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Geostatistics applied to forecasting metal prices Faulkner, Reginald Lloyd 1988

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G E O S T A T I S T I C S A P P L I E D TO FORECASTING METAL P R I C E S By Reginald Lloyd Faulkner .Sc., The University of B r i t i s h Columbia, 1974 A T H E S I S SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D S C I E N C E i n THE F A C U L T Y OF GRADUATE S T U D I E S ( Mining and Mineral Process Engineering ) We accept t h i s thesis as conforming to the required standard THE U N I V E R S I T Y OF B R I T I S H COLUMBIA MAY 1988 C o p y r i g h t Reginald Lloyd Faulkner, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M.<K.e./ft_( ^ T f r C e S S & * . £ i i * a . e j r i ^ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6(3/81> ABSTRACT The o b j e c t i v e of t h i s t h e s i s was to i n v e s t i g a t e the effectiveness of k r i g i n g as a predictor of future prices for copper, lead and zinc on the London Metal Exchange. The annual average metal p r i c e s from 1884 to 1986 were d e f l a t e d i n t o constant p r i c e series with reference to a base of 1984 prices. Analysis of the data showed that the requirement of s t a t i o n a r i t y was s a t i s f i e d i f the p r i c e s e r i e s were div i d e d i n t o three d i s t i n c t time periods v i z . 1884 to 1917; 1918 to 1953; 1954 to 1986. For copper each of the three time periods were studied i n d e t a i l , but for lead and zinc only the most recent period was included i n t h i s t h e s i s . Spherical models gave a good f i t to the experimental semi-variograms computed for each metal-time p e r i o d and were used to p r e d i c t future p r i c e s by ordinary k r i g i n g . Universal Kriging was applied to the most recent time period f o r each metal by f i t t i n g a polynomial curve to the price-time s e r i e s , computing experimental semi-variograms from the r e s i d u a l s and then f i t t i n g spherical models which were used to predict future p r i c e s . Within the most recent price-time series, a further subdivision was made by taking that portion of the period from the highest price to 1986. Experimental semi-variograms from the r e s i d u a l s from f i t t e d polynomial curves showed pure nugget e f f e c t and consequently extrapolation of the polynomial was used as the price predictor. The kriged and extrapolated future p r i c e estimates were compared to future i i prices estimated by a simple random walk using residual sums of squared differences. For four of the f i v e time seri e s analyzed, ordinary kriging produced the best future price estimates. For copper from 1918 to 1953 , the simple random walk was marginally better than ordinary k r i g i n g . This was probably due to the low p r i c e v a r i a b i l i t y i n t h i s period r e s u l t i n g from the Great Depression and government price controls associated with the Second World War and the Korean War. S p e c i f i c forecasts for 1985 and 1986 were most accurate for copper and lead by universal k r i g i n g and most accurate for zinc by ordinary k r i g i n g . The r e s u l t s are encouraging and f u t u r e i n v e s t i g a t i o n s should include: applying other k r i g i n g methods : analyzing d a i l y and monthly prices : comparing r e s u l t s with more sophisticated time serie s analysis techniques. TABLE OF CONTENTS Chapter page ABSTRACT i i . TABLE OF CONTENTS i v . LIST OF TABLES v i i . LIST OF FIGURES x. ACKNOWLEDGEMENT x i i . 1. INTRODUCTION 1. 2. LITERATURE SURVEY 4. 2.1 SURVEY 4. 2.2 METAL PRICE BEHAVIOUR AND PREDICTION 13. 2.3 SUMMARY 14. 3. THEORY OF REGIONALIZED VARIABLES 16. 3.1 THEORY OF RANDOM FUNCTIONS 16. 3.1.1 REALIZATIONS 17. 3.1.2 STATIONARITY 17. 3.1.3 INTRINSIC HYPOTHESIS 18. 3.1.4 QUASI-STATIONARITY AND QUASI-INTRINSIC HYPOTHESIS 19. 3.2 SEMI-VARIOGRAM 19. 3.2.1 PROPERTIES OF THE SEMI-VARIOGRAM 20. 3.3 SEMI-VARIOGRAM MODELS 22. 3.3.1 SPHERICAL MODEL 22. 3.3.2 LINEAR MODEL 23. 3.3.3 PURE NUGGET EFFECT 25. 3.4 KRIGING 25. 3.4.1 KRIGING ESTIMATOR 26. 3.4.2 ORDINARY KRIGING 27. 3.4.3 UNIVERSAL KRIGING 28. 3.4.4 MATRIX SOLUTION AND ESTIMATION VARIANCE 30. i v TABLE OF CONTENTS cont'd Chapter page 4. DATA AND METHODOLOGY 32. 4.1 DATA 32. 4.2 METHODOLOGY 34. 5. RESULTS AND DISCUSSION 38. 5.1 COPPER 38. 5.1.1 COPPER: VARIOGRAPHY; RESULTS 40. 5.1.2 COPPER: VARIOGRAPHY; DISCUSSION 43. 5.1.3 COPPER: KRIGING; RESULTS 1884 to 1917 45. 5.1.4 COPPER: KRIGING; RESULTS 1918 to 1953 46. 5.1.5 COPPER: KRIGING; RESULTS 1954 to 1986 47. 5.1.6 COPPER: KRIGING; DISCUSSION 52. 5.2 LEAD 62. 5.2.1 LEAD: VARIOGRAPHY; RESULTS 62. 5.2.2 LEAD: VARIOGRAPHY; DISCUSSION 62. 5.2.3 LEAD: KRIGING; RESULTS 1954 to 1986 64. 5.2.4 LEAD: KRIGING; DISCUSSION. 70. 5.3 ZINC 70. 5.3.1 ZINC: VARIOGRAPHY; RESULTS 71. 5.3.2 ZINC: VARIOGRAPHY; DISCUSSION 71. 5.3.3 ZINC: KRIGING; RESULTS 1954 to 1986 73. 5.3.4 ZINC: KRIGING; DISCUSSION 79. 5.4 SUMMARY 79. 6. CONCLUSIONS 83. 6.1 FURTHER WORK 83. 6.2 FORECASTING 1987 and 1988 84. BIBLIOGRAPHY 88. APPENDIX 1 LME CURRENT PRICES AND WHOLESALE PRICE INDEX 97. APPENDIX 2 EXPERIMENTAL SEMI-VARIOGRAMS 101. APPENDIX 3 SQUARED DIFFERENCES 113. v TABLE OF CONTENTS cont'd page APPENDIX 4 POLYNOMIAL CURVES 126. APPENDIX 5 ZINC PRICES FREQUENCIES 133. v i LIST OF TABLES Table page 1. SUMS OF SQUARED DIFFERENCES COPPER 53. 2. WEIGHTING FACTORS LINEAR 56. 3. THEORETICAL LINEAR, SPHERICAL SEMI-VARIANCES 59. 4. WEIGHTING FACTORS SPHERICAL 60. 5. SUMS OF SQUARED DIFFERENCES LEAD 69. 6. SUMS OF SQUARED DIFFERENCES ZINC 78. 7 . 1987 AND 1988 METAL PRICE FORECASTS 85. APPENDIX 1 97. 1. LME CURRENT PRICES AND WHOLESALE PRICE INDEX 98. APPENDIX 2 101. 1. EXPERIMENTAL SEMI-VARIOGRAM, COPPER 1884 to 1917 102. 2. EXPERIMENTAL SEMI-VARIOGRAM, COPPER 1918 to 1953 103. 3. EXPERIMENTAL SEMI-VARIOGRAM, COPPER 1954 to 1986 104. 4. EXPERIMENTAL SEMI-VARIOGRAM, COPPER 1954 to 1986 RESIDUALS 105. 5. EXPERIMENTAL SEMI-VARIOGRAM, COPPER 1973 to 1986 RESIDUALS 106. 6. EXPERIMENTAL SEMI-VARIOGRAM, LEAD 1954 to 1986 107. 7. EXPERIMENTAL SEMI-VARIOGRAM, LEAD 1954 to 1986 RESIDUALS 108. 8. EXPERIMENTAL SEMI-VARIOGRAM, LEAD 1979 to 1986 RESIDUALS 109. 9. EXPERIMENTAL SEMI-VARIOGRAM, ZINC 1954 to 1986 110. 10. EXPERIMENTAL SEMI-VARIOGRAM, ZINC 1954 to 1986 RESIDUALS 111. 11. EXPERIMENTAL SEMI-VARIOGRAM, ZINC 1974 to 1986 RESIDUALS 112. v i i LIST OF TABLES cont'd Table page APPENDIX 3 113. 1. SQUARED DIFFERENCES, COPPER 1885 to 1917 SPHERICAL, ORDINARY KRIGING 114. 2. SQUARED DIFFERENCES, COPPER 1885 to 1917 HOLE EFFECT, ORDINARY KRIGING 115. 3. SQUARED DIFFERENCES, COPPER 1918 to 1953 SPHERICAL, ORDINARY KRIGING 116. 4. SQUARED DIFFERENCES, COPPER 1954 to 1986 SPHERICAL, ORDINARY KRIGING 117. 5. SQUARED DIFFERENCES, COPPER 1954 to 1986 SPHERICAL, UNIVERSAL KRIGING 118. 6. SQUARED DIFFERENCES, COPPER 1973 to 1986 PURE NUGGET EFFECT 119. 7. SQUARED DIFFERENCES, LEAD 1954 to 1986 SPHERICAL, ORDINARY KRIGING 120. 8. SQUARED DIFFERENCES, LEAD 1954 to 1986 SPHERICAL, UNIVERSAL KRIGING 121. 9. SQUARED DIFFERENCES, LEAD 1979 to 1986 PURE NUGGET EFFECT 122. 10. SQUARED DIFFERENCES, ZINC 1954 to 1986 SPHERICAL, ORDINARY KRIGING 123. 11. SQUARED DIFFERENCES, ZINC 1954 to 1986 SPHERICAL, UNIVERSAL KRIGING 124. 12. SQUARED DIFFERENCES, ZINC 1974 to 1986 PURE NUGGET EFFECT 125. APPENDIX 4 126. 1. POLYNOMIAL CURVE, COPPER 1954 to 1986 127. 2. POLYNOMIAL CURVE, COPPER 1973 to 1986 128. 3. POLYNOMIAL CURVE, LEAD 1954 to 1986 129. 4. POLYNOMIAL CURVE, LEAD 1979 to 1986 130. 5. POLYNOMIAL CURVE, ZINC 1954 to 1986 131. 6. POLYNOMIAL CURVE, ZINC 1974 to 1986 132. v i i i LIST OF TABLES cont'd Table page APPENDIX 5 133. 1. FREQUENCIES OF ZINC PRICES 1954 to 1986 134. i x LIST OF FIGURES Figure page 1. THEORETICAL SPHERICAL SEMI-VARIOGRAM 24. 2. THEORETICAL LINEAR SEMI-VARIOGRAM 24. 3. PURE NUGGET EFFECT SEMI-VARIOGRAM 24. 4. LONDON METAL EXCHANGE ANNUAL COPPER PRICES 39. 5. EXPERIMENTAL SEMI-VARIOGRAMS OF LONDON METAL EXCHANGE COPPER PRICES, 1884 to 1917, 1918 to 1953, 1953 to 1986 39. 6. SEMI-VARIOGRAM OF LME 1884 to 1917 COPPER PRICES 42. 7. SEMI-VARIOGRAM OF LME 1918 to 1953 COPPER PRICES 42. 8. SEMI-VARIOGRAM OF LME 1954 to 1986 COPPER PRICES 42. 9. LME 1954 to 1986 CONSTANT COPPER PRICES: QUADRATIC 49. 10. RESIDUALS 1954 to 1986 LME COPPER PRICES 49. 11. SEMI-VARIOGRAM OF LME 1954 to 1986 COPPER PRICES: RESIDUALS 49. 12. LME 1973 to 1986 CONSTANT COPPER PRICES: QUADRATIC .... 51. 13. RESIDUALS 1973 to 1986 LME COPPER PRICES 51. 14. SEMI-VARIOGRAM OF LME 1973 to 1986 COPPER PRICES: RESIDUALS 51. 15. KRIGING WEIGHTS AS A FUNCTION OF A LINEAR SEMI-VARIOGRAM 55. 16. IDEALIZED SPHERICAL SEMI-VARIOGRAM MODEL 58. 17. WEIGHTING FACTORS 58. 18. LONDON METAL EXCHANGE ANNUAL LEAD PRICES 63. 19. SEMI-VARIOGRAM OF LME ANNUAL LEAD PRICES, 1954 to 1986 63. 20. LME 1954 to 1986 CONSTANT LEAD PRICES: QUADRATIC 65. 21. RESIDUALS 1954 to 1986 LME LEAD PRICES 65. 22. SEMI-VARIOGRAM OF LME 1954 to 1986 LEAD PRICES: RESIDUALS 65. 23. LME 1979 to 1986 CONSTANT LEAD PRICES: QUADRATIC 67. x LIST OF FIGURES cont'd Figure page 24. RESIDUALS 1979 to 1986 LME LEAD PRICES 67. 25. SEMI-VARIOGRAM OF LME 1979 to 1986 LEAD PRICES RESIDUALS 67. 26. LONDON METAL EXCHANGE ANNUAL ZINC PRICES 72. 27. SEMI-VARIOGRAM OF LME ANNUAL ZINC PRICES, 1954 to 1986 72. 28. LME 1954 to 1986 CONSTANT ZINC PRICES: LINEAR 74. 29. RESIDUALS 1954 to 1986 LME ZINC PRICES 74. 30. SEMI-VARIOGRAM OF LME 1954 to 1986 ZINC PRICES: RESIDUALS 74. 31. LME 1974 to 1986 CONSTANT ZINC PRICES: QUADRATIC 76. 32. RESIDUALS 1974 to 1986 LME ZINC PRICES 76. 33. SEMI-VARIOGRAM OF LME 1974 to 1986 ZINC PRICES: RESIDUALS 76. x i ACKNOWLEDGEMENTS I wish to express my gratitude to Alan J. Reed for his consideration, enthusiasm and timely advice. My gratitude i s also extended to the s t a f f and students of the Department of Mining and Mineral Process Engineering for t h e i r support and encouragement. In addition, I would l i k e to thank Dr. A. J. S i n c l a i r for h i s counsel. Most importantly, t h i s thesis i s dedicated to Peggy for support, encouragement, understanding and tolerance, l o these many years. x i i 1. INTRODUCTION It has been suggested (Journel and Huijbregts 1978) that metals p r i c e time series are one dimensional regionalized random var i a b l e s . This implies that g e o s t a t i s t i c a l methods could be used to estimate metal prices, but to the writers knowledge no such study has been published. This thesis i s an attempt to apply g e o s t a t i s t i c a l methods to estimate future metal prices, s p e c i f i c a l l y copper, lead and zinc pr i c e s . Metal p r i c e forecasts play an important r o l e i n economic planning, from large scale modelling of national economies to d e t a i l e d f e a s i b i l i t y studies of i n d i v i d u a l mineral deposits. The s i g n i f i c a n c e of such forecasts cannot be overestimated as a basis for investment decisions. Methods used to forecast prices can be c l a s s i f i e d into three broad categories. 1. Resource Approach. 2. Econometric Models. 3. Time Series Analysis. In the 'Resource Approach', future metal prices are related to the supply of metals. This approach determines the amount of metal i n known reserves and estimates pote n t i a l reserves, with s c a r c i t y implying higher p r i c e s and abundance lower. The problems encountered with t h i s approach are the accuracy with which reserves are known and the methods by which potential reserves are determined. 'Econometric Models' r e l a t e future metal pr i c e s to the supply and demand f o r goods and services. By modelling an 1 economy the future demands for a metal can be rela t e d to the future supply of that metal giving a r e l a t i o n s h i p from which a future p r i c e can be estimated. These models often use variables determined outside of the model which are themselves future estimates. 'Time Series Analysis' r e l a t e s future commodity prices to trends, cycles and random noise components i d e n t i f i e d i n past price s e r i e s . The analysis assumes that these same components w i l l continue to e x i s t i n the future with the same general c h a r a c t e r i s t i c s t h a t they have e x h i b i t e d i n the p a s t . G e o s t a t i s t i c a l methods of metal pr i c e forecasting f a l l within the category of time series analysis. Most studies of time series analysis applied to metal price forecasting attempt to model the underlying stochastic process t h a t c r e a t e d the time s e r i e s and then use the model to ex t r a p o l a t e future metal p r i c e s . G e o s t a t i s t i c s provides a mechanism f o r modelling the s t r u c t u r a l r e l a t i o n s h i p of a price s e r i e s and for applying the model to optimize forecasted prices by a v a r i e t y of methods known as kr i g i n g . Generally, g e o s t a t i s t i c s i s used to calcu l a t e ore reserves i n mining applications, though i t s usage i s expanding into other f i e l d s such as hydrology, geophysics, cartography and forestry. In mining, ore r e s e r v e c a l c u l a t i o n s are a two or three dimensional, volumetric problem. In metal price forecasting the problem i s reduced to one dimension and the fut u r e p r i c e e v e n t u a l l y w i l l be known p r e c i s e l y f o r comparison with an estimate. 2 Data presented i n t h i s thesis are h i s t o r i c a l prices for copper, lead and zinc from the London Metal Exchange i n Great B r i t a i n . These prices are deflated by the wholesale,producer pr i c e index for Great B r i t a i n and the r e s u l t i n g deflated prices form the data base for the g e o s t a t i s t i c a l analysis. Similar studies have been made on copper, lead, zinc, s i l v e r , gold and platinum pr i c e s from United States markets and s i l v e r and gold prices from markets i n Great B r i t a i n , but are not included i n t h i s t h e s i s . Comparisons are made between g e o s t a t i s t i c a l l y predicted prices, known prices and prices predicted by a r e l a t i v e l y simple random walk procedure. Although there are some problems, the re s u l t s are encouraging and suggestions are made for further work i n t h i s f i e l d . Time and money constraints have l i m i t e d the scope of t h i s study. 3 2. LITERATURE SURVEY The ultimate goal of time series analysis i s to forecast future values of the time s e r i e s . To t h i s end the a n a l y t i c a l techniques t r y to model the time series and extrapolate t h i s model into the future. The a n a l y t i c a l techniques investigate the structure of a discrete series of data observed successively i n time. In t h i s l i t e r a t u r e survey two threads are h i s t o r i c a l l y interwoven. One thread being the development of the techniques of time seri e s analysis. The other, the ap p l i c a t i o n of these techniques to economic time series and s p e c i f i c a l l y to price s e r i e s . 2.1 Survey S e c u l a r t r e n d f i t t i n g and harmonic a n a l y s i s are the e a r l i e s t forms of time series analysis. F i t t i n g a curve to the time s e r i e s i m p l i e s the s e c u l a r t r e n d i s the dominating c h a r a c t e r i s t i c of the time series with the e r r a t i c component ignored (Snyder 1933, 1940). Harmonic analysis implies a s t r i c t p e r i o d i c i t y expressed as a periodogram of the c y c l i c a l v a r i a t i o n v i s i b l e i n the time series (Schuster 1906). The application of harmonic a n a l y s i s to economic time series i s exemplified by B e v e r i d g e ' s (1922) a n a l y s i s of i n d i c e s of wheat p r i c e f l u c t u a t i o n s i n Western Europe for the years 1545 to 1844. Two papers by Yule (1921, 1926) investigated the e r r a t i c component of a time s e r i e s . By applying a variate difference method, moving average, s e r i a l c o r r e l a t i o n and correlograms he 4 d e s c r i b e d random s e r i e s and a systematic theory of random variables (Davis 1941). Most importantly he showed that l i n e a r operations on random series do not create random s e r i e s . Slutzky (1936) proved that "The summation of random causes generates a c y c l i c a l series which tends to imitate for a number of cycles a harmonic series of a r e l a t i v e l y small number of sine curves." In developing t h i s proof he s o l i d i f i e d the concept of moving average processes generating time s e r i e s that was introduced i n Yule's papers (Wold 1954; Granger and Hatanaka 1964). Yule (1929) showed that random pulses created a disturbed p e r i o d i c movement. He solved a second degree r e g r e s s i o n equation for Wolfer's sunspot numbers and proved i t was a damped harmonic f u n c t i o n . The i n v e s t i g a t i o n introduced the auto-regressive process of time seri e s generation (Wold 1954; Granger and Hatanaka 1964). The work of Bachelier (1900) and Pearson (1905, 1906) were the e a r l i e s t attempts at applying p r o b a b i l i t y theory to the idea of a random walk (Davis 1941; Yaglom 1962). I t was not u n t i l the 1930's with the work of A. N. Kolmogorov and A. Khinchine that rigourous p r o b a b i l i s t i c theories concerning random series were developed (Wold 1954; Yaglom 1962). From these seri e s the concepts of random functions and s t a t i o n a r i t y were created allowing s t a t i s t i c a l inferences to be made. Working (1934) and Jones (1937) investigated the use of f i r s t differences i n analyzing economic time s e r i e s . The former author proposed the use of a random-difference series as a type standard as the basis f o r s t a t i s t i c a l t e s t s of actual time 5 s e r i e s . The l a t t e r author developed a systematic application of the theory of runs whereby the signs of the f i r s t differences of a time ser i e s would determine whether the serie s was random, a random-difference series or a series that presented some type of in t e r n a l structure. Cowles and Jones (1937) applied t h i s theory to i n d i c e s of i n d u s t r i a l stock p r i c e s and found the series showed evidence of structure. Wold (1954) made a comprehensive study of the use of the moving average and auto-regressive processes to model time s e r i e s (Yaglom 1962). Of s p e c i f i c importance was the development of the decomposition theorem, Theorem 7. "He proved that any stationary process Xt can be uniquely represented as the sum of two mutually uncorrelated processes Xt = Dt + Yt where Dt i s l i n e a r l y deterministic and Yt i s a MA(CO) [moving average] process. The Y t component i s s a i d to be purely i n d e t e r m i n i s t i c . 11 (Granger and Newbold 1986). "Wold's decomposition has the nature of an existence theorem, but as soon as the deterministic component and the c o e f f i c i e n t s of the moving average are s p e c i f i e d i t becomes a l i n e a r p r e d i c t o r formula." (Whittle 1954). In the 1940's a s e r i e s of papers on i n t e r p o l a t i o n , extrapolation and smoothing of time series by A.N. Kolmogorov and N o r b e r t Wiener p r o v i d e d the wherewithal to develop forecasting techniques (Whittle 1954). "This p r e d i c t i o n theory i s concerned with the problem of predicting x t when the whole r e a l i z a t i o n i s available by means of a l i n e a r operation and the past values." (Hannan 1960). Their work plus that of Harald 6 Cramer also provided the basis f o r development of s p e c t r a l analysis (Whittle 1954; Granger and Hatanaka 1964). The papers by Working (1934), Jones (1937) and Kendall (1953) set the methodology for analyzing price time seri e s . Kendall (1953) found, by determining the s e r i a l c o r r e l a t i o n of the f i r s t differences of indices of i n d u s t r i a l share prices and cash wheat and spot cotton prices, that "The data behaved almost l i k e a wandering s e r i e s . " I t was also noted that aggregated index numbers behaved more systematically than t h e i r components. By determining that the changes or differences i n a price s e r i e s were independent and i d e n t i c a l l y d i s t r i b u t e d , the hypotheses of a random walk and the f a i r game market model postulating that the expectation of the speculator i s zero, was accepted (Bachelier 1900). Papers inv e s t i g a t i n g p r i c e behaviour showed mixed r e s u l t s with respect to accepting the random walk model. Osborne (1959) demonstrated that stock prices followed a Brownian motion. Larson (1960) showed that a low-weight moving average s t o c h a s t i c process generated random changes i n corn f u t u r e s p r i c e s . Alexander (1961) summarized past f i n d i n g s s t a t i n g " . . . i n speculative markets p r i c e changes appear to follow a random walk over time, but a move, once i n i t i a t e d , tends to p e r s i s t . " He introduced a f i l t e r i n g technique based on mechanical t r a d i n g r u l e s t h a t e x p l o i t e d t h i s movement, e x h i b i t i n g p r o f i t s . Houthakker (1961) formed a f i l t e r i n g technique based on "stop orders" for wheat and cotton futures and cotton spot p r i c e s f i n d i n g a p a t t e r n of p r i c e change behaviour contradictory to a random walk. 7 Yaglom (1962) presented an introduction to the theory of stationary random functions. He summarized previous t h e o r e t i c a l work and explained mathematically the problems of extrapolating and f i l t e r i n g random functions. Granger and Hatanaka (1964) gave one of the f i r s t comprehensive accounts of s p e c t r a l analysis as applicable to economic time s e r i e s . Granger and Morganstern (1963) using spectral analysis on New York stock price series showed that the simple random walk hypothesis of price behaviour f i t t e d the data. Fama (1965) t e s t i n g that successive stock price changes were i d e n t i c a l l y d i s t r i b u t e d and independent, used s e r i a l c o r r e l a t i o n , run analysis and f i l t e r i n g techniques. He concluded that the random walk model seemed to adequately describe stock price behaviour. The mixed acceptance of the random walk model and the existence of p r o f i t a b i l i t y resulted i n the t h e o r e t i c a l p o s i t i n g of systematic price movements with an expected gain of zero and pr i c e behaviour where the expected gain i s dependent on the r i s k accepted (Samuelson 1965; Mandelbrot 1966). C a r g i l l and Rausser (1969) applied spectral analysis to the f i r s t differences and f i r s t differences of the logarithmic transformation of futures contract prices for wheat, corn, oats and l i v e beef c a t t l e . They accepted the random walk model as a special case of the general stochastic martingale model. In t h i s general model the expected gain i s zero whether the prices behaved as a random walk or with systematic movements. Why prices behave i n the ways suggested has been related to the type, timing and use of information. Working presented a hypothesis of market expectation (Working 1949) and a theory of 8 anticipatory prices (Working 1958) based on these aspects and the random influence of information. Smidt (1968) f e l t i t was u n l i k e l y that information had a random e f f e c t on prices, but that the current price i n a competitive market should r e f l e c t a l l a v a i l a b l e information. He also proposed that uninformed p a r t i c i p a n t s or lags i n learning about new information created systematic dependence i n p r i c e s which could be u t i l i z e d by traders with superior knowledge to earn competitive returns. Fama (1970) hypothesized an e f f i c i e n t market as "A market i n which prices always ' f u l l y r e f l e c t ' available information . . ." A process of price formation i n terms of an expected return model was also s p e c i f i e d where " . . . the expected price c o n d i t i o n a l on a relevant information set i s equal to the previous p r i c e plus the expected return c o n d i t i o n a l on the relevant information set." There were three information sets defined: - weak form; h i s t o r i c a l p rice s e r i e s . - semi-strong form; h i s t o r i c a l p rice series and public information. - strong form; h i s t o r i c a l p rice series, public information and monopolistic access to relevant information. Special cases of the expected return model were the random walk model, the martingale model, and a sub-martingale model where the expected p r o f i t s were not greater than those received by buying and h o l d i n g f o r a s p e c i f i c time. Jensen (1978) reformulated the e f f i c i e n t market hypothesis as "A market i s e f f i c i e n t w i t h r e s p e c t to i n f o r m a t i o n s e t I t i f i t i s impossible to make economic p r o f i t s by trading on the basis of 9 the information set I t . . ." Economic p r o f i t s meant the r i s k adjusted returns net of a l l costs. Box and Jenkins (1976) addressed the problem of forecasting time s e r i e s by developing an approach to building, i d e n t i f y i n g , f i t t i n g and checking models of time s e r i e s . This approach uses mixed a u t o - r e g r e s s i v e and moving average p r o c e s s e s and integ r a t i o n to model stationary and non-stationary time series. I t i s an i t e r a t i v e approach where a t e n t a t i v e model i s i d e n t i f i e d , parameters of t h i s model estimated from f i t t i n g i t to the data and applying the p r i n c i p l e of parsimony and using diagnostic checks, to te s t the f i t of the model to the data. The model i s accepted i f no lack of f i t i s indicated otherwise the process i s repeated. Labys and Granger (1970) made an e m p i r i c a l study of commodity p r i c e s j o i n i n g " s t a t i s t i c a l and econometric theory with commodity price theory and forecasting." They found that the random walk model f i t a l l commodity price change seri e s . F u r t h e r m o r e , t h e y compared the f o r e c a s t i n g methods of econometric a n a l y s i s , e x p o n e n t i a l smoothing, Box-Jenkins predictors and some change, no change and average random walks. The random walk methods provided the better forecasts f o r soybean and cottonseed o i l , soybean meal, soybeans, rye and wheat p r i c e s . Stevenson and Bear (1970) concluded that the random walk model d i d not s a t i s f a c t o r i l y explain corn and soybean futures p r i c e s . Using s e r i a l c o r r e l a t i o n s , a n a l y s i s of runs and mechanical trading rules they found systematic price movements and made p r o f i t s . Labys, Rees and E l l i o t (1970) applied 10 spectral analysis to the f i r s t differences of cash and forward copper prices and found that the price series followed a random walk. Mixed r e s u l t s were encountered by C a r g i l l and Rausser (1972) when they investigated d a i l y c l o s i n g futures prices for l i v e beef c a t t l e , copper, corn, oats, frozen pork b e l l i e s , rye, soybeans and wheat. In c a l c u l a t i n g the a u t o - c o r r e l a t i o n function, spectral density and integrated periodogram for the commodities they rejected the random walk model for copper and the frozen pork b e l l i e s . Leuthold (1972) used spectral analysis and mechanical trading rules to investigate whether the random walk model applied to l i v e c a t t l e futures p r i c e s . He found that spectral analysis gave mixed r e s u l t s and applying the trading rules questioned whether accepting the random walk hypothesis precluded p r o f i t a b i l i t y as gross p r o f i t s were made. C a r g i l l and Rausser (1975) rejected the random walk model of p r i c e behaviour for corn, oats, soybeans, wheat, copper, l i v e c a t t l e and pork b e l l i e s futures, but concluded that t h i s did not r e j e c t the e f f i c i e n t market hypothesis. They used the theory of runs and auto-correlation function tests from the time domain and the inte g r a t e d periodogram and s p e c t r a l density function of the frequency domain of time series analysis. Labys and Thomas (1975) accepted the random walk model for futures prices of cocoa, coffee, sugar, s i l v e r , copper, t i n , lead, zinc and rubber using spectral analysis. I t i s known that prices and price differences are often not normally d i s t r i b u t e d and are non-stationary (Kendall 1953; Osborne 1959; Larson 1960; Houthakker 1961; Mandelbrot 1963; Fama 1965; F i e l i t z 1971; Boness, Chen and Jatusipitak 1974). 11 The v a g a r i t i e s of the acceptance of the random walk hypothesis of p r i c e behaviour have been attributed to applying a n a l y t i c a l techniques which require normal d i s t r i b u t i o n s and no trends to these pr i c e s and price differences (Mandelbrot 1963; C a r g i l l and Rausser 1975). Vagarities have also been att r i b u t e d to the fact t h a t mechanical t r a d i n g r u l e s have i n h e r e n t problems i n in t e r p r e t a t i o n of the r e s u l t s ( C a r g i l l and Rausser 1975) An explanation for the r e j e c t i o n of the random walk model i n some cases was discovered by Working (1960). He found that the f i r s t differences of a series of averages, even though the series may be a random chain, produced a s i g n i f i c a n t p o s i t i v e f i r s t order auto-correlation. Fama (1965) and Boness, Chen and Jatusipitak (1974) suggested the departure from normality i n the d i s t r i b u t i o n of prices and price differences was due to mixed d i s t r i b u t i o n s . To a l l e v i a t e these problems Osborne (1959) used logarithms of the price data. Granger and Morganstern (1963) found that the trend i n the variance was of no concern and that the lognormal transformations were not required. F i e l i t z (1974) showed t h a t the n a t u r a l l o garithms of p r i c e s and f i r s t differences of prices had trends i n mean and variance. He and Mandelbrot (1963) f e l t that the Paretian family of d i s t r i b u t i o n s best described the d i s t r i b u t i o n of the f i r s t differences of pr i c e s . It was determined by Solt and Swanson (1981) that the price differences on the London Gold Market and the New York S i l v e r Market were generated from normal d i s t r i b u t i o n s with a trend element. They found, by using lognormal transformed data, that the markets were not e f f i c i e n t , but the i n e f f i c i e n c i e s could not 12 be exploited for p r o f i t . Taylor (1980,1982a,1982b,1985) tested a random walk model and a p r i c e trend hypothesis of p r i c e behaviour. The random walk model was discovered to inadequately describe the process of price formation and the large number of po s i t i v e auto-correlation c o e f f i c i e n t s suggested trends i n the p r i c e s . He was not able to r e f u t e the e f f i c i e n t market hypothesis. Goss (1983) examined the semi-strong form and Peter B i r d (1985) the weak form i n e f f i c i e n c i e s of d a i l y cash and futures prices for copper, lead, t i n and zinc from the London Metal Exchange. I n e f f i c i e n c i e s i n the market were found i n the semi-strong and weak form sense. Granger and Newbold (1986) summarize techniques of time serie s analysis, forecasting and evaluation of the forecasts. S p e c i f i c a l l y they develop time series analysis of economic time serie s by modelling, forecasting using a model and evaluation of the forecasts and model. 2.2 Metal Price Behaviour and Prediction I n v e s t i g a t i o n s of metal p r i c e s o r i e n t e d towards p r i c e behaviour and pre d i c t i o n have been involved predominantly with trend determination and component a n a l y s i s . P h i l l i p s and Edwards (1976) derived an inverse r e l a t i o n s h i p between ore grade and metal price with ore grade f a l l i n g over time. Lee (1976) showed that i n terms of r e a l or constant prices, p r i c e trends i n d i f f e r e n t markets manifest s i m i l a r s t r u c t u r a l c h a r a c t e r i s t i c s . Wood, Werner and Azis (1977) using moving averages of deflated and converted London Metal Exchange price series determined an upward trending curve for copper. Petersen and Maxwell (1979) 13 suggested a general inverse r e l a t i o n s h i p between mineral prices and production with cost a c o n t r o l l i n g factor. They determined that three basic price trends were followed over time f a l l i n g , s t a b l e and r i s i n g . Slade ( 1982a, 1982b) emphasized the r e l a t i o n s h i p s of s c a r c i t y and investment with r e a l prices of natural resource commodities. She found increasing s c a r c i t y r e f l e c t e d i n concave quadratic price trends and the cycles seen were an e f f e c t of the timing of investment i n the industry. A l t e r n a t i v e l y Labys and Afrasabi (1983) suggested that cycles i n the United States copper market resulted from s h i f t s i n the demand for copper. D i f f e r e n t methods have been suggested and t r i e d i n p r e d i c t i n g metal price s . Etheridge (1978) stated that the r e a l p r i c e of a metal generally i s determined by the r e a l marginal u n i t cost of production. Therefore by determining the r e a l marginal unit cost of production future prices can be projected. Krige (1978) proposed that the trends i n r e a l prices and i n the r e l a t i v e purchasing powers of d i f f e r e n t currencies provided some o b j e c t i v i t y i n the long term estimation of p r i c e s . O'Leary and B u t l e r (1978) used F o u r i e r A n a l y s i s and Power Spectrum determinations on r e a l copper prices to predict future prices. Real copper price differences and the random walk model were used to generate future prices i n the study done by Rudenno (1982). 2.3 Summary A number of important p o i n t s can be drawn from t h i s l i t e r a t u r e survey. S i g n i f i c a n t to the analysis of metal price 14 time seri e s i s the data set used and the requirement of s p e c i f i c conditions that allow s t a t i s t i c a l inferences to be made. It i s from these s t a t i s t i c a l inferences that the forecasting of the time s e r i e s can occur. Assuming that the price r e f l e c t s a l l available information at a s p e c i f i c point i n time the simplest information set available i s a h i s t o r i c a l price s e r i e s . To be able to rel a t e one p r i c e to another i n the same series a common base, reference or support i s necessary. This i s e a s i l y achieved by de f l a t i n g the current prices i n the series to constant or r e a l p r i c e s . The random walk of price behaviour provides a hypothesis against which the behaviour of commodity prices i n a time series i s t e s t a b l e . A l t e r n a t i v e l y , future prices of a random walk p r i c e s e r i e s can be used to t e s t the a b i l i t y of d i f f e r e n t methods to forecast p r i c e s . The tes t i s whether these forecasts are better or worse estimates of future prices than the random walk. Looking at the p r i c e time s e r i e s as a r e s u l t of a st o c h a s t i c process i t i s necessary to make some s t a t i s t i c a l inferences concerning that process. The conditions which allow the inferences are rela t e d to the d i s t r i b u t i o n of the price data and trends or lack of trends i n the mean and/or variance of the data. 15 3. THEORY OF REGIONALIZED VARIABLES The Theory of Regionalized Variables, was developed by Matheron (1963,1971) to account for the s p a t i a l c o r r e l a t i o n of grades i n a mineral deposit. This theory q u a n t i f i e s the r e l a t i o n that grades of samples taken close together are more s i m i l a r than are those taken further apart. Therefore, a grade at a point i n space i s a function of the grades of other points nearby. By subs t i t u t i n g price for grade and a time series for a three dimensional mineral deposit, metal p r i c e s defined at discrete points i n time can be considered as a one dimensional regionalized variable (Journel and Huijbregts 1978). The time serie s of average metal prices over a s p e c i f i e d time i n t e r v a l can be thought of as a r e a l i z a t i o n of a one-dimensional random function. 3.1 Theory of Random Functions A random variable i s a quantity which takes a d i f f e r e n t numerical value for each r e p e t i t i o n of an experiment dependent on the p r o b a b i l i t y d i s t r i b u t i o n of the random variable. A s p e c i f i c n u m e r i c a l value can be c o n s i d e r e d as a s i n g l e r e a l i z a t i o n of the random variable. Because a set of random v a r i a b l e s can be considered a random f u n c t i o n , a set of r e a l i z a t i o n s of random v a r i a b l e s can be considered as a r e a l i z a t i o n of the random function (Yaglom 1962; Matheron 1971; Journel and Huijbregts 1978). To assume that a set of numerical values i s a r e a l i z a t i o n of a random f u n c t i o n r e q u i r e s t h a t a l l or p a r t of the 16 p r o b a b i l i t y law defining the random function be in f e r r e d . Since i t i s d i f f i c u l t to i n f e r the p r o b a b i l i t y law of a random function from a single r e a l i z a t i o n l i m i t e d to a f i n i t e number of points some assumptions must be made, such as second-order s t a t i o n a r i t y or the i n t r i n s i c h y p o t h e s i s i n order t h a t s t a t i s t i c a l inferences can be made. I t i s with one or more of these assumptions that the random and s t r u c t u r a l c h a r a c t e r i s t i c s of a regionalized variable can be determined and estimates with errors made. 3.1.1 Realizations In theory many r e a l i z a t i o n s of a random function are required before any s t a t i s t i c a l inferences can be made. By applying the appropriate hypothesis to a metal p r i c e time series, each p a i r of prices for d i f f e r e n t time separations can be considered as one r e a l i z a t i o n of the random f u n c t i o n . Therefore, only a single metal price time seri e s of the random function i s required to make any s t a t i s t i c a l inferences. 3.1.2 S t a t i o n a r i t y "The random function £; (t) w i l l be c a l l e d stationary i f a l l the finite-dimensional d i s t r i b u t i o n functions (1.3) defining £ (t) remain the same i f the whole group of points t x , t 2 , t n i s s h i f t e d along the time axis ..." (Yaglom 1962). This i s s t a t i o n a r i t y i n the strong sense. "In p r a c t i c a l applications strong s t a t i o n a r i t y i s v i r t u a l l y impossible to tes t for, and i t i s usual to work with the weaker form." (Granger and Newbold 1986). 17 Second-order s t a t i o n a r i t y , or s t a t i o n a r i t y i n the weak sense, i s defined f o r a time series as follows (Wold 1954; Yaglom 1962). 1. The mean value of the r e a l i z a t i o n of the random function e x i s t s and i s constant over the metal p r i c e time series and i s independent of any point i n time. 2. The covariance of the r e a l i z a t i o n of the random function e x i s t s and depends only on the time separating any two prices of a metal. The s t a t i o n a r i t y of the covariance implies the variance of the metal price s e r i e s e x i s t s and i s stationary and f i n i t e . 3.1.3 I n t r i n s i c Hypothesis The i n t r i n s i c h y p o t h e s i s allows a weakening of the conditions of second-order s t a t i o n a r i t y . If the variance of the r e a l i z a t i o n of the random f u n c t i o n does not e x i s t or i s i n f i n i t e l y increasing the variance of a set of metal p r i c e pairs for each time separation may be f i n i t e . The i n t r i n s i c hypothesis i s defined as follows (Matheron 1971; David 1977; Journel and Huijbregts C. J., 1978). 1. The mean value of the r e a l i z a t i o n of the random function e x i s t s and i s independent of any point i n time, but i s dependent on the time separation. 2. The semi-variogram ex i s t s and i s only dependent on the separation of metal prices i n time. The conditions of the i n t r i n s i c hypothesis allow a li n e a r d r i f t i n the mean and variance. A l i n e a r d r i f t i s permitted 18 because the mean and semi-variogram are a function of the time separation and not of any s p e c i f i c point i n time. If these conditions e x i s t then the i n t r i n s i c hypothesis applies, but the hypothesis of second-order s t a t i o n a r i t y does then both hypotheses apply. 3.1.4. Quasi-stationarity and Q u a s i - i n t r i n s i c Hypothesis Q u a s i - s t a t i o n a r i t y hypothesizes second-order s t a t i o n a r i t y over a l i m i t e d time i n t e r v a l and the q u a s i - i n t r i n s i c hypothesis assumes the i n t r i n s i c hypothesis applies only over a s p e c i f i c time. This can be interpreted as second-order s t a t i o n a r i t y or the i n t r i n s i c hypothesis applying only l o c a l l y over l i m i t e d time separations which can be estimated from the semi-variogram. 3.2 Semi-variogram The semi-variogram i s the mean squared difference i n metal pr i c e f o r various time separations (lags) of the pri c e s . For a metal p r i c e time series of annual average prices an experimental semi-variogram i s defined as follows: not. If the hypothesis of second-order s t a t i o n a r i t y exists, N(t) [ P(x A) - P(x. + t ) ] 2 for t = 1,...,n. 2 N(t) i = l where; X ( t ) the semi-variogram function of time separation t. N(t) the number of pairs of prices separated by time t. P(x ± ) the price at time x ±. 19 p ( x i + t ) = "*-he P r i c e at time (xt plus t ) . t = the time separation or s l a g ' . A p l o t of the e x p e r i m e n t a l semi-variogram f u n c t i o n generally produces a more-or-less sawtooth curve with a d i s t i n c t 'general pattern'. This general pattern i s assumed to be the underlying semi-variogram and i s f i t t e d by a mathematical model. The sawtooth character about the model i s assumed to be the sampling error. 3.2.1 Properties of the Semi-variogram There are four properties of the semi-variogram: behaviour near the o r i g i n , symmetry around the y axis, range of continuity and p o s i t i v e definiteness. The behaviour of the semi-variogram near the o r i g i n relates to the continuity and r e g u l a r i t y of the variance of the metal p r i c e s . Parabolic behaviour i s c h a r a c t e r i s t i c of a continuous and h i g h l y regular v a r i a b i l i t y over time. Linear behaviour implies continuous and average r e g u l a r i t y i n v a r i a b i l i t y over time. Discontinuous behaviour describes a lack of continuity or r e g u l a r i t y i n v a r i a b i l i t y for time separations less than one period. This l a t t e r type of behaviour i s c a l l e d the 'nugget e f f e c t ' and t h i s value i s equal to the experimental semi-variogram function at time separation 0. The 'range' i s the time i n t e r v a l corresponding to a l e v e l l i n g of the semi-variogram function. If the variance of the random function e x i s t s the covariance then also e x i s t s . The semi-variogram and the auto-covariance functions are related by the equation: 2 0 C ( t ) = C ( 0 ) - X ( t ) f o r t = l , . . . , n . w h e r e ; C ( t ) = a u t o - c o v a r i a n c e f u n c t i o n a t t i m e s e p a r a t i o n t . C ( 0 ) = a u t o - c o v a r i a n c e f u n c t i o n a t t i m e s e p a r a t i o n 0 w h i c h i s t h e v a r i a n c e o f t h e p r i c e s i n t h e t i m e s e r i e s . / ( t ) = s e m i - v a r i o g r a m f u n c t i o n a t t i m e s e p a r a t i o n t . T h e s e m i - v a r i o g r a m a n d a u t o - c o v a r i a n c e f u n c t i o n s a r e p o s i t i v e d e f i n i t e i f ; - t h e h y p o t h e s i s o f s e c o n d - o r d e r s t a t i o n a r i t y o r q u a s i - s t a t i o n a r i t y a p p l i e s . - t h e v a r i a n c e o f t h e ' s i l l ' o f t h e s e m i - v a r i o g r a m i s p o s i t i v e a n d f i n i t e o r a t t h e ' r a n g e ' o f t h e t r a n s i t i o n t o a ' s i l l ' t h e a u t o - c o v a r i a n c e i s z e r o o r p o s i t i v e a n d f i n i t e . The s e m i - v a r i o g r a m f u n c t i o n i s a c o n d i t i o n a l p o s i t i v e d e f i n i t e f u n c t i o n i f ; - t h e i n t r i n s i c h y p o t h e s i s o r q u a s i - i n t r i n s i c h y p o t h e s i s a p p l i e s . - a t t i m e s e p a r a t i o n 0 t h e a u t o - c o v a r i a n c e f u n c t i o n d o e s n o t e q u a l t h e v a r i a n c e o f t h e p r i c e s i n a t i m e s e r i e s , b u t t h e v a r i a n c e i s p o s i t i v e o r z e r o . I f t h e s e m i - v a r i o g r a m f u n c t i o n i s c o n d i t i o n a l l y p o s i t i v e d e f i n i t e i t s b e h a v i o u r a t i n f i n i t y i s s u c h t h a t i t i n c r e a s e s more s l o w l y a t i n f i n i t y t h a n d o e s t h e s q u a r e d t i m e s e p a r a t i o n s . I f t h e s e m i - v a r i o g r a m f u n c t i o n i n c r e a s e s a t l e a s t a s r a p i d l y a s t h e t i m e s e p a r a t i o n s q u a r e d f o r l a r g e t i m e s e p a r a t i o n s i t i s i n c o m p a t i b l e w i t h t h e i n t r i n s i c h y p o t h e s i s . T h e r e f o r e t h e r e i s 21 a trend or a d r i f t i n the mean, variance, semi-variogram or auto-covariance (Journel and Huijbregts 1978). 3.3 Semi-variogram Models If the hypothesis of second-order s t a t i o n a r i t y or quasi-s t a t i o n a r i t y i s s a t i s f i e d , the model of the semi-variogram increases continuously with some degree of r e g u l a r i t y to a ' s i l l ' . For second-order s t a t i o n a r i t y the ' s i l l ' i s equal to the variance of the metal prices minus the 'nugget e f f e c t ' . For qu a s i - s t a t i o n a r i t y the ' s i l l ' i s equal to a p o s i t i v e , f i n i t e variance minus the 'nugget e f f e c t ' . The time separation or 'lag' at which the increasing semi-variogram function i n t e r s e c t s the ' s i l l ' i s known as the 'range'. From t h i s point of in t e r s e c t i o n the model i s constant and equals the ' s i l l ' and the auto-covariance i s greater than or equal to zero. I f only the i n t r i n s i c or q u a s i - i n t r i n s i c hypothesis i s s a t i s f i e d and the semi-variogram function i s c o n d i t i o n a l l y p o s i t i v e d e f i n i t e , the model w i l l continuously increase as the size of the time separation increases. The variance of the metal pr i c e s i s dependent on the time separation and not any p o i n t i n time. Therefore no ' s i l l ' e x i s t s i n the semi-variogram. 