void	nag_tsa_multi_auto_corr_part (const double c0[], const double c[], Integer ns, Integer nl, Integer nk, double p[], double v0, double v[], double d[], double db[], double w[], double wb[], Integer nvp, NagError *fail)

3

Description

The input is a set of lagged autocovariance matrices

C_{0}, C_{1}, C_{2}, \dots, C_{m}

. These will generally be sample values such as are obtained from a multivariate time series using nag_tsa_multi_cross_corr (g13dmc).

The main calculation is the recursive determination of the coefficients in the finite lag (forward) prediction equation

x_{t} = Φ_{l, 1} x_{t - 1} + \dots + Φ_{l, l} x_{t - l} + e_{l, t}

and the associated backward prediction equation

x_{t - l - 1} = Ψ_{l, 1} x_{t - l} + \dots + Ψ_{l, l} x_{t - 1} + f_{l, t}

together with the covariance matrices

D_{l}

e_{l, t}

and

G_{l}

f_{l, t}

The recursive cycle, by which the order of the prediction equation is extended from

l

l + 1

, is to calculate

M_{l + 1} = C_{l + 1}^{T} - Φ_{l, 1} C_{l}^{T} - \dots - Φ_{l, l} C_{1}^{T}

(1)

then

Φ_{l + 1, l + 1} = M_{l + 1} D_{l}^{- 1}

Ψ_{l + 1, l + 1} = {M^{T}}_{l + 1} G_{l}^{- 1}

from which

Φ_{l + 1, j} = Φ_{l, j} - Φ_{l + 1, l + 1} Ψ_{l, l + 1 - j}, j = 1, 2, \dots, l

(2)

and

Ψ_{l + 1, j} = Ψ_{l, j} - Ψ_{l + 1, l + 1} Φ_{l, l + 1 - j}, j = 1, 2, \dots, l .

(3)

Finally,

D_{l + 1} = D_{l} - M_{l + 1} {Φ^{T}}_{l + 1, l + 1}

and

G_{l + 1} = G_{l} - {M^{T}}_{l + 1} {Ψ^{T}}_{l + 1, l + 1}

(Here

T

denotes the transpose of a matrix.)

The cycle is initialized by taking (for

l = 0

)

D_{0} = G_{0} = C_{0} .

In the step from

l = 0

1

, the above equations contain redundant terms and simplify. Thus (1) becomes

M_{1} = {C^{T}}_{1}

and neither (2) or (3) are needed.

Quantities useful in assessing the effectiveness of the prediction equation are generalized variance ratios

v_{l} = \det D_{l} / \det C_{0}, l = 1, 2, \dots

and multiple squared partial autocorrelations

p_{l}^{2} = 1 - v_{l} / v_{l - 1} .

4

References

Akaike H (1971) Autoregressive model fitting for control Ann. Inst. Statist. Math. 23 163–180

Whittle P (1963) On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral density matrix Biometrika 50 129–134

5

Arguments

1: $c0 [ns \times ns]$ – const doubleInput: On entry: contains the zero lag cross-covariances between the ns series as returned by nag_tsa_multi_cross_corr (g13dmc). (c0 is assumed to be symmetric, upper triangle only is used.)
2: $c [ns \times ns \times nl]$ – const doubleInput: On entry: the $k$ cross-covariances as returned by nag_tsa_multi_cross_corr (g13dmc).
3: $ns$ – IntegerInput: On entry: $k$ , the number of time series whose cross-covariances are supplied in c and c0.

Constraint: $ns \geq 1$ .
4: $nl$ – IntegerInput: On entry: $m$ , the maximum lag for which cross-covariances are supplied in c.

Constraint: $nl \geq 1$ .
5: $nk$ – IntegerInput: On entry: the number of lags to which partial auto-correlations are to be calculated.

Constraint: $1 \leq nk \leq nl$ .
6: $p [nk]$ – doubleOutput: On exit: the multiple squared partial autocorrelations from lags $1$ to nvp; that is, $p [l - 1]$ contains $p_{l}^{2}$ , for $l = 1, 2, \dots, nvp$ . For lags $nvp + 1$ to nk the elements of p are set to zero.
7: $v0$ – double *Output: On exit: the lag zero prediction error variance (equal to the determinant of c0).
8: $v [nk]$ – doubleOutput: On exit: the prediction error variance ratios from lags $1$ to nvp; that is, $v [l - 1]$ contains $v_{l}$ , for $l = 1, 2, \dots, nvp$ . For lags $nvp + 1$ to nk the elements of v are set to zero.
9: $d [ns \times ns \times nk]$ – doubleOutput: On exit: the prediction error variance matrices at lags $1$ to nvp, $d [(l - 1) k^{2} + (j - 1) k + i - 1]$ contains the $(i, j)$ th prediction error covariance of series $i$ and series $j$ at lag $l$ . Series $j$ leads series $i$ .
10: $db [ns \times ns]$ – doubleOutput: On exit: the backward prediction error variance matrix at lag nvp, $db [(j - 1) k + i - 1]$ contains the backward prediction error covariance of series $i$ and series $j$ .
11: $w [ns \times ns \times nk]$ – doubleOutput: On exit: the prediction coefficient matrices at lags $1$ to nvp, $w [(l - 1) k^{2} + (j - 1) k + i - 1]$ contains the $j$ th prediction coefficient of series $i$ at lag $l$ (i.e., the $(i, j)$ th element of $Φ_{L, l}$ ).
12: $wb [ns \times ns \times nk]$ – doubleOutput: On exit: the backward prediction coefficient matrices at lags $1$ to nvp, $wb [(l - 1) k^{2} + (j - 1) + i - 1]$ contains the $j$ th backward prediction coefficient of series $i$ at lag $l$ (i.e., the $(i, j)$ th element of $Ψ_{L, l}$ ).
13: $nvp$ – Integer *Output: On exit: the maximum lag, $L$ , for which calculation of p, v, d, db, w and wb was successful. If the function completes successfully nvp will equal nk.
14: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $nk = 〈value〉$ .
Constraint: $nk \geq 1$ .

On entry, $nl = 〈value〉$ .
Constraint: $nl \geq 1$ .

On entry, $ns = 〈value〉$ .
Constraint: $ns \geq 1$ .
NE_INT_2: On entry, $nk = 〈value〉$ and $nl = 〈value〉$ .
Constraint: $nk \leq nl$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_POS_DEF: At lag $nvp + 1 \leq nk$ , d is not positive definite, $nvp = 〈value〉$ and $nk = 〈value〉$ .

c0 is not positive definite.

7

Accuracy

The conditioning of the problem depends on the prediction error variance ratios. Very small values of these may indicate loss of accuracy in the computations.

8

Parallelism and Performance

nag_tsa_multi_auto_corr_part (g13dbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_tsa_multi_auto_corr_part (g13dbc) 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 time taken by nag_tsa_multi_auto_corr_part (g13dbc) is roughly proportional to

{nk}^{2} \times {ns}^{3}

If sample autocorrelation matrices are used as input, then the output will be relevant to the original series scaled by their standard deviations. If these autocorrelation matrices are produced by nag_tsa_multi_cross_corr (g13dmc), you must replace the diagonal elements of

C_{0}

(otherwise used to hold the series variances) by

1

10

Example

This example reads the autocovariance matrices for four series from lag

0

5

. It calls nag_tsa_multi_auto_corr_part (g13dbc) to calculate the multivariate partial autocorrelation function and other related matrices of statistics up to lag

3

. It prints the results.

NAG Library Function Document

nag_tsa_multi_auto_corr_part (g13dbc)

▸▿ 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

1

Purpose

2

Specification