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Wavelet transform modulus maxima

`hexp = wtmm(x)`

```
[hexp,tauq]
= wtmm(x)
```

`[___] = wtmm(x,'MinRegressionScale',scale)`

```
[hexp,tauq,structfunc]
= wtmm(___)
```

```
[localhexp,wt,wavscales]
= wtmm(x,'ScalingExponent','local')
```

`wtmm(___,'ScalingExponent','local')`

`[___] = wtmm(___,Name,Value)`

`[___] = wtmm(`

uses
only scales greater than or equal to `x`

,'MinRegressionScale',scale)`scale`

to estimate
the global Holder exponent. This syntax can include any of the output
arguments used in previous syntaxes.

`[`

also returns the multiresolution
structure functions, `hexp`

,`tauq`

,`structfunc`

]
= wtmm(___)`structfunc`

, for the global
Holder exponent estimate. This syntax can include any of the input
arguments used in previous syntaxes.

` [`

returns
the local Holder exponent estimates, the continuous wavelet transform `localhexp`

,`wt`

,`wavscales`

]
= wtmm(`x`

,'ScalingExponent','local')`wt`

,
and the scales, `wavscales`

, which are used to
calculate the CWT used in the `wtmm`

algorithm.
The wavelet used in the CWT is the second derivative of a Gaussian.

`wtmm(___,'ScalingExponent','local')`

with
no output arguments plots the wavelet maxima lines in the current
figure. Estimates of the local Holder exponents are displayed in a
table to the right of the plot.

`[___] = wtmm(___,`

returns
the Holder exponent and other specified outputs with additional options
specified by one or more `Name,Value`

)`Name,Value`

pair arguments.

The WTMM algorithm finds singularities in a signal by determining maxima. The algorithm first calculates the continuous wavelet transform using the second derivative of a Gaussian wavelet with 10 voices per octave. The wavelet that meets this criteria is the Mexican hat, or Ricker, wavelet. Then, the algorithm determines the modulus maxima for each scale. The WTMM is intended to be used with large data sets so that enough samples are available to determine maxima accurately.

The definition of the modulus maximum at point *x _{0}* and
scale

$$\left|Wf({s}_{0},x)\right|<\left|Wf({s}_{0},{x}_{0})\right|$$

where *x* is
either in the right or left neighborhood of *x _{0}*.
When

$$\left|Wf({s}_{0},x)\right|\le \left|Wf({s}_{0},{x}_{0})\right|$$

. The algorithm for finding additional maxima repeats for values in that scale. Then, the algorithm continues up through finer scales, checking whether the maxima align between scales. If a maximum converges to the finest scale, it is a true maximum and indicates a singularity at that point.

When each singularity is determined, the algorithm then estimates its Holder exponent. Holder exponents indicate the degree of differentiability for each singularity, which classifies the singularity strength. A Holder exponent less than or equal to 0 indicates a discontinuity at that location. Holder exponents greater than or equal to 1 indicate that the signal is differentiable at that location. Holder values between 0 and 1 indicate continuous, but not differentiable locations. They indicate how close the signal at that sample is to being differentiable. Holder exponents close to 0 indicate signal locations that are less differentiable than locations with exponents closer to 1. The signal is smoother at locations with higher local Holder exponents.

For signals with a few cusp-like singularities and Holder exponents that have large variation, you set the algorithm to return local Holder exponents, which provide individual values for each singularity. For signals with numerous Holder exponents that have relatively small variations, you set the algorithm to return a global Holder exponent. A global Holder exponent applies to the whole signal. For signals with many singularities, you can reduce the number of maxima found by limiting the algorithm to start at or regress to a specific minimum or maximum scale, respectively. For detailed information about the WTMM, see [1] and [3].

[1] Mallat, S., and W. L. Hwang. “Singularity
Detection and Processing with Wavelets.” *IEEE Transactions
on Information Theory*. Vol. 38, No. 2, March 1992, pp.
617–643.

[2] Wendt, H. and P. Abry. “Multifractality Tests Using
Bootstrapped Wavelet Leaders.” *IEEE Transactions on. Signal
Processing*. Vol. 55, No. 10, 2007, pp. 4811–4820.

[3] Arneodo, A., B. Audit, N. Decoster, J.-F.
Muzy, and C. Vaillant. “Wavelet-Based Multifractal Formalism:
Application to DNA Sequences, Satellite Images of the Cloud Structure
and Stock Market Data.” *The Science of Disasters:
Climate Disruptions, Heart Attacks, and Market Crashes*.
Bunde, A., J. Kropp, and H. J. Schellnhuber, Eds. 2002, pp. 26–102.