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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_tsa_multi_filter_arima (g13ba)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_tsa_multi_filter_arima (g13ba) filters a time series by an ARIMA model.

Syntax

[b, ifail] = g13ba(y, mr, par, cy, nb, 'ny', ny, 'nmr', nmr, 'npar', npar)
[b, ifail] = nag_tsa_multi_filter_arima(y, mr, par, cy, nb, 'ny', ny, 'nmr', nmr, 'npar', npar)

Description

From a given series y1,y2,…,yn, a new series b1,b2,…,bn is calculated using a supplied (filtering) ARIMA model. This model will be one which has previously been fitted to a series xt with residuals at. The equations defining bt in terms of yt are very similar to those by which at is obtained from xt. The only dissimilarity is that no constant correction is applied after differencing. This is because the series yt is generally distinct from the series xt with which the model is associated, though yt may be related to xt. Whilst it is appropriate to apply the ARIMA model to yt so as to preserve the same relationship between bt and at as exists between yt and xt, the constant term in the ARIMA model is inappropriate for yt. The consequence is that bt will not necessarily have zero mean.
The equations are precisely:
wt=∇d∇sDyt, (1)
the appropriate differencing of yt; both the seasonal and non-seasonal inverted autoregressive operations are then applied,
ut=wt-Φ1wt-s-⋯-ΦPwt-s×P (2)
vt=ut-ϕ1ut-1-⋯-ϕput-p (3)
followed by the inverted moving average operations
zt=vt+Θ1zt-s+⋯+ΘQzt-s×Q (4)
bt=zt+θ1bt-1+⋯+θqbt-q. (5)
Because the filtered series value bt depends on present and past values yt,yt-1,…, there is a problem arising from ignorance of y0,y-1,… which particularly affects calculation of the early values b1,b2,…, causing ‘transient errors’. The function allows two possibilities.
(i) The equations (1), (2) and (3) are applied from successively later time points so that all terms on their right-hand sides are known, with vt being defined for t=1+d+s×D+s×P,…,n. Equations (4) and (5) are then applied over the same range, taking any values on the right-hand side associated with previous time points to be zero.
This procedure may still however result in unacceptably large transient errors in early values of bt.
(ii) The unknown values y0,y-1,… are estimated by backforecasting. This requires that an ARIMA model distinct from that which has been supplied for filtering, should have been previously fitted to yt.
For efficiency, you are asked to supply both this ARIMA model for yt and a limited number of backforecasts which are prefixed to the known values of yt. Within the function further backforecasts of yt, and the series wt, ut, vt in (1), (2) and (3) are then easily calculated, and a set of linear equations solved for backforecasts of zt,bt for use in (4) and (5) in the case that q+Q>0.
Even if the best model for yt is not available, a very approximate guess such as
yt=c+et  
or
∇ yt=et  
can help to reduce the transients substantially.
The backforecasts which need to be prefixed to yt are of length Qy′=qy+sy×Qy, where qy and Qy are the non-seasonal and seasonal moving average orders and sy the seasonal period for the ARIMA model of yt. Thus you need not carry out the backforecasting exercise if Qy′=0. Otherwise, the series y1,y2,…,yn should be reversed to obtain yn,yn-1,…,y1 and nag_tsa_uni_arima_forcecast (g13aj) should be used to forecast Qy′ values, y^0,…,y^1-Qy′. The ARIMA model used is that fitted to yt (as a forward series) except that, if dy+Dy is odd, the constant should be changed in sign (to allow, for example, for the fact that a forward upward trend is a reversed downward trend). The ARIMA model for yt supplied to the filtering function must however have the appropriate constant for the forward series.
The series y^1-Qy′,…,y^0,y1,…,yn is then supplied to the function, and a corresponding set of values returned for bt.

