) VAR (14) 0 0 -0 -0 0 -0 OUT 00003552 00000548 00066667 00112719 03107864 02212007 .0 0 -0 -0 0 -0 00002100 .075E+01 0 04342 00002139 .022E+01 0 01167 00154883 -.022E+01 0 03176 04024149 .01237968 0 04395 05107220 .058E+01 0 00941 03282987 -.049E+01 0 01651 86024 06558 18527 00078 37030 45398 0999 7994 6696 9778 5469 5050 PART. CORR DEP VAR: VAR(IO) N: ADJUSTED SQUARED MULTIPLE R: VARIABLE COEFFICIENT CONSTANT VAR(5) VAR (6) VAR (4) VAR(16) VAR (15) VAR(14) 0 0 0 -0 -0 0 -0 34917888 00003552 00000548 00066667 00112719 03107864 02212007 41 MULTIPLE R: 0.841 SQUARED MULTIPLE R: 0.708 .657 STANDARD ERROR OF ESTIMATE: 0.38140815 STD ERROR 0 0 0 0 0 0 0 62100805 00002100 00002139 00154883 04024149 05107220 03282987 STD COEF TOLERANCE 0 0 0 -0 -0 0 -0 00000000 75201265 21962994 22379827 01237968 58119010 48579463 0 0 0 0 0 0 T P (2 TAIL) 0 56228 0434212 1 69123 0116716 0 25608 0317579 -0 43044 0439517 -0 02801 0094117 0 60852 0165150 -0 67378 0 0 0 0 0 0 0 57762 09994 79943 66960 97782 54688 50501 ANALYSIS OF VARIANCE SOURCE SUM-OF-SQUARES REGRESSION RESIDUAL 11.99853611 4.94605413 DURBIN-WATSON D STATISTIC FIRST ORDER AUTOCORRELATION F i n a l R e g r e s s i o n DF MEAN-SQUARE 6 34 F-RATIO 1.99975602 13.74665598 0.14547218 0.00000007 1.510 .240 E q u a t i o n : YIELD = 0.3491788 + 0.00003552 (CL,) + 0.00000548 (CL,) - 0.00066667 (CL,) - 0.00112719 (CH ) + 0.03107864 (CH ) - 0.02212007 (CH„) 3 6 Where CL and CH are the Cumulative Light and Cumulative Heat for 'a' weeks respectively. a a 131 EMPIRICAL MATH MODEL STATISTICAL RESULTS FOR 'ALTERNATIVE REGRESSION METHOD' Light & Heat (3,6,9 week) Combined Step-Wise Regression 1992 Equation for Yield Prediction (4 weeks ahead yield) Definition of Variables: VAR(9) VAR(3) VAR(4) VAR(5) VAR(6) VAR(7) VAR(8) = = = = = = = 4 weeks ahead Yield (kg/m 2) Cumulative Light for 3 weeks (W/m2) Cumulative Light for 6 weeks (W/m2) Cumulative Light for 9 weeks (W/m2) Cumulative Heat for 3 weeks (*C) Cumulative Heat for 6 weeks (*C) Cumulative Heat for 9 weeks (*C) A A A A EQUATION LAYOUT: VAR(9) = CONSTANT + a,VAR(3) + aVAR(4) + a VAR(5) + a VAR(6) + a VAR(7) + a VAR(8) 2 3 4 5 e Where 'CONSTANT' and a, to a are to be determined. 6 Step-Wise DEPENDENT VARIABLE Regression Results (Based on 1992 Data) VAR(9) MINIMUM TOLERANCE FOR ENTRY INTO MODEL = .010000 FORWARD STEPWISE WITH FIRST 9 VARIABLES FORCED IN MODEL ALPHA-TO-ENTER= .150 AND ALPHA-TO-REMOVE= .150 MAX # STEPS= STEP # 0 R= VARIABLE .000 RSQUARE= COEFFICIENT 14 .000 STD ERROR STD COEF TOLERANCE IN 1 CONSTANT OUT VAR(3) VAR (4) 4 VAR (5) VAR(6) VAR (7) 7 VAR(8) PART. CORR 0.81997755 0.71710795 0.58652376 0.55901034 0.50429635 0.45908945 STEP # 1 R= .820 RSQUARE= TERM ENTERED: VAR(3) VARIABLE .1E+0I .1E+01 .1E+01 .1E+01 .1E+01 .1E+01 .78E+02 .40E+02 .20E+02 .17E+02 .13E+02 .10E+02 0.0000 0.0000 0.0001 0.0002 0.0009 0.0029 .672 COEFFICIENT STD ERROR 0.00003977 0.00000450 STD COEF TOLERANCE F 'P' IN 1 CONSTANT 2 VAR(3) .082E+01 .1E+01 .78E+02 0.0000 OUT 3 4 5 6 7 PART. CORR VAR(4) VAR(5) VAR(6) VAR(7) VAR(8) -0-.34723606 -0.37278569 0.03765418 -0.04839489 -0.08879462 STEP # 2 R= . 