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.

2 Specification

#include <nag.h>

void	g13eac (Integer n, Integer m, Integer p, double s[], Integer tds, const double a[], Integer tda, const double b[], Integer tdb, const double q[], Integer tdq, const double c[], Integer tdc, const double r[], Integer tdr, double k[], Integer tdk, double h[], Integer tdh, double tol, NagError *fail)

The function may be called by the names: g13eac, nag_tsa_multi_kalman_sqrt_var or nag_kalman_sqrt_filt_cov_var.

3 Description

For the state space system defined by

\begin{array}{l} X_{i + 1} & = & A_{i} X_{i} + B_{i} W_{i} & var (W_{i}) = Q_{i} \\ Y_{i} & = & C_{i} X_{i} + V_{i} & var (V_{i}) = R_{i} \end{array}

the estimate of

X_{i}

given observations

Y_{1}

Y_{i - 1}

is denoted by

{\hat{X}}_{i ∣ i - 1}

with

var ({\hat{X}}_{i ∣ i - 1}) = P_{i ∣ i - 1} = S_{i} S_{i}^{T}

. g13eac performs one recursion of the square root covariance filter algorithm, summarised as follows:

\begin{matrix} (\begin{matrix} R_{i}^{1 / 2} & C_{i} S_{i} & 0 \\ 0 & A_{i} S_{i} & B_{i} Q_{i}^{1 / 2} \end{matrix}) U & = (\begin{matrix} H_{i}^{1 / 2} & 0 & 0 \\ G_{i} & S_{i + 1} & 0 \end{matrix}) \\ (Pre-array) & (Post-array) \end{matrix}

where

U

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

X

{\hat{X}}_{i ∣ i} = {\hat{X}}_{i ∣ i - 1} - K_{i} [C_{i} {\hat{X}}_{i ∣ i - 1} - Y_{i}]

(1)

where

K_{i}

is the Kalman gain matrix, whilst the time-update for

X

{\hat{X}}_{i + 1 ∣ i} = A_{i} {\hat{X}}_{i ∣ i} + D_{i} U_{i}

(2)

where

D_{i} U_{i}

represents any deterministic control used. The relationship between the Kalman gain matrix

K_{i}

and

G_{i}

is given by

A_{i} K_{i} = G_{i} {(H_{i}^{1 / 2})}^{−1}

The function returns the product of the matrices

A_{i}

and

K_{i}

represented as

{A K}_{i}

, and the state covariance matrix

P_{i ∣ i - 1}

factorized as

P_{i ∣ i - 1} = S_{i} S_{i}^{T}

(see the G13 Chapter Introduction for more information concerning the covariance filter).

4 References

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 Biometrika 66 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. Software 15 243–256

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

5 Arguments

1: $n$ – Integer Input: On entry: the actual state dimension, $n$ , i.e., the order of the matrices $S_{i}$ and $A_{i}$ .

Constraint: $n \geq 1$ .
2: $m$ – Integer Input: On entry: the actual input dimension, $m$ , i.e., the order of the matrix $Q_{i}^{1 / 2}$ .

Constraint: $m \geq 1$ .
3: $p$ – Integer Input: On entry: the actual output dimension, $p$ , i.e., the order of the matrix $R_{i}^{1 / 2}$ .

Constraint: $p \geq 1$ .
4: $s [n \times tds]$ – double Input/Output: Note: the $(i, j)$ th element of the matrix $S$ is stored in $s [(i - 1) \times tds + j - 1]$ .

On entry: the leading $n \times n$ lower triangular part of this array must contain $S_{i}$ , the left Cholesky factor of the state covariance matrix $P_{i ∣ i - 1}$ .

On exit: the leading $n \times n$ lower triangular part of this array contains $S_{i + 1}$ , the left Cholesky factor of the state covariance matrix $P_{i + 1 ∣ i}$ .
5: $tds$ – Integer Input: On entry: the stride separating matrix column elements in the array s.

Constraint: $tds \geq n$ .
6: $a [n \times tda]$ – const double Input: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(i - 1) \times tda + j - 1]$ .

On entry: the leading $n \times n$ part of this array must contain $A_{i}$ , the state transition matrix of the discrete system.
7: $tda$ – Integer Input: On entry: the stride separating matrix column elements in the array a.

