def getAlpha(self):
"""
Lag-1 autocorrelation for red noise
"""
self.alpha, var, mu2 = wavelet.ar1(self.signal)
if self.verbose:
print(
f"Variance of signal: {self.wavelet_spectra.feature.amplitude**2}"
)
print("Variance of red noise: ", var * self.feature.amplitude**2)
print("Mean square of red noise: ",
mu2 * self.feature.amplitude**2)
return self.alpha
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
## Define wavelet parameters
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,
def wavelet_transform(dat, mother, s0, dj, J, dt, lims=[20, 120], t0=0):
"""
Plot the continous wavelet transform for a given signal.
Make sure to detrend and normalize the data before calling this funcion.
This is a function wrapper around the pycwt simple_sample example with
some modifications.
----------
Args:
dat (Mandatory [array like]): input signal data.
mother (Mandatory [str]): the wavelet mother name.
s0 (Mandatory [float]): starting scale.
dj (Mandatory [float]): number of sub-octaves per octaves.
j (Mandatory [float]): powers of two with dj sub-octaves.
dt (Mandatory [float]): same frequency in the same unit as the input.
lims (Mandatory [list]): Period interval to integrate the local
power spectrum.
label (Mandatory [str]): the plot y-label.
title (Mandatory [str]): the plot title.
----------
Return:
No
"""
# also create a time array in years.
N = dat.size
t = np.arange(0, N) * dt + t0
# write the following code to detrend and normalize the input data by its
# standard deviation. Sometimes detrending is not necessary and simply
# removing the mean value is good enough. However, if your dataset has a
# well defined trend, such as the Mauna Loa CO\ :sub:`2` dataset available
# in the above mentioned website, it is strongly advised to perform
# detrending. Here, we fit a one-degree polynomial function and then
# subtract it from the
# original data.
p = np.polyfit(t - t0, dat, 1)
dat_notrend = dat - np.polyval(p, t - t0)
std = dat_notrend.std() # Standard deviation
var = std**2 # Variance
dat_norm = dat_notrend / std # Normalized dataset
alpha, _, _ = wavelet.ar1(dat) # Lag-1 autocorrelation for red noise
# the following routines perform the wavelet transform and inverse wavelet
# transform using the parameters defined above. Since we have normalized
# our input time-series, we multiply the inverse transform by the standard
# deviation.
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
dat_norm, dt, dj, s0, J, mother)
iwave = wavelet.icwt(wave, scales, dt, dj, mother) * std
# calculate the normalized wavelet and Fourier power spectra, as well as
# the Fourier equivalent periods for each wavelet scale.
power = (np.abs(wave))**2
fft_power = np.abs(fft)**2
period = 1 / freqs
# inverse transform but only considering lims
idx1 = np.argmin(np.abs(period - LIMS[0]))
idx2 = np.argmin(np.abs(period - LIMS[1]))
_wave = wave.copy()
_wave[0:idx1, :] = 0
igwave = wavelet.icwt(_wave, scales, dt, dj, mother) * std
# could stop at this point and plot our results. However we are also
# interested in the power spectra significance test. The power is
# significant where the ratio ``power / sig95 > 1``.
signif, fft_theor = wavelet.significance(1.0,
dt,
scales,
0,
alpha,
significance_level=0.95,
wavelet=mother)
sig95 = np.ones([1, N]) * signif[:, None]
sig95 = power / sig95
# calculate the global wavelet spectrum and determine its
# significance level.
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)
return t, dt, power, period, coi, sig95, iwave, igwave
avg1, avg2 = (2, 8) # Range of periods to average
slevel = 0.95 # Significance level
std = dat.std() # Standard deviation
std2 = std ** 2 # Variance
dat = (dat - dat.mean()) / std # Calculating anomaly and normalizing
N = dat.size # Number of measurements
time = numpy.arange(0, N) * ds.dt + ds.t0 # Time array in years
dj = 1 / 12 # Twelve sub-octaves per octaves
s0 = -1 # 2 * dt # Starting scale, here 6 months
J = -1 # 7 / dj # Seven powers of two with dj sub-octaves
# alpha = 0.0 # Lag-1 autocorrelation for white noise
try:
alpha, _, _ = wavelet.ar1(dat) # Lag-1 autocorrelation for red noise
except Warning:
# When the dataset is too short, or there is a strong trend, ar1 raises a
# warning. In this case, we assume a white noise background spectrum.
alpha = 1.0
mother = wavelet.Morlet(6) # Morlet mother wavelet with m=6
# The following routines perform the wavelet transform and siginificance
# analysis for the chosen data set.
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(dat, ds.dt, dj, s0, J,
mother)
iwave = wavelet.icwt(wave, scales, ds.dt, dj, mother)
# Normalized wavelet and Fourier power spectra
power = (numpy.abs(wave)) ** 2
# I. Continuous wavelet transform
# ===============================
# Calculate the CWT of both normalized time series. The function wavelet.cwt
# returns a a list with containing [wave, scales, freqs, coi, fft, fftfreqs]
# variables.
mother = wavelet.Morlet(6) # Morlet mother wavelet with m=6
slevel = 0.95 # Significance level
dj = 1/12 # Twelve sub-octaves per octaves
s0 = 2 * dt # Starting scale, here 6 months
J = 6 / dj # Seven powers of two with dj sub-octaves
if True:
alpha1, _, _ = wavelet.ar1(s1) # Lag-1 autocorrelation for red noise
alpha2, _, _ = wavelet.ar1(s2) # Lag-1 autocorrelation for red noise
else:
alpha1 = alpha2 = 0.0 # Lag-1 autocorrelation for white noise
# The following routines perform the wavelet transform and siginificance
# analysis for two data sets.
mother = 'morlet'
W1, scales1, freqs1, coi1, _, _ = wavelet.cwt(s1/std1, dt, dj, s0, J, mother)
signif1, fft_theor1 = wavelet.significance(1.0, dt, scales1, 0, alpha1,
significance_level=slevel,
wavelet=mother)
def wct_modified(y1,
y2,
dt,
dj=1 / 12,
s0=-1,
J=-1,
sig=True,
significance_level=0.95,
wavelet='morlet',
normalize=True,
**kwargs):
"""Wavelet coherence transform (WCT).
