naginterfaces.library.tsa.multi_inputmod_estim¶

naginterfaces.library.tsa.multi_inputmod_estim(mr, mt, para, xxy, kfc=1, kef=2, nit=1000, zsp=None, kpriv=0, io_manager=None)[source]¶

multi_inputmod_estim fits a multi-input model relating one output series to the input series with a choice of three different estimation criteria: nonlinear least squares, exact likelihood and marginal likelihood. When no input series are present, multi_inputmod_estim fits a univariate ARIMA model.

For full information please refer to the NAG Library document for g13be

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g13/g13bef.html

Parameters

mrint, array-like, shape $(7)$

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 $(ϕ)$ , moving average $(θ)$ , seasonal autoregressive $(Φ)$ and seasonal moving average $(Θ)$ 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 $(4, nser)$

The transfer function model orders $b$ , $p$ and $q$ of each of the input series. The order parameters for input series $i$ are 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, $r_{i} = 2$ for a transfer function input for which no allowance is to be made for pre-observation period effects, and $r_{i} = 3$ for a transfer function input for which pre-observation period effects will be treated by estimation of appropriate nuisance parameters.

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

parafloat, array-like, shape $(npara)$

Initial values of the multi-input model parameters. These are in order, firstly the ARIMA model parameters: $p$ values of $ϕ$ parameters, $q$ values of $θ$ parameters, $P$ values of $Φ$ parameters and $Q$ values of $Θ$ parameters. These are followed by initial values of the transfer function model parameters $ω_{0}, ω_{1}, \dots, ω_{q_{1}}$ , $δ_{1}, δ_{2}, \dots, δ_{p_{1}}$ for the first of any input series and similarly for each subsequent input series. The final component of $p a r a$ is the initial value of the constant $c$ , whether it is fixed or is to be estimated.

xxyfloat, array-like, shape $(nxxy, nser)$

The columns of $x x y$ must contain the $nxxy$ original, undifferenced values of each of the input series and the output series $x_{t}$ in that order.

kfcint, optional

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

kefint, optional

Indicates the likelihood option.

$k e f = 1$

Gives least squares.

$k e f = 2$

Gives exact likelihood.

$k e f = 3$

Gives marginal likelihood.

nitint, optional

The maximum required number of iterations.

$n i t = 0$

No change is made to any of the model parameters in array $p a r a$ except that the constant $c$ (if $k f c = 1$ ) and any $ω$ relating to simple input series are estimated. (Apart from these, estimates are always derived for the nuisance parameters relating to any backforecasts and any pre-observation period effects for transfer function inputs.)

zspNone or float, array-like, shape $(4)$ , optional

If $z s p is not N o n e$ , then $z s p$ must contain the four values used to control the strategy of the search procedure.

$z s p [0]$

Contains $α$ , the value used to constrain the magnitude of the search procedure steps.

$z s p [1]$

Contains $β$ , the multiplier which regulates the value of $α$ .

$z s p [2]$

Contains $δ$ , the value of the stationarity and invertibility test tolerance factor.

$z s p [3]$

Contains $γ$ , the value of the convergence criterion.

If $z s p is N o n e$ before entry, default values of $z s p$ are supplied by the function.

These are $0.01$ , $10.0$ , $1000.0$ and $m a x (100 \times machine precision, 0.0000001)$ , respectively.

kprivint, optional

Must not be set to $0$ , if it is required to monitor the course of the optimization. The course of the optimization is monitored by printing out at each iteration the iteration count ( $i t c$ ), the residual sum of squares ( $s$ ), the objective function ( $d$ ) and a description and value for each of the parameters in the $p a r a$ array. The descriptions are PHI for $ϕ$ , THETA for $θ$ , SPHI for $Φ$ , STHETA for $Θ$ , OMEGA/SI for $ω$ in a simple input, OMEGA for $ω$ in a transfer function input, DELTA for $δ$ and CONSTANT for $c$ . In addition SERIES 1, SERIES 2, etc. indicate the input series relevant to the OMEGA and DELTA parameters.

$k p r i v$ must be set to $0$ if the print-out of the above information is not required.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

parafloat, ndarray, shape $(npara)$

The latest values of the estimates of these parameters.

xxyfloat, ndarray, shape $(nxxy, nser)$

If $kzef = 0$ , $x x y$ remains unchanged on exit.

