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 arguments associated with the predictor of maximum order.

2 Specification

#include <nag.h>

void	g13acc (const double r[], Integer nk, Integer nl, double p[], double v[], double ar[], Integer nvl, NagError fail)

The function may be called by the names: g13acc, nag_tsa_uni_autocorr_part or nag_tsa_auto_corr_part.

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 g13abc.

The partial autocorrelation coefficient at lag

l

may be identified with the argument

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 arguments will in general alter as further terms are introduced into the equation (i.e., as

l

is increased).

The arguments 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{matrix} 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} \times (1 - p_{l + 1, l + 1}) (1 + p_{l + 1, l + 1}) . \end{matrix}

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 arguments 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 double Input: On entry: $r [k - 1]$ contains the autocorrelation coefficient relating to lag $k$ , for $k = 1, 2, \dots, K$ .
2: $nk$ – Integer Input: On entry: the number of lags, $K$ . The lags range from 1 to $K$ and do not include zero.

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

Constraint: $0 < nl \leq nk$ .
4: $p [nl]$ – double Output: On exit: $p [l - 1]$ contains the partial autocorrelation coefficient at lag $l$ , $p_{l, l}$ , for $l = 1, 2, \dots, nvl$ .
5: $v [nl]$ – double Output: On exit: $v [l - 1]$ contains the predictor error variance ratio $v_{l}$ , for $l = 1, 2, \dots, nvl$ .
6: $ar [nl]$ – double Output: On exit: the autoregressive arguments 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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $nk = 〈value〉$ while $nl = 〈value〉$ . These arguments must satisfy $nk \geq nl$ .
NE_CORR_NOT_POS_DEF: Recursion has been prematurely terminated; the supplied autocorrelation coefficients do not form a positive definite sequence. Parameter nvl returns the number of valid values computed.
NE_INT_ARG_LE: On entry, $nk = 〈value〉$ .
Constraint: $nk > 0$ .

On entry, $nl = 〈value〉$ .
Constraint: $nl > 0$ .
NE_INVALID_AUTOCO_COEF: On entry, the autocorrelation coefficient of lag 1 has an absolute value greater than or equal to 1.0; no recursions could be performed.

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

g13acc is not threaded in any implementation.

9 Further Comments

The time taken by g13acc is proportional to

{nvl}^{2}

10 Example

In the example below the input series is the set of 10 sample autocorrelation coefficients derived from the original series of sunspot numbers by g13abc example program. The results show 5 values of each of the three output arrays – partial autocorrelation coefficients, predictor error variance ratios and autoregressive arguments. All of these were valid.

g13ac: FL CL CPP AD

NAG CL Interfaceg13acc (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

NAG CL Interface
g13acc (uni_autocorr_part)