The WCT finds regions in time frequency space where the two time
series co-vary, but do not necessarily have high power.
Parameters
----------
y1, y2 : numpy.ndarray, list
Input signals.
dt : float
Sample spacing.
dj : float, optional
Spacing between discrete scales. Default value is 1/12.
Smaller values will result in better scale resolution, but
slower calculation and plot.
s0 : float, optional
Smallest scale of the wavelet. Default value is 2*dt.
J : float, optional
Number of scales less one. Scales range from s0 up to
s0 * 2**(J * dj), which gives a total of (J + 1) scales.
Default is J = (log2(N*dt/so))/dj.
significance_level (float, optional) :
Significance level to use. Default is 0.95.
normalize (boolean, optional) :
If set to true, normalizes CWT by the standard deviation of
the signals.
Returns
-------
TODO: Something TBA and TBC
See also
--------
cwt, xwt
"""
wavelet = pycwt.wavelet._check_parameter_wavelet(wavelet)
# Checking some input parameters
if s0 == -1:
# Number of scales
s0 = 2 * dt / wavelet.flambda()
if J == -1:
# Number of scales
J = np.int(np.round(np.log2(y1.size * dt / s0) / dj))
# Makes sure input signals are numpy arrays.
y1 = np.asarray(y1)
y2 = np.asarray(y2)
# Calculates the standard deviation of both input signals.
std1 = y1.std()
std2 = y2.std()
# Normalizes both signals, if appropriate.
if normalize:
y1_normal = (y1 - y1.mean()) / std1
y2_normal = (y2 - y2.mean()) / std2
else:
y1_normal = y1
y2_normal = y2
# Calculates the CWT of the time-series making sure the same parameters
# are used in both calculations.
_kwargs = dict(dj=dj, s0=s0, J=J, wavelet=wavelet)
W1, sj, freq, coi, _, _ = pycwt.cwt(y1_normal, dt, **_kwargs)
W2, sj, freq, coi, _, _ = pycwt.cwt(y2_normal, dt, **_kwargs)
scales1 = np.ones([1, y1.size]) * sj[:, None]
scales2 = np.ones([1, y2.size]) * sj[:, None]
# Smooth the wavelet spectra before truncating.
S1 = wavelet.smooth(np.abs(W1)**2 / scales1, dt, dj, sj)
S2 = wavelet.smooth(np.abs(W2)**2 / scales2, dt, dj, sj)
# Now the wavelet transform coherence
W12 = W1 * W2.conj()
scales = np.ones([1, y1.size]) * sj[:, None]
S12 = wavelet.smooth(W12 / scales, dt, dj, sj)
WCT = np.abs(S12)**2 / (S1 * S2)
aWCT = np.angle(W12)
# Calculate cross spectrum & its amplitude
WXS, WXA = W12, np.abs(S12)
# Calculates the significance using Monte Carlo simulations with 95%
# confidence as a function of scale.
if sig:
a1, b1, c1 = pycwt.ar1(y1)
a2, b2, c2 = pycwt.ar1(y2)
sig = pycwt.wct_significance(a1,
a2,
dt=dt,
dj=dj,
s0=s0,
J=J,
significance_level=significance_level,
wavelet=wavelet,
**kwargs)
else:
sig = np.asarray([0])
return WXS, WXA, WCT, aWCT, coi, freq, sig
def main():
# Then, we load the dataset and define some data related parameters. In this
# case, the first 19 lines of the data file contain meta-data, that we ignore,
# since we set them manually (*i.e.* title, units).
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
#%%
# We also create a time array in years.
N = dat.size
t = numpy.arange(0, N) * dt + t0
#%%
# We write the following code to detrend and normalize the input data by its
# standard deviation. Sometimes detrending is not necessary and simply
# removing the mean value is good enough. However, if your dataset has a well
# defined trend, such as the Mauna Loa CO\ :sub:`2` dataset available in the
# above mentioned website, it is strongly advised to perform detrending.
# Here, we fit a one-degree polynomial function and then subtract it from the
# original data.
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
#%%
# The next step is to define some parameters of our wavelet analysis. We
# select the mother wavelet, in this case the Morlet wavelet with
# :math:`\omega_0=6`.
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
#%%
# The following routines perform the wavelet transform and inverse wavelet
# transform using the parameters defined above. Since we have normalized our
# input time-series, we multiply the inverse transform by the standard
# deviation.
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(dat_norm, dt, dj, s0, J,
mother)
iwave = wavelet.icwt(wave, scales, dt, dj, mother) * std
#%%
# We calculate the normalized wavelet and Fourier power spectra, as well as
# the Fourier equivalent periods for each wavelet scale.
power = (numpy.abs(wave)) ** 2
fft_power = numpy.abs(fft) ** 2
period = 1 / freqs
#%%
# We could stop at this point and plot our results. However we are also
# interested in the power spectra significance test. The power is significant
# where the ratio ``power / sig95 > 1``.
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
#%%
# Then, we calculate the global wavelet spectrum and determine its
# significance level.
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)
#%%
# We also calculate the scale average between 2 years and 8 years, and its
# significance level.
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)
