欢迎大家来到IT世界,在知识的湖畔探索吧!

今天想搞一个时频分析，这个东西自己没有写过代码，用过别人的代码，但不知道怎么操作。百度一翻，得到一个现成的例子，又在CSDN上有人写了一段类似的代码，效果图一致，是一个开源的github上的代码。下面把开源的代码转发一下，希望对大家有帮助。

https://github.com/ct6502/wavelets(http://paos.colorado.edu/research/wavelets/)

作者分别写了Python版、matlab版、Fortran版的代码，这里只转一下Python代码。作者对代码做了详细的注释，有需要的自己看注释吧。 代码包含两部分：

waveletAnalysis.py

import matplotlib.pylab as plt
import matplotlib.ticker as ticker
from matplotlib.gridspec import GridSpec

import numpy as np

# from mpl_toolkits.axes_grid1 import make_axes_locatable

from waveletFunctions import wave_signif, wavelet

__author__ = 'Evgeniya Predybaylo'


# WAVETEST Example Python script for WAVELET, using NINO3 SST dataset
#
# See "http://paos.colorado.edu/research/wavelets/"
# The Matlab code written January 1998 by C. Torrence
# modified to Python by Evgeniya Predybaylo, December 2014
#
# Modified Oct 1999, changed Global Wavelet Spectrum (GWS) to be sideways,
#   changed all "log" to "log2", changed logarithmic axis on GWS to
#   a normal axis.
# ---------------------------------------------------------------------------

# READ THE DATA
sst = np.loadtxt('sst_nino3.dat')  # input SST time series
sst = sst - np.mean(sst)
variance = np.std(sst, ddof=1) ** 2
print("variance = ", variance)

# ----------C-O-M-P-U-T-A-T-I-O-N------S-T-A-R-T-S------H-E-R-E---------------

# normalize by standard deviation (not necessary, but makes it easier
# to compare with plot on Interactive Wavelet page, at
# "http://paos.colorado.edu/research/wavelets/plot/"
if 0:
    variance = 1.0
    sst = sst / np.std(sst, ddof=1)
n = len(sst)
dt = 0.25
time = np.arange(len(sst)) * dt + 1871.0  # construct time array
xlim = ([1870, 2000])  # plotting range
pad = 1  # pad the time series with zeroes (recommended)
dj = 0.25  # this will do 4 sub-octaves per octave
s0 = 2 * dt  # this says start at a scale of 6 months
j1 = 7 / dj  # this says do 7 powers-of-two with dj sub-octaves each
lag1 = 0.72  # lag-1 autocorrelation for red noise background
print("lag1 = ", lag1)
mother = 'MORLET'

# Wavelet transform:
wave, period, scale, coi = wavelet(sst, dt, pad, dj, s0, j1, mother)
power = (np.abs(wave)) ** 2  # compute wavelet power spectrum
global_ws = (np.sum(power, axis=1) / n)  # time-average over all times

# Significance levels:
signif = wave_signif(([variance]), dt=dt, sigtest=0, scale=scale,
    lag1=lag1, mother=mother)
# expand signif --> (J+1)x(N) array
sig95 = signif[:, np.newaxis].dot(np.ones(n)[np.newaxis, :])
sig95 = power / sig95  # where ratio > 1, power is significant

# Global wavelet spectrum & significance levels:
dof = n - scale  # the -scale corrects for padding at edges
global_signif = wave_signif(variance, dt=dt, scale=scale, sigtest=1,
    lag1=lag1, dof=dof, mother=mother)

# Scale-average between El Nino periods of 2--8 years
avg = np.logical_and(scale >= 2, scale < 8)
Cdelta = 0.776  # this is for the MORLET wavelet
# expand scale --> (J+1)x(N) array
scale_avg = scale[:, np.newaxis].dot(np.ones(n)[np.newaxis, :])
scale_avg = power / scale_avg  # [Eqn(24)]
scale_avg = dj * dt / Cdelta * sum(scale_avg[avg, :])  # [Eqn(24)]
scaleavg_signif = wave_signif(variance, dt=dt, scale=scale, sigtest=2,
    lag1=lag1, dof=([2, 7.9]), mother=mother)

