Linear stochastic models

  • N. T. Kottegoda


This chapter concerns types of linear models that are used to represent stochastic processes. The purpose is to generate likely future sequences of data for design and planning. In general, the models are formulated so that the current value of a variable is the weighted sum of past values and random numbers which represent unknown effects.


Autoregressive Model Serial Correlation Water Resource System Annual Flow Daily Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akaike, H. (1974). A new look at the statistical model identification. IEEE Trans. Autom. Control, 19, 716–23MathSciNetMATHCrossRefGoogle Scholar
  2. Anderson, B. D. O. (1971). A qualitative introduction to Wiener and Kalman–Bucy Filters. Proc. Inst. Radio Electron. Eng. Aust., March, 93–103Google Scholar
  3. Anderson, O. D. (1976). Time Series Analysis and Forecasting—The Box–Jenkins Approach, Butterworths, LondonGoogle Scholar
  4. Anderson, R. L. (1942). Distribution of the serial correlation coefficient. Ann. Math. Statist., 13, 1–13MathSciNetMATHCrossRefGoogle Scholar
  5. Anderson, T. W. (1957). Maximum likelihood estimates for a multivariate normal distribution when some observations are missing. J. Am. Statist. Assoc., 52, 200–3MathSciNetMATHCrossRefGoogle Scholar
  6. Beard, L. R. (1965). Use of interrelated records to simulate streamflow. J. Hydraul. Div., Am. Soc. Civ. Eng., 91 (HY5), 13–22.Google Scholar
  7. Beard, L. R. (1967). Simulation of daily streamflow. Proceedings of the International Hydrology Symposium, vol. 1, Fort Collins, Colorado, pp. 624–32Google Scholar
  8. Bernier, J. (1970). Inventaire des modèles des processus stochastiques applicables à la description des débits journaliers des rivières. Rev. Int. Statist. Inst., 38, 49–61CrossRefGoogle Scholar
  9. Borgman, L. E., and Amorocho, J. (1970). Some statistical problems in hydrology. Rev. Int. Statist. Inst., 38, 82–96CrossRefGoogle Scholar
  10. Box, G. E. P., and Cox, D. R. (1964). An analysis of transformations. J. R. Statist. Soc. B, 26, 211–52MathSciNetMATHGoogle Scholar
  11. Box, G. E. P., and Jenkins, G. M. (1976). Time Series Analysis: Forecasting and Control, revised edn, Holden Day, San Francisco, CaliforniaMATHGoogle Scholar
  12. Brewer, H. W. (1976). Identification of the noise in a Kalman filter. Control and Dynamic Systems, Advances in Theory and Applications, vol. 12 (ed. C. T. Leondes), Academic Press, New York, pp. 491–579Google Scholar
  13. Carlson, R. E., MacCormick, A. J. A., and Watts, D. G. (1970). Application of linear random models to four annual streamflow series. Water Resour. Res., 6, 1070–8CrossRefGoogle Scholar
  14. Chatfield, C., and Prothero, D. L. (1973). Box—Jenkins seasonal forecasting: problems in a case-study, with discussion. J. R. Statist. Soc. A, 136, 295–352CrossRefGoogle Scholar
  15. Clarke, R. T. (1974). The representation of a short period of experimental catchment data by a linear stochastic difference equation. Proceedings of the 1971 Warsaw Symposium on Mathematical Models in Hydrology, vol. 1, International Association of Scientific Hydrology, Paris, pp. 3–16Google Scholar
  16. Codner, G. P., and McMahon, T. A. (1973). Lognormal streamflow generation models re-examined. J. Hydraul. Div., Am. Soc. Civ. Eng., 99 (HY9), 1421–31Google Scholar
  17. Crosby, D. S., and Maddock, T., III(1970). Estimating coefficients of a flow generator from monotone samples of data. Water Resour. Res., 6, 1079–86CrossRefGoogle Scholar
  18. Davies, N., Triggs, C. M., and Newbold, P. (1977). Significance levels of the Box—Pierce portmanteau statistic in finite samples. Biometrika, 64, 517–22MathSciNetMATHCrossRefGoogle Scholar
