g13eac performs a combined measurement and time update of one iteration of the time-varying Kalman filter. The method employed for this update is the square root covariance filter with the system matrices in their original form.
The function may be called by the names: g13eac, nag_tsa_multi_kalman_sqrt_var or nag_kalman_sqrt_filt_cov_var.
3Description
For the state space system defined by
the estimate of given observations to is denoted by with . g13eac performs one recursion of the square root covariance filter algorithm, summarised as follows:
where is an orthogonal transformation triangularizing the pre-array. The triangularisation is carried out via Householder transformations exploiting the zero pattern in the pre-array. The measurement-update for the estimated state vector is
(1)
where is the Kalman gain matrix, whilst the time-update for is
(2)
where represents any deterministic control used. The relationship between the Kalman gain matrix and is given by
The function returns the product of the matrices and represented as , and the state covariance matrix factorized as (see the G13 Chapter Introduction for more information concerning the covariance filter).
4References
Anderson B D O and Moore J B (1979) Optimal Filtering Prentice–Hall
Harvey A C and Phillips G D A (1979) Maximum likelihood estimation of regression models with autoregressive — moving average disturbances Biometrika66 49–58
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
Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley
5Arguments
1: – IntegerInput
On entry: the actual state dimension, , i.e., the order of the matrices and .
Constraint:
.
2: – IntegerInput
On entry: the actual input dimension, , i.e., the order of the matrix .
Constraint:
.
3: – IntegerInput
On entry: the actual output dimension, , i.e., the order of the matrix .
Constraint:
.
4: – doubleInput/Output
Note: the th element of the matrix is stored in .
On entry: the leading lower triangular part of this array must contain , the left Cholesky factor of the state covariance matrix .
On exit: the leading lower triangular part of this array contains , the left Cholesky factor of the state covariance matrix .
5: – IntegerInput
On entry: the stride separating matrix column elements in the array s.
Constraint:
.
6: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: the leading part of this array must contain , the state transition matrix of the discrete system.
7: – IntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint:
.
8: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: if q is not NULL then the leading part of this array must contain the matrix , otherwise if q is NULL then the leading part of the array must contain the matrix . is the input weight matrix and is the noise covariance matrix.
9: – IntegerInput
On entry: the stride separating matrix column elements in the array b.
Constraint:
.
10: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: if the noise covariance matrix is to be supplied separately from the input weight matrix then the leading lower triangular part of this array must contain , the left Cholesky factor of the input process noise covariance matrix. If the noise covariance matrix is to be input with the weight matrix as then the array q must be set to NULL.
11: – IntegerInput
On entry: the stride separating matrix column elements in the array q.
On entry: the leading part of this array must contain , the output weight matrix of the discrete system.
13: – IntegerInput
On entry: the stride separating matrix column elements in the array c.
Constraint:
.
14: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: the leading lower triangular part of this array must contain , the left Cholesky factor of the measurement noise covariance matrix.
15: – IntegerInput
On entry: the stride separating matrix column elements in the array r.
Constraint:
.
16: – doubleOutput
Note: the th element of the matrix is stored in .
On exit: if k is not NULL then the leading part of k contains , the product of the Kalman filter gain matrix with the state transition matrix . If is not required then k must be set to NULL.
17: – IntegerInput
On entry: the stride separating matrix column elements in the array k.
On exit: if k is not NULL then the leading lower triangular part of this array contains . If k is NULL then h is not referenced and may be set to NULL.
19: – IntegerInput
On entry: the stride separating matrix column elements in the array h.
On entry: if both k and h are not NULL then tol is used to test for near singularity of the matrix . If you set tol to be less than then the tolerance is taken as , where is the machine precision. Otherwise, tol need not be set by you.
21: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
The use of the square root algorithm improves the stability of the computations.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g13eac 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
For this function two examples are presented. There is a single example program for g13eac, with a main program and the code to solve the two example problems is given in the functions ex1 and ex2.
Example 1 (ex1)
To apply three iterations of the Kalman filter (in square root covariance form) to the system . The same data is used for all three iterative steps.
Example 2 (ex2)
In the second example 2000 terms of an ARMA(1,1) time series (with and ) are generated using the function g05phc. The Kalman filter and optimization function e04ucc are then used to find the maximum likelihood estimate for the time series arguments and . The ARMA(1,1) time series is defined by