# ------------------------------------------------------ Plotting

# --- Plot time series
fig = plt.figure(figsize=(9, 10))
gs = GridSpec(3, 4, hspace=0.4, wspace=0.75)
plt.subplots_adjust(left=0.1, bottom=0.05, right=0.9, top=0.95,
                    wspace=0, hspace=0)
plt.subplot(gs[0, 0:3])
plt.plot(time, sst, 'k')
plt.xlim(xlim[:])
plt.xlabel('Time (year)')
plt.ylabel('NINO3 SST (\u00B0C)')
plt.title('a) NINO3 Sea Surface Temperature (seasonal)')

plt.text(time[-1] + 35, 0.5, 'Wavelet Analysis\nC. Torrence & G.P. Compo\n'
    'http://paos.colorado.edu/\nresearch/wavelets/',
    horizontalalignment='center', verticalalignment='center')

# --- Contour plot wavelet power spectrum
# plt3 = plt.subplot(3, 1, 2)
plt3 = plt.subplot(gs[1, 0:3])
levels = [0, 0.5, 1, 2, 4, 999]
# *** or use 'contour'
CS = plt.contourf(time, period, power, len(levels))
im = plt.contourf(CS, levels=levels,
    colors=['white', 'bisque', 'orange', 'orangered', 'darkred'])
plt.xlabel('Time (year)')
plt.ylabel('Period (years)')
plt.title('b) Wavelet Power Spectrum (contours at 0.5,1,2,4\u00B0C$^2$)')
plt.xlim(xlim[:])
# 95# significance contour, levels at -99 (fake) and 1 (95# signif)
plt.contour(time, period, sig95, [-99, 1], colors='k')
# cone-of-influence, anything "below" is dubious
plt.fill_between(time, coi * 0 + period[-1], coi, facecolor="none",
    edgecolor="#00000040", hatch='x')
plt.plot(time, coi, 'k')
# format y-scale
plt3.set_yscale('log', base=2, subs=None)
plt.ylim([np.min(period), np.max(period)])
ax = plt.gca().yaxis
ax.set_major_formatter(ticker.ScalarFormatter())
plt3.ticklabel_format(axis='y', style='plain')
plt3.invert_yaxis()
# set up the size and location of the colorbar
# position=fig.add_axes([0.5,0.36,0.2,0.01])
# plt.colorbar(im, cax=position, orientation='horizontal')
#   , fraction=0.05, pad=0.5)

# plt.subplots_adjust(right=0.7, top=0.9)

# --- Plot global wavelet spectrum
plt4 = plt.subplot(gs[1, -1])
plt.plot(global_ws, period)
plt.plot(global_signif, period, '--')
plt.xlabel('Power (\u00B0C$^2$)')
plt.title('c) Global Wavelet Spectrum')
plt.xlim([0, 1.25 * np.max(global_ws)])
# format y-scale
plt4.set_yscale('log', base=2, subs=None)
plt.ylim([np.min(period), np.max(period)])
ax = plt.gca().yaxis
ax.set_major_formatter(ticker.ScalarFormatter())
plt4.ticklabel_format(axis='y', style='plain')
plt4.invert_yaxis()

# --- Plot 2--8 yr scale-average time series
plt.subplot(gs[2, 0:3])
plt.plot(time, scale_avg, 'k')
plt.xlim(xlim[:])
plt.xlabel('Time (year)')
plt.ylabel('Avg variance (\u00B0C$^2$)')
plt.title('d) 2-8 yr Scale-average Time Series')
plt.plot(xlim, scaleavg_signif + [0, 0], '--')
plt.savefig('./wavelet.png',dpi=300)
#plt.show()

欢迎大家来到IT世界,在知识的湖畔探索吧!

