Tutorial for learning and working with Wavelets
Create new conda environment for Jupyter notebook
conda env create -f environment.yml
Activate new environment
conda activate wavelet_tutorial
Start JupyterLab and open in local browser at localhost:8888
jupyter lab
JupyterLab Issues and Solutions:
If the browser opens, but JupyterLab displays an empty page try `http://127.0.0.1:8888/lab` instead of the default `http://localhost:8888/lab`. If this does not work, try switching browsers (Firefox, Chrome, Safari, etc...)
If the browser opens, but JupyterLab prompts you for a password/token: Token authentication is enabled
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An example of the Jupyter output:
[I 2024-01-17 11:40:47.517 ServerApp] Jupyter Server 2.10.0 is running at:
[I 2024-01-17 11:40:47.517 ServerApp] http://localhost:8888/lab?token=b735c83f6f7a7d31a60cb773fc9bf3b392b14227396e26c3
[I 2024-01-17 11:40:47.517 ServerApp] http://127.0.0.1:8888/lab?token=b735c83f6f7a7d31a60cb773fc9bf3b392b14227396e26c3
[I 2024-01-17 11:40:47.517 ServerApp] Use Control-C to stop this server and shut down all kernels (twice to skip confirmation).
[C 2024-01-17 11:40:47.547 ServerApp]
To access the server, open this file in a browser:
file:///home/user/.local/share/jupyter/runtime/jpserver-6001-open.html
Or copy and paste one of these URLs:
http://localhost:8888/lab?token=b735c83f6f7a7d31a60cb773fc9bf3b392b14227396e26c3
http://127.0.0.1:8888/lab?token=b735c83f6f7a7d31a60cb773fc9bf3b392b14227396e26c3
Wavelets are a powerful tool to analyze time-series data and find the base frequencies that are creating a complex signal.
Sometimes also refered to as Windowed Fourier Transform
"A Practical Guide to Wavelet Analysis" (Torrence and Compo)
NCL Wavelet inputs: time series and significance levels
- 1D array data with length N
- mother wavelet: Morlet, Paul, DOG
- dt: sampling time (time between each y value)
- mother wavelet parameter
- Morlet: k0 (wave number) = 6 (default)
- Paul: m (order) = 4 (default)
- DOG: m (m-th derivative) = 2 (default)
- s0: smallest scale of wavelet, typically equal to 2*dt (twice the sampling time)
- Morlet: s0 = dt
- Paul: s0 = dt/4
- dj: spacing between discrete scales, typically dj = 0.25
- Smaller dj will increase resolution but increase computation time
- jtot: scales (integer number), range from s0 (2dt) to s02^((jtot-1)*dj)
- Max range set to smallest scale * 2 ^ ( (scale - 1) * space between scales)
- Commonly, rounded to integer, jtot = 1 + log((length data * sampling time)/spacing between scale)/log(2)
- (rounded to an integer) jtot = 1 + log((N*dt)/s0)/log(2)
- npad: number of points (including padding) used for wavelet transform, typically a power of 2
- Padding must be greater than/equal to length of data (N)
- If npad > N, zeroes are padded to end of time series
- Most commonly, npad = N (no padding)
- noise
- significance testing with noise
- noise = 0 for white noise background
- noise = 1 for red noise background
- significance testing
- isigtest = 0 for regular chi-square test (equ 18 in Torrence and Compo)
- isigtest = 1 for time-average test on global wavelet spectrum
- significance level
- typically, siglvl = 0.95
Returns:
- 3D array (wave transform) with dimensions 2 x jtot x N
- wave phase: 1D array jtot*N containing
- mean: scalar containing mean of input series
- stdev: 1D scalar containing STD fo input
- lag1: scalar containing lag-1 autcorrelation of input series
- r1: scalar of used in significance testing
- dof: degrees-of-freedom for significance testing
- scale: containing the wavelet scales used
- period: contains the Fourier period corresponding to scale
- gws: global wavelet spectrum
- coi: e-folding factor used for the cone of influence
- fft_theor: theoretical red-noise spectrum versus scale
- signif: significance levels versus scale
- cdelta: constant cdelta for mother wavelet
- psi0: psi(0) for the mother wavelet