naginterfaces.library.tsa.inhom_​iema

naginterfaces.library.tsa.inhom_iema(iema, t, tau, inter, sinit=None, pn=0, comm=None)

inhom_iema calculates the iterated exponential moving average for an inhomogeneous time series.

For full information please refer to the NAG Library document for g13me

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g13/g13mef.html

Parameters

iemafloat, array-like, shape $\left(\text{nb}\right)$

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

tfloat, array-like, shape $\left(\text{nb}\right)$

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

If $t_i\le t_{i-1}$, $errno$ = 31 will be returned, but inhom_iema will continue as if $t$ was strictly increasing by using the absolute value.

taufloat

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

interint, array-like, shape $\left(2\right)$

The type of interpolation used with $inter\left[0\right]$ indicating the interpolation method to use when calculating $\text{EMA}[\tau ,1;z]$ and $inter\left[1\right]$ the interpolation method to use when calculating $\text{EMA}[\tau ,j;z]$, $j>1$.

Three types of interpolation are possible:

$inter\left[i\right]=1$

Previous point, with $\nu =1$.

$inter\left[i\right]=2$

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

$inter\left[i\right]=3$

Next point, $\nu =\mu$.

Zumbach and Müller (2001) recommend that linear interpolation is used in second and subsequent iterations, i.e., $inter\left[1\right]=2$, irrespective of the interpolation method used at the first iteration, i.e., the value of $inter\left[0\right]$.

sinitNone or float, array-like, shape $(m+2)$, optional

If $pn=0$, the values used to start the iterative process, with

$sinit\left[0\right]=t_0$,

$sinit\left[1\right]=z_0$,

$sinit[\text{j}+1]=\text{EMA}[\tau ,\text{j};z]\left(t_0\right)$, for $\text{j}=1,2,\dots ,m$.

If $pn\ne 0$, $sinit$ is not referenced.

pnint, optional

$k$, the number of observations processed so far. On the first call to inhom_iema, 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 inhom_iema.

commNone or dict, communication object, optional, modified in place

Communication structure.

On initial entry: need not be set.

Returns

iemafloat, ndarray, shape $\left(\text{nb}\right)$

The iterated EMA, with $iema[i-1]=\text{EMA}[\tau ,m;z]\left(t_i\right)$.

pnint

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

Raises

NagValueError

(errno $11$)

On entry, $\text{nb}=⟨value⟩$.

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

(errno $32$)

On entry, $i=⟨value⟩$, $t[i-2]=⟨value⟩$ and $t[i-1]=⟨value⟩$.

Constraint: $t[i-1]\ne t[i-2]$ if linear interpolation is being used.

(errno $41$)

On entry, $tau=⟨value⟩$.

Constraint: $tau>0.0$.

(errno $42$)

On entry, $tau=⟨value⟩$.

On entry at previous call, $tau=⟨value⟩$.

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

(errno $51$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $52$)

On entry, $m=⟨value⟩$.

On entry at previous call, $m=⟨value⟩$.

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

(errno $71$)

On entry, $inter\left[0\right]=⟨value⟩$.

Constraint: $inter\left[0\right]=1$, $2$ or $3$.

(errno $72$)

On entry, $inter\left[1\right]=⟨value⟩$.

Constraint: $inter\left[1\right]=1$, $2$ or $3$.

(errno $73$)

On entry, $inter\left[0\right]=⟨value⟩$ and $inter\left[1\right]=⟨value⟩$.

On entry at previous call, $inter\left[0\right]=⟨value⟩$, $inter\left[1\right]=⟨value⟩$.

Constraint: if $pn\ne 0$, $inter$ must be unchanged since the previous call.

(errno $81$)

On entry, $pn=⟨value⟩$.

Constraint: $pn\ge 0$.

(errno $82$)

On entry, $pn=⟨value⟩$.

On exit from previous call, $pn=⟨value⟩$.

Constraint: if $pn>0$ then $pn$ must be unchanged since previous call.

(errno $91$)

$comm$[‘rcomm’] has been corrupted between calls.

(errno $101$)

On entry, $pn=0$, $\text{lrcomm}=⟨value⟩$ and $m=⟨value⟩$.

Constraint: if $pn=0$, $\text{lrcomm}=0$ or $\text{lrcomm}\ge m+20$.

(errno $102$)

On entry, $pn\ne 0$, $\text{lrcomm}=⟨value⟩$ and $m=⟨value⟩$.

Constraint: if $pn\ne 0$, $\text{lrcomm}\ge m+20$.

Warns

NagAlgorithmicWarning

(errno $31$)

On entry, $i=⟨value⟩$, $t[i-2]=⟨value⟩$ and $t[i-1]=⟨value⟩$.

Constraint: $t$ should be strictly increasing.

(errno $301$)

Truncation occurred to avoid overflow, check for extreme values in $t$, $iema$ or for $tau$.

Notes

inhom_iema 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 $(t_{\text{i}},z_i)$, for $\text{i}=1,2,\dots ,n$. 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

$$\frac{e^{-t_i/\tau }}{\tau }\text{.}$$

The exponential form of this kernel gives rise to the following iterative formula for the EMA operator (see Zumbach and Müller (2001)):

$$\text{EMA}[\tau ;z]\left(t_i\right)=\mu \text{EMA}[\tau ;z]\left(t_{i-1}\right)+(\nu -\mu )z_{i-1}+(1-\nu )z_i$$

where

$$\mu =e^{-\alpha }\text{and}\alpha =\frac{t_i-t_{i-1}}{\tau }\text{.}$$

The value of $\nu$ depends on the method of interpolation chosen. inhom_iema gives the option of three interpolation methods:

	Previous point:	$\nu =1$;
	Linear:	$\nu =(1-\mu )/\alpha$;
	Next point:	$\nu =\mu$.

The $m$-iterated exponential moving average, $\text{EMA}[\tau ,m;z]\left(t_i\right)$, $m>1$, is defined using the recursive formula:

$$\text{EMA}[\tau ,m;z]=\text{EMA}[\tau ;\text{EMA}[\tau ,m-1;z]]$$

with

$$\text{EMA}[\tau ,1;z]=\text{EMA}[\tau ;z]\text{.}$$

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 inhom_iema 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

See also

naginterfaces.library.examples.tsa.inhom_iema_ex.main()