waveletFunctions.py

欢迎大家来到IT世界,在知识的湖畔探索吧!# Copyright (C) 1995-2021, Christopher Torrence and Gilbert P.Compo
# Python version of the code is written by Evgeniya Predybaylo in 2014
# edited by Michael von Papen (FZ Juelich, INM-6), 2018, to include
# analysis at arbitrary frequencies
#
#   This software may be used, copied, or redistributed as long as it is not
#   sold and this copyright notice is reproduced on each copy made. This
#   routine is provided as is without any express or implied warranties
#   whatsoever.
#
# Notice: Please acknowledge the use of the above software in any publications:
#            Wavelet software was provided by C. Torrence and G. Compo,
#      and is available at URL: http://paos.colorado.edu/research/wavelets/''.
#
# Reference: Torrence, C. and G. P. Compo, 1998: A Practical Guide to
#            Wavelet Analysis. <I>Bull. Amer. Meteor. Soc.</I>, 79, 61-78.
#
# Please send a copy of such publications to either C. Torrence or G. Compo:
#  Dr. Christopher Torrence               Dr. Gilbert P. Compo
#  Research Systems, Inc.                 Climate Diagnostics Center
#  4990 Pearl East Circle                 325 Broadway R/CDC1
#  Boulder, CO 80301, USA                 Boulder, CO 80305-3328, USA
#  E-mail: chris[AT]rsinc[DOT]com         E-mail: compo[AT]colorado[DOT]edu
#
# ----------------------------------------------------------------------------


# # WAVELET  1D Wavelet transform with optional significance testing
#   wave, period, scale, coi = wavelet(Y, dt, pad, dj, s0, J1, mother, param)
#
#   Computes the wavelet transform of the vector Y (length N),
#   with sampling rate DT.
#
#   By default, the Morlet wavelet (k0=6) is used.
#   The wavelet basis is normalized to have total energy=1 at all scales.
#
# INPUTS:
#
#    Y = the time series of length N.
#    DT = amount of time between each Y value, i.e. the sampling time.
#
# OUTPUTS:
#
#    WAVE is the WAVELET transform of Y. This is a complex array
#    of dimensions (N,J1+1). FLOAT(WAVE) gives the WAVELET amplitude,
#    ATAN(IMAGINARY(WAVE),FLOAT(WAVE) gives the WAVELET phase.
#    The WAVELET power spectrum is ABS(WAVE)**2.
#    Its units are sigma**2 (the time series variance).
#
# OPTIONAL INPUTS:
#
# *** Note *** if none of the optional variables is set up, then the program
#   uses default values of -1.
#
#    PAD = if set to 1 (default is 0), pad time series with zeroes to get
#         N up to the next higher power of 2. This prevents wraparound
#         from the end of the time series to the beginning, and also
#         speeds up the FFT's used to do the wavelet transform.
#         This will not eliminate all edge effects (see COI below).
#
#    DJ = the spacing between discrete scales. Default is 0.25.
#         A smaller # will give better scale resolution, but be slower to plot.
#
#    S0 = the smallest scale of the wavelet.  Default is 2*DT.
#
#    J1 = the # of scales minus one. Scales range from S0 up to S0*2**(J1*DJ),
#        to give a total of (J1+1) scales. Default is J1 = (LOG2(N DT/S0))/DJ.
#
#    MOTHER = the mother wavelet function.
#             The choices are 'MORLET', 'PAUL', or 'DOG'
#
#    PARAM = the mother wavelet parameter.
#            For 'MORLET' this is k0 (wavenumber), default is 6.
#            For 'PAUL' this is m (order), default is 4.
#            For 'DOG' this is m (m-th derivative), default is 2.
#
#
# OPTIONAL OUTPUTS:
#
#    PERIOD = the vector of "Fourier" periods (in time units) that corresponds
#           to the SCALEs.
#
#    SCALE = the vector of scale indices, given by S0*2**(j*DJ), j=0...J1
#            where J1+1 is the total # of scales.
#
#    COI = if specified, then return the Cone-of-Influence, which is a vector
#        of N points that contains the maximum period of useful information
#        at that particular time.
#        Periods greater than this are subject to edge effects.

