Using state space representations, the author’s previous work on fitting continuous time autoregressions to unequally spaced univariate data is extended to several multivariate models of practical importance. Continuous time multivariate first order autoregressions and the limiting case of multivariate random walks are used to model multiple medical observations collected at unequally spaced time points. Mean levels can be included in these models as constants to be estimated or as random variables with prior variances. Maximum likelihood estimation of the unknown parameters allows the development of individual normal ranges so a physician can be alerted if a set of observations is out of line. Three examples based on medical data are presented. Extensions include optimal control when drug therapy is involved. Another application is transfer function estimation from unequally spaced data. A bicycle rider pedals against a load which varies as a stationary process. Various respiration gases are measured on a breath to breath basis which are unequally spaced in time. The transfer function of the body indicates the mechanisms of the body’s response to exercise.
- Covariance Matrix
- Random Walk
- Observational Error
- Random Input
- Space Time Point
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Akaike, H., Information theory and an extension of the maximum likelihood principle, Second International Symposium on Information Theory (B. N. Petrov and F. Csaki, Eds.), Akademia Kaido, Budapest, 267–281.
Box, G. E. P. and G. M. Jenkins, Time Series Analysis: forecasting and control. Revised Edition, Holden-Day, San Francisco, 1976.
Dennis, J. E. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, N. J. (1983).
Duncan, D. B, and S. D. Horn, Linear dynamic recursive estimation from the viewpoint of regression analysis, Journal of the American Statistical Association 67 (1972), 815–821.
Duncan, D. B. and R. H. Jones, Multiple regression with stationary errors. Journal of the American Statistical Association 61 (1966), 917–928.
Engeman, R. M., G. D. Swanson and R. H Jones, Input design for model discrimination: application to respiratory control during exercise. IEEE Transactions on Biomedical Engineering, BME-26 (1979), 579–585.
Graybill, F. A., Theory and Application of the Linear Model, Duxbury Press, North Scituate, Massachusetts, 1976.
Harris, E. K., Some theory of references values. I. Stratified (categorized) normal ranges and a method for following an individual’s clinical laboratory values. Clinical Chemistry 21 (1975), 1457–1464.
Harris, E. K., Some theory of references values. II. Comparison of some statistical models of intraindividual variation in blood constituents. Clinical Chemistry 22 (1976), 1343–1350.
Harris, E. K., Further applications of time series analysis to short series of biochemical measurements. Reference Values in Laboratory Medicine (Gräsbeck, R. and T. Alström, Eds.), John Wiley & Sons Ltd.(1981), 167-176.
Harris, E. K., T. Yasaka, M. R. Horton and G. Shakarji, Comparing multivariate and univariate subject-specific reference regions for blood constituents in healthy persons. Clinical Chemistry, 28 (1982), 422–426.
Harrison, P. J. and C. F. Stevens, Bayesian Forecasting. Journal of the Royal Statistical Society, Series B (Methodological) 38 (1976), 205–247.
Harvey, A. C. and G. D. A. Phillips, Maximum likelihood estimation of regression models with autoregressive-moving average disturbances. Biometrika 66 (1979), 49–58.
Jones, R. H., Exponential smoothing for multivariate time series. Journal of the Royal Statistical Society, Series B (Methodological) 28 (1966), 241–251.
Jones, R. H., Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics 22 (1980), 389–395.
Jones, R. H., Fitting a continuous time autoregression to discrete data. Applied Time Series Analysis II (Findley, D. F., Ed.), Academic Press (1981), 651-682.
R. H. Jones and P. V. Tryon, Estimating time from atomic clocks. Journal of research of the National Bureau of Standards, 88 (1983), 17–14.
Kalman, R. E., A new approach to linear filtering and prediction problems. Transactions of the ASME, Series D, Journal of Basic Engineering, 82 (1960), 35–45.
Kalman, R. E. and R. S. Bucy, New results in linear filtering and prediction theory. Transactions of the ASME, Series D, Journal of Basic Engineering, 83 (1961), 95–108.
Schnabel, R. B., J. E. Koontz and B. E. Weiss, A modular system of algorithms for unconstrained minimization, National Bureau of Standards, Boulder, Colorado (1983), University of Colorado Department of Computer Science Technical Report CU-CS-240-82, November, 1982.
Sherrill, D. L. and G. D. Swanson, Computered controlled cycle ergometer, IEEE Transactions on Biomedical Engineering, BME-28, (1981), 711–713.
P. V. Tryon and R. H. Jones, Estimation of parameters in models for cesium beam atomic clocks. Journal of Research of the National Bureau of Standards, 88 (1983), 3–16.
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Jones, R.H. (1984). Fitting Multivariate Models to Unequally Spaced Data. In: Parzen, E. (eds) Time Series Analysis of Irregularly Observed Data. Lecture Notes in Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9403-7_8
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