3.3.1 Spherical Model Under the conditions of a semi-variogram reaching a ' s i l l ' the behaviour of the f u n c t i o n can be c h a r a c t e r i z e d by a 'spherical model' (FIGURE 1). 22 r ( t> = c 3 t -2 a Co for t < a, = C + Co for t > a. where; Y(t) = the semi-variogram function at time separation t. C = the ' s i l l ' , a p o s i t i v e variance or variance of the metal prices minus the 'nugget e f f e c t ' , a = the 'range' of continuity and r e g u l a r i t y , the 'lag' at which the semi-variogram reaches a ' s i l l ' . t = the time separation or 'lag'. Co = the 'nugget e f f e c t ' whose value i s equal to the semi-variogram function at 'lag' 0. 3.3.2 Linear Model In the event the semi-variogram does not reach a " s i l l ' and no trend or d r i f t i s apparent the behaviour of the metal prices can be characterized by a 'linear model' (FIGURE 2). X(t) = m t + Co where; Y(t) = the semi-variogram function at time separation t. m = the slope of the l i n e through the points of the semi-variogram. t = time separation or 'lag'. Co = 'nugget e f f e c t ' whose value i s equal to the semi-variogram function at 'lag' 0. 2 3 T H E O R E T I C A L S P H E R I C A L S E M I - V A R I O G R A M FIGURE 1. I 2 3 4 3 * 7 8 Lagiyears Theoretical SPHERICAL semi-variogram with a SILL (C) = 9.0 (Price)1, a NUGGET EFFECT (Co) = 1.0 (Price)', and a RANGE (a) = 6.0 years. T H E O R E T I C A L L INEAR S E M I - V A R I O G R A M t 6 Slope — i i i i i i i i i "1 . . • 1 c 0 1 2 3 4 S 6 7 « 9 10 r i r i l P T 9 Log .yeors R I V J U K L Z. Theoretical LINEAR semi-variogram with a NUGGET EFFECT (Co) = 1.0 (Price) 1, and a SLOPE = 1.5 (Price)'/Year. P U R E N U G G E T E F F E C T S E M I - V A R I O G R A M - 9 a. ~ 6 Experimental Semi-variogram Population Variance FIGURE 3. 1 1 3 4 . 9 6 7 S 9 10 Lag iyaar t PURE NUGGET EFFECT semi-variogram represented by a horizontal line equal to the population variance of the metal prices. 24 3 . 3 . 3 Pure Nugget E f f e c t If the semi-variogram reaches a ' s i l l ' at less than one lag the 'nugget e f f e c t ' i s equal to the variance of the metal prices i n a time s e r i e s . This p r i c e behaviour i s a 'Pure Nugget E f f e c t ' (FIGURE 3 ) . 7 ( t ) = Co for a l l t = 1, . . . ,n. where; X ( t ) = the semi-variogram function at time separation t. Co = the 'nugget e f f e c t ' whose value i s equal to the variance of the metal prices, t = the time separation or 'lag'. This model implies an absence of c o r r e l a t i o n between prices f o r a l l a v a i l a b l e time separations. Further, the lack of c o r r e l a t i o n indicates the independence of the mean and variance of the metal p r i c e s with respect to the time separations. Therefore i n t h i s case, the best estimate of a future price i s the mean of the metal prices . 3 . 4 Kriging K r i g i n g i s a technique used to f i n d the best l i n e a r unbiased estimate of a regionalized variable such as the price of a metal at a point i n time using the temporal s e l e c t i o n of nearby known points i n time and t h e i r p r i c e s . To make the best l i n e a r estimate the kr i g i n g estimator must be unbiased and the estimation variance minimized. Both conditions are achieved through the assignment of a weighting factor to each known price used i n the kri g i n g neighbourhood. The weighting factors are r e l a t e d to the time separation of each known p r i c e i n the 2 5 neighbourhood with the point i n time being k r i g e d . The behaviour of the prices as a function of time i s defined by the model of the experimental semi-variogram of the metal price s e r i e s . 3.4.1 Kriging Estimator The k r i g i n g estimator i s a l i n e a r function of the prices i n a time series representing the kr i g i n g neighbourhood. n X ± P(x ± ) i= l where; Z*k = the kr i g i n g estimator for point x k i n time being kriged. n = the number of prices i n the k r i g i n g neighbourhood. X ± = the weighting factor applied to price x± . P(x ±) = the price at time (x ±) i n the k r i g i n g neighbourhood. The k r i g i n g neighbourhood i s defined i n part by the semi-variogram, but also somewhat a r b i t r a r i l y to provide a sound basis for estimation. The condition of unbiasedness implies that the estimated p r i c e on average should be equal to the true p r i c e for a point i n time. The condition of no bias i s achieved by requiring the sum of the weighting factors to be equal to 1. 2 6 3.4.2 Ordinary Kriging To minimize the estimation variance the weighting factors are chosen subject to the unbiased condition. The estimation v a r i a n c e equations can be w r i t t e n as a system of l i n e a r equations, the ordinary k r i g i n g system or OK, incorporating a Lagrangian m u l t i p l i e r related to the unbiasedness condition that a l l weights sum to 1 . n ]T Xi y ( P ( X I ) ; P ( X J ) ) + fi = y(P(x ± );P(x k )) J = 1 for i = 1 , . . n. where; X ± = the weighting factor for the price at time x± . Y( ) = the semi-variogram function which models the experimental semi-variogram of the pr i c e s i n a time s e r i e s . ( P ( ) ; P ( )) = two prices separated by time period t defined by times x ± , Xj. , and x k ; i e . X^ — X j — "t • H- = the Lagrangian m u l t i p l i e r used to induce unbiasedness. n = the number of prices i n the k r i g i n g neighbourhood. In matrix notation the ordinary k r i g i n g system i s defined as [ A ] [ B ] = [ C ] . 27 n £ X (P(x ± );P(Xj ) ) = [ A ] for i = 1, . . ., n. 3=1 [A] = y ( P ( X l );P( X l )) . . . / ( P ( X l );P(x n )) 1 • • • 7(P(x n );P( X l )) . . . T(P(x n );P(x n )) 1 1 . . . 1 0 n ]T A j +^t = [ B ] and y(P(x ± );P(x k )) + 1 = [ C ] 3=1 for i 1, n. [ B ] = [ C ] = 7(P( X l );P(x k )) ? ( P ( x n );P(x k )) 1 3.4.3 Universal Kriging To accommodate d r i f t i n a variable, additional equations must be incorporated into the kr i g i n g system to produce what i s known as the universal k r i g i n g system or UK. For a polynomial d r i f t of order m the kr i g i n g system i s comprised of n equations re l a t e d to the number of prices i n the k r i g i n g neighbourhood plus m equations describing the polynomial d r i f t of order m using Lagrangian m u l t i p l i e r s . The d r i f t i s assumed i n that a polynomial of order m i s f i t t e d to the pr i c e s e r i e s . The n equations are related to the t h e o r e t i c a l semi-variogram function 2 8 modelling the experimental semi-variogram of the residuals of the d r i f t . The residuals are the r e s u l t of subtracting the prices f i t t e d using the polynomial curve of degree n from the known pr i c e s . The Lagrangian m u l t i p l i e r related to the unbiased condition that a l l weights sum to 1 also i s included. The universal k r i g i n g system i s : n m n ]T \ ± y(P(x ± );P(x. )) + fjL0 + Y, X ( X J } 1 = j=l 1=1 j=l m y(P(x. );P(x k )) + 1 + Y <xk y 1 = 1 for i = 1, . . ., n. where the additional components are: fJL± = Lagrangian m u l t i p l i e r s for the polynomial d r i f t of degree m where 1 = 0 , . . ., m. (x.. J 1 = time X ; j to the power 1. (x k ) 1 = time x k to the power 1. 1 = the power of the polynomial of degree m with a range of 0, . . ., m. In matrix notation t h i s universal k r i g i n g system i s defined as [ A ] [ B ] = [ C ] where: n m n y ( p ( X . );p( X.)) + i + y V" ( X j ) x = [ A ] ^ P x. P(x  )  1 £ £j=l 1=0 j=l for i = 1, . . ., n. 2 9 [A] = y(P(x x );P(x 1 )) ... y ( p ( X l );P(x n )) l ( x ^ 1 ... ( X l y X(P(x n );P( X l )) ... y(P(x n );P(x n )) 1 ( x ^ 1 1 ... 1 0 0 ( x ^ 1 ... ( x n ) 1 0 0 (x x )" 0 0 0 0 ... 0 n m m £ Xj + ]T /xl = [B]: y(P(x ± );P(x k )) + 1 + £ = [C] j=l 1=0 1 = 1 for i= 1, . . ., n. [ B ] = \x [ C ] = y C P ^ );P(x k )) y(P(x n );P(x k )) 1 To use u n i v e r s a l k r i g i n g the o r i g i n of the coordinate system i s s h i f t e d to the time being kriged. The coordinate x k i n [C] then i s zero. The coordinates x x to x n i n [A] are then referenced to the o r i g i n x k. 3 . 4 . 4 Matrix Solution and Estimation Variance Since [ A ] and [ C ] are known the solu t i o n for [ B ] i s 3 0 [ B ] = [ A ] - 1 [ C ]. The weighting factors X ± , . . . , X n i n vector [ B ] are chosen optimally due to the condition of unbiasedness, minimization of the estimation variance and the semi-variogram function modelling the behaviour of the known variable such as prices i n a metal price time s e r i e s . The variance of estimation can be found by subs t i t u t i n g the derived values into the equations: - Ordinary Kriging n E V x k = = " 7(P(x k );P(x k )) + Y, X i r(P(x ± );P(x k )) + p i = l - Universal Kriging n m E v x k =<x\ = Y, x j );p(x k)) + A l o + Y / M V ) 1 j=i 1=1 where: EVx k = the variance of estimation of the point x k i n time. 31 4. DATA AND METHODOLOGY Data used i n the analysis of metal price behaviour and the forecasting of future metal prices consists of metal price time serie s and deflators from Great B r i t a i n . The time series and the metals investigated are annual average prices of monthly average pr i c e s for copper, lead and zinc from the London Metal Exchange f o r the years 1884 to 1986. A wholesale,producer price index i s used to deflate the current metals p r i c e series to constant or r e a l p r i c e s e r i e s . The methodology followed for ordinary k r i g i n g of each metal p r i c e time seri e s was to deflate current prices to constant p r i c e s . Then an experimental semi-variogram was calculated from each constant metal prices s e r i e s . If the experimental semi-variogram did not exhibit a "Pure Nugget E f f e c t " a t h e o r e t i c a l semi-variogram function was f i t t e d . A f i t t e d model indicated s t r u c t u r a l c o r r e l a t i o n between prices i n the time seri e s and i t then was used to krige future p r i c e s . These future prices were compared to deflated known prices and future prices predicted from a simple random walk. For universal k r i g i n g a polynomial curve was f i t t e d to the each of the constant p r i c e s e r i e s . Then a experimental semi-variogram was calculated f or each of the residual p r i c e s e r i e s (known price minus curve f i t t e d price) and the process continued as for ordinary k r i g i n g . 4.1 Data Metal p r i c e s i n Great B r i t a i n have undergone not only market pressures, but pressures r e l a t e d to devaluation and 32 r e s t r u c t u r i n g of the pound s t e r l i n g and changes i n the units of measurements and metal grades. The pound s t e r l i n g was devalued i n 1931, 1949 and 1967 and i n 1972 was restructured so that 100 new pence made a pound versus 240 old pence i n a pound. In 1970 prices quoted for copper, lead and zinc were changed from pounds s t e r l i n g per long ton to pounds s t e r l i n g per metric ton (2240 lbs. per long ton and 2204.6 lbs per metric ton). The data set of current prices r e f l e c t t h i s adjustment and are quoted i n pounds s t e r l i n g per metric ton. The grades of the metals for which prices were quoted on the London Metal Exchange have changed due to varying demands and supplies. - Copper: 1884 to 1888, Chile bars, spot quotation. : 1888 to 1902, Good Merchantable Brands, at least 94% copper. : 1902 to 1940, Standard Copper, rough copper assaying 99%. : 1940 to 1953, E l e c t r o l y t i c Copper, Ministry of Supply maximum pr i c e . : 1953 to 1962, Standard Copper, e l e c t r o l y t i c wire bar, monthly average settlement prices asked quotation for spot copper. : 1962 to 1986, E l e c t r o l y t i c Wire Bar. Lead : 1884 to 1940, Good Soft Pig Lead, spot quotation : 1940 to 1953, Soft Foreign Lead, Ministry of Supply , maximum pr i c e . : 1953 to 1961, Refined Pig Lead, assaying 99.97%. : 1961 to 1986, spot asked cash quotation. - Zinc : 1884 to 1940, Good Ordinary Brands, assaying 98%, spot quotation V i r g i n zinc. : 1940 to 1953, Ministry of Supply maximum pric e . : 1953 to 1961, V i r g i n zinc, assaying 98% zinc. : 1963 to 1986, spot quotation. Sources : American Bureau of Metal S t a t i s t i c s , Yearbooks. See APPENDIX 1 TABLE 1. Current Metal Prices. For Great B r i t a i n the deflator used on an annual basis i s an average of monthly averages. The wholesale prices index used f o r the p e r i o d 1884 to 1954 i s the Board of Trade a l l commodities index. In 1954 t h i s index was s p l i t into an index f o r output manufactured goods, home sales and an index for materials and fuel s purchased. For the period 1954 to 1986 on an annual basis the wholesale or producer p r i c e index for output of manufactured goods, home sales was used as the def l a t o r . Sources : M i t c h e l l B. R., 1980. : Central S t a t i s t i c a l O f f i c e ; Annual Abstract of S t a t i s t i c s . See APPENDIX 1 TABLE 1. Wholesale,Producer Price Index. 4.2 Methodology The same methods were used to analyze price behaviour and to p r e d i c t p r i c e s f o r each metal and time s e r i e s . They consisted of FORTRAN IV computer programs written by the author to deflate p r i c e data, calculate experimental semi-variograms and to krige future p r i c e s . The kri g i n g programs used the Univer s i t y of B r i t i s h Columbia's Computing Centre's subroutine FSLE to solve a system of l i n e a r equations i n matrix form. For universal k r i g i n g the assumed d r i f t was f i t t e d to the metals pr i c e s e r i e s using ordinary least squares, the OLSF subroutine from the Computing Centre. P l o t t i n g of the price data and semi-variograms was by the li c e n s e d program TELLAGRAF, an ISSCO product, and used the Computing Centre's QMSPLOT routine. A l l computations were performed on the U n i v e r s i t y of B r i t i s h Columbia's Amdahl V8 computer. Using the wholesale,producer price index a l l annual prices were deflated to constant 1984 prices. The equation used was: DPx± = Px± / ( Dx± / 100.00 ) for i = 1, . . .,n. where; DP X i = the deflated or constant price for time X i . Px± = the current price for time x ±. Dx± = the def l a t o r for time X i using as a basis 1984 = 100. n = the number of prices i n the time s e r i e s . The current prices were deflated to account for the v a r i a b i l i t y i n the "worth" of the pound s t e r l i n g from year to year. This d e f l a t i o n provides a constant support over the study period. An experimental semi-variogram was c a l c u l a t e d f o r each metal for deflated time series and the residuals from f i t t i n g a polynomial curve. The metal price time series used were: annual deflated copper, lead and zinc prices for 1884 to 1986. The experimental semi-variogram for each price s e r i e s was plotted by hand and i f a "Pure Nugget E f f e c t " was not exhibited a spherical or l i n e a r t h e o r e t i c a l semi-variogram function v i s u a l l y f i t t e d . From these models the parameters of 'nugget e f f e c t ' , ' s i l l ' , 'range' and/or 'slope' were taken. Two k r i g i n g methods were used to estimate p r i c e s for comparison with estimates from a simple random walk. These methods were; 1. Ordinary Kriging (OK). 2 . Universal Kriging (UK). Ordinary k r i g i n g assumed at least the q u a s i - i n t r i n s i c hypothesis applied whereas u n i v e r s a l k r i g i n g incorporated a d r i f t , the function of which must be assumed. Deflated prices were the basis f o r estimation. The simple random walk was based on the following; Pt - Pt _ ! = £ t where: t = a point i n time. t - 1 = a point i n time at t minus 1 time period. Pt = the pr i c e at time t. Pt - i = "the pr i c e at time t minus 1 time period. £ t = "the difference i n prices, such that E(£ t ) = 0 : the expectation of the difference i n price i s zero, and E(£ t,£ t _ x ) = 0 : the expectation of the c o r r e l a t i o n of price differences i s zero. This implied that the p r i c e differences are independent and i d e n t i c a l l y d i s t r i b u t e d . T h e r e f o r e , today's p r i c e was considered as the best estimate of tomorrow's pr i c e with the d i f f e r e n c e between the p r i c e s the e r r o r of estimation, the expectation of which i s zero. To t e s t the c a p a b i l i t y of g e o s t a t i s t i c s to estimate future prices a comparison was made between the kriged future prices 3 6 and the random walk prices using the sum of squared differences with respect to known prices. 1.0 -X ( P K i " P G I i = l i = l 100 = R where: i P K i G i RW i = time, for i = 1,...,T periods. = known metal price at time i . = kriged price at time i . = random walk price at time i . 2 i = l k r i g i n g residual sum of squares. z i = l ( P K i - P R W i ) = random walk residual sum of squares. R = percent r a t i o of residual sum of squares. A p o s i t i v e R indicated that k r i g i n g i s a better estimator of future p r i c e s than the random walk. A negative R s i g n i f i e d that k r i g i n g cannot estimate future p r i c e s better than a simple random walk. 37 5. RESULTS AND DISCUSSION The objective of t h i s thesis was to estimate future metal prices by g e o s t a t i s t i c a l methods. To do t h i s an experimental semi-variogram was calculated for each metal constant price time serie s and each residual constant price series from polynomial curve f i t t i n g . For each experimental semi-variogram not showing a 'Pure Nugget E f f e c t ' a t h e o r e t i c a l semi-variogram function was v i s u a l l y f i t t e d to model the s t r u c t u r a l c o r r e l a t i o n of the prices i n that time s e r i e s . The model was then used i n kriging future p r i c e s . To evaluate the a b i l i t y of g e o s t a t i s t i c s to estimate future p r i c e s the kriged future p r i c e s were compared to predicted prices determined from a simple random walk. The comparison was based on the residual sum of squared differences between the known prices and the estimated future prices for the time period analyzed. 5.1 Copper V i s u a l inspection of the London Metal Exchange constant copper pr i c e s (FIGURE 4) suggested the 1884 to 1986 time series can be divided into three separate stationary regimes. The proposed regimes were for the time periods 1884 to 1917, 1918 to 1953 and 1954 to 1986. Supporting t h i s d i v i s i o n were market f a c t o r s and the need f o r s t a t i o n a r i t y f o r g e o s t a t i s t i c a l applications. For the period 1884 to 1917 the market forces represented were those induced by expectations of a new century and by World 38 LONDON METAL EXCHANGE ANNUAL COPPER PRICES 3500 3000-c " 2500 L. • ^ 2000 CO c co *» c 3 o o_ 1500 1000 500-C u r r e n t Pr ices Constant Pr ices FIGURE 4 . I""" i i r i i | M i i i n i i i . i i i i i | I., i n . I. | , , , , , , , , , , | 1884 1901 1918 1935 1952 1969 1986 Years L o n d o n Meta l E x c h a n g e annua l a v e r a g e of m o n t h l y C O P P E R p r i c e s for 1884 to 1986 in c o n s t a n t a n d c u r r e n t p o u n d s s t e r l i n g / m e t r i c ton . The c u r r e n t p r i c e s a re d e f l a t e d to c o n s t a n t p r i c e s u s i n g a w h o l e s a l e , p r o d u c e r p r i c e i ndex wi th the b a s e 1984=100 . EXPERIMENTAL SEMI-VARIOGRAMS OF LME COPPER PRICES 1000000q 900000-j CM c H2 800000^ trie 700000^ e 600000^ c 500000-: ferll 400000^ CO • -o 300000^ (Pourr 200000-j 100000-1884 to 1917 . « 1917 to 1953 . 1954 to 1986 -/ : A / \ \ v / / \ —r-0 5 10 15 20 L a g : y e a r s 5. E x p e r i m e n t a l s e m i - v a r i o g r a m s of LME c o n s t a n t a n n u a l C O P P E R p r i c e s fo r 1884 to 1917, 1918 to 1953 a n d 1954 to 1986. They i n d i c a t e n o n - s t a t i o n a r i t y in the 1884 to 1986 c o n s t a n t c o p p e r p r i c e t ime s e r i e s in t h a t t h r e e r e g i m e s c a n be d i f f e r e n t i a t e d . 39 ~r~ 15 i 25 —I 30 FIGURE War I. The period 1918 to 1953 was i n i t i a t e d by the aftermath of World War I, sustained by the Great Depression and ended by the elimination of government control of the trading of metals that was prompted by World War II and the Korean War. From 1954 to 1986 post-war expansion and i n f l a t i o n a r y action and reaction d o m i n a t e d t h e market f o r c e s . C o n t r i b u t i n g t o t h e d i f f e r e n t i a t i o n of t h i s p e r i o d was the r e v i s i o n of the w h o l e s a l e , p r o d u c e r p r i c e i n d e x f o r G r e a t B r i t a i n and implementation of the new index i n 1954. The v i s u a l inspection of the 1884 to 1986 constant copper prices indicated a lack of s t a t i o n a r i t y i n the time series as trends were v i s i b l e i n the mean and variance. Experimental semi-variograms were calculated for the three regimes and when p l o t t e d showed non- s t a t i o n a r i t y i n that t h e i r semi-variances were s i g n i f i c a n t l y d i f f e r e n t (FIGURE 5). This coupled with the reasons for d i f f e r e n t i a t i n g the regimes allowed the three time periods to be treated separately. 5.1.1 Copper: Variography; Results The experimental semi-variogram (FIGURE 6, APPENDIX 2 TABLE 1) of the constant copper p r i c e s e r i e s f o r 1884 to 1917 increased continuously to a point (lag 4) where a constant l e v e l was assumed. A spherical semi-variogram function was f i t t e d to model the s t r u c t u r a l c o r r e l a t i o n of prices up to a lag of 4 years. The experimental semi-variogram showed a hole e f f e c t where the semi-variances formed a trough a f t e r reaching the peak at a lag of 4 years. Because the trough was r e l a t i v e l y shallow 40 a spherical semi-variogram function was though to be an adequate model. The parameters of the f i t t e d spherical model are: SILL C = 162819.8 (pounds sterling/tonne) 2 NUGGET EFFECT Co = 37500.0 (pounds sterling/tonne) 2 RANGE a 4.0 years. The s t r u c t u r a l r e l a t i o n s h i p between prices i n the 1918 to 1953 c o n s t a n t copper p r i c e s e r i e s was expressed i n the experimental semi-variogram (FIGURE 7, APPENDIX 2 TABLE 2). To model the re l a t i o n s h i p a spherical semi-variogram function was f i t t e d , representing c o r r e l a t i o n between prices f o r a range of 12 years. A minor hole e f f e c t was noted, but was not believed to s i g n i f i c a n t l y i n f l u e n c e the mod e l l i n g because of i t s shallowness. The parameters of the f i t t e d spherical model are: SILL C = 105500.0 (pounds sterling/tonne) 2 NUGGET EFFECT Co = 7500.0 (pounds sterling/tonne) 2 RANGE a = 12.0 years. The experimental semi-variogram of the constant copper pr i c e s e r i e s f or 1954 to 1986 (FIGURE 8, APPENDIX 2 TABLE 3) showed that the prices i n the time series were quite variable. This was evident as the f i t t e d spherical semi-variogram function had a large s i l l value. The parameters of the f i t t e d spherical model are: SILL C = 805818.7 (pounds sterling/tonne) 2 NUGGET EFFECT Co = 0.0 (pounds sterling/tonne) 2 RANGE a = 13.8 years. 