NAG Library Routine Document

g13dbf (multi_autocorr_part)

1
Purpose

g13dbf calculates the multivariate partial autocorrelation function of a multivariate time series.

2
Specification

Fortran Interface
Subroutine g13dbf ( c0, c, ldc0, ns, nl, nk, p, v0, v, d, db, w, wb, nvp, wa, iwa, ifail)
Integer, Intent (In):: ldc0, ns, nl, nk, iwa
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: nvp
Real (Kind=nag_wp), Intent (In):: c0(ldc0,ns), c(ldc0,ldc0,nl)
Real (Kind=nag_wp), Intent (Inout):: d(ldc0,ldc0,nk), db(ldc0,ns), w(ldc0,ldc0,nk), wb(ldc0,ldc0,nk)
Real (Kind=nag_wp), Intent (Out):: p(nk), v0, v(nk), wa(iwa)
C Header Interface
#include <nagmk26.h>
void  g13dbf_ (const double c0[], const double c[], const Integer *ldc0, const Integer *ns, const Integer *nl, const Integer *nk, double p[], double *v0, double v[], double d[], double db[], double w[], double wb[], Integer *nvp, double wa[], const Integer *iwa, Integer *ifail)

3
Description

The input is a set of lagged autocovariance matrices C0,C1,C2,,Cm. These will generally be sample values such as are obtained from a multivariate time series using g13dmf.
The main calculation is the recursive determination of the coefficients in the finite lag (forward) prediction equation
xt = Φl,1 xt-1 ++ Φl,l xt-l + el,t  
and the associated backward prediction equation
xt-l- 1=Ψl,1xt-l++Ψl,lxt- 1+fl,t  
together with the covariance matrices Dl of el,t and Gl of fl,t.
The recursive cycle, by which the order of the prediction equation is extended from l to l+1, is to calculate
Ml+1 = Cl+1T - Φ l,1 ClT -- Φl,l C1T (1)
then Φ l+1,l+1 = Ml+1 Dl-1 , Ψ l+1,l+1 = MT l+1 Gl-1
from which
Φl+1,j=Φl,j-Φl+1,l+1Ψl,l+1-j,   j=1,2,,l (2)
and
Ψl+1,j=Ψl,j-Ψl+1,l+1Φl,l+1-j,  j=1,2,,l. (3)
Finally, Dl+1=Dl-Ml+1ΦTl+1,l+1 and Gl+1=Gl-MTl+1ΨTl+1,l+1.
(Here T denotes the transpose of a matrix.)
The cycle is initialized by taking (for l=0)
D0=G0=C0.  
In the step from l=0 to 1, the above equations contain redundant terms and simplify. Thus (1) becomes M1=CT1 and neither (2) or (3) are needed.
Quantities useful in assessing the effectiveness of the prediction equation are generalized variance ratios
vl = detDl / detC0 ,   l=1,2,  
and multiple squared partial autocorrelations
pl2 = 1 - vl / 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:     c0ldc0ns – Real (Kind=nag_wp) arrayInput
On entry: contains the zero lag cross-covariances between the ns series as returned by g13dmf. (c0 is assumed to be symmetric, upper triangle only is used.)
2:     cldc0ldc0nl – Real (Kind=nag_wp) arrayInput
On entry: contains the cross-covariances at lags 1 to nl. cijk must contain the cross-covariance, cijk, of series i and series j at lag k. Series j leads series i.
3:     ldc0 – IntegerInput
On entry: the first dimension of the arrays c0, c, d, db, w and wb and the second dimension of the arrays c, d, w and wb as declared in the (sub)program from which g13dbf is called.
Constraint: ldc0maxns,1.
4:     ns – IntegerInput
On entry: k, the number of time series whose cross-covariances are supplied in c and c0.
Constraint: ns1.
5:     nl – IntegerInput
On entry: m, the maximum lag for which cross-covariances are supplied in c.
Constraint: nl1.
6:     nk – IntegerInput
On entry: the number of lags to which partial auto-correlations are to be calculated.
Constraint: 1nknl.
7:     pnk – Real (Kind=nag_wp) arrayOutput
On exit: the multiple squared partial autocorrelations from lags 1 to nvp; that is, pl contains pl2, for l=1,2,,nvp. For lags nvp+1 to nk the elements of p are set to zero.
8:     v0 – Real (Kind=nag_wp)Output
