modwtLayer

Maximal overlap discrete wavelet transform (MODWT) layer

Since R2022b

Description

A MODWT layer computes the MODWT and MODWT multiresolution analysis (MRA) of the input. Use of this layer requires Deep Learning Toolbox™.

Creation

Syntax

layer = modwtLayer

layer = modwtLayer(Name=Value)

Description

layer = modwtLayer creates a MODWT layer. By default, the layer computes the MODWTMRA to level 5 using the Daubechies least-asymmetric wavelet with four vanishing moments ('sym4') .

The input to modwtLayer must be a dlarray (Deep Learning Toolbox) object in "CBT" format. The size of the time dimension of the tensor input must be greater than or equal to 2^Level, where Level is the transform level of the MODWT. modwtLayer formats the output as "SCBT". For more information, see Layer Output Format.

Note

The object initializes the weights internally for use as wavelet filters in the MODWT. It is not recommended to initialize the weights directly.

layer = modwtLayer(Name=Value) creates a MODWT layer with properties specified by name-value arguments. For example, layer = modwtLayer(Wavelet="haar") creates a MODWT layer that uses the Haar wavelet. You can specify the wavelet and the level of decomposition, among others.

Properties

MODWT

Wavelet — Orthogonal wavelet

"sym4" (default) | character vector | string scalar

This property is read-only.

Name of an orthogonal wavelet used in the MODWT, specified as a character vector or a string scalar.

Orthogonal wavelets are designated as type 1 wavelets in the wavelet manager. Valid built-in orthogonal wavelet families begin with 'haar', 'dbN', 'fkN', 'coifN', 'blN', 'hanSR.LP', 'symN', 'vaid', or 'beyl'. Use waveinfo with the wavelet family short name to see supported values for any numeric suffixes and how to interpret those values. For example, waveinfo("han").

For a wavelet specified by wname, the associated lowpass and highpass filters Lo and Hi, respectively, are [~,~,Lo,Hi] = wfilters(wname).

Data Types: char | string

LowpassFilter,HighpassFilter — Initial wavelet filter pair

even-length real-valued vectors

This property is read-only.

Initial wavelet filter pair, specified as a pair of even-length real-valued vectors. The lengths of the vectors must be equal. LowpassFilter and HighpassFilter must correspond to the lowpass and highpass filters, respectively, associated with an orthogonal wavelet. You can use isorthwfb to determine orthogonality.

[~,~,Lo,Hi] = wfilters("db2");
[tf,checks] = isorthwfb(Lo,Hi);

If unspecified, both filters default to [] and modwtLayer uses Wavelet to determine the filters used in MODWT.

You cannot specify both a wavelet name and a wavelet filter pair.

Example: layer = modwtLayer('LowpassFilter',Lo,'HighpassFilter',Hi)

Data Types: single | double

Level — Transform level

5 (default) | positive integer

This property is read-only.

Transform level to compute the MODWT, specified as a positive integer less than or equal to floor(log2(N)), where N is the size of the layer input in the time dimension.

Data Types: single | double

Boundary — Boundary condition

"periodic" (default) | "reflection"

This property is read-only.

Boundary condition to use in the computation of the MODWT, specified as one of these:

• "periodic" — The signal is extended periodically along the time dimension. The number of wavelet coefficients equals the size of the signal in the time dimension.

• "reflection" — The signal is reflected symmetrically along the time dimension at the terminal end before computing the MODWT. The number of wavelet coefficients returned is twice the length of the input signal.

SelectedLevels — Selected levels

1:Level (default) | vector

This property is read-only.

Selected levels for modwtLayer to output, specified as a vector of positive integers less than or equal to Level.

Data Types: single | double

IncludeLowpass — Include lowpass coefficients

true or 1 (default) | false or 0

This property is read-only.

