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# wcoherence

Wavelet coherence and cross-spectrum

## Syntax

``wcoh = wcoherence(x,y)``
``````[wcoh,wcs] = wcoherence(x,y)``````
``````[wcoh,wcs,period] = wcoherence(x,y,ts)``````
`````` [wcoh,wcs,f] = wcoherence(x,y,fs)``````
``````[wcoh,wcs,f,coi] = wcoherence(___)``````
`````` [wcoh,wcs,period,coi] = wcoherence(___,ts)``````
``wcoherence(___) ``

## Description

example

````wcoh = wcoherence(x,y)` returns the magnitude-squared wavelet coherence, which is a measure of the correlation between signals `x` and `y` in the time-frequency plane. Wavelet coherence is useful for analyzing nonstationary signals. The inputs `x` and `y` must be equal length, 1-D, real-valued signals. The coherence is computed using the analytic Morlet wavelet.```
``````[wcoh,wcs] = wcoherence(x,y)``` returns the wavelet cross-spectrum of `x` and `y`. You can use the phase of the wavelet cross-spectrum values to identify the relative lag between the input signals.```

example

``````[wcoh,wcs,period] = wcoherence(x,y,ts)``` uses a `duration` `ts` with a positive, scalar input, as the sampling interval. The duration can be in years, days, hours, minutes, or seconds. `ts` is used to compute the scale-to period conversion, `period`. The `period` in an array of durations with the same time unit as specified in `ts````

example

`````` [wcoh,wcs,f] = wcoherence(x,y,fs)``` uses the positive sampling frequency, `fs`, to compute the scale-to-frequency conversion, `f`. The `fs` input is in Hz.```
``````[wcoh,wcs,f,coi] = wcoherence(___)``` returns the cone of influence, `coi`, for the wavelet coherence in cycles per sample. If you specify the sampling frequency, `fs`, the function returns both `f` and the cone of influence in Hz.```

example

`````` [wcoh,wcs,period,coi] = wcoherence(___,ts)``` returns the cone of influence, `coi`, in cycles per unit time. ```

example

````wcoherence(___) ` with no output arguments plots the wavelet coherence and cone of influence in the current figure. Due to the inverse relationship between frequency and period, a plot that uses the sampling interval is the inverse of a plot the uses the sampling frequency. For areas where the coherence exceeds 0.5, plots that use the sampling frequency display arrows to show the phase lag of `y` with respect to `x`. The arrows are spaced in time and scale. The direction of the arrows corresponds to the phase lag on the unit circle. For example, a vertical arrow indicates a π/2 or quarter-cycle phase lag. The corresponding lag in time depends on the duration of the cycle.```

## Examples

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Use default `wcoherence` settings to obtain the wavelet coherence between a sine wave with random noise and a frequency-modulated signal with decreasing frequency over time.

```t = linspace(0,1,1024); x = -sin(8*pi*t) + 0.4*randn(1,1024); x = x/max(abs(x)); y = wnoise('doppler',10); wcoh = wcoherence(x,y);```

The default coherence computation uses the Morlet wavelet, 12 voices per octave and smooths 12 scales. The default number of octaves is equal to `floor(log2(numel(x)))-1`, which in this case is 9.

Obtain the wavelet coherence data for two signals, specifying a sampling interval of 0.001 seconds. Both signals consist of two sine waves (10 Hz and 50 Hz) in white noise. The sine waves have different time supports.

Set the random number generator to its default settings for reproducibility. Then, create the two signals and obtain the coherence.

```rng default; t = 0:0.001:2; x = cos(2*pi*10*t).*(t>=0.5 & t<1.1)+ ... cos(2*pi*50*t).*(t>= 0.2 & t< 1.4)+0.25*randn(size(t)); y = sin(2*pi*10*t).*(t>=0.6 & t<1.2)+... sin(2*pi*50*t).*(t>= 0.4 & t<1.6)+ 0.35*randn(size(t)); [wcoh,~,period,coi] = wcoherence(x,y,seconds(0.001));```

