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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_tsa_multi_autocorr_part (g13db)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_tsa_multi_autocorr_part (g13db) calculates the multivariate partial autocorrelation function of a multivariate time series.

Syntax

[p, v0, v, d, db, w, wb, nvp, ifail] = g13db(c0, c, nl, nk, 'ns', ns)
[p, v0, v, d, db, w, wb, nvp, ifail] = nag_tsa_multi_autocorr_part(c0, c, nl, nk, 'ns', ns)

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 nag_tsa_multi_corrmat_cross (g13dm).
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 = det⁡Dl / det⁡C0 ,   l=1,2,…  
and multiple squared partial autocorrelations
pl2 = 1 - vl / v l-1 .  

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

Parameters

Compulsory Input Parameters

1:     c0ldc0ns – double array
ldc0, the first dimension of the array, must satisfy the constraint ldc0≥maxns,1.
Contains the zero lag cross-covariances between the ns series as returned by nag_tsa_multi_corrmat_cross (g13dm). (c0 is assumed to be symmetric, upper triangle only is used.)
2:     cldc0ldc0nl – double array
ldc0, the first dimension of the array, must satisfy the constraint ldc0≥maxns,1.
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:     nl – int64int32nag_int scalar
m, the maximum lag for which cross-covariances are supplied in c.
Constraint: nl≥1.
4:     nk – int64int32nag_int scalar
The number of lags to which partial auto-correlations are to be calculated.
Constraint: 1≤nk≤nl.

Optional Input Parameters

1:     ns – int64int32nag_int scalar
Default: the second dimension of the array c0.
k, the number of time series whose cross-covariances are supplied in c and c0.
Constraint: ns≥1.

Output Parameters

1:     pnk – double array
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.
2:     v0 – double scalar
The lag zero prediction error variance (equal to the determinant of c0).
3:     vnk – double array
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.
4:     dldc0ldc0nk – double array
ldc0=maxns,1.
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.
5:     dbldc0ns – double array
ldc0=maxns,1.
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.
6:     wldc0ldc0nk – double array
ldc0=maxns,1.
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.
7:     wbldc0ldc0nk – double array
ldc0=maxns,1.
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.
8:     nvp – int64int32nag_int scalar
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.
9:     ifail – int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   ifail=1
On entry,ldc0<1,
orns<1,
orns>ldc0,
ornl<1,
ornk<1,
ornk>nl,
oriwa<2×ns+1×ns.
   ifail=2
c0 is not positive definite.
v0, v, p, d, db, w, wb and nvp are set to zero.
W  ifail=3
At lag k=nvp+1≤nk, Dk was found not to be positive definite. Up to lag nvp, v0, v, p, d, w and wb contain the values calculated so far and from lag nvp+1 to lag nk the matrices contain zero. db contains the backward prediction coefficients for lag nvp.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

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.

Further Comments

The time taken by nag_tsa_multi_autocorr_part (g13db) 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 nag_tsa_multi_corrmat_cross (g13dm), you must replace the diagonal elements of C0 (otherwise used to hold the series variances) by 1.

Example

This example reads the autocovariance matrices for four series from lag 0 to 5. It calls nag_tsa_multi_autocorr_part (g13db) to calculate the multivariate partial autocorrelation function and other related matrices of statistics up to lag 3. It prints the results.

Open in the MATLAB editor: g13db_example

function g13db_example


fprintf('g13db example results\n\n');

% autocovariances
c0 = [0.0109,    -0.0077917,  0.0013004,  0.0012654;
     -0.0077917,  0.05704,    0.002418,   0.014409;
      0.0013004,  0.002418,   0.04396,   -0.021421;
      0.0012654,  0.014409,  -0.021421,   0.072289];
c(:,:,1) = ...
     [0.0045889,  0.0004651, -0.00013275, 0.0077531;
     -0.0024419, -0.011667,  -0.021956,  -0.0045803;
      0.001108,  -0.0080479,  0.013621,  -0.0085868;
     -0.00050614, 0.014045,  -0.0010087,  0.012269];
c(:,:,2) = ...
     [0.0018652, -0.0064389,  0.0088307, -0.0024808;
     -0.011865,   0.0072367, -0.019802,   0.0059069;
     -0.0080307,  0.014306,   0.014546,   0.01351;
     -0.0021791, -0.029528,  -0.015887,   0.00088308];
c(:,:,3) = ...
     [-8.055e-005,-0.0037759, 0.0075463, -0.0042276;
      0.0041447, -0.0037987,  0.0019332, -0.017564;
     -0.010582,   0.0067733,  0.0069832,  0.0061747;
      0.0041352, -0.016013,   0.017043,  -0.013412];
c(:,:,4) = ...
     [0.00076079,-0.0010134,  0.01187,   -0.0041651;
      0.0036014, -0.0036375, -0.025571,   0.0050218;
     -0.013924,   0.011718,  -0.0059088,  0.0059297;
      0.010739,  -0.014571,   0.013816,  -0.012588];
c(:,:,5) = ...
     [-0.00064365,-0.0044556, 0.0051334,  0.00071587;
      0.0063617,  0.00015217, 0.002727,  -0.0022261;
     -0.0085855,  0.0014468, -0.0028698,  0.0044384;
      0.0068339, -0.002179,   0.013759,   0.00028217];

nl = int64(5);
nk = int64(3);
ns = size(c0,1);