41 400000-1 CN *—v 330000-C o «s 300000-• < 230000-N • o> c 200000-w • 190000-CO M TJ 100000-c 3 -O O. 30000-w 0-S E M I - V A R I O G R A M O F L M E 1884 T O 1917 C O P P E R PRICES Semi-variogram Spherical Function 12 13 It 21 24 27 30 F i n i P F R L a g . y e a r . r ivjurcr. O. - r n e experimental semi-variogram can be fitted by a SPHERICAL f unction,SILL(C) = 162819.8(3/torine)1,NUGGET EFFECT(Co)=37500.0(S/tonne)' and RANGE(a) = 4years. S E M I - V A R I O G R A M O F L M E 1918 T O 1953 C O P P E R PRICES 210000-1 CM C 180000-o t-rle 190000-• 120000-o> c 9 0000-• CO M 60000-T> C 3 O a. 30000-0-S«m]-vorlogram Spherical Function FIGURE ' • The experimental semi-variogram can be fitted by a SPHERICAL function,SILL(C) = 105500.0(s/tonne)',NUGGET EFFECT(Co)=7500.0(£/tonne)J and RANGE(a) = 12years. S E M I - V A R I O G R A M O F L M E 1954 T O 1986 C O P P E R PRICES 5.1.2 Copper: Variography; Discussion The f i t t i n g of a spherical semi-variogram function to an experimental semi-variogram indicates that there i s a zone of c o r r e l a t i o n around the price at each point i n time. Within t h i s zone p r i c e s are c o r r e l a t e d and beyond the zone p r i c e s are uncorrelated. The degree of c o r r e l a t i o n i s re l a t e d to the 'range' of the zone of influence and the l e v e l or ' s i l l ' at which the t r a n s i t i o n from being correlated to being uncorrelated occurs. Furthermore, as the annual prices used i n t h i s thesis are a v e r a g e s of monthly p r i c e s the c o n t i n u i t y of the r e g u l a r i z a t i o n from monthly to annual prices i s expressed by the 'nugget e f f e c t ' . The 'range' of 4 years of the spherical model for the 1884 to 1917 copper constant price time series indicates that the pric e s r a p i d l y become more variable as t h e i r time separation (lag) increases (FIGURE 6). This v a r i a b i l i t y i s moderate as shown by the ' s i l l ' value (162819.8 (pounds ste r l i n g / t o n n e ) 2 ) that i s r e l a t e d to the v i s i b l e differences between the peaks and troughs of the constant price series (FIGURE 4). These peaks and troughs also seem to be somewhat systematic as v e r i f i e d by the hole e f f e c t i n the experimental semi-variogram. The large 'nugget e f f e c t ' (37500.0 (pounds ste r l i n g / t o n n e ) 2 ) s i g n i f i e s a di s c o n t i n u i t y i n the r e g u l a r i z a t i o n of the prices that may be due to seasonal v a r i a b i l i t y i n the monthly prices and/or a v a r i a b i l i t y i n monthly p r i c e s from year to year related to inconsistencies i n the copper supply. In contrast, the 'range' of 12 years of the spherical model for the 1918 to 1953 copper constant price time series shows 43 that the rate at which the prices become more variable i s slower (FIGURE 7). The v a r i a b i l i t y reached at t h i s 'range' i s also lower as expressed by the smaller ' s i l l ' value (105500.0(pounds st e r l i n g / t o n n e ) 2 ) and the lack of exaggerated peaks and troughs i n the constant p r i c e time series (FIGURE 4). The peaks and troughs that are apparent give r i s e to a subtle hole e f f e c t as shown i n the experimental semi-variogram (FIGURE 7). A small 'nugget e f f e c t ' shows the influence of government control, 1939 to 1953, and a very low demand for copper, 1930 to 1937. S i m i l a r to the s p h e r i c a l model f o r 1918 to 1953 time ser i e s , the 'range ' of the 1954 to 1986 copper constant price time seri e s spherical model i s large being 13.8 years (FIGURE 8) and implies that the rate at which prices become more variable i s high. Unlike the 1918 to 1953 time series, the v a r i a b i l i t y reached at the 'range' of t h i s time s e r i e s i s high as i s conveyed by the l a r g e ' s i l l ' v alue of 805818.7 (pounds sterl i n g / t o n n e ) 2 and as v i s i b l e i n the copper constant price time s e r i e s (FIGURE 4). The zero 'nugget e f f e c t ' for t h i s time s e r i e s suggests no seasonal e f f e c t s and a consistent copper supply. Though, the absence of a 'nugget e f f e c t ' i s possibly a function of the f i t t e d spherical model and the method of f i t t i n g the model. The c l a s s i f i c a t i o n of the v a r i a b i l i t y between prices i n the three regimes, 1884 to 1917; moderate, 1918 to 1953; low, and 1954 to 1986; high, supports the contention that the 1884 to 1986 copper constant price time series i s non-stationary. The three regimes are treated as being stationary. 44 5.1.3 Copper: Kriging; Results 1884 to 1917 A spherical semi-variogram function was used to model the s t r u c t u r a l c o r r e l a t i o n of prices within the 1884 to 1917 copper constant p r i c e time series (FIGURE 6). This model and kriging neighbourhoods of 4 and 5 years were used i n ordinary k r i g i n g to estimate prices for the years 1885 to 1917. A neighbourhood of 5 years was used to t e s t whether a d d i t i o n a l p r i c e s i n the k r i g i n g neighbourhood would give better estimates than the 'range' of 4 years of the model. The known p r i c e s , the kriged p r i c e estimates and t h e i r squared d i f f e r e n c e s and the random walk pr i c e estimates and t h e i r squared differences with the known prices are presented i n APPENDIX 3 TABLE 1. The sums of the squared differences and t h e i r percent r a t i o s , kriged versus random walk, are shown i n TABLE 1, A. With an apparent p e r i o d i c i t y of 6 years shown by the experimental semi-variogram of the 1884 to 1917 copper price time s e r i e s , a hole e f f e c t was used i n k r i g i n g estimates of future p r i c e s for t h i s price time s e r i e s . The hole e f f e c t was represented by semi-variances of the experimental semi-variogram for lags up to 8 years (FIGURE 6, APPENDIX 2 TABLE 1). These semi-variances were used as the elements of the ordinary kriging matrix f o r k r i g i n g neighbourhoods of 6, 7 and 8 years. Weighting factors were determined for the neighbourhood prices and future prices estimated. The kriged future price estimates using t h i s hole e f f e c t procedure and t h e i r squared differences with the known prices 4 5 are presented i n APPENDIX 3 TABLE 2. The sums of the squared differences and t h e i r percent r a t i o s are shown i n TABLE 1, A. The sums of squared differences f o r the 1884 to 1917 copper constant p r i c e time series (TABLE 1, A) showed three things: 1. Ordinary k r i g i n g gave better future price estimates for the years 1885 to 1917 than the simple random walk estimates, percent r a t i o s > + 12%. 2. Using the hole e f f e c t improved the estimation of future p r i c e s . 3. The r e s u l t s showed that the more prices i n the k r i g i n g neighbourhood the better the estimates of future prices over the 33 year period. 5.1.4 Copper: Kriging; Results 1918 to 1953 The s t r u c t u r a l c o r r e l a t i o n between prices i n the 1918 to 1953 copper constant p r i c e time s e r i e s was modelled by a spherical semi-variogram function (FIGURE 7). This model was used by ordinary k r i g i n g to estimate future prices for the years 1919 to 1953. Two k r i g i n g neighbourhoods were used, 12 and 13 years, with the l a t t e r t e s t i n g whether additional prices i n the k r i g i n g neighbourhood would give better estimates than the model 'range' of 12 years. Known prices and the kriged price estimates, t h e i r squared differences and the squared differences of the random walk price estimates and the known prices are presented i n APPENDIX 3 TABLE 3. The sums of the squared differences and t h e i r percent r a t i o s are shown i n TABLE 1, B. 46 The r e s u l t s of comparison between the sums of the squared d i f f e r e n c e s f o r ordinary k r i g i n g future p r i c e estimates and random walk pr i c e estimates (TABLE 1, B) for the 1918 to 1953 copper constant price time series were opposite to the res u l t s from the 1884 to 1917 se r i e s . 1. Ordinary k r i g i n g estimated future metal prices marginally poorer than the simple random walk for the years 1919 to 1953, percent r a t i o s < - 6.0%. 2. Estimates of future prices for the time seri e s 1919 to 1953 were not improved when a k r i g i n g neighbourhood larger than the 'range' of the spherical model was used. 5.1.5 Copper: Kriging; Results 1954 to 1986 A spherical semi-variogram function was used to model the s t r u c t u r a l c o r r e l a t i o n of prices i n the 1954 to 1986 copper constant p r i c e time series (FIGURE 8). Ordinary k r i g i n g used t h i s model and two kr i g i n g neighbourhoods, 4 and 14 years, to estimate future prices for the years 1955 to 1986. A kri g i n g neighbourhood of 4 years was used to tes t whether the size of the k r i g i n g neighbourhood g r e a t l y influenced the a b i l i t y of k r i g i n g to give better future price estimates than a simple random walk. The k r i g i n g r e s u l t s , the known prices and t h e i r squared differences are presented i n APPENDIX 3 TABLE 4. Also i n t h i s table are the random walk future price estimates and the squared differences between them and the known prices. The sums of the 4 7 squared differences and t h e i r percent r a t i o s are shown i n TABLE 1, C. Universal k r i g i n g was also undertaken for the 1954 to 1986 copper constant price time s e r i e s . A polynomial curve of order 2 (quadratic) was f i t t e d to the prices of t h i s time series (FIGURE 9). Residuals, known p r i c e minus the curve f i t t e d p r i c e , were calculated (FIGURE 10, APPENDIX 4 TABLE 1) and an experimental semi-variogram of the r e s i d u a l s was determined (FIGURE 11, APPENDIX 2 TABLE 4). A spherical semi-variogram function was f i t t e d to the experimental semi-variogram (FIGURE 11). SILL C = 333873.0 (pounds sterling/tonne) 2 NUGGET EFFECT Co = 60000.0 (pounds sterling/tonne) 2 RANGE a 7.0 years. The model and m = 2 equations were used by universal k r i g i n g to estimate prices for the years 1969 to 1986. This time series, 1969 to 1986, was chosen because i t was f e l t that with a 'range' of 7 years to adequately model a quadratic curve at l e a s t double the range was need f o r the k r i g i n g neighbourhood. Also a k r i g i n g neighbourhood of 15 years represented almost one-half of the t o t a l number of p r i c e s i n the time s e r i e s . Having determined a model and selected a k r i g i n g neighbourhood the procedure to estimate future prices was the same as that for ordinary k r i g i n g . The kriged future price estimates, the known prices and t h e i r squared differences and those of the simple random walk are presented i n APPENDIX 3 TABLE 5. The sums of the squared 48 L M E 1954 T O 1986 A N N U A L C O N S T A N T C O P P E R PRICES 3600 o 3000 . 2400 CO c leoo * * 1200 -a c 3 o eoo Constant Prices Quadrat ic Curve 1993 FIGURE 9. •—i—1—•—i—•—•—i—•—>—i—•—•—i—'—'—i—•—'—i—•—'—i—'—•—i—'—•—i—•—• 1996 1939 1962 1969 1968 1971 1974 1977 1980 1983 Years London Metal Exchange constant 1954 to 1986 COPPER prices fitted with a quadratic curve, -30575.2 + 785.7 T - 4.7 T*. R E S I D U A L 1954 T O 1986 L M E C O P P E R PRICES GO in «• •o o -1000 1993 FIGURE 10. 1996 1962 1974 1977 1980 1983 1969 1968 1971 Year* The constant COPPER price residuals for 1954 to 1986, the known prices minus the quadratic curve prices. S E M I - V A R I O G R A M O F L M E 1954 T O 1986 C O P P E R PRICES 900000-1 Semi-variogram . Spherical Function -t— FIGURE 11. Semi-variogram of the residuals fiHed by a SPHERICAL f unction,SILL(C) = 805818.7(3/ionne)1,NUGGET EFFECT(Co)=0.0(ytonne)» and RANGE(a) = 13.8y«ars. 49 differences for the time series 1969 to 1986 and the percent r a t i o s are shown i n TABLE 1, C. A polynomial curve of order 2 (quadratic) was also f i t t e d to 1973 to 1986 constant copper prices of the 1954 to 1986 time s e r i e s (FIGURE 12). This was to see whether a short trend determination would improve the future price estimation. The residuals were determined (FIGURE 13, APPENDIX 4 TABLE 2) and an experimental semi-variogram calculated (FIGURE 14, APPENDIX 2 TABLE 5). This experimental semi-variogram evinced a 'Pure Nugget E f f e c t ' which implied that the best estimate of a future price was the price determined from the quadratic equation. Prices estimated from the quadratic equation for 1985 and 1986, the known prices and t h e i r squared differences and the random walk estimates and the related squared differences are shown i n APPENDIX 3 TABLE 6. The sums of the squared differences and t h e i r percent r a t i o s are shown i n TABLE 1C. The r e s u l t s of the sums of squared differences comparisons between the random walk, ordinary k r i g i n g and universal k r i g i n g future p r i c e estimates (TABLE 1, C) for the 1954 to 1986 copper constant price time series were: 1. Ordinary k r i g i n g gave a better estimate of future prices than the simple random walk and universal kriging, percent r a t i o s > + 0.7%. 2. Better estimates of future prices occurred when a large k r i g i n g neighbourhood was used, 2 the 'range'. 3a. Universal k r i g i n g with a quadratic curve f i t t e d to the 1954 to 1986 time ser i e s gave the worst 50 L M E 1973 T O 1986 A N N U A L C O N S T A N T C O P P E R PRICES 3200-] 3000-c 2800-e t- 2600-V 2400-2200-• 2000-w c 1800-1600-u 1400-© 1/1 •• 1200-1000-T> c 800-3 o 600-a. 400-200-1972 FIGURE 1 2 . Conttant Priest Quadrat ic Curve 1974 1976 1978 19(0 1962 1984 Year * London Metal Exchange constant 1973 to 1986 COPPER prices fitted with a quadratic curve, 185572.6 - 3692.8 T + 18.5 T'. R E S I D U A L 1973 T O 1986 L M E C O P P E R PRICES 1986 400-i- 200-0> -200 -800 1972 FIGURE 1 3 . i • 1 • r 1974 1976 1976 I960 1982 1964 Year* The constant COPPER price residuals for 1973 to 1986, the known prices minus the quadratic curve prices. S E M I - V A R I O G R A M O F L M E 1973 T O 1986 C O P P E R PRICES 1986 d *— c o 140000 120000 T 100000 CO c o 1/1 80000-60000 . 40000 o 20000 Semi-var lo gram Population Variance FIGURE 1 4 . - 1 — 1 2 3 4 9 6 7 8 9 10 II 12 L a g i y e o r t Experimental semi-variogram of the residuals showing a PURE NUGGET EFFECT. 51 estimates for 1969 to 1986, percent r a t i o s < - 3 0 % . 3b. Universal k r i g i n g with a quadratic curve f i t t e d to the 1954 to 1986 time seri e s gave the best estimate of the future price for 1985 and 1986, percent r a t i o > + 75%. 4. The quadratic curve f i t t e d to the 1973 to 1986 copper constant price time series, the f i t t e d prices, gave the worst future estimates of the 1985 and 1986 prices, percent r a t i o s < - 60%. 5.1.6 Copper: Kriging; Discussion From the r e s u l t s i t can be suggested that ordinary k r i g i n g i s generally a better estimator of future p r i c e s than i s the simple random walk. Though i t appears that the a b i l i t y to estimate future prices i s rel a t e d to the v a r i a b i l i t y between pric e s i n the time s e r i e s . Where ordinary k r i g i n g appears to excel i s when the v a r i a b i l i t y between p r i c e s i s moderate, s i m i l a r to that of the 1884 to 1917 constant copper time series, to highly variable, the 1954 to 1986 time s e r i e s . In the case of a lower v a r i a b i l i t y between prices, the 1918 to 1953 time s e r i e s , the random walk estimates of f u t u r e p r i c e s are apparently better. The k r i g i n g neighbourhood i s an important aspect i n the a b i l i t y to estimate future p r i c e s . Being that k r i g i n g i s an averaging method, the l a r g e r the neighbourhood when the v a r i a b i l i t y between p r i c e s i s moderate to high the better ordinary k r i g i n g can estimate future p r i c e s . Averaging dampens 52 TABLE 1. *** SUMS OF SQUARED DIFFERENCES COPPER *** V a r i a b l e : COPPER, London Metal Exchange D e f l a t o r : WHOLESALE,PRODUCER PRICE INDEX 1984=100 Time S e r i e s : YEARLY, 1954 to 1986 TABLE A : 1885 to 1917 S p h e r i c a l Model RANDOM ORDINARY ORDINARY WALK KRIGING . KRIGING N = 4 N = 5 ( £ / m . t . ) 2 ( £ / m . t . ) 2 ( £ / m . t . ) 2 6247672.9 5476779.7 5307291.2 % RATIO +12.3 +15.0 Hole E f f e c t Model ORDINARY ORDINARY ORDINARY KRIGING KRIGING KRIGING N = 6 N = 7 N = 8 (£ / m . t . ) 2 ( £ / m . t . ) 2 ( £ / m . t . ) 2 5237646.0 5213216.5 5208069.3 +16.2 + 16.56 + 16.64 TABLE B : 1919 to 1953 S p h e r i c a l Model RANDOM ORDINARY ORDINARY WALK KRIGING KRIGING N = 12 N = 13 ( £ / m . t . ) 2 ( £ / m . t . ) 2 ( £ / m . t . ) 2 1324785.3 1405791.7 1430671.3 % RATIO - 6 . 1 - 8.0 TABLE C : 1955 to 1986 S p h e r i c a l Model YEARS RANDOM ORDINARY ORDINARY UNIVERSAL POLYNOMIAL WALK KRIGING KRIGING KRIGING CURVE N = 4 N = 14 N = 15 FITTING (£/m.t. ) 2 (£/m.t. ) 2 ( £ / m . t . ) 2 (£/m.t . ) 2 (£/m.t. ) 1 955-1968 3279083. 9 3246191. 4 3234317.5 — 1969-1984 4614828. 3 4592104. 1 4352199.3 6889939 .8 --1985-1986 39165. 6 38452. 6 53236.2 9351 .2 62898.8 SUM 7933077. 8 7876748. 1 7639753.0 6899291 .0 62898.8 % RATIO + 0.7 + 3.7 - 48. 2 - 60.6 N: K r i g i n g Neighbourhood; m.t.: M e t r i c Tons, 53 the v a r i a b i l i t y between prices and more prices used i n kriging increases the dampening e f f e c t . A suggestion from t h i s l i n e of thought i s that using experimental semi-variograms with low v a r i a b i l i t y a smaller k r i g i n g neighbourhood would give better estimates. From the above reasoning i t can be shown that the simple random walk used here i s a sp e c i a l case of ordinary kriging. The random walk says that the best estimate of tomorrow's price i s today's p r i c e . This occurs because only today's price i s viewed as influencing tomorrow's p r i c e . In the same way by l i m i t i n g the k r i g i n g neighbourhood to only one time period ordinary k r i g i n g produces a simple random walk. This occurs because of the condition that the sum of the weighting factors must equal 1; for examples note the 1885, 1919 and 1955 kriged future prices i n APPENDIX 3 TABLES 1, 3 and 4 respectively. The weighting factors given to the prices i n the k r i g i n g neighbourhood are determined from the continuous function f i t t e d to the experimental semi-variogram and re l a t e d to the number of prices i n the k r i g i n g neighbourhood. Ordinary k r i g i n g using a l i n e a r semi-variogram function c a l c u l a t e s weighting f a c t o r s dependent on the slope of the l i n e a r model. This dependence i s shown i n FIGURE 15. Weighting factors have been calculated using each of the l i n e a r models for a k r i g i n g neighbourhood of 11 periods (FIGURE 15,TABLE 2). As the slope of the l i n e a r model decreases the number of non-zero weighting f a c t o r s increases (FIGURES 15b, c and d). It can be deduced that for a slope of zero, a 'Pure Nugget E f f e c t ' would occur as the weighting factors would a l l be equal. For an extremely steep 54 FIGURE 15. KRIGING WEIGHTS AS A FUNCTION OF A LINEAR SEMIVARIOGRAM OPJ_- OJ091 OJNIT)!/PERIOD. Co 3 4 7 Lag iper lodt 13 Theoretical LINEAR semi-vasograms with a NUGGET EFFECT = 1.0 (unit)' and SLOPES = 5.5, 1.0 and 0.091 (unlt)'/perlod. (c) to Nelghbourhoodiperlodt Kriging weights from ordinary kriging using a LINEAR semi-varlogram with a nugget effect = 1.0 (unit)' and a SLOPE = 1.0 (unlt)l/p«rlod for a neighbourhood of 11 periods. (b) -»-10 Neighbour ho o d . p e r l o d i i-o.»-u 0 0.8-u O 0.7-u. OJ c 0.6-"Z. X 0.3-• 0.4-0 0.3-e H o'.J-vi 0.1-0-Kriging weights from ordinary kriging using a LINEAR semi-varlogram with a nugget effect = 1.0 (unit)' and a SLOPE = 5.5 (unlt)Vperlod for a neighbourhood of 11 periods. (d) 3 6 7 N e l g h b o u r h o o d i p e r l o d l Kriging weights from ordinary kriging uslnga LINEAR seml-varlogram with a nugget effect = 1.0 (unit)2 and a SLOPE = 0.091 (unit)Vp e r I°d for a neighbourhood of 11 periods. TABLE 2. *** WEIGHTING FACTORS *** T h e o r e t i c a l L i n e a r Semi-variogram Weighting F a c t o r s . L i n e a r Semi-variograms : FIGURE 15. 1. NUGGET EFFECT (Co) = 1.0 (UNIT) 2 SLOPE (A) 2. NUGGET EFFECT (Co) SLOPE (A) 3. NUGGET EFFECT (Co) SLOPE (A) 5.5 (UNIT) /PERIOD 1.0 (UNIT) 2 1.0 (UNIT) 2/PERIOD 2 = 1.0 (UNIT) = 0.091 (UNIT) 2/PERIOD WEIGHTING FACTORS from o r d i n a r y k r i g i n g using a k r i g i n g neighbourhood of 11 years f o r e c a s t i n g the succeeding p e r i o d . LINEAR SEMI-VARIOGRAM : P e r i o d 1. 2. 1 0.923 0.732 2 0.071 0.196 3 0.006 0.053 4 0.000 0.014 5 0.000 0.004 6 0.000 0.001 7 0.000 0.000 8 -0.000 0.000 9 0.000 0.000 10 -0.000 -0.000 11 0.000 0.000 3. 0.345 0.226 1 48 097 064 042 028 019 0.013 0.010 0.008 0, 0, 0. 0, 0. 0, 56 slope a weighting factor of 1 would be given to the f i r s t point i n the k r i g i n g neighbourhood r e s u l t i n g i n a random walk. It has been stated that a spherical semi-variogram function i s l i n e a r for up to about two t h i r d s of i t s 'range' (David 1977). This i s shown i n FIGURE 16 where a t h e o r e t i c a l l i n e a r and a spherical semi-variogram function have been intersected (TABLE 3 ). U s i n g the s p h e r i c a l model and a k r i g i n g neighbourhood of 20 periods weighting factors were produced by ordinary k r i g i n g and are shown as zones of p o s i t i v e and negative weights i n FIGURE 17 (TABLE 4). Zoning of the weights implies the f i r s t sequence of p o s i t i v e weights i s associated with the l i n e a r portion of the model, the negative weights the curvature and the next sequence of p o s i t i v e weights the ' s i l l ' (FIGURE 16). This suggests that changing the slope of the l i n e a r portion of a spherical model would change the values of the weighting f a c t o r s and the d i s t r i b u t i o n of the negative and p o s i t i v e weights. Changes i n slope occur when the ' s i l l ' , 'nugget e f f e c t ' and 'range' of the spherical model are varied i n d i v i d u a l l y or i n combinations. Negative weights have been explained i n mining situations as representing a screening e f f e c t . This e f f e c t occurs when one point that i s nearer to the point being kriged i s i n front of a point that i s further away (Baafi, Kim and Szidarovszky 1986). Journel (1986) states that negative weights are no e v i l and are expected whenever a continuous s p a t i a l model i s considered. As stated the weighting f a c t o r s are a function of the continuous semi-variogram function f i t t e d to the experimental semi-variogram. T h i s f u n c t i o n converts a d i s c r e t e semi-57 IDEALIZED SPHERICAL SEMI-VARIOGRAM MODEL C M / Slope - 1.0 ( u n i O V p o r i o d +IVE W E I G H T S - I V E W E I G H T S ! c ! +IVE ; W E I G H T S ^ w , a i I I i 1 1 1 1 1 1 1 1 C c L a g : p e r i o d s FIGURE 16. I d e a l i z e d S P H E R I C A L s e m i - v a r i o g r a m s i l l (C) = 9 .0 (un i t ) 2 , n u g g e t e f f e c t (Co) = 1.0 (un i t ) 2 , a n d r a n g e (a) = 13.4 p e r i o d s s h o w i n g l i n e a r i n f l u e n c e a n d z o n e s of + ive a n d —ive we igh t s f r o m o r d i n a r y k r i g i n g a n d a n e i g h b o u r h o o d of 2 0 p e r i o d s , note FIGURE 17 b e l o w . WEIGHTING FACTORS 0.7-2 FIGURE 17. 