00055 �8eb8f33682dd4ffedb872005d8a4b1

54

Hembree & Zimmer

quite a while for discussion of the Kalman filter to start appearing in the statistics literaturę. Several papers, Harrison and Stevens (1971, 1976), Duncan and Horn (1972), Ledolter (1979) and Meinhold and Singpurwalla (1983), madę the connections between Kalman filtering and Bayesian recursive least squares forecasting for the dynamie linear model. Although this connection was madę very early on in the engineering literaturę, Ho and Lee (1964), these papers did serve to introduce the Kalman filter to statisticians in a familiar notational setting.

It was not until very recently that the application of Kalman filtering to statistical process control was madę. Phadke (1981) used a Kalman filter approach to evaluate ąuality plans. Crowder (1986) developed an adaptive Kalman filter for control charting applications.

A derivation of the Kalman filter is presented here as a vehicle for developing the notation that is needed later. The derivation given below relies on developing the appropriate likelihood and prior densities and then utilizing standard Bayes Normal-Normal conjugate prior theory to derive the corresponding posterior distribution.

Let the State portion of the system model satisfy the linear equation

^x, = <**,-. + ^w,

where 0 is an nxn State transition matrix and w is w-vector white noise where W/ ~ N(0, Q,). This may be either a continuous-time or discrete process. The state eąuation, is propagated from some initial condition Xo. Since it may not be known a priori, \q is modeled as a random vector such that xo~ N(x₀, P₀)

where x₀ and P₀ are the prior mean and covariance.

The system output is observed by noisy measurements modeled by

y, = Hx, +▼,

where y is an m-vector discrete time measurement process, H is an mxn measurement matrix, x is the w-vector State process and v is w-vector discrete time white noise where v,~ N(0, R,). It is further assumed that Xo, w, and v are uncorrelated.

Given the assumed model above, the likelihood function is Gaussian. In particular, the conditional likelihood density for the current measurement given complete knowledge of the current State and previous measurement data is given by