naginterfaces.library.tsa.uni_​arima_​estim_​easy

naginterfaces.library.tsa.uni_arima_estim_easy(mr, par, c, x, nppc, kfc=1, kpiv=0, nit=100, ires=None, io_manager=None)

uni_arima_estim_easy is an easy-to-use version of uni_arima_estim(). It fits a seasonal autoregressive integrated moving average (ARIMA) model to an observed time series, using a nonlinear least squares procedure incorporating backforecasting. Parameter estimates are obtained, together with appropriate standard errors. The residual series is returned, and information for use in forecasting the time series is produced for use in uni_arima_update() and uni_arima_forecast_state().

The estimation procedure is iterative, starting with initial parameter values such as may be obtained using uni_arima_prelim(). It continues until a specified convergence criterion is satisfied or until a specified number of iterations have been carried out. The progress of the iteration can be monitored by means of an optional printing facility.

For full information please refer to the NAG Library document for g13af

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13aff.html

Parameters

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

The orders vector $(p,d,q,P,D,Q,s)$ of the ARIMA model whose parameters are to be estimated. $p$, $q$, $P$ 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.

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

The initial estimates of the $p$ values of the $\varphi$ parameters, the $q$ values of the $\theta$ parameters, the $P$ values of the $\Phi$ parameters and the $Q$ values of the $\Theta$ parameters, in that order.

cfloat

If $kfc=0$, $c$ must contain the expected value, $c$, of the differenced series.

If $kfc=1$, $c$ must contain an initial estimate of $c$.

Therefore, if $c$ and $kfc$ are both zero on entry, there is no constant correction.

xfloat, array-like, shape $\left(\text{nx}\right)$

The $n$ values of the original undifferenced time series.

nppcint

The number of $\varphi$, $\theta$, $\Phi$, $\Theta$ and $c$ parameters to be estimated. $nppc=p+q+P+Q+1$ if the constant is being estimated and $nppc=p+q+P+Q$ if not.

kfcint, optional

Must be set to $1$ if the constant, $c$, is to be estimated and $0$ if it is to be held fixed at its initial value.

kpivint, optional

Must be nonzero if the progress of the optimization is to be monitored using the built-in printing facility. Otherwise $kpiv$ must contain zero. If selected, monitoring output will be sent to the file object associated with the advisory I/O unit (see FileObjManager). For each iteration, the heading

G13AFZ MONITORING OUTPUT - ITERATION n

followed by the argument values, and residual sum of squares, are printed.

nitint, optional

The maximum number of iterations to be performed.

iresNone or int, optional

Note: if this argument is None then a default value will be used, determined as follows: $15\times (\text{q}+\text{Q}\times \text{s})+11\times \text{nx}+13\times (\text{npar}+kfc)+8\times (\text{p}+\text{P}\times \text{s})+12+2\times {(\text{q}+\text{Q}\times \text{s}+\text{npar}+kfc)}^2$.

The dimension of the array $res$.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

parfloat, ndarray, shape $\left(\text{npar}\right)$

Contains the latest values of the estimates of these parameters.

cfloat

If $kfc=0$, $c$ is unchanged.

If $kfc=1$, $c$ contains the latest estimate of $c$.

sfloat

The residual sum of squares after the latest series of parameter estimates has been incorporated into the model. If the function exits with a faulty input parameter, $s$ contains zero.

ndfint

The number of degrees of freedom associated with $s$,

$ndf=n-d-D\times s-p-q-P-Q-kfc$.

sdfloat, ndarray, shape $\left(nppc\right)$

The standard deviations corresponding to the parameters in the model ($p$ autoregressive, $q$ moving average, $P$ seasonal autoregressive, $Q$ seasonal moving average and $c$, if estimated, in that order). If the function exits with $\text{errno}$ containing a value other than $0$ or $9$, or if the required number of iterations is zero, the contents of $sd$ will be indeterminate.

cmfloat, ndarray, shape $(nppc,nppc)$

The correlation coefficients associated with each pair of the $nppc$ parameters. These are held in the first $nppc$ rows and the first $nppc$ columns of $cm$. These correlation coefficients are indeterminate if $\text{errno}$ contains on exit a value other than $0$ or $9$, or if the required number of iterations is zero.

stfloat, ndarray, shape $\left(\text{nx}\right)$

The value of the state set in its first $nst$ elements. If the function exits with $\text{errno}$ containing a value other than $0$ or $9$, the contents of $st$ will be indeterminate.

nstint

The size of the state set. $nst=P\times s+D\times s+d+q+max(p,Q\times s)$.

$nst$ should be used subsequently in uni_arima_update() and uni_arima_forecast_state() as the dimension of $st$.

itcint

The number of iterations performed.

isfint, ndarray, shape $\left(4\right)$

The first four elements of $isf$ contain success/failure indicators, one for each of the four types of parameter in the model (autoregressive, moving average, seasonal autoregressive, seasonal moving average), in that order.