If $kzef \neq 0$ , the columns of $x x y$ hold the corresponding values of the input component series $z_{t}$ in place of $x_{t}$ and the output noise component $n_{t}$ in place of $y_{t}$ , in that order.

zspfloat, ndarray, shape $(4)$

Contains the values, default or otherwise, used by the function.

itcint

The number of iterations carried out.

$i t c = - 1$

Indicates that the only estimates obtained up to this point have been for the nuisance parameters relating to backforecasts, unless the marginal likelihood option is used, in which case estimates have also been obtained for simple input coefficients $ω$ and for the constant $c$ (if $k f c = 1$ ). This value of $i t c$ usually indicates a failure in a consequent step of estimating transfer function input pre-observation period nuisance parameters.

$i t c = 0$

Indicates that estimates have been obtained up to this point for the constant $c$ (if $k f c = 1$ ), for simple input coefficients $ω$ and for the nuisance parameters relating to the backforecasts and to transfer function input pre-observation period effects.

sdfloat, ndarray, shape $(npara)$

The $npara$ values of the standard deviations corresponding to each of the parameters in $p a r a$ . When the constant is fixed its standard deviation is returned as zero. When the values of $p a r a$ are valid, the values of $s d$ are usually also valid. However, if an exit value of $e r r n o$ = 3, 8 or 10, then the contents of $s d$ will be indeterminate.

cmfloat, ndarray, shape $(npara, npara)$

The first $npara$ rows and columns of $c m$ contain the correlation coefficients relating to each pair of parameters in $p a r a$ . All coefficients relating to the constant will be zero if the constant is fixed. The contents of $c m$ will be indeterminate under the same conditions as $s d$ .

sfloat

The residual sum of squares, $S$ , at the latest set of valid parameter estimates.

dfloat

The objective function, $D$ , at the latest set of valid parameter estimates.

ndfint

The number of degrees of freedom associated with $S$ .

resfloat, ndarray, shape $(nxxy)$

The values of the residuals relating to the differenced values of the output series. The remainder of the first $nxxy$ terms in the array will be zero.

sttffloat, ndarray, shape $(n s t t f)$

The $n s t t f$ values of the state set array.

nsttfint

The number of values in the state set array $s t t f$ .

Raises

NagValueError

(errno $1$ )

On entry, $nser = 1$ and there are no parameters in the model.

(errno $1$ )

On entry, $nser = ⟨ v a l u e ⟩$ .

Constraint: $nser \geq 1$ .

(errno $1$ )

On entry, $n i t = ⟨ v a l u e ⟩$ .

Constraint: $n i t \geq 0$ .

(errno $1$ )

On entry, $k e f = ⟨ v a l u e ⟩$ .

Constraint: $k e f = 1$ , $2$ or $3$ .

(errno $1$ )

On entry, $k f c = ⟨ v a l u e ⟩$ .

Constraint: $k f c = 0$ or $1$ .

(errno $2$ )

On entry, $n d f = ⟨ v a l u e ⟩$ .

Constraint: $n d f > 0$ .

(errno $2$ )

On entry, $npara = ⟨ v a l u e ⟩$ and $k f c = ⟨ v a l u e ⟩$ .

Constraint: $npara$ , $k f c$ , $m r$ and $m t$ must be consistent.

(errno $2$ )

On entry, $i = ⟨ v a l u e ⟩$ and $m t [3, i - 1] = ⟨ v a l u e ⟩$ .

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

(errno $2$ )

The orders vector $m r$ is invalid.

(errno $3$ )

On entry, or during execution, one or more sets of delta parameters do not satisfy the stationarity or invertibility conditions.

(errno $4$ )

On entry, $z s p [3] = ⟨ v a l u e ⟩$ .

Constraint: $0.0 \leq z s p [3] < 1.0$ .

(errno $4$ )

On entry, $z s p [2] = ⟨ v a l u e ⟩$ .

Constraint: $z s p [2] \geq 1.0$ .

(errno $4$ )

On entry, $z s p [1] = ⟨ v a l u e ⟩$ .

Constraint: $z s p [1] > 1.0$ .

(errno $4$ )

On entry, $z s p [0] = ⟨ v a l u e ⟩$ .

