NAG Toolbox: nag_tsa_inhom_iema_all (g13mf)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{z}\left(\mathbf{nb}\right)$ – double array

$z_{\mathit{i}}$, the current block of observations, for $\mathit{i}=k+1,\dots ,k+b$, where $k$ is the number of observations processed so far, i.e., the value supplied in pn on entry.

Constraint: if $\mathbf{ftype}=1$ or $2$ and $\mathbf{p}<0.0$, $\mathbf{z}\left(\mathit{i}\right)\ne 0$, for $\mathit{i}=1,2,\dots ,\mathbf{nb}$.

2:     $\mathrm{t}\left(\mathbf{nb}\right)$ – double array

$t_{\mathit{i}}$, the times for the current block of observations, for $\mathit{i}=k+1,\dots ,k+b$, where $k$ is the number of observations processed so far, i.e., the value supplied in pn on entry.

If $t_i\le t_{i-1}$, $\mathbf{ifail}=\mathbf{61}$ will be returned, but nag_tsa_inhom_iema_all (g13mf) will continue as if $t$ was strictly increasing by using the absolute value.

3:     $\mathrm{tau}$ – double scalar

$\tau$, the parameter controlling the rate of decay. $\tau$ must be sufficiently large that $e^{-\alpha }$, $\alpha =\left(t_i-t_{i-1}\right)/\tau$ can be calculated without overflowing, for all $i$.

Constraint: $\mathbf{tau}>0.0$.

4:     $\mathrm{m1}$ – int64int32nag_int scalar

The minimum number of times the EMA operator is to be iterated.

Constraint: $\mathbf{m1}\ge 1$.

5:     $\mathrm{m2}$ – int64int32nag_int scalar

The maximum number of times the EMA operator is to be iterated. Therefore nag_tsa_inhom_iema_all (g13mf) returns $\text{EMA}\left[\tau ,m;y\right]$, for $m=\mathbf{m1},\mathbf{m1}+1,\dots ,\mathbf{m2}$.

Constraint: $\mathbf{m2}\ge \mathbf{m1}$.

6:     $\mathrm{sinit}\left(\mathbf{m2}+2\right)$ – double array

If $\mathbf{pn}=0$, the values used to start the iterative process, with

• $\mathbf{sinit}\left(1\right)=t_0$,
• $\mathbf{sinit}\left(2\right)=y_0$,
• $\mathbf{sinit}\left(j+2\right)=\text{EMA}\left[\tau ,j;y\right]\left(t_0\right)$, $j=1,2,\dots ,\mathbf{m2}$.

If $\mathbf{pn}\ne 0$ then sinit is not referenced.

Constraint: if $\mathbf{ftype}\ne 1$, $\mathbf{sinit}\left(\mathit{j}\right)\ge 0$, for $\mathit{j}=2,3,\dots ,\mathbf{m2}+2$.

7:     $\mathrm{inter}\left(2\right)$ – int64int32nag_int array

The type of interpolation used with $\mathbf{inter}\left(1\right)$ indicating the interpolation method to use when calculating $\text{EMA}\left[\tau ,1;z\right]$ and $\mathbf{inter}\left(2\right)$ the interpolation method to use when calculating $\text{EMA}\left[\tau ,j;z\right]$, $j>1$.

Three types of interpolation are possible:

$\mathbf{inter}\left(i\right)=1$

Previous point, with $\nu =1$.

$\mathbf{inter}\left(i\right)=2$

Linear, with $\nu =\left(1-\mu \right)/\alpha$.

$\mathbf{inter}\left(i\right)=3$

Next point, $\nu =\mu$.

Zumbach and Müller (2001) recommend that linear interpolation is used in second and subsequent iterations, i.e., $\mathbf{inter}\left(2\right)=2$, irrespective of the interpolation method used at the first iteration, i.e., the value of $\mathbf{inter}\left(1\right)$.

Constraint: $\mathbf{inter}\left(\mathit{i}\right)=1$, $2$ or $3$, for $\mathit{i}=1,2$.

8:     $\mathrm{ftype}$ – int64int32nag_int scalar

The function type used to define the relationship between $y$ and $z$ when calculating $\text{EMA}\left[\tau ,1;y\right]$. Three functions are provided:

$\mathbf{ftype}=1$

The identity function, with $y_i={z_i}^{\left[p\right]}$.

