NAG Library Routine Document

G13EBF

INTEGER	N, M, L, LDS, LDQ, LDM, IWK(M), IFAIL
REAL (KIND=nag_wp)	A(LDS,N), B(LDS,L), Q(LDQ,), C(LDM,N), R(LDM,M), S(LDS,N), K(LDS,M), H(LDM,M), U(LDS,), TOL, WK((N+M)*(N+M+L))
LOGICAL	STQ
CHARACTER(1)	TRANSF

3 Description

The Kalman filter arises from the state space model given by

\begin{array}{l} X_{i + 1} = A X_{i} + B W_{i}, & Var (W_{i}) = Q_{i} \\ Y_{i} = C X_{i} + V_{i}, & Var (V_{i}) = R_{i} \end{array}

where

X_{i}

is the state vector of length

n

at time

i

Y_{i}

is the observation vector of length

m

at time

i

and

W_{i}

of length

l

and

V_{i}

of length

m

are the independent state noise and measurement noise respectively. The matrices

A, B

and

C

are time invariant.

The estimate of

X_{i}

given observations

Y_{1}

Y_{i - 1}

is denoted by

{\hat{X}}_{i ∣ i - 1}

with state covariance matrix

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

while the estimate of

X_{i}

given observations

Y_{1}

Y_{i}

is denoted by

{\hat{X}}_{i ∣ i}

with covariance matrix

Var ({\hat{X}}_{i ∣ i}) = P_{i ∣ i}

. The update of the estimate,

{\hat{X}}_{i ∣ i - 1}

, from time

i

to time

(i + 1)

is computed in two stages. First, the measurement-update is given by

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

(1)

where

K_{i} = P_{i ∣ i} C^{T} {[C P_{i ∣ i} C^{T} + R_{i}]}^{- 1}

is the Kalman gain matrix. The second stage is the time-update for

X

, which is given by

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

(2)

where

D_{i} U_{i}

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

(\begin{matrix} R_{i}^{1 / 2} & 0 & C S_{i} \\ 0 & B Q_{i}^{1 / 2} & A S_{i} \end{matrix}) U = (\begin{matrix} H_{i}^{1 / 2} & 0 & 0 \\ G_{i} & S_{i + 1} & 0 \end{matrix})

where

U

is an orthogonal transformation triangularizing the left-hand pre-array to produce the right-hand post-array. The triangularization is carried out via Householder transformations exploiting the zero pattern of the pre-array. The relationship between the Kalman gain matrix

K_{i}

and

G_{i}

is given by

A K_{i} = G_{i} {(H_{i}^{1 / 2})}^{- 1} .

In order to exploit the invariant parts of the model to simplify the computation of

U

the results for the transformed state space

U^{*} X

are computed where

U^{*}

is the transformation that reduces the matrix pair

(A, C)

to lower observer Hessenberg form. That is, the matrix

U^{*}

is computed such that the compound matrix

[\begin{matrix} C U^{* T} \\ U^{*} A U^{* T} \end{matrix}]

is a lower trapezoidal matrix. Further the matrix

B

is transformed to

U^{*} B

. These transformations need only be computed once at the start of a series, and G13EBF will, optionally, compute them. G13EBF returns transformed matrices

U^{*} A U^{* T}

U^{*} B

C U^{* T}

and

U^{*} A K_{i}

, the Cholesky factor of the updated transformed state covariance matrix

S_{i + 1}^{*}

(where

U^{*} P_{i + 1 ∣ i} U^{* T} = S_{i + 1}^{*} S_{i + 1}^{* T}

) and the matrix

H_{i}^{1 / 2}

, valid for both transformed and original models, which is used in the computation of the likelihood for the model. Note that the covariance matrices

Q_{i}

and

R_{i}

can be time-varying.

4 References

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

5 Parameters

1: $TRANSF$ – CHARACTER(1)Input

On entry: indicates whether to transform the input matrix pair

(A, C)

to lower observer Hessenberg form. The transformation will only be required on the first call to G13EBF.

$TRANSF ='T'$: The matrices in arrays A and C are transformed to lower observer Hessenberg form and the matrices in B and S are transformed as described in Section 3.
$TRANSF ='H'$: The matrices in arrays A, C and B should be as returned from a previous call to G13EBF with $TRANSF ='T'$ .

