NAG Library Routine Document

g13mff (inhom_iema_all)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{sorder}$ – IntegerInput

On entry: determines the storage order of output returned in iema.

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

2:     $\mathbf{nb}$ – IntegerInput

On entry: $b$, 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 g13mff.

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

3:     $\mathbf{z}\left(\mathbf{nb}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $z_{\mathit{i}}$, 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 $\mathit{i}=1,2,\dots ,\mathbf{nb}$.

4:     $\mathbf{iema}\left(\mathbf{ldiema},*\right)$ – Real (Kind=nag_wp) arrayOutput

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

On exit: 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.

5:     $\mathbf{ldiema}$ – IntegerInput

On entry: the first dimension of the array iema as declared in the (sub)program from which g13mff is called.

Constraints:

• if $\mathbf{sorder}=1$, $\mathbf{ldiema}\ge \mathbf{nb}$;
• otherwise $\mathbf{ldiema}\ge \mathbf{m2}-\mathbf{m1}+1$.

6:     $\mathbf{t}\left(\mathbf{nb}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $t_{\mathit{i}}$, 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 g13mff will continue as if $t$ was strictly increasing by using the absolute value.

7:     $\mathbf{tau}$ – Real (Kind=nag_wp)Input

On entry: $\tau$, 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: $\mathbf{tau}>0.0$.

8:     $\mathbf{m1}$ – IntegerInput

On entry: the minimum number of times the EMA operator is to be iterated.

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

9:     $\mathbf{m2}$ – IntegerInput

On entry: the maximum number of times the EMA operator is to be iterated. Therefore g13mff returns $\text{EMA}\left[\tau ,m;y\right]$, for $m=\mathbf{m1},\mathbf{m1}+1,\dots ,\mathbf{m2}$.

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

10:   $\mathbf{sinit}\left(\mathbf{m2}+2\right)$ – Real (Kind=nag_wp) arrayInput

On entry: 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 $\mathit{j}=2,3,\dots ,\mathbf{m2}+2$.

11:   $\mathbf{inter}\left(2\right)$ – Integer arrayInput

On entry: 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 $\mathit{i}=1,2$.

12:   $\mathbf{ftype}$ – IntegerInput

On entry: 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 $3$.

13:   $\mathbf{p}$ – Real (Kind=nag_wp)Input/Output

On entry: $p$, the power used in the transformation function.

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

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

14:   $\mathbf{x}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

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

On entry: 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 $\mathit{i}=1,2,\dots ,\mathbf{nb}$.

15:   $\mathbf{pn}$ – IntegerInput/Output

On entry: $k$, the number of observations processed so far. On the first call to g13mff, 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 g13mff.

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

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

16:   $\mathbf{rcomm}\left(\mathbf{lrcomm}\right)$ – Real (Kind=nag_wp) arrayCommunication Array

On entry: communication array, used to store information between calls to g13mff. If $\mathbf{lrcomm}=0$, rcomm is not referenced, pn must be set to $0$ and all the data must be supplied in one go.

17:   $\mathbf{lrcomm}$ – IntegerInput

On entry: the dimension of the array rcomm as declared in the (sub)program from which g13mff is called.

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

18:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=11$

On entry, $\mathbf{sorder}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{sorder}=1$ or $2$.

$\mathbf{ifail}=21$

On entry, $\mathbf{nb}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nb}\ge 0$.

$\mathbf{ifail}=51$

On entry, $\mathbf{sorder}=1$, $\mathbf{ldiema}=〈\mathit{\text{value}}〉$ and $\mathbf{nb}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldiema}\ge \mathbf{nb}$.

On entry, $\mathbf{sorder}=2$, $\mathbf{ldiema}=〈\mathit{\text{value}}〉$ and $\mathbf{m2}-\mathbf{m1}+1=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldiema}\ge \mathbf{m2}-\mathbf{m1}+1$.

$\mathbf{ifail}=61$

On entry, $i=〈\mathit{\text{value}}〉$, $\mathbf{t}\left(i-1\right)=〈\mathit{\text{value}}〉$ and $\mathbf{t}\left(i\right)=〈\mathit{\text{value}}〉$.
Constraint: t should be strictly increasing.

$\mathbf{ifail}=62$

On entry, $i=〈\mathit{\text{value}}〉$, $\mathbf{t}\left(i-1\right)=〈\mathit{\text{value}}〉$ and $\mathbf{t}\left(i\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{t}\left(i\right)\ne \mathbf{t}\left(i-1\right)$ if linear interpolation is being used.

$\mathbf{ifail}=71$

On entry, $\mathbf{tau}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{tau}>0.0$.