9 10 11 Periods T I I I I I I I I 12 13 1-4 15 16 17 18 19 20 K r i g i n g w e i g h t s f r o m o r d i n a r y k r i g i n g u s i n g the a b o v e S P H E R I C A L m o d e l a n d a k r i g i n g n e i g h b o u r h o o d of 2 0 p e r i o d s . TABLE 3. *** THEORETICAL LINEAR, SPHERICAL SEMI-VARIANCES *** T h e o r e t i c a l L i n e a r S e m i - v a r i o g r a m : FIGURE 15. P a r a m e t e r s : NUGGET EFFECT (Co) = 1.0 ( U N I T ) 2 : SLOPE (A) T h e o r e t i c a l S p h e r i c a l S e m i - v a r i o g r a m 1.0 (UNIT) /PERIOD FIGURE 16. P a r a m e t e r s : S I L L (C) = 9,0 : NUGGET EFFECT (Co) = 1.0 : RANGE (a ) = 13.4 SEMI--VARIANCES LAGS ; LINEAR SPHERICAL 1 2.0 2.006 2 3.0 3.000 3 4.0 3.972 4 5.0 4.910 5 6.0 5.804 6 7.0 6.641 7 8.0 7.411 8 9.0 8. 102 9 10.0 8.704 10 11.0 9.204 1 1 12.0 9.593 1 2 9.858 1 3 9.988 14 10.000 59 T A B L E 4 . * * * WEIGHTING FACTORS * * * T h e o r e t i c a l S p h e r i c a l S e m i - v a r i o g r a m W e i g h t i n g F a c t o r s . S p h e r i c a l S e m i - v a r i o g r a m : F IGURE 16 . P a r a m e t e r s : S I L L (C) = 9 . 0 (UNIT) : NUGGET E F F E C T (Co) = 1.0 (UNIT) : RANGE (a ) = 1 3 . 4 YEARS WEIGHTING FACTORS f r o m o r d i n a r y k r i g i n g u s i n g k r i g i n g n e i g h b o u r h o o d s o f 2 t o 20 y e a r s f o r e c a s t i n g t h e s u c c e e d i n g p e r r i o d . NEIGHBOURHOOD : P e r i o d 2 3 4 5 6 7 8 1 0 . 748 0 . 7 2 9 0 . 727 0 . 726 0 . 7 2 6 0 . 7 2 5 0 . 7 2 4 2 0 . 2 5 2 0 . 1 9 7 0 . 191 0 . 190 0 . 1 8 9 0 . 189 0 . 189 3 0 . 0 7 4 0 . 052 0 . 047 0 . 0 4 6 0 . 0 4 6 0 . 0 4 6 4 0 . 031 0 . 014 0 . 0 0 9 0 . 0 0 7 0 . 0 0 7 5 0 . 023 0 . 0 0 5 -•0.001 - 0 . 0 0 3 6 0 . 0 2 6 0 . 0 0 3 - 0 . 0 0 4 7 0.031 0 . 0 0 4 8 0 . 0 3 7 NEIGHBOURHOOD • • P e r i o d 9 10 1 1 12 13 14 15 20 1 0 . 722 0 . 720 0 . 717 0 . 713 0 . 7 0 9 0 . 7 0 3 0 . 6 9 9 0 . 6 9 3 2 0 . 189 0 . 1 8 9 0 .188 0 .188 0 . 187 0 . 186 0 . 1 8 5 0 .181 3 0 . 046 0 . 0 4 6 0 . 046 0 . 046 0 . 0 4 7 0 . 0 4 7 0 . 0 4 7 0 . 0 4 5 4 0 . 007 0 . 008 0 .008 0 .008 0 . 0 0 9 0 . 0 0 9 0 . 0 1 0 0 . 0 0 8 5 • - 0 . 003 - 0 . 0 0 3 - 0 . 003 - 0 . 002 - 0 . 0 0 2 - 0 . 0 0 1 - 0 . 0 0 1 - 0 . 0 0 2 6 - 0 . 006 - 0 . 0 0 6 - 0 .006 - 0 . 0 0 5 - 0 . 0 0 5 - 0 . 0 0 4 - 0 . 0 0 4 - 0 . 0 0 5 7 • - 0 . 004 - 0 . 0 0 6 - 0 .007 - 0 . 0 0 7 - 0 . 0 0 6 - 0 . 0 0 6 - 0 . 0 0 5 - 0 . 0 0 5 8 0 . 005 - 0 . 004 - 0 . 007 - 0 . 0 0 7 - 0 . 0 0 7 - 0 . 0 0 7 - 0 . 0 0 6 - 0 . 0 0 5 9 0 . 043 0 . 0 0 7 - 0 .004 - 0 . 0 0 7 - 0 . 0 0 7 - 0 . 0 0 7 - 0 . 0 0 7 - 0 . 0 0 6 1 0 0 . 050 0 . 0 0 9 - 0 .004 - 0 . 0 0 7 - 0 . 0 0 8 - 0 . 0 0 8 - 0 . 0 0 7 1 1 0 .057 0 . 010 - 0 . 0 0 4 - 0 . 0 0 7 - 0 . 0 0 8 - 0 . 0 0 7 12 0 . 0 6 6 0 . 0 1 2 - 0 . 0 0 2 - 0 . 0 0 5 - 0 . 0 0 4 1 3 0 . 0 7 5 0 . 0 2 0 0 . 0 1 0 0 . 0 0 8 14 0 . 0 7 7 0.041 0 . 0 3 2 1 5 0 . 0 5 2 0 . 0 1 9 16 0 . 0 0 8 17 0 . 0 0 3 18 0 . 0 0 3 19 0 . 0 0 9 20 0 . 0 3 3 60 variogram to a continuous one and has a smoothing e f f e c t . In the case of using the semi-variances from the experimental semi-variogram to model the hole e f f e c t i n the 1884 to 1917 copper constant p r i c e time series that smoothing does not occur. But t h i s technique gives a better estimate of the future prices than that using a continuous function. This suggests that smoothing by using a continuous function produces worse estimates when discrete equally spaced prices are being estimated. In t h i s context the semi-variances of an experimental semi-variogram which shows a 'Pure Nugget E f f e c t ' may be used i n estimating f u t u r e p r i c e s . Therefore, continuous f u n c t i o n s are more a p p l i c a b l e when the p r i c e s of p o i n t s intermediate to the discrete evenly spaced time series are wanted. Universal k r i g i n g produces the worst estimates of future prices when applied over the whole time seri e s investigated. The problem l i e s with the method of curve f i t t i n g as ordinary le a s t squares give the greatest importance to the f i r s t and l a s t few p r i c e s i n the time s e r i e s . This r e s u l t s i n very good estimates for the l a s t few prices i n the time series (TABLE 1 C). By f i t t i n g a curve to the prices for 1973 to 1986 i t was shown that the experimental semi-variogram was a 'Pure Nugget E f f e c t ' implying that the best estimates of the 1985 and 1986 p r i c e s are the curve f i t t e d p r i c e s . This and the previous discussion suggests that a curve f i t t e d over a long time series would allow universal k r i g i n g to give better estimates of future pri c e s only at the end of the time seri e s and for one or two periods beyond the s e r i e s . 61 5.2 Lead Following the reasoning outlined f or copper the 1884 to 1986 constant lead price time series (FIGURE 18) was separated into three regimes. These regimes were 1884 to 1917, 1918 to 1953 and 1954 to 1986. Only the 1954 to 1986 pri c e time series was investigated as i t represents the basis for forecasting lead prices f o r 1987 and beyond. 5.2.1 Lead: Variography; Results The experimental semi-variogram (FIGURE 19, APPENDIX 2 TABLE 6) of the constant lead price time series for 1954 to 1986 rose continuously f or 5 lags then deteriorated i n t o a 'Pure Nugget E f f e c t ' . A spherical semi-variogram function was f i t t e d to the experimental semi-variogram to model the s t r u c t u r a l c o r r e l a t i o n between the prices f or 5 lags and the apparent lack of c o r r e l a t i o n beyond. The parameters of the f i t t e d spherical model are: SILL C = 23673.8 (pounds sterling/tonne) 2 NUGGET EFFECT Co = 2000.0 (pounds sterling/tonne) 2 RANGE a = 5.0 years. 5.2.2 Lead: Variography; Discussion The 'range' of 5 years of the spherical model f i t t e d to the experimental semi-variogram for the 1954 to 1986 lead constant pr i c e time seri e s indicates that the rate at which prices become more var i a b l e i s f a i r l y high as the lag increases (FIGURE 19). The p r i c e s i n the time series (FIGURE 18) indicate a sequence of peaks and troughs which do not c o n t a i n dramatic changes LONDON METAL EXCHANGE ANNUAL LEAD PRICES 1400 1200-1 c w 1000 L. « O ) c to -o c 3 O Q -800-600 400-200-C u r r e n t Pr ices C o n s t a n t Pr ices FIGURE 18. I ' 1 ' " | d u n l i n 1864 1901 1918 1935 1952 1969 Years L o n d o n Metal E x c h a n g e a n n u a l a v e r a g e of m o n t h l y L E A D p r i c e s for 1884 to 1986 in c o n s t a n t a n d c u r r e n t p o u n d s s t e r l i n g / m e t r i c t on . T h e c u r r e n t p r i c e s a re d e f l a t e d to c o n s t a n t p r i c e s u s i n g a w h o l e s a l e , p r o d u c e r p r i c e index wi th the b a s e 1 9 8 4 = 1 0 0 . SEMI-VARIOGRAM OF LME ANNUAL LEAD PRICES 19B6 63000 CM 54000 c t, 45000 ^ 36000 C O c to 27000 -D 18000 c o ~ 9000 S e m i - v a r i o g r a m • S p h e r i c a l Funct ion <-i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — r 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 0 1 2 3 4 5 6 7 L a g : y e a r s FIGURE 19. S e m i - v a r i o g r a m of L M E c o n s t a n t a n n u a l L E A D p r i c e s fo r 1954 to 1986 The e x p e r i m e n t a l s e m i - v a r i o g r a m c a n be f i t t e d by a S P H E R I C A L f u n c t i o n w i th a SILL (C) = 2 3 6 7 3 . 8 ( £ / t o n n e ) 2 , N U G G E T E F F E C T (Co) = 2 0 0 0 . 0 ( V t o n n e ) 2 a n d a R A N G E (a) = 5 .0 y e a r s . 63 s u g g e s t i n g t h a t t h e p r i c e s a r e m o d e r a t e l y v a r i a b l e . S i g n i f i c a n t l y the ' s i l l ' plus the 'nugget e f f e c t ' equals the variance of the price population for t h i s time ser i e s (25673.8 (pounds s t e r l i n g / t o n n e ) 2 ) . The 'nugget e f f e c t ' i s considered as implying s l i g h t seasonal supply and demand e f f e c t s . An a l t e r n a t i v e i n t e r p r e t a t i o n to the lack of s t r u c t u r a l c o r r e l a t i o n represented by the experimental semi-variogram after the 5th lag i s that a cycle of approximately 4 years occurs s t a r t i n g at the 5 th lag. This apparent p e r i o d i c i t y has not investigated. 5.2.3 Lead: Kriging; Results 1954 to 1986 The s p h e r i c a l semi-variogram f u n c t i o n m o d e l l i n g the s t r u c t u r a l c o r r e l a t i o n of prices i n the 1954 to 1986 time series and k r i g i n g neighbourhoods of 5 and 6 years were used by ordinary k r i g i n g to estimate future prices for the years 1955 to 1986. The known prices, the ordinary kriged p r i c e estimates, the random walk p r i c e f o r e c a s t s and t h e i r r e s p e c t i v e squared differences are presented i n APPENDIX 4 TABLE 7. The sums of the squared differences and t h e i r percent r a t i o s , kriged versus the random walk, are shown i n TABLE 5. Future prices for 1966 to 1986 were also estimated using universal k r i g i n g . A polynomial curve of order 2 (quadratic) was f i t t e d to the prices of the 1954 to 1986 serie s (FIGURE 20). The residuals, known pri c e minus the curve f i t t e d p r i c e, (FIGURE 21, APPENDIX 4 TABLE 3) were calculated and an experimental 64 LME 1954 TO 1986 ANNUAL CONSTANT LEAD PRICES 1000 o o. 800 $ 600 cn 400 200 Constant Prices Quadrat ic Curve 1933 FIGURE 20. •—i—r—•—i—1—1—i—1—•—i—•—1—i—•—•—i—•—•—i—•—•—i—•—•—i—•—1—i— 1936 1939 1962 1969 1968 19/1 1974 1977 1980 1963 Years London Metal Exchange constant 1954 to 1986 LEAD prices fitted with a quadratic curve, -1961.3 + 65.1 T - 0.4 T. RESIDUAL 1954 TO 1986 LME LEAD PRICES 1986 FIGURE 21. i—•—'—r 19 3 3 19 9 6 19 9 9 19 6 2 19 6 3 19 6 6 1971 1974 19 77 19 8 0 19 8 3 Years The constant LEAD price residuals for 1954 to 1986, the known prices minus the quadratic curve prices. SEMI-VARIOGRAM OF LME 1954 TO 1986 LEAD PRICES 40000-1 Seml-varlograoi Spherical Function 1 • 1 1 W 1 —1—1—1—1—1—1— *e.. >.. •e. • FIGURE 2 2 - s 6 m i _ v a r i o g r a m o f t n e r e siduals fitted by a SPHERICAL function,SILL(C) = 20118.4(£ftonne)»,NUGGET EFFECT(Co)=4000.0(£/tonne)» and RANGE(a)=6.0years. 65 semi-variogram determined (FIGURE 22, APPENDIX 2 TABLE 7). A spherical semi-variogram function was f i t t e d (FIGURE 22). SILL C = 20118.4 (pounds sterling/tonne) 2 NUGGET EFFECT Co = 4000.0 (pounds sterling/tonne) 2 RANGE a 6.0 years. A k r i g i n g neighbourhood of 12 years was used as s i x of the years represented the 'range' of the model and i t was f e l t that at l e a s t double the range was needed to adequately represent the f i t t e d quadratic curve. Therefore, only the prices f o r 1966 to 1986 were estimated. The estimation procedures have been discussed previously. The u n i v e r s a l l y kriged future price estimates, the known prices, the random walk estimates and the appropriate squared differences are presented i n APPENDIX 3 TABLE 8. The sums of the squared differences and t h e i r percent r a t i o s are shown i n TABLE 5. To f u r t h e r t e s t the a b i l i t y of u n i v e r s a l k r i g i n g to estimate future prices a polynomial of order 2 (quadratic) was f i t t e d to the 1979 to 1986 constant lead p r i c e time series (FIGURE 23). The residuals (FIGURE 24, APPENDIX 4 TABLE 4) and an experimental semi-variogram of the r e s i d u a l s (FIGURE 25, APPENDIX 2 TABLE 8) were calculated. This experimental semi-variogram exhibited a 'Pure Nugget E f f e c t ' which implied that universal k r i g i n g would not give the best p r i c e estimates, but t h a t the best estimate of a f u t u r e p r i c e was the p r i c e determined from the quadratic equation. P r i c e s estimated f o r 1985 and 1986 from the quadratic equation and the random walk, the known prices and the squared 6 6 L M E 1979 T O 1986 A N N U A L C O N S T A N T L E A D PRICES 3 o a. • 00 600 -400 200-Cont lanl Prices Quadrat ic Curve 1978 FIGURE 2 3 . 1980 1984 1982 Years London Metal Exchange constant 1979 to 1986 LEAD prices fitted with a quadratic curve, 191973.8 - 3762.7 T + 18.6 T. RESIDUAL 1979 T O 1986 L M E L E A D PRICES 1986 100 CO c -so -100 1978 FIGURE 24 . 1980 1962 1984 Years The constant LEAD price residuals for 1979 to 1986, the known prices minus the quadratic curve prices. S E M I - V A R I O G R A M O F L M E 1973 T O 1986 L E A D PRICES 10000 £ 6000 0> 4000 2000 o a. Semi-variogram Population Variance FIGURE 2 5 . Lag ,years Experimental semi-variogram of the residuals showing a PURE NUGGET EFFECT. 67 differences are presented i n APPENDIX 3 TABLE 9. The sums of the squared differences and the percent r a t i o s are shown i n TABLE 5. The r e s u l t s of the sums of squared differences comparisons between the random walk, ordinary kriging, universal k r i g i n g and polynomial curve price estimates (TABLE 5) for the 1954 to 1986 constant lead p r i c e time series were: 1. Ordinary kriging gave better prices estimates over the time series investigated than the random walk and universal kriging, percent r a t i o + 7.4%. 2. The better estimates of future prices occurred when a kr i g i n g neighbourhood equivalent to the 'range' of the spherical model was used. By increasing the k r i g i n g neighbourhood from 5 to 6 years the percent r a t i o goes from + 7.4% down to - 3.9%. 3a. Universal kriging with a neighbourhood of 12 years gave the worst future price estimates for the 1966 to 1986 time series, percent r a t i o -48.8%. 3b. S i g n i f i c a n t l y universal k r i g i n g using the polynomial curve f i t t e d to the 1954 to 1986 time series gave the best estimates for 1985 and 1986, percent r a t i o + 75.1%. 4. The 1985 and 1986 price estimates from the equation of the quadratic curve f i t t e d to the 1979 to 1986 lead constant price time ser i e s were worse estimates than those from the random walk, percent r a t i o - 4.4 %. 6 8 TABLE 5. *** SUMS OF SQUARED DIFFERENCES LEAD *** V a r i a b l e : LEAD, London Metal Exchange D e f l a t o r : WHOLESALE,PRODUCER PRICE INDEX 1984=100 Time S e r i e s : YEARLY, 1954 to 1986 S p h e r i c a l Model YEARS RANDOM ORDINARY ORDINARY UNIVERSAL WALK KRIGING KRIGING KRIGING N = 5 N = 6 N = 12 ( £/m. t . ) 2 ( £ / m . t . ) 2 ( £ / m . t . ) 2 ( £/ m . t . ) 2 1 955-1 965 117934.7 140330.7 146074.6 1966-1984 465323.2 393855.6 440398.7 696358.7 1985-1986 3309.4 9044.9 23012.9 822.8 SUM 586567.3 543231.2 609486.2 6899291.0 % RATIO + 7.4 - 3.9 - 48.8 N : K r i g i n g Neighbourhood > • m. t . : M e t r i c Tons. CURVE FITTING (£/m.t.) 3455.3 62898.8 - 4.4 69 5.2.4 Lead: Kriging; Discussion These r e s u l t s support the discussion presented for copper. Ordinary k r i g i n g i s generally better at estimating future prices than the random walk. Furthermore, when the v a r i a b i l i t y between prices i n a time series i s moderate and a kr i g i n g neighbourhood equivalent to the 'range' of the model i s used ordinary kriging excels. The experimental semi-variogram for lead, 1954 to 1986 (FIGURE 22), indicates less v a r i a b i l i t y and a slower rate of change i n v a r i a b i l i t y than that for copper, 1884 to 1917 (FIGURE 7) supporting the contention that a r e l a t i v e l y smaller, 'range' of the f i t t e d model, k r i g i n g neighbourhood gives a better estimate. Again universal k r i g i n g gives the worst estimates when forecasts are over a large time series, but i s very good when i t uses the curve f i t t e d over the large time to estimate pri c e s at the end of that s e r i e s . 5.3 Zinc For the reasons previously discussed the 1884 to 1986 constant zinc p r i c e time series (FIGURE 26) was pa r t i t i o n e d into three regimes and l i k e lead only the l a s t time series, 1954 to 1986, was investigated. Unlike lead prices for t h i s time series constant zinc prices indicated o u t l i e r s . O u t l i e r s were defined as those prices which occur as separate populations when the frequencies of the prices i n the time series were determined. For z i n c two constant prices were determined to be o u t l i e r s (APPENDIX 5 TABLE 1): : 1973 1395.3 pounds sterling/tonne : 1974 1729.1 pounds sterling/tonne. 70 O u t l i e r s are s i g n i f i c a n t c o n t r i b u t o r s to n o n - s t a t i o n a r i t y therefore these two prices were extracted from the pri c e series and t h e s m a l l e r d a t a s e t was used i n s e m i - v a r i o g r a m determinations for ordinary k r i g i n g . For universal k r i g i n g the pric e s e r i e s used included the o u t l i e r s . 5.3.1 Zinc: Variography; Results The experimental semi-variogram (FIGURE 27, APPENDIX 2 TABLE 9) of the constant zinc price time ser i e s f or 1954 to 1986, without the o u t l i e r s , increased continuously to a peak at a lag of 6 years where a constant l e v e l was assumed. This experimental semi-variogram was f i t t e d with a spherical semi-variogram function to model the s t r u c t u r a l c o r r e l a t i o n of prices up to a lag of 5.8 years. The semi-variances of the experimental semi-variogram indicated e r r a t i c v a r i a b i l i t y between prices that were greater than 6 years apart. The parameters of the f i t t e d spherical model are: SILL C = 16655.9 (pounds sterling/tonne) 2 NUGGET EFFECT Co = 0.0 (pounds sterling/tonne) 2 RANGE a 5.8 years. 5.3.2 Zinc: Variography; Discussion The 'range' of 5.8 years of the spherical model f i t t e d to the experimental semi-variogram f o r the 1954 to 1986 zinc constant p r i c e time series, without o u t l i e r s , shows that the rate at which prices become more variable as the lag increases i s high (FIGURE 27). The prices i n t h i s pruned time series show l i t t l e v a r i a b i l i t y as there are no excessive peaks or troughs 71 LONDON METAL EXCHANGE ANNUAL ZINC PRICES 2100 1800 c « 1500 ^ 1200 CO c « 900 C O c 3 O 600 300 C u r r e n t Pr ices Constan t Pr ices 1884 1901 1918 1952 1969 FIGURE 26 . 1935 Years L o n d o n Metal E x c h a n g e a n n u a l a v e r a g e of m o n t h l y ZINC p r i c e s fo r 1884 to 1986 in c o n s t a n t a n d c u r r e n t p o u n d s s t e r l i n g / m e t r i c t o n . The c u r r e n t p r i c e s a re d e f l a t e d to c o n s t a n t p r i c e s u s i n g a w h o l e s a l e , p r o d u c e r p r i c e index w i th the b a s e 1 9 8 4 = 1 0 0 . SEMI-VARIOGRAM OF LME ANNUAL ZINC PRICES 1986 20000 CN c 16000 .O S e m i - v a r i o g r a m • * S p h e r i c a l Funct ion i • -4 1 I * — I 12000 CO c e 8000 oo c 3 o 4000--1 -12 —r-15 — r~ 18 —r-21 I 24 27 30 Lag=y e a r s FIGURE 27 . S e m i - v a r i o g r a m of LME c o n s t a n t a n n u a l ZINC p r i c e s f o r 1954 to 1986. The e x p e r i m e n t a l s e m i - v a r i o g r a m c a n be f i t t e d by a S P H E R I C A L f u n c t i o n w i th a SILL (C) = 16655 .9 ( £ / tonne) z ,NUGGET E F F E C T (Co) = 0 .0 ( f i / f o n n e ^ a n d a R A N G E (a) = 5 .8 y e a r s . 72 (FIGURE 26). The zero 'nugget e f f e c t ' implies continuity i n the r e g u l a r i z a t i o n of the monthly prices that are averaged to make up the annual p r i c e s . The experimental semi-variogram of the constant zinc prices i s very sawtoothed a f t e r a lag of 6 years. Also indicated i s a downward trend i n the semi-variances. It i s assumed that these features resulted from using a data set containing gaps. This assumption i s made as the sawtoothedness i s l i k e n e d to experimental semi-variograms from mining data which i s not evenly spaced. The downward trend i s considered a function of the v a r i a b i l i t y between prices and t h e i r spacing, where prices f u r t h e r apart are more s i m i l a r . This experimental semi-variogram i s considered as being moderate to highly var i a b l e . 5.3.3 Zinc: Kriging; Results 1954 to 1986 The spherical model of the s t r u c t u r a l c o r r e l a t i o n between prices within the 1954 to 1986 time series, without o u t l i e r s , (FIGURE 27) and a k r i g i n g neighbourhood of 6 years were used by ordinary k r i g i n g to estimate future prices for the years 1955 to 1986. The known prices, the kriged price estimates and the random walk pr i c e estimates and t h e i r squared differences are presented i n APPENDIX 3 TABLE 10. The sums of the squared differences and t h e i r percent r a t i o s , kriged versus the random walk, are shown i n TABLE 6. For u n i v e r s a l k r i g i n g a polynomial curve of order 1 ( l i n e a r ) was f i t t e d to the 1954 to 1986 p r i c e time series, with o u t l i e r s , (FIGURE 28). Residuals, known p r i c e minus curve 73 L M E 1954 T O 1986 A N N U A L C O N S T A N T Z I N C PRICES 2000 1600 ^ 1200 800 400-Conitant Priest Linear Curve 1993 FIGURE 28. 1936 1999 1 9 6 2 1974 1977 1980 1983 1969 1968 1971 Years London Metal Exchange constant 1954 to 1986 ZINC prices fitted with a linear curve, +479.4 + 1.9 T. RESIDUAL 1954 T O 1986 L M E Z INC PRICES 1986 1100-1 1000-e 0 900-H> 800-u 700-• < 600-300-c 400-300-e vt 200-TJ 100-C 3 0-O a. -100-- 2 0 0 -- 3 0 0 -1993 FIGURE 29. 1996 —I ' 1*83 9 39 196 2 19 6 3 19 6 8 1971 1974 19 7 7 1981 Years The constant ZINC price residuals for 1954 to 1986, the known prices minus the linear curve prices. S E M I - V A R I O G R A M O F L M E 1954 T O 1986 Z I N C PRICES 1986 140000-c 120000 o a — 60000-«n TJ C 3 O 40000-20000-Seml-var logram Spherical Function —r~ 12 13 L a g i y e a r t —i— 24 — Co »7 FIGURE 3 0 . S e m j _ v a r i o g r a m o f th e residuals fitted by a SPHERICAL function,SILL(C)= 90370.0(ytonne)»,NUGGET EFFECTCCo^SOOO.OCVtonne)1 and RANGE(a)=6.0years. 7 4 f i t t e d p r i c e , (FIGURE 29, APPENDIX 4 TABLE 5) and the experimental semi-variogram of the r e s i d u a l s (FIGURE 30, APPENDIX 2 TABLE 10) were c a l c u l a t e d . A s p h e r i c a l semi-variogram f u n c t i o n was f i t t e d t o the experimental semi-variogram. SILL C = 90370.0 (pounds sterling/tonne) 2 NUGGET EFFECT Co = 5000.0 (pounds sterling/tonne) 2 RANGE a = 6.0 years. Using the model and a kr i g i n g neighbourhood of 12 years prices were kriged fo the years 1966 to 1986. This s i z e of kriging neighbourhood was used to incorporate the f i t t e d l i n e a r d r i f t and allow i t to a f f e c t the kr i g i n g estimates. It was noted that the experimental semi-variogram of the residuals (FIGURE 30) showed a downward trend a f t e r a s p e c i f i c lag period, 7 years. The suggestion that t h i s downward trend was the r e s u l t of increasing s i m i l a r i t y between prices further apart was also proposed for t h i s case. It was e a s i l y seen that t h i s was the case as the residuals (FIGURE 29) showed more v a r i a b i l i t y over short periods than over longer ones. The u n i v e r s a l kriged future p r i c e estimates, the known prices and t h e i r squared differences and those of the simple random walk are presented i n APPENDIX 3 TABLE 11. The sums of the squared differences for the time series 1966 to 1986 and the percent r a t i o s , kriged versus the random walk, are shown i n TABLE 6. A polynomial curve of order 2 (quadratic) was f i t t e d to the 1974 to 1986 constant zinc prices of the 1954 to 1986 time serie s (FIGURE 31). The residuals (FIGURE 32, APPENDIX 4 TABLE 7 5 L M E 1974 T O 1986 A N N U A L C O N S T A N T ZINC PRICES 197-4 FIGURE 31. 