References

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

Parameters

Compulsory Input Parameters

1:     yny – double array
The Qy′ backforecasts, starting with backforecast at time 1-Qy′ to backforecast at time 0, followed by the time series starting at time 1, where Qy′=mr10+mr13×mr14. If there are no backforecasts, either because the ARIMA model for the time series is not known, or because it is known but has no moving average terms, then the time series starts at the beginning of y.
2:     mrnmr – int64int32nag_int array
The orders vector for the filtering model, followed by the orders vector for the ARIMA model for the time series if the latter is known. The orders appear in the standard sequence p,d,q,P,D,Q,s as given in the G13 Chapter Introduction. If the ARIMA model for the time series is supplied, then the function will assume that the first Qy′ values of the array y are backforecasts.
Constraints:
the filtering model is restricted in the following ways:
  • mr1+mr3+mr4+mr6>0, i.e., filtering by a model which contains only differencing terms is not permitted;
  • mrk≥0, for k=1,2,…,7;
  • if mr7=0, mr4+mr5+mr6=0;
  • if mr7≠0, mr4+mr5+mr6≠0;
  • mr7≠1.
the ARIMA model for the time series is restricted in the following ways:
  • mrk≥0, for k=8,9,…,14;
  • if mr14=0, mr11+mr12+mr13=0;
  • if mr14≠0, mr11+mr12+mr13≠0;
  • mr14≠1.
3:     parnpar – double array
The parameters of the filtering model, followed by the parameters of the ARIMA model for the time series, if supplied. Within each model the parameters are in the standard order of non-seasonal AR and MA followed by seasonal AR and MA.
4:     cy – double scalar
If the ARIMA model is known (i.e., nmr=14), cy must specify the constant term of the ARIMA model for the time series. If this model is not known (i.e., nmr=7), then cy is not used.
5:     nb – int64int32nag_int scalar
The dimension of the array b. in addition to holding the returned filtered series, b is also used as an intermediate work array if the ARIMA model for the time series was known.
Constraints:
  • if nmr=14, nb≥ny+maxK3,K1+K2;
  • if nmr=7, nb≥ny.
Where
  • K1=mr1+mr4×mr7;
  • K2=mr2+mr5×mr7;
  • K3=mr3+mr6×mr7.

Optional Input Parameters

1:     ny – int64int32nag_int scalar
Default: the dimension of the array y.
The total number of backforecasts and time series data points in array y.
Constraint: ny≥max1+Qy′,npar.
2:     nmr – int64int32nag_int scalar
Default: the dimension of the array mr.
The number of values specified in the array mr. It takes the value 7 if no ARIMA model for the time series is supplied but otherwise it takes the value 14. Thus nmr acts as an indicator as to whether backforecasting can be carried out.
Constraint: nmr=7 or 14.
3:     npar – int64int32nag_int scalar
Default: the dimension of the array par.
The total number of parameters held in array par.
Constraints:
  • if nmr=7, npar=mr1+mr3+mr4+mr6;
  • if nmr=14, npar=mr1+mr3+mr4+mr6+ mr8+mr10+mr11+mr13.
Note: the first constraint (i.e., mr1+mr3+mr4+mr6>0) on the orders of the filtering model, in argument mr, ensures that npar>0.

Output Parameters

1:     bnb – double array
The filtered output series. If the ARIMA model for the time series was known, and hence Qy′ backforecasts were supplied in y, then b contains Qy′ ‘filtered’ backforecasts followed by the filtered series. Otherwise, the filtered series begins at the start of b just as the original series began at the start of y. In either case, if the value of the series at time t is held in yt, then the filtered value at time t is held in bt.
2:     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,nmr≠7 and nmr≠14.
   ifail=2
On entry,the orders vector mr does not satisfy the constraints given in Arguments.
   ifail=3
On entry,npar is inconsistent with the contents of mr (see Arguments).
   ifail=4
On entry,ny is too small to successfully carry out the requested filtering, (see Arguments).
   ifail=5
On entry,the work array wa is too small.
   ifail=6
On entry,the array b is too small.
   ifail=7
The orders vector for the filtering model is invalid.
   ifail=8
The orders vector for the ARIMA model is invalid. (Only occurs if nmr=14.)
   ifail=9
The initial values of the filtered series are indeterminate for the given models.
   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

Accuracy and stability are high except when the MA parameters are close to the invertibility boundary.