Constraint: $z s p [0] > 0.0$ .

(errno $8$ )

Unable to calculate the latest parameter estimates.

(errno $9$ )

Failure in inversion of second derivative matrix.

(errno $10$ )

On entry, or during execution, one or more sets of ARIMA parameters do not satisfy the stationarity or invertibility conditions.

(errno $11$ )

On entry, $isttf$ too small.

Constraint: $isttf \geq (P \times s) + d + (D \times s) + q + m a x (p, Q \times s) + ncg$ .

(errno $12$ )

The routine has failed to converge after $⟨ v a l u e ⟩$ iterations.

(errno $13$ )

On entry, $n i t = ⟨ v a l u e ⟩$ .

$isttf$ is too small and the process failed to converge after $n i t$ iterations.

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.

The Multi-input Model

The output series $y_{t}$ , for $t = 1, 2, \dots, n$ , is assumed to be the sum of (unobserved) components $z_{i, t}$ which are due respectively to the inputs $x_{i, t}$ , for $i = 1, 2, \dots, m$ .

Thus $y_{t} = z_{1, t} + \dots + z_{m, t} + n_{t}$ where $n_{t}$ is the error, or output noise component.

A typical component $z_{t}$ may be either

a simple regression component, $z_{t} = ω x_{t}$ (here $x_{t}$ is called a simple input), or
a transfer function model component which allows for the effect of lagged values of the variable, related to $x_{t}$ by

$z_{t} = δ_{1} z_{t - 1} + δ_{2} z_{t - 2} + \dots + δ_{p} z_{t - p} + ω_{0} x_{t - b} - ω_{1} x_{t - b - 1} - \dots - ω_{q} x_{t - b - q} .$

The noise $n_{t}$ is assumed to follow a (possibly seasonal) ARIMA model, i.e., may be represented in terms of an uncorrelated series, $a_{t}$ , by the hierarchy of equations

$\nabla^{d} \nabla_{s}^{D} n_{t} = c + w_{t}$
$w_{t} = Φ_{1} w_{t - s} + Φ_{2} w_{t - 2 \times s} + \dots + Φ_{P} w_{t - P \times s} + e_{t} - Θ_{1} e_{t - s} - Θ_{2} e_{t - 2 \times s} - \dots - Θ_{Q} e_{t - Q \times s}$
$e_{t} = ϕ_{1} e_{t - 1} + ϕ_{2} e_{t - 2} + \dots + ϕ_{p} e_{t - p} + a_{t} - θ_{1} a_{t - 1} - θ_{2} a_{t - 2} - \dots - θ_{q} a_{t - q}$

as outlined in Notes for uni_arima_estim.

Note: the orders $p, q$ appearing in each of the transfer function models and the ARIMA model are not necessarily the same; $\nabla^{d} \nabla_{s}^{D} n_{t}$ is the result of applying non-seasonal differencing of order $d$ and seasonal differencing of seasonality $s$ and order $D$ to the series $n_{t}$ : the differenced series is then of length $N = n - d - s \times D$ ; the constant term parameter $c$ may optionally be held fixed at its initial value (usually, but not necessarily zero) rather than being estimated.

For the purpose of defining an estimation criterion it is assumed that the series $a_{t}$ is a sequence of independent Normal variates having mean $0$ and variance $σ_{a}^{2}$ . An allowance has to be made for the effects of unobserved data prior to the observation period. For the noise component an allowance is always made using a form of backforecasting.

For each transfer function input, you have to decide what values are to be assumed for the pre-period terms $z_{0}, z_{- 1}, \dots, z_{1 - p}$ and $x_{0}, x_{- 1}, \dots, x_{1 - b - q}$ which are in theory necessary to re-create the component series $z_{1}, z_{2}, \dots, z_{n}$ , during the estimation procedure.

The first choice is to assume that all these values are zero. In this case, in order to avoid undesirable transient distortion of the early values $z_{1}, z_{2}, \dots$ , you are advised first to correct the input series $x_{t}$ by subtracting from all the terms a suitable constant to make the early values $x_{1}, x_{2}, \dots$ , close to zero. The series mean $¯ x$ is one possibility, but for a series with strong trend the constant might be simply $x_{1}$ .