197* 1978 i960 I9S2 198-4 Years London Metal Exchange constant 1974 to 1986 ZINC prices f i t ted with a quadrat ic curve, 155148.3 - 3131.7 T + 15.8 T». RESIDUAL 1974 T O 1986 L M E Z INC PRICES -400-1 - 3 0 0 -1974 FIGURE 32. 1976 1978 I960 19*2 1964 Years The constant ZINC price residuals for 1974 to 1986, the known pr ices minus the quadratic curve pr ices. S E M I - V A R I O G R A M O F L M E 1974 T O 1986 Z INC PRICES 1986 36000- Seml-var logram Population Variance % 30000-— 24000-e e 18000-12000-6000-FIGURE 33. 1 2 3 4 9 6 7 8 9 ID Logiyear t Experimental s e m i - v a r i o g r a m of the residuals showing a PURE NUGGET EFFECT. 76 6) and the experimental semi-variogram of the residuals (FIGURE 33, APPENDIX 2 TABLE 11) were calculated. This experimental semi-variogram exhibited a 'Pure Nugget E f f e c t ' which implied that the best estimate of a future p r i c e would not be from k r i g i n g , but from the quadratic equation. Prices estimated from the quadratic equation for 1985 and 1986, the random walk estimates, the known p r i c e s and the rel a t e d squared differences are presented i n APPENDIX 3 TABLE 12. The sums of the squared differences and t h e i r percent r a t i o s are i n TABLE 6. The r e s u l t s of the sums of the squared d i f f e r e n c e s comparisons for future constant zinc price estimates for the years 1966 to 1986 were: 1. Ordinary k r i g i n g gave better future p r i c e estimates over the whole period estimated, percent r a t i o + 2.2%, but the random walk was better from 1966 to 1972, percent r a t i o - 78.1 %. 2. Universal k r i g i n g was worse at estimating future prices than the random walk for whatever period considered, percent r a t i o - 53.4%. 3. The prices from the quadratic equation of the curve f i t t e d to the 1974 to 1986 constant zinc p r i c e time seri e s were not the worst estimates of 1985 and 1986 prices versus the random walk (TABLE 6 universal k r i g i n g ) , percent r a t i o - 132.8%, but su r p r i s i n g l y ordinary k r i g i n g was the better estimator, percent r a t i o + 20.9%. 77 TABLE 6. *** SUMS OF SQUARED DIFFERENCES ZINC *** V a r i a b l e : ZINC, L o n d o n M e t a l E x c h a n g e D e f l a t o r : WHOLESALE,PRODUCER PRICE INDEX 1984=100 Time S e r i e s : YEARLY, 1954 t o 1986 S p h e r i c a l M o d e l YEARS RANDOM RANDOM ORDINARY UNIVERSAL POLYNOMIAL WALK WALK KRIGING KRIGING CURVE N = 6 N = 12 F I T T I N G ( £/m. t . ) 2 ( £ / m . t . ) 2 ( £ / m . t . ) 2 ( £ / m . t . ) 2 ( £ / m . t . ) 2 1 955-1 965 125728.0 125728.0 124842.6 — --1 966-1 972 15855.0 15855.0 28240.9 30966.0 — 1 973-1975 1370681.5 OUTLIERS 2012387.4 — 1 976-1 984 1 1 4565.3 114565.3 101237.4 348563.2 1 985-1 986 20401.6 20401 .6 16128.5 134738.3 47491.6 SUM 1647231.4 276549.9 270449.4 2526654.9 47491.6 % RATIO • + 2.2 - 53.4 - 132.8 N : K r i g i n g N e i g h b o u r h o o d m. t . : M e t r i c T o n s . 7 8 5.3.4 Zinc: Kriging; Discussion Again ordinary k r i g i n g i s supported as a better estimator of future prices than i s the random walk and again universal k r i g i n g i s the worst. Universal k r i g i n g i s not even vindicated i n estimating the 1985 and 1986 prices as previously shown. This i s a r e s u l t of the type of curve f i t t e d , as the l i n e a r model increases over time because of the influence of the high 1973 and 1974 prices while the actual price has decreased for those years. Since only one kr i g i n g neighbourhood (N=6) i s investigated no s p e c i f i c remarks can be addressed to the subject of the appropriate k r i g i n g neighborhood to use. I t i s suggested that the rate that p r i c e v a r i a b i l i t y increases with increasing lag i n t e r v a l s and the v a r i a b i l i t y shown i n the 1954 to 1986 zinc constant p r i c e time seri e s imply a moderate to high v a r i a b i l i t y and therefore the appropriate k r i g i n g neighbourhood i s s l i g h t l y larger than the 'range' of the spherical model. 5.4 Summary Market f a c t o r s and the requirement of s t a t i o n a r i t y for g e o s t a t i s t i c a l applications were the reasons for p a r t i t i o n i n g each 1884 to 1986 metal price time series i n t o three regimes. The p a r t i t i o n e d regimes were 1884 to 1917, 1918 to 1953 and 1954 to 1986. Only for copper were a l l three regimes investigated as they represented d i f f e r e n t environments and presented d i f f e r e n t problems that were comparable being based on copper constant p r i c e s . With t h i s foundation the subsequent investigations into lead and zinc constant price time seri e s were l i m i t e d to the 79 1954 to 1986 time s e r i e s . Another reason to l i m i t these i n v e s t i g a t i o n s was that the purpose of the t h e s i s was to i n v e s t i g a t e e s t i m a t i n g future p r i c e s and the 1954 to 1986 constant price times series investigations l a i d the framework for estimating prices i n 1987 and beyond. S i g n i f i c a n t to the zinc constant p r i c e time series investigated was the occurrence of o u t l i e r s and t h e i r treatment. Spherical semi-variogram functions were f i t t e d to those experimental semi-variograms that did not show a 'Pure Nugget E f f e c t ' . In a l l cases the experimental semi-variograms showed a smooth, not e r r a t i c , increase to a ' s i l l ' value over the 'range' of the spherical model. This smoothness was contrasted to the sawtoothed habit expressed by experimental semi-variograms from miming data and was re l a t e d to the even spacing and lack of gaps i n the metals constant price time s e r i e s . The ' s i l l ' and the 'range' of the spherical models were rel a t e d to the v a r i a b i l i t y shown by the prices i n the time series with the d i f f e r e n t metals and time s e r i e s given d i f f e r e n t c l a s s i f i c a t i o n s : Copper, 1918 to 1953, low v a r i a b i l i t y Lead, 1954 to 1986, low to moderate v a r i a b i l i t y Copper, 1884 to 1917, moderate v a r i a b i l i t y Zinc, 1954 to 1986, moderate to high v a r i a b i l i t y Copper, 1954 to 1986, high v a r i a b i l i t y . These s u b j e c t i v e d e s i g n a t i o n s e x p r e s s e d the degree of v a r i a b i l i t y between p r i c e s and allowed r e l a t i o n s h i p s to be developed between the price time series the spherical models and the k r i g i n g r e s u l t s . Not unexpectedly the spherical models of the residuals from f i t t i n g polynomial curves to the price time 8 0 s e r i e s were s i m i l a r i n shape and value to the spherical models of the o r i g i n a l constant price time s e r i e s . What was unexpected was the experimental semi-variograms of the r e s i d u a l s from f i t t i n g polynomial curves to the sequences of prices i n the 1954 to 1986 time s e r i e s e x h i b i t i n g 'Pure Nugget E f f e c t s ' . The reason for t h i s e f f e c t was related to the l i m i t e d number of prices used i n f i t t i n g the curves, from 8 to 14 prices depending on the metal. Ordinary k r i g i n g was generally found to give better future p r i c e estimates than the random walk over the time s e r i e s studied. Interestingly the simple random walk method of future p r i c e estimation was found to be a s p e c i a l case of ordinary k r i g i n g . This became obvious when the random walk gave better estimates f o r the copper 1918 to 1953 constant p r i c e time s e r i e s . I t was obvious because the ordinary k r i g i n g r e s u l t s suggested that for prices with low v a r i a b i l i t y a small kriging neighbourhood would give better future price estimates than a large one. Confirmation occurred when weighting factors were developed f o r a t h e o r e t i c a l semi-variogram f u n c t i o n using ordinary k r i g i n g and very small k r i g i n g neighbourhoods. For moderately variable prices i n a time series ordinary kriging neighbourhoods equal to the range of the spherical model or s l i g h t l y larger were suggested. Highly variable prices i n a time s e r i e s used to estimate future prices required a kriging neighbourhood larger than the 'range' of the spherical model to allow ordinary k r i g i n g to give an adequate estimate. U n i v e r s a l k r i g i n g produced the worst f u t u r e p r i c e estimates. This i n a b i l i t y was thought to be r e l a t e d to the type 81 of curve f i t t i n g technique used and the c i r c u l a r problem that the semi-variogram i s required to estimate the d r i f t and the d r i f t i s required to determine the semi-variogram. Universal k r i g i n g stands out when used to estimate future prices near the end of the time series, s p e c i f i c a l l y when a quadratic or higher order polynomial curve was f i t t e d to the whole time s e r i e s . 8 2 6. CONCLUSIONS This study i s an attempt to apply g e o s t a t i s t i c a l methods to estimate future p r i c e s . To tes t the a b i l i t y of g e o s t a t i s t i c s to estimate future prices a sum of squared differences comparison i s made between the kriged estimations and the future prices estimated from a simple random walk. The conclusions of t h i s study are: 1. The s t r u c t u r a l c o r r e l a t i o n between metal prices i n a h i s t o r i c a l time series can be represented by a one-dimensional experimental semi-variogram and f i t t e d with a t h e o r e t i c a l semi-variogram function. 2. Future metal prices can be forecast by the g e o s t a t i s t i c a l technique of kr i g i n g . In the t e s t s done ordinary k r i g i n g generally gives better estimates of future prices than does a simple random walk. 3. The r e s u l t s obtained from t h i s study are encouraging and further investigations i n to applying g e o s t a t i s t i c s to metal price forecasting are warranted. 6.1 Further Work Areas of in v e s t i g a t i o n that are of i n t e r e s t : - How accurately can d a i l y prices be forecast and from d a i l y price series can weekly, monthly and yearly prices be accurately forecast ? This question also applies to using weekly and monthly price s e r i e s . 83 - How would transforming the prices series by normalizing t h e i r d i s t r i b u t i o n s e f f e c t the forecasts ? - Having studied the o u t l i e r e f f e c t how would the r e s u l t s of t r e a t i n g the o u t l i e r s by truncation or transformation a f f e c t the forecasts ? - Would t e s t i n g the parameters of the f i t t e d t h e o r e t i c a l semi-variogram functions by forecasting prices within the series and determining the error of estimation improve the models ? - Would forecasting the prices further into the future than one period be more accurate i f the preceding forecast p r i c e was included i n the k r i g i n g neighbourhood ? - Having j u s t touched on the aspects of r e l a t i n g weighting factors, k r i g i n g neighbourhoods, t h e o r e t i c a l semi-variogram functions and k r i g i n g methods to the nearest p r i c e forecasts, expand the study into t h i s area. 6.2 Forecasting 1987 and 1988 Prices Based on the r e s u l t s of t h i s study pr i c e s are forecast for London Metal Exchange copper, lead and zinc for 1987 and 1988 (TABLE 7). I t must be pointed out that the forecasts for 1987 and 1988 are deflated or constant prices referenced to 1984. 84 TABLE 7. *** 1987 AND 1988 METAL PRICE FORECASTS *** LONDON METAL EXCHANGE: COPPER: : Time S e r i e s , YEARLY; 1954 TO 1986 : RANDOM WALK: : FORECASTS; Year P r i c e E r r o r £/tonne 1987 849.4 490.3 1988 849.4 490.3 : ORDINARY KRIGING: K r i g i n g Neighbourhood 14 y e a r s . : SPHERICAL Model; SILL C = 805818.7 (£/tonne) 2 ; NUGGET EFFECT Co = 0.0 (£/tonne) 2 ; RANGE a = 13.8 y e a r s . : FORECASTS; Year P r i c e K r i g i n g E r r o r £/tonne 1987 990.2 412.0 1988 1121.4 574.7 : UNIVERSAL KRIGING: K r i g i n g Neighbourhood 15 y e a r s . : SPHERICAL Model; SILL C = 805818.7 (£/tonne) 2 ; NUGGET EFFECT Co = 0.0 (£/tonne) 2 ; RANGE a = 13.8 y e a r s . : FORECASTS; Year P r i c e K r i g i n g E r r o r £/tonne 1987 740.4 590.1 1988 1032.6 724.7 : Time S e r i e s , YEARLY; 1973 to 1986 : QUADRATIC EQUATION; 185572.6 - 3692.8 T + 18.5 T 2 : FORECASTS; Year P r i c e £/tonne 1987 1221.4 1988 1387.4 85 TABLE 7. cont'd .*** 1 987 AND 1988 METAL PRICE FORECASTS *** LONDON METAL EXCHANGE: LEAD: : Time Series, YEARLY; 1954 TO 1986 : RANDOM WALK: : FORECASTS; Year Price Error £/tonne 1987 251.1 133.3 1988 251.1 133.3 : ORDINARY KRIGING: Kriging Neighbourhood 5 years. : SPHERICAL Model; SILL C = 23673.8 (£/tonne) 2 ; NUGGET EFFECT Co = 2000.0 (£/tonne) 2 ; RANGE a 5.0 years. : FORECASTS; Year Price Kriging Error £/tonne 1987 266.9 129.4 1988 261.5 168.0 : UNIVERSAL KRIGING: Kriging Neighbourhood 12 years. : SPHERICAL Model; SILL C =20118.4 (£/tonne) 2 ; NUGGET EFFECT Co = 4000.0 (£/tonne) 2 ; RANGE a = 6.0 years. : FORECASTS; Year Price Kriging Error £/tonne 1987 155.7 160.4 1988 141.5 198.3 : Time Series, YEARLY; 1979 to 1986 : QUADRATIC EQUATION; 191973.8 - 3762.8 T + 18.6 T 2 : FORECASTS; Year Price £/tonne 1987 1384.8 1988 1500.9 86 T A B L E 7 . c o n t ' d * * * 1987 AND 1988 METAL PR ICE FORECASTS * * * LONDON METAL EXCHANGE: Z I N C : : T i m e S e r i e s , Y E A R L Y ; 1954 TO 1986 : RANDOM WALK: : F O R E C A S T S ; Y e a r P r i c e E r r o r £ / t o n n e 1987 4 6 5 . 9 2 1 4 . 7 1988 4 6 5 . 9 2 1 4 . 7 : ORDINARY K R I G I N G : K r i g i n g N e i g h b o u r h o o d 6 y e a r s . : S P H E R I C A L M o d e l ; S I L L C = 1 6 6 5 5 . 9 ( £ / t o n n e ) ; NUGGET E F F E C T Co = 0 . 0 ( £ / t o n n e ) ; RANGE a = 5 .8 y e a r s . : F O R E C A S T S ; Y e a r P r i c e K r i g i n g E r r o r £ / t o n n e 1987 4 6 5 . 3 8 9 . 5 1988 4 4 9 . 4 123 .0 : U N I V E R S A L K R I G I N G : K r i g i n g N e i g h b o u r h o o d 12 y e a r s . : S P H E R I C A L M o d e l ; S I L L C = 9 0 3 7 0 . 0 ( £ / t o n n e ) ; NUGGET E F F E C T Co = 5 0 0 0 . 0 ( £ / t o n n e ) ; RANGE a 6 . 0 y e a r s . : F O R E C A S T S ; Y e a r P r i c e K r i g i n g E r r o r £ / t o n n e 1987 4 8 0 . 2 2 8 8 . 4 1988 4 8 6 . 8 3 5 3 . 2 : T i m e S e r i e s , Y E A R L Y ; 1974 t o 1986 : QUADRATIC E Q U A T I O N ; 1 5 5 1 4 8 . 3 - 3 1 3 1 . 7 T + 1 5 . 8 T 2 : F O R E C A S T S ; Y e a r P r i c e £ / t o n n e 1987 8 3 7 . 5 1988 1017 .5 87 BIBLIOGRAPHY Agterberg F.P., 1974. 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A, Vol. 226. pp. 267 - 298. 9 6 A P P E N D I X 1 L M E C U R R E N T P R I C E S A N D W H O L E S A L E P R I C E I N D E X 97 T A B L E 1 . * * * CURRENT METAL P R I C E S AND WHOLESALE,PRODUCER P R I C E INDEX M e t a l P r i c e s F r o m : GREAT B R I T A I N , LONDON METAL EXCHANGE P r i c e I n d e x F r o m : GREAT B R I T A I N T i m e S e r i e s , Y E A R L Y : 1884 t o 1986 T IME COPPER LEAD Z INC WHOLESALE P R I C E INDEX ( Y e a r ) ( £ / t o n n e ) ( £ / t o n n e ) ( £ / t o n n e ) 1984=100 1884 5 3 . 0 2 4 1 1 . 1 2 2 1 4 . 2 1 8 3 .2 1885 4 2 . 8 6 2 1 1 . 3 1 8 1 3 . 7 7 5 3 . 0 1886 3 9 . 4 5 0 1 3 . 0 2 9 1 4 . 0 2 9 2 . 8 1887 4 5 . 2 9 4 1 2 . 3 4 4 1 4 . 9 6 0 2 . 7 1888 8 0 . 2 7 5 1 3 . 3 5 3 1 7 . 8 0 2 2 . 8 1889 4 8 . 9 4 7 1 2 . 1 3 4 1 9 . 4 6 7 2 . 9 1890 5 3 . 4 0 6 1 3 . 0 0 4 2 2 . 8 8 3 2 . 9 1891 5 0 . 6 5 4 1 2 . 0 9 8 2 2 . 8 8 7 3 . 0 1892 4 4 . 9 3 7 1 0 . 3 1 3 2 0 . 5 0 0 2 . 8 1893 4 3 . 0 8 4 9 . 5 3 9 1 7 . 1 2 9 2 . 8 1894 3 9 . 7 2 9 9 .321 1 5 . 1 8 5 2 . 6 1895 4 2 . 3 0 0 1 0 . 3 2 6 1 4 . 3 7 7 2 . 5 1896 4 6 . 1 6 3 1 1 . 1 1 8 1 6 . 3 3 0 2 . 4 1897 4 8 . 3 5 3 1 2 . 1 7 2 1 7 . 2 1 6 2 . 5 1898 5 1 . 0 1 0 1 2 . 7 7 8 2 0 . 1 1 5 2 . 6 1899 7 2 . 5 2 3 1 4 . 6 9 7 2 4 . 4 6 5 2 . 5 1900 7 2 . 4 6 2 1 6 . 7 1 9 1 9 . 9 5 4 2 . 8 1901 6 5 . 9 2 5 1 2 . 3 2 3 1 6 . 7 6 0 2 . 7 1902 5 1 . 6 3 1 1 1 . 0 8 4 1 8 . 2 5 2 2 . 7 1903 5 7 . 0 5 4 1 1 . 3 9 6 2 0 . 6 3 9 2 . 7 1904 5 7 . 9 5 3 1 1 . 7 9 4 2 2 . 2 3 4 2 . 7 1905 6 8 . 3 6 8 1 3 . 5 0 2 2 5 . 0 3 1 2 . 7 1906 8 5 . 9 0 3 1 7 . 0 9 6 2 6 . 5 9 3 2 . 8 1907 8 5 . 6 3 3 1 8 . 7 3 3 2 3 . 3 9 6 2 . 9 1908 5 8 . 9 5 6 1 3 . 2 2 7 1 9 . 8 4 5 2 . 9 1909 5 7 . 8 0 4 1 2 . 8 3 6 2 1 . 8 3 5 2 . 9 1910 5 6 . 1 5 3 1 2 . 7 1 6 2 2 . 6 8 6 3 . 0 1911 5 5 . 0 8 9 1 3 . 7 4 9 2 4 . 8 8 2 3 . 0 1912 7 1 . 7 9 0 1 7 . 6 4 6 2 6 . 0 0 4 3 .2 1913 6 7 . 2 5 6 1 8 . 4 4 7 2 2 . 3 8 7 3 . 2 1914 6 0 . 5 5 2 1 8 . 7 7 5 2 2 . 1 8 8 3 .2 1915 7 1 . 3 8 6 2 2 . 5 5 5 6 6 . 4 8 6 4 . 0 1916 1 1 4 . 2 2 6 3 0 . 8 6 4 7 0 . 9 3 3 5 .2 1917 1 2 2 . 9 1 9 3 0 . 0 1 8 5 1 . 5 8 5 6 . 7 1918 1 1 3 . 7 0 5 2 9 . 6 2 5 5 3 . 3 2 4 7 .4 1919 8 9 . 3 6 2 2 8 . 1 3 8 4 2 . 2 0 2 8 . 2 1920 9 5 . 9 4 0 3 7 . 2 3 4 4 3 . 6 7 1 1 0 . 2 1921 6 8 . 2 6 1 2 2 . 3 9 3 2 5 . 4 3 7 6 . 6 1922 6 1 . 1 4 2 2 3 . 7 1 6 2 9 . 5 2 9 5 . 3 1923 6 4 . 8 0 0 2 3 . 7 6 6 3 2 . 5 3 6 5 . 3 1924 6 2 . 1 5 2 3 3 . 8 7 7 3 3 . 1 9 5 5 . 5 1925 6 0 . 9 4 2 3 5 . 8 5 4 3 6 . 0 4 6 5 . 3 98 T A B L E 1 . c o n t ' d * * * CURRENT METAL P R I C E S AND WHOLESALE,PRODUCER P R I C E INDEX M e t a l P r i c e s F r o m : GREAT B R I T A I N , LONDON METAL EXCHANGE P r i c e I n d e x F r o m : GREAT B R I T A I N T i m e S e r i e s , YEARLY : 1884 t o 1986 T I M E COPPER LEAD Z INC WHOLESALE P R I C E ( Y e a r ) ( £ / t o n n e ) ( £ / t o n n e ) ( x / t o n n e ) 1984=100 1926 5 7 . 0 5 5 3 0 . 5 8 4 3 3 . 5 6 6 4 . 9 1927 5 4 . 7 7 4 2 3 . 8 1 0 2 8 . 0 6 3 4 . 7 1928 6 2 . 6 9 7 2 0 . 7 2 7 2 4 . 8 8 5 4 . 9 1929 7 4 . 2 2 5 2 2 . 8 7 9 2 4 . 3 9 8 4 . 5 1930 5 3 . 7 4 8 1 7 . 7 9 1 1 6 . 3 0 8 4 . 0 1931 3 7 . 6 3 8 1 2 . 7 5 3 1 2 . 0 2 2 3 . 5 1932 3 1 . 1 8 2 1 1 . 7 2 5 1 3 . 3 3 1 3 . 4 1933 3 2 . 0 1 0 1 1 . 4 8 6 1 5 . 4 1 9 3 . 4 1934 29.. 803 1 0 . 7 6 2 1 3 . 4 4 1 3 . 5 1935 3 1 . 3 6 4 1 4 . 0 1 3 1 3 . 8 6 0 3 . 5 1936 3 7 . 8 3 4 1 7 . 3 2 1 1 4 . 6 8 4 3 . 7 1937 5 3 . 6 0 5 2 2 . 9 5 8 2 1 . 9 0 6 4 . 3 1938 4 0 . 0 6 4 1 5 . 0 2 5 1 3 . 7 6 9 4 . 0 1939 4 2 . 0 1 5 1 4 . 4 7 5 1 3 . 6 5 3 4 .1 1940 6 1 . 0 2 1 2 4 . 6 0 5 2 5 . 3 4 3 5 .4 1941 6 1 . 0 2 1 2 4 . 6 0 5 2 5 . 3 4 3 6 .1 1942 6 1 . 0 2 1 2 4 . 6 0 5 2 5 . 3 4 3 6 . 3 1943 6 1 . 0 2 1 2 4 . 6 0 5 2 5 . 3 4 3 6 . 5 1944 6 1 . 0 2 1 2 4 . 6 0 5 2 5 . 3 4 3 6 . 6 1945 6 1 . 0 2 1 2 7 . 3 4 1 2 8 . 3 5 5 6 . 7 1946 7 5 . 9 5 1 4 7 . 2 5 2 4 2 . 1 8 3 7 . 0 1947 1 2 8 . 4 7 8 8 3 . 6 5 7 6 8 . 8 9 4 7 . 6 1948 1 3 1 . 8 8 3 9 3 . 9 9 2 7 8 . 7 6 6 8 . 7 1949 1 3 0 . 9 3 8 1 0 1 . 5 6 0 8 6 . 1 3 4 9 .1 1950 1 7 5 . 9 5 8 1 0 4 . 7 3 5 1 1 7 . 3 7 6 1 0 . 4 1951 2 1 6 . 8 8 1 1 5 9 . 4 8 6 1 6 8 . 9 7 9 1 2 . 6 1952 2 5 5 . 3 7 7 1 3 4 . 5 9 4 1 4 7 . 0 5 2 1 3 . 0 1953 2 5 2 . 2 2 7 9 0 . 0 5 1 7 3 . 5 2 0 1 3 . 0 1954 2 4 5 . 4 1 1 9 4 . 9 2 6 7 7 . 0 3 0 1 3 . 1 1955 3 4 6 . 7 1 2 1 0 4 . 1 9 2 8 9 . 2 3 6 1 3 . 4 1956 3 2 3 . 8 8 3 1 1 4 . 4 7 6 9 6 . 1 7 0 1 4 . 0 1957 2 1 6 . 1 7 3 9 5 . 1 1 3 8 0 . 2 9 2 1 4 . 5 1958 1 9 4 . 7 2 1 7 1 . 6 3 3 6 4 . 8 5 9 1 4 . 6 1959 2 3 4 . 0 7 6 6 9 . 6 6 1 8 0 . 9 3 4 1 4 . 6 1960 2 4 2 . 1 1 8 7 1 . 0 0 6 8 7 . 8 8 6 1 4 . 8 1961 2 2 6 . 1 6 2 6 3 . 1 9 8 7 6 . 5 0 1 1 5 . 2 1962 2 3 0 . 4 0 2 5 5 . 3 9 9 6 6 . 3 9 3 1 5 . 5 1963 2 3 1 . 0 6 7 6 2 . 4 3 6 7 5 . 4 9 2 1 5 . 7 1964 3 4 7 . 3 0 5 9 9 . 6 5 1 1 1 6 . 2 5 9 1 6 . 2 1965 4 6 2 . 4 5 3 1 1 3 . 1 8 4 1 1 1 . 1 1 7 1 6 . 8 1966 5 4 5 . 7 1 3 9 3 . 6 4 7 1 0 0 . 3 8 1 1 7 . 2 1967 4 1 0 . 7 4 6 8 2 . 4 4 0 9 8 . 8 4 3 1 7 . 4 99 T A B L E 1 . c o n t ' d * * * CURRENT METAL P R I C E S AND WHOLESALE,PRODUCER P R I C E INDEX * * * M e t a l P r i c e s F r o m : GREAT B R I T A I N , LONDON METAL EXCHANGE P r i c e I n d e x F r o m : GREAT B R I T A I N T i m e S e r i e s , YEARLY : 1884 t o 1986 T I M E COPPER LEAD Z INC WHOLESALE P R I C E INDEX ( Y e a r ) ( £ / t o n n e ) ( £ / t o n n e ) ( £ / t o n n e ) 1984=100 1968 5 1 5 . 6 9 9 1 0 0 . 1 8 8 1 0 9 . 4 1 9 1 8 . 1 1969 6 1 1 . 4 4 1 1 2 0 . 7 6 2 1 1 9 . 2 3 6 1 8 . 8 1970 5 7 8 . 6 1 6 1 2 4 . 4 3 0 1 2 1 . 1 7 5 2 0 . 1 1971 4 4 4 . 4 3 0 1 0 3 . 7 9 0 1 2 6 . 9 6 0 2 1 . 9 1972 4 2 7 . 9 6 0 1 2 0 . 5 8 0 1 5 0 . 9 0 0 2 3 . 0 1973 7 2 6 . 8 2 0 1 7 4 . 3 4 0 3 4 4 . 6 3 0 2 4 . 7 1974 8 7 7 . 0 0 0 2 5 2 . 4 8 0 5 2 7 . 3 8 0 3 0 . 5 1975 5 5 6 . 8 1 0 1 8 5 . 3 8 0 3 3 5 . 3 4 0 3 7 . 9 1976 7 8 2 . 4 0 0 2 5 0 . 4 4 0 3 9 4 . 6 0 0 4 4 . 4 1977 7 5 0 . 2 5 0 3 5 3 . 7 9 0 3 3 7 . 7 6 0 5 3 . 2 1978 7 1 0 . 5 0 0 3 4 2 . 5 0 0 3 0 8 . 7 9 0 5 8 . 0 1979 9 3 6 . 1 9 8 5 6 6 . 4 1 5 3 4 9 . 9 8 9 6 5 . 1 1980 9 4 1 . 3 3 4 3 9 0 . 7 0 2 3 2 6 . 8 8 7 7 5 . 7 1981 8 6 4 . 6 1 2 3 6 2 . 1 7 2 4 2 3 . 4 7 3 8 2 . 9 1982 8 4 6 . 7 3 4 3 1 0 . 7 1 0 4 2 5 . 1 0 8 8 9 . 3 1983 1 0 4 9 . 3 4 0 2 8 0 . 2 9 4 5 0 5 . 0 5 1 9 4 . 2 1984 1 0 3 2 . 6 8 4 3 3 2 . 0 7 3 6 6 7 . 8 3 7 1 0 0 . 0 1985 1 1 0 4 . 4 0 4 3 0 3 . 5 9 8 5 9 4 . 6 6 0 1 0 5 . 5 1986 9 3 6 . 9 1 8 2 7 6 . 9 8 2 5 1 3 . 9 3 4 1 1 0 . 3 100 APPENDIX 2 EXPERIMENTAL SEMI-VARIOGRAMS 1 0 1 T A B L E 1. * * * EXPERIMENTAL SEMI-VARIOGRAM * * * V a r i a b l e : C O P P E R , L o n d o n M e t a l E x c h a n g e , F IGURE 6 . D e f l a t o r : WHOLESALE,PRODUCER PRICE INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1884 t o 1917 P o p u l a t i o n : 34 Mean : 2 0 2 6 . 0 p o u n d s ; s t e r l i n g / t o n n e V a r i a n c e : 1 9 1 4 1 8 . 0 ( £ / t o n n e ) 2 L a g S p a c i n g : 1 YEAR Number o f L a g s : 30 LAG NUMBER OF S E M I - VARIANCE C O - V A R I A N C E ( Y e a r s ) P R I C E PAIRS ( £ / t o n n e ) 2 ( £ / t o n n e ) 2 1 33 9 4 6 5 8 . 6 1 0 2 9 8 4 . 0 2 32 149283 .2 4 6 7 9 8 . 7 3 31 191657 .7 1 8 0 6 . 6 4 30 2 0 0 3 1 9 . 8 - 3 3 0 1 . 6 5 29 176591 .8 1 6 0 5 8 . 2 6 28 152554 .4 4 5 3 7 9 . 8 7 27 155479 .6 4 8 1 3 9 . 0 8 26 188183 .8 1 8 8 7 6 . 3 9 25 223833 .1 - 1 3 8 4 5 . 6 10 24 2 5 2 9 7 8 . 9 - 4 0 7 6 0 . 0 1 1 23 2 1 6 0 0 7 . 5 - 1 2 0 4 8 .8 1 2 22 2 3 1 5 9 5 . 2 - 3 8 3 7 5 . 9 13 21 2 4 7 4 0 2 . 2 - 4 6 4 4 5 . 3 1 4 20 3 1 0 5 3 9 . 6 - 1 0 0 2 6 4 . 7 1 5 19 2 9 8 2 3 0 . 6 - 7 3 3 0 5 . 1 1 6 18 2 5 2 8 4 1 . 7 - 4 0 8 7 2 . 8 17 17 197793.1 1 6 4 2 0 . 3 18 1 6 176026 .4 3 9 7 3 7 . 6 19 15 170829 .0 45709 .1 20 1 4 2 7 1 3 5 5 . 6 - 4 5 7 8 3 . 6 21 13 2 9 8 7 8 9 . 2 - 5 8 0 6 3 . 0 22 12 2 8 1 5 8 5 . 6 - 4 0 8 1 6 . 1 23 1 1 202857 .1 - 2 2 1 6 0 . 8 24 10 8 5 4 5 5 . 5 3 6 4 4 0 . 7 25 9 9 7 4 3 4 . 0 2 4 4 5 0 . 0 26 8 137215 .7 - 1 3 8 5 6 . 2 27 7 189453 .0 - 5 8 7 9 5 . 3 28 6 126065 .8 4 4 9 8 8 . 4 29 5 188902 .7 - 2 2 3 4 5 . 1 30 4 103399 .8 - 1 2 0 9 3 . 3 102 T A B L E 2 . * * * EXPERIMENTAL SEMI-VARIOGRAM * * * V a r i a b l e : C O P P E R , L o n d o n M e t a l E x c h a n g e , F IGURE 7 D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1918 t o 1953 P o p u l a t i o n Mean V a r i a n c e 36 1 2 1 5 . 5 p o u n d s s t e r l i n g / t o n n e 9 2 1 0 0 . 0 ( £ / t o n n e ) 2 La-g S p a c i n g : 1 YEAR Number o f L a g s : 30 LAG ( Y e a r s ) NUMBER OF P R I C E PAIRS S E M I - V A R I A N C E ( £ / t o n n e ) 2 C O - V A R I A N C E ( £ / t o n n e ) 2 1 35 18926 .3 6 8 6 0 2 . 0 2 34 3 8 2 4 9 . 6 4 2 5 5 2 . 3 3 33 47571 .3 3 0 7 0 0 . 2 4 32 5 8 4 2 6 . 8 18320 .7 5 31 6 8 1 9 4 . 7 1 0 1 5 2 . 5 6 30 8 0 1 4 0 . 2 - 9 4 0 . 9 7 29 9 1 8 6 8 . 2 - 1 3 8 3 2 . 0 8 28 9 9 2 3 4 . 6 - 1 8 6 9 3 . 4 9 27 1 0 3 8 5 3 . 