g13dk:: Time Series Analysis (NAG Toolbox)

Purpose

nag_tsa_multi_varma_update (g13dk) accepts a sequence of new observations in a multivariate time series and updates both the forecasts and the standard deviations of the forecast errors. A call to nag_tsa_multi_varma_forecast (g13dj) must be made prior to calling this function in order to calculate the elements of a reference vector together with a set of forecasts and their standard errors. On a successful exit from nag_tsa_multi_varma_update (g13dk) the reference vector is updated so that should future series values become available these forecasts may be updated by recalling nag_tsa_multi_varma_update (g13dk).

Syntax

Description

Let

Z_{t} = {(z_{1 t}, z_{2 t}, \dots, z_{k t})}^{T}

, for

t = 1, 2, \dots, n

, denote a

k

-dimensional time series for which forecasts of

{\hat{Z}}_{n + 1}, {\hat{Z}}_{n + 2}, \dots, {\hat{Z}}_{n + l_{\max}}

have been computed using nag_tsa_multi_varma_forecast (g13dj). Given

m

further observations

Z_{n + 1}, Z_{n + 2}, \dots, Z_{n + m}

, where

m < l_{\max}

, nag_tsa_multi_varma_update (g13dk) updates the forecasts of

Z_{n + m + 1}, Z_{n + m + 2}, \dots, Z_{n + l_{\max}}

and their corresponding standard errors.

nag_tsa_multi_varma_update (g13dk) uses a multivariate version of the procedure described in Box and Jenkins (1976). The forecasts are updated using the

ψ

weights, computed in nag_tsa_multi_varma_forecast (g13dj). If

Z_{t}^{*}

denotes the transformed value of

Z_{t}

and

{\hat{Z}}_{t}^{*} (l)

denotes the forecast of

Z_{t + l}^{*}

from time

t

with a lead of

l

(that is the forecast of

Z_{t + l}^{*}

given observations

Z_{t}^{*}, Z_{t - 1}^{*}, \dots

), then

{\hat{Z}}_{t + 1}^{*} (l) = τ + ψ_{l} ε_{t + 1} + ψ_{l + 1} ε_{t} + ψ_{l + 2} ε_{t - 1} + \dots

and

{\hat{Z}}_{t}^{*} (l + 1) = τ + ψ_{l + 1} ε_{t} + ψ_{l + 2} ε_{t - 1} + \dots

where

τ

is a constant vector of length

k

involving the differencing parameters and the mean vector

μ

. By subtraction we obtain

{\hat{Z}}_{t + 1}^{*} (l) = {\hat{Z}}_{t}^{*} (l + 1) + ψ_{l} ε_{t + 1} .

Estimates of the residuals corresponding to the new observations are also computed as

ε_{n + l} = Z_{n + l}^{*} - {\hat{Z}}_{n}^{*} (l)

, for

l = 1, 2, \dots, m

. These may be of use in checking that the new observations conform to the previously fitted model.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

If a further

m^{*}

observations,

Z_{n + mlast + 1}, Z_{n + mlast + 2}, \dots, Z_{n + mlast + m^{*}}

, become available, then forecasts of

Z_{n + mlast + m^{*} + 1}, Z_{n + mlast + m^{*} + 2}, \dots, Z_{n + l_{\max}}

may be updated by recalling nag_tsa_multi_varma_update (g13dk) with

m = m^{*}

. Note that m and the contents of the array z are the only quantities which need updating; mlast is updated on exit from the previous call. On a successful exit, v contains estimates of

ε_{n + mlast + 1}, ε_{n + mlast + 2}, \dots, ε_{n + mlast + m^{*}}

; columns

mlast + 1, mlast + 2, \dots, mlast + m^{*}

of predz contain the new observed values

Z_{n + mlast + 1}, Z_{n + mlast + 2}, \dots, Z_{n + mlast + m^{*}}

and columns

mlast + 1, mlast + 2, \dots, mlast + m^{*}

of sefz are set to zero.

Example

This example shows how to update the forecasts of two series each of length

48

. No transformation has been used and no differencing applied to either of the series. nag_tsa_multi_varma_estimate (g13dd) is first called to fit an AR(1) model to the series.

