naginterfaces.library.tsa.multi_​transf_​prelim

naginterfaces.library.tsa.multi_transf_prelim(r0, r, nna, s)

multi_transf_prelim calculates preliminary estimates of the parameters of a transfer function model.

For full information please refer to the NAG Library document for g13bd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13bdf.html

Parameters

r0float

The cross-correlation between the two series at lag $0$, $r_{xy}\left(0\right)$.

rfloat, array-like, shape $\left(\text{nl}\right)$

The cross-correlations between the two series at lags $1$ to $L$, $r_{xy}\left(\text{l}\right)$, for $\text{l}=1,2,\dots ,L$.

nnaint, array-like, shape $\left(3\right)$

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).

sfloat

The ratio of the standard deviation of the $y$ series to that of the $x$ series, $s_y/s_x$.

Returns

wdsfloat, ndarray, shape $(:)$

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 $0.0$.

isfint, ndarray, shape $\left(2\right)$

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.

Raises

NagValueError

(errno $1$)

On entry, $\text{nwds}=⟨value⟩$, $nna\left[1\right]=⟨value⟩$ and $nna\left[2\right]=⟨value⟩$.

Constraint: $\text{nwds}=nna\left[1\right]+nna\left[2\right]+1$.

(errno $1$)

On entry, $s=⟨value⟩$.

Constraint: $s>0.0$.

(errno $1$)

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

Constraint: $-1.0\le r\left[\text{i}\right]\le 1.0$.

(errno $1$)

On entry, $r0=⟨value⟩$.

Constraint: $-1.0\le r0\le 1.0$.

(errno $1$)

On entry, $\text{nl}=⟨value⟩$, $nna\left[0\right]=⟨value⟩$, $nna\left[1\right]=⟨value⟩$ and $nna\left[2\right]=⟨value⟩$.

Constraint: $\text{nl}\ge max(nna\left[0\right]+nna\left[1\right]+nna\left[2\right],1)$.

(errno $1$)

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

Constraint: $nna[i-1]\ge 0$.

Notes

multi_transf_prelim calculates estimates of parameters ${\delta }_1,{\delta }_2,\dots ,{\delta }_p$, ${\omega }_0,{\omega }_1,\dots ,{\omega }_q$ in the transfer function model

$$y_t={\delta }_1{y}_{t-1}+{\delta }_2{y}_{t-2}+\cdots +{\delta }_p{y}_{t-p}+{\omega }_0{x}_{t-b}-{\omega }_1{x}_{t-b-1}-\cdots -{\omega }_q{x}_{t-b-q}$$

given cross-correlations between the series $x_t$ and lagged values of $y_t$:

$$r_{xy}\left(l\right)\text{,}l=0,1,\dots ,L$$

and the ratio of standard deviations $s_y/s_x$, as supplied by multi_xcorr().

It is assumed that the series $x_t$ 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 $b_t$ and $a_t$, as defined in the description of multi_filter_arima(), should be used in place of those between $y_t$ and $x_t$.

The estimates are obtained by solving for ${\delta }_1,{\delta }_2,\dots ,{\delta }_p$ the equations

$$r_{xy}(b+q+j)={\delta }_1{r}_{xy}(b+q+j-1)+\cdots +{\delta }_p{r}_{xy}(b+q+j-p)\text{,}j=1,2,\dots ,p$$

then calculating

$${\omega }_i=\pm \left(s_y/s_x\right)[r_{xy}(b+i)-{\delta }_1{r}_{xy}(b+i-1)-\cdots -{\delta }_p{r}_{xy}(b+i-p)]\text{,}i=0,1,\dots ,q$$

where the ‘$+$’ is used for ${\omega }_0$ and ‘$-$’ for ${\omega }_i$, $i>0$.

Any value of $r_{xy}\left(l\right)$ arising in these equations for $l<b$ is taken as zero. The parameters ${\delta }_1,{\delta }_2,\dots ,{\delta }_p$ are checked as to whether they satisfy the stability criterion.

References

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