84 4 RSQUARE TERM ENTERED: VAR(4) ! VARIABLE 0.09725 0.27466 0.56170 0.58919 0.62907 5.07284 5.97175 0.05253 0.08686 0.29404 0.0303 0.0194 0.8200 0.7699 0.5909 .712 COEFFICIENT STD ERROR 0.00006913 -0.00001641 0.00001372 -0.00000729 STD COEF TOLERANCE • p. IN 1 CONSTANT 2 VAR(3) 3 VAR(4) OUT 4 5 6 7 VAR{5) VAR(6) VAR (7) VAR(8) .25E+02 5.07284 0.0000 0.0303 0.03967 0.55079 0.56169 0.58369 0.77089 0.31185 0.03047 0.00088 0.3858 0.5800 0.8624 0.9765 PART. CORR -0.14479167 0.09267208 0.02907883 0.00493994 STEP # 3 R= .847 RSQUARE= TERM ENTERED: VAR(5) VARIABLE .143E+01 0.09725 -.064E+01 0.09725 COEFFICIENT .718 STD ERROR STD COEF TOLERANCE IN CONSTANT VAR(3) VAR (4) 4 VAR (5) OUT 0.00005730 -0.00000079 -0.00000705 0.0,0001927 .118E+01 0.04965 8.84633 0.0052 -0.00001923 .03076607 0.01405 • 0.00170 0.9674 -0.00000803 -.039E+01 0.03967 0.77089 0.3858 PART. CORR 5 VAR(6) 6 VAR(7) 7 VAR(8) 0.08444639 0.01526350 0.00355501 STEP # 4 R= .848 RSQUARE= TERM ENTERED: VAR(6) VARIABLE COEFFICIENT 0.54852 0.55640 0.58364 0.25138 0.00816 0.00044 0.6192 0.9286 0.9833 .720 STD ERROR STD COEF TOLERANCE .p. IN 1 2 3 4 5 CONSTANT VAR (3) VAR (4) VAR(5) VAR(6) OUT 0.00005673 -0.00000189 -0.00000679 0.00665415 0.00001950 .117E+01 0.04948 0.00001956 .07355506 0.01387 0.00000814 -.038E+01 0.03951 0.01327160 .06055901 0.54852 8.46182 0.00938 0.69702 0.25138 0.0063 0.9234 0.4095 0.6192 0.05696 0.12118 1.46242 0.86748 0.2349 0.3582 6.40679 0.02336 0.94485 1.70881 1.46242 0.0162 0.8794 0.3379 0.1999 0.2349 PART. CORR 6 VAR (7) 7 VAR (8) -0.20307304 -0.15773190 STEP # 5 R= .855 RSQUARE= TERM ENTERED: VAR(7) VARIABLE COEFFICIENT .731 STD ERROR STD COEF TOLERANCE IN CONSTANT VAR (3) VAR (4) VAR(5) VAR(6) VAR(7) 0.00005066 0.00000303 -0.00000791 0.05386681 -0.03271546 0.00002001 .104E+01 0.04637 0.00001985 .012E+01 0.01328 -0.00000814 -.044E+01 0.03900 0.04120735 .049E+01 0.05616 -0.02705307 -.045E+01 0.05696 133 OUT PART. CORR 7 VAR(8) 0.03349897 STEP # 6 R= .855 RSQUARE= TERM ENTERED: VAR(8) VARIABLE . . 0.03169 , 0.03707 0.8485 .732 COEFFICIENT STD ERROR STD COEF TOLERANCE F 'P' IN 1 2 3 4 5 6 7 CONSTANT VAR (3) VAR (4) VAR(5) VAR(6) VAR (7) VAR (8) OUT 0 0 -0 0 -0 0 00005016 00000414 00000853 05684290 04159460 00611141 0 0 -0 0 00002047 '. 103E+01 0 04561 00002094 .016E+01 0 01229 00000886 -.047E+01 0 03380 04456955 .052E+01 0 04940 05366335 -.057E+01 0 01490 0 03174018 .09750775 0 03169 •-o 6 00256 0 03906 0 '92609 1 62658 0 60078 0 03707 0 0 0 0 0 0 0198 84 45 3429 2111 4 438 8485 PART. CORR none DEP VAR: VAR(9) N: ADJUSTED SQUARED MULTIPLE R: VARIABLE COEFFICIENT CONSTANT VAR (3) VAR(4) VAR(5) VAR(6) VAR(7) VAR(8) 0 0 0 -0 '0 -0 0 46699163 00005016 00000414 00000853 05684290 04159460 00611141 40 MULTIPLE R: 0.855 SQUARED MULTIPLE R: 0.732 .683 STANDARD ERROR OF'ESTIMATE: 0.36795680 