Further Comments

If an ARIMA model is supplied, a local workspace array of fixed length is allocated internally by nag_tsa_multi_filter_arima (g13ba). The total size of this array amounts to K integer elements, where K is the expression defined in the description of the argument wa.
The time taken by nag_tsa_multi_filter_arima (g13ba) is approximately proportional to
ny×mr1+mr3+mr4+mr6,  
with an appreciable fixed increase if an ARIMA model is supplied for the time series.

Example

This example reads a time series of length 296. It reads the univariate ARIMA 4,0,2,0,0,0,0 model and the ARIMA filtering 3,0,0,0,0,0,0 model for the series. Two initial backforecasts are required and these are calculated by a call to nag_tsa_uni_arima_forcecast (g13aj) . The backforecasts are inserted at the start of the series and nag_tsa_multi_filter_arima (g13ba) is called to perform the calculations.

Open in the MATLAB editor: g13ba_example

function g13ba_example


fprintf('g13ba example results\n\n');

% orders
mrx = [int64(4);0;2;0;0;0;0];

% Number of backforecasts
nbf = 2;

% data
nx = 296;
y  = zeros(nx+nbf,1);
y(nbf+1:nbf+nx) = ...
  [  53.8; 53.6; 53.5; 53.5; 53.4; 53.1; 52.7; 52.4; 52.2; 52.0; 52.0;
     52.4; 53.0; 54.0; 54.9; 56.0; 56.8; 56.8; 56.4; 55.7; 55.0; 54.3;
     53.2; 52.3; 51.6; 51.2; 50.8; 50.5; 50.0; 49.2; 48.4; 47.9; 47.6;
     47.5; 47.5; 47.6; 48.1; 49.0; 50.0; 51.1; 51.8; 51.9; 51.7; 51.2;
     50.0; 48.3; 47.0; 45.8; 45.6; 46.0; 46.9; 47.8; 48.2; 48.3; 47.9;
     47.2; 47.2; 48.1; 49.4; 50.6; 51.5; 51.6; 51.2; 50.5; 50.1; 49.8;
     49.6; 49.4; 49.3; 49.2; 49.3; 49.7; 50.3; 51.3; 52.8; 54.4; 56.0;
     56.9; 57.5; 57.3; 56.6; 56.0; 55.4; 55.4; 56.4; 57.2; 58.0; 58.4;
     58.4; 58.1; 57.7; 57.0; 56.0; 54.7; 53.2; 52.1; 51.6; 51.0; 50.5;
     50.4; 51.0; 51.8; 52.4; 53.0; 53.4; 53.6; 53.7; 53.8; 53.8; 53.8;
     53.3; 53.0; 52.9; 53.4; 54.6; 56.4; 58.0; 59.4; 60.2; 60.0; 59.4;
     58.4; 57.6; 56.9; 56.4; 56.0; 55.7; 55.3; 55.0; 54.4; 53.7; 52.8;
     51.6; 50.6; 49.4; 48.8; 48.5; 48.7; 49.2; 49.8; 50.4; 50.7; 50.9;
     50.7; 50.5; 50.4; 50.2; 50.4; 51.2; 52.3; 53.2; 53.9; 54.1; 54.0;
     53.6; 53.2; 53.0; 52.8; 52.3; 51.9; 51.6; 51.6; 51.4; 51.2; 50.7;
     50.0; 49.4; 49.3; 49.7; 50.6; 51.8; 53.0; 54.0; 55.3; 55.9; 55.9;
     54.6; 53.5; 52.4; 52.1; 52.3; 53.0; 53.8; 54.6; 55.4; 55.9; 55.9;
     55.2; 54.4; 53.7; 53.6; 53.6; 53.2; 52.5; 52.0; 51.4; 51.0; 50.9;
     52.4; 53.5; 55.6; 58.0; 59.5; 60.0; 60.4; 60.5; 60.2; 59.7; 59.0;
     57.6; 56.4; 55.2; 54.5; 54.1; 54.1; 54.4; 55.5; 56.2; 57.0; 57.3;
     57.4; 57.0; 56.4; 55.9; 55.5; 55.3; 55.2; 55.4; 56.0; 56.5; 57.1;
     57.3; 56.8; 55.6; 55.0; 54.1; 54.3; 55.3; 56.4; 57.2; 57.8; 58.3;
     58.6; 58.8; 58.8; 58.6; 58.0; 57.4; 57.0; 56.4; 56.3; 56.4; 56.4;
     56.0; 55.2; 54.0; 53.0; 52.0; 51.6; 51.6; 51.1; 50.4; 50.0; 50.0;
     52.0; 54.0; 55.1; 54.5; 52.8; 51.4; 50.8; 51.2; 52.0; 52.8; 53.8;
     54.5; 54.9; 54.9; 54.8; 54.4; 53.7; 53.3; 52.8; 52.6; 52.6; 53.0;
     54.3; 56.0; 57.0; 58.0; 58.6; 58.5; 58.3; 57.8; 57.3; 57.0];