#%%
# Finally, we plot our results in four different subplots containing the
# (i) original series anomaly and the inverse wavelet transform; (ii) the
# wavelet power spectrum (iii) the global wavelet and Fourier spectra ; and
# (iv) the range averaged wavelet spectrum. In all sub-plots the significance
# levels are either included as dotted lines or as filled contour lines.
# 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))
def takewav_makefig(dd,moornum):
if moornum==8:
dt=1
dat=pd.read_csv(dd,header=12,sep='\s*')
date=unique([datetime.datetime(int(dat.ix[ii,0]),
int(dat.ix[ii,1]),
int(dat.ix[ii,2]),
int(dat.ix[ii,3])) for ii in range(len(dat))])
utest=array(dat.ix[:,6]/100)
vtest=array(dat.ix[:,7]/100)
nomd=int(nanmean(array(dat.ix[:,5])))
dat=utest**2+vtest**2
savetit='M1-'+str(nomd)+'m'
else:
dataset=xr.open_dataset(dd)
date=dataset['TIME']
ke=dataset['UCUR']**2+dataset['VCUR']**2
dat=ke.values.flatten()
dt=0.5
nomd=int(dataset.geospatial_vertical_min)
savetit=dataset.platform_code[-3:]+'-'+str(nomd)+'m'
dat[isnan(dat)]=nanmean(dat)
alpha, _, _ = wavelet.ar1(dat) # Lag-1 autocorrelation for red noise
N=len(dat)
#in hours
t = numpy.arange(0, N) * dt
std=dat.std()
var=std**2
dat_norm=dat/std
# The following routines perform the wavelet transform and inverse wavelet transform using the parameters defined above. Since we have normalized our input time-series, we multiply the inverse transform by the standard deviation.
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(dat_norm,dt, dj, s0, J, mother)
iwave = wavelet.icwt(wave, scales, dt, dj, mother) * std
# We calculate the normalized wavelet and Fourier power spectra, as well as the Fourier equivalent periods for each wavelet scale.
power = (numpy.abs(wave)) ** 2
fft_power = numpy.abs(fft) ** 2
period = 1 / freqs
# Optionally, we could also rectify the power spectrum according to the suggestions proposed by Liu et al. (2007)[2]
power /= scales[:, None]
# We could stop at this point and plot our results. However we are also interested in the power spectra significance test. The power is significant where the ratio power / sig95 > 1.
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
# Then, we calculate the global wavelet spectrum and determine its significance level.
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)
# We also calculate the scale average between pmin and pmax, and its significance level.
f,dx = pyplot.subplots(6,1,figsize=(12,12),sharex=True)
bands=[1,2,8,16,48,128,512]
for ii in range(len(bands)-1):
pmin=bands[ii]
pmax=bands[ii+1]
sel = find((period >= pmin) & (period < pmax))
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)
dx[ii].axhline(scale_avg_signif, color='C'+str(ii), linestyle='--', linewidth=1.)
dx[ii].plot(date, scale_avg, '-', color='C'+str(ii), linewidth=1.5,label='{}--{} hour band'.format(pmin,pmax))
[dx[ii].axvline(dd,color=clist[jj],linewidth=3) for jj,dd in enumerate(dlist)]
dx[ii].legend()
dx[0].set_title('Scale-averaged power: '+savetit)
dx[3].set_ylabel(r'Average variance [{}]'.format(units))
if moornum ==8:
dx[0].set_xlim(date[0],date[-1])
else:
dx[0].set_xlim(date[0].values,date[-1].values)
savefig(figdir+'ScaleSep_'+savetit+'.png',bbox_inches='tight')
pmin=2
pmax=24
sel = find((period >= pmin) & (period < pmax))
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)
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(date, dat, linewidth=1.5, color=[0.5, 0.5, 0.5])
ax.plot(date, iwave, 'k-', linewidth=1,zorder=100)
if moornum ==8:
ax.set_xlim(date[0],date[-1])
else:
ax.set_xlim(date[0].values,date[-1].values)
# 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])
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('{} Wavelet Power Spectrum ({})'.format(label, mother.name))
bx.set_ylabel('Period (hours)')
#
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)
bx.set_xticklabels('')
bx.set_xlim(t.min(),t.max())
# 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('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)
spowdic={}
spowdic['sig']=scale_avg_signif
if moornum==8:
spowdic['date']=date
else:
spowdic['date']=date.values
spowdic['spow']=scale_avg
# 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(date, scale_avg, 'k-', linewidth=1.5)
dx.set_title('{}--{} hour scale-averaged power'.format(pmin,pmax))
# [dx.axvline(dd,color=clist[ii],linewidth=3) for ii,dd in enumerate(dlist)]
# dx.set_xlabel('Time (hours)')
dx.set_ylabel(r'Average variance [{}]'.format(units))
if moornum ==8:
dx.set_xlim(date[0],date[-1])
else:
dx.set_xlim(date[0].values,date[-1].values)
fig.suptitle(savetit)
savefig(figdir+'Wavelet_'+savetit+'.png',bbox_inches='tight')
return nomd,spowdic
def cwt(signal, t, obspy=None):
# from __future__ import division
import numpy
from matplotlib import pyplot
import pycwt as wavelet
from pycwt.helpers import find
signal = signal[10000:11000]
t = t[10000:11000]
url = 'http://paos.colorado.edu/research/wavelets/wave_idl/nino3sst.txt'
