Wavelet analysis is a new time-frequency localization analysis method developed on the basis of Fourier analysis. The basic idea of wavelet analysis is to use a cluster of wavelet function systems to represent or approximate a certain signal or function.
The principle of wavelet analysis involves Fourier transform, and there are many kinds of wavelet transforms, which are a little complicated. But it doesn't matter if you don't know the principle, as long as you can apply and explain it.
In time series analysis, wavelet analysis is mainly used for denoising and filtering of time series, calculation of information coefficient and fractal dimension, detection of mutation points and identification of periodic components, and analysis of multiple time scales.
Wavelet analysis is usually based on Matlab software. For details, you can refer to the teaching of drawing wavelet coefficient contour maps and wavelet variance maps .
Most people don't know how to use matlab, and it's slow to install and get started. It's better to try Python to realize wavelet analysis.
Recently, I used wavelet analysis for time series analysis in my project. Starting from scratch, the learning process is still a bit painful. After learning, I have been able to explain the output graph initially. I hereby record it for your reference.
When I was studying, I wanted to find someone to do it in a certain treasure, but the price was 600 yuan, which aroused my desire to conquer and earned myself 600 yuan, hahaha.
Python is easy to learn and easy to implement. PyWavelets is a free and open-source library for wavelet transforms in Python. There are almost no explanations for wavelet analysis using python for example analysis on the entire network. The following step-by-step introduction will save you 600 yuan.
pip install pycwt #安装
The main features of PyWavelets are:
- 1D, 2D and nD forward and inverse discrete wavelet transforms (DWT and IDWT)
- 1D, 2D and nD multilevel DWT and IDWT
- One-dimensional and two-dimensional steady-state wavelet transform (Undecimated Wavelet Transform)
- 1D and 2D wavelet packet decomposition and reconstruction
- 1D continuous wavelet transform
- Computes approximations to wavelet and scaling functions
- More than 100 built-in wavelet filters and support for custom wavelets
- Single and Double Precision Computing
- real and complex calculations
- The Matlab Wavelet Toolbox ™ compatible result
PyWavelets gives an example that I feel is not clearly explained.
from __future__ import division
import numpy
from matplotlib import pyplot
import pycwt as wavelet
from pycwt.helpers import find
url = 'http://paos.colorado.edu/research/wavelets/wave_idl/nino3sst.txt'
dat = numpy.genfromtxt(url, skip_header=19)
title = 'NINO3 Sea Surface Temperature'
label = 'NINO3 SST'
units = 'degC'
t0 = 1871.0
dt = 0.25 # In years
N = dat.size
t = numpy.arange(0, N) * dt + t0
p = numpy.polyfit(t - t0, dat, 1)
dat_notrend = dat - numpy.polyval(p, t - t0)
std = dat_notrend.std() # Standard deviation
var = std ** 2 # Variance
dat_norm = dat_notrend / std # Normalized dataset
mother = wavelet.Morlet(6)
s0 = 2 * dt # Starting scale, in this case 2 * 0.25 years = 6 months
dj = 1 / 12 # Twelve sub-octaves per octaves
J = 7 / dj # Seven powers of two with dj sub-octaves
alpha, _, _ = wavelet.ar1(dat) # Lag-1 autocorrelation for red noise
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(dat_norm, dt, dj, s0, J, mother)
iwave = wavelet.icwt(wave, scales, dt, dj, mother) * std
power = (numpy.abs(wave)) ** 2
fft_power = numpy.abs(fft) ** 2
period = 1 / freqs
power /= scales[:, None]
signif, fft_theor = wavelet.significance(1.0, dt, scales, 0, alpha,
significance_level=0.95,
wavelet=mother)
sig95 = numpy.ones([1, N]) * signif[:, None]
sig95 = power / sig95
glbl_power = power.mean(axis=1)
dof = N - scales # Correction for padding at edges
glbl_signif, tmp = wavelet.significance(var, dt, scales, 1, alpha,
significance_level=0.95, dof=dof,
wavelet=mother)
sel = find((period >= 2) & (period < 8))
Cdelta = mother.cdelta
scale_avg = (scales * numpy.ones((N, 1))).transpose()
scale_avg = power / scale_avg # As in Torrence and Compo (1998) equation 24
scale_avg = var * dj * dt / Cdelta * scale_avg[sel, :].sum(axis=0)
scale_avg_signif, tmp = wavelet.significance(var, dt, scales, 2, alpha,
significance_level=0.95,
dof=[scales[sel[0]],
scales[sel[-1]]],
wavelet=mother)
# Prepare the figure
pyplot.close('all')
pyplot.ioff()
figprops = dict(figsize=(11, 8), dpi=72)
fig = pyplot.figure(**figprops)