Include lowpass coefficients, specified as a numeric or logical 1 (true) or 0 (false). If specified as true, the MODWT layer includes the Levelth level lowpass (scaling) coefficients in the MODWT, or Levelth level smooth in the MODWTMRA.

Data Types: logical

AggregateLevels — Aggregate selected levels

false or 0 (default) | true or 1

This property is read-only.

Aggregate selected levels, specified as a numeric or logical 1 (true) or 0 (false). If specified as true, the MODWT layer aggregates the selected levels and lowpass level (if IncludeLowpass is true) of each input channel by summation. If AggregateLevels is true, the size of the output in the spatial dimension is 1. For more information, Compare modwtLayer Output with modwt and modwtmra Outputs.

Data Types: logical

Algorithm — Algorithm

"MODWTMRA" (default) | "MODWT"

This property is read-only.

Algorithm modwtLayer uses to compute the output, specified as one of these:

• "MODWTMRA" — Compute the maximal overlap discrete wavelet transform based multiresolution analysis.

• "MODWT" — Compute the wavelet coefficients of the maximal overlap discrete wavelet transform.

For more information, see Comparing MODWT and MODWTMRA.

Layer

WeightLearnRateFactor — Multiplier for weight learning rate

0 (default) | nonnegative scalar

Multiplier for weight learning rate, specified as a nonnegative scalar. If not specified, this property defaults to zero, resulting in weights that do not update with training. You can also set this property using the setLearnRateFactor (Deep Learning Toolbox) function.

The learnable parameter 'Weights' in modwtLayer is a two-row matrix of the current filter pair. The first row is the lowpass filter and the second row is the highpass filter. By default, the weights are the lowpass and highpass filters associated with the default wavelet and do not update.

Data Types: single | double

Name — Layer name

'' (default) | character vector | string scalar

Layer name, specified as a character vector or a string scalar. For Layer array input, the trainNetwork (Deep Learning Toolbox), assembleNetwork (Deep Learning Toolbox), layerGraph (Deep Learning Toolbox), and dlnetwork (Deep Learning Toolbox) functions automatically assign names to layers with the name ''.

Data Types: char | string

NumInputs — Number of inputs

1 (default)

This property is read-only.

Number of inputs of the layer. This layer accepts a single input only.

Data Types: double

InputNames — Input names

{"in"} (default)

This property is read-only.

Input names of the layer. This layer accepts a single input only.

Data Types: cell

NumOutputs — Number of outputs

1 (default)

This property is read-only.

Number of outputs of the layer. This layer has a single output only.

Data Types: double

OutputNames — Output names

{'out'} (default)

This property is read-only.

Output names of the layer. This layer has a single output only.

Data Types: cell

Examples

Use modwtLayer in Deep Learning Network

Create a MODWT layer to compute the multiresolution analysis for the input signal. Use a coiflet wavelet with order 5. Set the transform level to 8. Only keep the details at levels 3, 5, and 7, and the approximation.

layer = modwtLayer(Wavelet="coif5",Level=8, ...
    SelectedLevels=[3,5,7],Name="MODWT");

Create a dlnetwork object containing a sequence input layer, a MODWT layer, and an LSTM layer. For a level-8 decomposition, set the minimum sequence length to 2^8 samples. To work with an LSTM layer, a flatten layer is also needed before the LSTM layer to collapse the spatial dimension into the channel dimension.

mLength=2^8;
sqLayer = sequenceInputLayer(1,Name="input",MinLength=mLength);
layers = [sqLayer
    layer
    flattenLayer
    lstmLayer(10,Name="LSTM")
    ];
dlnet = dlnetwork(layers);

Run a batch of 10 random single-channel signals through the dlnetwork object. Inspect the size and dimensions of the output. The flatten layer has collapsed the spatial dimension.

dataout = dlnet.forward(dlarray(randn(1,10,2000,'single'),'CBT'));
size(dataout)

ans = 1×3

          10          10        2000

dims(dataout)

ans = 
'CBT'