On exit: the lag zero prediction error variance (equal to the determinant of c0).
9:     vnk – Real (Kind=nag_wp) arrayOutput
On exit: the prediction error variance ratios from lags 1 to nvp; that is, vl contains vl, for l=1,2,,nvp. For lags nvp+1 to nk the elements of v are set to zero.
10:   dldc0ldc0nk – Real (Kind=nag_wp) arrayOutput
On exit: the prediction error variance matrices at lags 1 to nvp.
Element i,j,k of d contains the prediction error covariance of series i and series j at lag k, for k=1,2,,nvp. Series j leads series i; that is, the i,jth element of Dk. For lags nvp+1 to nk the elements of d are set to zero.
11:   dbldc0ns – Real (Kind=nag_wp) arrayOutput
On exit: the backward prediction error variance matrix at lag nvp.
dbij contains the backward prediction error covariance of series i and series j; that is, the i,jth element of the Gk, where k=nvp.
12:   wldc0ldc0nk – Real (Kind=nag_wp) arrayOutput
On exit: the prediction coefficient matrices at lags 1 to nvp.
wijl contains the jth prediction coefficient of series i at lag l; that is, the i,jth element of Φkl, where k=nvp, for l=1,2,,nvp. For lags nvp+1 to nk the elements of w are set to zero.
13:   wbldc0ldc0nk – Real (Kind=nag_wp) arrayOutput
On exit: the backward prediction coefficient matrices at lags 1 to nvp.
wbijl contains the jth backward prediction coefficient of series i at lag l; that is, the i,jth element of Ψkl, where k=nvp, for l=1,2,,nvp. For lags nvp+1 to nk the elements of wb are set to zero.
14:   nvp – IntegerOutput
On exit: the maximum lag, L, for which calculation of p, v, d, db, w and wb was successful. If the routine completes successfully nvp will equal nk.
15:   waiwa – Real (Kind=nag_wp) arrayWorkspace
16:   iwa – IntegerInput
On entry: the dimension of the array wa as declared in the (sub)program from which g13dbf is called.
Constraint: iwa2×ns+1×ns.
17:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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 or -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, iwa=value.
Constraint: iwa1.
On entry, iwa=value and the minimum size required=value.
Constraint: iwa2×ns+1×ns.
On entry, ldc0=value.
Constraint: ldc01.
On entry, nk=value.
Constraint: nk1.
On entry, nk=value and nl=value.
Constraint: nknl.
On entry, nl=value.
Constraint: nl1.
On entry, ns=value.
Constraint: ns1.
On entry, ns=value and ldc0=value.
Constraint: nsldc0.
ifail=2
c0 is not positive definite. The arguments v0, v, p, d, db, w, wb and nvp are set to zero.
ifail=3
For nvp=value, at lag k=nvp+1, Dk was found not to be positive definite.
Up to lag k-1, arguments v0, v, p, d, w and wb contain the values calculated so far. From lag k they contain zero. The argument db contains the backward prediction coefficients for lag k-1.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

g13dbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13dbf 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

The time taken by g13dbf is roughly proportional to nk2×ns3.
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 g13dmf, you must replace the diagonal elements of C0 (otherwise used to hold the series variances) by 1.

10
Example

This example reads the autocovariance matrices for four series from lag 0 to 5. It calls g13dbf to calculate the multivariate partial autocorrelation function and other related matrices of statistics up to lag 3. It prints the results.

10.1
Program Text

Program Text (g13dbfe.f90)

10.2
Program Data

Program Data (g13dbfe.d)

10.3
Program Results

Program Results (g13dbfe.r)