import numpy as np

from scipy.optimize import fminbound
from scipy.special._ufuncs import gamma, gammainc

__author__ = 'Evgeniya Predybaylo, Michael von Papen'


def wavelet(Y, dt, pad=0, dj=-1, s0=-1, J1=-1, mother=-1, param=-1, freq=None):
    n1 = len(Y)

    if s0 == -1:
        s0 = 2 * dt
    if dj == -1:
        dj = 1. / 4.
    if J1 == -1:
        J1 = np.fix((np.log(n1 * dt / s0) / np.log(2)) / dj)
    if mother == -1:
        mother = 'MORLET'

    # construct time series to analyze, pad if necessary
    x = Y - np.mean(Y)
    if pad == 1:
        # power of 2 nearest to N
        base2 = np.fix(np.log(n1) / np.log(2) + 0.4999)
        nzeroes = (2 ** (base2 + 1) - n1).astype(np.int64)
        x = np.concatenate((x, np.zeros(nzeroes)))

    n = len(x)

    # construct wavenumber array used in transform [Eqn(5)]
    kplus = np.arange(1, int(n / 2) + 1)
    kplus = (kplus * 2 * np.pi / (n * dt))
    kminus = np.arange(1, int((n - 1) / 2) + 1)
    kminus = np.sort((-kminus * 2 * np.pi / (n * dt)))
    k = np.concatenate(([0.], kplus, kminus))

    # compute FFT of the (padded) time series
    f = np.fft.fft(x)  # [Eqn(3)]

    # construct SCALE array & empty PERIOD & WAVE arrays
    if mother.upper() == 'MORLET':
        if param == -1:
            param = 6.
        fourier_factor = 4 * np.pi / (param + np.sqrt(2 + param**2))
    elif mother.upper() == 'PAUL':
        if param == -1:
            param = 4.
        fourier_factor = 4 * np.pi / (2 * param + 1)
    elif mother.upper() == 'DOG':
        if param == -1:
            param = 2.
        fourier_factor = 2 * np.pi * np.sqrt(2. / (2 * param + 1))
    else:
        fourier_factor = np.nan

    if freq is None:
        j = np.arange(0, J1 + 1)
        scale = s0 * 2. ** (j * dj)
        freq = 1. / (fourier_factor * scale)
        period = 1. / freq
    else:
        scale = 1. / (fourier_factor * freq)
        period = 1. / freq
    # define the wavelet array
    wave = np.zeros(shape=(len(scale), n), dtype=complex)

    # loop through all scales and compute transform
    for a1 in range(0, len(scale)):
        daughter, fourier_factor, coi, _ = \
            wave_bases(mother, k, scale[a1], param)
        wave[a1, :] = np.fft.ifft(f * daughter)  # wavelet transform[Eqn(4)]

    # COI [Sec.3g]
    coi = coi * dt * np.concatenate((
        np.insert(np.arange(int((n1 + 1) / 2) - 1), [0], [1E-5]),
        np.insert(np.flipud(np.arange(0, int(n1 / 2) - 1)), [-1], [1E-5])))
    wave = wave[:, :n1]  # get rid of padding before returning

    return wave, period, scale, coi


# --------------------------------------------------------------------------
# WAVE_BASES  1D Wavelet functions Morlet, Paul, or DOG
#
#  DAUGHTER,FOURIER_FACTOR,COI,DOFMIN = wave_bases(MOTHER,K,SCALE,PARAM)
#
#   Computes the wavelet function as a function of Fourier frequency,
#   used for the wavelet transform in Fourier space.
#   (This program is called automatically by WAVELET)
#
# INPUTS:
#
#    MOTHER = a string, equal to 'MORLET' or 'PAUL' or 'DOG'
#    K = a vector, the Fourier frequencies at which to calculate the wavelet
#    SCALE = a number, the wavelet scale
#    PARAM = the nondimensional parameter for the wavelet function
#
# OUTPUTS:
#
#    DAUGHTER = a vector, the wavelet function
#    FOURIER_FACTOR = the ratio of Fourier period to scale
#    COI = a number, the cone-of-influence size at the scale
#    DOFMIN = a number, degrees of freedom for each point in the wavelet power
#             (either 2 for Morlet and Paul, or 1 for the DOG)

def wave_bases(mother, k, scale, param):
    n = len(k)
    kplus = np.array(k > 0., dtype=float)

    if mother == 'MORLET':  # -----------------------------------  Morlet

        if param == -1:
            param = 6.

