different kinds of measurement, but a feature common to all four is use of advanced statistical methodology. Four of the seven authors are mathematical statisticians, illustrating the value of collaboration of scientists and statisticians in measurement science today. We plan to devote some future issues to single topics of special interest. Thus we are contemplating an issue on non-destructive evaluation, another on sensors, and a third on inverse scattering problems. We shall welcome suggestions and critical comments from our readers. Board of Editors Executive Editors Donald R. Johnson (Natl. Measurement Lab.) Churchill Eisenhart (Mathematics) John W. Cooper (Physics) Sharon G. Lias (Chemistry) Donald G. Eitzen (Engineering) Howard J. M. Hanley (Boulder Lab.) Journal of Research National Bureau of Standards JOURNAL OF RESEARCH of the National Bureau of Standards Vol. 88, No. 1, January-February 1983 Estimation of Parameters in Models Peter V. Tryon* and Richard H. Jones** National Bureau of Standards, Boulder, CO 80303 August 2, 1982 This paper is intended to serve as an introduction to the use of the Kalman filter in modeling atomic clocks and obtaining maximum likelihood estimates of the model paramaters from data on an ensemble of clocks. Tests for the validity of the model and confidence intervals for the parameter estimates are discussed. Techniques for dealing with unequally spaced and partially or completely missing multivariate data are described. The existence of deterministic frequency drifts in clocks is established and estimates of the drifts are obtained. Key words: atomic clocks; Kalman filter; maximum likelihood; missing observations; random walks; state 1. Introduction The recursive updating of least squares estimates that would later become a special case of the Kalman "filter" was apparently known to Gauss [1],' and published in the modern statistical literature by Plackett [2]. Kalman's contribution [3] was the generalization to dynamic systems. Kalman's filter has found immense application in diverse areas of engineering. In classical applications, recursive estimates are obtained of the "state" of the system but parameters appearing elsewhere in the state space representation for the system and the mathematical model for the system must be considered known. In our application, Kalman's recursive equations allow us to compute the likelihood function for given values of parameters occurring anywhere in the state space representation. Nonlinear optimization techniques can then be used to find the maximum likelihood estimates. The recursive residuals, or innovations, may be examined to judge the adequacy-of-fit of the model, and generalized likelihood ratio tests may be used to test the significance of model parameters such as frequency drifts and obtain confidence intervals for the estimated parameters. Kalman filter techniques make it quite easy to deal with unequally spaced and completely or partially missing multivariate data. We describe these methods in this paper. The next article in this issue, "Estimating Time from Atomic Clocks," by Jones and Tryon [4] describes statistical procedures for detecting clock errors and allowing for the insertion or deletion of clocks in the ensemble during the parameter estimation process. It also describes the development of a time scale algorithm based on the model developed in this paper. 2. Models For Atomic Clocks A cesium atomic clock is a feedback control device whose frequency locks onto the fundamental resonance of the cesium atom at (ideally) 9,192,631,770 Hz, which defines the second. Frequency bias due to fundamental instrumentation limitations and environmental effects are determined by calibration against primary frequency standards. Stochastic fluctuations arise from shot noise in the cesium beam and the probabilistic nature of quantum *Deceased. Dr. Tryon served with the Center for Applied Mathematics, National Engineering Laboratory. **Division of Biometrics, Box B-119, School of Medicine, University of Colorado, Denver, CO 80262. Figures in brackets indicate the literature references at the end of this paper. different kinds of measurement, but a feature common to all four is use of advanced statistical methodology. Four of the seven authors are mathematical statisticians, illustrating the value of collaboration of scientists and statisticians in measurement science today. We plan to devote some future issues to single topics of special interest. Thus we are contemplating an issue on non-destructive evaluation, another on sensors, and a third on inverse scattering problems. We shall welcome suggestions and critical comments from our readers. Board of Editors Executive Editors Donald R. Johnson (Natl. Measurement Lab.) Churchill Eisenhart (Mathematics) John W. Cooper (Physics) Journal of Research National Bureau of Standards JOURNAL OF RESEARCH of the National Bureau of Standards Vol. 88, No. 1, January-February 