NAG Toolbox: nag_tsa_inhom_ma (g13mg)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{ma}\left(\mathbf{nb}\right)$ – 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.

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 $t_i\le t_{i-1}$, $\mathbf{ifail}=\mathbf{31}$ will be returned, but nag_tsa_inhom_ma (g13mg) will continue as if $t$ was strictly increasing by using the absolute value. The lagged difference, $t_i-t_{i-1}$ must be sufficiently small that $e^{-\alpha }$, $\alpha =\left(t_i-t_{i-1}\right)/\tilde{\tau }$ can be calculated without overflowing, for all $i$.

3:     $\mathrm{tau}$ – double scalar

$\tau$, the parameter controlling the rate of decay. $\tau$ must be sufficiently large that $e^{-\alpha }$, $\alpha =\left(t_i-t_{i-1}\right)/\tilde{\tau }$ can be calculated without overflowing, for all $i$, where $\tilde{\tau }=\frac{2\tau }{m_2+m_1}$.

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

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

$m_1$, the iteration of the EMA operator at which the sum is started.

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

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

$m_2$, the iteration of the EMA operator at which the sum is ended.

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

6:     $\mathrm{sinit}\left(:\right)$ – double array

The dimension of the array sinit must be at least $2\times \mathbf{m2}+3$ if $\mathbf{ftype}=3$ or $5$, and at least $\mathbf{m2}+2$ otherwise

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

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

In addition, if $\mathbf{ftype}=3$ or $5$ then

• $\mathbf{sinit}\left(\mathbf{m2}+3\right)=z_0$,
• $\mathbf{sinit}\left(\mathbf{m2}+\mathit{j}+2\right)=\text{EMA}\left[\tau ,\mathit{j};z\right]\left(t_0\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{m2}$.

i.e., initial values based on the original data $z$ as opposed to the transformed data $y$

If $\mathbf{pn}\ne 0$, 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 $y_i={z_i}^{\left[p\right]}$.

$\mathbf{ftype}=2$ or $4$

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

$\mathbf{ftype}=3$ or $5$

The absolute difference, with $y_i={\left|z_i-\text{MA}\left[\tau ,m;y\right]\left(t_i\right)\right|}^p$.

If $\mathbf{ftype}=4$ or $5$ then the resulting vector of averages is scaled by $p^{-1}$ as described in ma.

Constraint: $\mathbf{ftype}=1$, $2$, $3$, $4$ or $5$.

9:     $\mathrm{p}$ – double scalar

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

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

Optional Input Parameters

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

Default: the dimension of the arrays ma, t. (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 ma and t can vary; therefore nb can change between calls to nag_tsa_inhom_ma (g13mg).

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

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

Default: $0$

$k$, the number of observations processed so far. On the first call to nag_tsa_inhom_ma (g13mg), 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_ma (g13mg).

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

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

Communication array, used to store information between calls to nag_tsa_inhom_ma (g13mg). On the first call to nag_tsa_inhom_ma (g13mg), or if all the data is provided in one go, rcomm need not be provided.

Output Parameters

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

The moving average:

if $\mathbf{ftype}=4$ or $5$

$\mathbf{ma}\left(i\right)={\left\{\text{MA}\left[\tau ,m_1,m_2;y\right]\left(t_i\right)\right\}}^{1/p}$,

otherwise

$\mathbf{ma}\left(i\right)=\text{MA}\left[\tau ,m_1,m_2;y\right]\left(t_i\right)$.

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

Either the moving average or exponential moving average, depending on the value of ftype.

if $\mathbf{ftype}=3$ or $5$

$\mathbf{wma}\left(i\right)=\text{MA}\left[\tau ;y\right]\left(t_i\right)$

otherwise

$\mathbf{wma}\left(i\right)=\text{EMA}\left[\tilde{\tau };y\right]\left(t_i\right)$.

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

Communication array, used to store information between calls to nag_tsa_inhom_ma (g13mg).

6:     $\mathrm{ifail}$ – 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: $\mathbf{nb}\ge 0$.

W  $\mathbf{ifail}=31$

Constraint: t should be strictly increasing.

