Abstract
The application of the continuous state space model to unequally spaced sequence data is discussed and illustrated. The continuous model implies a discrete model for the observed data. Practical expressions for relevant discrete model quantities are given. These quantities are required for the digital processing of the data and in particular for the application of the Kalman and smoothing filter and related calculations. Applications illustrate the procedures.
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Anderson, B. D. O. and Moore, J. B.: 1979, Optimal Filtering, Prentice-Hall, Englewood Cliffs.
Ansley, C. F., Kohn, R. and Wong, C. M.: 1993, Nonparametric spline regression with prior information, Biometrika 80(1), 75–88.
Ansley, C. F. and Wecker, W. F.: 1993, ‘Extensions and examples of the signal extraction approach to regression’, in Proc. Conf. Applied Time Series Analysis of Economic Data, A. Zellner (ed.) Bureau of the Census, Washington, D.C., pp. 181–192.
Arnold, L.: Stochastic Differential Equations: Theory and Applications, John Wiley, New York, 1974.
Bellman, R.: 1997, Introduction to Matrix Analysis, 2nd edn, SIAM, Philadelphia.
Besse, P. C., Cardot, H. and Ferraty F.: 1997, Simultaneous nonparametric regressions of unbalanced longitudinal data, Comput. Statist. Data Anal. 24, 255–270.
Box, G. E. P., Jenkins, G.M., and Reinsel, G. C.: Time Series Analysis: Forecasting and Control, 3rd edn, Prentice-Hall, Englewood Cliffs.
De Jong, P.: 1989, Smoothing and interpolation with the state-space model, J. Am. Statist. Assoc. 84, 1085–1088.
De Jong, P.: 1991a, The diffuse Kalman filter, Ann. Statist. 19(2), 1073–1083.
De Jong, P.: 1991b, Stable algorithms for the state-space model, J. Time Series Anal. 12(2), 143–158.
De Jong, P. and Chu-Chun-Lin, S.: 1994, Stationary and nonstationary state space models, J. Time Series Anal. 15(1), 151–166.
De Jong, P. and Penzer, J. R.: 1998, Diagnosing shocks in time series, J. Am. Statist. Assoc. 93, 796–806.
Denison, D. G. T., Mallick, B. K. and Smith, A. F. M.: 1998, Automatic Bayesian curve fitting, J. Royal Statist. Soc.-B 60(2), 333–350.
Draper, N. and Smith, H.: Applied Regression Analysis, 3rd edn, John Wiley, New York.
Green, P. G. and Silverman B.W.: 1994, Nonparametric Regression and Generalized Linear Models, Chapman and Hall, London.
Hall, P. and Titterington D. M.: 1992, Edge-preserving and peak-preserving smoothing, Technometrics 34(4), 429–440.
Harvey, A. C.: 1989, Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.
Harvey, A. C. and Stock J. H.: 1994, Estimation, smoothing, interpolation and distribution for structural time-series models in continuous time, in Models, Methods and Applications of Econometrics, P. C. B. Philips (ed.), Basil Blackwell, Oxford, pp. 55–70.
Heckman, N. E. and Ramsay J. O.: 2000, Penalized regression with model-based penalties, Canad. J. Statist. (to appear).
Kohn, R. and Ansley, C. F.: 1987a, A New algorithm for spline smoothing based on smoothing a stochastic process, SIAM J. Sci. Statist. Computat. 8(1), 33–48.
Kohn, R. and Ansley, C. F.: 1987b, Signal extraction for finite nonstationary time series, Biometrika 74(2), 411–421.
Kohn, R. and Ansley, C. F.: 1989, A fast algorithm for signal extraction, influence and crossvalidation in state space models, Biometrika 76(1), 65–79.
Lavielle, M.: 1999, Detection of multiple changes in a sequence of dependent variables, Stoch. Proc. Appl. 83(1), 79–102.
Ramsay, J. O. and Silverman, B. W.: 1997, Functional Data Analysis, Springer-Verlag, New York.
Rosenberg, B.: 1973, Random coefficient models: the analysis of a cross-section of time series by stochastically convergent parameter regression, Ann. Eco. Social Measure. 2, 399–428.
Shilov, G.: 1971, Linear Algebra, Prentice Hall, Englewood Cliffs.
Stewart, G.: Introduction to Matrix Computations, Academic Press, New York.
Wahba, G.: 1978, Improper priors, spline smoothing, and the problems of guarding against model errors in regression, J. Royal Statist. Soc.-Series B 40, 364–372.
Wahba, G.: 1990, Spline Models for Observational Data, 2nd edn, SIAM, Philadelphia.
Wecker, W. and Ansley, C. F.: 1983, The signal extraction approach to nonlinear regression and spline smoothing, J. Am. Statist. Assoc. 78, 81–89.
Weinert, H. R., Byrd, R. and Sidhu, G.: 1980, A stochastic framework for recursive computation of spline functions: part II, smoothing splines, J. Optim. Theory Appl. 30(2), 255–268.
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Jong, P.d., Mazzi, S. Modeling and Smoothing Unequally Spaced Sequence Data. Statistical Inference for Stochastic Processes 4, 53–71 (2001). https://doi.org/10.1023/A:1017510420686
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DOI: https://doi.org/10.1023/A:1017510420686