void	g13mgc (Integer nb, double ma[], const double t[], double tau, Integer m1, Integer m2, const double sinit[], const Nag_TS_Interpolation inter[], Nag_TS_Transform ftype, double p, Integer pn, double wma[], double rcomm[], NagError *fail)

The function may be called by the names: g13mgc or nag_tsa_inhom_ma.

3 Description

g13mgc 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

(t_{i}, z_{i})

, for

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

τ

is defined as

MA [τ, m_{1}, m_{2}; y] (t_{i}) = \frac{1}{m_{2} - m_{1} + 1} \sum_{j = m_{1}}^{m_{2}} EMA [\tilde{τ}, j; y] (t_{i})

(1)

where

\tilde{τ} = \frac{2 τ}{m_{2} + m_{1}}

m_{1}

and

m_{2}

are user-supplied integers controlling the amount of lag and smoothing respectively, with

m_{2} \geq m_{1}

and

EMA (\cdot)

is the iterated exponential moving average operator.

The iterated exponential moving average,

EMA [\tilde{τ}, m; y] (t_{i})

, is defined using the recursive formula:

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

with

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

and

EMA [\tilde{τ}; y] (t_{i}) = μ EMA [\tilde{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}}{\tilde{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. g13mgc 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:	$y_{i} = {\|z_{i} - MA [τ, m_{1}, m_{2}; z] (t_{i})\|}^{p}$ .

where the notation

[p]

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 g13mgc, a few of which are:

(i)Moving Average (MA), as defined in (1) (obtained by setting $ftype = Nag_Identity$ and $p = 1$ ).
(ii)Moving Norm (MNorm), defined as
$MNorm (τ, m, p; z) = {MA [τ, 1, m; {|z|}^{p}]}^{1 / p}$
(obtained by setting $ftype = Nag_AbsValScaled$ , $m1 = 1$ and $m2 = m$ ).
(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 $ftype = Nag_AbsDiff$ , $m1 = 1$ and $m2 = m$ ).
(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 $ftype = Nag_AbsDiffScaled$ , $m1 = 1$ and $m2 = m$ ).

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 g13mgc 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: $nb$ – 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 ma and t can vary; therefore nb can change between calls to g13mgc.

Constraint:

nb \geq 0

2: $ma [nb]$ – double Input/Output

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.

On exit: the moving average:

if $ftype = Nag_AbsValScaled$ or $Nag_AbsDiffScaled$: $ma [i - 1] = {\{MA [τ, m_{1}, m_{2}; y] (t_{i})\}}^{1 / p}$ ,
otherwise: $ma [i - 1] = MA [τ, m_{1}, m_{2}; y] (t_{i})$ .

3: $t [nb]$ – 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 g13mgc 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^{- α}

α = (t_{i} - t_{i - 1}) / \tilde{τ}

can be calculated without overflowing, for all

i

4: $tau$ – 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}) / \tilde{τ}

can be calculated without overflowing, for all

i

, where

\tilde{τ} = \frac{2 τ}{m_{2} + m_{1}}

Constraint:

tau > 0.0

5: $m1$ – Integer Input

On entry:

m_{1}

, the iteration of the EMA operator at which the sum is started.

Constraint:

m1 \geq 1

6: $m2$ – Integer Input

On entry:

m_{2}

, the iteration of the EMA operator at which the sum is ended.

Constraint:

m2 \geq m1

7: $sinit [\dim]$ – const double Input

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

$2 \times m2 + 3$ when $ftype = Nag_AbsDiff$ or $Nag_AbsDiffScaled$ ;
$m2 + 2$ when $ftype = Nag_Identity$ , $Nag_AbsVal$ or $Nag_AbsValScaled$ ;
sinit may be NULL when $pn \neq 0$ .

On entry: if

pn = 0

, the values used to start the iterative process, with

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

In addition, if

ftype = Nag_AbsDiff

Nag_AbsDiffScaled

then

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

i.e., initial values based on the original data

z

as opposed to the transformed data

y

pn \neq 0

, 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

8: $inter [2]$ – 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

9: $ftype$ – 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 $y_{i} = {z_{i}}^{[p]}$ .
$ftype = Nag_AbsVal$ or $Nag_AbsValScaled$: The absolute value, with $y_{i} = {|z_{i}|}^{p}$ .
$ftype = Nag_AbsDiff$ or $Nag_AbsDiffScaled$: The absolute difference, with $y_{i} = {|z_{i} - MA [τ, m; y] (t_{i})|}^{p}$ .

ftype = Nag_AbsValScaled

Nag_AbsDiffScaled

then the resulting vector of averages is scaled by

p^{- 1}

as described in ma.

Constraint:

ftype = Nag_Identity

Nag_AbsVal

Nag_AbsDiff

Nag_AbsValScaled

Nag_AbsDiffScaled

10: $p$ – 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

11: $pn$ – Integer * Input/Output

On entry:

k

, the number of observations processed so far. On the first call to g13mgc, 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 g13mgc.

On exit:

k + b

, the updated number of observations processed so far.

Constraint:

pn \geq 0

12: $wma [nb]$ – double Output

On exit: either the moving average or exponential moving average, depending on the value of ftype.

if $ftype = Nag_AbsDiff$ or $Nag_AbsDiffScaled$: $wma [i - 1] = MA [τ; y] (t_{i})$
otherwise: $wma [i - 1] = EMA [\tilde{τ}; y] (t_{i})$ .

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

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

$2 \times m2 + 20$ , when $rcomm is not NULL$ ;
$0$ , otherwise.

On entry: communication array, used to store information between calls to g13mgc. If

rcomm is NULL

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

14: $fail$ – 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_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, $i = 〈value〉$ , $ma [i - 1] = 〈value〉$ and $p = 〈value〉$ .
Constraint: if $ftype = Nag_Identity$ , $Nag_AbsVal$ or $Nag_AbsValScaled$ and $ma [i - 1] = 0$ for any $i$ then $p > 0.0$ .

On entry, $i = 〈value〉$ , $ma [i - 1] = 〈value〉$ , $wma [i - 1] = 〈value〉$ and $p = 〈value〉$ .
Constraint: if $p < 0.0$ , $ma [i - 1] - wma [i - 1] \neq 0.0$ , for any $i$ .

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, ma or for tau. Results are returned using the truncated values.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g13mgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g13mgc 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 m_{2}

real elements are internally allocated by g13mgc. If

ftype = Nag_AbsDiff

Nag_AbsDiffScaled

then a further nb real elements are also allocated.

The more data you supply to g13mgc in one call, i.e., the larger nb is, the more efficient the function 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 ma, t or tau are supplied.

10 Example

The example reads in a simulated time series,

(t, z)

and calculates the moving average. The data is supplied in three blocks of differing sizes.

g13mg: FL CL CPP AD

NAG CL Interfaceg13mgc (inhom_​ma)

▸▿ 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
g13mgc (inhom_ma)