Constraint:

TRANSF ='T'

'H'

2: $N$ – INTEGERInput

On entry:

n

, the size of the state vector.

Constraint:

N \geq 1

3: $M$ – INTEGERInput

On entry:

m

, the size of the observation vector.

Constraint:

M \geq 1

4: $L$ – INTEGERInput

On entry:

l

, the dimension of the state noise.

Constraint:

L \geq 1

5: $A (LDS, N)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

TRANSF ='T'

, the state transition matrix,

A

TRANSF ='H'

, the transformed matrix as returned by a previous call to G13EBF with

TRANSF ='T'

On exit: if

TRANSF ='T'

, the transformed matrix,

U^{*} A U^{* T}

, otherwise A is unchanged.

6: $LDS$ – INTEGERInput

On entry: the first dimension of the arrays A, B, S, K and U as declared in the (sub)program from which G13EBF is called.

Constraint:

LDS \geq N

7: $B (LDS, L)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

TRANSF ='T'

, the noise coefficient matrix

B

TRANSF ='H'

, the transformed matrix as returned by a previous call to G13EBF with

TRANSF ='T'

On exit: if

TRANSF ='T'

, the transformed matrix,

U^{*} B

, otherwise B is unchanged.

8: $STQ$ – LOGICALInput

On entry: if

STQ = .TRUE.

, the state noise covariance matrix

Q_{i}

is assumed to be the identity matrix. Otherwise the lower triangular Cholesky factor,

Q_{i}^{1 / 2}

, must be provided in Q.

9: $Q (LDQ *)$ – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array Q must be at least

L

STQ = .FALSE.

and at least

1

STQ = .TRUE.

On entry: if

STQ = .FALSE.

, Q must contain the lower triangular Cholesky factor of the state noise covariance matrix,

Q_{i}^{1 / 2}

. Otherwise Q is not referenced.

10: $LDQ$ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which G13EBF is called.

Constraints:

if $STQ = .FALSE.$ , $LDQ \geq L$ ;
otherwise $LDQ \geq 1$ .

11: $C (LDM, N)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

TRANSF ='T'

, the measurement coefficient matrix,

C

TRANSF ='H'

, the transformed matrix as returned by a previous call to G13EBF with

TRANSF ='T'

On exit: if

TRANSF ='T'

, the transformed matrix,

C U^{* T}

, otherwise C is unchanged.

12: $LDM$ – INTEGERInput

On entry: the first dimension of the arrays C, R and H as declared in the (sub)program from which G13EBF is called.

Constraint:

LDM \geq M

13: $R (LDM, M)$ – REAL (KIND=nag_wp) arrayInput

On entry: the lower triangular Cholesky factor of the measurement noise covariance matrix

R_{i}^{1 / 2}

14: $S (LDS, N)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

TRANSF ='T'

the lower triangular Cholesky factor of the state covariance matrix,

S_{i}

TRANSF ='H'

the lower triangular Cholesky factor of the covariance matrix of the transformed state vector

S_{i}^{*}

as returned from a previous call to G13EBF with

TRANSF ='T'

On exit: the lower triangular Cholesky factor of the transformed state covariance matrix,

S_{i + 1}^{*}

15: $K (LDS, M)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the Kalman gain matrix for the transformed state vector premultiplied by the state transformed transition matrix,

U^{*} A K_{i}

16: $H (LDM, M)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the lower triangular matrix

H_{i}^{1 / 2}

17: $U (LDS *)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array U must be at least

N

TRANSF ='T'

, and at least

1

otherwise.

On exit: if

TRANSF ='T'

the

n

n

transformation matrix

U^{*}

, otherwise U is not referenced.

18: $TOL$ – REAL (KIND=nag_wp)Input

On entry: the tolerance used to test for the singularity of

H_{i}^{1 / 2}

. If

0.0 \leq TOL < m^{2} \times machine precision

, then

m^{2} \times machine precision

is used instead. The inverse of the condition number of

H^{1 / 2}

is estimated by a call to F07TGF (DTRCON). If this estimate is less than TOL then

H^{1 / 2}

is assumed to be singular.