The second choice is to treat the unknown pre-period terms as nuisance parameters and estimate them along with the other parameters. This choice should be used with caution. For example, if $p = 1$ and $b = q = 0$ , it is equivalent to fitting to the data a decaying geometric curve of the form $A δ^{t}$ , for $t = 1, 2, \dots,$ , along with the other inputs, this being the form of the transient. If the output $y_{t}$ contains a strong trend of this form, which is not otherwise represented in the model, it will have a tendency to influence the estimate of $δ$ away from the value appropriate to the transfer function model.

In most applications the first choice should be adequate, with the option possibly being used as a refinement at the end of the modelling process. The number of nuisance parameters is then $m a x (p, b + q)$ , with a corresponding loss of degrees of freedom in the residuals. If you align the input $x_{t}$ with the output by using in its place the shifted series $x_{t - b}$ , then setting $b = 0$ in the transfer function model, there is some improvement in efficiency. On some occasions when the model contains two or more inputs, each with estimation of pre-period nuisance parameters, these parameters may be co-linear and lead to failure of the function. The option must then be ‘switched off’ for one or more inputs.

The Estimation Criterion

This is a measure of how well a proposed set of parameters in the transfer function and noise ARIMA models matches the data. The estimation function searches for parameter values which minimize this criterion. For a proposed set of parameter values it is derived by calculating

the components $z_{1, t}, z_{2, t}, \dots, z_{m, t}$ as the responses to the input series $x_{1, t}, x_{2, t} \dots, x_{m, t}$ using the equations (a) or (b) above,
the discrepancy between the output and the sum of these components, as the noise

$n_{t} = y_{t} - (z_{1, t} + z_{2, t} + \dots + z_{m, t}),$
the residual series $a_{t}$ from $n_{t}$ by reversing the recursive equations (i), (ii) and (iii) above.

This last step again requires treatment of the effect of unknown pre-period values of $n_{t}$ and other terms in the equations regenerating $a_{t}$ . This is identical to the treatment given in Notes for uni_arima_estim, and leads to a criterion which is a sum of squares function $S$ , of the residuals $a_{t}$ . It may be shown that the finite algorithm presented there is equivalent to taking the infinite set of past values $n_{0}, n_{- 1}, n_{- 2}, \dots$ , as (linear) nuisance parameters. There is no loss of degrees of freedom however, because the sum of squares function $S$ may be expressed as including the corresponding set of past residuals; see page 273 of Box and Jenkins (1976), who prove that

S = n \sum - \infty a_{t}^{2} .

The function $D = S$ is the first of the three possible criteria, and is quite adequate for moderate to long series with no seasonal parameters. The second is the exact likelihood criterion which considers the past set $n_{0}, n_{- 1}, n_{- 2}$ not as simple nuisance parameters, but as unobserved random variables with known distribution. Calculation of the likelihood of the observed set $n_{1}, n_{2}, \dots, n_{n}$ requires theoretical integration over the range of the past set. Fortunately this yields a criterion of the form $D = M \times S$ (whose minimization is equivalent to maximizing the exact likelihood of the data), where $S$ is exactly as before, and the multiplier $M$ is a function calculated from the ARIMA model parameters. The value of $M$ is always $\geq 1$ , and $M$ tends to $1$ for any fixed parameter set as the sample size $n$ tends to $\infty$ . There is a moderate computational overhead in using this option, but its use avoids appreciable bias in the ARIMA model parameters and yields a better conditioned estimation problem.

The third criterion of marginal likelihood treats the coefficients of the simple inputs in a manner analogous to that given to the past set $n_{0}, n_{- 1}, n_{- 2}, \dots$ . These coefficients, together with the constant term $c$ used to represent the mean of $w_{t}$ , are in effect treated as random variables with highly dispersed distributions. This leads to the criterion $D = M \times S$ again, but with a different value of $M$ which now depends on the simple input series values $x_{t}$ . In the presence of a moderate to large number of simple inputs, the marginal likelihood criterion can counteract bias in the ARIMA model parameters which is caused by estimation of the simple inputs. This is particularly important in relatively short series.

multi_inputmod_estim can be used with no input series present, to estimate a univariate ARIMA model for the output alone. The marginal likelihood criterion is then distinct from exact likelihood only if a constant term is being estimated in the model, because this is treated as an implicit simple input.

The Estimation Procedure

This is the minimization of the estimation criterion or objective function $D$ (for deviance). The function uses an extension of the algorithm of Marquardt (1963). The step size in the minimization is inversely related to a parameter $α$ , which is increased or decreased by a factor $β$ at successive iterations, depending on the progress of the minimization. Convergence is deemed to have occurred if the fractional reduction of $D$ in successive iterations is less than a value $γ$ , while $α < 1$ .

Certain model parameters (in fact all excluding the $ω$ s) are subject to stability constraints which are checked throughout to within a specified tolerance multiple $δ$ of machine accuracy. Using the least squares criterion, the minimization may halt prematurely when some parameters ‘stick’ at a constraint boundary. This can happen particularly with short seasonal series (with a small number of whole seasons). It will not happen using the exact likelihood criterion, although convergence to a point on the boundary may sometimes be rather slow, because the criterion function may be very flat in such a region. There is also a smaller risk of a premature halt at a constraint boundary when marginal likelihood is used.

A positive, or zero number of iterations can be specified. In either case, the value $D$ of the objective function at iteration zero is presented at the initial parameter values, except for estimation of any pre-period terms for the input series, backforecasts for the noise series, and the coefficients of any simple inputs, and the constant term (unless this is held fixed).

At any later iteration, the value of $D$ is computed after re-estimation of the backforecasts to their optimal values, corresponding to the model parameters presented at that iteration. This is not true for any pre-period terms for the input series which, although they are updated from the previous iteration, may not be precisely optimal for the parameter values presented, unless convergence of those parameters has occurred. However, in the case of marginal likelihood being specified, the coefficients of the simple inputs and the constant term are also re-estimated together with the backforecasts at each iteration, to values which are optimal for the other parameter values presented.

Further Results

The residual variance is taken as $erv = \frac{S}{df}$ , where $df = N -$ (total number of parameters estimated), is the residual degrees of freedom. The pre-period nuisance parameters for the input series are included in the reduction of $df$ , as is the constant if it is estimated.

The covariance matrix of the vector of model parameter estimates is given by

erv \times H^{- 1}

where $H$ is the linearized least squares matrix taken from the final iteration of the algorithm of Marquardt. From this expression are derived the vector of standard deviations, and the correlation matrix of parameter estimates. These are approximations which are only valid asymptotically, and must be treated with great caution when the parameter estimates are close to their constraint boundaries.

The residual series $a_{t}$ is available upon completion of the iterations over the range $t = 1 + d + s \times D, \dots, n$ corresponding to the differenced noise series $w_{t}$ .

Because of the algorithm used for backforecasting, these are only true residuals for $t \geq 1 + q + s \times Q - p - s \times P - d - s \times D$ , provided this is positive. Estimation of pre-period terms for the inputs will also tend to reduce the magnitude of the early residuals, sometimes severely.

The model component series $z_{1, t}, \dots, z_{m, t}$ and $n_{t}$ may optionally be returned in place of the supplied series values, in order to assess the effects of the various inputs on the output.

Forecasting Information

For the purpose of constructing forecasts of the output series at future time points $t = n + 1, n + 2, \dots$ using multi_inputmod_forecast_state(), it is not necessary to use the whole set of observations $y_{t}$ and $x_{1, t}, x_{2, t}, \dots, x_{m, t}$ , for $t = 1, 2, \dots, m$ . It is sufficient to retain a limited set of quantities constituting the ‘state set’ as follows: for each series which appears with lagged subscripts in equations (a), (b), (i), (ii) and (iii) above, include the values at times $n + 1 - k$ for $k = 1$ up to the maximum lag associated with that series in the equations. Note that (i) implicitly includes past values of $n_{t}$ and intermediate differences of $n_{t}$ such as $\nabla^{d - 1} \nabla_{s}^{D}$ .

If later observations of the series become available, it is possible to update the state set (without re-estimating the model) using multi_inputmod_update(). If time series data is supplied with a previously estimated model, it is possible to construct the state set (and forecasts) using multi_inputmod_forecast().

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

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naginterfaces.library.tsa.multi_inputmod_estim¶

naginterfaces.library.tsa.multi_​inputmod_​estim¶

naginterfaces.library.tsa.multi_inputmod_estim¶