% Calculate multivariate partial autocorrelation function
[p, v0, v, d, db, w, wb, nvp, ifail] = ...
  g13db( ...
	 c0, c, nl, nk);

%   Display results
fprintf('Number of valid parameters = %10d\n\n', nvp);
fprintf('Multivariate partial autocorrelations\n');
for j = 1:5:nk
  fprintf('%12.5f', p(j:min(j+4,nk)));
  fprintf('\n');
end
fprintf('\nZero lag predictor error variance determinant\n');
fprintf('followed by error variance ratios\n');
fprintf('%12.5f\n', v0);
for j = 1:5:nk
  fprintf('%12.5f', v(j:min(j+4,nk)));
  fprintf('\n');
end
fprintf('\nPrediction error variances\n');
for k = 1:nk
  fprintf('\nLag = %4d\n', k);
  disp(d(1:ns,1:ns,k));
end
fprintf('\nLast backward prediction error variances\n\n');
fprintf('Lag = %4d\n', nvp);
disp(db(1:ns,1:ns));
fprintf('\nPrediction coefficients\n');
for k = 1:nk
  fprintf('\nLag = %4d\n', k);
  disp(w(1:ns,1:ns,k));
end
fprintf('\nBackward prediction coefficients\n');
for k = 1:nk
  fprintf('\nLag = %4d\n', k);
  disp(wb(1:ns,1:ns,k));
end



g13db example results

Number of valid parameters =          3

Multivariate partial autocorrelations
     0.64498     0.92669     0.84300

Zero lag predictor error variance determinant
followed by error variance ratios
     0.00000
     0.35502     0.02603     0.00409

Prediction error variances

Lag =    1
    0.0081   -0.0051    0.0016   -0.0003
   -0.0051    0.0409    0.0076    0.0184
    0.0016    0.0076    0.0383   -0.0189
   -0.0003    0.0184   -0.0189    0.0676


Lag =    2
    0.0035   -0.0009   -0.0007   -0.0011
   -0.0009    0.0195    0.0053    0.0057
   -0.0007    0.0053    0.0190   -0.0107
   -0.0011    0.0057   -0.0107    0.0406


Lag =    3
    0.0030   -0.0009   -0.0005    0.0007
   -0.0009    0.0182    0.0087    0.0025
   -0.0005    0.0087    0.0093   -0.0022
    0.0007    0.0025   -0.0022    0.0225


Last backward prediction error variances

Lag =    3
    0.0033   -0.0039   -0.0011    0.0059
   -0.0039    0.0189    0.0035   -0.0033
   -0.0011    0.0035    0.0100   -0.0105
    0.0059   -0.0033   -0.0105    0.0334


Prediction coefficients

Lag =    1
    0.8186    0.2340   -0.1710    0.0926
    0.0674   -0.4872   -0.1406    0.0429
    0.1504    0.1192   -0.3672   -0.4209
   -0.7097    0.0300    0.5978    0.3461


Lag =    2
   -0.3405   -0.1337    0.4061   -0.0218
   -1.2757   -0.1359   -0.6578   -0.1127
   -0.4544    0.1938    0.6342    0.3392
   -0.4324   -0.5485   -0.6290    0.1667


Lag =    3
    0.1644    0.1386    0.0129    0.0346
    0.3929    0.0741   -0.0880   -0.1536
   -1.2924   -0.2449    0.3023    0.3944
    0.8977   -0.3904    0.2515   -0.2830


Backward prediction coefficients

Lag =    1
    0.4154    0.0615    0.1532    0.0508
    0.1237   -0.2647   -0.2272    0.4850
   -0.8693   -0.4737    0.3792    0.1381
    1.3078   -0.0918   -1.4540   -0.2197


Lag =    2
   -0.0674   -0.1226   -0.1367   -0.0973
   -1.2480    0.0309    0.5171   -0.2892
    0.9804   -0.2019    0.1631   -0.1087
   -1.6839   -0.7459    0.5290    0.4158


Lag =    3
    0.0379    0.1049   -0.2164    0.0801
    0.7539    0.2260   -0.2566   -0.4745
   -0.0034    0.0564   -0.0882    0.1272
    0.5502   -0.4123    0.7165   -0.1457
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