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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_tsa_multi_transf_prelim (g13bd)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_tsa_multi_transf_prelim (g13bd) calculates preliminary estimates of the parameters of a transfer function model.


[wds, isf, ifail] = g13bd(r0, r, nna, s, 'nl', nl)
[wds, isf, ifail] = nag_tsa_multi_transf_prelim(r0, r, nna, s, 'nl', nl)


nag_tsa_multi_transf_prelim (g13bd) calculates estimates of parameters δ1,δ2,,δp, ω0,ω1,,ωq in the transfer function model
given cross-correlations between the series xt and lagged values of yt:
rxyl,  l=0,1,,L  
and the ratio of standard deviations sy/sx, as supplied by nag_tsa_multi_xcorr (g13bc).
It is assumed that the series xt used to calculate the cross-correlations is a sample from a time series with true autocorrelations of zero. Otherwise the cross-correlations between the series bt and at, as defined in the description of nag_tsa_multi_filter_arima (g13ba), should be used in place of those between yt and xt.
The estimates are obtained by solving for δ1,δ2,,δp the equations
rxyb+q+j=δ1rxyb+q+j-1++δprxyb+q+j-p,  j=1,2,,p  
then calculating
ωi = ± sy / sx rxy b+i - δ1 rxy b+i- 1 - - δp rxy b+i-p ,   i= 0,1,,q  
where the ‘+’ is used for ω0 and ‘-’ for ωi, i>0.
Any value of rxyl arising in these equations for l<b is taken as zero. The parameters δ1,δ2,,δp are checked as to whether they satisfy the stability criterion.


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


Compulsory Input Parameters

1:     r0 – double scalar
The cross-correlation between the two series at lag 0, rxy0.
Constraint: -1.0r01.0.
2:     rnl – double array
The cross-correlations between the two series at lags 1 to L, rxyl, for l=1,2,,L.
Constraint: -1.0ri1.0, for i=1,2,,nl.
3:     nna3 int64int32nag_int array
The transfer function model orders in the standard form b,q,p (i.e., delay time, number of moving-average MA-like followed by number of autoregressive AR-like parameters).
Constraint: nnai0, for i=1,2,3.
4:     s – double scalar
The ratio of the standard deviation of the y series to that of the x series, sy/sx.
Constraint: s>0.0.

Optional Input Parameters

1:     nl int64int32nag_int scalar
Default: the dimension of the array r.
L, the number of lagged cross-correlations in the array r.
Constraint: nlmaxnna1+nna2+nna3,1.

Output Parameters

1:     wdsnwds – double array
The preliminary estimates of the parameters of the transfer function model in the order of q+1 MA-like parameters followed by the p AR-like parameters. If the estimation of either type of parameter fails then these arguments are set to 0.0.
2:     isf2 int64int32nag_int array
Indicators of the success of the estimation of MA-like and AR-like parameters respectively. A value 0 indicates that there are no parameters of that type to be estimated. A value of 1 or -1 indicates that there are parameters of that type in the model and the estimation of that type has been successful or unsuccessful respectively. Note that there is always at least one MA-like parameter in the model.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,nnai<0, for i=1,2,3,
orr0<-1.0 or r0>1.0,
orri<-1.0 or ri>1.0, for some i=1,2,,nl,
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Equations used in the computations may become unstable, in which case results are reset to zero with array isf values set accordingly.

Further Comments

If nna3>0,a local workspace array of fixed length is allocated internally by nag_tsa_multi_transf_prelim (g13bd). The total size of this array amounts to nna3 integer elements and nna3×nna3+1 double elements.
The time taken by nag_tsa_multi_transf_prelim (g13bd) is roughly proportional to nwds3.


This example reads the cross-correlations between two series at lags 0 to 6. It then reads a 3,2,1 transfer function model and calculates and prints the preliminary estimates of the parameters of the model.
function g13bd_example

fprintf('g13bd example results\n\n');

% Cross-correlation at lag 0
r0  = -0.0155;
% Other cross-correlations
r   = [0.0339; -0.0374; -0.2895; -0.3430; -0.4518; -0.2787];

% transfer function model orders
nna = [int64(3);2;1];

%Standard deviation ratio
s   = 1.9256;

% Calculate parameter estimates
[wds, isf, ifail] = g13bd( ...
                           r0, r, nna, s);

% Display results
fprintf('Success/failure indicator       = %4d%4d\n\n', isf(1:2));
fprintf('Transfer function model B, Q, P = %4d%4d%4d\n\n', nna(1:3));
disp('Parameter initial estimates');

g13bd example results

Success/failure indicator       =    1   1

Transfer function model B, Q, P =    3   2   1

Parameter initial estimates
   -0.5575    0.3166    0.4626    0.6169

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Chapter Contents
Chapter Introduction
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