dat = numpy.genfromtxt(url, skip_header=19)
title = 'DICARDIA'
label = 'DICARDIA SST'
units = 'degC'
t0 = 1871.0
dt = 0.25 # In years
N = signal.shape[0]
print(N)
p = numpy.polyfit(t, signal, 1)
dat_notrend = signal - numpy.polyval(p, t)
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(
signal) # 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, signal, '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()
def parse_frames(image_file, sig=0.95):
"""
"""
cap = cv2.VideoCapture(image_file)
if verbose: print("Video successfully loaded")
FRAME_COUNT = int(cap.get(cv2.CAP_PROP_FRAME_COUNT))
FPS = cap.get(cv2.CAP_PROP_FPS)
if verbose > 1:
FRAME_HEIGHT = cap.get(cv2.CAP_PROP_FRAME_HEIGHT)
FRAME_WIDTH = cap.get(cv2.CAP_PROP_FRAME_WIDTH)
print(
"INFO: \n Frame count: ",
FRAME_COUNT,
"\n",
"FPS: ",
FPS,
" \n",
"FRAME_HEIGHT: ",
FRAME_HEIGHT,
" \n",
"FRAME_WIDTH: ",
FRAME_WIDTH,
" \n",
)
directory = os.getcwd(
) + '\\analysis\\{}_{}_{}_{}({})_{}_{}_scaled\\'.format(
date, trial_type, name, wavelet, order, per_min, per_max)
if not os.path.exists(directory):
os.makedirs(directory)
made = False
frame_idx = 0
idx = 0
dropped = 0
skip = True
thresh = None
df_wav = pd.DataFrame()
df_auc = pd.DataFrame()
df_for = pd.DataFrame()
df_pow = pd.DataFrame()
for i in range(FRAME_COUNT):
a, img = cap.read()
if a:
frame_idx += 1
if made == False:
#first we need to manually determine the boundaries and angle
res = bg.manual_format(img)
#print(res)
x, y, w, h, angle = res
horizon_begin = x
horizon_end = x + w
vert_begin = y
vert_end = y + h
#scale_array = np.zeros((FRAME_COUNT, abs(horizon_begin - horizon_end)))
#area_time = np.zeros((FRAME_COUNT))
#df[']
print("Now Select the Red dot")
red_res = bg.manual_format(img, stop_sign=True)
red_x, red_y, red_w, red_h = red_res
box_h_begin = red_x
box_h_end = red_x + red_w
box_v_begin = red_y
box_v_end = red_y + red_h
made = True
#dims = (vert_begin, vert_end, horizon_begin, horizon_end)
real_time = i / FPS
rows, cols, chs = img.shape
M = cv2.getRotationMatrix2D((cols / 2, rows / 2), angle, 1)
rot_img = cv2.warpAffine(img, M, (cols, rows))
roi = rot_img[vert_begin:vert_end, horizon_begin:horizon_end, :]
red_box = img[box_v_begin:box_v_end, box_h_begin:box_h_end, 2]
if thresh == None:
thresh = np.mean(red_box)
#print(np.mean(red_box))
percent_drop = 1 - (np.mean(red_box) / thresh)
print(percent_drop)
if percent_drop >= 0.18:
#cv2.imshow("Red Image", red_box)
#cv2.waitKey(0)
skip = False
if skip:
if verbose >= 1:
print('Frame is skipped {} / {}'.format(
frame_idx, FRAME_COUNT))
continue
if verbose >= 1:
print('Processing frame {} / {}'.format(
frame_idx, FRAME_COUNT))
idx += 1
begin_code, data_line = extract_frame(roi)
#We need to detrend the data before sending it away
N = len(data_line)
dt = su / N
t = np.arange(0, N) * dt
t = t - np.mean(t)
var, std, dat_norm = detrend(data_line)
###################################################################
if wavelet == 'DOG':
mother = cwt.DOG(order)
elif wavelet == 'Paul':
mother = cwt.Paul(order)
elif wavelet == 'Morlet':
mother = cwt.Morlet(order)
elif wavelet == 'MexicanHat':
mother = cwt.MexicanHat(order)
s0 = 4 * dt
try:
alpha, _, _ = cwt.ar1(dat_norm)
except:
alpha = 0.95
wave, scales, freqs, coi, fft, fftfreqs = cwt.cwt(
dat_norm, dt, dj, s0, J, mother)
iwave = cwt.icwt(
wave, scales, dt, dj,
mother) * std #This is a reconstruction of the wave
power = (np.abs(wave))**2 #This is the power spectra
fft_power = np.abs(fft)**2 #This is the fourier power
period = 1 / freqs #This is the periods of the wavelet analysis in cm
power /= scales[:,
None] #This is an option suggested by Liu et. al.
#Next we calculate the significance of the power spectra. Significane where power / sig95 > 1
signif, fft_theor = cwt.significance(1.0,
dt,
scales,
0,
alpha,
significance_level=0.95,
wavelet=mother)
sig95 = np.ones([1, N]) * signif[:, None]
sig95 = power / sig95
#This is the significance of the global wave
glbl_power = power.mean(axis=1)
dof = N - scales # Correction for padding at edges
glbl_signif, tmp = cwt.significance(var,
dt,
scales,
1,
alpha,
significance_level=0.95,
dof=dof,
wavelet=mother)
sel = find((period >= per_min) & (period < per_max))
Cdelta = mother.cdelta
scale_avg = (scales * np.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_array[i,:] = scale_array[i,:]/np.max(scale_array[i,:])
#data_array[i,:] = data_array[i,:]/np.max(data_array[i,:])
scale_avg = var * dj * dt / Cdelta * scale_avg[sel, :].sum(axis=0)
scale_avg_signif, tmp = cwt.significance(
var,
dt,
scales,
2,
alpha,
significance_level=0.95,
dof=[scales[sel[0]], scales[sel[-1]]],
wavelet=mother)
Yticks = 2**np.arange(np.ceil(np.log2(period.min())),
np.ceil(np.log2(period.max())))
plt.close('all')
plt.ioff()
figprops = dict(figsize=(11, 8), dpi=72)
fig = plt.figure(**figprops)
wx = plt.axes([0.77, 0.75, 0.2, 0.2])
imz = 0
for idxy in range(0, len(period), 10):
wx.plot(t, mother.psi(t / period[idxy]) + imz, linewidth=1.5)
imz += 1
wx.xaxis.set_ticklabels([])
#wx.set_ylim([-10,10])