        k0 = np.copy(param)
        # calc psi_0(s omega) from Table 1
        expnt = -(scale * k - k0) ** 2 / 2. * kplus
        norm = np.sqrt(scale * k[1]) * (np.pi ** (-0.25)) * np.sqrt(n)
        daughter = norm * np.exp(expnt)
        daughter = daughter * kplus  # Heaviside step function
        # Scale-->Fourier [Sec.3h]
        fourier_factor = (4 * np.pi) / (k0 + np.sqrt(2 + k0 ** 2))
        coi = fourier_factor / np.sqrt(2)  # Cone-of-influence [Sec.3g]
        dofmin = 2  # Degrees of freedom
    elif mother == 'PAUL':  # --------------------------------  Paul
        if param == -1:
            param = 4.
        m = param
        # calc psi_0(s omega) from Table 1
        expnt = -scale * k * kplus
        norm_bottom = np.sqrt(m * np.prod(np.arange(1, (2 * m))))
        norm = np.sqrt(scale * k[1]) * (2 ** m / norm_bottom) * np.sqrt(n)
        daughter = norm * ((scale * k) ** m) * np.exp(expnt) * kplus
        fourier_factor = 4 * np.pi / (2 * m + 1)
        coi = fourier_factor * np.sqrt(2)
        dofmin = 2
    elif mother == 'DOG':  # --------------------------------  DOG
        if param == -1:
            param = 2.
        m = param
        # calc psi_0(s omega) from Table 1
        expnt = -(scale * k) ** 2 / 2.0
        norm = np.sqrt(scale * k[1] / gamma(m + 0.5)) * np.sqrt(n)
        daughter = -norm * (1j ** m) * ((scale * k) ** m) * np.exp(expnt)
        fourier_factor = 2 * np.pi * np.sqrt(2. / (2 * m + 1))
        coi = fourier_factor / np.sqrt(2)
        dofmin = 1
    else:
        print('Mother must be one of MORLET, PAUL, DOG')

    return daughter, fourier_factor, coi, dofmin


# --------------------------------------------------------------------------
# WAVE_SIGNIF  Significance testing for the 1D Wavelet transform WAVELET
#
#   SIGNIF = wave_signif(Y,DT,SCALE,SIGTEST,LAG1,SIGLVL,DOF,MOTHER,PARAM)
#
# INPUTS:
#
#    Y = the time series, or, the VARIANCE of the time series.
#        (If this is a single number, it is assumed to be the variance...)
#    DT = amount of time between each Y value, i.e. the sampling time.
#    SCALE = the vector of scale indices, from previous call to WAVELET.
#
#
# OUTPUTS:
#
#    SIGNIF = significance levels as a function of SCALE
#    FFT_THEOR = output theoretical red-noise spectrum as fn of PERIOD
#
#
# OPTIONAL INPUTS:
#    SIGTEST = 0, 1, or 2.    If omitted, then assume 0.
#
#         If 0 (the default), then just do a regular chi-square test,
#             i.e. Eqn (18) from Torrence & Compo.
#         If 1, then do a "time-average" test, i.e. Eqn (23).
#             In this case, DOF should be set to NA, the number
#             of local wavelet spectra that were averaged together.
#             For the Global Wavelet Spectrum, this would be NA=N,
#             where N is the number of points in your time series.
#         If 2, then do a "scale-average" test, i.e. Eqns (25)-(28).
#             In this case, DOF should be set to a
#             two-element vector [S1,S2], which gives the scale
#             range that was averaged together.
#             e.g. if one scale-averaged scales between 2 and 8,
#             then DOF=[2,8].
#
#    LAG1 = LAG 1 Autocorrelation, used for SIGNIF levels. Default is 0.0
#
#    SIGLVL = significance level to use. Default is 0.95
#
#    DOF = degrees-of-freedom for signif test.
#         IF SIGTEST=0, then (automatically) DOF = 2 (or 1 for MOTHER='DOG')
#         IF SIGTEST=1, then DOF = NA, the number of times averaged together.
#         IF SIGTEST=2, then DOF = [S1,S2], the range of scales averaged.
#
#       Note: IF SIGTEST=1, then DOF can be a vector (same length as SCALEs),
#            in which case NA is assumed to vary with SCALE.
#            This allows one to average different numbers of times
#            together at different scales, or to take into account
#            things like the Cone of Influence.
#            See discussion following Eqn (23) in Torrence & Compo.
#
#    GWS = global wavelet spectrum, a vector of the same length as scale.
#          If input then this is used as the theoretical background spectrum,
#          rather than white or red noise.

