NAG Library Routine Document

g13acf calculates partial autocorrelation coefficients given a set of autocorrelation coefficients. It also calculates the predictor error variance ratios for increasing order of finite lag autoregressive predictor, and the autoregressive parameters associated with the predictor of maximum order.

2

Specification

Fortran Interface

Subroutine g13acf (

r, nk, nl, p, v, ar, nvl, ifail)

Integer, Intent (In)	::	nk, nl
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	nvl
Real (Kind=nag_wp), Intent (In)	::	r(nk)
Real (Kind=nag_wp), Intent (Out)	::	p(nl), v(nl), ar(nl)

C Header Interface

#include <nagmk26.h>

void	g13acf_ (const double r[], const Integer nk, const Integer nl, double p[], double v[], double ar[], Integer nvl, Integer ifail)

3

Description

The data consist of values of autocorrelation coefficients

r_{1}, r_{2}, \dots, r_{K}

, relating to lags

1, 2, \dots, K

. These will generally (but not necessarily) be sample values such as may be obtained from a time series

x_{t}

using g13abf.

The partial autocorrelation coefficient at lag

l

may be identified with the parameter

p_{l, l}

in the autoregression

x_{t} = c_{l} + p_{l, 1} x_{t - 1} + p_{l, 2} x_{t - 2} + \dots + p_{l, l} x_{t - l} + e_{l, t}

where

e_{l, t}

is the predictor error.

The first subscript

l

p_{l, l}

and

e_{l, t}

emphasizes the fact that the parameters will in general alter as further terms are introduced into the equation (i.e., as

l

is increased).

The parameters are determined from the autocorrelation coefficients by the Yule–Walker equations

r_{i} = p_{l, 1} r_{i - 1} + p_{l, 2} r_{i - 2} + \dots + p_{l, l} r_{i - l}, i = 1, 2, \dots, l

taking

r_{j} = r_{|j|}

when

j < 0

, and

r_{0} = 1

The predictor error variance ratio

v_{l} = var (e_{l, t}) / var (x_{t})

is defined by

v_{l} = 1 - p_{l, 1} r_{1} - p_{l, 2} r_{2} - \dots - p_{l, l} r_{l} .

The above sets of equations are solved by a recursive method (the Durbin–Levinson algorithm). The recursive cycle applied for

l = 1, 2, \dots, (L - 1)

, where

L

is the number of partial autocorrelation coefficients required, is initialized by setting

p_{1, 1} = r_{1}

and

v_{1} = 1 - r_{1}^{2}

Then

\begin{array}{lcl} p_{l + 1, l + 1} & = & (r_{l + 1} - p_{l, 1} r_{l} - p_{l, 2} r_{l - 1} - \dots - p_{l, l} r_{1}) / v_{l} \\ p_{l + 1, j} & = & p_{l, j} - p_{l + 1, l + 1} p_{l, l + 1 - j}, j = 1, 2, \dots, l \\ v_{l + 1} & = & v_{l} (1 - p_{l + 1, l + 1}) (1 + p_{l + 1, l + 1}) . \end{array}

If the condition

|p_{l, l}| \geq 1

occurs, say when

l = l_{0}

, it indicates that the supplied autocorrelation coefficients do not form a positive definite sequence (see Hannan (1960)), and the recursion is not continued. The autoregressive parameters are overwritten at each recursive step, so that upon completion the only available values are

p_{L j}

, for

j = 1, 2, \dots, L

, or

p_{l_{0} - 1, j}

if the recursion has been prematurely halted.

4

References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day

Durbin J (1960) The fitting of time series models Rev. Inst. Internat. Stat. 28 233

Hannan E J (1960) Time Series Analysis Methuen

5

Arguments

1: $r (nk)$ – Real (Kind=nag_wp) arrayInput: On entry: the autocorrelation coefficient relating to lag $k$ , for $k = 1, 2, \dots, K$ .
2: $nk$ – IntegerInput: On entry: $K$ , the number of lags. The lags range from $1$ to $K$ and do not include zero.

Constraint: $nk > 0$ .
3: $nl$ – IntegerInput: On entry: $L$ , the number of partial autocorrelation coefficients required.

Constraint: $0 < nl \leq nk$ .
4: $p (nl)$ – Real (Kind=nag_wp) arrayOutput: On exit: $p (l)$ contains the partial autocorrelation coefficient at lag $l$ , $p_{l, l}$ , for $l = 1, 2, \dots, nvl$ .
5: $v (nl)$ – Real (Kind=nag_wp) arrayOutput: On exit: $v (l)$ contains the predictor error variance ratio $v_{l}$ , for $l = 1, 2, \dots, nvl$ .
6: $ar (nl)$ – Real (Kind=nag_wp) arrayOutput: On exit: the autoregressive parameters of maximum order, i.e., $p_{L j}$ if $ifail = 0$ , or $p_{l_{0} - 1, j}$ if $ifail = 3$ , for $j = 1, 2, \dots, nvl$ .
7: $nvl$ – IntegerOutput: On exit: the number of valid values in each of p, v and ar. Thus in the case of premature termination at iteration $l_{0}$ (see Section 3), nvl is returned as $l_{0} - 1$ .
8: $ifail$ – 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

- 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, $nk = 〈value〉$ .
Constraint: $nk > 0$ .

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

On entry, $nl = 〈value〉$ .
Constraint: $nl > 0$ .

$ifail = 2$: On entry, the autocorrelation coefficient of lag $1$ has absolute value greater than or equal to $1$ .

$ifail = 3$: The autocorrelation coefficients do not form a positive definite sequence. The number of valid values computed are returned in nvl (see Section 3 for more details).

$ifail = - 99$: 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 computations are believed to be stable.

8

Parallelism and Performance

g13acf is not threaded in any implementation.

9

Further Comments

The time taken by g13acf is proportional to

{(nvl)}^{2}

10

Example

This example uses an input series of

10

sample autocorrelation coefficients derived from the original series of sunspot numbers generated by the g13abf example program. The results show five values of each of the three output arrays: partial autocorrelation coefficients, predictor error variance ratios and autoregressive parameters. All of these were valid.

This plot shows the partial autocorrelations for all possible lag values. Reference lines are given at

\pm z_{0.975} / \sqrt{n}

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

G13 (tsa) Chapter Contents

G13 (tsa) Chapter Introduction

NAG Library Routine Document

g13acf (uni_autocorr_part)

▸▿ 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