void	g13mfc (Nag_OrderType order, Integer nb, const double z[], double iema[], Integer pdiema, const double t[], double tau, Integer m1, Integer m2, const double sinit[], const Nag_TS_Interpolation inter[], Nag_TS_Transform ftype, double p, const double x[], Integer pn, double rcomm[], NagError *fail)

The function may be called by the names: g13mfc or nag_tsa_inhom_iema_all.

3 Description

g13mfc 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_{i}, z_{i})

, for

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

τ

, is an average operator, with the exponentially decaying kernel given by

\frac{e^{- t_{i} / 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] (t_{i}) = μ EMA [τ; y] (t_{i - 1}) + (ν - μ) y_{i - 1} + (1 - ν) y_{i}

where

μ = e^{- α} and α = \frac{t_{i} - t_{i - 1}}{τ} .

The value of

ν

depends on the method of interpolation chosen and the relationship between

y

and the input series

z

depends on the transformation function chosen. g13mfc gives the option of three interpolation methods:

1.	Previous point:	$ν = 1$ ;
2.	Linear:	$ν = (1 - μ) / α$ ;
3.	Next point:	$ν = μ$ .

and three transformation functions:

1.	Identity:	$y_{i} = {z_{i}}^{[p]}$ ;
2.	Absolute value:	$y_{i} = {\| z_{i} \|}^{p}$ ;
3.	Absolute difference:	$y_{i} = {\| z_{i} - x_{i} \|}^{p}$ ;

where the notation

[p]

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,

(t_{i}, z_{i}, x_{i})

The

m

-iterated exponential moving average,

EMA [τ, m; y] (t_{i})

, is defined using the recursive formula:

EMA [τ, m; y] (t_{i}) = EMA [τ; EMA [τ, m - 1; y] (t_{i})] (t_{i})

with

EMA [τ, 1; y] (t_{i}) = EMA [τ; y] (t_{i}) .

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 g13mfc called multiple times.

4 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

5 Arguments

1: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $nb$ – Integer Input

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 g13mfc.

Constraint:

nb \geq 0

3: $z [nb]$ – const double Input

On entry:

z_{i}

, the current block of observations, for

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

ftype = Nag_Identity

Nag_AbsVal

and

p < 0.0

z [i - 1] \neq 0

, for

i = 1, 2, \dots, nb

4: $iema [\dim]$ – double Output

Note: where

IEMA (i, j)

appears in this document, it refers to the array element

$iema [(j - 1) \times pdiema + i - 1]$ when $order = Nag_ColMajor$ ;
$iema [(i - 1) \times pdiema + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the iterated exponential moving average.

order = Nag_ColMajor

IEMA (i, j) = EMA [τ, j + m1 - 1; y] (t_{i + k})

order = Nag_RowMajor

IEMA (j, i) = EMA [τ, j + m1 - 1; y] (t_{i + k})

For

i = 1, 2, \dots, nb

j = 1, 2, \dots, m2 - m1 + 1

and

k

is the number of observations processed so far, i.e., the value supplied in pn on entry.

5: $pdiema$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array iema.

Constraints:

if $order = Nag_ColMajor$ , $pdiema \geq nb$ ;
otherwise $pdiema \geq m2 - m1 + 1$ .

6: $t [nb]$ – const double Input

On entry:

t_{i}

, the times for the current block of observations, for

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.

t_{i} \leq t_{i - 1}

fail . code =

NE_NOT_STRICTLY_INCREASING will be returned, but g13mfc will continue as if

t

was strictly increasing by using the absolute value.

7: $tau$ – double Input

On entry:

τ

, the parameter controlling the rate of decay.

τ

must be sufficiently large that

e^{- α}

α = (t_{i} - t_{i - 1}) / τ

can be calculated without overflowing, for all

i

Constraint:

tau > 0.0

8: $m1$ – Integer Input

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

Constraint:

m1 \geq 1

9: $m2$ – Integer Input

On entry: the maximum number of times the EMA operator is to be iterated. Therefore, g13mfc returns

EMA [τ, m; y]

, for

m = m1, m1 + 1, \dots, m2

Constraint:

m2 \geq m1

10: $sinit [m2 + 2]$ – const double Input

On entry: if

pn = 0

, the values used to start the iterative process, with

$sinit [0] = t_{0}$ ,
$sinit [1] = y_{0}$ ,
$sinit [j + 1] = EMA [τ, j; y] (t_{0})$ , $j = 1, 2, \dots, m2$ .

pn \neq 0

then sinit is not referenced and may be NULL.