Constraint: $tda \geq n$ .
8: $b [n \times tdb]$ – const double Input: Note: the $(i, j)$ th element of the matrix $B$ is stored in $b [(i - 1) \times tdb + j - 1]$ .

On entry: if q is not NULL then the leading $n \times m$ part of this array must contain the matrix $B_{i}$ , otherwise if q is NULL then the leading $n \times m$ part of the array must contain the matrix $B_{i} Q_{i}^{1 / 2}$ . $B_{i}$ is the input weight matrix and $Q_{i}$ is the noise covariance matrix.
9: $tdb$ – Integer Input: On entry: the stride separating matrix column elements in the array b.

Constraint: $tdb \geq m$ .
10: $q [m \times tdq]$ – const double Input: Note: the $(i, j)$ th element of the matrix $Q$ is stored in $q [(i - 1) \times tdq + j - 1]$ .

On entry: if the noise covariance matrix is to be supplied separately from the input weight matrix then the leading $m \times m$ lower triangular part of this array must contain $Q_{i}^{1 / 2}$ , 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 $B_{i} Q_{i}^{1 / 2}$ then the array q must be set to NULL.
11: $tdq$ – Integer Input: On entry: the stride separating matrix column elements in the array q.

Constraint: $tdq \geq m$ if q is defined.
12: $c [p \times tdc]$ – const double Input: Note: the $(i, j)$ th element of the matrix $C$ is stored in $c [(i - 1) \times tdc + j - 1]$ .

On entry: the leading $p \times n$ part of this array must contain $C_{i}$ , the output weight matrix of the discrete system.
13: $tdc$ – Integer Input: On entry: the stride separating matrix column elements in the array c.

Constraint: $tdc \geq n$ .
14: $r [p \times tdr]$ – const double Input: Note: the $(i, j)$ th element of the matrix $R$ is stored in $r [(i - 1) \times tdr + j - 1]$ .

On entry: the leading $p \times p$ lower triangular part of this array must contain $R_{i}^{1 / 2}$ , the left Cholesky factor of the measurement noise covariance matrix.
15: $tdr$ – Integer Input: On entry: the stride separating matrix column elements in the array r.

Constraint: $tdr \geq p$ .
16: $k [n \times tdk]$ – double Output: Note: the $(i, j)$ th element of the matrix $K$ is stored in $k [(i - 1) \times tdk + j - 1]$ .

On exit: if k is not NULL then the leading $n \times p$ part of k contains ${A K}_{i}$ , the product of the Kalman filter gain matrix $K_{i}$ with the state transition matrix $A_{i}$ . If $A K_{i}$ is not required then k must be set to NULL.
17: $tdk$ – Integer Input: On entry: the stride separating matrix column elements in the array k.

Constraint: if k is not NULL, $tdk \geq p$
18: $h [p \times tdh]$ – double Output: Note: the $(i, j)$ th element of the matrix $H$ is stored in $h [(i - 1) \times tdh + j - 1]$ .

On exit: if k is not NULL then the leading $p \times p$ lower triangular part of this array contains $H_{i}^{1 / 2}$ . If k is NULL then h is not referenced and may be set to NULL.
19: $tdh$ – Integer Input: On entry: the stride separating matrix column elements in the array h.

Constraint: if both k and h are not NULL, $tdh \geq p$
20: $tol$ – double Input: On entry: if both k and h are not NULL then tol is used to test for near singularity of the matrix $H_{i}^{1 / 2}$ . If you set tol to be less than $p^{2} ε$ then the tolerance is taken as $p^{2} ε$ , where $ε$ is the machine precision. Otherwise, tol need not be set by you.
21: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry $tda = ⟨ value ⟩$ while $n = ⟨ value ⟩$ . These arguments must satisfy $tda \geq n$ .

On entry $tdb = ⟨ value ⟩$ while $m = ⟨ value ⟩$ . These arguments must satisfy $tdb \geq m$ .

On entry $tdc = ⟨ value ⟩$ while $n = ⟨ value ⟩$ . These arguments must satisfy $tdc \geq n$ .

On entry $tdh = ⟨ value ⟩$ while $p = ⟨ value ⟩$ . These arguments must satisfy $tdh \geq p$ .