STD ERROR 0 0 0 0 0 0 0 70285733 00002047 00002094 00000886 04456955 05366335 03174018 STD COEF TOLERANCE 0 1 0 -0 0 -0 0 00000000 03422418 16075081 47190541 51732383 57251194 09750775 0 0 0 0 0 0 T P(2 TAIL) 0 66442 0456141 2 45001 0122872 0 19764 0338014 -0 96234 0494018 1 27538 0148984 -0 77510 0316941 0 19255 0 0 0 0 0 0 0 51104 01976 84454 34288 21108 4 4380 84850 ANALYSIS OF VARIANCE SOURCE SUM-OF-SQUARES REGRESSION RESIDUAL 12.18925463 4.46794287 DF MEAN-SQUARE 6 33 F-RAT IO 2.03154244 15.00486965 0.13539221 0.00000003 Final Regression Equation: YIELD = 0.46699163 + 0.00005016(CLj) + 0.00000414 (CL,) - 0.00000853 (CL,) + 0.05684290 (CH ) 0.04159460 (CH ) + 0.00611141 (CH ) 3 6 9 Where CL and CH are the Cumulative Light and Cumulative Heat for 'a' weeks respectively. a a 134 APPENDIX E 135 EMPIRICAL MATH MODEL (HUMIDITY TRIAL) STEP-WISE STATISTICAL RESULTS FOR: Light & Heat & Humidity (3, 6, 9 week) Combined Step-Wise Regression 1992 Equation for Tomato Yield Prediction Definition of Variables: VAR( 9) = Yield (kg / m 2) VAR( 3) = Cumulative Light for 3 weeks ( W / m 2) VAR( 4) = Cumulative Light for 6 weeks ( WI m 2) VAR( 5) = Cumulative Light for 9 weeks ( W / m 2) VAR( 6) = Cumulative Heat for 3 weeks (*C) VAR( 7) = Cumulative Heat for 6 weeks ( *C) VAR( 8) = Cumulative Heat for 9 weeks (*C) VAR(19)= Humidity, Vapor Pressure Deficit Avg. for 3 weeks (kPa) VAR(20)= Humidity, Vapor Pressure Deficit Avg. for 6 weeks (kPa) VAR(21)= Humidity, Vapor Pressure Deficit Avg. for 9 weeks (kPa) A A A A EQUATION LAYOUT: VAR(9) = CONSTANT + a VAR(3) + a VAR(4) + a VAR(5) + a VAR(6) + a VAR(7) + a VAR(8) + a VAR(19) + a VAR(20) + a VAR(21) 1 6 STEP # 2 7 1 R= 3 8 820 RSQUARE= 4 s 9 673 TERM ENTERED: VAR(3) VARIABLE COEFFICIENT STD ERROR 0 00003767 0.00000405 STD COEF TOLERANCE IN 1 CONSTANT 2 VAR(3) OUT 3 4 5 6 7 8 9 10 PART. VAR(4) VAR(5) VAR(6) VAR(7) VAR(8) VAR(19) VAR(20) VAR(21) 0 0 0 0 0 0 0 0 .082E+01 1E+01 .86E+02 0 0000 07492 21251 46353 52028 57098 98058 94214 94064 .15E+02• .15E+02 8.72209 5.47287 4.67641 3.74035 0.21120 0.29214 0 0 0 0 0 0 0 0 CORR 52099545 51599741 41882792 34316870 31997085 28913879 07158819 08411291 ' . 0 0 0 0 0 0 0 0 0003 0004 0052 0243 0365 0600 6483 5918 STEP # TERM ENTERED: 2 R= .873 RSQUARE= .762 VAR(4) VARIABLE COEFFICIENT STD ERROR STD COEF TOLERANCE IN 1 CONSTANT 2 VAR(3) 3 VAR(4) OUT 4 5 6 7 8 9 10 -0.00001041 0.00002623 PART. VAR(5) VAR(6) VAR(7) VAR(8) VAR(19) VAR(20) VAR(21) STEP # TERM ENTERED: -0.00001279 - . 0 2 3 E + 0 1 0.07492 0.00000671 .109E+01 0.07492 0.66273 .15E+02 0.4203 0.0003 0.03056 0.44360 0.48968 0.53551 0.97720 0.94195 0.91989 0.44414 6.43334 2.94410 2.23421 6.55608 0.22749 0.00254 0.5090 0.0152 0.0939 0.1428 0.0143 0.6360 0.9601 CORR 0.10479260 0.37222300 0.26183291 0.23000082 0.37526138 0.07520101 0.00796328 3 R= .874 RSQUARE= .764 VAR(5) VARIABLE COEFFICIENT STD ERROR - 0 00000220 0 00001524 0 00000493 - 0 00001783 0 00001782 0 00000739 STD COEF