NAG Library Function Document

nag_tsa_inhom_iema_all (g13mfc)

+− 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

1 Purpose

nag_tsa_inhom_iema_all (g13mfc) calculates the iterated exponential moving average for an inhomogeneous time series, returning the intermediate results.

2 Specification

#include <nag.h>

#include <nagg13.h>

void

nag_tsa_inhom_iema_all (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)

3 Description

nag_tsa_inhom_iema_all (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. nag_tsa_inhom_iema_all (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 nag_tsa_inhom_iema_all (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_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: 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 nag_tsa_inhom_iema_all (g13mfc).

Constraint:

nb \geq 0

3: z[nb] – const doubleInput

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[ $\dim$ ] – doubleOutput

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

pdiema \times

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 – IntegerInput

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 doubleInput

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 nag_tsa_inhom_iema_all (g13mfc) will continue as if

t

was strictly increasing by using the absolute value.

7: tau – doubleInput

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 – IntegerInput

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

Constraint:

m1 \geq 1

9: m2 – IntegerInput

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

EMA [τ, m; y]

, for

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

Constraint:

m2 \geq m1

10: sinit[ $m2 + 2$ ] – const doubleInput

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_InterpolationInput

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_TransformInput

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[ $nb$ ] – const doubleInput

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 nag_tsa_inhom_iema_all (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 nag_tsa_inhom_iema_all (g13mfc).

On exit:

k + b

, the updated number of observations processed so far.

Constraint:

pn \geq 0

16: rcomm[ $m2 + 20$ ] – doubleCommunication Array

On entry: communication array, used to store information between calls to nag_tsa_inhom_iema_all (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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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

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

nag_tsa_inhom_iema_all (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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

Approximately

4 \times m2

real elements are internally allocated by nag_tsa_inhom_iema_all (g13mfc).

The more data you supply to nag_tsa_inhom_iema_all (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

NAG Library Function Documentnag_tsa_inhom_iema_all (g13mfc)

+− 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 Library Function Document

nag_tsa_inhom_iema_all (g13mfc)