   $\mathbf{ifail}=32$

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

   $\mathbf{ifail}=41$

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

   $\mathbf{ifail}=42$

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

   $\mathbf{ifail}=51$

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

   $\mathbf{ifail}=52$

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

   $\mathbf{ifail}=61$

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

   $\mathbf{ifail}=62$

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

   $\mathbf{ifail}=71$

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

   $\mathbf{ifail}=81$

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

   $\mathbf{ifail}=82$

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

   $\mathbf{ifail}=83$

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

   $\mathbf{ifail}=91$

Constraint: $\mathbf{ftype}=1$, $2$, $3$, $4$ or $5$.

   $\mathbf{ifail}=92$

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

   $\mathbf{ifail}=101$

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

   $\mathbf{ifail}=102$

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

   $\mathbf{ifail}=103$

Constraint: if $\mathbf{ftype}=1$, $2$ or $4$ and $\mathbf{ma}\left(i\right)=0$ for any $i$ then $\mathbf{p}>0.0$.

   $\mathbf{ifail}=104$

Constraint: if $\mathbf{p}<0.0$, $\mathbf{ma}\left(i\right)-\mathbf{wma}\left(i\right)\ne 0.0$, for any $i$.

   $\mathbf{ifail}=105$

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

   $\mathbf{ifail}=111$

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

   $\mathbf{ifail}=112$

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

   $\mathbf{ifail}=131$

rcomm has been corrupted between calls.

   $\mathbf{ifail}=141$

Constraint: if $\mathbf{pn}=0$, $\mathit{lrcomm}=0$ or $\mathit{lrcomm}\ge 2\mathbf{m2}+20$.

   $\mathbf{ifail}=142$

Constraint: if $\mathbf{pn}\ne 0$, $\mathit{lrcomm}\ge 2\mathbf{m2}+20$.

W  $\mathbf{ifail}=301$

Truncation occurred to avoid overflow, check for extreme values in t, ma 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: g13mg_example

function g13mg_example


fprintf('g13mg example results\n\n');

m1 = int64(1);
m2 = int64(2);
ftype = int64(1);
p = 1;
inter = [int64(3); 2];
tau = 2;
sinit = zeros(8, 1);
nb = [5, 10, 15];
rcomm = zeros(20+2*m2, 1);
t = cell(3, 1);
z = cell(3, 1);
t{1}  = [ 7.5;  8.2; 18.1; 22.8; 25.8];
ma{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];
ma{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];
ma{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('             Time           MA\n');

% Loop over each block of data.
for i = 1:numel(nb)
  if i == 1
    % Initialise the moving average operator for this block of data
    [ma{i}, p, pn, wma, rcomm, ifail] = ...
    g13mg( ...
           ma{i}, t{i}, tau, m1, m2, sinit, inter, ftype, p, 'rcomm', rcomm);
  else
    % Update the moving average operator for this block of data
    [ma{i}, p, pn, wma, rcomm, ifail] = ...
    g13mg( ...
           ma{i}, t{i}, tau, m1, m2, sinit, inter, ftype, p, ...
           'pn', pn, 'rcomm', rcomm);
  end

  % Display the results for this block of data
  for j=1:nb(i)
    fprintf('%3d    %10.1f    %10.3f\n', pn-nb(i)+j, t{i}(j), ma{i}(j));
  end
  fprintf('\n');
end

g13mg example results

             Time           MA
  1           7.5         0.545
  2           8.2         0.567
  3          18.1         0.786
  4          22.8         0.214
  5          25.8         0.187

  6          26.8         0.192
  7          31.1         0.444
  8          38.4         0.680
  9          45.9         0.155
 10          48.2         0.298
 11          48.9         0.406
 12          57.9         0.777
 13          58.5         0.677
 14          63.9         0.258
 15          65.2         0.351

 16          66.6         0.291
 17          67.4         0.289
 18          69.3         0.572
 19          69.9         0.593
 20          73.0         0.244
 21          75.6         0.532
 22          77.0         0.715
 23          84.7         0.618
 24          86.8         0.426
 25          88.0         0.284
 26          88.5         0.240
 27          91.0         0.332
 28          93.0         0.723
 29          93.7         0.814
 30          94.0         0.744