# First sub-plot, the original time series anomaly and inverse wavelet
# transform.
ax = plt.axes([0.1, 0.75, 0.65, 0.2])
ax.plot(t,
data_line - np.mean(data_line),
'k',
label="Original Data")
ax.plot(t,
iwave,
'-',
linewidth=1,
color=[0.5, 0.5, 0.5],
label="Reconstructed wave")
ax.plot(t,
dat_norm,
'--k',
linewidth=1.5,
color=[0.5, 0.5, 0.5],
label="Denoised Wave")
ax.set_title(
'a) {:10.2f} from beginning of trial.'.format(real_time))
ax.set_ylabel(r'{} [{}]'.format("Amplitude", unit))
ax.legend(loc=1)
ax.set_ylim([-200, 200])
#If the non-serrated section, bounds are 200 -
# 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 = plt.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]
cont = bx.contourf(t,
np.log2(period),
np.log2(power),
np.log2(levels),
extend='both',
cmap=plt.cm.viridis)
extent = [t.min(), t.max(), 0, max(period)]
bx.contour(t,
np.log2(period),
sig95, [-99, 1],
colors='k',
linewidths=2,
extent=extent)
bx.fill(np.concatenate(
[t, t[-1:] + dt, t[-1:] + dt, t[:1] - dt, t[:1] - dt]),
np.concatenate([
np.log2(coi), [1e-9],
np.log2(period[-1:]),
np.log2(period[-1:]), [1e-9]
]),
'k',
alpha=0.3,
hatch='x')
bx.set_title(
'b) {} Octaves Wavelet Power Spectrum [{}({})]'.format(
octaves, mother.name, order))
bx.set_ylabel('Period (cm)')
#
Yticks = 2**np.arange(np.ceil(np.log2(period.min())),
np.ceil(np.log2(period.max())))
bx.set_yticks(np.log2(Yticks))
bx.set_yticklabels(Yticks)
cbar = fig.colorbar(cont, ax=bx)
# Third sub-plot, the global wavelet and Fourier power spectra and theoretical
# noise spectra. Note that period scale is logarithmic.
cx = plt.axes([0.77, 0.37, 0.2, 0.28], sharey=bx)
cx.plot(glbl_signif, np.log2(period), 'k--')
cx.plot(var * fft_theor, np.log2(period), '--', color='#cccccc')
cx.plot(var * fft_power,
np.log2(1. / fftfreqs),
'-',
color='#cccccc',
linewidth=1.)
cx.plot(var * glbl_power, np.log2(period), 'k-', linewidth=1.5)
cx.set_title('c) Global Wavelet Spectrum')
cx.set_xlabel(r'Power [({})^2]'.format(unit))
#cx.set_xlim([0, (var*fft_theor).max()])
plt.xscale('log')
cx.set_ylim(np.log2([period.min(), period.max()]))
cx.set_yticks(np.log2(Yticks))
cx.set_yticklabels(Yticks)
#if sig_array == []:
yvals = np.linspace(Yticks.min(), Yticks.max(), len(period))
plt.xscale('linear')
plt.setp(cx.get_yticklabels(), visible=False)
# Fourth sub-plot, the scale averaged wavelet spectrum.
dx = plt.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) {}-{}cm scale-averaged power'.format(
per_min, per_max))
dx.set_xlabel('Distance from center(cm)')
dx.set_ylabel(r'Average variance [{}]'.format(unit))
#dx.set_ylim([0,500])
ax.set_xlim([t.min(), t.max()])
#plt.savefig(directory+'{}_analysis_frame-{}.png'.format(name, idx), bbox = 'tight')
if verbose >= 2:
print('*' * int((i / FRAME_COUNT) * 100))
df_wav[real_time] = (pd.Series(dat_norm, index=t))
df_pow[real_time] = (pd.Series(var * glbl_power,
index=np.log2(period)))
df_for[real_time] = (pd.Series(var * fft_power,
index=np.log2(1. / fftfreqs)))
df_auc[real_time] = [np.trapz(data_line)]
else:
print("Frame #{} has dropped".format(i))
dropped += 1
if verbose >= 1: print('All images saved')
if verbose >= 1:
print("{:10.2f} % of the frames have dropped".format(
(dropped / FRAME_COUNT) * 100))
#Plotting and saving tyhe
row, cols = df_pow.shape
time = np.arange(0, cols) / FPS
plt.close('all')
plt.ioff()
plt.contourf(time, df_pow.index.tolist(), df_pow)
plt.contour(time, df_pow.index.tolist(), df_pow)
plt.title("Global Power over Time")
plt.ylabel("Period[cm]")
plt.xlabel("Time")
cax = plt.gca()
#plt.xscale('log')
cax.set_ylim(np.log2([period.min(), period.max()]))
cax.set_yticks(np.log2(Yticks))
cax.set_yticklabels(Yticks)
plt.savefig(directory + '{}_global_power-{}.png'.format(name, idx),
bbox='tight')
row, cols = df_for.shape
time = np.arange(0, cols) / FPS
plt.close('all')
plt.ioff()
plt.contourf(time, df_for.index.tolist(), df_for)
plt.contour(time, df_for.index.tolist(), df_for)
plt.title("Fourier Power over Time")
plt.ylabel("Period[cm]")
plt.xlabel("Time")
cax = plt.gca()
#plt.xscale('log')
cax.set_ylim(np.log2([period.min(), period.max()]))
cax.set_yticks(np.log2(Yticks))
cax.set_yticklabels(Yticks)
plt.savefig(directory + '{}_fourier_power-{}.png'.format(name, idx),
bbox='tight')
plt.close('all')
plt.ioff()
rows, cols = df_auc.shape
time = np.arange(0, cols) / FPS
plt.plot(time, df_auc.T)
plt.xlabel("Time")
plt.ylabel("Area under the curve in cm")
plt.title("Area under the curve over time")
plt.savefig(directory + '{}_area_under_curve-{}.png'.format(name, idx),
bbox='tight')
df_wav['Mean'] = df_wav.mean(axis=1)
df_pow['Mean'] = df_pow.mean(axis=1)
df_for['Mean'] = df_for.mean(axis=1)