  19. Delleur, J. W., Tao, P. C., and Kavvas, M. L. (1976). An evaluation of the practicality and complexity of some rainfall and runoff time series models. Water Resour. Res., 12, 953–70CrossRefGoogle Scholar
  20. Doi, K. (1978). Physical reality of shot noise model for the short-term variability of cyg X-1. Nature (London), 275, 197–8CrossRefGoogle Scholar
  21. Dooge, J. C. I. (1972). Mathematical models of hydrologic systems. Modelling of Water Resource Systems, vol 1 (ed. A. K. Biswas), Harvest House, Montreal, pp. 170–88Google Scholar
  22. Duong, N., Winn, C. B., and Johnson, G. R. (1975). Modern control concepts in hydrology. IEEE Trans. Syst. Manage. Cybern., 5, 46–53CrossRefGoogle Scholar
  23. Durbin, J. (1960). The fitting of time-series models. Rev. Int. Statist. Inst., 28, 233–44MATHCrossRefGoogle Scholar
  24. Ekern, S. (1976). Forecasting with adaptive filtering: a critical re-examination. Oper. Res. Q., 27, 705–15MATHCrossRefGoogle Scholar
  25. Fiering, M. B. (1964). Multivariate technique for synthetic hydrology. J. Hydraul. Div., Am. Soc. Civ. Eng., 90 (HY5), 43–60Google Scholar
  26. Fiering, M. B. (1967). Streamflow Synthesis, Macmillan, LondonCrossRefGoogle Scholar
  27. Fiering, M. B. (1968). Schemes for handling inconsistent matrices. Water Resour. Res., 4, 291–7CrossRefGoogle Scholar
  28. Gelb, A. (ed.) (1974): Applied Optimal Estimation, Massachusetts Institute of Technology Press, Cambridge, MassachusettsGoogle Scholar
  29. Granger, C. W. J., and Morris, M. J. (1976). Time series modelling and interpretation. J. R. Statist. Soc. A, 139, 246–57MathSciNetCrossRefGoogle Scholar
  30. Green, N. M. D. (1973). A synthetic model for daily streamflow. J. Hydrol., 20, 351–64CrossRefGoogle Scholar
  31. Gupta, V. L., and Fordham, J. W. (1972). Streamflow synthesis—a case study. J. Hydraul. Div., Am. Soc. Civ. Eng., 98 (HY6), 1049–55Google Scholar
  32. Hamlin, M. J., and Kottegoda, N. T. (1971). Extending the record of the Terne. J. Hydrol., 12, 100–16CrossRefGoogle Scholar
  33. Hamlin, M. J.,(1974). The preparation of a data set for hydrological system analysis.Design of Water Resources Projects with Inadequate Data, vol. 1, Symposium, Madrid, 1973, Int. Assoc. Sci. Hydrol. Publ., No. 108, pp. 163–77Google Scholar
  34. Hamlin, M. J., Kottegoda, N. T., and Kitson, T. (1976). Control of a river system with two reservoirs. 1974 Symposium on the Control of Water Resource Systems, Haiffa, Israel, J. Hydrol., 28, 155–73Google Scholar
  35. Harrison, P. J., and Stevens, C. F. (1976). Bayesian forecasting. J. R. Statist. Soc. B, 38, 205–47MathSciNetMATHGoogle Scholar
  36. Hinkley, D. (1977). On quick choice of power transformation. Appl. Statist., 26, 67–9CrossRefGoogle Scholar
  37. Hipel, K. W., McLeod, A. I., and Lennox, W. C. (1977). Advances in Box–Jenkins modelling, 1, model construction. Water Resour. Res., 13, 567–75CrossRefGoogle Scholar
  38. Hovanessian, S. A. (1976). Computational Mathematics in Engineering, Lexington Books, Lexington, MassachusettsGoogle Scholar
  39. Jazwinski, A. H. (1969). Adaptive filtering. Automatica, 5, 475–85MATHCrossRefGoogle Scholar
  40. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory, Academic Press, New YorkMATHGoogle Scholar
  41. Jenkins, R. H., and Watts, D. G. (1968). Spectral Analysis and its Applications, Holden Day, San Francisco, CaliforniaMATHGoogle Scholar
  42. Jettmar, R. U., and Young, G. K. (1975). Hydrologic estimation and economic regret. Water. Resour. Res., 11, 648–56CrossRefGoogle Scholar
  43. Kahan, J. P. (1974). A method for maintaining cross and serial correlations and the coefficient of skewness under generation in the linear bivariate regression model. Water Resour. Res., 10, 1245–8CrossRefGoogle Scholar