TOLERANCE F 'P' IN 1 CONSTANT 2 VAR (3) 3 VAR(4) 4 VAR (5) JUT 5 6 7 8 9 10 PART. VAR(6) VAR(7) VAR(8) VAR(19) VAR(20) VAR(21) STEP # TERM ENTERED: VARIABLE 0 0 0 0 0 -0 4 R= .04782375 0 03909 .063E+01 0 01077 .029E+01 0 03056 0 01517 0 73185 0 44414 0 9026 0 3974 0 5090 6 2 1 5 0 0 11890 74058 96132 90342 13483 01133 0 0 0 0 0 0 F 1 CORR 36826236 25623690 21882036 36258689 05869590 01704183 0 0 0 0 0 0 .892 RSQUARE= 44176 48704 52530 88100 91519 86984 0178 1059 1693 0198 7155 9158 .796 VAR(6) COEFFICIENT STD ERROR STD COEF TOLERANCE P' T H 1 2 3 4 5 CONSTANT VAR(3) VAR(4) VAR(5) VAR(6) OUT 6 7 8 9 10 -0 0 0 0 PART. VAR(7) VAR(8) VAR(19) VAR(20) VAR(21) -0 -0 0 0 0 00000697 00001438 00000382 01910167 - 0 . 0 0 0 0 1 6 9 0 - . 0 1 5 E + 0 1 0 03858 0.00001678 .060E+01 0 01077 0.00000697 .023E+01 0 03044 0.00772209 .027E+01 0 44176 0 .17012 0 .73421 0 .29924 6 .11890 0 .6823 0 . 3968 0 .5875 0 .0178 7 .39734 4 .65334 9 .29007 8 .88867 .11E+02 0 .0098 0 .0374 0 .0042 0 .0050 0 .0020 CORR 40366647 33029798 44322527 43539595 47428924 0 0 0 0 0 03388 08036 86712 51259 33216 137 STEP # 5 R= .911 RSQUARE= .829 TERM ENTERED: VAR(7) VARIABLE COEFFICIENT STD ERROR STD COEF TOLERANCE ' P' IN 1 2 3 4 5 6 CONSTANT VAR(3) VAR(4) VAR(5) VAR(6) VAR(7) -0 0 0 0 -0 CUT 7 8 9 10 00001305 00001548 00000457 09029463 03759139 -0 0 0 0 -0 00001582 --.028E+01 0 03781 00001556 .064E+01 0 01076 00000647 .027E+01 0 03038 02713664 .127E+01 0 03073 01382135 --.099E+01 0 03388 0.68038 0.98936 0.49934 .11E+02 7.39734 0 0 0 0 0 4146 3262 4841 0020 0098 0.22666 .23E+02 9.26293 6.83292 0 0 0 0 6368 0000 0043 0129 81047 17767 23875 69657 62155 22666 0 0 0 0 0 0 3738 2849 6280 0036 1139 6368 IT. CORR VAR(8) VAR(19) VAR(20) VAR(21) STEP # 0 0 0 0 6 R= 0 0 0 0 911 RSQUARE= TERM ENTERED: VAR(8) VARIABLE 07802977 62193128 44746344 39482351 COEFFICIENT 01576 80748 51031 29592 .831 STD ERROR STD COEF TOLERANCE IN 1 2 3 4 5 6 7 CONSTANT VAR(3) VAR(4) VAR(5) VAR (6) VAR(7) VAR(8) OUT -0 0 0 0 -0 0 00001474 00001801 00000341 09749132 05104955 00704359 -0 0 0 0 -0 0 00001638 --.032E+01 0 03603 00001660 .075E+01 0 00965 00000698 .020E+01 0 02666 03130811 .137E+01 0 02357 03152920 --.134E+01 0 00665 01479472 .026E+01 0 01576 0 1 0 9 2 0 PART. CORR 8 VAR(19) 9 VAR(20) 10 VAR(21) STEP # 0.62017524 0.44201762 0.41669313 7 R= 0. 80438 0.49757 0.28649 .946 RSQUARE= .23E+02 8.74159 7 . 56419 0.0000 0.0055 0.0093 5.64930 7.33993 0.58794 .24E+02 5.47093 0.09215 .23E+02 0 0 0 0 0 0 0 0.16425 5.59960 0.6877 0.0236 .896 TERM ENTERED: VAR(19) VARIABLE COEFFICIENT STD ERROR STD COEF TOLERANCE IN CONSTANT VAR(3) VAR(4) VAR(5) VAR(6) VAR(7) VAR(8) VAR(19) OUT 9 VAR(20) 10 VAR(21) -0 0 -0 0 -0 0 0 00003216 -0 00001353 --.070E+01 0 03338 00003744 0 00001382 . 