$\mathbf{ftype}=2$

The absolute value, with $y_i={\left|z_i\right|}^p$.

$\mathbf{ftype}=3$

The absolute difference, with $y_i={\left|z_i-x_i\right|}^p$, where the vector $x$ is supplied in x.

Constraint: $\mathbf{ftype}=1$, $2$ or $3$.

9:     $\mathrm{p}$ – double scalar

$p$, the power used in the transformation function.

Constraint: $\mathbf{p}\ne 0$.

10:   $\mathrm{x}\left(:\right)$ – double array

The dimension of the array x must be at least $\mathbf{nb}$ if $\mathbf{ftype}=3$

If $\mathbf{ftype}=3$, $x_i$, the vector used to shift the current block of observations, for $\mathit{i}=k+1,\dots ,k+b$, where $k$ is the number of observations processed so far, i.e., the value supplied in pn on entry.

If $\mathbf{ftype}\ne 3$ then x is not referenced.

Constraint: if $\mathbf{ftype}=3$ and $\mathbf{p}<0$, $\mathbf{x}\left(\mathit{i}\right)\ne \mathbf{z}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nb}$.

Optional Input Parameters

1:     $\mathrm{sorder}$ – int64int32nag_int scalar

Default: $1$

Determines the storage order of output returned in iema.

Constraint: $\mathbf{sorder}=1$ or $2$.

2:     $\mathrm{nb}$ – int64int32nag_int scalar

Default: the dimension of the arrays z, t, x. (An error is raised if these dimensions are not equal.)

$b$, the number of observations in the current block of data. At each call the size of the block of data supplied in z, t and x can vary; therefore nb can change between calls to nag_tsa_inhom_iema_all (g13mf).

Constraint: $\mathbf{nb}\ge 0$.

3:     $\mathrm{pn}$ – int64int32nag_int scalar

Default: $0$

$k$, the number of observations processed so far. On the first call to nag_tsa_inhom_iema_all (g13mf), or when starting to summarise a new dataset, pn must be set to $0$. On subsequent calls it must be the same value as returned by the last call to nag_tsa_inhom_iema_all (g13mf).

Constraint: $\mathbf{pn}\ge 0$.

4:     $\mathrm{rcomm}\left(\mathit{lrcomm}\right)$ – double array

Communication array, used to store information between calls to nag_tsa_inhom_iema_all (g13mf). On the first call to nag_tsa_inhom_iema_all (g13mf), or if all the data is provided in one go, rcomm need not be provided.

Output Parameters

1:     $\mathrm{iema}\left(\mathit{ldiema},*\right)$ – double array

Note: the second dimension of the array iema must be at least $\mathbf{m2}-\mathbf{m1}+1$ if $\mathbf{sorder}=1$, otherwise at least nb.

The iterated exponential moving average.

If $\mathbf{sorder}=1$, $\mathbf{iema}\left(i,j\right)=\text{EMA}\left[\tau ,j+\mathbf{m1}-1;y\right]\left(t_{i+k}\right)$.

If $\mathbf{sorder}=2$, $\mathbf{iema}\left(j,i\right)=\text{EMA}\left[\tau ,j+\mathbf{m1}-1;y\right]\left(t_{i+k}\right)$.

For $i=1,2,\dots ,\mathbf{nb}$, $j=1,2,\dots ,\mathbf{m2}-\mathbf{m1}+1$ and $k$ is the number of observations processed so far, i.e., the value supplied in pn on entry.

2:     $\mathrm{p}$ – double scalar

If $\mathbf{ftype}=1$, then $\left[p\right]$, the actual power used in the transformation function is returned, otherwise p is unchanged.

3:     $\mathrm{pn}$ – int64int32nag_int scalar

Default: $0$

$k+b$, the updated number of observations processed so far.

4:     $\mathrm{rcomm}\left(\mathit{lrcomm}\right)$ – double array

Communication array, used to store information between calls to nag_tsa_inhom_iema_all (g13mf).

5:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=11$

Constraint: $\mathbf{sorder}=1$ or $2$.

   $\mathbf{ifail}=21$

Constraint: $\mathbf{nb}\ge 0$.

