naginterfaces.library.tsa.multi_​diff

naginterfaces.library.tsa.multi_diff(z, tr, dord, delta)

multi_diff differences and/or transforms a multivariate time series. It is intended to be used prior to multi_varma_estimate() to fit a vector autoregressive moving average (VARMA) model to the differenced/transformed series.

For full information please refer to the NAG Library document for g13dl

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g13/g13dlf.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 $\left(k\right)$

The order of differencing for each series, ${\text{d}}_1,{\text{d}}_2,\dots ,{\text{d}}_k$.

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 $\text{j}=1,2,\dots ,{\text{d}}_i$.

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

Returns

wfloat, ndarray, shape $(k,nd)$

$w[\text{i}-1,\text{t}-1]$ contains the value of $w_{\text{i},\text{t}+\text{d}}$, for $\text{t}=1,2,\dots ,n-\text{d}$, for $\text{i}=1,2,\dots ,k$.

ndint

The number of differenced values, $n-\text{d}$, in the series, where $\text{d}=max\left(dord\right[i-1\left]\right)$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 1$.

(errno $2$)

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

Constraint: $dord<n$.

(errno $2$)

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

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

(errno $3$)

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

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

(errno $4$)

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

Notes

For certain time series it may first be necessary to difference the original data to obtain a stationary series before calculating autocorrelations, etc. This function also allows you to apply either a square root or a log transformation to the original time series to stabilize the variance if required.

If the order of differencing required for the $i$th series is ${\text{d}}_i$, then the differencing operator 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, $B{Z}_t=Z_{t-1}$. Let $\text{d}$ denote the maximum of the orders of differencing, ${\text{d}}_i$, over the $k$ series. The function computes values of the differenced/transformed series $W_{\text{t}}={(w_{1\text{t}},w_{2\text{t}},\dots ,w_{\text{k}\text{t}})}^T$, for $\text{t}=\text{d}+1,\dots ,n$, as follows:

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

where $z_{it}^{\ast }$ are the transformed values of the original $k$-dimensional time series $Z_t={(z_{1t},z_{2t},\dots ,z_{kt})}^T$.

The differencing parameters ${\delta }_{ij}$, for $i=1,2,\dots ,k$ and $j=1,2,\dots ,{\text{d}}_i$, must be supplied by you. If the $i$th series does not require differencing, then ${\text{d}}_i=0$.

References

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

Wei, W W S, 1990, Time Series Analysis: Univariate and Multivariate Methods, Addison–Wesley