naginterfaces.library.tsa.multi_​varma_​forecast

naginterfaces.library.tsa.multi_varma_forecast(z, tr, dord, delta, ip, iq, mean, par, qq, v, lmax, lref)

multi_varma_forecast computes forecasts of a multivariate time series. It is assumed that a vector ARMA model has already been fitted to the appropriately differenced/transformed time series using multi_varma_estimate(). The standard deviations of the forecast errors are also returned. A reference vector is set up so that, should future series values become available, the forecasts and their standard errors may be updated by calling multi_varma_update().

For full information please refer to the NAG Library document for g13dj

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g13/g13djf.html

Parameters

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

$z[\text{i},\text{t}]$ must contain, $z_{\text{i}\text{t}}$, the $\text{i}$th component of $Z_{\text{t}}$, for $\text{t}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,k$.

trstr, length 1, array-like, shape $\left(k\right)$

$tr[\text{i}-1]$ indicates whether the $\text{i}$th time series is to be transformed, for $\text{i}=1,2,\dots ,k$.

$tr[i-1]=\text{'N'}$

No transformation is used.

$tr[i-1]=\text{'L'}$

A log transformation is used.

$tr[i-1]=\text{'S'}$

A square root transformation is used.

dordint, array-like, shape $\left(k\right)$

$dord[i-1]$ must specify, ${\text{d}}_i$, the order of differencing required for the $i$th series.

deltafloat, array-like, shape $(k,\text{d})$

If $dord[i-1]>0$, then $delta[\text{i}-1,\text{j}-1]$ must be set equal to ${\delta }_{\text{i}\text{j}}$, for $\text{i}=1,2,\dots ,k$, for $\text{j}=1,2,\dots ,{\text{d}}_{\text{i}}$.

If $\text{d}=0$, $delta$ is not referenced.

ipint

$p$, the number of AR parameter matrices.

iqint

$q$, the number of MA parameter matrices.

meanstr, length 1

$mean=\text{'M'}$, if components of $\mu$ have been estimated and $mean=\text{'Z'}$, if all elements of $\mu$ are to be taken as zero.

parfloat, array-like, shape $\left(\text{lpar}\right)$

Must contain the parameter estimates read in row by row in the order ${\varphi }_1,{\varphi }_2,\dots ,{\varphi }_p$, ${\theta }_1,{\theta }_2,\dots ,{\theta }_q$, $\mu$.

Thus,

if $ip>0$, $par[(\text{l}-1)\times k\times k+(\text{i}-1)\times k+\text{j}-1]$ must be set equal to an estimate of the $(\text{i},\text{j})$th element of ${\varphi }_{\text{l}}$, for $\text{j}=1,2,\dots ,k$, for $\text{i}=1,2,\dots ,k$, for $\text{l}=1,2,\dots ,p$;

if $iq>0$, $par[p\times k\times k+(\text{l}-1)\times k\times k+(\text{i}-1)\times k+\text{j}-1]$ must be set equal to an estimate of the $(\text{i},\text{j})$th element of ${\theta }_{\text{l}}$, for $\text{j}=1,2,\dots ,k$, for $\text{i}=1,2,\dots ,k$, for $\text{l}=1,2,\dots ,q$;

if $mean=\text{'M'}$, $par[(p+q)\times k\times k+\text{i}-1]$ must be set equal to an estimate of the $\text{i}$th component of $\mu$, for $\text{i}=1,2,\dots ,k$.

qqfloat, array-like, shape $(k,k)$

$qq[i-1,j-1]$ must contain an estimate of the $(i,j)$th element of $\Sigma$. The lower triangle only is needed.

vfloat, array-like, shape $(k,n-\text{d})$

$v[\text{i}-1,\text{t}-1]$ must contain an estimate of the $\text{i}$th component of ${ϵ}_{\text{t}+\text{d}}$, for $\text{t}=1,2,\dots ,n-\text{d}$, for $\text{i}=1,2,\dots ,k$.

If $q=0$, $v$ is not used.

lmaxint

The number, $l_{max}$, of forecasts required.

lrefint

The dimension of the array $ref$.