Constraint: if

ftype \neq Nag_Identity

sinit [j - 1] \geq 0

, for

j = 2, 3, \dots, m2 + 2

11: $inter [2]$ – const Nag_TS_Interpolation Input

On entry: the type of interpolation used with

inter [0]

indicating the interpolation method to use when calculating

EMA [τ, 1; z]

and

inter [1]

the interpolation method to use when calculating

EMA [τ, j; z]

j > 1

Three types of interpolation are possible:

$inter [i] = Nag_PreviousPoint$: Previous point, with $ν = 1$ .
$inter [i] = Nag_Linear$: Linear, with $ν = (1 - μ) / α$ .
$inter [i] = Nag_NextPoint$: Next point, $ν = μ$ .

Zumbach and Müller (2001) recommend that linear interpolation is used in second and subsequent iterations, i.e.,

inter [1] = Nag_Linear

, irrespective of the interpolation method used at the first iteration, i.e., the value of

inter [0]

Constraint:

inter [i - 1] = Nag_PreviousPoint

Nag_Linear

Nag_NextPoint

, for

i = 1, 2

12: $ftype$ – Nag_TS_Transform Input

On entry: the function type used to define the relationship between

y

and

z

when calculating

EMA [τ, 1; y]

. Three functions are provided:

$ftype = Nag_Identity$: The identity function, with $y_{i} = {z_{i}}^{[p]}$ .
$ftype = Nag_AbsVal$: The absolute value, with $y_{i} = {| z_{i} |}^{p}$ .
$ftype = Nag_AbsDiff$: The absolute difference, with $y_{i} = {| z_{i} - x_{i} |}^{p}$ , where the vector $x$ is supplied in x.

Constraint:

ftype = Nag_Identity

Nag_AbsVal

Nag_AbsDiff

13: $p$ – double * Input/Output

On entry:

p

, the power used in the transformation function.

On exit: if

ftype = Nag_Identity

, then

[p]

, the actual power used in the transformation function is returned, otherwise p is unchanged.

Constraint:

p \neq 0

14: $x [\dim]$ – const double Input

Note: the dimension, dim, of the array x must be at least

$nb$ when $ftype = Nag_AbsDiff$ .

On entry: if

ftype = Nag_AbsDiff

x_{i}

, the vector used to shift the current block of observations, for

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.

ftype \neq Nag_AbsDiff

then x is not referenced and may be NULL.

Constraint: if

ftype = Nag_AbsDiff

and

p < 0

x [i - 1] \neq z [i - 1]

, for

i = 1, 2, \dots, nb

15: $pn$ – Integer * Input/Output

On entry:

k

, the number of observations processed so far. On the first call to g13mfc, 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 g13mfc.

On exit:

k + b

, the updated number of observations processed so far.

Constraint:

pn \geq 0

16: $rcomm [\dim]$ – double Communication Array

Note: the dimension, dim, of the array rcomm must be at least

$m2 + 20$ , when $rcomm is not NULL$ .

On entry: communication array, used to store information between calls to g13mfc. If

rcomm is NULL

then pn must be set to zero and all the data must be supplied in one go.

17: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE: On entry, $order = Nag_ColMajor$ , $pdiema = ⟨ value ⟩$ and $nb = ⟨ value ⟩$ .
Constraint: $pdiema \geq nb$ .

On entry, $order = Nag_RowMajor$ , $pdiema = ⟨ value ⟩$ and $m2 - m1 + 1 = ⟨ value ⟩$ .
Constraint: $pdiema \geq m2 - m1 + 1$ .
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_ILLEGAL_COMM: rcomm has been corrupted between calls.
NE_INT: On entry, $m1 = ⟨ value ⟩$ .
Constraint: $m1 \geq 1$ .