5 - 2 2 1 3 2 . 1 10 26 9 3 0 9 7 . 4 - 9 3 5 2 . 4 1 1 25 8 5 5 6 8 . 2 - 1 2 5 . 1 12 24 9 2 1 4 3 . 6 - 1 0 5 5 9 . 4 13 23 1 0 2 2 6 9 . 6 - 1 9 3 8 4 . 2 14 22 115436 .7 - 2 9 2 9 5 . 5 15 21 123037 .4 - 3 4 9 0 7 . 6 16 20 1 30280.1 - 4 0 1 2 4 . 3 17 19 132562 .7 - 3 9 4 0 0 . 2 18 18 118722 .4 - 2 2 3 3 8 . 8 19 17 112015 .8 - 1 4 0 2 3 . 3 20 16 111145 .7 - 1 3 0 3 5 . 1 21 15 9 5 5 0 9 . 4 5 1 3 2 . 6 22 14 7 6 2 1 6 . 2 2 6 3 4 3 . 2 23 13 7 0 8 8 4 . 9 3 7 8 7 1 . 7 24 12 8 1 5 7 6 . 2 3 5 2 2 2 . 9 25 1 1 1 1 5 3 0 0 . 5 7288 .1 26 10 145936 .2 - 1 2 8 7 4 . 3 27 9 1 7 2 3 6 5 . 5 - 2 9 0 7 6 . 0 28 8 1 8 2 4 2 0 . 5 -24031 .7 29 7 1 7 2 4 9 5 . 6 12834 .2 30 6 1 9 4 2 5 7 . 6 - 7 7 4 7 . 8 1 0 3 T A B L E 3 . * * * EXPERIMENTAL SEMI-VARIOGRAM * * * V a r i a b l e : C O P P E R , L o n d o n M e t a l E x c h a n g e , F IGURE 8 . D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1954 t o 1986 P o p u l a t i o n : 33 Mean : 1 8 4 8 . 0 p o u n d s s t e r l i n g / t o n n e V a r i a n c e : 4 9 9 2 3 0 . 0 ( £ / t o n n e ) 2 L a g S p a c i n g : 1 YEAR Number o f L a g s : 30 L A G NUMBER OF S E M I - V A R I A N C E C O - V A R I A N C E ( Y e a r s ) P R I C E PAIRS ( £ / t o n n e ) 2 ( £ / t o n n e ) 2 1 32 1 2 3 9 6 0 . 5 387411 . 1 2 31 2 4 3 1 3 9 . 4 2 6 2 9 4 8 . 8 3 30 2 5 2 5 0 1 . 4 2 5 7 6 8 5 . 7 4 29 2 6 3 8 4 9 . 9 2 5 2 0 2 3 . 6 5 28 3 5 6 8 3 0 . 0 153094 .4 6 27 4 9 2 9 5 6 . 7 1 6 4 5 5 . 6 7 26 579598 .1 - 6 3 1 9 0 . 3 8 25 6 2 1 9 7 9 . 5 - 9 5 7 6 7 . 6 9 24 6 7 7 7 5 5 . 0 - 1 4 7 4 7 0 . 6 10 23 7 3 8 8 6 6 . 7 - 2 0 1 2 9 1 . 8 1 1 22 840594 .1 - 2 8 0 9 2 8 . 0 12 21 9 2 3 6 9 0 . 6 - 3 4 9 4 0 3 . 4 1 3 20 8 0 5 8 1 8 . 7 - 2 4 6 1 7 6 . 3 1 4 19 7 1 8 5 4 2 . 7 - 1 5 8 3 1 6 . 2 1 5 18 793695 .1 - 2 1 4 8 9 9 . 7 16 17 842334 .1 - 2 6 4 5 3 6 . 4 17 16 7 0 5 5 0 1 . 0 - 1 4 5 3 8 6 . 4 18 15 5 1 6 5 8 7 . 9 - 7 4 7 . 4 19 14 4 8 4 0 5 8 . 6 11190 .8 20 13 4 9 6 7 8 2 . 6 - 2 6 2 3 0 . 7 21 12 3 1 9 6 4 8 . 2 2 1 2 5 5 . 7 22 1 1 2 5 3 7 5 0 . 9 2 0 4 0 4 . 6 23 10 2 2 3 5 9 8 . 9 4 9 9 5 9 . 7 24 9 2 4 6 6 1 1 . 0 6 7 8 3 7 . 4 25 8 3 1 4 5 4 3 . 5 3 5 4 7 5 . 5 26 7 414774 .1 - 3 2 2 0 . 6 27 6 4 7 2 9 8 9 . 5 3 2 5 3 . 2 28 5 5 0 9 9 0 5 . 5 3 9 8 7 6 . 6 29 4 6 2 6 2 0 1 . 6 3 1 0 9 0 . 0 30 3 8 7 0 5 7 9 . 5 - 2 7 4 4 . 9 104 T A B L E 4 . * * * EXPERIMENTAL SEMI-VARIOGRAM RESIDUALS * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l P o p u l a t i o n Mean V a r i a n c e L a g S p a c i n g : 1 YEAR COPPER, L o n d o n M e t a l E x c h a n g e , F IGURE 1 1 . WHOLESALE,PRODUCER P R I C E INDEX 1984=100 Y E A R L Y , 1954 t o 1986 QUADRATIC 33 0 .0 p o u n d s s t e r l i n g / t o n n e 2 7 0 3 9 2 . 6 ( £ / t o n n e ) 2 Number o f L a g s : 30 LAG ( Y e a r s ) NUMBER OF P R I C E PAIRS S E M I - V A R I A N C E ( £ / t o n n e ) C O - V A R I A N C E ( £ / t o n n e ) 1 32 122493 .4 1 5 8 5 9 7 . 8 2 31 2 4 3 4 2 2 . 0 2 7 1 4 7 . 6 3 30 2 5 2 6 0 4 . 9 2 1 3 2 5 . 4 4 29 2 5 0 3 6 1 . 3 30961 .0 5 28 3 1 0 8 1 4 . 2 - 2 9 3 5 1 . 4 6 27 4 0 6 2 4 9 . 7 - 1 2 3 0 0 4 . 7 7 26 4 4 4 6 0 0 . 9 - 1 5 7 3 6 3 . 3 8 25 4 2 7 3 1 8 . 7 - 1 3 8 1 7 0 . 8 9 24 3 9 3 8 3 7 . 0 - 1 0 9 7 3 4 . 0 10 23 3 5 2 1 4 4 . 2 - 7 4 6 6 4 . 1 1 1 22 3 7 6 2 8 3 . 4 - 8 9 3 8 3 . 7 1 2 21 3 6 9 9 8 7 . 4 - 8 1 9 9 3 . 5 13 20 2 8 2 9 4 8 . 8 -4671 . 5 14 19 2 1 2 4 0 4 . 8 6 9 9 8 6 . 0 1 5 18 2 3 2 6 5 7 . 7 5 3 3 9 3 . 1 1 6 17 2 5 8 2 5 3 . 9 16931 .2 17 16 180671.1 93669 .1 18 1 5 7 7 0 6 5 . 5 1 8 6 8 4 8 . 7 19 14 9 6 0 2 6 . 4 1 7 1 4 4 8 . 2 20 13 167634.1 8 2 6 7 2 . 6 21 12 1 9 9 5 5 0 . 6 - 1 2 3 9 9 . 5 22 1 1 2 6 4 6 4 2 . 6 - 8 8 2 6 3 . 2 23 10 3 0 9 6 5 4 . 6 - 1 1 8 3 9 9 . 5 24 9 2 8 2 6 1 7 . 6 - 9 4 6 4 7 . 8 25 8 2 8 4 9 5 8 . 2 - 1 1 0 5 5 3 . 4 26 7 3 0 7 8 4 5 . 2 - 1 4 3 8 4 4 . 8 27 6 3 0 5 6 6 4 . 8 - 1 3 7 3 0 0 . 7 28 5 256921 .0 - 8 5 6 5 6 . 5 29 4 1 5 1 6 7 4 . 5 -26511 .7 30 3 8 3 9 3 3 . 3 2 5 2 3 4 . 3 1 0 5 T A B L E 5 . * * * EXPERIMENTAL SEMI-VARIOGRAM RESIDUALS * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l P o p u l a t i o n Mean V a r i a n c e C O P P E R , L o n d o n M e t a l E x c h a n g e , F I G U R E 14, WHOLESALE,PRODUCER P R I C E INDEX 1984=100 Y E A R L Y , 1973 t o 1986 QUADRATIC 13 0 . 0 p o u n d s s t e r l i n g / t o n n e 6 2 9 9 8 . 6 ( £ / t o n n e ) 2 L a g S p a c i n g : 1 YEAR Number o f L a g s : 30 LAG NUMBER OF S E M I - V A R I A N C E C O - V A R I A N C E ( Y e a r s ) P R I C E PAIRS ( £ / t o n n e ) 2 ( £ / t o n n e ) 2 1 13 7 3 1 6 3 . 2 - 9 1 7 7 . 9 2 12 67129 .1 - 5 3 3 6 . 7 3 1 1 5 2 5 2 9 . 7 - 5 2 5 2 . 2 4 10 7 1 2 9 9 . 5 - 2 2 9 1 3 . 6 5 9 5 9 4 1 7 . 8 - 5 4 8 5 . 9 6 8 5 9 1 7 1 . 0 9 1 9 . 4 7 7 69646 .1 - 6 3 1 6 . 0 8 6 8 2 1 0 3 . 9 - 1 5 4 3 5 . 1 9 5 6 4 4 3 0 . 2 1 8 6 1 8 . 4 10 4 7 9 3 1 3 . 0 2 0 0 7 5 . 2 1 1 3 54793 .1 9 5 3 6 3 . 5 12 2 1 2 4 7 2 6 . 4 - 4 1 0 8 5 . 9 13 1 8 4 8 5 4 . 7 0 . 0 106 T A B L E 6 . * * * EXPERIMENTAL SEMI-VARIOGRAM * * * V a r i a b l e : L E A D , L o n d o n M e t a l E x c h a n g e , F IGURE 18 . D e f l a t o r : WHOLESALE,PRODUCER PRICE INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1954 t o 1986 P o p u l a t i o n : 33 Mean : 5 4 2 . 3 p o u n d s s t e r l i n g / t o n n e V a r i a n c e : 2 5 6 7 3 . 8 ( £ / t o n n e ) z L a g S p a c i n g : 1 YEAR Number o f L a g s : 30 L A G NUMBER OF S E M I - V A R I A N C E C O - V A R I A N C E ( Y e a r s ) P R I C E PAIRS ( £ / t o n n e ) 2 ( £ / t o n n e ) 2 1 32 9 1 6 5 . 2 1 6 0 1 7 . 6 2 31 16503.1 7 4 6 2 . 1 3 30 19581 .5 3 3 8 3 . 0 4 29 2 1 3 2 2 . 7 1469 .8 5 28 2 1 9 3 5 . 2 1 0 9 2 . 2 6 27 3 0 5 7 2 . 3 - 7 2 4 8 . 3 7 26 3 5 4 1 9 . 5 - 1 1 4 9 7 . 2 8 25 2 7 0 6 6 . 3 - 4 8 7 0 . 7 9 24 2 2 5 4 5 . 6 - 3 0 8 . 1 10 23 2 3 2 8 0 . 0 - 1 1 7 1 . 3 1 1 22 2 9 7 1 9 . 8 - 6 8 7 0 . 0 1 2 21 3 2 8 8 7 . 7 - 9 2 3 5 . 1 . 13 20 2 5 4 3 2 . 3 - 2 5 2 5 . 8 1 4 19 2 1 2 7 2 . 4 2 0 3 3 . 6 15 18 2 5 8 0 5 . 4 - 1 3 5 2 . 6 1 6 1 7 3 2 2 9 9 . 3 - 6 8 9 1 . 7 1 7 16 3 0 0 2 9 . 5 - 3 1 1 8 . 1 18 15 2 4 7 7 7 . 9 3 6 2 7 . 6 19 14 2 3 8 1 3 . 5 6 8 2 7 . 0 20 13 2 4 9 8 5 . 6 7 4 3 4 . 9 21 12 2 3 9 2 5 . 0 8 7 6 3 . 0 22 1 1 14306 .9 2 0 7 0 6 . 6 23 10 7 3 4 7 . 7 3 1 5 9 6 . 3 24 9 14866.1 2 8 1 5 5 . 9 25 8 2 8 2 3 0 . 4 2 0 5 8 6 . 2 26 7 4 4 4 2 3 . 4 8 9 3 6 . 6 27 6 6 1 2 5 3 . 3 5 0 7 9 . 0 28 5 8 0 1 0 3 . 9 3912 .1 29 4 103193.1 1 5 0 8 . 3 30 3 1 1 9 1 6 3 . 9 - 1 8 9 1 . 6 1 0 7 T A B L E 7 . * * * EXPERIMENTAL SEMI-VARIOGRAM RESIDUALS * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l P o p u l a t i o n Mean V a r i a n c e L a g S p a c i n g : 1 YEAR L E A D , L o n d o n M e t a l E x c h a n g e , F I G U R E 2 2 . WHOLESALE,PRODUCER P R I C E INDEX 1984=100 Y E A R L Y , 1954 t o 1986 QUADRATIC 33 0 . 0 p o u n d s s t e r l i n g / t o n n e 2 0 5 1 5 . 9 ( £ / t o n n e ) 2 L A G ( Y e a r s ) NUMBER OF P R I C E PAIRS Number o f L a g s : 30 S E M I - V A R I A N C E ( £ / t o n n e ) C O - V A R I A N C E ( £ / t o n n e ) 1 32 9 0 7 1 . 4 1 1 9 4 1 . 6 2 31 16171 .2 4 5 9 7 . 2 3 30 18944 .0 1 5 8 2 . 2 4 29 2 0 0 3 0 . 6 9 6 8 . 8 5 28 19709 .6 1 7 1 2 . 3 6 27 2 7 4 2 5 . 9 - 5 8 5 7 . 3 7 26 3 1 4 9 6 . 6 - 9 6 8 9 . 9 8 25 2 4 1 1 8 . 4 - 5 1 0 9 . 9 9 24 19196 .3 - 6 9 0 . 7 10 23 2 0 0 6 6 . 2 - 2 1 5 9 . 5 1 1 22 2 6 2 3 5 . 5 - 7 7 7 4 . 6 12 21 2 8 5 0 4 . 6 - 9 4 7 3 . 0 1 3 20 2 3 1 4 9 . 9 - 4 7 0 3 . 4 1 4 19 2 0 2 7 5 . 9 - 1 2 9 8 . 1 1 5 18 2 4 3 0 8 . 3 - 4 4 9 2 . 9 1 6 1 7 2 9 5 3 4 . 6 - 9 4 3 4 . 1 17 16 2 7 7 4 7 . 7 -6241 . 9 18 1 5 23483 .1 - 1 7 5 . 7 19 1 4 2 1 1 6 5 . 3 3 0 1 8 . 3 20 1 3 2 1 0 7 9 . 5 4 0 6 5 . 3 21 1 2 18321 .0 4 8 7 4 . 6 22 1 1 10698 .3 1 5 5 0 1 . 0 23 1 0 4 4 7 8 . 3 25153 .1 24 9 6 0 1 4 . 4 2 2 2 3 4 . 0 25 8 10698 .3 1 6 0 2 7 . 4 26 7 13653 .2 4 5 4 3 . 0 27 6 2 0 6 3 0 . 9 1 2 0 8 . 5 28 5 2 6 9 9 0 . 7 1 1 8 1 . 3 29 4 3 4 7 6 4 . 3 951 .6 30 3 4 1 2 6 0 . 6 - 8 8 3 . 4 108 T A B L E 8 . * * * EXPERIMENTAL SEMI-VARIOGRAM RESIDUALS * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l P o p u l a t i o n Mean V a r i a n c e L a g S p a c i n g L E A D , L o n d o n M e t a l E x c h a n g e , F IGURE 2 5 . WHOLESALE,PRODUCER P R I C E INDEX 1984=100 Y E A R L Y , 1979 t o 1986 QUADRATIC 8 p o u n d s s t e r l i n g / t o n n e 2 0 . 0 3 1 4 2 . 3 ( £ / t o n n e ) 1 YEAR L A G ( Y e a r s ) NUMBER OF P R I C E PAIRS Number o f L a g s : 30 S E M I - V A R I A N C E ( £ / t o n n e ) C O - V A R I A N C E ( £ / t o n n e ) 1 7 3 4 1 7 . 2 - 4 5 6 . 0 2 6 3657.1 - 1 2 2 7 . 5 3 5 3433.1 - 1 0 2 2 . 9 4 4 4 7 7 1 . 2 - 1 9 2 0 . 9 5 3 2 4 6 7 . 2 2 8 7 7 . 7 6 2 1139 .9 6 7 9 2 . 9 7 1 8 7 5 8 . 5 0 . 0 1 0 9 T A B L E 9 . * * * EXPERIMENTAL SEMI-VARIOGRAM * * * V a r i a b l e : Z I N C , L o n d o n M e t a l E x c h a n g e , F IGURE 2 7 . D e f l a t o r : WHOLESALE,PRODUCER PRICE INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1954 t o 1986 P o p u l a t i o n Mean V a r i a n c e 31 5 8 8 . 3 p o u n d s s t e r l i n g / t o n n e 1 1 8 3 2 . 4 ( £ / t o n n e ) 2 L a g S p a c i n g : 1 YEAR Number o f L a g s : 30 L A G ( Y e a r s ) NUMBER OF P R I C E PAIRS S E M I - V A R I A N C E ( £ / t o n n e ) 2 C O - V A R I A N C E ( £ / t o n n e ) 1 29 4 7 6 8 . 6 6 2 6 0 . 7 2 27 10710 .0 - 8 6 2 . 9 3 26 1 2 2 3 7 . 9 - 6 1 4 . 2 4 25 13848 .4 - 1 5 5 . 3 5 24 15267 .0 - 1 8 2 2 . 9 6 23 16778 .6 - 3 5 1 8 . 1 7 22 16533 .2 - 3 4 1 3 . 2 8 21 13599 .4 221 .4 9 20 13302 .4 6 6 . 9 10 19 16523 .3 - 3 0 4 1 . 6 1 1 18 11913 .5 - 4 7 7 . 6 12 1 7 9 1 8 9 . 3 2 1 4 . 0 1 3 17 15045 .2 - 4 9 9 0 . 0 1 4 1 7 16996 .0 - 6 3 7 5 . 5 15 1 6 13306 .6 - 2 6 5 0 . 6 1 6 1 5 12695 .0 - 1 3 1 3 . 4 17 14 15796 .2 - 3 1 8 4 . 4 18 1 3 16328 .7 - 2 8 3 8 . 1 19 12 10296 .3 4 9 8 7 . 6 20 1 2 6 6 9 9 . 3 9 1 1 3 . 3 21 1 2 11698 .3 3 6 5 7 . 2 22 1 1 11823 .2 - 2 3 5 . 3 23 10 5 8 3 4 . 3 1 0 0 . 5 24 9 5 4 6 4 . 4 7 2 7 . 2 25 8 7377 .1 - 1 1 8 7 . 6 26 7 11405 .4 - 4 7 6 7 . 4 27 6 8 3 2 0 . 6 - 2 1 5 3 . 6 28 5 3 0 3 1 . 7 5 5 4 5 . 9 29 4 3 1 9 9 . 7 4 0 7 2 . 6 30 3 1 0 9 4 4 . 6 - 5 0 2 3 . 2 110 T A B L E 10 . * * * EXPERIMENTAL SEMI-VARIOGRAM RESIDUALS * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l P o p u l a t i o n Mean V a r i a n c e L a g S p a c i n g : 1 YEAR Z I N C , L o n d o n M e t a l E x c h a n g e , F IGURE 3 0 . WHOLESALE,PRODUCER PRICE INDEX 1984=100 Y E A R L Y , 1954 t o 1986 L I N E A R 33 0 . 0 p o u n d s s t e r l i n g / t o n n e 6 6 4 5 9 . 4 ( £ / t o n n e ) 2 Number o f L a g s : 30 L A G ( Y e a r s ) NUMBER OF P R I C E PAIRS S E M I - V A R I A N C E ( £ / t o n n e ) ' C O - V A R I A N C E ( £ / t o n n e ) ' 1 32 2 5 7 4 7 . 9 4 3 4 2 2 . 6 2 31 5 4 2 5 5 . 7 1 6 0 3 4 . 2 3 30 6 7 3 8 4 . 6 4 8 2 4 . 8 4 29 7 8 5 5 3 . 2 - 4 6 0 3 . 2 5 28 8 4 4 2 9 . 9 - 9 4 1 1 . 6 6 27 9 5 3 5 6 . 3 - 1 8 5 7 6 . 6 7 26 9 9 2 9 2 . 7 - 2 0 8 7 7 . 0 8 25 9 5 8 5 2 . 5 - 1 5 0 2 1 . 8 9 24 92019.1 - 9 3 8 9 . 2 10 23 9 4 9 0 3 . 6 - 9 6 9 6 . 4 1 1 22 109502 .9 - 2 1 8 7 4 . 7 1 2 21 121114 .2 - 3 0 6 6 3 . 6 13 20 8 6 5 6 3 . 8 - 1 8 1 8 9 . 1 1 4 1 9 6 4 9 3 4 . 8 - 5 2 6 2 . 8 1 5 18 7 2 0 4 4 . 7 - 9 0 6 9 . 3 16 1 7 77026.1 - 1 0 5 9 7 . 2 17 1 6 6 8 6 5 4 . 7 2 8 0 5 . 2 18 15 6 3 4 1 7 . 8 1 3 7 8 0 . 8 19 1 4 6 7 3 1 2 . 2 1 5 5 2 4 . 4 20 13 5 2 8 5 9 . 6 1 1 4 6 1 . 6 21 1 2 11668 .0 4 4 0 5 . 2 22 1 1 12613 .7 3 4 0 . 5 23 10 7 4 5 1 . 5 530 .1 24 9 8 0 2 6 . 8 9 9 7 . 0 25 8 10956 .6 - 1 0 5 8 . 8 26 7 15781 .3 - 4 7 3 4 . 4 27 6 1 2 0 4 7 . 5 - 1 9 8 6 . 9 28 5 6 9 6 9 . 8 5 7 4 4 . 6 29 4 8 4 1 8 . 9 4 2 0 6 . 4 30 3 17319 .8 - 4 9 2 0 . 1 111 T A B L E 1 1 . * * * EXPERIMENTAL SEMI-VARIOGRAM RESIDUALS * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l P o p u l a t i o n Mean Z I N C , L o n d o n M e t a l E x c h a n g e , F IGURE 3 3 . WHOLESALE,PRODUCER P R I C E INDEX 1984=100 Y E A R L Y , 1974 t o 1986 QUADRATIC 13 0 . 0 p o u n d s s t e r l i n g / t o n n e V a r i a n c e : 2 5 6 4 0 . 8 ( £ / t o n n e ) 2 L a g S p a c i n g : 1 YEAR Number o f L a g s : 30 L A G NUMBER OF S E M I - V A R I A N C E C O - V A R I A N C E ( Y e a r s ) P R I C E PAIRS ( £ / t o n n e ) 2 ( £ / t o n n e ) 2 1 12 2 2 6 1 9 . 2 - 1 6 0 3 . 1 2 1 1 1 8 9 2 4 . 3 7 3 9 . 8 3 10 2 3 2 1 7 . 2 - 4 1 1 9 . 8 4 9 2 8 5 1 5 . 8 - 9 5 5 9 . 4 5 8 26010 .1 - 4 7 2 1 . 9 6 7 3 3 3 5 3 . 4 - 9 9 3 8 . 3 7 6 3 0 6 3 2 . 2 - 1 9 1 8 . 9 8 5 30446 .1 3 8 8 3 . 8 9 4 3 3 3 9 0 . 5 7 4 1 2 . 0 10 3 19987 .8 39431 .2 1 1 2 3 2 1 7 1 . 0 6 5 6 0 8 . 6 1 2 1 162790 .4 0 . 0 112 APPENDIX 3 SQUARED DIFFERENCES 1 1 3 T A B L E 1. * * * SQUARED D I F F E R E N C E S COPPER 1885 TO 1917 * * * V a r i a b l e : COPPER, L o n d o n M e t a l E x c h a n g e D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1884 t o 1917 V a r i o g r a m : S P H E R I C A L , F IGURE 6 . K r i g i n g : ORDINARY KRIGING YEAR P R I C E RANDOM ( D I F F . ) 2 KRIGING ( D I F F . ) 2 K R I G I N G ( D I F F . ) 2 WALK N = 4 N = 5 ( £ / m . t . ) ( £ / m . t .) ( £ / m . t . ) 2 ( £ / m . t . ) ( £ / m . t . ) 2 ( £ / m . t . ) ( £ / m . t • ) : 1884 1657 .0 - - " — — — — 1885 1428 . 7 1657 . 0 5 2 1 2 0 . 9 1657 . 0 5 2 1 2 0 . 9 1 6 5 7 . 0 52120 . 9 1886 1408 . 9 1428 . 7 3 9 2 . 0 1481 . 7 5 2 9 9 . 8 1 481 . 7 5299 .8 1887 1677 . 6 1408 . 9 7 2 1 9 9 . 7 1446 . 1 5 3 5 9 2 . 2 1 4 4 6 . 1 5 3 9 2 . 2 1888 2867 . 0 1677 . 6 1414672 . 3 1645 . 4 1492306 . 5 1 6 4 5 . 4 1492306 . 5 1889 1687 .8 2 8 6 7 . 0 1390512 . 6 2 4 6 8 . 4 6 0 9 3 3 6 . 4 2 4 7 4 . 9 619526 .4 1890 1841 . 6 1687 . 8 2 3 6 5 4 . 4 1744 . 4 9 4 4 7 . 8 1 7 5 6 . 3 7276 . 1 1891 1688 . 5 1841 . 6 2 3 4 3 9 . 6 1723 . 6 1232 . 0 1 8 0 0 . 2 12476 . 9 1892 1604 . 9 1688 . 5 6 9 8 9 . 0 1766 . 3 2 6 0 5 0 . 0 1 921 . 0 99919 .2 1893 1538 . 7 1604 . 9 4 3 8 2 . 4 1844. 8 9 3 6 9 7 . 2 1 6 2 9 . 7 8281 .0 1 894 1528 .0 1538 . 7 1 14. 5 1 5 8 7 . 0 3481 . 0 1 601 . 9 5461 .2 1895 1692 .0 1528 . 0 2 6 8 9 6 . 0 1 5 9 4 . 0 9 6 0 4 . 0 1 5 5 9 . 0 •17698 .0 1896 1923 . 5 1692 . 0 5 3 5 9 2 . 2 1 6 7 2 . 2 6 3 1 5 1 . 7 1 6 5 8 . 6 70172 .0 1897 1934 . 1 1923 . 5 112 . 4 1826 . 6 11556 . 2 1 8 2 7 . 4 1 1384 . 9 1898 1961 . 9 1934 . 1 7 7 2 . 8 1842 . 2 14328 . 1 1 8 5 7 . 2 10959 .8 1899 2900 . 9 1961 . 9 8 8 1 7 2 1 . 0 1865 . 7 1071639 . 0 1 9 0 9 . 2 983468 . 9 1 900 2587 . 9 2 9 0 0 . 9 9 7 9 6 9 . 0 2 5 4 6 . 0 1755 . 6 2621 . 3 1115 .6 1 901 2441 . 7 2 5 8 7 . 9 2 1 3 7 4 . 4 2471 . 4 8 8 2 . 1 2 4 9 6 . 6 3014 . 0 1902 1912 . 3 2441 . 7 2 8 0 2 6 4 . 4 2331 . 1 175393 . 4 2 3 7 0 . 2 209672 . 4 1903 21 1 3 . 1 1912 . 3 4 0 3 2 0 . 6 2 0 1 9 . 1 8 8 3 6 . 0 2 1 5 1 . 0 1436 .4 1 904 21 46 .4 2 1 1 3 . 1 1 1 0 8 . 9 2 2 5 7 . 9 12432 . 2 2181 . 5 1232 .0 1905 2532 . 1 2 1 4 6 . 4 148764 . 5 2 2 4 3 . 1 8 3 5 2 1 . 0 2 1 9 6 . 4 112694 . 5 1 906 3068 . 0 2 5 3 2 . 1 2 7 8 1 8 8 . 8 2 4 4 8 . 8 3 8 3 4 0 8 . 6 2 3 7 7 . 0 477481 .0 1 907 2952 . 9 3 0 6 8 . 0 13248 . 0 2 7 6 4 . 0 3 5 6 8 3 . 2 2 8 3 3 . 7 1 4208 . 6 1 908 2033 .0 2 9 5 2 . 9 8 4 6 2 1 6 . 0 2 7 7 3 . 2 5 4 7 8 9 6 . 0 2 8 1 4 . 2 610273 . 4 1909 1993 .2 2 0 3 3 . 0 1584 . 0 2 1 5 9 . 0 2 7 4 8 9 . 6 2 2 2 2 . 9 52762 . 1 1910 1871 .8 1 9 9 3 . 2 14738 . 0 2 1 4 0 . 6 7 2 2 5 3 . 4 2 1 9 9 . 3 107256 .2 1911 1836 .3 1871 . 8 1 2 6 0 . 2 2 1 5 9 . 0 104135 . 3 2 0 8 3 . 9 61305 .8 1912 2243 .4 1936 . 3 165730 . 4 2 0 5 2 . 1 3 6 5 9 5 . 7 1 8 7 6 . 8 134395 .6 1913 2101 . 7 2 2 4 3 . 4 2 0 0 7 8 . 9 21 5 5 . 6 2 9 0 5 . 2 21 5 4 . 4 2777 . 3 1914 1892 .2 2101 . 7 4 3 8 9 0 . 2 2 0 8 7 . 5 3 8 1 4 2 . 1 2 0 7 2 . 7 32580 .2 1915 1784 . 6 1892 . 2 1 1 5 7 7 . 8 1901 . 6 13689 . 0 1 9 0 4 . 8 1 4448 .0 1916 21 96 .6 1784 . 6 169744 . 0 1825 . 5 137715 . 2 1881 . 4 99351 .0 1917 1834 . 6 2 1 9 6 . 6 131044 . 0 2 1 6 2 . 8 107715 . 2 2 1 3 6 . 0 90842 .0 6 2 4 7 6 7 2 . 9 5 4 7 6 7 7 9 . 7 5307291 .2 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m . t . : M e t r i c T o n s . 1 1 4 T A B L E 2 . * * * SQUARED D I F F E R E N C E S COPPER 1885 TO 1917 * * * V a r i a b l e : C O P P E R , L o n d o n M e t a l E x c h a n g e D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1884 t o 1917 V a r i o g r a m : HOLE E F F E C T , F IGURE 6 . K r i g i n g : ORDINARY KRIGING YEAR P R I C E KRIGING ( D I F F . ) 2 KRIGING ( D I F F . ) 2 K R I G I N G ( D I F F . ) 2 N = 6 N = 7 N = 8 ( S/m.t . ) ( £ / m . t .) ( £ / m . t . ) 2 ( V m . t . ) ( £ / m . t . ) 2 ( £ / m . t . ) ( £ / m . t . ) 1884 1657 .0 - - - — —• — 1885 1428 . 7 1657 . 0 5 2 1 2 0 . 9 1657 . 0 5 2 1 2 0 . 9 1657 .0 52120 .9 1-886 1 408 . 9 1476 . 9 4624 . 0 1476 . 9 4 6 2 4 . 0 1 476 .9 4624 .0 1887 1677 .6 1420 . 8 6 5 9 4 6 . 2 1420 . 8 6 5 9 4 6 . 2 1 420 .8 65946 .2 1888 2867 .0 1 6 1 7 . 9 1560250 . 8 1617 . 9 1560250 . 8 1617 .9 1560250 .8 1889 1687 .8 2 4 8 7 . 2 6 3 9 0 4 0 . 4 2 4 8 7 . 2 6 3 9 0 4 0 . 4 2487 .2 639040 .4 1890 1841 . 6 1807 . 7 1 145 . 1 1807 . 7 1 145 . 1 1807 .7 1 1 45 . 1 1891 1688 . 5 1610 . 5 6081 . 9 1600 . 3 7 7 7 3 . 6 1600 .3 7 7 3 . 69 1892 1 604 . 9 1630 . 3 6 4 6 . 7 1 6 0 7 . 7 7 . 9 1600 .7 17 . 5 1893 1 538 .7 1829 . 8 8 4 7 3 9 . 7 1787 . 6 61971 . 0 1 792 .3 64300 . 1 1894 1528 .0 1871 . 0 117603 . 6 1813 . 7 8 1 5 9 8 . 1 1839 .6 97046 . 1 1895 1692 .0 1 621 . 7 4 9 4 5 . 3 1697 . 1 1 6 6 . 4 1 721 . 1 844 .6 1896 1923 . 5 1735 . 3 3 5 4 1 0 . 8 1713 . 8 4 3 9 4 3 . 5 1 636 .4 82356 .2 1897 1 934 . 1 1847 . 1 7 5 6 6 . 0 1834 . 4 9 9 5 0 . 2 1841 .6 8554 . 3 1898 1 961 . 9 1840 . 9 14641 . 6 1 8 1 7 . 6 20821 . 2 1813 .6 21996 .7 1899 2900 . 9 1821 . 8 1 164411 . 0 1796 . 2 1 2 2 0 4 2 5 . 0 1801 .0 1209790 .0 1900 2587 . 9 2401 . 3 3 4 8 4 0 . 8 2 3 5 8 . 1 52841 . 5 2369 . 1 47875 . 1 1901 2441 .7 2401 . 6 1606. 9 2 3 3 5 . 9 11194 . 3 2362 .3 6301 .7 1902 1912 . 3 2 2 6 4 . 4 124032 . 4 2 2 0 9 . •7 8 8 4 5 7 . 1 2249 . 1 1 13490 .8 1903 21 13 . 1 1937 . 8 3 0 7 2 5 . 2 1910 . 4 4 1 0 9 9 . 8 1939 .6 30096 . 1 1904 2146 . 4 2 1 6 6 . 1 3 8 9 . 1 2 1 3 8 . 1 6 8 . 3 2141 .9 19 . 9 1905 2532 . 1 2 4 1 5 . 3 13643. 2 2351 . 8 32511 . 4 2361 .4 291 53 . 5 1906 3068 .0 2 5 5 2 . 9 2 6 5 2 4 0 . 1 2551 . 0 2 6 7 2 1 8 . 1 2596 .4 222333 .6 1907 2952 . 9 2 7 9 4 . 1 2 5 1 9 0 . 7 2 7 6 7 . 7 3 4 2 7 4 . 6 2748 .9 41598 . 5 1908 2033 .0 2 6 4 7 . 5 4 1 1 5 1 3 . 4 2 6 4 8 . 2 3 7 8 4 9 5 . 9 2650 . 1 380873 .7 1909 1 993 .2 2 1 1 7 . 4 15415 . 8 2 0 6 8 . 6 5 6 8 3 . 5 2061 .5 4665 .8 1910 1871 .8 2 0 4 5 . 7 3 0 2 3 9 . 1 2 0 2 6 . 2 2 3 8 5 7 . 3 2045 .9 30312 . 5 191 1 1836 . 3 2 2 0 8 . 8 138782 . 9 2 1 7 4 . 9 1 1 4 6 5 5 . 6 2173 .2 113503 .2 1912 2243 .4 2301 . 6 3 3 8 7 . 1 2 2 7 2 . 0 8 1 3 . 7 2285 .4 1758 . 9 1913 2101 .7 2 3 7 6 . 5 7 5 4 9 0 . 7 2 3 8 4 . 8 8 0 1 3 3 . 1 2395 .9 86501 .4 1914 1892 .2 2131 . 2 5 7 1 0 5 . 8 2 1 5 9 . 2 7 1 2 8 0 . 0 2129 .4 56236 . 5 1915 1784 .6 1933 . 3 2 2 0 8 4 . 9 1 9 1 3 . 5 1 6 5 9 5 . 5 1858 .7 5485 .3 1916 2196 .6 1810 . 5 148122 . 6 1800 . 1 1 5 7 2 6 6 . 8 1 7 9 7 8 . 0 158955 .7 1917 1834 .6 2 1 1 7 . 2 7 9 8 6 6 . 1 2 0 9 3 . 4 6 6 9 9 0 . 8 2085 .8 63105 .8 5 2 3 7 8 4 6 . 0 5 2 1 3 2 1 6 . 5 5208069 .3 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m . t . : M e t r i c T o n s . 115 T A B L E 3 . * * * SQUARED D I F F E R E N C E S COPPER 1919 TO 1953 * * * V a r i a b l e : COPPER, L o n d o n M e t a l E x c h a n g e D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1918 t o 1953 V a r i o g r a m : S P H E R I C A L , F IGURE 7 . K r i g i n g : ORDINARY KRIGING YEAR P R I C E RANDOM ( D I F F . ) KRIGING ( D I F F . ) KRIGING ( D I F F . ) WALK N = 12 N = 1 3 •-( £ / m . t . ) ( £ / m . t . ) ( £ / m . t . ) 2 ( £ / m . t . ) ( £ / m . t . ) 2 ( £ / m . t . ) ( £ / m . t . ) 1918 1 536 . 6 - - — — — — 1919 1 089 .8 1536 . 6 199630 . 2 1536 . 6 199630 . 9 1 5 3 6 . 6 199630 . 2 1920 940 .6 1089 . 8 2 2 2 6 0 . 6 1 172 . 9 5 3 9 6 3 . 3 1 172 . 9 5 3 9 6 3 . 3 1921 1 034 . 3 9 4 0 . 6 8 7 7 9 . 7 9 8 9 . 7 1989 . 2 9 8 9 . 7 1 9 8 9 . 2 1922 1 153 . 6 1034 . 3 14232 . 5 1034 . 2 14256 . 4 1 0 3 4 . 2 14256 . 4 1923 1 222 .6 1 1 5 3 . 6 4761 . 0 1141 . 6 6561 . 0 1 1 4 1 . 6 6561 . 0 1 924 1 1 30 .0 1222 . 6 8 5 7 4 . 8 1219 . 4 7 9 9 2 . 4 1 2 1 9 . 4 7 9 9 2 . 4 1925 1 149 .8 1 130 . 0 3 9 2 . 0 1161 . 6 139 . 2 1 1 6 1 . 6 139 . 2 1926 1 1 64 .4 1 1 4 9 . 8 2 1 3 . 2 1 169 . 3 2 4 . 0 1 169 . 3 2 4 . 0 1 927 1 165 .4 1164 . 4 1 . 0 1 185 . 0 3 8 4 . 2 1 185 . 0 3 8 4 . 2 1 928 1279 . 5 1 1 6 5 . 4 13018 . 8 1191 . 6 7 7 2 6 . 4 1 1 9 1 . 6 7 7 2 6 . 4 1 929 1 649 .4 1279 . 5 136826 . 0 1286 . 0 129888 . 2 1 2 8 6 . 0 129888 . 2 1 930 1 343 .7 1649 . 4 9 3 4 5 2 . 5 1595 . 2 6 3 2 5 2 . 2 1 5 9 5 . 2 6 3 2 5 2 . 2 1 931 1 073 .4 1343 . 7 7 1 9 8 4 . 9 1402 . 4 106929 . 0 1 3 6 9 . 7 8 6 6 1 2 . 5 1 932 917 . 1 1075 . 4 2 5 0 5 8 . 9 1 120 . 6 4 1 4 1 2 . 2 1 1 1 1 . 1 3 7 6 3 6 . 0 1 933 941 . 5 917.1 5 9 5 . 4 9 4 3 . 6 4 . 4 9 5 0 . 3 7 7 . 4 1 934 851 . 5 941 . 5 8 1 0 0 . 0 941 . 7 8 1 5 4 . 1 9 4 9 . 2 9 5 4 5 . 3 1 935 896 . 1 851 . 5 1989 . 2 8 8 0 . 8 2 3 4 . 1 8 8 2 . 2 193 . 2 1 936 1 022 . 5 8 9 6 . 1 15977 . 0 9 0 4 . 4 T 3 9 4 7 . 5 8 9 6 . 0 16002 . 2 1 937 1246 .6 1022 . 5 5 0 2 2 0 . 8 1000 . 8 6 0 4 1 7 . 6 1 001 . 7 5 9 9 7 6 . 0 1 938 1001 .6 1246 . 6 6 0 0 2 5 . 0 1 196 . 3 3 7 9 0 8 . 1 1 198 . 7 3 8 8 4 8 . 4 1 939 1024 .6 1 001 . 6 5 2 9 . 0 1045 . 3 4 3 6 . 8 1 0 4 7 . 1 5 0 6 . 2 1 940 1 130 .0 1024 . 6 11109 . 2 1042 . 5 7 6 5 6 . 2 1 0 5 4 . 0 5 7 7 6 . 0 1 941 1000 .3 1130 . 0 16822 . 1 1 146 . 3 2 1 3 1 6 . 0 1 1 6 7 . 5 2 7 9 5 5 . 8 1 942 968 .6 1000 . 3 1004. 9 1079 . 6 12321 . 0 1 0 5 2 . 3 7 0 0 5 . 7 1 943 938 .8 9 6 8 . 6 8 8 8 . 0 1 0 0 7 . 7 4 7 4 7 . 2 9 8 5 . 5 2 1 8 0 . 9 1 944 924 .6 9 3 8 . 8 201 . 6 951 . 5 7 2 3 . 6 9 3 9 . 7 2 2 8 . 0 1945 910 .8 9 2 4 . 6 190. 4 9 2 2 . 6 1 3 9 . 2 9 2 2 . 9 146 . 4 1946 1085 .0 9 1 0 . 8 3 0 3 4 5 . 6 9 0 6 . 6 3 1 8 2 6 . 6 9 0 0 . 7 3 3 9 6 6 . 5 1947 1690 . 5 1085 . 0 3 6 6 6 3 0 . 2 1 0 3 3 . 8 4 3 1 2 5 4 . 9 1039 . 9 4 2 3 2 8 0 . 4 1948 1515 . 9 1690 . 5 3 0 4 8 5 . 2 1536 . 4 4 2 0 . 2 1552 . 6 1 3 4 6 . 9 1 949 1438 . 9 1515 . 9 5 9 2 9 . 0 1509 . 6 4 9 9 8 . 5 1525 . 9 7 5 6 9 . 0 1950 1691 . 9 1438 . 9 6 4 0 0 9 . 0 1448 . 9 5 9 0 4 9 . 0 1434 . 2 6 6 4 0 9 . 3 1951 1 721 . 3 1691 . 9 8 6 4 . 4 1613 . 7 1 1 5 7 7 . 8 1620 . 4 10180 . 8 1952 1 964 .4 1721 . 3 5 9 0 9 7 . 6 1673 . 3 8 4 7 3 9 . 2 1682 . 9 7 9 2 4 2 . 2 1953 1 940 .2 1964 . 4 5 8 5 . 6 1872 . 0 4651 . 2 1867 . 4 5 2 9 9 . 8 13424785 .3 1 4 0 5 7 9 1 . 7 1 4 3 0 6 7 1 . 3 D I F F . , D i f f e r e n c e s ; N , K r i g i n g N e i g h b o u r h o o d ; m . t . , M e t r i c T o n s 116 TABLE 4. *** SQUARED DIFFERENCES COPPER 1955 TO 1986 *** V a r i a b l e : COPPER, London M e t a l E x c h a n g e D e f l a t o r : WHOLESALE,PRODUCER PRICE INDEX 1984=100 Time S e r i e s : YEARLY, 1954 t o 1986 V a r i o g r a m : SPHERICAL, FIGURE 8. K r i g i n g : ORDINARY KRIGING YEAR P R I C E RANDOM ( D I F F . ) 2 KRIGING ( D I F F . ) 2 K R I G I N G ( D I F F . ) 2 WALK N = 4 N = 14 ( £/m. t . ) (£/m.t .) (£/m.t. ) : 2(£/m. t . )(£/m.t . ) : 2(£/m. t . ) ( £/m.t. ) 1954 1873. 4 - - - — — _ 1955 2 5 8 7 . 4 1873. 4 509796. 0 1873. 4 509796 .0 1873. 4 5 0 9 7 9 6 . 0 1956 2 3 1 3 . 4 2587. 4 75076. 0 2583 . 6 73008 .0 2583. 6 7300 8 . 0 1957 1490. 8 231 3. 4 676670. 8 2307. 4 666835 .6 2307. 4 6 6 6 8 3 5 . 