1983 Estimation of Parameters in Models Peter V. Tryon* and Richard H. Jones** National Bureau of Standards, Boulder, CO 80303 August 2, 1982 This paper is intended to serve as an introduction to the use of the Kalman filter in modeling atomic clocks and obtaining maximum likelihood estimates of the model paramaters from data on an ensemble of clocks. Tests for the validity of the model and confidence intervals for the parameter estimates are discussed. Techniques for dealing with unequally spaced and partially or completely missing multivariate data are described. The existence of deterministic frequency drifts in clocks is established and estimates of the drifts are obtained. Key words: atomic clocks; Kalman filter; maximum likelihood; missing observations; random walks; state 1. Introduction The recursive updating of least squares estimates that would later become a special case of the Kalman “filter" was apparently known to Gauss [1],' and published in the modern statistical literature by Plackett [2]. Kalman's contribution [3] was the generalization to dynamic systems. Kalman's filter has found immense application in diverse areas of engineering. In classical applications, recursive estimates are obtained of the "state" of the system but parameters appearing elsewhere in the state space representation for the system and the mathematical model for the system must be considered known. In our application, Kalman's recursive equations allow us to compute the likelihood function for given values of parameters occurring anywhere in the state space representation. Nonlinear optimization techniques can then be used to find the maximum likelihood estimates. The recursive residuals, or innovations, may be examined to judge the adequacy-of-fit of the model, and generalized likelihood ratio tests may be used to test the significance of model parameters such as frequency drifts and obtain confidence intervals for the estimated parameters. Kalman filter techniques make it quite easy to deal with unequally spaced and completely or partially missing multivariate data. We describe these methods in this paper. The next article in this issue, "Estimating Time from Atomic Clocks," by Jones and Tryon [4] describes statistical procedures for detecting clock errors and allowing for the insertion or deletion of clocks in the ensemble during the parameter estimation process. It also describes the development of a time scale algorithm based on the model developed in this paper. 2. Models For Atomic Clocks A cesium atomic clock is a feedback control device whose frequency locks onto the fundamental resonance of the cesium atom at (ideally) 9,192,631,770 Hz, which defines the second. Frequency bias due to fundamental instrumentation limitations and environmental effects are determined by calibration against primary frequency standards. Stochastic fluctuations arise from shot noise in the cesium beam and the probabilistic nature of quantum *Deceased. Dr. Tryon served with the Center for Applied Mathematics, National Engineering Laboratory. mechanical transition rates which cause white noise in the frequency of the clock which is integrated into a random walk in time. In addition, empirical studies have demonstrated that frequency wanders independently as a random walk, introducing the integral of a random walk into time. Finally, there is considerable international debate over the inclusion of linear drift in frequency to account for break-in and aging. Such drifts could be strictly constant over a long time period, or allowed to change slowly as a third random walk. The methods developed in this paper provide, for the first time, valid statistical tests of these hypotheses. x(t) is the clock's deviation from "perfect" time at sample point t in nanoseconds; d(t) is the time interval in days between sample points t and t-l; y(t) is the clock's frequency deviation from perfect frequency at sample point t in nanoseconds/day; w(t) is the drift in frequency in nanoseconds/day2; ε(t), n(t), and a(t) are mutually independent white zero mean Gaussian random variables with standard deviations √d(t)o., √d(T)o,, and √d(t)o., respectively, in nanoseconds, nanoseconds/day, and nanoseconds/day2. The Vd(t) arises because the variance of a random walk is proportional to the time interval. If σ = 0, w(t) is a constant, w, representing a deterministic linear trend. If both σ = 0 and w = 0, the model is drift-free. The purpose of this study is to evaluate the validity of these models, and to obtain generalized likelihood ratio tests for the significance of the drift parameters and maximum likelihood estimates of the parameters o., Ong and w (or o., if appropriate), for each clock in the ensemble. 3. The Kalman State Space Model for a Clock Ensemble The combination of clocks into a Kalman state space model will be explained by considering the special case of a three-clock model. The state transition equation is |