naginterfaces.library.tsa.multi_​inputmod_​forecast

naginterfaces.library.tsa.multi_inputmod_forecast(mr, mt, para, kfc, nev, nfv, xxy, kzef, rmsxy, mrx, parx, isttf, io_manager=None)

multi_inputmod_forecast produces forecasts of a time series (the output series) which depends on one or more other (input) series via a previously estimated multi-input model for which the state set information is not available. The future values of the input series must be supplied. In contrast with multi_inputmod_forecast_state() the original past values of the input and output series are required. Standard errors of the forecasts are produced. If future values of some of the input series have been obtained as forecasts using ARIMA models for those series, this may be allowed for in the calculation of the standard errors.

For full information please refer to the NAG Library document for g13bj

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g13/g13bjf.html

Parameters

mrint, array-like, shape $\left(7\right)$

The orders vector $(p,d,q,P,D,Q,s)$ of the ARIMA model for the output noise component.

$p$, $q$, $P$ and $Q$ refer respectively to the number of autoregressive $\left(\varphi \right)$, moving average $\left(\theta \right)$, seasonal autoregressive $\left(\Phi \right)$ and seasonal moving average $\left(\Theta \right)$ parameters.

$d$, $D$ and $s$ refer respectively to the order of non-seasonal differencing, the order of seasonal differencing and the seasonal period.

mtint, array-like, shape $(4,\text{nser})$

The transfer function model orders $b$, $p$ and $q$ of each of the input series. The data for input series $i$ is held in column $i$. Row 1 holds the value $b_i$, row 2 holds the value $q_i$ and row 3 holds the value $p_i$.

For a simple input, $b_i=q_i=p_i=0$.

Row 4 holds the value $r_i$, where $r_i=1$ for a simple input, and $r_i=2$ or $3$ for a transfer function input.

The choice $r_i=3$ leads to estimation of the pre-period input effects as nuisance parameters, and $r_i=2$ suppresses this estimation.

This choice may affect the returned forecasts and the state set.

When $r_i=1$, any nonzero contents of rows 1, 2 and 3 of column $i$ are ignored.

parafloat, array-like, shape $\left(\text{npara}\right)$

Estimates of the multi-input model parameters. These are in order, firstly the ARIMA model parameters: $p$ values of $\varphi$ parameters, $q$ values of $\theta$ parameters, $P$ values of $\Phi$ parameters, $Q$ values of $\Theta$ parameters.

These are followed by the transfer function model parameter values ${\omega }_0,{\omega }_1,\dots ,{\omega }_{q_1}$, ${\delta }_1,\dots ,{\delta }_{p_1}$ for the first of any input series and similarly for each subsequent input series.

The final component of $para$ is the value of the constant $c$.

kfcint

Must be set to $1$ if the constant was estimated when the model was fitted, and $0$ if it was held at a fixed value. This only affects the degrees of freedom used in calculating the estimated residual variance.

nevint

The number of original (undifferenced) values in each of the input and output time series.

nfvint

The number of forecast values of the output series required.

xxyfloat, array-like, shape $((nev+nfv),\text{nser})$

The columns of $xxy$ must contain in the first $nev$ places, the past values of each of the input and output series, in that order. In the next $nfv$ places, the columns relating to the input series (i.e., columns $1$ to $\text{nser}-1$) contain the future values of the input series which are necessary for construction of the forecasts of the output series $y$.

kzefint

Must be set to $0$ if the relevant $nfv$ values of the forecasts ($fva$) are to be held in the output series column ($\text{nser}$) of $xxy$ (which is otherwise unchanged) on exit, and must not be set to $0$ if the values of the input component series $z_t$ and the values of the output noise component $n_t$ are to overwrite the contents of $xxy$ on exit.

rmsxyfloat, array-like, shape $\left(\text{nser}\right)$

The first $(\text{nser}-1)$ elements of $rmsxy$ must contain the estimated residual variance of the input series ARIMA models. In the case of those inputs for which no ARIMA model is available or its effects are to be excluded in the calculation of forecast standard errors, the corresponding entry of $rmsxy$ should be set to $0$.

mrxint, array-like, shape $(7,\text{nser})$

The orders array for each of the input series ARIMA models. Thus, column $i$ contains values of $p$, $d$, $q$, $P$, $D$, $Q$, $s$ for input series $i$. In the case of those inputs for which no ARIMA model is available, the corresponding orders should be set to $0$.

parxfloat, array-like, shape $(:,\text{nser})$

Values of the parameters ($\varphi$, $\theta$, $\Phi$, and $\Theta$) for each of the input series ARIMA models.