% Parameters
parx = [2.42;     -2.38;     1.16;     -0.23;     0.31;     -0.47];

% Get back forecasts
x(nx:-1:1) = y(nbf+1:nbf+nx);

kfc = int64(1);
cx  = 0;

% Problem sizes
ist = int64(6);
ifv = int64(nbf);

% Apply ARIMA model
[rms, st, nst, fva, fsd, isf, ifail] = ...
  g13aj( ...
         mrx, parx, cx, kfc, x, ist, ifv, ifv);

% Put back forecasts at start of y
y(1:nbf) = fva(nbf:-1:1);

% Add filter model orders and params to start
mr  = [int64(3); 0; 0; 0; 0; 0; 0; mrx];
par = [1.97;     -1.37;     0.34; parx];
cy  = cx;

% Filter series
ny = nx + nbf;
nb = int64(ny+4);
[b, ifail] = g13ba( ...
                    y, mr, par, cy, nb);

% Display results
fprintf('                 Original        Filtered\n');
fprintf('Backforecasts    y-series         series\n');
ival = [-nbf:-1]';
fprintf('%8d%17.4f%15.4f\n', [ival y(1:nbf) b(1:nbf)]');
fprintf('\n%16s%16s%16s%16s\n','Filtered','Filtered','Filtered','Filtered');
fprintf('%15s%16s%16s%16s\n',  'series',  'series',  'series',  'series');
ivar = [1:nx]';
result = [ivar b(nbf+1:ny)];
for j = 1:4:nx
  fprintf('%7d%9.4f', result(j:min(j+3,nx),:)');
  fprintf('\n');
end



g13ba example results

                 Original        Filtered
Backforecasts    y-series         series
      -2          49.9807         3.4222
      -1          52.6714         3.0809