df_auc['Mean'] = df_auc.mean(axis=1)
df_wav['Standard Deviation'] = df_wav.std(axis=1)
df_pow['Standard Deviation'] = df_pow.std(axis=1)
df_for['Standard Deviation'] = df_for.std(axis=1)
df_auc['Standard Deviation'] = df_auc.std(axis=1)
##[Writing analysis to excel]##############################################
print("Writing files")
writer = pd.ExcelWriter(directory + "analysis{}.xlsx".format(trial_name))
df_wav.to_excel(writer, "Raw Waveforms")
df_auc.to_excel(writer, "Area Under the Curve")
df_for.to_excel(writer, "Fourier Spectra")
df_pow.to_excel(writer, "Global Power Spectra")
writer.save()
##[Writing means to a single file]#########################################
#filename = 'C:\\pyscripts\\wavelet_analysis\\Overall_Analysis.xlsx'
#append_data(filename, df_pow['Mean'].values, str(trial_name), Yticks)
##[Plotting mean power and foruier]########################################
plt.close('all')
plt.ioff()
plt.plot(df_pow['Mean'], df_pow.index.tolist(), label="Global Power")
plt.plot(df_for['Mean'], df_for.index.tolist(), label="Fourier Power")
plt.title("Global Power averaged over Time")
plt.ylabel("Period[cm]")
plt.xlabel("Power[cm^2]")
cax = plt.gca()
#plt.xscale('log')
cax.set_ylim(np.log2([period.min(), period.max()]))
cax.set_yticks(np.log2(Yticks))
cax.set_yticklabels(Yticks)
plt.legend()
plt.savefig(directory + '{}_both_{}.png'.format(name, idx), bbox='tight')
plt.close('all')
plt.ioff()
plt.plot(df_pow['Mean'], df_pow.index.tolist(), label="Global Power")
plt.title("Global Power averaged over Time")
plt.ylabel("Period[cm]")
plt.xlabel("Power[cm^2]")
cax = plt.gca()
#plt.xscale('log')
cax.set_ylim(np.log2([period.min(), period.max()]))
cax.set_yticks(np.log2(Yticks))
cax.set_yticklabels(Yticks)
plt.legend()
plt.savefig(directory + '{}_global_power_{}.png'.format(name, idx),
bbox='tight')
plt.close('all')
plt.ioff()
plt.plot(df_for['Mean'], df_for.index.tolist(), label="Fourier Power")
plt.title("Fourier averaged over Time")
plt.ylabel("Period[cm]")
plt.xlabel("Power[cm^2]")
cax = plt.gca()
#plt.xscale('log')
cax.set_ylim(np.log2([period.min(), period.max()]))
cax.set_yticks(np.log2(Yticks))
cax.set_yticklabels(Yticks)
plt.legend()
plt.savefig(directory + '{}_fourier_{}.png'.format(name, idx),
bbox='tight')
cap.release()
return directory
std1 = s1.std()
std2 = s2.std()
# I. Continuous wavelet transform
# ===============================
# Calculate the CWT of both normalized time series. The function wavelet.cwt
# returns a a list with containing [wave, scales, freqs, coi, fft, fftfreqs]
# variables.
mother = wavelet.Morlet(6) # Morlet mother wavelet with m=6
slevel = 0.95 # Significance level
dj = 1/12 # Twelve sub-octaves per octaves
s0 = -1 # 2 * dt # Starting scale, here 6 months
J = -1 # 7 / dj # Seven powers of two with dj sub-octaves
if True:
alpha1, _, _ = wavelet.ar1(s1) # Lag-1 autocorrelation for red noise
alpha2, _, _ = wavelet.ar1(s2) # Lag-1 autocorrelation for red noise
else:
alpha1 = alpha2 = 0.0 # Lag-1 autocorrelation for white noise
# The following routines perform the wavelet transform and siginificance
# analysis for two data sets.
W1, scales1, freqs1, coi1, _, _ = wavelet.cwt(s1/std1, dt, dj, s0, J, mother)
signif1, fft_theor1 = wavelet.significance(1.0, dt, scales1, 0, alpha1,
significance_level=slevel,
wavelet=mother)
W2, scales2, freqs2, coi2, _, _ = wavelet.cwt(s2/std2, dt, dj, s0, J, mother)
signif2, fft_theor2 = wavelet.significance(1.0, dt, scales2, 0, alpha2,
significance_level=slevel,
wavelet=mother)
# HW8-2
"""
for ii in i:
eta[ii] = np.sin(2 * np.pi * ii / 20) + np.sin(2 * np.pi * ii / 10) / 2
"""
fig, sub = plt.subplots(figsize=(10, 4))
sub.plot(i, eta, ls="-", c="k", lw=1)
sub.set_xlim(min(i), max(i))
sub.set_xticks(np.arange(0, 200 + 20, 20))
sub.set_xlabel("i", fontdict={"weight": "bold"})
sub.set_ylabel("Eta", fontdict={"weight": "bold"})
sub.set_title("Data", fontdict={"weight": "bold"})
mother = wavelet.Morlet(f0=6)
alpha, _, _ = wavelet.ar1(eta)
dj = 0.25
s0 = 2 * dt
J = 7 / dj
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(signal = eta, dt = dt,\
dj = dj, s0 = s0, J = J,\
wavelet = mother)
power = np.abs(wave)**2
fft_power = np.abs(fft)**2
period = 1 / freqs
signif, fft_theor = wavelet.significance(signal = 1.0, dt = dt,\
scales = scales, alpha = alpha,\
significance_level = 0.95,\
def plot_wavelet(t,
dat,
dt,
pl,
pr,
period_pltlim=None,
ax=None,
ax2=None,
stscale=2,
siglev=0.95,
cmap='viridis',
title='',
levels=None,
label='',
units='',
tunits='',
sav_img=False):
import pycwt as wavelet
from pycwt.helpers import find
import numpy as np
import matplotlib.pyplot as plt
from copy import copy
import numpy.ma as ma
t_ = copy(t)
t0 = t[0]