def wave_signif(Y, dt, scale, sigtest=0, lag1=0.0, siglvl=0.95,
                dof=None, mother='MORLET', param=None, gws=None):
    n1 = len(np.atleast_1d(Y))
    J1 = len(scale) - 1
    dj = np.log2(scale[1] / scale[0])

    if n1 == 1:
        variance = Y
    else:
        variance = np.std(Y) ** 2

    # get the appropriate parameters [see Table(2)]
    if mother == 'MORLET':  # ----------------------------------  Morlet
        empir = ([2., -1, -1, -1])
        if param is None:
            param = 6.
            empir[1:] = ([0.776, 2.32, 0.60])
        k0 = param
        # Scale-->Fourier [Sec.3h]
        fourier_factor = (4 * np.pi) / (k0 + np.sqrt(2 + k0 ** 2))
    elif mother == 'PAUL':
        empir = ([2, -1, -1, -1])
        if param is None:
            param = 4
            empir[1:] = ([1.132, 1.17, 1.5])
        m = param
        fourier_factor = (4 * np.pi) / (2 * m + 1)
    elif mother == 'DOG':  # -------------------------------------Paul
        empir = ([1., -1, -1, -1])
        if param is None:
            param = 2.
            empir[1:] = ([3.541, 1.43, 1.4])
        elif param == 6:  # --------------------------------------DOG
            empir[1:] = ([1.966, 1.37, 0.97])
        m = param
        fourier_factor = 2 * np.pi * np.sqrt(2. / (2 * m + 1))
    else:
        print('Mother must be one of MORLET, PAUL, DOG')

    period = scale * fourier_factor
    dofmin = empir[0]  # Degrees of freedom with no smoothing
    Cdelta = empir[1]  # reconstruction factor
    gamma_fac = empir[2]  # time-decorrelation factor
    dj0 = empir[3]  # scale-decorrelation factor

    freq = dt / period  # normalized frequency

    if gws is not None:   # use global-wavelet as background spectrum
        fft_theor = gws
    else:
        # [Eqn(16)]
        fft_theor = (1 - lag1 ** 2) / \
            (1 - 2 * lag1 * np.cos(freq * 2 * np.pi) + lag1 ** 2)
        fft_theor = variance * fft_theor  # include time-series variance

