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Wavelet coherence and cross-spectrum

`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(___) `

returns
the magnitude-squared wavelet coherence, which is a measure of the
correlation between signals `wcoh`

= wcoherence(`x`

,`y`

)`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.

`[`

uses
a `wcoh`

,`wcs`

,`period`

]
= wcoherence(`x`

,`y`

,`ts`

)`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`

`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.

[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.