On entry $tdk = ⟨ value ⟩$ while $p = ⟨ value ⟩$ . These arguments must satisfy $tdk \geq p$ .

On entry $tdq = ⟨ value ⟩$ while $m = ⟨ value ⟩$ . These arguments must satisfy $tdq \geq m$ .

On entry $tdr = ⟨ value ⟩$ while $p = ⟨ value ⟩$ . These arguments must satisfy $tdr \geq p$ .

On entry, $tds = ⟨ value ⟩$ while $n = ⟨ value ⟩$ . These arguments must satisfy $tds \geq n$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $p = ⟨ value ⟩$ .
Constraint: $p \geq 1$ .
NE_MAT_SINGULAR: The matrix sqrt( $H$ ) is singular.
NE_NULL_ARRAY: Array h has null address.

7 Accuracy

The use of the square root algorithm improves the stability of the computations.

8 Parallelism 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.

9 Further Comments

The algorithm requires

\frac{7}{6} n^{3} + n^{2} (\frac{5}{2} p + m) + n (\frac{1}{2} m^{2} + p^{2})

operations and is backward stable (see Vanbegin et al. (1989)).

10 Example

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

(A_{i}, B_{i}, C_{i})

. 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

σ^{2} = 1.0, θ = 0.9

and

ϕ = 0.4

) 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

θ

and

ϕ

. The ARMA(1,1) time series is defined by

y_{k} = ϕ y_{k - 1} + ε_{k} - θ ε_{k - 1}

This has the following state space representation (Harvey and Phillips (1979))

\begin{array}{l} x_{k} = (\begin{matrix} ϕ & 1 \\ 0 & 0 \end{matrix}) x_{k - 1} + (\begin{matrix} 1 \\ - θ \end{matrix}) ε_{k} \\ y_{k} = (\begin{matrix} 1 & 0 \end{matrix}) x_{k} \end{array}

where the state vector

x_{k} = (\begin{matrix} y_{k} \\ - θ ε_{k} \end{matrix})

and

ε_{k}

is uncorrelated white noise with zero mean and variance

σ^{2}

, i.e.,

E [ε_{k}] = 0, E [ε_{k} ε_{k}] = σ^{2}, E [y_{k} ε_{k}] = σ^{2} ​ and ​ E [ε_{k} ε_{k - 1}] = 0 .

Since

σ^{2} = 1

we arrive at the following Kalman Filter matrices

\begin{array}{l} A_{k} = (\begin{matrix} ϕ & 1 \\ 0 & 0 \end{matrix}), B_{k} = (\begin{matrix} 1 \\ - θ \end{matrix}) \\ C_{k} = (\begin{matrix} 1 & 0 \end{matrix}), Q_{k} = 0 ​ and ​ R_{k} = 1 . \end{array}

The initial estimates for the state vector,

x_{1 ∣ 0}

, and state covariance matrix,

P_{1 ∣ 0}

, are:

x_{1 ∣ 0} = E [x_{k}] = 0 ​ and ​ P_{1 ∣ 0} = E [x_{k} x_{k}^{T}] = (\begin{matrix} E [y_{k} y_{k}] & - θ E [y_{k} ε_{k}] \\ - θ E [y_{k} ε_{k}] & θ^{2} E [ε_{k} ε_{k}] \end{matrix}) .

Since

E [y_{k} y_{k}] = γ_{\circ} = \frac{(1 + θ^{2} - 2 ϕ θ) σ^{2}}{(1 - ϕ^{2})}

(Wei (1990))

P_{1 ∣ 0} = (\begin{matrix} γ_{\circ} & - θ \\ - θ & θ^{2} \end{matrix}) .

Using

P_{1 ∣ 0} = S_{1 ∣ 0} S_{1 ∣ 0}^{T}

gives an initial Cholesky ‘square root’ of

S_{1 ∣ 0} = (\begin{matrix} \sqrt{γ_{\circ}} & 0 \\ \frac{- θ}{\sqrt{γ_{\circ}}} & θ \sqrt{\frac{γ_{\circ} - 1}{γ_{\circ}}} \end{matrix}) .

g13ea: FL CL CPP AD PY MB

NAG CL Interfaceg13eac (multi_​kalman_​sqrt_​var)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g13eac (multi_kalman_sqrt_var)