        Filtered        Filtered        Filtered        Filtered
         series          series          series          series
      1   2.9813      2   2.7803      3   3.7057      4   3.2450
      5   3.0760      6   3.0070      7   3.0610      8   3.1720
      9   3.1170     10   3.0360     11   3.2580     12   3.4520
     13   3.3320     14   3.6980     15   3.3140     16   3.8070
     17   3.3330     18   2.9580     19   3.2800     20   3.0960
     21   3.2270     22   3.0830     23   2.6410     24   3.1870
     25   2.9910     26   3.1110     27   2.8460     28   3.0240
     29   2.7030     30   2.6130     31   2.8060     32   2.9560
     33   2.8170     34   2.8950     35   2.8510     36   2.9160
     37   3.2530     38   3.3050     39   3.1830     40   3.3760
     41   2.9730     42   2.8610     43   3.0490     44   2.8420
     45   2.3190     46   2.3660     47   2.9410     48   2.3810
     49   3.3420     50   2.9340     51   3.1800     52   2.9230
     53   2.6470     54   2.8860     55   2.5310     56   2.6200
     57   3.4170     58   3.4940     59   3.2590     60   3.1310
     61   3.1420     62   2.6710     63   2.8990     64   2.8180
     65   3.2150     66   2.8800     67   2.9610     68   2.8800
     69   3.0020     70   2.8930     71   3.1210     72   3.2210
     73   3.2040     74   3.5360     75   3.7520     76   3.5630
     77   3.7260     78   3.1560     79   3.6310     80   2.9380
     81   3.1480     82   3.4490     83   3.1400     84   3.7380
     85   4.1200     86   3.1540     87   3.7480     88   3.3280
     89   3.3640     90   3.3400     91   3.3950     92   3.0720
     93   3.0050     94   2.8520     95   2.7810     96   3.1950
     97   3.2490     98   2.6370     99   3.0080    100   3.2410
    101   3.5570    102   3.2080    103   3.0880    104   3.3980
    105   3.1660    106   3.1960    107   3.2460    108   3.2870
    109   3.1590    110   3.2620    111   2.7280    112   3.4130
    113   3.2190    114   3.6750    115   3.8550    116   4.0100
    117   3.5380    118   3.8440    119   3.4660    120   3.0640
    121   3.4780    122   3.1140    123   3.5300    124   3.2400
    125   3.3630    126   3.2610    127   3.3020    128   3.1150
    129   3.3280    130   2.8730    131   3.0800    132   2.8390
    133   2.6570    134   3.0260    135   2.4580    136   3.2600
    137   2.8380    138   3.2150    139   3.1140    140   3.1050
    141   3.1400    142   2.9100    143   3.1370    144   2.7500
    145   3.1160    146   3.0680    147   2.8590    148   3.3840
    149   3.5500    150   3.4160    151   3.1770    152   3.3390
    153   3.0190    154   3.1780    155   3.0110    156   3.1940
    157   3.2680    158   3.0500    159   2.8060    160   3.1850
    161   3.0560    162   3.2690    163   2.7940    164   3.0900
    165   2.7100    166   2.7890    167   2.9510    168   3.2440
    169   3.2570    170   3.4360    171   3.4450    172   3.3780
    173   3.3520    174   3.9180    175   2.9190    176   3.1780
    177   2.2580    178   3.5150    179   2.8010    180   3.6030
    181   3.2610    182   3.5300    183   3.3270    184   3.4420
    185   3.5240    186   3.2720    187   3.1110    188   2.8240
    189   3.2330    190   3.1500    191   3.5710    192   3.0810
    193   2.7820    194   2.9040    195   3.2350    196   2.7970
    197   3.1320    198   3.1680    199   4.5210    200   2.6650
    201   4.6870    202   3.9470    203   3.2220    204   3.3410
    205   3.9950    206   3.4820    207   3.3630    208   3.4550
    209   3.2950    210   2.6910    211   3.4600    212   2.9440
    213   3.4400    214   3.1830    215   3.4200    216   3.4100
    217   4.0550    218   2.9990    219   3.8250    220   3.1340
    221   3.5010    222   3.0430    223   3.2660    224   3.3660
    225   3.2650    226   3.3720    227   3.2880    228   3.5470
    229   3.6840    230   3.3100    231   3.6790    232   3.1780
    233   2.9360    234   2.7910    235   3.8020    236   2.6100
    237   4.1690    238   3.7460    239   3.4560    240   3.3910
    241   3.5820    242   3.6220    243   3.4870    244   3.5770
    245   3.4240    246   3.3960    247   3.1220    248   3.4300
    249   3.4580    250   3.0280    251   3.7660    252   3.3770
    253   3.2470    254   3.0180    255   2.9720    256   2.8000
    257   3.2040    258   2.8020    259   3.4100    260   3.1680
    261   2.4600    262   2.8810    263   3.1750    264   3.1740
    265   4.8640    266   3.0600    267   2.9600    268   2.2530
    269   2.5620    270   3.3150    271   3.3480    272   3.5900
    273   3.2560    274   3.2320    275   3.6160    276   3.1700
    277   3.2890    278   3.1200    279   3.3300    280   2.9910
    281   2.9420    282   3.4070    283   2.8720    284   3.3470
    285   3.1920    286   3.4880    287   4.0680    288   3.7550
    289   3.0510    290   3.9680    291   3.3900    292   3.1380
    293   3.6170    294   3.1700    295   3.4150    296   3.4830
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