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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_tsa_inhom_ma (g13mg)

## Purpose

nag_tsa_inhom_ma (g13mg) provides a moving average, moving norm, moving variance and moving standard deviation operator for an inhomogeneous time series.

## Syntax

[ma, p, pn, wma, rcomm, ifail] = g13mg(ma, t, tau, m1, m2, sinit, inter, ftype, p, 'nb', nb, 'pn', pn, 'rcomm', rcomm)
[ma, p, pn, wma, rcomm, ifail] = nag_tsa_inhom_ma(ma, t, tau, m1, m2, sinit, inter, ftype, p, 'nb', nb, 'pn', pn, 'rcomm', rcomm)

## Description

nag_tsa_inhom_ma (g13mg) provides a number of operators 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 $\left({t}_{\mathit{i}},{z}_{i}\right)$, for $\mathit{i}=1,2,\dots ,n$. Time $t$ can be measured in any arbitrary units, as long as all elements of $t$ use the same units.
The main operator available, the moving average (MA), with parameter $\tau$ is defined as
 $MA τ, m1, m2; y ti = 1 m2 - m1 +1 ∑ j=m1 m2 EMA τ~, j; y ti$ (1)
where $\stackrel{~}{\tau }=\frac{2\tau }{{m}_{2}+{m}_{1}}$, ${m}_{1}$ and ${m}_{2}$ are user-supplied integers controlling the amount of lag and smoothing respectively, with ${m}_{2}\ge {m}_{1}$ and $\text{EMA}\left(·\right)$ is the iterated exponential moving average operator.
The iterated exponential moving average, $\text{EMA}\left[\stackrel{~}{\tau },m;y\right]\left({t}_{i}\right)$, is defined using the recursive formula:
 $EMA τ~,m ; y ti = EMA τ~ ; EMA τ~,m-1 ; y ti ti$
with
 $EMA τ~,1;y ti = EMA τ~;y ti$
and
 $EMA τ~ ; y ti = μ ⁢ EMA τ~;y ti-1 + ν-μ ⁢ yi-1 + 1-ν ⁢ yi$
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_ma (g13mg) 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: ${y}_{i}={\left|{z}_{i}-\text{MA}\left[\tau ,{m}_{1},{m}_{2};z\right]\left({t}_{i}\right)\right|}^{p}$.
where the notation $\left[p\right]$ is used to denote the integer nearest to $p$. In addition, if either the absolute value or absolute difference transformation are used then the resulting moving average can be scaled by ${p}^{-1}$.
The various parameter options allow a number of different operators to be applied by nag_tsa_inhom_ma (g13mg), a few of which are:
(i) Moving Average (MA), as defined in (1) (obtained by setting ${\mathbf{ftype}}=1$ and ${\mathbf{p}}=1$).
(ii) Moving Norm (MNorm), defined as
 $MNorm τ,m,p;z = MA τ,1,m; z p 1 / p$
(obtained by setting ${\mathbf{ftype}}=4$, ${\mathbf{m1}}=1$ and ${\mathbf{m2}}=m$).
(iii) Moving Variance (MVar), defined as
 $MVar τ,m,p;z = MA τ,1,m; z - MA τ,1,m;z p$
(obtained by setting ${\mathbf{ftype}}=3$, ${\mathbf{m1}}=1$ and ${\mathbf{m2}}=m$).
(iv) Moving Standard Deviation (MSD), defined as
 $MSD τ,m,p;z = MA τ,1,m; z - MA τ,1,m;z p 1 / p$
(obtained by setting ${\mathbf{ftype}}=5$, ${\mathbf{m1}}=1$ and ${\mathbf{m2}}=m$).
For large datasets or where all the data is not available at the same time, $z$ and $t$ can be split into arbitrary sized blocks and nag_tsa_inhom_ma (g13mg) 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:     $\mathrm{ma}\left({\mathbf{nb}}\right)$ – 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 ${e}^{-\alpha }$, $\alpha =\left({t}_{i}-{t}_{i-1}\right)/\stackrel{~}{\tau }$ 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 ${e}^{-\alpha }$, $\alpha =\left({t}_{i}-{t}_{i-1}\right)/\stackrel{~}{\tau }$ can be calculated without overflowing, for all $i$, where $\stackrel{~}{\tau }=\frac{2\tau }{{m}_{2}+{m}_{1}}$.
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 $2×{\mathbf{m2}}+3$ 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
${\mathbf{wma}}\left(i\right)=\text{EMA}\left[\stackrel{~}{\tau };y\right]\left({t}_{i}\right)$.
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

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: ${\mathbf{nb}}\ge 0$.
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.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

Approximately $4{m}_{2}$ real elements are internally allocated by nag_tsa_inhom_ma (g13mg). If ${\mathbf{ftype}}=3$ or $5$ then a further nb real elements are also allocated.
The more data you supply to nag_tsa_inhom_ma (g13mg) in one call, i.e., the larger nb is, the more efficient the function 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 ma, t or tau are supplied.

## Example

The example reads in a simulated time series, $\left(t,z\right)$ and calculates the moving average. The data is supplied in three blocks of differing sizes.
```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

```