Use the `pcolor` command to plot the coherence and cone of influence.

```period = seconds(period); coi = seconds(coi); h = pcolor(t,log2(period),wcoh); h.EdgeColor = 'none'; ax = gca; ytick=round(pow2(ax.YTick),3); ax.YTickLabel=ytick; ax.XLabel.String='Time'; ax.YLabel.String='Period'; ax.Title.String = 'Wavelet Coherence'; hcol = colorbar; hcol.Label.String = 'Magnitude-Squared Coherence'; hold on; plot(ax,t,log2(coi),'w--','linewidth',2);```

Use `wcoherence(x,y,seconds(0.001))` without any outputs arguments. This plot includes the phase arrows and the cone of influence.

`wcoherence(x,y,seconds(0.001));`

Obtain the wavelet coherence for two signals, specifying a sampling frequency of 1000 Hz. Both signals consist of two sine waves (10 Hz and 50 Hz) in white noise. The sine waves have different time supports.

Set the random number generator to its default settings for reproducibility. Then, create the two signals and obtain the coherence.

```rng default; t = 0:0.001:2; x = cos(2*pi*10*t).*(t>=0.5 & t<1.1)+... cos(2*pi*50*t).*(t>= 0.2 & t< 1.4)+0.25*randn(size(t)); y = sin(2*pi*10*t).*(t>=0.6 & t<1.2)+... sin(2*pi*50*t).*(t>= 0.4 & t<1.6)+ 0.35*randn(size(t)); wcoherence(x,y,1000);```

This coherence plot is flipped with respect to the coherence plot in the previous example, which specifies a sampling interval instead of a sampling frequency.

Obtain the scale-to-freqeuncy conversion output in `f`.

`[wcoh,wcs,f] = wcoherence(x,y,1000);`

Obtain the wavelet coherence for two signals. Both signals consist of two sine waves (10 Hz and 50 Hz) in white noise. Use the default number of scales to smooth. This value is equivalent to the number of voices per octave. Both values default to 12.

Set the random number generator to its default settings for reproducibility. Then, create the two signals and obtain the coherence.

```rng default; t = 0:0.001:2; x = cos(2*pi*10*t).*(t>=0.5 & t<1.1)+ ... cos(2*pi*50*t).*(t>= 0.2 & t< 1.4)+0.25*randn(size(t)); y = sin(2*pi*10*t).*(t>=0.6 & t<1.2)+... sin(2*pi*50*t).*(t>= 0.4 & t<1.6)+ 0.35*randn(size(t)); wcoherence(x,y);```

Set the number of scales to smooth to 16. The increased smoothing causes reduced low frequency resolution.

`wcoherence(x,y,'NumScalesToSmooth',16);`

Compare the effects of using different phase display thresholds on the wavelet coherence.

Plot the wavelet coherence between the El Nino time series and the All India Average Rainfall Index. The data are sampled monthly. Specify the sampling interval as 1/12 of a year to display the periods in years. Use the default phase display threshold of 0.5, which shows phase arrows only where the coherence is greater than or equal to 0.5.

```load ninoairdata; wcoherence(nino,air,years(1/12));```

Set the phase display threshold to 0.7. The number of phase arrows decreases.

`wcoherence(nino,air,years(1/12),'PhaseDisplayThreshold',0.7);`

## Input Arguments

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Input signal, specified as a vector of real values. `x` must be a 1-D, real-valued signal. The two input signals, `x` and `y`, must be the same length and must have at least four samples.

Input signal, specified as vector of real values. `y` must be a 1-D, real-valued signal. The two input signals, `x` and `y`, must be the same length and must have at least four samples.

Sampling interval, also known as the sampling period, specified as a `duration` with positive scalar input. Valid durations are `years`, `days`, `hours`, `seconds`, and `minutes`. You can also use the `duration` function to specify `ts`. You cannot use calendar durations (`caldays`, `calweeks`, `calmonths`, `calquarters`, or `calyears`). You cannot specify both `ts` and `fs`.

Sampling frequency, specified as a positive scalar.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `'PhaseDisplayThreshold',0.7;` specifies the threshold for displaying phase vectors.

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Number of voices per octave to use in the wavelet coherence, specified as an integer from 10 to 32.