Compare modwtLayer Output with modwt and modwtmra Outputs

Load the Espiga3 electroencephalogram (EEG) dataset. The data consists of 23 channels of EEG sampled at 200 Hz. There are 995 samples in each channel. Save the multisignal as a dlarray, specifying the dimensions in order. dlarray permutes the array dimensions to the "CBT" shape expected by a deep learning network.

load Espiga3
[N,nch] = size(Espiga3);
x = dlarray(Espiga3,"TCB");

Use modwt and modwtmra to obtain the MODWT and MRA of the multisignal down to level 6. By default, modwt and modwtmra use the sym4 wavelet.

lev = 6;
wt = modwt(Espiga3,lev);
mra = modwtmra(wt);

Compare with modwt

Create a MODWT layer that can be used with the data. Set the transform level to 6. Specify the layer to use MODWT to compute the output. By default, the layer uses the sym4 wavelet.

mlayer = modwtLayer(Level=lev,Algorithm="MODWT");

Create a two-layer dlnetwork object containing a sequence input layer and the MODWT layer you just created. Treat each channel as a feature. For a level-6 decomposition, set the minimum sequence length to 2^6.

mLength = mlayer.Level;
sqInput = sequenceInputLayer(nch,MinLength=2^mLength);
layers = [sqInput
    mlayer];
dlnet = dlnetwork(layers);

Run the EEG data through the forward method of the network.

dataout = forward(dlnet,x);

The modwt and modwtmra functions return the MODWT and MRA of a multichannel signal as a 3-D array. The first, second, and third dimensions of the array correspond to the wavelet decomposition level, signal length, and channel, respectively. Convert the network output to a numeric array. Permute the dimensions of the network output to match the function output. Compare the network output with the modwt output.

q = extractdata(dataout);
q = permute(q,[1 4 2 3]);
max(abs(q(:)-wt(:)))

ans = 8.4402e-05

Choose a MODWT result from modwtLayer. Compare with the corresponding channel in the EEG data. Plot each level of the modwtLayer output. Different levels contain information about the signal in different frequency ranges. The levels are not time aligned with the original signal because the layer uses the MODWT algorithm.

channel = 10;
t = 100:400;
subplot(lev+2,1,1)
plot(t,Espiga3(t,channel))
ylabel("Original EEG")
for k=2:lev+1
    subplot(lev+2,1,k)
    plot(t,q(k-1,t,channel))
    ylabel(["Level ",k-1," of MODWT"])
end
subplot(lev+2,1,lev+2)
plot(t,q(lev+1,t,channel))
ylabel(["Scaling","Coefficients","of MODWT"])
set(gcf,Position=[0 0 500 700])

Compare with modwtmra

Create a second network similar to the first network, except this time specify that modwtLayer use the MODWTMRA algorithm and aggregate the fourth, fifth, and sixth levels. Do not include the lowpass level in the aggregation.

sLevels = [4 5 6];
mlayer = modwtLayer(Level=lev, ...
    SelectedLevels=sLevels, ...
    IncludeLowpass=0, ...
    AggregateLevels=1);
layers = [sqInput
    mlayer];
dlnet2 = dlnetwork(layers);

Run the EEG data through the forward method of the network. Convert the network output to a numeric array. Permute the dimensions as done previously.

dataout = forward(dlnet2,x);
q = extractdata(dataout);
q = permute(q,[1 4 3 2]);

Aggregate the fourth, fifth, and sixth levels of the MRA. Compare with the network output.

mraAggregate = sum(mra(sLevels,:,:));
max(abs(q(:)-mraAggregate(:)))

ans = 2.1036e-04

Inspect a MODWTMRA result from the layer. Compare with the corresponding channel in the EEG data. By choosing only the fourth, fifth, and six levels, and not including the lowpass component, the layer removes several high and low frequency components from the signal. The transformed signal is smoother than the original signal and the low frequency components are removed so that the offset is closer to 0. The output is time aligned with the original signal because the layer uses the default MODWTMRA algorithm. Depending on your goal, preserving time alignment can be useful.

channel = 10;
t = 100:400;
figure
hold on
plot(t, Espiga3(t,channel))
plot(t,q(1,t,1,channel))
hold off
legend(["Original EEG", "Layer Output"], ...
    Location="northwest")

Comparing MODWT and MODWTMRA

This example demonstrates the differences between the MODWT and MODWTMRA. The MODWT partitions a signal's energy across detail coefficients and scaling coefficients. The MODWTMRA projects a signal onto wavelet subspaces and a scaling subspace.

Choose the sym6 wavelet. Load and plot an electrocardiogram (ECG) signal. The sampling frequency for the ECG signal is 180 hertz. The data are taken from Percival and Walden (2000), p.125 (data originally provided by William Constantine and Per Reinhall, University of Washington).

load wecg
t = (0:numel(wecg)-1)/180;
wv = 'sym6';
plot(t,wecg)
grid on
title(['Signal Length = ',num2str(numel(wecg))])
xlabel('Time (s)')
ylabel('Amplitude')

Take the MODWT of the signal.

wtecg = modwt(wecg,wv);

The input data are samples of a function $$f(x)$$ evaluated at $$N$$ time points. The function can be expressed as a linear combination of the scaling function $$\varphi (x)$$ and wavelet $$\psi (x)$$ at varying scales and translations: $$f(x)=\sum _{k=0}^{N-1}{c}_k{2}^{-J_0/2}\varphi (2^{-J_0}x-k)+\sum _{j=1}^{J_0}{f}_j(x)$$, where $$f_j(x)=\sum _{k=0}^{N-1}{d}_{j,k}{2}^{-j/2}\psi (2^{-j}x-k)$$ and $$J_0$$ is the number of levels of wavelet decomposition. The first sum is the coarse scale approximation of the signal, and the $$f_j(x)$$ are the details at successive scales. MODWT returns the $$N$$ coefficients $$\{c_k\}$$ and the $$(J_0\times N)$$ detail coefficients $$\{d_{j,k}\}$$ of the expansion. Each row in wtecg contains the coefficients at a different scale.

When taking the MODWT of a signal of length $$N$$, there are $$\text{floor}({\log }_2(N))$$ levels of decomposition by default. Detail coefficients are produced at each level. Scaling coefficients are returned only for the final level. In this example, $$N=2048$$, $$J_0=\text{floor}(\log 2(2048))=11$$, and the number of rows in wtecg is $$J_0+1=11+1=12$$.

The MODWT partitions the energy across the various scales and scaling coefficients: $${||X||}^2=\sum _{j=1}^{J_0}{||W_j||}^2+{||V_{J_0}||}^2$$, where $$X$$ is the input data, $$W_j$$ are the detail coefficients at scale $$j$$, and $$V_{J_0}$$ are the final-level scaling coefficients.

Compute the energy at each scale, and evaluate their sum.

energy_by_scales = sum(wtecg.^2,2);
Levels = {'D1';'D2';'D3';'D4';'D5';'D6';...
    'D7';'D8';'D9';'D10';'D11';'A11'};
energy_table = table(Levels,energy_by_scales);
disp(energy_table)

    Levels     energy_by_scales
    _______    ________________

    {'D1' }         14.063     
    {'D2' }         20.612     
    {'D3' }         37.716     
    {'D4' }         25.123     
    {'D5' }         17.437     
    {'D6' }         8.9852     
    {'D7' }         1.2906     
    {'D8' }         4.7278     
    {'D9' }         12.205     
    {'D10'}         76.428     
    {'D11'}         76.268     
    {'A11'}         3.4192     

energy_total = varfun(@sum,energy_table(:,2))

energy_total=table
    sum_energy_by_scales
    ____________________

           298.28       

Confirm the MODWT is energy-preserving by computing the energy of the signal and comparing it with the sum of the energies over all scales.