6 1958 1333. 7 1490. 8 24680. 4 1486. 9 23470 .2 1486. 9 234 7 0 . 2 1959 1603. 3 1333. 7 72684. 2 1 332. 7 73224 .4 1347. 7 6 5 3 3 1 . 4 1 960 1635. 9 1603. 3 1 063. 3 1 603. 4 1056 .2 1616. 5 376. 4 1 961 1487. 9 1635. 9 21904. 0 1637. 2 22290 .5 1635. 4 21 7 5 6 . 2 1962 1486. 5 1487. 9 2. 0 1491 . 6 26 .0 1484. 1 5. 8 1963 1471 . 8 1486. 5 216. 1 1492. 3 420 .2 1487. 5 2 4 6 . 5 1964 21 43. 9 1471 . 8 451718. 4 1480. 0 440896 .0 1474. 2 44 8 4 9 8 . 1 1965 2 7 5 2 . 7 21 43. 9 370637. 4 2143. 9 370637 .4 2140. 3 3 7 5 0 3 3 . 8 1 966 31 72. 7 2752. 7 176400. 0 2738. 6 188442 .8 2742. 4 185158. 1 1 967 2360. 6 31 72. 7 659506. 4 3141 . 3 609492 .5 3 1 5 3 . 3 6 2 8 3 7 3 . 3 1968 2 8 4 9 . 2 2360. 6 238730. 0 2344. 5 254722 . 1 2350. 9 2 4 8 3 0 2 . 9 1969 3 2 5 2 . 3 2849. 2 162489. 6 2882. 6 136678 . 1 2848. 4 163135. 2 1 970 2878. 7 3252. 3 139577. 0 3251 . 1 138681 .8 3257. 8 143716. 8 1 971 2 0 2 9 . 4 2878. 7 721310. 5 2813. 6 614969 .6 2868. 7 70 4 4 2 4 . 5 1 972 1860. 7 2029. 4 28459. 7 1970. 4 12034 . 1 2031 . 6 29206. 8 1 973 294 2 . 6 1860. 7 1170507. 6 1825. 6 1247689 .0 1867. 6 1155625. 0 1974 2 8 7 5 . 4 2942. 6 451 5. 8 2872. 6 7 .8 2951 . 9 5852. 2 1 975 1469. 2 2875. 4 1977398. 4 2787. 2 1737124 .0 2867. 0 1953844. 8 1976 1762. 3 1469. 2 85849. 0 1415. 7 120062 .2 1460. 4 91 0 8 3 . 2 1977 1410. 2 1762. 2 123904. 0 1692. 4 76636 .8 1776. 6 134249. 0 1978 1225. 0 1410. 2 34299. 0 1406. 6 32978 .6 1431 . 4 4 2 6 0 1 . 0 1979 1438. 1 1225. 0 45411. 6 1282. 6 24180 .2 1225. 1 4 5 3 6 9 . 0 1980 1243. 5 1 438. 1 37869. 2 1531 . 6 83001 .6 1444. 5 4 0 4 0 1 . 0 1981 1043. 0 1243. 5 40200. 2 1277. 1 54849 .6 1245. 2 40 9 2 5 . 3 1982 9 4 8 . 2 1043. 0 8987. 0 1131 .3 33525 .6 1042. 7 8930 . 2 1983 1113. 9 948. 2 27456. 5 1084. 4 870 .2 953. 9 256 0 0 . 0 1 984 1032. 7 1113. 9 6593. 4 1222. 2 35910 .2 1117. 2 7140. 2 1985 1046. 8 1032. 7 198. 8 1075. 0 795 .2 1032. 9 1 9 3. 2 1986 8 4 9 . 4 1046. 8 38966. 8 1078. 4 52441 .0 1045. 0 382 5 9 . 4 7933077. 7 7876747 .9 7 6 3 9 7 5 2 . 9 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m.t. : M e t r i c T o n s . 117 TABLE 5. *** SQUARED DIFFERENCES COPPER 1969 TO 1986 *** V a r i a b l e : COPPER, London M e t a l E x c h a n g e D e f l a t o r : WHOLESALE,PRODUCER PRI C E INDEX 1984=100 Time S e r i e s : YEARLY, 1954 t o 1986 P o l y n o m i a l : QUADRATIC V a r i o g r a m : SPHERICAL, FIGURE 11. K r i g i n g : UNIVERSAL KRIGING YEAR PRICE RANDOM ( D I F F . ) 2 KRIGING ( D I F F . ) 2 WALK N = 15 ( £/m. t . ) (£/m.t.) (£/m.t. ) 2 (£/m.t.) (£/m.t. ) 2 1969 3252. 3 2849.2 162489 .6 3013.9 56856 .6 1970 2878. 7 3252.3 139577 .0 3628.8 561886 .0 1971 2029. 4 2878.7 721310 .5 3260.0 1514394 .0 1972 1860. 7 2029.4 28459 .7 2098.7 56643 .5 1973 2942. 6 1860.7 1170507 .6 1673.6 1610235 .0 1974 2 8 7 5 . 4 2942.6 4515 .8 2808.4 4495 .9 1975 1469. 2 2875.4 1977398 .4 2924.9 2119107 .0 1976 1762. 3 1469.2 85849 .0 1345.5 173585 .7 1977 1410. 2 1762.2 123904 .0 1332.6 6023 .9 1978 1 225. 0 1410.2 34299 .0 896.5 1 07892 .6 1979 1438. 1 1225.0 4541 1 .6 730.3 500974 .3 1980 1243. 5 1438.1 37869 .2 1 120.4 1 51 54 .9 1 981 1043. 0 1243.5 40200 .2 1195.9 23386 .0 1 982 948. 2 1043.0 8987 .0 817.8 16987 .8 1983 1113. 9 948.2 27456 .5 765.5 1 21437 . 1 1 984 1032. 7 1113.9 6593 .4 1062.3 880 . 1 1985 1046. 8 1032.7 1 98 .8 1013.5 1112 .3 1986 849. 4 1046.8 38966 .8 940.2 8238 .8 4653993 .6 6899291 .0 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m.t. : M e t r i c T o n s . 118 T A B L E 6 . * * * SQUARED D I F F E R E N C E S COPPER 1985 TO 1986 * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l V a r i o g r a m K r i g i n g C O P P E R , L o n d o n M e t a l E x c h a n g e WHOLESALE,PRODUCER P R I C E INDEX 1984=100 Y E A R L Y , 1973 t o 1986 QUADRATIC PURE NUGGET E F F E C T , F IGURE 14. n o t a p p l i c a b l e YEAR 1985 1986 D I F F . N m . t . P R I C E RANDOM WALK ( £ / m . t . ) ( £ / m . t . ) 1 0 4 6 . 8 8 4 9 . 4 1032 .7 1046 .8 ( D I F F . ) ( £ / m . t . ) 2 198 .8 3 8 9 6 6 . 8 3 9 1 6 5 . 6 D i f f e r e n c e s . K r i g i n g N e i g h b o u r h o o d . M e t r i c T o n s . ( D I F F . ) ( £ / m . t . ) ( £ / m . t . ) 2 CURVE F I T T E D 1 0 0 4 . 4 1096 .6 1 8 0 3 . 3 6 1 0 9 5 . 5 6 2 8 9 8 . 8 1 1 9 TABLE 7. *** SQUARED DIFFERENCES LEAD 1955 TO 1986 *** V a r i a b l e : LEAD, London M e t a l E x c h a n g e D e f l a t o r : WHOLESALE,PRODUCER PRICE INDEX 1984=100 T i m e S e r i e s : YEARLY, 1954 t o 1986 V a r i o g r a m : SPHERICAL, FIGURE 19. K r i g i n g : ORDINARY KRIGING YEAR P R I C E RANDOM ( D I F F . ) KRIGING ( D I F F . ) KRIGING ( D I F F . ) WALK N = 5 N = 6 (£/m.t. ) (£/m.t (£/m.t. > 2 (£ /m. t . )(£/m.t. > 2 (£/m. t . ) (£/m.t. ) 1954 724. 6 - - - - - -1 955 777. 5 724. 6 2798. 4 724. 6 2798. 4 724. 6 27 9 8 . 4 1956 817. 7 777. 5 1616. 0 770. 0 2275. 3 770. 0 2 2 7 5 . 3 1957 655. 9 817. 7 26179. 2 807. 0 22831 . 2 807. 0 22831 . 2 1958 490. 6 655. 9 27624. 1 672. 1 32942. 2 672. 1 3 2 9 4 2 . 2 1959 477. 1 490. 6 182. 2 525. 9 2381 . 4 525. 9 2381 . 4 1 960 •479. 8 477. 1 7. 3 516. 8 1369. 0 514. 7 1218. 0 1 961 415. 8 479. 8 4096. 0 533. 3 13806. 2 535. 1 14232. 5 1 962 357. 4 415. 8 341 0. 6 457. 8 10080. 2 489. 2 17371. 2 1 963 397. 7 357. 4 1624. 1 432. 0 1 176. 5 406. 6 79. 2 1 964 615. 1 397. 7 47262. 8 406. 3 43597. 4 409. 8 42 1 4 8 . 1 1 965 673. 7 615. 1 3434. 0 589. 6 7072. 8 585. 4 77 9 6 . 9 1966 544. 5 67 3 . 7 16692. 6 640. 5 9216. 0 644. 7 10040. 0 1 967 473 . 8 544. 5 4998. 5 52 2 . 1 2332. 9 528. 0 2937. 6 1 968 553. 5 473. 8 6352. 1 456. 4 9428. 4 447. 8 11172. 5 1969 6 4 2 . 3 553. 5 . 7885. 4 556. 4 7378. 8 523. 3 14161. 0 1 970 619. 0 642. 3 542. 9 648. 9 894. 0 640. 8 457. 2 1 971 473. 9 619. 0 21054. 0 612. 9 19321. 0 631 . 9 24964. 0 1 972 524. 3 473 . 9 2540. 2 476. 8 2256. 2 488. 0 1317. 7 1 973 705. 8 524. 3 32942. 2 520. 7 34262. 0 508. 3 39 0 0 6 . 2 197 4 827. 8 705. 8 14884. 0 691 . 5 18577. 7 675. 4 2 3 2 2 5 . 8 1975 489. 1 827. 8 114717. 7 800. 7 97094. 6 800. 1 9 6 7 2 1 . 0 1 976 564. 0 489. 1 5610. 0 501 . 5 3906. 2 525. 1 1513. 2 1 977 665. 0 564. 0 10201. 0 541 . 9 15153. 6 532. 5 17556. 2 1 978 590. 5 665 . 0 5550. 2 666. 3 5745. 6 637. 0 21 62. 2 1 979 870. 1 590. 5 78176. 2 639. 4 53222. 5 628. 6 58 3 2 2 . 2 1 980 516. 1 870. 1 125245. 2 799. 4 80258. 9 841 . 5 105885. 2 1981 436. 9 516. 1 6272. 6 541 . 4 10920. 2 531 . 5 8 9 4 9 . 2 1 982 347. 9 436. 9 7921 . 0 466. 0 13947. 6 457. 6 12030. 2 1 983 297. 5 347. 9 2540. 2 378. 3 6528. 6 392. 5 9025. 0 1 984 332. 1 297. 5 1 197. 2 390. 2 3410. 6 362. 6 930. 2 1985 287. 8 332. 1 1962. 5 358. 6 5012. 6 417. 0 16692. 6 1986 251 . 1 287. 8 1346. 9 314. 6 4032. 2 330. 6 6320. 2 586567. 3 543231 . 2 6094 8 6 . 2 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m.t. : M e t r i c T o n s . 1 2 0 T A B L E 8 . * * * SQUARED D I F F E R E N C E S L E A D 1966 TO 1986 * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l V a r i o g r a m K r i g i n g L E A D , L o n d o n M e t a l E x c h a n g e WHOLESALE,PRODUCER P R I C E INDEX Y E A R L Y , 1954 t o 1986 QUADRATIC S P H E R I C A L , F IGURE 2 2 . UNIVERSAL KRIGING 1984=100 YEAR P R I C E RANDOM ( D I 1 F . ) 2 KRIGING ( D I F F . ) 2 WALK N = 12 ( £ / m . t . ) ( £ / m . t . ) ( £ / m . t . ) 2 ( £ / m . t . ) ( £ / m . t . ) 2 1966 5 4 4 . 5 6 7 3 . 7 1 967 4 7 3 . 8 5 4 4 . 5 1968 5 5 3 . 5 4 7 3 . 8 1969 6 4 2 . 3 5 5 3 . 5 1970 6 1 9 . 0 6 4 2 . 3 1 971 4 7 3 . 9 6 1 9 . 0 1 972 5 2 4 . 3 4 7 3 . 9 1 973 7 0 5 . 8 5 2 4 . 3 1974 8 2 7 . 8 7 0 5 . 8 1 975 489 .1 8 2 7 . 8 1976 5 6 4 . 0 4 8 9 . 1 1 977 6 6 5 . 0 5 6 4 . 0 1 978 5 9 0 . 5 6 6 5 . 0 1 979 8 7 0 . 1 5 9 0 . 5 1980 516 .1 8 7 0 . 1 1 981 4 3 6 . 9 516.1 1 982 3 4 7 . 9 4 3 6 . 9 1 983 2 9 7 . 5 3 4 7 . 9 1 984 3 3 2 . 1 2 9 7 . 5 1 985 2 8 7 . 8 3 3 2 . 1 1986 251 . 1 2 8 7 . 8 16692 . 6 7 5 7 . 9 4 5 5 4 8 . 4 4 9 9 8 . 5 641 . 9 2 8 2 4 5 . 0 6 3 5 2 . 1 5 2 9 . 9 5 5 7 . 1 7 8 8 5 . 4 5 7 3 . 5 4 7 4 3 . 5 5 4 2 . 9 681 . 0 3841 . 7 2 1 0 5 4 . 0 6 8 4 . 1 4 4 1 9 2 . 3 2 5 4 0 . 2 4 8 5 . 2 1 5 2 6 . 1 3 2 9 4 2 . 2 4 6 4 . 8 5 8 0 8 7 . 7 14884 . 0 6 7 2 . 1 2 4 2 3 6 . 0 114717 . 7 8 8 0 . 7 1 5 3 3 5 1 . 6 5 6 1 0 . 0 5 4 2 . 2 4 7 7 . 7 10201 . 0 5 4 8 . 1 1 3 6 6 7 . 2 5 5 5 0 . 2 6 3 7 . 5 2 2 0 8 . 4 7 8 1 7 6 . 2 5 7 3 . 3 8 8 0 7 3 . 6 125245 . 2 9 3 5 . 8 176094 . 1 6 2 7 2 . 6 5 5 9 . 8 1 5 1 1 3 . 8 7921 . 0 3 5 2 . 1 1 7 . 1 2 5 4 0 . 2 1 9 8 . 6 9 7 8 6 . 6 1 197 . 2 169 . 0 2 6 5 9 0 . 9 1962 . 5 2 7 4 . 5 1 7 6 . 5 1 3 4 6 . 9 2 7 6 . 1 6 4 6 . 2 4 6 8 6 3 2 . 6 697181 . 5 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m . t . : M e t r i c T o n s . 1 2 1 T A B L E 9 . * * * SQUARED D I F F E R E N C E S LEAD 1985 TO 1986 * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l V a r i o g r a m K r i g i n g L E A D , L o n d o n M e t a l E x c h a n g e WHOLESALE,PRODUCER PRICE INDEX 1984=100 Y E A R L Y , 1979 t o 1986 QUADRATIC PURE NUGGET E F F E C T , F IGURE 2 5 . n o t a p p l i c a b l e YEAR P R I C E RANDOM WALK ( D I F F . )' ( £ / m . t . ) ( £ / m . t . ) ( £ / m . t . ) ' CURVE ( D I F F . ) F I T T E D ( £ / m . t . ) ( £ / m . t . ) 2 1 985 1 986 2 8 7 . 8 251 .1 3 3 2 . 1 2 8 7 . 8 1 9 6 2 . 5 1 3 4 6 . 9 2 6 2 . 2 3 0 4 . 0 6 5 5 . 4 2 7 9 8 . 4 3 3 0 9 . 4 3 4 5 3 . 8 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m . t . : M e t r i c T o n s . 1 2 2 T A B L E 10 . * * * SQUARED D I F F E R E N C E S ZINC 1955 TO 1986 * * * V a r i a b l e : Z I N C , L o n d o n M e t a l E x c h a n g e , no o u t l i e r s . D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1954 t o 1986 V a r i o g r a m : S P H E R I C A L , F IGURE 2 7 . K r i g i n g : ORDINARY KRIGING 'EAR P R I C E RANDOM ( D I F F . ) 2 KRIGING ( D I F F . ) 2 WALK N = 6 ( S/m.t.) ( S/m.t. ) U / m . t . ) 2 ( £ / m . t . ) ( £ / m . t . ) 2 1954 5 8 8 . 0 — — _ • 1 955 6 6 5 . 9 5 8 8 . 0 6 0 6 8 . 4 5 8 8 . 0 6 0 6 8 . 4 1956 6 8 6 . 9 6 6 5 . 9 441 . 0 6 6 3 . 6 5 4 2 . 9 1 957 5 5 3 . 7 6 8 6 . 9 1 7 7 4 2 . 2 681 . 5 16332 .8 1958 4 4 4 . 2 5 5 3 . 7 1 1 9 9 0 . 2 550 .1 1214 .8 1959 5 5 4 . 3 4 4 4 . 2 1 2 1 2 2 . 0 4 4 7 . 0 11513 .3 1960 5 9 3 . 8 5 5 4 . 3 1 5 6 0 . 2 5 5 4 . 5 1 5 4 4 . 5 1961 5 0 3 . 3 5 9 3 . 8 8 1 9 0 . 2 6 0 8 . 4 11046 .0 1962 4 2 8 . 3 5 0 3 . 3 5 6 2 5 . 0 5 3 2 . 7 10899 .4 1963 4 8 0 . 8 4 2 8 . 3 2 7 5 6 . 2 441 . 1 1576.1 1964 7 1 7 . 6 4 8 0 . 8 5 6 0 7 4 . 2 4 6 9 . 1 6 1 7 5 2 . 2 1965 661 . 4 7 1 7 . 6 3 1 5 8 . 4 7 0 9 . 9 2 3 5 2 . 2 1966 5 3 8 . 6 661 . 4 6 0 5 2 . 8 6 6 4 . 7 15901 .2 1967 5 6 8 . 1 5 8 3 . 6 2 4 0 . 2 5 7 3 . 5 2 9 . 2 1968 6 0 4 . 5 568 .1 1 3 2 5 . 0 5 4 0 . 0 4 1 6 0 . 2 1969 6 3 4 . 2 6 0 4 . 5 8 8 2 . 1 5 7 8 . 6 3 0 9 1 . 4 1970 6 0 2 . 9 6 3 4 . 2 9 7 9 . 7 651 . 3 2 3 4 2 . 6 1971 5 7 9 . 7 6 0 2 . 9 5 3 8 . 2 6 1 4 . 2 1190 .2 1 972 6 5 6 . 1 5 7 9 . 7 5 8 3 7 . 0 5 7 8 . 5 1526.1 1973 6 5 6 . 1 6 4 5 . 6 1974 6 4 8 . 3 1975 8 8 4 . 8 6 5 8 . 5 5 1 2 0 2 . 2 1 976 8 8 8 . 7 8 8 4 . 8 1 5 . 2 8 5 7 . 0 1004 .2 1 977 6 3 4 . 9 8 8 8 . 7 6 4 4 1 4 . 4 8 4 8 . 9 45822.1 1978 5 3 2 . 4 6 3 4 . 9 1 0 5 0 6 . 2 6 2 0 . 1 7 6 9 2 . 3 1979 5 3 7 . 6 5 3 2 . 4 2 7 . 0 5 5 2 . 1 2 1 0 . 5 1980 431 .8 5 3 7 . 6 1 1 1 9 3 . 6 5 6 9 . 4 18941 .5 1981 5 1 0 . 8 431 .8 6 2 4 1 . 0 4 8 5 . 6 6 3 5 . 0 1982 4 7 6 . 0 5 1 0 . 8 1 2 1 1 . 0 5 7 2 . 6 9 3 3 1 . 6 1983 536 .1 4 7 6 . 0 3 6 1 2 . 0 5 0 0 . 5 1267 .4 1 984 6 6 7 . 8 536 .1 1 7 3 4 4 . 9 5 4 0 . 0 16332 .8 1985 5 6 3 . 7 6 6 7 . 8 10836 .8 6 6 4 . 0 10060.1 1986 4 6 5 . 9 5 6 3 . 7 9 5 6 4 . 8 5 4 3 . 8 6 0 6 8 . 4 2 7 6 5 4 9 . 9 2 7 0 4 4 9 . 4 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . 123 T A B L E 1 1 . * * * SQUARED D I F F E R E N C E S ZINC 1966 TO 1986 * * * V a r i a b l e : Z I N C , L o n d o n M e t a l E x c h a n g e , o u t l i e r s i n c l u d e d D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1966 t o 1986 P o l y n o m i a l : L I N E A R V a r i o g r a m : S P H E R I C A L , F IGURE 3 0 . K r i g i n g : UN IVERSAL KRIGING YEAR P R I C E RANDOM WALK ( £ / m . t . ) ( £ / m . t . ) 1966 5 3 8 . 6 661 .4 1967 568 .1 5 8 3 . 6 1968 6 0 4 . 5 568.1 1969 6 3 4 . 2 6 0 4 . 5 1 970 6 0 2 . 9 6 3 4 . 2 1 971 5 7 9 . 7 6 0 2 . 9 1 972 656 .1 5 7 9 . 7 1 973 1 3 9 5 . 3 656.1 1974 1729.1 1395 .3 1975 8 8 4 . 8 1729.1 1976 8 8 8 . 7 8 8 4 . 8 1977 6 3 4 . 9 8 8 8 . 7 1 978 5 3 2 . 4 6 3 4 . 9 1979 5 3 7 . 6 5 3 2 . 4 1980 431 .8 5 3 7 . 6 1981 5 1 0 . 8 431 .8 1 982 4 7 6 . 0 5 1 0 . 8 1 983 5 3 6 . 1 4 7 6 . 0 1 984 6 6 7 . 8 536.1 1985 5 6 3 . 7 6 6 7 . 8 1 986 4 6 5 . 9 5 6 3 . 7 ( D I F F . ) 2 KRIGING ( D I F F . > 2 N = 12 ( £ / m . t . ) 2 ( £ / m . t . ) ( £ / m . t . > 2 6 0 5 2 . 8 7 2 0 . 4 18715 . 4 2 4 0 . 2 6 2 6 . 4 3399 . 9 1 3 2 5 . 0 5 8 8 . 3 263 . 4 8 8 2 . 1 5 9 8 . 4 1283 .2 9 7 9 . 7 6 3 4 . 8 1018 .8 5 3 8 . 2 6 3 2 . 0 2737 .2 5 8 3 7 . 0 5 9 6 . 5 3548 .0 5 4 6 4 1 6 . 6 6 5 4 . 9 548096 .2 111422 . 4 1616 . 3 12725 .2 7 1 2 8 4 2 . 5 2 0 8 9 . 6 1451566 .0 1 5 . 2 9 5 9 . 7 5030 . 3 6 4 4 1 4 . 4 8 2 2 . 8 35296 .3 10506 . 2 441 . 9 8197 . 1 2 7 . 0 3 5 0 . 4 35063 .3 1 1 193 . 6 461 . 8 897 .2 6241 . 0 3 1 4 . 6 38490 .2 1211 . 0 3 4 3 . 7 17501 . 9 3 6 1 2 . 0 2 5 8 . 9 76867 . 6 17344 . 9 3 6 8 . 8 89401 . 4 10836 . 8 7 6 8 . 1 41817 .8 9 5 6 4 . 8 7 7 0 . 8 92920 . 4 1521503 .4 2 4 8 4 8 3 6 . 5 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m . t . : M e t r i c T o n s . 1 2 4 T A B L E 12 . * * * SQUARED D I F F E R E N C E S ZINC 1985 TO 1986 * * * V a r i a b l e D e f l a t o r T i m e S e r i e s P o l y n o m i a l V a r i o g r a m K r i g i n g Z I N C , L o n d o n M e t a l E x c h a n g e WHOLESALE,PRODUCER P R I C E INDEX 1984=100 Y E A R L Y , 1974 t o 1986 QUADRATIC PURE NUGGET E F F E C T , F I G U R E 3 3 . n o t a p p l i c a b l e YEAR 1985 1986 P R I C E 5 6 3 . 7 4 6 5 . 9 RANDOM WALK ( D I F F . ) CURVE F I T T E D ( D I F F . ) ( £ / m . t . ) ( £ / m . t . ) ( £ / m . t . ) ( £ / m . t . ) ( £ / m . t . ) 6 6 7 . 8 5 6 3 . 7 10836 .8 9 5 6 4 . 8 5 6 7 . 2 6 8 3 . 8 1 2 . 9 4 7 4 7 8 . 7 2 0 4 0 1 . 6 4 7 4 9 1 . 6 D I F F . : D i f f e r e n c e s . N : K r i g i n g N e i g h b o u r h o o d . m . t . : M e t r i c T o n s . 1 2 5 APPENDIX 4 POLYNOMIAL CURVES T A B L E 1. * * * POLYNOMIAL CURVE F I T T I N G COPPER 1954 TO 1986 * * * V a r i a b l e : COPPER, L o n d o n M e t a l E x c h a n g e , F I G U R E S 9 , 10 . D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1954 t o 1986 P o p u l a t i o n : 33 F i t t e d C u r v e : POLYNOMIAL OF ORDER K = 2 , QUADRATIC S E R I E S YEAR P R I C E F I T T E D P R I C E RESIDUAL T ( £ / t o n n e ) ( £ / t o n n e ) ( £ / t o n n e ) 71 1954 1873 .4 1 5 6 1 . 0 3 1 2 . 4 72 1955 2 5 8 7 . 4 1 6 7 5 . 9 9 1 1 . 5 73 1956 2 3 1 3 . 4 1 7 8 1 . 4 5 3 2 . 1 74 1957 1490 .8 1 8 7 7 . 5 - 3 8 6 . 7 75 1958 1 3 3 3 . 7 1 9 6 4 . 2 - 6 3 0 . 5 76 1959 1 6 0 3 . 3 2 0 4 1 . 6 - 4 3 8 . 3 77 1960 1635 .9 2 1 0 9 . 6 - 4 7 3 . 6 78 1961 1 4 8 7 . 9 2 1 6 8 . 2 - 6 8 0 . 3 79 1962 1 4 8 6 . 5 2 2 1 7 . 4 - 7 3 0 . 9 80 1963 1471 .8 2 2 5 7 . 2 - 7 8 5 . 4 81 1964 2 1 4 3 . 9 2 2 8 7 . 6 - 1 4 3 . 8 82 1965 2 7 5 2 . 7 2 3 0 8 . 7 4 4 4 . 0 83 1966 3 1 7 2 . 7 2 3 2 0 . 4 8 5 2 . 3 84 1967 2 3 6 0 . 6 2 3 2 2 . 7 3 7 . 9 85 1968 2 8 4 9 . 2 2 3 1 5 . 6 5 3 3 . 5 86 1969 3 2 5 2 . 3 2 2 9 9 . 2 9 5 3 . 2 87 1970 2 8 7 8 . 7 2 2 7 3 . 3 6 0 5 . 4 88 1971 2 0 2 9 . 4 2 2 3 8 . 1 - 2 0 8 . 7 89 1972 1 8 6 0 . 7 2 1 9 3 . 5 - 3 3 2 . 8 90 1973 2 9 4 2 . 6 2 1 3 9 . 5 8 0 3 . 1 91 1974 2 8 7 5 . 4 2 0 7 6 . 1 7 9 9 . 3 92 1975 1469 .2 2 0 0 3 . 4 - 5 3 4 . 2 93 1976 1762 1 9 2 1 . 2 - 1 5 9 . 1 94 1977 1410 .2 1 8 2 9 . 7 - 4 1 9 . 5 95 1978 1225 .0 1728 .8 - 5 0 3 . 8 96 1979 1438.1 1 6 1 8 . 5 - 1 8 0 . 5 97 1980 1 2 4 3 . 5 1 4 9 8 . 9 - 2 5 5 . 4 98 1981 1043 .0 1 3 6 9 . 8 - 3 2 6 . 9 99 1982 9 4 8 . 2 1231 .4 - 2 8 3 . 2 100 1983 1 1 1 3 . 9 1 0 8 3 . 6 3 0 . 3 101 1984 1032 .7 9 2 6 . 4 1 0 6 . 2 102 1985 1046 .8 7 5 9 . 9 2 8 7 . 0 103 1986 8 4 9 . 4 5 8 3 . 9 2 6 5 . 5 QUADRATIC EQUATION : - 3 0 5 7 5 . 2 + 7 8 5 . 7 T - 4 . 7 T 2 T = S e r i e s V a l u e . 1 2 7 TABLE 2. *** POLYNOMIAL CURVE F I T T I N G COPPER 1973 TO 1986 *** V a r i a b l e : COPPER, L o n d o n M e t a l E x c h a n g e , FIGURES 12, 13. D e f l a t o r : WHOLESALE,PRODUCER PRI C E INDEX 1984=100 Time S e r i e s : YEARLY, 1973 t o 1986 P o p u l a t i o n : 14 F i t t e d C u r v e : POLYNOMIAL OF ORDER K = 2, QUADRATIC SERIES YEAR PRICE FITTED PRICE RESIDUAL T ( £/tonne) (£/tonne) (£/tonne) 90 1973 2942.6 2777.8 164.8 91 1974 2875.4 2426.9 448.5 92 1975 1469.2 2113.0 -643.8 93 1976 1762.2 1835.9 -73.8 94 1977 1410.2 1595.8 -185.6 95 1 978 1225.0 1392.7 -167.7 96 1979 1 438. 1 1226.4 211.7 97 1 980 1243.5 1097.1 146.4 98 1981 1043.0 1004.7 38.3 99 1982 948.2 949.2 -1 .0 1 00 1983 1113.9 930.7 183.3 101 1984 1032.7 949.0 83.6 1 02 1985 1046.8 1004.4 42.5 1 03 1 986 849.4 1096.6 -247.2 QUADRATIC EQUATION: 185572.6 -- 3692.8 T + 18.5 T 2 T = S e r i e s V a l u e . 1 2 8 T A B L E 3 . * * * POLYNOMIAL CURVE F I T T I N G LEAD 1954 TO 1986 * * * V a r i a b l e : L E A D , L o n d o n M e t a l E x c h a n g e , F IGURES 2 0 , 2 1 . D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1954 t o 1986 P o p u l a t i o n : 33 F i t t e d C u r v e : POLYNOMIAL OF ORDER K = 2 , QUADRATIC S E R I E S YEAR P R I C E F I T T E D P R I C E RESIDUA] T ' ( £ / t o n n e ) ( £ / t o n n e ) ( £ / t o n n e 71 1954 7 2 4 . 6 5 8 0 . 9 1 4 3 . 7 72 1955 7 7 7 . 5 5 8 7 . 0 1 9 0 . 5 73 1956 8 1 7 . 7 5 9 2 . 3 2 2 5 . 4 74 1957 6 5 5 . 9 5 9 6 . 8 5 9 . 2 75 1958 4 9 0 . 6 6 0 0 . 4 - 1 0 9 . 8 76 1959 477 .1 6 0 3 . 2 - 1 2 6 . 1 77 1960 4 7 9 . 8 6 0 5 . 2 - 1 2 5 . 4 78 1961 4 1 5 . 8 6 0 6 . 4 - 1 9 0 . 6 79 1 962 3 5 7 . 4 6 0 6 . 7 - 2 4 9 . 3 80 1963 3 9 7 . 7 6 0 6 . 2 - 2 0 8 . 5 81 1964 615.1 6 0 4 . 9 1 0 . 2 82 1965 6 7 3 . 7 6 0 2 . 8 7 0 . 9 83 1 966 5 4 4 . 5 5 9 9 . 8 - 5 5 . 3 84 1 967 4 7 3 . 8 5 9 6 . 0 - 1 2 2 . 2 85 1968 5 5 3 . 5 591 .4 - 3 7 . 9 86 1969 6 4 2 . 3 5 8 6 . 0 5 6 . 4 87 1970 6 1 9 . 0 5 7 9 . 7 3 9 . 3 88 1971 4 7 3 . 9 5 7 2 . 6 - 9 8 . 7 89 1972 5 2 4 . 3 5 6 4 . 7 - 4 0 . 4 90 1973 7 0 5 . 8 5 5 6 . 0 1 4 9 . 9 91 1974 8 2 7 . 8 5 4 6 . 4 281 . 4 92 1975 4 8 9 . 1 5 3 6 . 0 - 4 6 . 9 93 1976 5 6 4 . 0 5 2 4 . 8 3 9 . 2 94 1977 6 6 5 . 0 5 1 2 . 8 1 5 2 . 2 95 1978 5 9 0 . 5 4 9 9 . 9 9 0 . 6 96 1979 8 7 0 . 1 4 8 6 . 2 3 8 3 . 8 97 1980 516.1 471 .7 4 4 . 4 98 1 981 4 3 6 . 9 4 5 6 . 4 - 1 9 . 5 99 1982 3 4 7 . 9 4 4 0 . 2 - 9 2 . 3 100 1983 2 9 7 . 5 4 2 3 . 2 • - 1 2 5 . 7 101 1 984 3 3 2 . 1 4 0 5 . 4 - 7 3 . 3 1 02 1985 2 8 7 . 8 3 6 8 . 8 - 9 9 . 0 1 03 1 986 251 . 1 3 6 7 . 3 - 1 1 6 . 2 QUADRATIC EQUATION : - 1 9 6 1 . 3 + 65 .1 T - 0 . 4 T 2 T = S e r i e s V a l u e . 1 2 9 T A B L E 4 . * * * POLYNOMIAL CURVE F I T T I N G LEAD 1979 TO 1986 * * * V a r i a b l e : L E A D , L o n d o n M e t a l E x c h a n g e , F I G U R E S 2 3 , 2 4 . D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1979 t o 1986 P o p u l a t i o n : 8 F i t t e d C u r v e : POLYNOMIAL OF ORDER K = 2 , QUADRATIC S E R I E S YEAR P R I C E F I T T E D P R I C E R E S I D U A L T ( £ / t o n n e ) ( £ / t o n n e ) ( £ / t o n n e ) 96 1979 870 .1 7 9 0 . 6 7 9 . 4 97 1980 516 .1 6 0 9 . 8 - 9 3 . 7 98 1981 4 3 6 . 9 4 6 6 . 0 - 2 9 . 2 99 1982 3 4 7 . 9 3 5 9 . 4 - 1 1 . 5 100 1983 2 9 7 . 5 2 8 9 . 9 7 . 7 101 1984 3 3 2 . 1 2 5 7 . 5 7 4 . 6 1 02 1985 2 8 7 . 8 2 6 2 . 2 2 5 . 6 1 03 1986 251 .1 3 0 4 . 0 - 5 2 . 9 QUADRATIC EQUATION: 1 9 0 9 7 3 . 8 - 3 7 6 2 . 7 T + 1 8 . 6 T T = S e r i e s V a l u e . 1 3 0 T A B L E 5 . * * * POLYNOMIAL CURVE F I T T I N G ZINC 1954 TO 1986 * * * V a r i a b l e : Z I N C , L o n d o n M e t a l E x c h a n g e , F IGURES 2 8 , 2 9 . D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1954 t o 1986 P o p u l a t i o n : 33 F i t t e d C u r v e : POLYNOMIAL OF ORDER K = 1, L I N E A R S E R I E S YEAR PRICE F I T T E D P R I C E RESIDUAL T ( £ / t o n n e ) ( £ / t o n n e ) ( £ / t o n n e ) 71 1954 5 8 8 . 0 6 1 6 . 5 - 2 8 . 5 72 1955 6 5 6 . 9 6 1 8 . 4 4 7 . 5 73 1956 6 8 6 . 9 6 2 0 . 3 6 6 . 6 74 1957 5 5 3 . 7 6 2 2 . 3 - 6 8 . 5 75 1958 4 4 4 . 2 6 2 4 . 2 - 1 8 0 . 0 76 1959 5 5 4 . 3 626.1 -71 .8 77 1960 5 9 3 . 8 6 2 8 . 1 - 3 4 . 2 78 1961 5 0 3 . 3 6 3 0 . 0 - 1 2 6 . 7 79 1962 4 2 8 . 3 631 . 9 - 2 0 3 . 6 80 1963 4 8 0 . 8 6 3 3 . 8 - 1 5 3 . 0 81 1964 7 1 7 . 6 6 3 5 . 8 81 . 9 82 1965 661 .4 6 3 7 . 7 2 3 . 7 83 1966 5 8 3 . 6 6 3 9 . 6 - 5 6 . 0 84 1967 568.1 641 .6 - 7 3 . 5 85 1968 6 0 4 . 5 6 4 3 . 5 - 3 9 . 0 86 1969 6 3 4 . 2 6 4 5 . 4 - 1 1 . 2 87 1 970 6 0 2 . 9 6 4 7 . 4 - 4 4 . 5 88 1 971 5 7 9 . 7 6 4 9 . 3 - 6 9 . 6 89 1972 656.1 6 5 1 . 2 4 . 9 90 1 973 1395 .3 6 5 3 . 1 7 4 2 . 1 91 1974 1729.1 6 5 5 . 1 1074 .0 92 1975 8 8 4 . 8 6 5 7 . 0 2 2 7 . 8 93 1976 8 8 8 . 7 6 5 8 . 9 2 2 9 . 8 94 1 977 6 3 4 . 9 6 6 0 . 9 - 2 6 . 0 95 1 978 5 3 2 . 4 6 6 2 . 8 - 1 3 0 . 4 96 1 979 5 3 7 . 6 6 6 4 . 7 - 1 2 7 . 1 97 1980 431 .8 6 6 6 . 7 - 2 3 4 . 8 98 1981 5 1 0 . 8 6 6 8 . 6 - 1 5 7 . 8 99 1 982 4 7 6 . 0 6 7 0 . 5 - 1 9 4 . 5 100 1 983 5 3 6 . 1 6 7 2 . 4 - 1 3 6 . 3 101 1984 6 6 7 . 8 6 7 4 . 4 - 6 . 5 1 02 1985 5 6 3 . 7 6 7 6 . 3 - 1 1 2 . 6 1 03 1986 4 6 5 . 9 6 7 8 . 2 - 2 1 2 . 3 QUADRATIC EQUATION : + 4 7 9 . 4 + 1 . 9 T T = S e r i e s V a l u e . 1 3 1 T A B L E 6 . * * * POLYNOMIAL CURVE F I T T I N G ZINC 1974 TO 1986 * * * V a r i a b l e : Z I N C , L o n d o n M e t a l E x c h a n g e , F IGURES 3 1 , 3 2 . D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1974 t o 1986 P o p u l a t i o n : 13 F i t t e d C u r v e : POLYNOMIAL OF ORDER K = 2 , QUADRATIC S E R I E S YEAR P R I C E F I T T E D PRICE RESIDUAL T ( £ / t o n n e ) ( £ / t o n n e ) ( £ / t o n n e ) 91 1974 1729.1 1 3 7 6 . 4 3 5 2 . 7 92 1975 8 8 4 . 8 1 1 4 4 . 4 - 2 5 9 . 6 93 1976 8 8 8 . 7 944 .1 - 5 5 . 3 94 1977 6 3 4 . 9 7 7 5 . 4 - 1 4 0 . 5 95 1978 5 3 2 . 4 6 3 8 . 5 - 1 0 6 . 1 96 1979 5 3 7 . 6 5 3 3 . 2 4 . 4 97 1980 431 .8 4 5 9 . 7 - 2 7 . 9 98 1981 5 1 0 . 8 4 1 7 . 8 9 3 . 0 99 1982 4 7 6 . 0 4 0 7 . 6 6 8 . 4 100 1983 536.1 4 2 9 . 1 1 0 7 . 0 101 1984 6 6 7 . 8 4 8 2 . 4 1 8 5 . 5 1 02 1985 5 6 3 . 7 5 6 7 . 2 - 3 . 6 103 1986 4 6 5 . 9 6 8 3 . 8 - 2 1 7 . 9 QUADRATIC E Q U A T I O N : 1 5 5 1 4 8 . 3 -- 3 1 3 1 . 7 T + 15 .8 T 2 T = S e r i e s V a l u e . 132 APPENDIX 5 ZINC PRICES FREQUENCIES 1 3 3 T A B L E 1. * * * FREQUENCIES OF ZINC PRICES 1954 TO 1986 * * * V a r i a b l e : Z I N C , L o n d o n M e t a l E x c h a n g e D e f l a t o r : WHOLESALE,PRODUCER P R I C E INDEX 1984=100 T i m e S e r i e s : Y E A R L Y , 1954 t o 1986 Number o f V a l u e s : 33 Mean V a r i a n c e 6 4 7 . 4 p o u n d s s t e r l i n g / t o n n e 66799 .1 ( p o u n d s s t e r l i n g / t o n n e ) C L A S S ( £ / t o n n e ) 0 . 0 5 0 . 0 1 5 0 . 0 2 5 0 . 0 3 5 0 . 0 4 5 0 . 0 5 5 0 . 0 6 5 0 . 0 7 5 0 . 0 8 5 0 . 0 9 5 0 . 0 1 0 5 0 . 0 1 1 5 0 . 0 1 2 5 0 . 0 FREQUENCY PERCENT 6 13 9 1 2 18 .2 3 9 . 4 2 7 . 3 3 . 0 6 . 0 CUMMULATIVE PERCENT 18 .2 5 7 . 6 8 4 . 9 8 7 . 9 9 3 . 9 9 3 . 9 9 3 . 9 9 3 . 9 9 3 . 9 1350 1 450 1 550 1650 1 750 ,0 .0 ,0 ,0 ,0 OUTLIERS 3 . 0 3 . 0 9 6 . 9 9 6 . 9 9 6 . 9 9 6 . 9 9 9 . 9 1 3 4 

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