# First sub-plot, the original time series anomaly and inverse wavelet
# transform.
ax = pyplot.axes([0.1, 0.75, 0.65, 0.2])
ax.plot(t, iwave, '-', linewidth=1, color=[0.5, 0.5, 0.5])
ax.plot(t, dat, 'k', linewidth=1.5)
ax.set_title('a) {}'.format(title))
ax.set_ylabel(r'{} [{}]'.format(label, units))
# Second sub-plot, the normalized wavelet power spectrum and significance
# level contour lines and cone of influece hatched area. Note that period
# scale is logarithmic.
bx = pyplot.axes([0.1, 0.37, 0.65, 0.28], sharex=ax)
levels = [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
bx.contourf(t, numpy.log2(period), numpy.log2(power), numpy.log2(levels),
extend='both', cmap=pyplot.cm.viridis)
extent = [t.min(), t.max(), 0, max(period)]
bx.contour(t, numpy.log2(period), sig95, [-99, 1], colors='k', linewidths=2,
extent=extent)
bx.fill(numpy.concatenate([t, t[-1:] + dt, t[-1:] + dt,
t[:1] - dt, t[:1] - dt]),
numpy.concatenate([numpy.log2(coi), [1e-9], numpy.log2(period[-1:]),
numpy.log2(period[-1:]), [1e-9]]),
'k', alpha=0.3, hatch='x')
bx.set_title('b) {} Wavelet Power Spectrum ({})'.format(label, mother.name))
bx.set_ylabel('Period (years)')
#
Yticks = 2 ** numpy.arange(numpy.ceil(numpy.log2(period.min())),
numpy.ceil(numpy.log2(period.max())))
bx.set_yticks(numpy.log2(Yticks))
bx.set_yticklabels(Yticks)
# Third sub-plot, the global wavelet and Fourier power spectra and theoretical
# noise spectra. Note that period scale is logarithmic.
cx = pyplot.axes([0.77, 0.37, 0.2, 0.28], sharey=bx)
cx.plot(glbl_signif, numpy.log2(period), 'k--')
cx.plot(var * fft_theor, numpy.log2(period), '--', color='#cccccc')
cx.plot(var * fft_power, numpy.log2(1./fftfreqs), '-', color='#cccccc',
linewidth=1.)
cx.plot(var * glbl_power, numpy.log2(period), 'k-', linewidth=1.5)
cx.set_title('c) Global Wavelet Spectrum')
cx.set_xlabel(r'Power [({})^2]'.format(units))
cx.set_xlim([0, glbl_power.max() + var])
cx.set_ylim(numpy.log2([period.min(), period.max()]))
cx.set_yticks(numpy.log2(Yticks))
cx.set_yticklabels(Yticks)
pyplot.setp(cx.get_yticklabels(), visible=False)
# Fourth sub-plot, the scale averaged wavelet spectrum.
dx = pyplot.axes([0.1, 0.07, 0.65, 0.2], sharex=ax)
dx.axhline(scale_avg_signif, color='k', linestyle='--', linewidth=1.)
dx.plot(t, scale_avg, 'k-', linewidth=1.5)
dx.set_title('d) {}--{} year scale-averaged power'.format(2, 8))
dx.set_xlabel('Time (year)')
dx.set_ylabel(r'Average variance [{}]'.format(units))
ax.set_xlim([t.min(), t.max()])
pyplot.show()