# print(Time(t[-1:], format='plot_date').iso)
# We also create a time array in years.
N = dat.size
t = np.arange(0, N) * dt + t0
# print(Time(t[-1:], format='plot_date').iso)
# We write the following code to detrend and normalize the input data by its
# standard deviation. Sometimes detrending is not necessary and simply
# removing the mean value is good enough. However, if your dataset has a well
# defined trend, such as the Mauna Loa CO\ :sub:`2` dataset available in the
# above mentioned website, it is strongly advised to perform detrending.
# Here, we fit a one-degree polynomial function and then subtract it from the
# original data.
p = np.polyfit(t - t0, dat, 1)
dat_notrend = dat - np.polyval(p, t - t0)
std = dat_notrend.std() # Standard deviation
var = std**2 # Variance
dat_norm = dat_notrend / std # Normalized dataset
# The next step is to define some parameters of our wavelet analysis. We
# select the mother wavelet, in this case the Morlet wavelet with
# :math:`\omega_0=6`.
mother = wavelet.Morlet(6)
s0 = stscale * dt # Starting scale, in this case 2 * 0.25 years = 6 months
dj = 1 / 12 # Twelve sub-octaves per octaves
J = -1 # 7 / dj # Seven powers of two with dj sub-octaves
alpha, _, _ = wavelet.ar1(dat) # Lag-1 autocorrelation for red noise
# The following routines perform the wavelet transform and inverse wavelet
# transform using the parameters defined above. Since we have normalized our
# input time-series, we multiply the inverse transform by the standard
# deviation.
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
dat_norm, dt, dj, s0, J, mother)
iwave = wavelet.icwt(wave, scales, dt, dj, mother) * std
# We calculate the normalized wavelet and Fourier power spectra, as well as
# the Fourier equivalent periods for each wavelet scale.
power = (np.abs(wave))**2
fft_power = np.abs(fft)**2
period = 1 / freqs
# We could stop at this point and plot our results. However we are also
# interested in the power spectra significance test. The power is significant
# where the ratio ``power / sig95 > 1``.
signif, fft_theor = wavelet.significance(1.0,
dt,
scales,
0,
alpha,
significance_level=siglev,
wavelet=mother)
sig95 = np.ones([1, N]) * signif[:, None]
sig95 = power / sig95
# Then, we calculate the global wavelet spectrum and determine its
# significance level.
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=siglev,
dof=dof,
wavelet=mother)
# We also calculate the scale average between 2 years and 8 years, and its
# significance level.
sel = find((period >= pl) & (period < pr))
Cdelta = mother.cdelta
scale_avg = (scales * np.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=siglev,
dof=[scales[sel[0]], scales[sel[-1]]],
wavelet=mother)
# levels = [0.25, 0.5, 1, 2, 4, 8, 16,32]
if levels is None:
levels = np.linspace(0.0, 128., 256)
# ax.contourf(t, np.log2(period), np.log2(power), np.log2(levels), extend='both', cmap=plt.cm.viridis)
im = ax.contourf(t_,
np.array(period) * 24 * 60,
power,
levels,
extend='both',
cmap=cmap,
zorder=-20)
# for pathcoll in im.collections:
# pathcoll.set_rasterized(True)
ax.set_rasterization_zorder(-10)
# im = ax.pcolormesh(t_, np.array(period) * 24 * 60, power,vmax=32.,vmin=0, cmap=cmap)
# im = ax.contourf(t, np.array(period)*24*60, np.log2(power), np.log2(levels), extend='both', cmap=cmap)
extent = [t_.min(), t_.max(), 0, max(period) * 24 * 60]
# ax.contour(t, np.log2(period), sig95, [-99, 1], colors='k', linewidths=1, extent=extent)
CS = ax.contour(t_,
np.array(period) * 24 * 60,
sig95 * siglev, [-99, 1.0 * siglev],
colors='k',
linewidths=1,
extent=extent)
ax.clabel(CS, inline=1, fmt='%1.3f')
ax.fill(np.concatenate(
[t_, t_[-1:] + dt, t_[-1:] + dt, t_[:1] - dt, t_[:1] - dt]),
np.concatenate([
np.array(coi), [2**(1e-9)],
np.array(period[-1:]),
np.array(period[-1:]), [2**(1e-9)]
]) * 24 * 60,
color='k',
alpha=0.75,
edgecolor='None',
facecolor='k',
hatch='x')
# ### not Matplotlib does not display hatching when rendering to pdf. Here is a workaround.
# ax.fill(np.concatenate([t_, t_[-1:] + dt, t_[-1:] + dt, t_[:1] - dt, t_[:1] - dt]),
# np.concatenate(
# [np.array(coi), [2 ** (1e-9)], np.array(period[-1:]), np.array(period[-1:]),
# [2 ** (1e-9)]]) * 24 * 60,
# color='None', alpha=1.0, edgecolor='k', hatch='x')
# ax.set_title('b) {} Wavelet Power Spectrum ({})'.format(label, mother.name))
#
# ax.set_rasterization_zorder(20)
# Yticks = np.arange(np.ceil(np.array(period.min()*24*60)), np.ceil(np.array(period.max()*24*60)))
# ax.set_yticks(np.array(Yticks))
# ax.set_yticklabels(Yticks)
ax2.plot(glbl_signif, np.array(period) * 24 * 60, 'k--')
