nag_tsa_uni_arima_forecast_state (g13ah) produces forecasts of a time series, given a time series model which has already been fitted to the time series using nag_tsa_uni_arima_estim (g13ae) or nag_tsa_uni_arima_estim_easy (g13af). The original observations are not required, since nag_tsa_uni_arima_forecast_state (g13ah) uses as input either the original state set produced by nag_tsa_uni_arima_estim (g13ae) or nag_tsa_uni_arima_estim_easy (g13af) or the state set updated by a series of new observations using nag_tsa_uni_arima_update (g13ag). Standard errors of the forecasts are also provided.

Syntax

[fva, fsd, ifail] = g13ah(st, mr, par, c, rms, nfv, 'nst', nst, 'npar', npar)

[fva, fsd, ifail] = nag_tsa_uni_arima_forecast_state(st, mr, par, c, rms, nfv, 'nst', nst, 'npar', npar)

Description

The original time series is

x_{t}

, for

t = 1, 2, \dots, n

and parameters have been fitted to the model of this time series using nag_tsa_uni_arima_estim (g13ae) or nag_tsa_uni_arima_estim_easy (g13af).

Forecasts of

x_{t}

, for

t = n + 1, \dots, n + L

, are calculated in five stages, as follows:

(i)	set $a_{t} = 0$ for $t = N + 1, N + 2, \dots, N + L$ , where $N = n - d - (D \times s)$ is the number of differenced values in the series;
(ii)	calculate the values of $e_{t}$ , for $t = N + 1, \dots, N + L$ , and $e_{t} = ϕ_{1} \times e_{t - 1} + \dots + ϕ_{p} \times e_{t - p} + a_{t} - θ_{1} \times a_{t - 1} - \dots - θ_{q} \times a_{t - q}$ ;
(iii)	calculate the values of $w_{t}$ , for $t = N + 1, \dots, N + L$ , where $w_{t} = Φ_{1} \times w_{t - s} + \dots + Φ_{P} \times w_{t - s \times P} + e_{t} - Θ_{1} \times e_{t - s} - \dots - Θ_{Q} \times e_{t - s \times Q}$ and $w_{t}$ for $t \leq N$ are the first $s \times P$ values in the state set, corrected for the constant;
(iv)	add the constant term $c$ to give the differenced series $\nabla^{d} \nabla_{s}^{D} x_{t} = w_{t} + c$ , for $t = N + 1, \dots, N + L$ ;
(v)	the differencing operations are reversed to reconstitute $x_{t}$ , for $t = n + 1, \dots, n + L$ .

The standard errors of these forecasts are given by

s_{t} = {[V \times (ψ_{0}^{2} + ψ_{1}^{2} + \dots + ψ_{t - n - 1}^{2})]}^{1 / 2}

, for

t = n + 1, \dots, n + L

, where

ψ_{0} = 1

V

is the residual variance of

a_{t}

, and

ψ_{j}

is the coefficient expressing the dependence of

x_{t}

a_{t - j}

To calculate

ψ_{j}

, for

j = 1, 2, \dots, (L - 1)

, the following device is used.

A copy of the state set is initialized to zero throughout and the calculations outlined above for the construction of forecasts are carried out with the settings

a_{N + 1} = 1

, and

a_{t} = 0

, for

t = N + 2, \dots, N + L

The resulting quantities corresponding to the sequence

x_{N + 1}, x_{N + 2}, \dots, x_{N + L}

are precisely

1

ψ_{1}, ψ_{2}, \dots, ψ_{L - 1}

The supplied time series model is used throughout these calculations, with the exception that the constant term

c

is taken to be zero.

References

None.

Parameters

Compulsory Input Parameters

1: $st (nst)$ – double array

The state set derived from nag_tsa_uni_arima_estim (g13ae) or nag_tsa_uni_arima_estim_easy (g13af) originally, or as modified using earlier calls of nag_tsa_uni_arima_update (g13ag).

2: $mr (7)$ – int64int32nag_int array

The orders vector

(p, d, q, P, D, Q, s)

of the ARIMA model, in the usual notation.

Constraints:

$p, d, q, P, D, Q, s \geq 0$ ;
$p + q + P + Q > 0$ ;
$s \neq 1$ ;
if $s = 0$ , $P + D + Q = 0$ ;
if $s > 1$ , $P + D + Q > 0$ .

3: $par (npar)$ – double array

The estimates of the

p

values of the

ϕ

parameters, the

q

values of the

θ

parameters, the

P

values of the

Φ

parameters and the

Q

values of the

Θ

parameters which specify the model and which were output originally by nag_tsa_uni_arima_estim (g13ae) or nag_tsa_uni_arima_estim_easy (g13af).

