and X(t) and (t) are redefined from the clock model in the obvious way. This model would be suitable for estimation purposes but not for a time scale algorithm. With the drift in the state vector we can enter the uncertainty of its estimation in the initial state covariance matrix. It is then propagated through to the uncertainty in time. The simplified model above assumes w is known without error.

8. Conclusion

The Kalman filter is a powerful tool for maximum likelihood model fitting and parameter estimation in time series analysis. We have established the validity of the proposed models for clock behavior and obtained precise estimates of clock parameters. The existence of deterministic frequency drifts over a period of a year has been demonstrated in clocks at the beginning and end of their life span.

The primary disadvantage of these methods is the cost of function evaluation in the nonlinear optimization process. The analysis of a year of data on seven clocks takes 10 minutes on a CDC Cyber 170/750 computer and 10 hours on the DEC 11/70. Use of state-of-the-art optimization codes is essential. We use an optimization package especially designed for maximum likelihood applications. It is adapted from the package written by Weiss [7], based on algorithms by Dennis and Schnabel [8]. The code used on the CDC Cyber 170/750 is part of a preliminary version of the National Bureau of Standard's STARPAC library [9].

The authors would like to thank James A. Barnes and David W. Allan of the National Bureau of Standards, Time and Frequency Division. They not only brought the problem to our attention and suggested the mathematical model, but collaborated with us on a continuous basis. This study was partially supported by the NBS National Engineering Laboratory.

9. References

[1] Gauss C. F. Theoria combinationis observationum erroribus minimis obnoxiae. Werke, 4. Gottingen. 1821. (Collected works 1873)

[2] Plackett, R. L. Some theorems in least squares. Biometrika, 19: 149-157; 1950.

[3] Kalman, R. E. A new approach to linear filtering and prediction problems. ASME Transactions, Part D, Journal of Basic Engineering, 82: 35-45; 1960.

[4] Jones, R. H.; Tryon, P. V. Estimating time from atomic clocks. Second Symposium on Atomic Time Scale Algorithms, June 23-25, 1982, Boulder, CO.

[5] Graybill, F. A. Theory and application of the linear model. North Scituate: Duxbury Press; 1976.

[6] Jones, R. H. Maximum likelihood fitting of ARMA models to time series with missing observations. Technometics, 22: 389-395; 1980.

[7] Weiss, B. E. A modular software package for solving unconstrained nonlinear optimization problems. Master's thesis, University of Colorado at Boulder; 1980.

[8] Dennis, J. E.; Schnabel, R. B. Numerical methods for unconstrained optimization and nonlinear equations. New Jersey: Prentice Hall; 1982.

[9] Donaldson, J. R.; Tryon, P. V. STARPAC: The standards time series and regression package. Unpublished-Documentation available from Janet P. Donaldson, NBS-714, 325 Broadway, Boulder, CO

80303.

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Figures la-f and 2a-f follow.