Each indicator has the interpretation:

$-2$	On entry parameters of this type have initial estimates which do not satisfy the stationarity or invertibility test conditions.
$-1$	The search procedure has failed to converge because the latest set of parameter estimates of this type is invalid.
$0$	No parameter of this type is in the model.
$1$	Valid final estimates for parameters of this type have been obtained.

resfloat, ndarray, shape $\left(ires\right)$

The first $nres$ elements of $res$ contain the model residuals derived from the differenced series. If the function exits with $\text{errno}$ holding a value other than $0$ or $9$, these elements of $res$ will be indeterminate. The rest of the array $res$ is used as workspace.

nresint

The number of model residuals returned in $res$.

Raises

NagValueError

(errno $1$)

On entry, $kfc=⟨value⟩$.

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

(errno $1$)

The orders vector $mr$ is invalid.

(errno $1$)

On entry, $\text{npar}=⟨value⟩$.

Constraint: $\text{npar}=p+q+P+Q$.

(errno $2$)

The model is over-parameterised.

(errno $3$)

On entry, $nit=⟨value⟩$.

Constraint: $nit\ge 0$.

(errno $4$)

On entry, $\text{nx}=⟨value⟩$ and minimum required size for the state set $\text{array}=⟨value⟩$.

(errno $5$)

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

Constraint: $ires\ge 15\times Q^{\prime }+11n+13\times nppc+8\times P^{\prime }+12+2\times {(Q^{\prime }+nppc)}^2$.

Warns

NagAlgorithmicWarning

(errno $7$)

A failure in the search procedure has occurred.

(errno $8$)

The inversion of the Hessian matrix in the calculation of the covariance matrix of the parameter estimates has failed.

(errno $9$)

Failure whilst calculating backforecasts.

(errno $10$)

Satisfactory parameter estimates could not be obtained for all parameter types in the model.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The time series $x_1,x_2,\dots ,x_n$ supplied to the function is assumed to follow a seasonal autoregressive integrated moving average (ARIMA) model defined as follows:

$${\nabla }^d{\nabla }_s^D{x}_t-c=w_t\text{,}$$

where ${\nabla }^d{\nabla }_s^D{x}_t$ is the result of applying non-seasonal differencing of order $d$ and seasonal differencing of seasonality $s$ and order $D$ to the series $x_t$, as outlined in the description of uni_diff(). The differenced series is then of length $N=n-d^{\prime }$, where $d^{\prime }=d+(D\times s)$ is the generalized order of differencing. The scalar $c$ is the expected value of the differenced series, and the series $w_1,w_2,\dots ,w_N$ follows a zero-mean stationary autoregressive moving average (ARMA) model defined by a pair of recurrence equations. These express $w_t$ in terms of an uncorrelated series $a_t$, via an intermediate series $e_t$. The first equation describes the seasonal structure:

$$w_t={\Phi }_1{w}_{t-s}+{\Phi }_2{w}_{t-2\times s}+\cdots +{\Phi }_P{w}_{t-P\times s}+e_t-{\Theta }_1{e}_{t-s}-{\Theta }_2{e}_{t-2\times s}-\cdots -{\Theta }_Q{e}_{t-Q\times s}\text{.}$$

The second equation describes the non-seasonal structure. If the model is purely non-seasonal the first equation is redundant and $e_t$ above is equated with $w_t$:

$$e_t={\varphi }_1{e}_{t-1}+{\varphi }_2{e}_{t-2}+\cdots +{\varphi }_p{e}_{t-p}+a_t-{\theta }_1{a}_{t-1}-{\theta }_2{a}_{t-2}-\cdots -{\theta }_q{a}_{t-q}\text{.}$$

Estimates of the model parameters defined by

$$\begin{array}{c}\begin{array}{c}{\varphi }_1,{\varphi }_2,\dots ,{\varphi }_p,{\theta }_1,{\theta }_2,\dots ,{\theta }_q\text{,}\\ {\Phi }_1,{\Phi }_2,\dots ,{\Phi }_P,{\Theta }_1,{\Theta }_2,\dots ,{\Theta }_Q\end{array}\end{array}$$

and (optionally) $c$ are obtained by minimizing a quadratic form in the vector $w={(w_1,w_2,\dots ,w_N)}^{\prime }$.

The minimization process is iterative, iterations being performed until convergence is achieved (see Notes for uni_arima_estim for full details), or until the user-specified maximum number of iterations are completed.

The final values of the residual sum of squares and the parameter estimates are used to obtain asymptotic approximations to the standard deviations of the parameters, and the correlation matrix for the parameters. The ‘state set’ array of information required by forecasting is also returned.

Note: if the maximum number of iterations are performed without convergence, these quantities may not be reliable. In this case, the sequence of iterates should be checked, using the optional monitoring function, to verify that convergence is adequate for practical purposes.

References

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

Marquardt, D W, 1963, An algorithm for least squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math. (11), 431