Number of scales to smooth in time and scale, specified as a positive integer. This value must be less than one half the total number of scales. The function uses a moving average filter to smooth across scale. If you do not specify the number of scales to smooth, the value of `NumScalesToSmooth` defaults to the number of voices per octave. If your coherence is noisy, you can specify a larger `NumScalesToSmooth` value to smooth the coherence more.

Number of octaves to use in the wavelet coherence, specified as a positive integer between 1 and `floor(log2(numel(x)))-1`. If you do not need to examine lower frequency values, use a smaller `NumOctaves` value.

Threshold for displaying phase vectors, specified as a real scalar between 0 and 1. This function displays phase vectors for regions with coherence greater than or equal to the specified threshold value. Lowering the threshold value displays more phase vectors. If you use `wcoherence` with any output arguments, the `PhaseDisplayThreshold` value is ignored.

## Output Arguments

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Wavelet coherence, returned as a matrix. The coherence is computed using the analytic Morlet wavelet over logarithmic scales, with a default value of 12 voices per octave. The default number of octaves is equal to `floor(log2(numel(x)))-1`. If you do not specify a sampling interval, sampling frequency is assumed.

Wavelet cross-spectrum, returned as a matrix of complex values. You can use the phase of the wavelet cross-spectrum values to identify the relative lag between the input signals.

Data Types: `double`
Complex Number Support: Yes

Scale-to-period conversion, returned as an array of durations. The conversion values are computed from the sampling period specified in `ts`. Each `period` element has the same format as `ts`.

Scale-to-frequency conversion, returned as a vector. The vector contains the peak frequency values for the wavelets used to compute the coherence. If you want to output `f`, but do not specify a sampling frequency input, `fs`, the returned wavelet coherence is in cycles per sample.

Cone of influence for the wavelet coherence, returned as either an array of doubles or array of durations. The cone of influence indicates where edge effects occur in the coherence data. If you specify a sampling frequency, `fs`, the cone of influence is in Hz. If you specify a sampling interval or period, `ts`, the cone of influence is in periods. Due to the edge effects, give less credence to areas of apparent high coherence that are outside or overlap the cone of influence. The cone of influence is indicated by a dashed line.

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### Wavelet Cross Spectrum

The wavelet cross- spectrum is a measure of the distribution of power of two signals.

The wavelet cross spectrum of two time series, x and y, is:

`${C}_{xy}\left(a,b\right)=S\left({C}_{x}^{*}\left(a,b\right){C}_{y}\left(a,b\right)\right)$`

Cx(a,b) and Cy(a,b) denote the continuous wavelet transforms of x and y at scales a and positions b. The superscript * is the complex conjugate, and S is a smoothing operator in time and scale.

For real-valued time series, the wavelet cross-spectrum is real-valued if you use a real-valued analyzing wavelet, and complex-valued if you use a complex-valued analyzing wavelet.

### Wavelet Coherence

Wavelet coherence is a measure of the correlation between two signals.

The wavelet coherence of two time series x and y is:

`$\frac{{|S\left({C}_{x}^{*}\left(a,b\right){C}_{y}\left(a,b\right)\right)|}^{2}}{S\left(|{C}_{x}\left(a,b\right){|}^{2}S\left(|{C}_{y}\left(a,b\right){|}^{2}}$`

Cx(a,b) and Cy(a,b) denote the continuous wavelet transforms of x and y at scales a and positions b. The superscript * is the complex conjugate and S is a smoothing operator in time and scale.

For real-valued time series, the wavelet coherence is real-valued if you use a real-valued analyzing wavelet, and complex-valued if you use a complex-valued analyzing wavelet.

## References

[1] Grinsted, A, J., C. Moore, and S. Jevrejeva. “Application of the cross wavelet transform and wavelet coherence to geophysical time series.” Nonlinear Processes in Geophysics. Vol. 11, Issue 5/6, 2004, pp. 561–566.

[2] Maraun, D., J. Kurths, and M. Holschneider. "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation and significance testing.” Physical Review E 75. 2007, pp. 016707-1–016707-14.

[3] Torrence, C., and P. Webster. "Interdecadal changes in the ESNO-Monsoon System." Journal of Climate. Vol. 12, 1999, pp. 2679–2690.