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.
The data consist of values of autocorrelation coefficients , relating to lags . These will generally (but not necessarily) be sample values such as may be obtained from a time series using nag_tsa_auto_corr (g13abc).
The partial autocorrelation coefficient at lag may be identified with the parameter in the autoregression
where is the predictor error.
The first subscript of and emphasizes the fact that the parameters will in general alter as further terms are introduced into the equation (i.e., as is increased).
The parameters are determined from the autocorrelation coefficients by the Yule–Walker equations
taking when , and .
The predictor error variance ratio is defined by
The above sets of equations are solved by a recursive method (the Durbin–Levinson algorithm). The recursive cycle applied for , where is the number of partial autocorrelation coefficients required, is initialized by setting and .
If the condition occurs, say when , 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 , for , or if the recursion has been prematurely halted.
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
– const doubleInput
On entry: the autocorrelation coefficient relating to lag
, for .
On entry: , the number of lags. The lags range from to and do not include zero.
On entry: , the number of partial autocorrelation coefficients required.
On exit: contains the partial autocorrelation coefficient at lag , , for .
On exit: contains the predictor error variance ratio , for .
On exit: the autoregressive parameters of maximum order, i.e.,
if NE_NOERROR, or if NE_CORR_NOT_POS_DEF, for .
– 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 (see Section 3), nvl is returned as .
– NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).
6 Error Indicators and Warnings
The autocorrelation coefficients do not form a positive definite sequence.
On entry, .
On entry, .
On entry, and .
The computations are believed to be stable.
8 Parallelism and Performance
nag_tsa_auto_corr_part (g13acc) is not threaded in any implementation.
9 Further Comments
The time taken by nag_tsa_auto_corr_part (g13acc) is proportional to .
This example uses an input series of 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.