155E+01 0 00881 00000444 -0 00000579 --.026E+01 0 02448 12470081 0 02555106 .176E+01 0 02238 05877324 • -0 02512748 --.155E+01 0 00662 00357863 0 01178866 .013E+01 0 01570 69266441 0 14602588 .028E+01 0 80438 0229 0103 4482 0000 0250 7632 0000 PART. CORR -0.06834448 -0.37137923 0.20402 0.06060 138 STEP # 8 R= TERM ENTERED: VAR(20) VARIABLE .947 RSQUARE= 896 STD ERROR COEFFICIENT STD COEF TOLERANCE IN CONSTANT VAR(3) VAR(4) VAR(5) VAR(6) VAR(7) VAR(8) VAR(19) VAR(20) OUT -0 0 -0 0 -0 0 0 -0 0228 0104 4496 0000 0265 7170 0022 6877 00001380 -.072E+01 0 03284 00001418 .159E+01 0 00856 00000586 -.027E+01 0 02448 .178E+01 0 02184 02617240 02691998 -.164E+01 0 00590 .016E+01 0 01524 01210936 23073661 .031E+01 0 32983 44960168 .04886636 0 20402 5 .67163 7 .33172 0 .58473 .23E+02 5 .36601 0 .13353 11E+02 0 .16425 0 0 0 0 0 0 0 0 0 05480 5 .36484 0 0267 8 .29517 7 .04483 0 .30470 30E+02 6 .10671 0 .70497 17E+02 .09480 5 .36484 0 0 0 0 0 0 0 0 0 -0 36916807 9 R= TERM ENTERED: VAR(21) VARIABLE -0 0 -0 0 -0 0 0 -0 vr. CORR 10 VAR(21) STEP # 00003287 00003840 00000448 12635311 06235919 00442500 76449037 18221414 .954 RSQUARE= COEFFICIENT .910 STD ERROR STD COEF TOLERANCE IN CONSTANT VAR(3) VAR(4) VAR(5) VAR(6) VAR(7) VAR(8) 8 VAR(19) 9 VAR(20) 10 VAR(21) OUT -0 0 0 0 -0 -0 1 0 -2 00003803 -0 00001320 -.083E+01 0 03190 .148E+01 0 00850 00003563 0 00001342 .021E+01 0 01749 00000361 0 00000654 0 02503952 .192E+01 0 02121 13616252. 06272876 -0 02538418 -.165E+01 0 00590 01112060 -0 01324475 -.041E+01 0 01132 29369637 0 31549768 .053E+01 0 15685 13726317 0 44581701 .03681137 0 18449 61590639 -1 12938969 -.051E+01 0 05480 PART. CORR THE SUBSET MODEL INCLUDES THE FOLLOWING PREDICTORS: CONSTANT VAR(3) VAR(4) VAR(5) VAR(6) VAR(7) VAR(8) VAR(19) VAR(20) VAR(21) 6 0068 0120 5846 0000 0186 4070 0002 7600 0267 139 DEP VAR: VAR(9) N: 44 MULTIPLE R: 0.954 SQUARED MULTIPLE R: 0.910 ADJUSTED SQUARED MULTIPLE R: .887 STANDARD ERROR OF ESTIMATE: 0.22643645 VARIABLE COEFFICIENT CONSTANT VAR(3) VAR (4) VAR(5) VAR (6) VAR (7) VAR(8) VAR(19) VAR(20) VAR(21) 1 -0 0 0 0 -0 -0 1 0 -2 35770256 00003803 00003563 00000361 13616252 06272876 01112060 29369637 13726317 61590639 STD ERROR 0 0 0 0 0 0 0 0 0 1 84447452 00001320 00001342 00000654 02503952 02538418 01324475 31549768 44581701 12938969 ANALYSIS SOURCE SUM-OF-SQUARES REGRESSION RESIDUAL 17.69865677 1.74329777 DURBIN-WATSON D S T A T I S T I C FIRST ORDER AUTOCORRELATION RESIDUALS STD COEF TOLERANCE 0 -0 1 0 1 -1 -0 0 0 -0 00000000 82807586 0 0319035 47884509 0 0084953 21434412 0 0174902 91743544 0 0212117 65176495 0 0059029 40521191 0 0113229 53170290 0 1568509 03681137 0 1844946 50810503 0 0548028 T 1 -2 2 0 5 -2 -0 4 0 -2 P(2 60775 88013 65421 55199 43791 47118 83962 10049 30789 31621 TAIL) 0 0 0 0 0 0 0 0 0 0 11714 00683 01200 58456 00000 01864 40698 00024 76004 02671 OF VARIANCE DF MEAN-SQUARE F-RATIO 9 34 1.96651742 0. 05127346 38.35351215 0.00000000 2.100 -.083 HAVE BEEN SAVED F i n a l R e g r e s s i o n E q u a t i o nforL i g h t H e a t & H u m i d i t y (3. 