Performs Kalman filtering and evaluates the likelihood function for the state‑space model.

The type double function is imsls_d_kalman.

Function imsls_f_kalman is based on a recursive algorithm given by Kalman (1960), which has come to be known as the Kalman filter. The underlying model is known as the state‑space model. The model is specified stage by stage where the stages generally correspond to time points at which the observations become available. The function imsls_f_kalman avoids many of the computations and storage requirements that would be necessary if one were to process all the data at the end of each stage in order to estimate the state vector. This is accomplished by using previous computations and retaining in storage only those items essential for processing of future observations.

The notation used here follows that of Sallas and Harville (1981). Let yk (input in y using optional argument IMSLS_UPDATE) be the nk × 1 vector of observations that become available at time k. The subscript k is used here rather than t, which is more customary in time series, to emphasize that the model is expressed in stages k = 1, 2, … and that these stages need not correspond to equally spaced time points. In fact, they need not correspond to time points of any kind. The observation equation for the state‑space model is

Here, Zk (input in z using optional argument IMSLS_UPDATE) is an nk × q known matrix and bk is the q × 1 state vector. The state vector bk is allowed to change with time in accordance with the state equation

starting with b1 = μ1 + w1.

The change in the state vector from time k to k + 1 is explained in part by the transition matrix T k+1 (the identity matrix by default, or optionally input using IMSLS_T), which is assumed known. It is assumed that the q-dimensional wks (k = 1, 2,…) are independently distributed multivariate normal with mean vector 0 and variance-covariance matrix σ 2Qk, that the nk-dimensional eks (k = 1, 2,…) are independently distributed multivariate normal with mean vector 0 and variance-covariance matrix σ 2Rk, and that the wks and eks are independent of each other. Here, μ1 is the mean of b1 and is assumed known, σ 2 is an unknown positive scalar. Qk+1(input in q) and Rk (input in r) are assumed known.

Denote the estimator of the realization of the state vector bk given the observations y1, y2, …, yj by

By definition, the mean squared error matrix for

is

At the time of the k-th invocation, we have

and Ck|k-1 which were computed from the (k-1)-st invocation, input in b and covb, respectively. During the k-th invocation, function imsls_f_kalman computes the filtered estimate

along with Ck|k. These quantities are given by the update equations:

where

and where

Here, vk (stored in v) is the one-step-ahead prediction error, and σ 2Hk is the variance-covariance matrix for vk. Hk is stored in covv. The “start-up values” needed on the first invocation of imsls_f_kalman are

and C 1|0 = Q1 input via b and covb, respectively. Computations for the k-th invocation are completed by imsls_f_kalman computing the one-step-ahead estimate

along with Ck+1|k given by the prediction equations:

If both the filtered estimates and one-step-ahead estimates are needed by the user at each time point, imsls_f_kalman can be invoked twice for each time point—first without IMSLS_T and IMSLS_Q to produce

and Ck|k, and second without IMSLS_UPDATE to produce

and Ck+1|k. (Without IMSLS_T and IMSLS_Q, the prediction equations are skipped. Without IMSLS_UPDATE, the update equations are skipped.)

Often, one desires the estimate of the state vector more than one-step-ahead, i.e., an estimate of

is needed where k > j + 1. At time j, imsls_f_kalman is invoked with IMSLS_UPDATE to compute

Subsequent invocations of imsls_f_kalman without IMSLS_UPDATE can compute

Computations for

and Ck|j assume the variance-covariance matrices of the errors in the observation equation and state equation are known up to an unknown positive scalar multiplier, σ 2. The maximum likelihood estimate of σ 2 based on the observations y1, y2, …, ym, is given by

where

N and SS are the input/output arguments n and ss.

If σ 2 is known, the Rks and Qks can be input as the variance-covariance matrices exactly. The earlier discussion is then simplified by letting σ 2 = 1.

In practice, the matrices Tk, Qk, and Rk are generally not completely known. They may be known functions of an unknown parameter vector θ. In this case, imsls_f_kalman can be used in conjunction with an optimization program (see function imsl_f_min_uncon_multivar, IMSL C/Math/Library, Chapter 8, Optimization) to obtain a maximum likelihood estimate of θ. The natural logarithm of the likelihood function for y1, y2, …, ym differs by no more than an additive constant from

(Harvey 1981, page 14, equation 2.21).

Here,

(stored in alndet) is the natural logarithm of the determinant of V where σ 2V is the variance-covariance matrix of the observations.

Minimization of -2L(θ, σ 2; y1, y2, ..., ym) over all θ and σ 2 produces maximum likelihood estimates. Equivalently, minimization of -2Lc(θ;y1, y2, ..., ym) where

produces maximum likelihood estimates

The minimization of -2Lc(θ;y1, y2, ..., ym) instead of -2L(θ, σ 2; y1, y2, ..., ym), reduces the dimension of the minimization problem by one. The two optimization problems are equivalent since

minimizes -2L(θ, σ 2; y1, y2, ..., ym) for all θ, consequently,

can be substituted for σ 2 in L(θ, σ 2; y1, y2, ..., ym) to give a function that differs by no more than an additive constant from Lc(θ;y1, y2, ..., ym).

The earlier discussion assumed Hk to be nonsingular. If Hk is singular, a modification for singular distributions described by Rao (1973, pages 527–528) is used. The necessary changes in the preceding discussion are as follows:

Maximum likelihood estimation of parameters in the Kalman filter is discussed by Sallas and Harville (1988) and Harvey (1981, pages 111–113).

Function imsls_f_kalman is used to compute the filtered estimates and one-step-ahead estimates for a scalar problem discussed by Harvey (1981, pages 116‑117). The observation equation and state equation are given by

where the eks are identically and independently distributed normal with mean 0 and variance σ 2, the wks are identically and independently distributed normal with mean 0 and variance 4σ 2, and b1is distributed normal with mean 4 and variance 16σ 2. Two invocations of imsls_f_kalman are needed for each time point in order to compute the filtered estimate and the one-step-ahead estimate. The first invocation does not use the optional arguments IMSLS_T and IMSLS_Q so that the prediction equations are skipped in the computations. The update equations are skipped in the computations in the second invocation.

This example also computes the one-step-ahead prediction errors. Harvey (1981, page 117) contains a misprint for the value v4 that he gives as 1.197. The correct value of v4 = 1.003 is computed by imsls_f_kalman.

Function imsls_f_kalman is used with function imsl_f_min_uncon_multivar, (see IMSL C/Math/Library, Chapter 8, Optimization) to find a maximum likelihood estimate of the parameter θ in a MA(1) time series represented by yk = ɛk - θɛk-1. Function imsls_f_random_arma (see Chapter 12, Random Number Generation) is used to generate 200 random observations from an MA(1) time series with θ = 0.5 and σ 2 = 1.

The MA(1) time series is cast as a state‑space model of the following form (see Harvey 1981, pages 103–104, 112):

where the two-dimensional wks are independently distributed bivariate normal with mean 0 and variance σ 2 Qk and

The warning error that is printed as part of the output is not serious and indicates that imsl_f_min_uncon_multivar (see C/Math/Library, Chapter 8, Optimization) is generally used for multi-parameter minimization.