   $\mathbf{ifail}=51$

Constraint: $\mathit{ldiema}\ge \mathbf{m2}-\mathbf{m1}+1$.

Constraint: $\mathit{ldiema}\ge \mathbf{nb}$.

W  $\mathbf{ifail}=61$

Constraint: t should be strictly increasing.

   $\mathbf{ifail}=62$

Constraint: $\mathbf{t}\left(i\right)\ne \mathbf{t}\left(i-1\right)$ if linear interpolation is being used.

   $\mathbf{ifail}=71$

Constraint: $\mathbf{tau}>0.0$.

   $\mathbf{ifail}=72$

Constraint: if $\mathbf{pn}>0$ then tau must be unchanged since previous call.

   $\mathbf{ifail}=81$

Constraint: $\mathbf{m1}\ge 1$.

   $\mathbf{ifail}=82$

Constraint: if $\mathbf{pn}>0$ then m1 must be unchanged since previous call.

   $\mathbf{ifail}=91$

Constraint: $\mathbf{m2}\ge \mathbf{m1}$.

   $\mathbf{ifail}=92$

Constraint: if $\mathbf{pn}>0$ then m2 must be unchanged since previous call.

   $\mathbf{ifail}=101$

Constraint: if $\mathbf{ftype}\ne 1$, $\mathbf{sinit}\left(\mathit{j}\right)\ge 0.0$, for $\mathit{j}=2,3,\dots ,\mathbf{m2}+2$.

   $\mathbf{ifail}=111$

Constraint: $\mathbf{inter}\left(1\right)=1$, $2$ or $3$.

   $\mathbf{ifail}=112$

Constraint: $\mathbf{inter}\left(2\right)=1$, $2$ or $3$.

   $\mathbf{ifail}=113$

Constraint: if $\mathbf{pn}\ne 0$, inter must be unchanged since the last call.

   $\mathbf{ifail}=121$

Constraint: $\mathbf{ftype}=1$, $2$ or $3$.

   $\mathbf{ifail}=122$

Constraint: if $\mathbf{pn}\ne 0$, ftype must be unchanged since the previous call.

   $\mathbf{ifail}=131$

Constraint: absolute value of p must be representable as an integer.

   $\mathbf{ifail}=132$

Constraint: if $\mathbf{ftype}\ne 1$, $\mathbf{p}\ne 0.0$. If $\mathbf{ftype}=1$, the nearest integer to $\mathbf{p}$ must not be $0$.

   $\mathbf{ifail}=133$

Constraint: if $\mathbf{ftype}=1$ or $2$ and $\mathbf{z}\left(i\right)=0$ for any $i$ then $\mathbf{p}>0.0$.

   $\mathbf{ifail}=134$

Constraint: if $\mathbf{ftype}=3$ and $\mathbf{z}\left(i\right)=\mathbf{x}\left(i\right)$ for any $i$ then $\mathbf{p}>0.0$.

   $\mathbf{ifail}=135$

Constraint: if $\mathbf{pn}>0$ then p must be unchanged since previous call.

   $\mathbf{ifail}=151$

Constraint: $\mathbf{pn}\ge 0$.

   $\mathbf{ifail}=152$

Constraint: if $\mathbf{pn}>0$ then pn must be unchanged since previous call.

   $\mathbf{ifail}=161$

rcomm has been corrupted between calls.

   $\mathbf{ifail}=171$

Constraint: if $\mathbf{pn}=0$, $\mathit{lrcomm}=0$ or $\mathit{lrcomm}\ge \mathbf{m2}+20$.

   $\mathbf{ifail}=172$

Constraint: if $\mathbf{pn}\ne 0$ then $\mathit{lrcomm}\ge \mathbf{m2}+20$.