6, Yield (kg/ m ) = 1.3577 - 0.00003803 (CL ) + 0.00003563 (CL ) + 0.00000361 (CL ) + 0.1361 (CH ) - 0.0627 (CH ) - 0.0111 (CH ) + 1.293 (VPD ) + 0.1372 (VPD ) - 2.6159 (VPD ) 2 3 3 6 6 6 9 9 Where C L , C H , and VPD are the Cumulative Light, Cumulative Heat, and Average Vapor Pressure Deficit (VPD) for 'a' weeks respectively. a a a 9 3 140 1994 Predicted Yield - 4 W e e k s A h e a d 3 Light, 3 Heat, & 3 Humidity (VPD) Week Number A c t u a l Yield Figure A - 2 2 - e - 3L, 3 H , 3 V P D , E q n . Predicted yield for 4 w e e k s a h e a d using 3 light, 3 heat, 3 humidity variables. (CL3,6,9, C H 3 , 6 , 9 , V P D 3,6,9)
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The prediction and validation of greenhouse tomato yield using mathematical models and expert systems Tang, Winston C. 1995
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Title | The prediction and validation of greenhouse tomato yield using mathematical models and expert systems |
Creator |
Tang, Winston C. |
Date Issued | 1995 |
Description | Greenhouse tomato yields were predicted using two mathematical models developed in this study - the empirical and deterministic models. Weekly yield predictions for an entire growing season were compared with actual yields and results from an expert system model developed by the Agassiz Research Station. The deterministic math model involved using first principle equations of photosynthesis and respiration to simulate crop growth. Utilizing a known tomato yield conversion factor, net photosynthesis rates (Pnet) were converted to weekly yield predictions and compared with actual recorded yields. A deterministic model using two week cumulations of Pnet converted to yield was used successfully to predict actual tomato yields 6 weeks ahead of time with a root-mean-square-error (RMSE) of 0.38 kg/m2. The empirical math model employed regression techniques to fit historical greenhouse climate data to recorded yields. Correlations between light, temperature, and weekly tomato yields were derived into equations to predict yields for future growing seasons. An empirical model cumulating 3, 6, and 9 weeks of light and temperature data was developed to predict yields 4 weeks ahead of time with a RMSE of 0.45 kg/m2. When one-week-ahead predictions from the Agassiz expert system model were compared with actual recorded yields a RMSE of 0.401 kg/m2 was calculated. The expert system model utilizing trend recognition techniques was also used as a comparison with the two math models. When compared and ranked for prediction accuracy, application flexibility, and user-friendliness, the expert system was chosen as the overall best model for tomato yield prediction. |
Extent | 9451978 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-01-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0058533 |
URI | http://hdl.handle.net/2429/3678 |
Degree |
Master of Applied Science - MASc |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1995-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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