Returns

qqfloat, ndarray, shape $(k,k)$

If $errno$ != 1, then the upper triangle is set equal to the lower triangle.

predzfloat, ndarray, shape $(k,lmax)$

$predz[\text{i}-1,\text{l}-1]$ contains the forecast of $z_{\text{i},n+\text{l}}$, for $\text{l}=1,2,\dots ,{\text{l}}_{max}$, for $\text{i}=1,2,\dots ,k$.

sefzfloat, ndarray, shape $(k,lmax)$

$sefz[\text{i}-1,\text{l}-1]$ contains an estimate of the standard error of the forecast of $z_{\text{i},n+\text{l}}$, for $\text{l}=1,2,\dots ,{\text{l}}_{max}$, for $\text{i}=1,2,\dots ,k$.

reffloat, ndarray, shape $\left(lref\right)$

The reference vector which may be used to update forecasts using multi_varma_update(). The first $(lmax-1)\times k\times k$ elements contain the $\psi$ weight matrices, ${\psi }_1,{\psi }_2,\dots ,{\psi }_{l_{max}-1}$. The next $k\times lmax$ elements contain the forecasts of the transformed series ${\hat{Z}}_{n+1}^{\ast },{\hat{Z}}_{n+2}^{\ast },\dots ,{\hat{Z}}_{n+l_{max}}^{\ast }$ and the next $k\times lmax$ contain the variances of the forecasts of the transformed variables. The last $\text{k}$ elements are used to store the transformations for the series.

Raises

NagValueError

(errno $1$)

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

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

(errno $1$)

On entry, $i=⟨value⟩$, $dord[i-1]=⟨value⟩$ and $n-max(ip,iq)=⟨value⟩$.

Constraint: $0\le dord[\text{i}-1]<n-max(ip,iq)$.

(errno $1$)

On entry, $lmax=⟨value⟩$.

Constraint: $lmax\ge 1$.

(errno $1$)

On entry, number of observations $=⟨value⟩$ and number of parameters $=⟨value⟩$.

Constraint: number of $\text{observations}\ge \text{number}$ of parameters.

(errno $1$)

On entry, $\text{lpar}$ is too small: $\text{lpar}=⟨value⟩$ and the minimum size $\text{required}=⟨value⟩$.

(errno $1$)

On entry, $mean$ is an invalid character.

Constraint: $mean=\text{'M'}$ or $\text{'Z'}$.

(errno $1$)

On entry, $iq=⟨value⟩$.

Constraint: $iq\ge 0$.

(errno $1$)

On entry, $ip=⟨value⟩$.

Constraint: $ip\ge 0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 3$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 1$.

(errno $2$)

On entry, $i=⟨value⟩$ and $tr[i-1]$ is invalid.

Constraint: $tr[i-1]=\text{'N'}$, $\text{'L'}$ or $\text{'S'}$.

(errno $3$)

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

(errno $4$)

On entry, the MA parameter matrices are outside the invertibility region.

(errno $4$)

On entry, the AR parameter matrices are outside the stationarity region.

(errno $4$)

On entry, the covariance matrix $qq$ is not positive definite.

(errno $5$)

An excessive number of iterations were needed to evaluate the eigenvalues of the matrices used to test for stationarity and invertibility.

(errno $6$)

The covariance matrix may be nearly non-positive definite.

(errno $7$)

The forecasts will overflow if computed.

Notes

Let the vector $Z_{\text{t}}={(z_{1\text{t}},z_{2\text{t}},\dots ,z_{k\text{t}})}^T$, for $\text{t}=1,2,\dots ,n$, denote a $k$-dimensional time series for which forecasts of $Z_{n+1},Z_{n+2},\dots ,Z_{n+l_{max}}$ are required. Let $W_t={(w_{1t},w_{2t},\dots ,w_{kt})}^T$ be defined as follows:

$$w_{it}={\delta }_i\left(B\right)z_{it}^{\ast }\text{,}i=1,2,\dots ,k\text{,}$$

where ${\delta }_i\left(B\right)$ is the differencing operator applied to the $i$th series and where $z_{it}^{\ast }$ is equal to either $z_{it}$, $\sqrt{z_{it}}$ or $loge\left(z_{it}\right)$ depending on whether or not a transformation was required to stabilize the variance before fitting the model.

If the order of differencing required for the $i$th series is ${\text{d}}_i$, then the differencing operator for the $i$th series is defined by ${\delta }_i\left(B\right)=1-{\delta }_{i1}B-{\delta }_{i2}{B}^2-\cdots -{\delta }_{i{\text{d}}_i}{B}^{{\text{d}}_i}$ where $B$ is the backward shift operator; that is, $B{Z}_t=Z_{t-1}$. The differencing parameters ${\delta }_{\text{i}\text{j}}$, for $\text{j}=1,2,\dots ,{\text{d}}_{\text{i}}$, for $\text{i}=1,2,\dots ,k$, must be supplied by you. If the $i$th series does not require differencing, then ${\text{d}}_i=0$.

