naginterfaces.library.tsa.multi_​varma_​update

naginterfaces.library.tsa.multi_varma_update(mlast, z, ref, predz, sefz)

multi_varma_update 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 multi_varma_forecast() 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 multi_varma_update the reference vector is updated so that should future series values become available these forecasts may be updated by recalling multi_varma_update.

For full information please refer to the NAG Library document for g13dk

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g13/g13dkf.html

Parameters

mlastint

On the first call to multi_varma_update, since calling multi_varma_forecast(), $mlast$ 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 multi_varma_update.

zfloat, array-like, shape $(k,m)$

$z[\text{i}-1,\text{j}-1]$ must contain the value of $z_{\text{i},n+mlast+\text{j}}$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,k$, and where $n$ is the number of observations in the time series in the last call made to multi_varma_forecast().

reffloat, array-like, shape $\left(\text{lref}\right)$

Must contain the first $(\text{lmax}-1)\times k\times k+2\times k\times \text{lmax}+k$ elements of the reference vector as returned on a successful exit from multi_varma_forecast() (or a previous call to multi_varma_update).

predzfloat, array-like, shape $(k,\text{lmax})$

Nonupdated values are kept intact.

sefzfloat, array-like, shape $(k,\text{lmax})$

Nonupdated values are kept intact.

Returns

mlastint

Is incremented by $m$ to indicate that $mlast+m$ observations have now been used to update the forecasts since the last call to multi_varma_forecast().

$mlast$ must not be changed between calls to multi_varma_update, unless a call to multi_varma_forecast() has been made between the calls in which case $mlast$ should be reset to $0$.

reffloat, ndarray, shape $\left(\text{lref}\right)$

The elements of $ref$ are updated. The first $(\text{lmax}-1)\times k\times k$ elements store the $\psi$ weights ${\psi }_1,{\psi }_2,\dots ,{\psi }_{l_{max}-1}$. The next $k\times \text{lmax}$ elements contain the forecasts of the transformed series and the next $k\times \text{lmax}$ elements contain the variances of the forecasts of the transformed variables; see multi_varma_forecast(). The last $\text{k}$ elements are not updated.

vfloat, ndarray, shape $(k,m)$

$v[\text{i}-1,\text{j}-1]$ contains an estimate of the $\text{i}$th component of ${ϵ}_{n+mlast+\text{j}}$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,k$.

predzfloat, ndarray, shape $(k,\text{lmax})$

$predz[\text{i}-1,\text{j}-1]$ contains the updated forecast of $z_{\text{i},n+\text{j}}$, for $\text{j}=mlast+m+1,\dots ,l_{max}$, for $\text{i}=1,2,\dots ,k$.

The columns of $predz$ corresponding to the new observations since the last call to either multi_varma_forecast() or multi_varma_update are set equal to the corresponding columns of $z$.

sefzfloat, ndarray, shape $(k,\text{lmax})$

$sefz[\text{i}-1,\text{j}-1]$ contains an estimate of the standard error of the corresponding element of $predz$, for $\text{j}=mlast+m+1,\dots ,l_{max}$, for $\text{i}=1,2,\dots ,k$.

The columns of $sefz$ corresponding to the new observations since the last call to either multi_varma_forecast() or multi_varma_update are set equal to zero.

Raises

NagValueError

(errno $1$)

On entry, $\text{lref}=⟨value⟩$ and the minimum size $\text{required}=⟨value⟩$.

Constraint: $\text{lref}\ge (\text{lmax}-1)\times k\times k+2\times k\times \text{lmax}+k$.

(errno $1$)

On entry, $mlast=⟨value⟩$.

Constraint: $mlast\ge 0$.

(errno $1$)

On entry, $m=⟨value⟩$, $\text{lmax}=⟨value⟩$ and $mlast=⟨value⟩$.

Constraint: $m<\text{lmax}-mlast$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m>0$.

(errno $1$)

On entry, $\text{lmax}=⟨value⟩$.

Constraint: $\text{lmax}\ge 2$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 1$.

(errno $2$)

On entry, some of the elements of the array $ref$ have been corrupted.

(errno $3$)

On entry, one (or more) of the transformations requested is invalid.

(errno $4$)

The updated forecasts will overflow if computed.

Notes

Let $Z_{\text{t}}={(z_{1\text{t}},z_{2\text{t}},\dots ,z_{k\text{t}})}^T$, for $\text{t}=1,2,\dots ,n$, denote a $k$-dimensional time series for which forecasts of ${\hat{Z}}_{n+1},{\hat{Z}}_{n+2},\dots ,{\hat{Z}}_{n+l_{max}}$ have been computed using multi_varma_forecast(). Given $m$ further observations $Z_{n+1},Z_{n+2},\dots ,Z_{n+m}$, where $m<l_{max}$, multi_varma_update updates the forecasts of $Z_{n+m+1},Z_{n+m+2},\dots ,Z_{n+l_{max}}$ and their corresponding standard errors.

multi_varma_update uses a multivariate version of the procedure described in Box and Jenkins (1976). The forecasts are updated using the $\psi$ weights, computed in multi_varma_forecast(). If $Z_t^{\ast }$ denotes the transformed value of $Z_t$ and ${\hat{Z}}_t^{\ast }\left(l\right)$ denotes the forecast of $Z_{t+l}^{\ast }$ from time $t$ with a lead of $l$ (that is the forecast of $Z_{t+l}^{\ast }$ given observations $Z_t^{\ast },Z_{t-1}^{\ast },\dots$), then

$${\hat{Z}}_{t+1}^{\ast }\left(l\right)=\tau +{\psi }_l{ϵ}_{t+1}+{\psi }_{l+1}{ϵ}_t+{\psi }_{l+2}{ϵ}_{t-1}+\cdots$$

and

$${\hat{Z}}_t^{\ast }(l+1)=\tau +{\psi }_{l+1}{ϵ}_t+{\psi }_{l+2}{ϵ}_{t-1}+\cdots$$

where $\tau$ is a constant vector of length $k$ involving the differencing parameters and the mean vector $\mu$. By subtraction we obtain

$${\hat{Z}}_{t+1}^{\ast }\left(l\right)={\hat{Z}}_t^{\ast }(l+1)+{\psi }_l{ϵ}_{t+1}\text{.}$$

Estimates of the residuals corresponding to the new observations are also computed as ${ϵ}_{n+\text{l}}=Z_{n+\text{l}}^{\ast }-{\hat{Z}}_n^{\ast }\left(\text{l}\right)$, for $\text{l}=1,2,\dots ,m$. 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 multi_varma_update may be called again should future series values become available, see Further Comments.

When a transformation has been used the forecasts and their standard errors are suitably modified to give results in terms of the original series $Z_t$; see Granger and Newbold (1976).

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