W  $\mathbf{ifail}=301$

Truncation occurred to avoid overflow, check for extreme values in t, z, x or for tau. Results are returned using the truncated values.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g13mf_example

function g13mf_example


fprintf('g13mf example results\n\n');

m1    = int64(2);
m2    = int64(6);
ftype = int64(1);
p     = 1;
inter = [int64(3); 2];
tau   = 2;
sinit = zeros(8, 1);
nb    = [5, 10, 15];
rcomm = zeros(20+m2, 1);
x     = [];
t     = cell(3, 1);
z     = cell(3, 1);

t{1} = [ 7.5;  8.2; 18.1; 22.8; 25.8];
z{1} = [ 0.6;  0.6;  0.8;  0.1;  0.2];
t{2} = [26.8; 31.1; 38.4; 45.9; 48.2; 48.9; 57.9; 58.5; 63.9; 65.2];
z{2} = [0.2;   0.5;  0.7;  0.1;  0.4;  0.7;  0.8;  0.3;  0.2;  0.5];
t{3} = [66.6; 67.4; 69.3; 69.9; 73.0; 75.6; 77.0; 84.7; 86.8; 88.0; ...
        88.5; 91.0; 93.0; 93.7; 94.0];
z{3} = [ 0.2;  0.3;  0.8;  0.6;  0.1;  0.7;  0.9;  0.6;  0.3;  0.1;  ...
         0.1;  0.4;  1.0;  1.0;  0.1];

fprintf('%41s\n%17s', 'Iteration', 'Time');
fprintf('%10d', [2:6]);
fprintf('\n');

% Loop over each block of data.
miema = m2-m1+1;
iema = cell(numel(nb), 1);
fmt = '%3d%14.1f  %10.3f%10.3f%10.3f%10.3f%10.3f\n';
for i = 1:numel(nb)
  if i == 1
    % Initialise the iterated EMA
    [iema{i}, p, pn, rcomm, ifail] = ...
    g13mf( ...
           z{i}, t{i}, tau, m1, m2, sinit, inter, ftype, p, x, 'rcomm', rcomm);
  else
    % Update the iterated EMA for this block of data
    [iema{i}, p, pn, rcomm, ifail] = ...
    g13mf( ...
           z{i}, t{i}, tau, m1, m2, sinit, inter, ftype, p, x, ...
           'pn', pn, 'rcomm', rcomm);
  end

  % Display the results for this block of data
  for j=1:nb(i)
    fprintf(fmt, pn-nb(i)+j, t{i}(j), iema{i}(j, 1:miema));
  end
  fprintf('\n');
end

g13mf example results

                                Iteration
             Time         2         3         4         5         6
  1           7.5       0.433     0.320     0.237     0.175     0.130
  2           8.2       0.479     0.361     0.268     0.198     0.147
  3          18.1       0.756     0.700     0.631     0.558     0.485
  4          22.8       0.406     0.535     0.592     0.600     0.577
  5          25.8       0.232     0.351     0.459     0.530     0.561

  6          26.8       0.217     0.301     0.406     0.491     0.540
  7          31.1       0.357     0.309     0.318     0.364     0.422
  8          38.4       0.630     0.556     0.490     0.445     0.425
  9          45.9       0.263     0.357     0.407     0.428     0.432
 10          48.2       0.241     0.284     0.343     0.388     0.413
 11          48.9       0.279     0.277     0.325     0.372     0.403
 12          57.9       0.713     0.617     0.543     0.496     0.469
 13          58.5       0.717     0.643     0.566     0.511     0.478
 14          63.9       0.385     0.495     0.541     0.546     0.531
 15          65.2       0.346     0.432     0.502     0.533     0.535

 16          66.6       0.330     0.384     0.453     0.504     0.526
 17          67.4       0.315     0.364     0.427     0.483     0.515
 18          69.3       0.409     0.367     0.389     0.435     0.478
 19          69.9       0.459     0.385     0.386     0.423     0.465
 20          73.0       0.377     0.403     0.394     0.398     0.419
 21          75.6       0.411     0.399     0.399     0.397     0.403
 22          77.0       0.536     0.440     0.410     0.401     0.401
 23          84.7       0.632     0.606     0.563     0.524     0.493
 24          86.8       0.538     0.587     0.583     0.557     0.526
 25          88.0       0.444     0.542     0.574     0.567     0.542
 26          88.5       0.401     0.515     0.564     0.567     0.548
 27          91.0       0.331     0.404     0.481     0.529     0.545
 28          93.0       0.495     0.418     0.438     0.483     0.518
 29          93.7       0.585     0.455     0.438     0.469     0.506
 30          94.0       0.612     0.475     0.441     0.465     0.500