On entry, $nb = ⟨ value ⟩$ .
Constraint: $nb \geq 0$ .

On entry, $pn = ⟨ value ⟩$ .
Constraint: $pn \geq 0$ .
NE_INT_2: On entry, $m1 = ⟨ value ⟩$ and $m2 = ⟨ value ⟩$ .
Constraint: $m2 \geq m1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING: On entry, $i = ⟨ value ⟩$ , $t [i - 2] = ⟨ value ⟩$ and $t [i - 1] = ⟨ value ⟩$ .
Constraint: t should be strictly increasing.
NE_PREV_CALL: If $pn > 0$ then ftype must be unchanged since previous call.

If $pn > 0$ then inter must be unchanged since previous call.

On entry, $m1 = ⟨ value ⟩$ .
On entry at previous call, $m1 = ⟨ value ⟩$ .
Constraint: if $pn > 0$ then m1 must be unchanged since previous call.

On entry, $m2 = ⟨ value ⟩$ .
On entry at previous call, $m2 = ⟨ value ⟩$ .
Constraint: if $pn > 0$ then m2 must be unchanged since previous call.

On entry, $p = ⟨ value ⟩$ .
On exit from previous call, $p = ⟨ value ⟩$ .
Constraint: if $pn > 0$ then p must be unchanged since previous call.

On entry, $pn = ⟨ value ⟩$ .
On exit from previous call, $pn = ⟨ value ⟩$ .
Constraint: if $pn > 0$ then pn must be unchanged since previous call.

On entry, $tau = ⟨ value ⟩$ .
On entry at previous call, $tau = ⟨ value ⟩$ .
Constraint: if $pn > 0$ then tau must be unchanged since previous call.
NE_REAL: On entry, $i = ⟨ value ⟩$ , $z [i - 1] = ⟨ value ⟩$ and $p = ⟨ value ⟩$ .
Constraint: if $ftype = Nag_Identity$ or $Nag_AbsVal$ and $z [i] = 0$ for any $i$ then $p > 0.0$ .

On entry, $i = ⟨ value ⟩$ , $z [i - 1] = ⟨ value ⟩$ , $x [i - 1] = ⟨ value ⟩$ and $p = ⟨ value ⟩$ .
Constraint: if $ftype = Nag_AbsDiff$ and $z [i] = x [i]$ for any $i$ then $p > 0.0$ .

On entry, $p = ⟨ value ⟩$ .
Constraint: absolute value of p must be representable as an integer.

On entry, $p = ⟨ value ⟩$ .
Constraint: if $ftype \neq Nag_Identity$ , $p \neq 0.0$ . If $ftype = Nag_Identity$ , the nearest integer to $p$ must not be $0$ .

On entry, $tau = ⟨ value ⟩$ .
Constraint: $tau > 0.0$ .
NE_REAL_ARRAY: On entry, $ftype \neq Nag_Identity$ , $j = ⟨ value ⟩$ and $sinit [j - 1] = ⟨ value ⟩$ .
Constraint: if $ftype \neq Nag_Identity$ , $sinit [j - 1] \geq 0.0$ , for $j = 2, 3, \dots, m2 + 2$ .

On entry, $i = ⟨ value ⟩$ , $t [i - 2] = ⟨ value ⟩$ and $t [i - 1] = ⟨ value ⟩$ .
Constraint: $t [i - 1] \neq t [i - 2]$ if linear interpolation is being used.
NW_OVERFLOW_WARN: Truncation occurred to avoid overflow, check for extreme values in t, z, x or for tau. Results are returned using the truncated values.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g13mfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g13mfc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Approximately

4 \times m2

real elements are internally allocated by g13mfc.

The more data you supply to g13mfc in one call, i.e., the larger nb is, the more efficient the routine will be.

Checks are made during the calculation of

α

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

fail . code =

NW_OVERFLOW_WARN is returned. This should not occur in standard usage and will only occur if extreme values of z, t, x or tau are supplied.

10 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

g13mf: FL CL CPP AD PY MB

NAG CL Interfaceg13mfc (inhom_​iema_​all)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g13mfc (inhom_iema_all)