  44. Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. J. Basic Eng., Trans. Am. Soc. Mech. Eng. D, 82,35–45CrossRefGoogle Scholar
  45. Kalman, R. E., and Bucy, R. S. (1961). New results in linear filtering and prediction theory. J. Basic Eng., Trans. Am. Soc. Mech. Eng. D, 83, 95–108MathSciNetCrossRefGoogle Scholar
  46. Kendall, M. G. (1976). Time Series, 2nd edn, Griffin, LondonMATHGoogle Scholar
  47. Kottegoda, N. T. (1970). Statistical methods of river flow synthesis for water resources assessment, with discussion. Proc. Inst. Civ..Eng., paper 7339S, suppl. XVIII, 415–42Google Scholar
  48. Kottegoda, N. T. (1972). Stochastic five daily stream flow model. J. Hydraul. Div., Am. Soc.Civ. Eng., 98 (HY9), 1469–85Google Scholar
  49. Kottegoda, N. T., and Elgy, J.(1977). Infilling missing flow data. Proceedings of the 3rd International Hydrology Symposium, Fort Collins, Colorado preprintsGoogle Scholar
  50. Kottegoda, N. T., and Yevjevich, V. (1977). Preservation of correlation in generated hydrologic samples through two-station models. J. Hydrol., 33, 99–121CrossRefGoogle Scholar
  51. Lawrance, A. J. (1976). A reconsideration of the Fiering two-station model. J. Hydrol., 29, 77–85CrossRefGoogle Scholar
  52. Lawrance, A. J., and Kottegoda, N. T. (1977). Stochastic modelling of riverflow time series, with discussion. J. R. Statist. Soc. A, 140, 1–47CrossRefGoogle Scholar
  53. Ledolter,J.(1978). The analysis of multivariate time series applied to problems in hydrology. J. Hydrol. 36, 327–52CrossRefGoogle Scholar
  54. Lettenmaier, D. P., and Burges, S. J. (1976). Use of state estimation techniques in water resource system modelling. Water Resour. Bull., 12, 83–99. (1977). Discussion. Water Resour. Bull., 13, 161–7, 1289–91CrossRefGoogle Scholar
  55. McClave, J. T. (1978). Estimating the order of autoregressive models: the max x2 method. J. Am. Statist. Assoc. 73, 122–8MATHGoogle Scholar
  56. McGregor, J. R. (1962). The approximate distribution of the correlation between two stationary linear Markov series. Biometrika, 49 379–88Google Scholar
  57. McKerchar, A. I., and Delleur, J. W. (1974). Application of seasonal parametric linear stochastic models to monthly flow data. Water Resour. Res., 10, 246–55CrossRefGoogle Scholar
  58. McMichael, F. C., and Hunter, J. S. (1972). Stochastic modelling of temperature and flow in rivers. Water Resour. Res., 8, 87–98CrossRefGoogle Scholar
  59. Maissis, A. H. (1977). Optimal filtering techniques for hydrological forecasting. J. Hydrol., 33, 319–30CrossRefGoogle Scholar
  60. Makhoul,J.(1975). Linear prediction: a tutorial review. Proc. IEEE, 63 561–80CrossRefGoogle Scholar
  61. Mandelbrot, B. B. (1976). Note on the definition and the stationarity of fractional gaussian noise. J. Hydrol., 30, 407–9CrossRefGoogle Scholar
  62. Markov, A. A. (1907). Investigation of a noteworthy case of dependent trials. Izvestiya Rossiiskoi Akademii Nauk,1Google Scholar
  63. Marriott, F. H. C., and Pope, J. A. (1954). Bias in the estimation of autocorrelations. Biometrika, 41, 390–402MathSciNetMATHCrossRefGoogle Scholar
  64. Matalas, N. C. (1967). Mathematical assessment of synthetic hydrology. Water Resour. Res., 3, 937–45CrossRefGoogle Scholar
  65. Mejia, J. M., and Rodriguez-Iturbe, I. R. (1974). Correlation links between normal and log normal processes. Water Resour. Res., 10, 689–90CrossRefGoogle Scholar
  66. Mejia, J. M., and Rousselle, J. (1976). Disaggregration models in hydrology revisited. Water Resour. Res., 12, 185–6CrossRefGoogle Scholar
  67. Moran, P. A. P. (1970). Simulation and evaluation of complex water systems operations. Water Resour. Res., 6, 1737–42CrossRefGoogle Scholar
