NAG Library Function Document

nag_tsa_auto_corr_part (g13acc)

+− 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

nag_tsa_auto_corr_part (g13acc) 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

#include <nag.h>

#include <nagg13.h>

void	nag_tsa_auto_corr_part (const double r[], Integer nk, Integer nl, double p[], double v[], double ar[], Integer nvl, NagError fail)

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 nag_tsa_auto_corr (g13abc).

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] – const doubleInput: 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] – doubleOutput: On exit: $p [l - 1]$ contains the partial autocorrelation coefficient at lag $l$ , $p_{l, l}$ , for $l = 1, 2, \dots, nvl$ .
5: v[nl] – doubleOutput: On exit: $v [l - 1]$ contains the predictor error variance ratio $v_{l}$ , for $l = 1, 2, \dots, nvl$ .
6: ar[nl] – doubleOutput: On exit: the autoregressive parameters of maximum order, i.e., $p_{L j}$ if $fail . code =$ NE_NOERROR, or $p_{l_{0} - 1, j}$ if $fail . code =$ NE_CORR_NOT_POS_DEF, for $j = 1, 2, \dots, nvl$ .
7: nvl – Integer *Output: 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: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_CORR_NOT_POS_DEF: The autocorrelation coefficients do not form a positive definite sequence.
NE_INT: On entry, $nk = ⟨value⟩$ .
Constraint: $nk > 0$ .
On entry, $nl = ⟨value⟩$ .
Constraint: $nl > 0$ .
NE_INT_2: On entry, $nk = ⟨value⟩$ and $nl = ⟨value⟩$ .
Constraint: $nk \geq nl$ .

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_tsa_auto_corr_part (g13acc) 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 nag_tsa_auto_corr (g13abc) 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