$W_t$ is assumed to follow a multivariate ARMA model of the form:

$$W_t-\mu ={\varphi }_1(W_{t-1}-\mu )+{\varphi }_2(W_{t-2}-\mu )+\cdots +{\varphi }_p(W_{t-p}-\mu )+{ϵ}_t-{\theta }_1{ϵ}_{t-1}-\cdots -{\theta }_q{ϵ}_{t-q}\text{,}$$

where ${ϵ}_{\text{t}}={({ϵ}_{1\text{t}},{ϵ}_{2\text{t}},\dots ,{ϵ}_{k\text{t}})}^T$, for $\text{t}=1,2,\dots ,n$, is a vector of $k$ residual series assumed to be Normally distributed with zero mean and positive definite covariance matrix $\Sigma$. The components of ${ϵ}_t$ are assumed to be uncorrelated at non-simultaneous lags. The ${\varphi }_i$ and ${\theta }_j$ are $k\times k$ matrices of parameters. The matrices ${\varphi }_{\text{i}}$, for $\text{i}=1,2,\dots ,p$, are the autoregressive (AR) parameter matrices, and the matrices ${\theta }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$, the moving average (MA) parameter matrices. The parameters in the model are thus the $p$ ($k\times k$) $\varphi$-matrices, the $q$ ($k\times k$) $\theta$-matrices, the mean vector $\mu$ and the residual error covariance matrix $\Sigma$. The ARMA model (1) must be both stationary and invertible; see uni_arma_roots() for a method of checking these conditions.

The ARMA model (1) may be rewritten as

$$\varphi \left(B\right)(\delta \left(B\right)Z_t^{\ast }-\mu )=\theta \left(B\right){ϵ}_t\text{,}$$

where $\varphi \left(B\right)$ and $\theta \left(B\right)$ are the autoregressive and moving average polynomials and $\delta \left(B\right)$ denotes the $k\times k$ diagonal matrix whose $i$th diagonal elements is ${\delta }_i\left(B\right)$ and $Z_t^{\ast }={(z_{1t}^{\ast },z_{2t}^{\ast }\dots z_{kt}^{\ast })}^T$.

This may be rewritten as

$$\varphi \left(B\right)\delta \left(B\right)Z_t^{\ast }=\varphi \left(B\right)\mu +\theta \left(B\right){ϵ}_t$$

or

$$Z_t^{\ast }=\tau +\psi \left(B\right){ϵ}_t=\tau +{ϵ}_t+{\psi }_1{ϵ}_{t-1}+{\psi }_2{ϵ}_{t-2}+\cdots$$

where $\psi \left(B\right)={\delta }^{-1}\left(B\right){\varphi }^{-1}\left(B\right)\theta \left(B\right)$ and $\tau ={\delta }^{-1}\left(B\right)\mu$ is a vector of length $k$.

Forecasts are computed using a multivariate version of the procedure described in Box and Jenkins (1976). If ${\hat{Z}}_n^{\ast }\left(l\right)$ denotes the forecast of $Z_{n+l}^{\ast }$, then ${\hat{Z}}_n^{\ast }\left(l\right)$ is taken to be that linear function of $Z_n^{\ast },Z_{n-1}^{\ast },\dots$ which minimizes the elements of $E\left\{e_n\left(l\right)e_n^{\prime }\left(l\right)\right\}$ where $e_n\left(l\right)=Z_{n+l}^{\ast }-{\hat{Z}}_n^{\ast }\left(l\right)$ is the forecast error. ${\hat{Z}}_n^{\ast }\left(l\right)$ is referred to as the linear minimum mean square error forecast of $Z_{n+l}^{\ast }$.

The linear predictor which minimizes the mean square error may be expressed as

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

The forecast error at $t$ for lead $l$ is then

$$e_n\left(l\right)=Z_{n+l}^{\ast }-{\hat{Z}}_n^{\ast }\left(l\right)={ϵ}_{n+l}+{\psi }_1{ϵ}_{n+l-1}+{\psi }_2{ϵ}_{n+l-2}+\cdots +{\psi }_{l-1}{ϵ}_{n+1}\text{.}$$

Let $\text{d}=max\left({\text{d}}_{\text{i}}\right)$, for $\text{i}=1,2,\dots ,k$. Unless $q=0$ the function requires estimates of ${ϵ}_{\text{t}}$, for $\text{t}=\text{d}+1,\dots ,n$, which are obtainable from multi_varma_estimate(). The terms ${ϵ}_{\text{t}}$ are assumed to be zero, for $\text{t}=n+1,\dots ,n+l_{max}$. You may use multi_varma_update() to update these $l_{max}$ forecasts should further observations, $Z_{n+1},Z_{n+2},\dots$, become available. Note that when $l_{max}$ or more further observations are available then multi_varma_forecast must be used to produce new forecasts for $Z_{n+l_{max}+1},Z_{n+l_{max}+2},\dots$, should they be required.

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