The routine may be called by the names g13eaf or nagf_tsa_multi_kalman_sqrt_var.
The Kalman filter arises from the state space model given by:
where is the state vector of length at time , is the observation vector of length at time , and of length and of length are the independent state noise and measurement noise respectively.
The estimate of given observations to is denoted by with state covariance matrix , while the estimate of given observations to is denoted by with covariance matrix . The update of the estimate, , from time to time , is computed in two stages. First, the measurement-update is given by
where is the Kalman gain matrix. The second stage is the time-update for which is given by
where represents any deterministic control used.
The square root covariance filter algorithm provides a stable method for computing the Kalman gain matrix and the state covariance matrix. The algorithm can be summarised as
where is an orthogonal transformation triangularizing the left-hand pre-array to produce the right-hand post-array. The relationship between the Kalman gain matrix, , and is given by
g13eaf requires the input of the lower triangular Cholesky factors of the noise covariance matrices and, optionally, and the lower triangular Cholesky factor of the current state covariance matrix, , and returns the product of the matrices and , , the Cholesky factor of the updated state covariance matrix and the matrix used in the computation of the likelihood for the model.
Vanbegin M, van Dooren P and Verhaegen M H G (1989) Algorithm 675: FORTRAN subroutines for computing the square root covariance filter and square root information filter in dense or Hessenberg forms ACM Trans. Math. Software15 243–256
Verhaegen M H G and van Dooren P (1986) Numerical aspects of different Kalman filter implementations IEEE Trans. Auto. Contr.AC-31 907–917
1: – IntegerInput
On entry: , the size of the state vector.
2: – IntegerInput
On entry: , the size of the observation vector.
3: – IntegerInput
On entry: , the dimension of the state noise.
4: – Real (Kind=nag_wp) arrayInput
On entry: the state transition matrix, .
5: – IntegerInput
On entry: the first dimension of the arrays a, b, s and k as declared in the (sub)program from which g13eaf is called.
6: – Real (Kind=nag_wp) arrayInput
On entry: the noise coefficient matrix .
7: – LogicalInput
On entry: if , the state noise covariance matrix is assumed to be the identity matrix. Otherwise the lower triangular Cholesky factor, , must be provided in q.
8: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array q
must be at least
On entry: if , q must contain the lower triangular Cholesky factor of the state noise covariance matrix, . Otherwise q is not referenced.
9: – IntegerInput
On entry: the first dimension of the array q as declared in the (sub)program from which g13eaf is called.
if , ;
10: – Real (Kind=nag_wp) arrayInput
On entry: the measurement coefficient matrix, .
11: – IntegerInput
On entry: the first dimension of the arrays c, r and h as declared in the (sub)program from which g13eaf is called.
12: – Real (Kind=nag_wp) arrayInput
On entry: the lower triangular Cholesky factor of the measurement noise covariance matrix .
13: – Real (Kind=nag_wp) arrayInput/Output
On entry: the lower triangular Cholesky factor of the state covariance matrix, .
On exit: the lower triangular Cholesky factor of the state covariance matrix, .
14: – Real (Kind=nag_wp) arrayOutput
On exit: the Kalman gain matrix, , premultiplied by the state transition matrix, , .
15: – Real (Kind=nag_wp) arrayOutput
On exit: the lower triangular matrix .
16: – Real (Kind=nag_wp)Input
On entry: the tolerance used to test for the singularity of . If , then is used instead. The inverse of the condition number of is estimated by a call to f07tgf. If this estimate is less than tol then is assumed to be singular.
17: – Integer arrayWorkspace
18: – Real (Kind=nag_wp) arrayWorkspace
19: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
On entry, and .
On entry, .
On entry, and .
On entry, and .
On entry, .
On entry, .
On entry, .
The matrix is singular.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The use of the square root algorithm improves the stability of the computations as compared with the direct coding of the Kalman filter. The accuracy will depend on the model.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g13eaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13eaf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
For models with time-invariant and , g13ebf can be used.
The estimate of the state vector can be computed from by
are the independent one step prediction residuals. The required matrix-vector multiplications can be performed by f06paf.
If and are independent multivariate
Normal variates then the log-likelihood for observations is given by
where is a constant.
The Cholesky factors of the covariance matrices can be computed using f07fdf.
Note that the model
can be specified either with b set to the identity matrix and and the matrix input in q or with and b set to .
This example first inputs the number of updates to be computed and the problem sizes. The initial state vector and state covariance matrix are input followed by the model matrices and optionally . The Cholesky factors of the covariance matrices can be computed if required. The model matrices can be input at each update or only once at the first step. At each update the observed values are input and the residuals are computed and printed and the estimate of the state vector, , and the deviance are updated. The deviance is log-likelihood ignoring the constant. After the final update the state covariance matrix is computed from s and printed along with final estimate of the state vector and the value of the deviance.
The data is for a two-dimensional time series to which a VARMA has been fitted. For the specification of a VARMA model as a state space model see the G13 Chapter Introduction. The initial value of , , is the solution to
For convenience, the mean of each series is input before the first update and subtracted from the observations before the measurement update is computed.