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 <nag.h>

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

The routine may be called by the names g13acf or nagf_tsa_uni_autocorr_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 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) array Input: On entry: the autocorrelation coefficient relating to lag $k$ , for $k = 1, 2, \dots, K$ .
2: $nk$ – Integer Input: On entry: $K$ , the number of lags. The lags range from $1$ to $K$ and do not include zero.

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

Constraint: $0 < nl \leq nk$ .
4: $p (nl)$ – Real (Kind=nag_wp) array Output: 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) array Output: 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) array Output: 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: $nvl$ – 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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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.