μ

is to be estimated and

ϕ_{1} (2, 1)

constrained to be zero. A call to nag_tsa_multi_varma_forecast (g13dj) is then made in order to compute forecasts of the next five series values. After one new observation becomes available the four forecasts are updated. A further observation becomes available and the three forecasts are updated.

function g13dk_example


fprintf('g13dk example results\n\n');

% Series
z = [-1.490 -1.620  5.200  6.230  6.210  5.860  4.090  3.180  ...
      2.620  1.490  1.170  0.850 -0.350  0.240  2.440  2.580  ...
      2.040  0.400  2.260  3.340  5.090  5.000  4.780  4.110  ...
      3.450  1.650  1.290  4.090  6.320  7.500  3.890  1.580  ...
      5.210  5.250  4.930  7.380  5.870  5.810  9.680  9.070  ...
      7.290  7.840  7.550  7.320  7.970  7.760  7.000  8.350;
      7.340  6.350  6.960  8.540  6.620  4.970  4.550  4.810  ...
      4.750  4.760 10.880 10.010 11.620 10.360  6.400  6.240  ...
      7.930  4.040  3.730  5.600  5.350  6.810  8.270  7.680  ...
      6.650  6.080 10.250  9.140 17.750 13.300  9.630  6.800  ...
      4.080  5.060  4.940  6.650  7.940 10.760 11.890  5.850  ...
      9.010  7.500 10.020 10.380  8.150  8.370 10.730 12.140];
[k,n] = size(z);

% Difference /transform series
tr    = {'N'; 'N'};
id    = [int64(0);0];
delta = [0;  0];
[w, nd, ifail] = g13dl( ...
			z, tr, id, delta);

% VARMA info
ip    = int64(1);
iq    = int64(0);
mean_p = true;

% Initial parameter estimates and free parameter flags
par    = zeros(6, 1);
parhld = [false;  false;  true;  false;  false;  false];

% Exact likelihood
exact  = true;
% control parameters
iprint = int64(-1);
cgetol = 0.0001;
ishow  = int64(0);

qq = [0, 0; 0, 0];

% Fit VARMA
[par, qq, ~, ~, v, ~, ~, ifail] = ...
  g13dd( ...
	 ip, iq, mean_p, par, qq, w, parhld, exact, iprint, cgetol, ...
	 ishow, 'n', nd);

% Perform forecast
lmax = int64(5);
lref = int64(150);
mean_p = 'M';
[qq, predz, sefz, ref, ifail] = ...
  g13dj( ...
	 z, tr, id, delta, ip, iq, mean_p, par, qq, v, lmax, lref);

% Display results
g13dk_print(k,n,lmax,predz,sefz);

% Update forecasts
m = int64([1; 1]);
z = [ 8.1  8.5;
     10.2 10.0];
mlast = int64(0);
for j = 1:numel(m)
  [mlast, ref, ~, predz, sefz, ifail] = ...
    g13dk( ...
	   int64(k), mlast, z(:,j), ref, predz, sefz);
  % Display results
  g13dk_print(k,n+mlast,lmax,predz,sefz);
end



function g13dk_print(k,n,lmax,predz,sefz)
  fprintf('\n Forecast Summary Table\n');
  fprintf(' ----------------------\n\n');
  fprintf(' Forecast origin is set at t = %4d\n\n', n);
  loop = lmax/5;
  if mod(lmax,5)~=0
    loop = loop + 1;
  end
  for j = 1:loop
    i2 = (j-1)*5;
    l2 = min(i2+5,lmax);
    fprintf('Lead Time %14s',' ');
    fprintf('%7d', [i2+1:l2]);
    fprintf('\n\n');
    for i = 1:k
      fprintf('Series %d : Forecast       ', i);
      fprintf('%7.2f',predz(i,i2+1:l2));
      fprintf('\n%8s : Standard Error ', ' ');
      fprintf('%7.2f',sefz(i,i2+1:l2));
      fprintf('\n');
    end
  end

g13dk example results


 Forecast Summary Table
 ----------------------

 Forecast origin is set at t =   48

Lead Time                     1      2      3      4      5

Series 1 : Forecast          7.82   7.28   6.77   6.33   5.95
         : Standard Error    1.72   2.23   2.51   2.68   2.79
Series 2 : Forecast         10.31   9.25   8.65   8.30   8.10
         : Standard Error    2.32   2.68   2.78   2.82   2.83

 Forecast Summary Table
 ----------------------

 Forecast origin is set at t =   49

Lead Time                     1      2      3      4      5

Series 1 : Forecast          8.10   7.49   6.94   6.46   6.06
         : Standard Error    0.00   1.72   2.23   2.51   2.68
Series 2 : Forecast         10.20   9.19   8.61   8.28   8.08
         : Standard Error    0.00   2.32   2.68   2.78   2.82

 Forecast Summary Table
 ----------------------

 Forecast origin is set at t =   50

Lead Time                     1      2      3      4      5

Series 1 : Forecast          8.10   8.50   7.80   7.18   6.65
         : Standard Error    0.00   0.00   1.72   2.23   2.51
Series 2 : Forecast         10.20  10.00   9.08   8.54   8.24
         : Standard Error    0.00   0.00   2.32   2.68   2.78

On entry,	$k < 1$ ,
or	$lmax < 2$ ,
or	$m \leq 0$ ,
or	$mlast + m \geq lmax$ ,
or	$mlast < 0$ ,
or	$kmax < k$ ,
or	$lref < (lmax - 1) \times k \times k + 2 \times k \times lmax + k$ .

NAG Toolbox: nag_tsa_multi_varma_update (g13dk)

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