4: $c$ – double scalar

c

, the value of the model constant. This will have been output by nag_tsa_uni_arima_estim (g13ae) or nag_tsa_uni_arima_estim_easy (g13af).

5: $rms$ – double scalar

V

, the residual variance associated with the model.

If nag_tsa_uni_arima_estim_easy (g13af) was used to estimate the model, rms should be set to

s / ndf

, where s and ndf were output by nag_tsa_uni_arima_estim_easy (g13af).

If nag_tsa_uni_arima_estim (g13ae) was used to estimate the model, rms should be set to

s / icount (5)

, where s and

icount (5)

were output by nag_tsa_uni_arima_estim (g13ae).

Constraint:

rms \geq 0.0

6: $nfv$ – int64int32nag_int scalar

L

, the required number of forecasts.

Constraint:

nfv > 0

Optional Input Parameters

1: $nst$ – int64int32nag_int scalar: Default: the dimension of the array st.
The number of values in the state set array st.

Constraint: $nst = P \times s + D \times s + d + q + \max (p, Q \times s)$ . (As returned by nag_tsa_uni_arima_estim (g13ae) or nag_tsa_uni_arima_estim_easy (g13af)).
2: $npar$ – int64int32nag_int scalar: Default: the dimension of the array par.
The number of $ϕ$ , $θ$ , $Φ$ and $Θ$ parameters in the model.

Constraint: $npar = p + q + P + Q$ .

Output Parameters

1: $fva (nfv)$ – double array: nfv forecast values relating to the original undifferenced series.
2: $fsd (nfv)$ – double array: The standard errors associated with each of the nfv forecast values in fva.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$npar \neq p + q + P + Q$ ,
or	the orders vector mr is invalid (check the constraints given in Arguments).

$ifail = 2$

On entry,

nst \neq P \times s + D \times s + d + q + \max (Q \times s, p)

$ifail = 3$

On entry,

nfv \leq 0

$ifail = 4$

On entry,

nwa < 4 \times npar + 3 \times nst

$ifail = 5$

On entry,

rms < 0.0

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The computations are believed to be stable.

Further Comments

The time taken by nag_tsa_uni_arima_forecast_state (g13ah) is approximately proportional to

nfv \times npar

Example

The following program is based on the data derived in the example used to illustrate nag_tsa_uni_arima_update (g13ag).

These consist of a set of orders indicating that there are two moving average parameters (one non-seasonal, and one seasonal with periodicity

12

The model constant is zero.

The state set contains

26

values.

In addition the residual mean-square derived when the model was originally fitted is given.

Twelve forecasts and their associated errors are obtained.

Open in the MATLAB editor: g13ah_example

function g13ah_example


fprintf('g13ah example results\n\n');

% state set from g13ae, g13af or g13ag
st = [0.0660;    -0.0513;     0.1715;    -0.0249;     0.0588;
      0.1167;     0.1493;     0.0199;    -0.1884;    -0.1289;
     -0.1172;     0.1122;     6.0039;     0.0443;    -0.0070;
      0.0252;     0.0020;     0.0353;    -0.0460;     0.0374;
      0.0151;    -0.0237;     0.0031;     0.0188;     0.0066;
      0.0125];

% Orders
mr = [int64(0);1;1;0;1;1;12];

% Parameter estimates
par = [0.327; 0.6262];

% residual variance
rms = 0.0014;

c = 0;
nfv = int64(12);

% Produce forecasts
[fva, fsd, ifail] = g13ah( ...
                           st, mr, par, c, rms, nfv);

fprintf('The required %4d forecast values are as follows\n', nfv);
for j = 1:8:nfv
  fprintf('%8.4f', fva(j:min(j+7,nfv)));
  fprintf('\n');
end
fprintf('\nThe standard deviations corresponding to the forecasts are\n');
for j = 1:8:nfv
  fprintf('%8.4f', fsd(j:min(j+7,nfv)));
  fprintf('\n');
end

g13ah example results

The required   12 forecast values are as follows
  6.0381  5.9912  6.1469  6.1207  6.1574  6.3029  6.4288  6.4392
  6.2657  6.1348  6.0059  6.1139

The standard deviations corresponding to the forecasts are
  0.0374  0.0451  0.0517  0.0575  0.0627  0.0676  0.0721  0.0764
  0.0805  0.0843  0.0880  0.0915

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox