waveletScattering

Wavelet time scattering

Description

Use the waveletScattering object to create a network for a wavelet time scattering decomposition using the Gabor (analytic Morlet) wavelet. The network uses wavelets and a lowpass scaling function to generate low-variance representations of real-valued time series data. Wavelet time scattering yields representations insensitive to translations in the input signal without sacrificing class discriminability. You can use the representations as inputs to a classifier. You can specify the duration of translation invariance and the number of wavelet filters per octave. The scattering network also supports time × channel × batch (T×C×B) inputs.

Creation

Syntax

sf = waveletScattering

sf = waveletScattering(Name,Value)

Description

sf = waveletScattering creates a wavelet time scattering network with two filter banks. The first filter bank has a quality factor of eight wavelets per octave. The second filter bank has a quality factor of one wavelet per octave. By default, waveletScattering assumes a signal input length of 1024 samples. The scale invariance length is 512 samples. By default, waveletScattering uses periodic boundary conditions.

sf = waveletScattering(Name,Value) creates a network for wavelet scattering, sf, with properties specified by one or more Name,Value pair arguments. Properties can be specified in any order as Name1,Value1,...,NameN,ValueN. Enclose each property name in quotes.

Note

With the exception of OversamplingFactor, after creation you cannot change a property value of an existing scattering network. For example, if you have a network sf with a SignalLength of 2000, you must create a second network sf2 for a signal with 2001 samples. You cannot assign a different SignalLength to sf.

Properties

SignalLength — Signal length

1024 (default) | positive integer ≥ 16

Signal length in samples, specified as a positive integer ≥ 16. If the input to the scattering network is a row vector, SignalLength must match the number of columns in the input data. If the input to the scattering network is a column vector, matrix, or 3-D array, SignalLength must match the number of rows in the data.

Data Types: double

SamplingFrequency — Sampling frequency

1 (default) | positive scalar

Sampling frequency in hertz, specified as a positive scalar. If unspecified, frequencies are in cycles/sample and the Nyquist frequency is ½.

Data Types: double

InvarianceScale — Scattering transform invariance scale

one-half of SignalLength (default) | positive scalar

Scattering transform invariance scale, specified as a positive scalar. InvarianceScale specifies the translation invariance of the scattering transform. If you do not specify SamplingFrequency, InvarianceScale is measured in samples. If you specify SamplingFrequency, InvarianceScale is measured in seconds. By default, InvarianceScale is one-half the SignalLength in samples.

InvarianceScale cannot exceed SignalLength in samples.

Example: sf = waveletScattering('SignalLength',1000,'SamplingFrequency',200,'InvarianceScale',5) has the largest possible InvarianceScale.

Data Types: double

QualityFactors — Scattering filter bank Q factors

[8 1] (default) | positive integer | vector of positive integers

Scattering filter bank Q factors, specified as a positive integer or a vector of positive integers. A filter bank Q factor is the number of wavelet filters per octave. Quality factors cannot exceed 32 and must be greater than or equal to 1.

If QualityFactors is specified as a vector, the elements of QualityFactors must be strictly decreasing.

Example: sf = waveletScattering('QualityFactors',[8 2 1]) creates a wavelet scattering network with three filter banks.

Data Types: double

Boundary — Signal extension method

'periodic' (default) | 'reflection'

Signal extension method to apply at the boundary:

• 'periodic' — Extend signal periodically to length 2^ceil(log2(N)), where N is the signal length.

• 'reflection' — Extend signal by reflection to length 2^ceil(log2(2 N)), where N is the signal length.

The signal is extended to match the length of the wavelet filters. The length of the filters are powers of two.

The signal extension method is for internal operations. Results are downsampled back onto the scale of the original signal before being returned.

Precision — Precision of scattering decomposition

'double' (default) | 'single'

Precision of scattering decomposition:

• 'double' — Double precision

• 'single' — Single precision

All calculations involving the wavelet scattering network are carried out in Precision. Wavelet scattering functions such as featureMatrix and filterbank return outputs such as filters in Precision. The precision of the output of the scatteringTransform function does not exceed the precision of sf.

Note

• If you construct a scattering network with double-precision filters and apply the network to single-precision data, the filters are cast internally to single-precision. Subsequent filtering is done with single precision until a new network is created regardless of input data type.

• Specifying Precision as 'single' at construction is especially useful in code generation because it eliminates the need to create an extra copy of the scattering filter banks.

OversamplingFactor — Oversampling factor

0 (default) | nonnegative integer | Inf

Oversampling factor, specified as a nonnegative integer or Inf. The factor specifies how much the scattering coefficients are oversampled with respect to the critically downsampled values. The factor is on a log₂ scale. By default, OversamplingFactor is set to 0, which corresponds to critically downsampling the coefficients. You can use numCoefficients to determine the number of coefficients obtained for a scattering network. To obtain a fully undecimated scattering transform, set OversamplingFactor to Inf.

Setting OversamplingFactor to a value that would result in more coefficients than samples is equivalent to setting OversamplingFactor to Inf. Increasing the OversamplingFactor significantly increases the computational complexity and memory requirements of the scattering transform.

Example: If sf = waveletScattering('OversamplingFactor',2), the scattering transform returns 22 times as many coefficients for each scattering path with respect to the critically sampled number.

OptimizePath — Optimize scattering transform logical

false or 0 (default) | true or 1

Optimize scattering transform logical which determines whether the scattering transform reduces the number of scattering paths to compute based on a bandwidth consideration, specified as a numeric or logical 1 (true) or 0 (false).

If you specify OptimizePath as true, the scattering transform excludes scattering paths of order 2 and greater which do not satisfy the following criterion: The center frequency minus ½ the 3-dB bandwidth of the wavelet filter in the (i+1)th filter bank must overlap 0 (DC) plus ½ the 3-dB bandwidth of the wavelet filter in the ith filter bank. If this criterion is not satisfied, the higher-order path is excluded. Setting OptimizePath to true can significantly reduce the number of scattering paths and computational complexity of the scattering transform for most networks.

You can use paths to determine which and how many scattering paths are computed.

Object Functions

scatteringTransform	Wavelet 1-D scattering transform
featureMatrix	Scattering feature matrix
log	Natural logarithm of scattering transform
filterbank	Wavelet time scattering filter banks
littlewoodPaleySum	Littlewood-Paley sum
scattergram	Visualize scattering or scalogram coefficients
centerFrequencies	Wavelet scattering bandpass center frequencies
numorders	Number of scattering orders
numfilterbanks	Number of scattering filter banks
numCoefficients	Number of wavelet scattering coefficients
paths	Scattering network paths

Examples

Wavelet Time Scattering with Default Values

Create a wavelet time scattering network with default values.

sf = waveletScattering

sf = 
  waveletScattering with properties:

          SignalLength: 1024
       InvarianceScale: 512
        QualityFactors: [8 1]
              Boundary: 'periodic'
     SamplingFrequency: 1
             Precision: 'double'
    OversamplingFactor: 0
          OptimizePath: 0

Plot the wavelet filters used in the first and second filter banks.

[filters,f] = filterbank(sf);
plot(f,filters{2}.psift)
title('First Filter Bank')
xlabel('Cycles/Sample')
ylabel('Magnitude')
grid on

figure
plot(f,filters{3}.psift)
title('Second Filter Bank')
xlabel('Cycles/Sample')
ylabel('Magnitude')
grid on

Plot the Littlewood-Paley sums of the filter banks.

[lpsum,f] = littlewoodPaleySum(sf);
figure
plot(f,lpsum)
legend('1st Filter Bank','2nd Filter Bank')
xlabel('Cycles/Sample')
grid on

Apply Wavelet Time Scattering Network

This example shows how to create and apply a wavelet time scattering network with three filter banks to data.

Load in a data set. Create a scattering network with three filter banks that can be applied to the data.

load handel
disp(['Data Sampling Frequency: ',num2str(Fs),' Hz'])

Data Sampling Frequency: 8192 Hz

sf = waveletScattering('SignalLength',numel(y),...
    'SamplingFrequency',Fs,...
    'QualityFactors',[4 2 1])

sf = 
  waveletScattering with properties:

          SignalLength: 73113
       InvarianceScale: 4.4625
        QualityFactors: [4 2 1]
              Boundary: 'periodic'
     SamplingFrequency: 8192
             Precision: 'double'
    OversamplingFactor: 0
          OptimizePath: 0

Inspect the network. Plot the wavelet filters used in the third filter bank.

[filters,f] = filterbank(sf);
plot(f,filters{4}.psift)
title('Third Filter Bank')
xlabel('Hertz')
ylabel('Magnitude')
grid on

Plot the Littlewood-Paley sums of the three filter banks.

[lpsum,f] = littlewoodPaleySum(sf);
figure
plot(f,lpsum)
xlabel('Hertz')
grid on
legend('1st Filter Bank','2nd Filter Bank','3rd Filter Bank')

Calculate the wavelet 1-D scattering transform of the data for sf. Visualize the scattergram of the scalogram coefficients for the first filter bank.

[S,U] = scatteringTransform(sf,y);
figure
scattergram(sf,U,'FilterBank',1)

More About

Time Windows

You set the invariance scale based on how much insensitivity to time-shifts in the data you want. Equivalently, the invariance scale is the length of the Gaussian smoothing function the scattering transform convolves with the scalogram coefficients. The longer the Gaussian is in time (samples), the narrower the support (bandwidth) of its Fourier transform. You can downsample the output of the smoothing operation. In the context of wavelet scattering, the term "time windows" refers to the number of samples obtained after downsampling the output of the smoothing operation.

When you use the default waveletScattering network, the scattering transform downsamples the output of the smoothing operation as much as possible without causing aliasing. If you want to downsample less than the maximum amount, set the oversampling factor of the network to a nonzero value. That, in turn, affects the number of time windows, because you will keep more samples from the output of the application of the Gaussian smoothing filter to the scalograms. By changing the oversampling factor, you can obtain more time windows for the same given invariance scale.

References

[1] Andén, Joakim, and Stéphane Mallat. “Deep Scattering Spectrum.” IEEE Transactions on Signal Processing 62, no. 16 (August 2014): 4114–28. https://doi.org/10.1109/TSP.2014.2326991.

[2] Mallat, Stéphane. “Group Invariant Scattering.” Communications on Pure and Applied Mathematics 65, no. 10 (October 2012): 1331–98. https://doi.org/10.1002/cpa.21413.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• If specified, 'Precision' and 'Boundary' values must be coder.Constant.

• QualityFactors cannot be variable sized.

• waveletScattering is a handle class, so you must use an entry-point function.

• Filters are not cast when the network is compiled with single-precision data. The recommended workflow is to use the underlying data type of the signal when creating the network in the entry-point function.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2018b

R2021a: waveletScattering property Decimate has been removed

The waveletScattering property Decimate has been removed. Use the property OversamplingFactor instead.

Functionality	What Happens When You Use This Functionality?	Use This Instead	Compatibility Considerations
Decimate property	Errors	OversamplingFactor	Replace all instances of 'Decimate',true with 'OversamplingFactor',0. Replace all instances of 'Decimate',false with 'OversamplingFactor',Inf.

R2021a: Highest wavelet center frequency is computed using geometric mean

Starting in R2021a, the highest wavelet center frequency is computed using the geometric mean. The method for determining how to space linearly those frequencies lower than the invariance scale has also changed. These changes improve the Littlewood-Paley sums of the resulting filter banks.

Center frequencies are logarithmically spaced from the highest frequency to the frequency that corresponds to the invariance scale. Starting in R2021a, depending on scattering network parameters such as the invariance scale, the number of filters you obtain may be different than in previous releases. The method for applying the filters to compute the scattering and scalogram coefficients has not changed.