# NAG Library Function Document

## 1Purpose

nag_tsa_multi_auto_corr_part (g13dbc) calculates the multivariate partial autocorrelation function of a multivariate time series.

## 2Specification

 #include #include
 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)

## 3Description

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
 $xt = Φl,1 xt-1 +⋯+ Φl,l xt-l + el,t$
and the associated backward prediction equation
 $xt-l- 1=Ψl,1xt-l+⋯+Ψl,lxt- 1+fl,t$
together with the covariance matrices ${D}_{l}$ of ${e}_{l,t}$ and ${G}_{l}$ of ${f}_{l,t}$.
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 ${\Phi }_{l+1,l+1}={M}_{l+1}{D}_{l}^{-1}$, $\text{ }{\Psi }_{l+1,l+1}={{M}^{\mathrm{T}}}_{l+1}{G}_{l}^{-1}$
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, ${D}_{l+1}={D}_{l}-{M}_{l+1}{{\Phi }^{\mathrm{T}}}_{l+1,l+1}$ and ${G}_{l+1}={G}_{l}-{{M}^{\mathrm{T}}}_{l+1}{{\Psi }^{\mathrm{T}}}_{l+1,l+1}$.
(Here $\mathrm{T}$ 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 ${M}_{1}={{C}^{\mathrm{T}}}_{1}$ 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
 $pl2 = 1 - vl / v l-1 .$

## 4References

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

## 5Arguments

1:    $\mathbf{c0}\left[{\mathbf{ns}}×{\mathbf{ns}}\right]$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:    $\mathbf{c}\left[{\mathbf{ns}}×{\mathbf{ns}}×{\mathbf{nl}}\right]$const doubleInput
On entry: the $k$ cross-covariances as returned by nag_tsa_multi_cross_corr (g13dmc).
3:    $\mathbf{ns}$IntegerInput
On entry: $k$, the number of time series whose cross-covariances are supplied in c and c0.
Constraint: ${\mathbf{ns}}\ge 1$.
4:    $\mathbf{nl}$IntegerInput
On entry: $m$, the maximum lag for which cross-covariances are supplied in c.
Constraint: ${\mathbf{nl}}\ge 1$.
5:    $\mathbf{nk}$IntegerInput
On entry: the number of lags to which partial auto-correlations are to be calculated.
Constraint: $1\le {\mathbf{nk}}\le {\mathbf{nl}}$.
6:    $\mathbf{p}\left[{\mathbf{nk}}\right]$doubleOutput
On exit: the multiple squared partial autocorrelations from lags $1$ to nvp; that is, ${\mathbf{p}}\left[l-1\right]$ contains ${p}_{\mathit{l}}^{2}$, for $\mathit{l}=1,2,\dots ,{\mathbf{nvp}}$. For lags ${\mathbf{nvp}}+1$ to nk the elements of p are set to zero.
7:    $\mathbf{v0}$double *Output
On exit: the lag zero prediction error variance (equal to the determinant of c0).
8:    $\mathbf{v}\left[{\mathbf{nk}}\right]$doubleOutput
On exit: the prediction error variance ratios from lags $1$ to nvp; that is, ${\mathbf{v}}\left[\mathit{l}-1\right]$ contains ${v}_{\mathit{l}}$, for $\mathit{l}=1,2,\dots ,{\mathbf{nvp}}$. For lags ${\mathbf{nvp}}+1$ to nk the elements of v are set to zero.
9:    $\mathbf{d}\left[{\mathbf{ns}}×{\mathbf{ns}}×{\mathbf{nk}}\right]$doubleOutput
On exit: the prediction error variance matrices at lags $1$ to nvp, ${\mathbf{d}}\left[\left(l-1\right){k}^{2}+\left(j-1\right)k+i-1\right]$ contains the $\left(i,j\right)$th prediction error covariance of series $i$ and series $j$ at lag $l$. Series $j$ leads series $i$.
10:  $\mathbf{db}\left[{\mathbf{ns}}×{\mathbf{ns}}\right]$doubleOutput
On exit: the backward prediction error variance matrix at lag nvp, ${\mathbf{db}}\left[\left(j-1\right)k+i-1\right]$ contains the backward prediction error covariance of series $i$ and series $j$.
11:  $\mathbf{w}\left[{\mathbf{ns}}×{\mathbf{ns}}×{\mathbf{nk}}\right]$doubleOutput
On exit: the prediction coefficient matrices at lags $1$ to nvp, ${\mathbf{w}}\left[\left(l-1\right){k}^{2}+\left(j-1\right)k+i-1\right]$ contains the $j$th prediction coefficient of series $i$ at lag $l$ (i.e., the $\left(i,j\right)$th element of ${\Phi }_{L,l}$).
12:  $\mathbf{wb}\left[{\mathbf{ns}}×{\mathbf{ns}}×{\mathbf{nk}}\right]$doubleOutput
On exit: the backward prediction coefficient matrices at lags $1$ to nvp, ${\mathbf{wb}}\left[\left(l-1\right){k}^{2}+\left(j-1\right)+i-1\right]$ contains the $j$th backward prediction coefficient of series $i$ at lag $l$ (i.e., the $\left(i,j\right)$th element of ${\Psi }_{L,l}$).
13:  $\mathbf{nvp}$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:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{nk}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nk}}\ge 1$.
On entry, ${\mathbf{nl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nl}}\ge 1$.
On entry, ${\mathbf{ns}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ns}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{nk}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nk}}\le {\mathbf{nl}}$.
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 ${\mathbf{nvp}}+1\le {\mathbf{nk}}$, d is not positive definite, ${\mathbf{nvp}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nk}}=〈\mathit{\text{value}}〉$.
c0 is not positive definite.

## 7Accuracy

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.

## 8Parallelism 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.

The time taken by nag_tsa_multi_auto_corr_part (g13dbc) is roughly proportional to ${{\mathbf{nk}}}^{2}×{{\mathbf{ns}}}^{3}$.
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 $1$.

## 10Example

This example reads the autocovariance matrices for four series from lag $0$ to $5$. 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.

### 10.1Program Text

Program Text (g13dbce.c)

### 10.2Program Data

Program Data (g13dbce.d)

### 10.3Program Results

Program Results (g13dbce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017