    signif = fft_theor
    if dof is None:
        dof = dofmin

    if sigtest == 0:  # no smoothing, DOF=dofmin [Sec.4]
        dof = dofmin
        chisquare = chisquare_inv(siglvl, dof) / dof
        signif = fft_theor * chisquare  # [Eqn(18)]
    elif sigtest == 1:  # time-averaged significance
        if len(np.atleast_1d(dof)) == 1:
            dof = np.zeros(J1) + dof
        dof[dof < 1] = 1
        # [Eqn(23)]
        dof = dofmin * np.sqrt(1 + (dof * dt / gamma_fac / scale) ** 2)
        dof[dof < dofmin] = dofmin   # minimum DOF is dofmin
        for a1 in range(0, J1 + 1):
            chisquare = chisquare_inv(siglvl, dof[a1]) / dof[a1]
            signif[a1] = fft_theor[a1] * chisquare
    elif sigtest == 2:  # time-averaged significance
        if len(dof) != 2:
            print('ERROR: DOF must be set to [S1,S2],'
                ' the range of scale-averages')
        if Cdelta == -1:
            print('ERROR: Cdelta & dj0 not defined'
                  ' for ' + mother + ' with param = ' + str(param))

        s1 = dof[0]
        s2 = dof[1]
        avg = np.logical_and(scale >= 2, scale < 8)  # scales between S1 & S2
        navg = np.sum(np.array(np.logical_and(scale >= 2, scale < 8),
            dtype=int))
        if navg == 0:
            print('ERROR: No valid scales between ' + s1 + ' and ' + s2)
        Savg = 1. / np.sum(1. / scale[avg])  # [Eqn(25)]
        Smid = np.exp((np.log(s1) + np.log(s2)) / 2.)  # power-of-two midpoint
        dof = (dofmin * navg * Savg / Smid) * \
            np.sqrt(1 + (navg * dj / dj0) ** 2)  # [Eqn(28)]
        fft_theor = Savg * np.sum(fft_theor[avg] / scale[avg])  # [Eqn(27)]
        chisquare = chisquare_inv(siglvl, dof) / dof
        signif = (dj * dt / Cdelta / Savg) * fft_theor * chisquare  # [Eqn(26)]
    else:
        print('ERROR: sigtest must be either 0, 1, or 2')

    return signif


# --------------------------------------------------------------------------
# CHISQUARE_INV  Inverse of chi-square cumulative distribution function (cdf).
#
#   X = chisquare_inv(P,V) returns the inverse of chi-square cdf with V
#   degrees of freedom at fraction P.
#   This means that P*100 percent of the distribution lies between 0 and X.
#
#   To check, the answer should satisfy:   P==gammainc(X/2,V/2)

# Uses FMIN and CHISQUARE_SOLVE

def chisquare_inv(P, V):

    if (1 - P) < 1E-4:
        print('P must be < 0.9999')

    if P == 0.95 and V == 2:  # this is a no-brainer
        X = 5.9915
        return X

    MINN = 0.01  # hopefully this is small enough
    MAXX = 1  # actually starts at 10 (see while loop below)
    X = 1
    TOLERANCE = 1E-4  # this should be accurate enough

    while (X + TOLERANCE) >= MAXX:  # should only need to loop thru once
        MAXX = MAXX * 10.
    # this calculates value for X, NORMALIZED by V
        X = fminbound(chisquare_solve, MINN, MAXX, args=(P, V), xtol=TOLERANCE)
        MINN = MAXX

    X = X * V  # put back in the goofy V factor

    return X  # end of code


# --------------------------------------------------------------------------
# CHISQUARE_SOLVE  Internal function used by CHISQUARE_INV
    #
    #   PDIFF=chisquare_solve(XGUESS,P,V)  Given XGUESS, a percentile P,
    #   and degrees-of-freedom V, return the difference between
    #   calculated percentile and P.

    # Uses GAMMAINC
    #
    # Written January 1998 by C. Torrence

    # extra factor of V is necessary because X is Normalized

def chisquare_solve(XGUESS, P, V):

    PGUESS = gammainc(V / 2, V * XGUESS / 2)  # incomplete Gamma function

    PDIFF = np.abs(PGUESS - P)            # error in calculated P

    TOL = 1E-4
    if PGUESS >= 1 - TOL:  # if P is very close to 1 (i.e. a bad guess)
        PDIFF = XGUESS   # then just assign some big number like XGUESS

    return PDIFF

效果如图

另外一个版本成图效果：

如果需要代码，关注，后台发“wavelet”，除了代码，还有一个数据文件。