NAG Library Function Document

nag_mlmodwt (c09dcc)

+− 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_mlmodwt (c09dcc) computes the one-dimensional multi-level maximal overlap discrete wavelet transform (MODWT). The initialization function nag_wfilt (c09aac) must be called first to set up the MODWT options.

2 Specification

#include <nag.h>

#include <nagc09.h>

void	nag_mlmodwt (Integer n, const double x[], Nag_WaveletCoefficients keepa, Integer lenc, double c[], Integer nwl, Integer na, Integer icomm[], NagError fail)

3 Description

nag_mlmodwt (c09dcc) computes the multi-level MODWT for a data set,

x_{i}

, for

i = 1, 2, \dots, n

, in one dimension. For a chosen number of levels,

n_{l}

, with

n_{l} \leq l_{\max}

, where

l_{\max}

is returned by the initialization function nag_wfilt (c09aac) in nwlmax, the transform is returned as a set of coefficients for the different levels stored in a single array. Periodic reflection is currently the only available end extension method to reduce the edge effects caused by finite data sets.

The argument keepa can be set to retain both approximation and detail coefficients at each level resulting in

n_{l} \times (n_{a} + n_{d})

coefficients being returned in the output array, c, where

n_{a}

is the number of approximation coefficients and

n_{d}

is the number of detail coefficients. Otherwise, only the detail coefficients are stored for each level along with the approximation coefficients for the final level, in which case the length of the output array, c, is

n_{a} + n_{l} \times n_{d}

. In the present implementation, for simplicity,

n_{a}

and

n_{d}

are chosen to be equal by adding zero padding to the wavelet filters where necessary.

4 References

Percival D B and Walden A T (2000) Wavelet Methods for Time Series Analysis Cambridge University Press

5 Arguments

1: n – IntegerInput

On entry: the number of elements,

n

, in the data array

x

Constraint: this must be the same as the value n passed to the initialization function nag_wfilt (c09aac).

2: x[n] – const doubleInput

On entry: x contains the input dataset

x_{i}

, for

i = 1, 2, \dots, n

3: keepa – Nag_WaveletCoefficientsInput

On entry: determines whether the approximation coefficients are stored in array c for every level of the computed transform or else only for the final level. In both cases, the detail coefficients are stored in c for every level computed.

$keepa = Nag_StoreAll$: Retain approximation coefficients for all levels computed.
$keepa = Nag_StoreFinal$: Retain approximation coefficients for only the final level computed.

Constraint:

keepa = Nag_StoreAll

Nag_StoreFinal

4: lenc – IntegerInput

On entry: the dimension of the array c. c must be large enough to contain the number of wavelet coefficients.

keepa = Nag_StoreFinal

, the total number of coefficients,

n_{c}

, is returned in nwc by the call to the initialization function nag_wfilt (c09aac) and corresponds to the MODWT being continued for the maximum number of levels possible for the given data set. When the number of levels,

n_{l}

, is chosen to be less than the maximum, then the number of stored coefficients is correspondingly smaller and lenc can be reduced by noting that

n_{d}

detail coefficients are stored at each level, with the storage increased at the final level to contain the

n_{a}

approximation coefficients.

keepa = Nag_StoreAll

n_{d}

detail coefficients and

n_{a}

approximation coefficients are stored for each level computed, requiring

lenc \geq n_{l} \times (n_{a} + n_{d}) = 2 \times n_{l} \times n_{a}

, since the numbers of stored approximation and detail coefficients are equal. The number of approximation (or detail) coefficients at each level,

n_{a}

, is returned in na.

Constraints:

if $keepa = Nag_StoreFinal$ , $lenc \geq (n_{l} + 1) \times n_{a}$ ;
if $keepa = Nag_StoreAll$ , $lenc \geq 2 \times n_{l} \times n_{a}$ .

5: c[lenc] – doubleOutput

On exit: the coefficients of a multi-level wavelet transform of the dataset.

The coefficients are stored in c as follows:

keepa = Nag_StoreFinal

$C (1 : n_{a})$: Contains the level $n_{l}$ approximation coefficients;
$C (n_{a} + (i - 1) \times n_{d} + 1 : n_{a} + i \times n_{d})$: Contains the level $(n_{l} - i + 1)$ detail coefficients, for $i = 1, 2, \dots, n_{l}$ ;

keepa = Nag_StoreAll

$C ((i - 1) \times n_{a} + 1 : i \times n_{a})$: Contains the level $(n_{l} - i + 1)$ approximation coefficients, for $i = 1, 2, \dots, n_{l}$ ;
$C (n_{l} \times n_{a} + (i - 1) \times n_{d} + 1 : n_{l} \times n_{a} + i \times n_{d})$: Contains the level i detail coefficients, for $i = 1, 2, \dots, n_{l}$ ;

The values

n_{a}

and

n_{d}

denote the numbers of approximation and detail coefficients respectively, which are equal and returned in na.

6: nwl – IntegerInput

On entry: the number of levels,

n_{l}

, in the multi-level resolution to be performed.

Constraint:

1 \leq nwl \leq l_{\max}

, where

l_{\max}

is the value returned in nwlmax (the maximum number of levels) by the call to the initialization function nag_wfilt (c09aac).

7: na – Integer *Output

On exit: na contains the number of approximation coefficients,

n_{a}

, at each level which is equal to the number of detail coefficients,

n_{d}

. With periodic end extension (

mode = Nag_Periodic

in nag_wfilt (c09aac)) this is the same as the length, n, of the data array, x.

8: icomm[ $100$ ] – IntegerCommunication Array

On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wfilt (c09aac).

On exit: contains additional information on the computed transform.

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ARRAY_DIM_LEN: On entry, lenc is set too small: $lenc = ⟨value⟩$ .
Constraint: $lenc \geq ⟨value⟩$ .
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INITIALIZATION: On entry, n is inconsistent with the value passed to the initialization function: $n = ⟨value⟩$ , n should be $⟨value⟩$ .
On entry, nwl is larger than the maximum number of levels returned by the initialization function: $nwl = ⟨value⟩$ , maximum = $⟨value⟩$ .
On entry, the initialization function nag_wfilt (c09aac) has not been called first or it has not been called with $wtrans = Nag_MODWTMulti$ , or the communication array icomm has become corrupted.
NE_INT: On entry, $nwl = ⟨value⟩$ .
Constraint: $nwl \geq 1$ .
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.

7 Accuracy

The accuracy of the wavelet transform depends only on the floating-point operations used in the convolution and downsampling and should thus be close to machine precision.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The wavelet coefficients at each level can be extracted from the output array c using the information contained in na on exit.

10 Example

A set of data values (

n = 64

) is decomposed using the MODWT over two levels and then the inverse (nag_imlmodwt (c09ddc)) is applied to restore the original data set.

NAG Library Function Documentnag_mlmodwt (c09dcc)