Thus column $i$ contains $mrx[0,i-1]$ values of $\varphi$, $mrx[2,i-1]$ values of $\theta$, $mrx[3,i-1]$ values of $\Phi$ and $mrx[5,i-1]$ values of $\Theta$, in that order.

Values in the columns relating to those input series for which no ARIMA model is available are ignored.

isttfint

The dimension of the array $sttf$.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

parafloat, ndarray, shape $\left(\text{npara}\right)$

The parameter values may be updated using an additional iteration in the estimation process.

xxyfloat, ndarray, shape $((nev+nfv),\text{nser})$

If $kzef=0$ then $xxy$ is unchanged except that the relevant $nfv$ values in the column relating to the output series (column $\text{nser}$) contain the forecast values ($fva$), but if $kzef\ne 0$ then the columns of $xxy$ contain the corresponding values of the input component series $z_t$ and the values of the output noise component $n_t$, in that order.

rmsxyfloat, ndarray, shape $\left(\text{nser}\right)$

$rmsxy[\text{nser}-1]$ contains the estimated residual variance of the output noise ARIMA model which is calculated from the supplied series. Otherwise $rmsxy$ is unchanged.

mrxint, ndarray, shape $(7,\text{nser})$

Unchanged, except for column $\text{nser}$ which is used as workspace.

fvafloat, ndarray, shape $\left(nfv\right)$

The required forecast values for the output series. (Note that these are also output in column $\text{nser}$ of $xxy$ if $kzef=0$.)

fsdfloat, ndarray, shape $\left(nfv\right)$

The standard errors for each of the forecast values.

sttffloat, ndarray, shape $\left(isttf\right)$

The $nsttf$ values of the state set based on the first $nev$ sets of (past) values of the input and output series.

nsttfint

The number of values in the state set array $sttf$.

Raises

NagValueError

(errno $1$)

On entry, $nfv=⟨value⟩$.

Constraint: $nfv>0$.

(errno $1$)

On entry, $\text{ldxxy}=⟨value⟩$, $nev=⟨value⟩$ and $nfv=⟨value⟩$.

Constraint: $\text{ldxxy}\ge nev+nfv$.

(errno $1$)

On entry, $kfc=⟨value⟩$.

Constraint: $kfc=0$ or $1$.

(errno $2$)

The orders vector $mr$ is invalid.

(errno $2$)

$\text{npara}$ is inconsistent with $mr$ and $mt$.

(errno $2$)

Insufficient degrees of freedom to solve the problem.

(errno $2$)

On entry, $\text{npara}=⟨value⟩$ and the expected $\text{value}=⟨value⟩$.

Constraint: $\text{npara}$, $mr$ and $mt$ must be consistent.

(errno $2$)

On entry, $i=⟨value⟩$ and $mt[3,i-1]=⟨value⟩$.

Constraint: $mt[3,i-1]=1$, $2$ or $3$.

(errno $3$)

One or more sets of delta parameters do not satisfy the stationarity or invertibility conditions.

(errno $8$)

Unable to calculate the latest parameter estimates.

(errno $9$)

Failure in inversion of second derivative matrix.

(errno $10$)

One or more sets of ARIMA parameters do not satisfy the stationarity or invertibility conditions.

(errno $11$)

On entry, $isttf=⟨value⟩$.

Constraint: $isttf\ge (P\times s)+d+(D\times s)+q+max(p,Q\times s)+\text{ncf}$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

multi_inputmod_forecast has two stages. The first stage is essentially the same as a call to the model estimation function multi_inputmod_estim(), with zero iterations. In particular, all the parameters remain unchanged in the supplied input series transfer function models and output noise series ARIMA model. The internal nuisance parameters associated with the pre-observation period effects of the input series are estimated where requested, and so are any backforecasts of the output noise series. The output components $z_t$ and $n_t$, and residuals $a_t$ are calculated exactly as in Notes for multi_inputmod_estim, and the state set for forecasting is constituted.

The second stage is essentially the same as a call to the forecasting function multi_inputmod_forecast_state(). The same information is required, and the same information is returned.

Use of multi_inputmod_forecast should be confined to situations in which the state set for forecasting is unknown. Forecasting from the original data is relatively expensive because it requires recalculation of the state set. multi_inputmod_forecast returns the state set for use in producing further forecasts using multi_inputmod_forecast_state(), or for updating the state set using multi_inputmod_update().

References

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