$\mathbf{ifail}=72$

On entry, $\mathbf{tau}=〈\mathit{\text{value}}〉$.
On entry at previous call, $\mathbf{tau}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}>0$ then tau must be unchanged since previous call.

$\mathbf{ifail}=81$

On entry, $\mathbf{m1}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m1}\ge 1$.

$\mathbf{ifail}=82$

On entry, $\mathbf{m1}=〈\mathit{\text{value}}〉$.
On entry at previous call, $\mathbf{m1}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}>0$ then m1 must be unchanged since previous call.

$\mathbf{ifail}=91$

On entry, $\mathbf{m1}=〈\mathit{\text{value}}〉$ and $\mathbf{m2}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m2}\ge \mathbf{m1}$.

$\mathbf{ifail}=92$

On entry, $\mathbf{m2}=〈\mathit{\text{value}}〉$.
On entry at previous call, $\mathbf{m2}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}>0$ then m2 must be unchanged since previous call.

$\mathbf{ifail}=101$

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

$\mathbf{ifail}=111$

On entry, $\mathbf{inter}\left(1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{inter}\left(1\right)=1$, $2$ or $3$.

$\mathbf{ifail}=112$

On entry, $\mathbf{inter}\left(2\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{inter}\left(2\right)=1$, $2$ or $3$.

$\mathbf{ifail}=113$

On entry, $\mathbf{inter}\left(1\right)=〈\mathit{\text{value}}〉$ and $\mathbf{inter}\left(2\right)=〈\mathit{\text{value}}〉$.
On entry at previous call, $\mathbf{inter}\left(1\right)=〈\mathit{\text{value}}〉$, $\mathbf{inter}\left(2\right)=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}\ne 0$, inter must be unchanged since the last call.

$\mathbf{ifail}=121$

On entry, $\mathbf{ftype}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ftype}=1$, $2$ or $3$.

$\mathbf{ifail}=122$

On entry, $\mathbf{ftype}=〈\mathit{\text{value}}〉$, On entry at previous call, $\mathbf{ftype}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}\ne 0$, ftype must be unchanged since the previous call.

$\mathbf{ifail}=131$

On entry, $\mathbf{p}=〈\mathit{\text{value}}〉$.
Constraint: absolute value of p must be representable as an integer.

$\mathbf{ifail}=132$

On entry, $\mathbf{p}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{ftype}\ne 1$, $\mathbf{p}\ne 0.0$. If $\mathbf{ftype}=1$, the nearest integer to $\mathbf{p}$ must not be $0$.

$\mathbf{ifail}=133$

On entry, $i=〈\mathit{\text{value}}〉$, $\mathbf{z}\left(i\right)=〈\mathit{\text{value}}〉$ and $\mathbf{p}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{ftype}=1$ or $2$ and $\mathbf{z}\left(i\right)=0$ for any $i$ then $\mathbf{p}>0.0$.

$\mathbf{ifail}=134$

On entry, $i=〈\mathit{\text{value}}〉$, $\mathbf{z}\left(i\right)=〈\mathit{\text{value}}〉$, $\mathbf{x}\left(i\right)=〈\mathit{\text{value}}〉$ and $\mathbf{p}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{ftype}=3$ and $\mathbf{z}\left(i\right)=\mathbf{x}\left(i\right)$ for any $i$ then $\mathbf{p}>0.0$.

$\mathbf{ifail}=135$

On entry, $\mathbf{p}=〈\mathit{\text{value}}〉$.
On exit from previous call, $\mathbf{p}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}>0$ then p must be unchanged since previous call.

$\mathbf{ifail}=151$

On entry, $\mathbf{pn}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pn}\ge 0$.

$\mathbf{ifail}=152$

On entry, $\mathbf{pn}=〈\mathit{\text{value}}〉$.
On exit from previous call, $\mathbf{pn}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}>0$ then pn must be unchanged since previous call.

$\mathbf{ifail}=161$

rcomm has been corrupted between calls.

$\mathbf{ifail}=171$

On entry, $\mathbf{pn}=0$, $\mathbf{lrcomm}=〈\mathit{\text{value}}〉$ and $\mathbf{m2}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}=0$, $\mathbf{lrcomm}=0$ or $\mathbf{lrcomm}\ge \mathbf{m2}+20$.

$\mathbf{ifail}=172$

On entry, $\mathbf{pn}\ne 0$, $\mathbf{lrcomm}=〈\mathit{\text{value}}〉$ and $\mathbf{m2}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{pn}\ne 0$ then $\mathbf{lrcomm}\ge \mathbf{m2}+20$.

$\mathbf{ifail}=301$

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.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

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

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (g13mffe.f90)

10.2

Program Data

Program Data (g13mffe.d)

10.3

Program Results

Program Results (g13mffe.r)