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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_tsa_inhom_iema_all (g13mf)

## Purpose

nag_tsa_inhom_iema_all (g13mf) calculates the iterated exponential moving average for an inhomogeneous time series, returning the intermediate results.

## Syntax

[iema, p, pn, rcomm, ifail] = g13mf(z, t, tau, m1, m2, sinit, inter, ftype, p, x, 'sorder', sorder, 'nb', nb, 'pn', pn, 'rcomm', rcomm)
[iema, p, pn, rcomm, ifail] = nag_tsa_inhom_iema_all(z, t, tau, m1, m2, sinit, inter, ftype, p, x, 'sorder', sorder, 'nb', nb, 'pn', pn, 'rcomm', rcomm)

## Description

nag_tsa_inhom_iema_all (g13mf) calculates the iterated exponential moving average for an inhomogeneous time series. The time series is represented by two vectors of length $n$: a vector of times, $t$; and a vector of values, $z$. Each element of the time series is therefore composed of the pair of scalar values $\left({t}_{\mathit{i}},{z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$. Time can be measured in any arbitrary units, as long as all elements of $t$ use the same units.
The exponential moving average (EMA), with parameter $\tau$, is an average operator, with the exponentially decaying kernel given by
 $e -ti/τ τ .$
The exponential form of this kernel gives rise to the following iterative formula (Zumbach and Müller (2001)) for the EMA operator:
 $EMA τ ; y ti = μ ⁢ EMA τ;y ti-1 + ν-μ ⁢ yi-1 + 1-ν ⁢ yi$
where
 $μ = e-α and α = ti - ti-1 τ .$
The value of $\nu$ depends on the method of interpolation chosen and the relationship between $y$ and the input series $z$ depends on the transformation function chosen. nag_tsa_inhom_iema_all (g13mf) gives the option of three interpolation methods:
 1 Previous point: $\nu =1$; 2 Linear: $\nu =\left(1-\mu \right)/\alpha$; 3 Next point: $\nu =\mu$.
and three transformation functions:
 1 Identity: ${y}_{i}={{z}_{i}}^{\left[p\right]}$; 2 Absolute value: ${y}_{i}={\left|{z}_{i}\right|}^{p}$; 3 Absolute difference: ${y}_{i}={\left|{z}_{i}-{x}_{i}\right|}^{p}$;
where the notation $\left[p\right]$ is used to denote the integer nearest to $p$. In the case of the absolute difference $x$ is a user-supplied vector of length $n$ and therefore each element of the time series is composed of the triplet of scalar values, $\left({t}_{i},{z}_{i},{x}_{i}\right)$.
The $m$-iterated exponential moving average, $\text{EMA}\left[\tau ,m;y\right]\left({t}_{i}\right)$, is defined using the recursive formula:
 $EMA τ,m ; y ti = EMA τ ; EMA τ,m-1 ; y ti ti$
with
 $EMA τ,1;y ti = EMA τ;y ti .$
For large datasets or where all the data is not available at the same time, $z,t$ and, where required, $x$ can be split into arbitrary sized blocks and nag_tsa_inhom_iema_all (g13mf) called multiple times.

## References

Dacorogna M M, Gencay R, Müller U, Olsen R B and Pictet O V (2001) An Introduction to High-frequency Finance Academic Press
Zumbach G O and Müller U A (2001) Operators on inhomogeneous time series International Journal of Theoretical and Applied Finance 4(1) 147–178

## 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

Errors or warnings detected by the function:

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.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

Approximately $4×{\mathbf{m2}}$ real elements are internally allocated by nag_tsa_inhom_iema_all (g13mf).
The more data you supply to nag_tsa_inhom_iema_all (g13mf) in one call, i.e., the larger nb is, the more efficient the routine will be, particularly if the function is being run using more than one thread.
Checks are made during the calculation of $\alpha$ and ${y}_{i}$ to avoid overflow. If a potential overflow is detected the offending value is replaced with a large positive or negative value, as appropriate, and the calculations performed based on the replacement values. In such cases ${\mathbf{ifail}}={\mathbf{301}}$ is returned. This should not occur in standard usage and will only occur if extreme values of z, t, x or tau are supplied.

## Example

This example reads in three blocks of simulated data from an inhomogeneous time series, then calculates and prints the iterated EMA for $m$ between $2$ and $6$.
```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

```