  68. Moss, M. E. (1972). Reduction of uncertainties in autocorrelation by the use of physical models. Proceedings of the International Symposium on Uncertainties in Hydrologic and Water Resource Systems, vol. 1, 11–14Google Scholar
  69. Moss, M. E. December 1972, University of Arizona, Tucson, pp. 203–29Google Scholar
  70. Nelson, C. R. (1976). The interpretation of R2 in autoregressive moving average time series models. Am. Statist., 30, 175–80MATHGoogle Scholar
  71. O’Donnell, T., Hall, M. J., and O’Connell, P. E. (1972). Some applications of stochastic hydrological models. Modelling of Water Resource Systems, vol. 1 (ed. A. K. Biswas), Harvest House, Montreal, pp. 250–62Google Scholar
  72. Parzen, E. (1962). Stochastic Processes, Holden Day, San Francisco, CaliforniaMATHGoogle Scholar
  73. Payne, K., Neuman, W. R., and Kerri, K. D. (1969). Daily streamflow simulation. J. Hydraul. Div., Am. Soc. Civ. Eng., 95 (HY4), 1163–79Google Scholar
  74. Pegram, G. G. S., and James, W. (1972). Multilag multivariate autoregressive model for the generation of operational hydrology. Water Resour. Res., 8, 1074–6CrossRefGoogle Scholar
  75. Phatarfod, R. M. (1976). Some aspects of stochastic reservoir theory. J. Hydrol., 30, 199–217CrossRefGoogle Scholar
  76. Quenouille, M. H. (1949). Approximate tests of correlation in time series. J. R. Statist. Soc. B, 11, 68–84MathSciNetMATHGoogle Scholar
  77. Quenouille, M. H. (1956). Notes on bias in estimation. Biometrika, 43, 353–60MathSciNetMATHCrossRefGoogle Scholar
  78. Quenouille, M. H. (1957). The Analysis of Multiple Time Series, Statistical Monographs No. 1, Griffin, LondonGoogle Scholar
  79. Quimpo, R. G. (1968). Stochastic analysis of daily riverflows. J. Hydraul. Div., Am. Soc. Civ. Eng., 94 (HY1), 43–57Google Scholar
  80. Rao, R. A., and Kashyap, R. L. (1974). Stochastic difference equation modelling of hydrologic processes. Proceedings of the 1971 Warsaw Symposium on Mathematical Models in Hydrology, vol. 1, International Association of Scientific Hydrology, Paris, pp. 140–50Google Scholar
  81. Roesner, L. A., and Yevjevich, V. M. (1966). Mathematical models for time series of monthly precipitation and monthly runoff. Colo. St. Univ., Fort Collins, Hydrol. Papers, No. 15Google Scholar
  82. Salas, J. D., and Pegram, G. G. S. (1977). A seasonal multivariate multilag autoregressive model in hydrology. Proceedings of the 3rd International Hydrology Symposium, Fort Collins, Colorado, preprintsGoogle Scholar
  83. Sorenson, H. W. (1968). Controllability and observability of linear, stochastic, time-discrete control systems. Adv. Control Syst., 6, 95–158MathSciNetMATHGoogle Scholar
  84. Sorenson, H. W. (1970). Least squares estimation: from Gauss to Kalman. IEEE Spectrum, 7, 63–8CrossRefGoogle Scholar
  85. Swerling, P. (1959). A proposed stagewise differential correction procedure for satellite tracking and prediction, J. Astronaut. Sci., 6; Rand Corp., Santa Monica, Calif., Rep., No. P-1292Google Scholar
  86. Szöllösi-Nagy, A. (1975). An adaptive identification and prediction algorithm for the real-time forecasting of hydrologic time series. Proceedings of the International Symposium and Workshop on the Application of Mathematical Models in Hydrology and Water Resource Systems, International Association of Scientific Hydrology, Bratislava, preprintsGoogle Scholar
  87. Szöllösi-Nagy,A.(1976). An adaptive identification and prediction algorithm for the real time forecasting of hydrological time series. Hydrol. Sci. Bull., 21, 163–76CrossRefGoogle Scholar
  88. Thomas, H. A., and Fiering, M. B. (1962). Mathematical synthesis of streamflow sequences for the analysis of river basins by simulation. Design of Water Resource Systems (eds A. Maass et al.), Harvard University Press, Cambridge, Massachusetts, chapter 12, pp. 459–93Google Scholar