Suggested value:

TOL = 0.0

Constraint:

TOL \geq 0.0

19: $IWK (M)$ – INTEGER arrayWorkspace

20: $WK ((N + M) \times (N + M + L))$ – REAL (KIND=nag_wp) arrayWorkspace

21: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$TRANSF \neq'T'$ or $'H'$ ,
or	$N < 1$ ,
or	$M < 1$ ,
or	$L < 1$ ,
or	$LDS < N$ ,
or	$LDM < M$ ,
or	$STQ = .TRUE.$ and $LDQ < 1$ ,
or	$STQ = .FALSE.$ and $LDQ < L$ ,
or	$TOL < 0.0$ .

$IFAIL = 2$: The matrix $H_{i}^{1 / 2}$ is singular.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

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.

8 Parallelism and Performance

G13EBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

G13EBF 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.

9 Further Comments

For models with time-varying

A, B

and

C

, G13EAF can be used.

The initial estimate of the transformed state vector can be computed from the estimate of the original state vector

{\hat{X}}_{1 ∣ 0}

, say, by premultiplying it by

U^{*}

as returned by G13EBF with

TRANSF ='T'

; that is,

{\hat{X}}_{1 ∣ 0}^{*} = U^{*} {\hat{X}}_{1 ∣ 0}

. The estimate of the transformed state vector

{\hat{X}}_{i + 1 ∣ i}^{*}

can be computed from the previous value

{\hat{X}}_{i ∣ i - 1}^{*}

{\hat{X}}_{i + 1 ∣ i}^{*} = (U^{*} A U^{* T}) {\hat{X}}_{i ∣ i - 1}^{*} + (U^{*} A K_{i}) r_{i}

where

r_{i} = Y_{i} - (C U^{* T}) {\hat{X}}_{i ∣ i - 1}^{*}

are the independent one-step prediction residuals for both the transformed and original model. The estimate of the original state vector can be computed from the transformed state vector as

U^{* T} {\hat{X}}_{1 + 1 ∣ i}^{*}

. The required matrix-vector multiplications can be performed by F06PAF (DGEMV).

W_{i}

and

V_{i}

are independent multivariate Normal variates then the log-likelihood for observations

i = 1, 2, \dots, t

is given by

l (θ) = κ - \frac{1}{2} \sum_{i = 1}^{t} l n (\det (H_{i})) - \frac{1}{2} \sum_{i = 1}^{t} {(Y_{i} - C_{i} X_{i ∣ i - 1})}^{T} H_{i}^{- 1} (Y_{i} - C_{i} X_{i ∣ i - 1})

where

κ

is a constant.

The Cholesky factors of the covariance matrices can be computed using F07FDF (DPOTRF).

Note that the model

\begin{array}{l} X_{i + 1} = A X_{i} + W_{i}, & Var (W_{i}) = Q_{i} \\ Y_{i} = C X_{i} + V_{i}, & Var (V_{i}) = R_{i} \end{array}

can be specified either with B set to the identity matrix and

STQ = .FALSE.

and the matrix

Q^{1 / 2}

input in Q or with

STQ = .TRUE.

and B set to

Q^{1 / 2}

The algorithm requires

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

operations and is backward stable (see Verhaegen and van Dooren (1986)). The transformation to lower observer Hessenberg form requires

O ((n + m) n^{2})

operations.

10 Example

This example first inputs the number of updates to be computed and the problem sizes. The initial state vector and the Cholesky factor of the state covariance matrix are input followed by the model matrices

A, B, C, R^{1 / 2}

and optionally

Q^{1 / 2}

(the Cholesky factors of the covariance matrices being input). At the first update the matrices are transformed using the

TRANSF ='T'

option and the initial value of the state vector is transformed. At each update the observed values are input and the residuals are computed and printed and the estimate of the transformed state vector,

{\hat{U}}^{*} X_{i ∣ i - 1}

, and the deviance are updated. The deviance is

- 2 \times log-likelihood

ignoring the constant. After the final update the estimate of the state vector is computed from the transformed state vector and the state covariance matrix is computed from S and these are printed along with the value of the deviance.

The data is for a two-dimensional time series to which a VARMA

(1, 1)

has been fitted. For the specification of a VARMA model as a state space model see the G13 Chapter Introduction. The means of the two series are included as additional states that do not change over time. The initial value of

P

P_{0}

, is the solution to

P_{0} = A P_{0} A^{T} + B Q B^{T} .

NAG Library Routine DocumentG13EBF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

G13EBF