nag_tsa_varma_update (g13dkc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_tsa_varma_update (g13dkc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_tsa_varma_update (g13dkc) accepts a sequence of new observations in a multivariate time series and updates both the forecasts and the standard deviations of the forecast errors. A call to nag_tsa_varma_forecast (g13djc) must be made prior to calling this function in order to calculate the elements of a reference vector together with a set of forecasts and their standard errors. On a successful exit from nag_tsa_varma_update (g13dkc) the reference vector is updated so that should future series values become available these forecasts may be updated by recalling nag_tsa_varma_update (g13dkc).

2  Specification

#include <nag.h>
#include <nagg13.h>
void  nag_tsa_varma_update (Integer k, Integer lmax, Integer m, Integer *mlast, const double z[], Integer kmax, double ref[], Integer lref, double v[], double predz[], double sefz[], NagError *fail)

3  Description

Let Zt = z1t,z2t,,zktT , for t=1,2,,n, denote a k-dimensional time series for which forecasts of Z^n+1,Z^n+2,, Z^n+lmax have been computed using nag_tsa_varma_forecast (g13djc). Given m further observations Zn+1,Zn+2,,Zn+m, where m<lmax, nag_tsa_varma_update (g13dkc) updates the forecasts of Zn+m+1,Zn+m+2,,Zn+lmax and their corresponding standard errors.
nag_tsa_varma_update (g13dkc) uses a multivariate version of the procedure described in Box and Jenkins (1976). The forecasts are updated using the ψ weights, computed in nag_tsa_varma_forecast (g13djc). If Zt* denotes the transformed value of Zt and Z^t*l denotes the forecast of Zt+l* from time t with a lead of l (that is the forecast of Zt+l* given observations Zt*,Zt-1*,), then
Z^t+1*l=τ+ψlεt+1+ψl+1εt+ψl+2εt-1+
and
Z^t*l+ 1=τ+ψl+ 1εt+ψl+ 2εt- 1+
where τ is a constant vector of length k involving the differencing parameters and the mean vector μ. By subtraction we obtain
Z^t+1*l=Z^t*l+1+ψlεt+1.
Estimates of the residuals corresponding to the new observations are also computed as εn+l=Zn+l*-Z^n*l, for l=1,2,,m. These may be of use in checking that the new observations conform to the previously fitted model.
On a successful exit, the reference array is updated so that nag_tsa_varma_update (g13dkc) may be called again should future series values become available, see Section 8.
When a transformation has been used the forecasts and their standard errors are suitably modified to give results in terms of the original series Zt; see Granger and Newbold (1976).

4  References

Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day
Granger C W J and Newbold P (1976) Forecasting transformed series J. Roy. Statist. Soc. Ser. B 38 189–203
Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley

5  Arguments

The quantities k, lmax, kmax, ref and lref from nag_tsa_varma_forecast (g13djc) are suitable for input to nag_tsa_varma_update (g13dkc).
1:     kIntegerInput
On entry: k, the dimension of the multivariate time series.
Constraint: k1.
2:     lmaxIntegerInput
On entry: the number, lmax, of forecasts requested in the call to nag_tsa_varma_forecast (g13djc).
Constraint: lmax2.
3:     mIntegerInput
On entry: m, the number of new observations available since the last call to either nag_tsa_varma_forecast (g13djc) or nag_tsa_varma_update (g13dkc). The number of new observations since the last call to nag_tsa_varma_forecast (g13djc) is then m+mlast.
Constraint: 0<m<lmax-mlast.
4:     mlastInteger *Input/Output
On entry: on the first call to nag_tsa_varma_update (g13dkc), since calling
nag_tsa_varma_forecast (g13djc), mlast must be set to 0 to indicate that no new observations have yet been used to update the forecasts; on subsequent calls mlast must contain the value of mlast as output on the previous call to nag_tsa_varma_update (g13dkc).
On exit: is incremented by m to indicate that mlast+m observations have now been used to update the forecasts since the last call to nag_tsa_varma_forecast (g13djc).
mlast must not be changed between calls to nag_tsa_varma_update (g13dkc), unless a call to nag_tsa_varma_forecast (g13djc) has been made between the calls in which case mlast should be reset to 0.
Constraint: 0mlast<lmax-m.
5:     z[kmax×m]const doubleInput
On entry: z[kmax×j-1+i-1] must contain the value of zi,n+mlast+j, for i=1,2,,k and j=1,2,,m, and where n is the number of observations in the time series in the last call made to nag_tsa_varma_forecast (g13djc).
Constraint: if the transformation defined in tr in nag_tsa_varma_forecast (g13djc) for the ith series is the log transformation, then z[kmax×j-1+i-1]>0.0, and if it is the square-root transformation, then z[kmax×j-1+i-1]0.0, for j=1,2,,m and i=1,2,,k.
6:     kmaxIntegerInput
On entry: the first dimension of the arrays z, predz, sefz and v.
Constraint: kmaxk.
7:     ref[lref]doubleInput/Output
On entry: must contain the first lmax-1×k×k+2×k×lmax+k elements of the reference vector as returned on a successful exit from nag_tsa_varma_forecast (g13djc) (or a previous call to nag_tsa_varma_update (g13dkc)).
On exit: the elements of ref are updated. The first lmax-1×k×k elements store the ψ weights ψ1,ψ2,,ψlmax-1. The next k×lmax elements contain the forecasts of the transformed series and the next k×lmax elements contain the variances of the forecasts of the transformed variables; see nag_tsa_varma_forecast (g13djc). The last k elements are not updated.
8:     lrefIntegerInput
On entry: the dimension of the array ref.
Constraint: lreflmax-1×k×k+2×k×lmax+k.
9:     v[kmax×m]doubleOutput
On exit: v[kmax×j-1+i-1] contains an estimate of the ith component of εn+mlast+j, for i=1,2,,k and j=1,2,,m.
10:   predz[kmax×lmax]doubleInput/Output
On entry: nonupdated values are kept intact.
On exit: predz[kmax×j-1+i-1] contains the updated forecast of zi,n+j, for i=1,2,,k and j=mlast+m+1,,lmax.
The columns of predz corresponding to the new observations since the last call to either nag_tsa_varma_forecast (g13djc) or nag_tsa_varma_update (g13dkc) are set equal to the corresponding columns of z.
11:   sefz[kmax×lmax]doubleInput/Output
On entry: nonupdated values are kept intact.
On exit: sefz[kmax×j-1+i-1] contains an estimate of the standard error of the corresponding element of predz, for i=1,2,,k and j=mlast+m+1,,lmax.
The columns of sefz corresponding to the new observations since the last call to either nag_tsa_varma_forecast (g13djc) or nag_tsa_varma_update (g13dkc) are set equal to zero.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, k=value.
Constraint: k1.
On entry, lmax=value.
Constraint: lmax2.
On entry, lref is too small: lref=value but must be at least value.
On entry, m=value.
Constraint: m>0.
On entry, mlast=value.
Constraint: mlast0.
NE_INT_2
On entry, kmax=value and k=value.
Constraint: kmaxk.
NE_INT_3
On entry, lmax=value, m=value and mlast=value.
Constraint: 0<m<lmax-mlast.
On entry, lmax=value, m=value and mlast=value.
Constraint: 0mlast<lmax-m.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REF_VEC
On entry, some of the elements of the array ref have been corrupted.
NE_RESULT_OVERFLOW
The updated forecasts will overflow if computed.
NE_TRANSFORMATION
On entry, one (or more) of the transformations requested is invalid.

7  Accuracy

The matrix computations are believed to be stable.

8  Further Comments

If a further m* observations, Zn+mlast+1,Zn+mlast+2,, Zn+mlast+m*, become available, then forecasts of Zn+mlast+m*+1,Zn+mlast+m*+2,, Zn+lmax may be updated by recalling nag_tsa_varma_update (g13dkc) with m=m*. Note that m and the contents of the array z are the only quantities which need updating; mlast is updated on exit from the previous call. On a successful exit, v contains estimates of εn+mlast+1,εn+mlast+2,, εn+mlast+m*; columns mlast+1,mlast+2,,mlast+m* of predz contain the new observed values Zn+mlast+1,Zn+mlast+2,, Zn+mlast+m* and columns mlast+1,mlast+2,,mlast+m* of sefz are set to zero.

9  Example

This example shows how to update the forecasts of two series each of length 48. No transformation has been used and no differencing applied to either of the series. nag_tsa_varma_estimate (g13ddc) is first called to fit an AR(1) model to the series. μ is to be estimated and ϕ12,1 constrained to be zero. A call to nag_tsa_varma_forecast (g13djc) is then made in order to compute forecasts of the next five series values. After one new observation becomes available the four forecasts are updated. A further observation becomes available and the three forecasts are updated.

9.1  Program Text

Program Text (g13dkce.c)

9.2  Program Data

Program Data (g13dkce.d)

9.3  Program Results

Program Results (g13dkce.r)


nag_tsa_varma_update (g13dkc) (PDF version)
g13 Chapter Contents
g13 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012