energy_ecg = sum(wecg.^2);
max(abs(energy_total.sum_energy_by_scales-energy_ecg))

ans = 7.4402e-10

Take the MODWTMRA of the signal.

mraecg = modwtmra(wtecg,wv);

MODWTMRA returns the projections of the function $$f(x)$$ onto the various wavelet subspaces and final scaling space. That is, MODWTMRA returns $$\sum _{k=0}^{N-1}{c}_k{2}^{-J_0/2}\varphi (2^{-J_0}x-k)$$ and the $$J_0$$-many $$\{f_j(x)\}$$ evaluated at $$N$$ time points. Each row in mraecg is a projection of $$f(x)$$ onto a different subspace. This means the original signal can be recovered by adding all the projections. This is not true in the case of the MODWT. Adding the coefficients in wtecg will not recover the original signal.

Choose a time point, add the projections of $$f(x)$$ evaluated at that time point, and compare with the original signal.

time_point = 1000;
abs(sum(mraecg(:,time_point))-wecg(time_point))

ans = 3.0846e-13

Confirm that, unlike MODWT, MODWTMRA is not an energy-preserving transform.

energy_ecg = sum(wecg.^2);
energy_mra_scales = sum(mraecg.^2,2);
energy_mra = sum(energy_mra_scales);
max(abs(energy_mra-energy_ecg))

ans = 115.7053

The MODWTMRA is a zero-phase filtering of the signal. Features will be time-aligned. Show this by plotting the original signal and one of its projections. To better illustrate the alignment, zoom in.

plot(t,wecg,'b')
hold on
plot(t,mraecg(4,:),'-')
hold off
grid on
xlim([4 8])
legend('Signal','Projection','Location','northwest')
xlabel('Time (s)')
ylabel('Amplitude')

Make a similar plot using the MODWT coefficients at the same scale. Features will not be time-aligned. The MODWT is not a zero-phase filtering of the input.

plot(t,wecg,'b')
hold on
plot(t,wtecg(4,:),'-')
hold off
grid on
xlim([4 8])
legend('Signal','Coefficients','Location','northwest')
xlabel('Time (s)')
ylabel('Amplitude')

More About

Layer Output Format

modwtLayer formats the output as "SCBT", a sequence of 1-D images where the image height corresponds to the level of the wavelet transform, the second dimension corresponds to the channel, the third dimension corresponds to the batch, and the fourth dimension corresponds to time. The kth row, where k ≤ Level, contains the kth level detail of the signal. The (Level+1)th row contains the Levelth level smooth of the signal.

• You can feed the output of modwtLayer unchanged to a 1-D convolutional layer when you want to convolve along the level ("S") dimension. For more information, see convolution1dLayer (Deep Learning Toolbox).

• To feed the output of modwtLayer to a 1-D convolutional layer when you want to convolve along the time ("T") dimension, you must place a flatten layer after the modwtLayer. For more information, see flattenLayer (Deep Learning Toolbox).

• You can feed the output of modwtLayer unchanged to a 2-D convolutional layer when you want to convolve along the level ("S") and time ("T") dimensions jointly. For more information, see convolution2dLayer (Deep Learning Toolbox).

• To use modwtLayer as part of a recurrent neural network, you must place a flatten layer after the modwtLayer. For more information, see lstmLayer (Deep Learning Toolbox) and gruLayer (Deep Learning Toolbox).

• To use the output of modwtLayer with a fully connected layer as part of a classification workflow, you must reduce the time ("T") dimension of the output so that it is of size 1. To reduce the time dimension of the output, place a global pooling layer before the fully connected layer. For more information, see globalAveragePooling2dLayer (Deep Learning Toolbox) and fullyConnectedLayer (Deep Learning Toolbox).

Version History

Introduced in R2022b