  89. Thompson, H. E., and Tiao, G. C. (1971). Analysis of telephone data: a case study of forecasting seasonal time series. Bell J. Econ. Manage. Sci., 2, 5 15–41CrossRefGoogle Scholar
  90. Tillotson, A., and Cluckie, I. D. (1977). Linear models for flow estimation and prediction. Proceedings of the 3rd International Hydrology Symposium, Fort Collins, Colorado, preprintsGoogle Scholar
  91. Todini, E., and Bouillot, D. (1976). A rainfall run-off Kalman filter model. System Simulation in Water Resources (ed. G. C. Vansteenkiste), North-Holland, AmsterdamGoogle Scholar
  92. Toyoda, J., Toriumi, N., and Inoue, Y. (1969). An adaptive predictor of river flow for on line control of water resource systems. Automatica, 4, 175–81CrossRefGoogle Scholar
  93. Treiber, B., and Plate, E. J. (1975). A stochastic model for the simulation of daily flows. Proceedings of the International Symposium and Workshop on the Application of Mathematical Models in Hydrology and Water Resource Systems, International Association of Scientific Hydrology, Bratislava, preprintsGoogle Scholar
  94. Valencia, R. D., and Schaake, J. C., Jr. (1973). Disaggregation processes in stochastic hydrology. Water Resour. Res., 9, 580–5CrossRefGoogle Scholar
  95. Walker, Sir Gilbert (1931). On periodicity in series of related terms. Proc. R. Soc. London A, 131, 518–32MATHCrossRefGoogle Scholar
  96. Weiss, G. (1977). Shot noise models for synthetic generation of multisite daily streamflow data. Water Resour. Res., 13, 101–8CrossRefGoogle Scholar
  97. Wheelwright,S.C., and Makridakis, S. (1973). An examination of the use of adaptive filtering in forecasting. Oper. Res. Q., 24, 55–64MATHCrossRefGoogle Scholar
  98. Wheelwright,S.C.,(1977). Forecasting Methods for Management 2nd edn, Wiley, New YorkGoogle Scholar
  99. Wiener, N. (1949). The Extrapolation, Interpolation and Smoothing of Stationary Time Series, Wiley, New YorkMATHGoogle Scholar
  100. Wold, H. (1954). A Study in the Analysis of Stationary Time Series, 2nd edn, Almqvist and Wiksell, StockholmMATHGoogle Scholar
  101. Yakowitz, S. J. (1973). A stochastic model for daily river flows in an arid region. Water Resour. Res., 9, 1271–85CrossRefGoogle Scholar
  102. Yevjevich, V. M. (1963). Fluctuations of wet and dry years, part I, research data assembly and mathematical models. Colo. St. Univ., Fort Collins, Hydrol. Papers, No. 1Google Scholar
  103. Yevjevich, V. M. (1966). Stochastic problems in the design of reservoirs. Water Research (eds A. V. Kneese and S. C. Smith), John Hopkins Press, Baltimore, pp. 375–411.Google Scholar
  104. Yevjevich, V. M. (1972). Structural analysis of hydrologic time series, Colo. St. Univ., Fort Collins, Hydrol. Papers, No. 56Google Scholar
  105. Young, G. K. (1968). Discussion of ‘Mathematical assessment of synthetic hydrology’ by N. C. Matalas. Water Resour. Res., 4, 681–3CrossRefGoogle Scholar
  106. Young, G. K., and Pisano, W. C. (1968). Operational hydrology using residuals. J. Hydraul. Div., Am. Soc. Civ. Eng., 94 (HY4), 909–23Google Scholar
  107. Young, P. (1974). Recursive approaches to time series analysis. J. Inst. Math. Its Appl., 10, 209–29Google Scholar
  108. Yule, G. U. (1921). On the time correlation problem with special preference to the variate-difference correlation method. J. R. Statist. Soc., 84, 497–537CrossRefGoogle Scholar
  109. Yevjevich, V. M. (1927). On a method of investigating periodicities in disturbed series, with special reference to Wolfer’s sunspot numbers. Philos. Trans. A, 226, 267–98CrossRefGoogle Scholar

Copyright information

© N. T. Kottegoda 1980

Authors and Affiliations

  • N. T. Kottegoda
    • 1
  1. 1.Department of Civil EngineeringUniversity of BirminghamUK

Personalised recommendations