# ax2.plot(var * fft_theor, np.array(period) * 24 * 60, '--', color='#cccccc')
# ax2.plot(var * fft_power, np.array(1. / fftfreqs) * 24 * 60, '-', color='#cccccc',
# linewidth=1.)
ax2.plot(var * glbl_power, np.array(period) * 24 * 60, 'k-', linewidth=1)
mperiod = ma.masked_outside(np.array(period), period_pltlim[0],
period_pltlim[1])
mpower = ma.masked_array(var * glbl_power, mask=mperiod.mask)
# ax2.set_title('c) Global Wavelet Spectrum')
ax2.set_xlabel(r'Power'.format(units))
ax2.set_xlim([0, mpower.compressed().max() + var])
# print(glbl_power)
# ax2.set_ylim(np.array([period.min(), period.max()]))
# ax2.set_yticks(np.array(Yticks))
# ax2.set_yticklabels(Yticks)
plt.setp(ax2.get_yticklabels(), visible=False)
if period_pltlim:
ax.set_ylim(np.array(period_pltlim) * 24 * 60)
else:
ax.set_ylim(np.array([period.min(), period.max()]) * 24 * 60)
return im
def get_graph_from_file(in_filepath, out_folder, out_filename):
# Get data
# TODO there are differents formats of file
# TODO implement differents parsers by parameters of function
p1 = numpy.genfromtxt(in_filepath)
# TODO fix this shit
dat = p1
title = 'NINO3 Sea Surface Temperature'
label = 'NINO3 SST'
units = 'degC'
# Values for calculations
# TODO spike about args
t0 = 12.0 # start time
dt = 0.5 # step of differentiation - in minutes
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(dpi=144)
fig = pyplot.figure(**figprops)
# 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])
levels = [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
bx.contourf(t,
period,
numpy.log2(power),
numpy.log2(levels),
extend='both',
cmap=pyplot.cm.viridis)
extent = [t.min(), t.max(), 0, max(period)]
bx.contour(t,
period,
sig95, [-99, 1],
colors='k',
linewidths=2,
extent=extent)
bx.set_title('{} Wavelet Power Spectrum ({})'.format(label, mother.name))
bx.set_ylabel('Period (minutes)')
#
#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)
bx.set_ylim([2, 20])
# Save graph to file
# TODO implement
#pyplot.savefig('{}/{}.png'.format(out_folder, out_filename))
# ----------------------------------------------
# or show the graph
pyplot.show()
dat = mei
N = dat.size
t = np.arange(0, N) * dt + t0
p = np.polyfit(t - t0, dat, 1)
dat_notrend = dat - np.polyval(p, t - t0)
std = dat_notrend.std()
var = std**2
dat_norm = dat_notrend / std
mother = wavelet.Morlet(6)
s0 = 2 * dt
dj = 1 / 12
J = 7 / dj
alpha, _, _ = wavelet.ar1(dat)
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 = (np.abs(wave))**2
fft_power = np.abs(fft)**2
period = 1 / freqs
power /= scales[:, None]
signif, fft_theor = wavelet.significance(1.0,
dt,
scales,
0,
def graph_wavelet(data_xs, title, lims, font = 11, params = default_params):
a_lims, b_lims, d_lims = lims
plt.rcParams.update({'font.size': font})
return_data = {}
N = len(data_xs)
dt = (2*params['per_pixel'])/N #This is how much cm each pixel equals
t = np.arange(0, N) * dt
t = t - np.mean(t)
t0 = 0
per_min = params['min_per']
per_max = params['max_per']
units = params['units']
sx = params['sx']
octaves = params['octaves']
dj = 1/params['suboctaves'] #suboctaves
order = params['order']
var, std, dat_norm = detrend(data_xs)
mother = cwt.DOG(order) #This is the Mother Wavelet
s0 = sx * dt #This is the starting scale, which in out case is two pixels or 0.04cm/40um\
J = octaves/dj #This is powers of two with dj suboctaves
return_data['var'] = var
return_data['std'] = std
try:
alpha, _, _ = cwt.ar1(dat_norm) #This calculates the Lag-1 autocorrelation for red noise
except:
alpha = 0.95
wave, scales, freqs, coi, fft, fftfreqs = cwt.cwt(dat_norm, dt, dj, s0, J,
mother)
return_data['scales'] = scales
return_data['freqs'] = freqs
return_data['fft'] = fft
iwave = cwt.icwt(wave, scales, dt, dj, mother) * std
power = (np.abs(wave)) ** 2
fft_power = np.abs(fft) ** 2
period = 1 / freqs
power /= scales[:, None] #This is an option suggested by Liu et. al.
#Next we calculate the significance of the power spectra. Significane where power / sig95 > 1
signif, fft_theor = cwt.significance(1.0, dt, scales, 0, alpha,
significance_level=0.95,
wavelet=mother)
sig95 = np.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 = cwt.significance(var, dt, scales, 1, alpha,
significance_level=0.95, dof=dof,
wavelet=mother)
sel = find((period >= per_min) & (period < per_max))
Cdelta = mother.cdelta
scale_avg = (scales * np.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 = cwt.significance(var, dt, scales, 2, alpha,
significance_level=0.95,
dof=[scales[sel[0]],
scales[sel[-1]]],
wavelet=mother)
# Prepare the figure
plt.close('all')
plt.ioff()
figprops = dict(figsize=(11, 11), dpi=72)
fig = plt.figure(**figprops)
wx = plt.axes([0.77, 0.75, 0.2, 0.2])
imz = 0
for idxy in range(0,len(period), 10):
wx.plot(t, mother.psi(t / period[idxy]) + imz, linewidth = 1.5)
imz+=1
wx.xaxis.set_ticklabels([])
ax = plt.axes([0.1, 0.75, 0.65, 0.2])
ax.plot(t, data_xs, 'k', linewidth=1.5)
ax.plot(t, iwave, '-', linewidth=1, color=[0.5, 0.5, 0.5])
ax.plot(t, dat_norm, '--', linewidth=1.5, color=[0.5, 0.5, 0.5])
if a_lims != None:
ax.set_ylim([-a_lims, a_lims])
ax.set_title('a) {}'.format(title))
ax.set_ylabel(r'Displacement [{}]'.format(units))
#ax.set_ylim([-20,20])
bx = plt.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, np.log2(period), np.log2(power), np.log2(levels),
extend='both', cmap=plt.cm.viridis)
extent = [t.min(), t.max(), 0, max(period)]
bx.contour(t, np.log2(period), sig95, [-99, 1], colors='k', linewidths=2,
extent=extent)
bx.fill(np.concatenate([t, t[-1:] + dt, t[-1:] + dt,
t[:1] - dt, t[:1] - dt]),
np.concatenate([np.log2(coi), [1e-9], np.log2(period[-1:]),
np.log2(period[-1:]), [1e-9]]),
'k', alpha=0.3, hatch='x')
bx.set_title('b) {} Octaves Wavelet Power Spectrum [{}({})]'.format(octaves, mother.name, order))
bx.set_ylabel('Period (cm)')
#
Yticks = 2 ** np.arange(np.ceil(np.log2(period.min())),
np.ceil(np.log2(period.max())))
bx.set_yticks(np.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 = plt.axes([0.77, 0.37, 0.2, 0.28], sharey=bx)
cx.plot(glbl_signif, np.log2(period), 'k--')
cx.plot(var * fft_theor, np.log2(period), '--', color='#cccccc')
cx.plot(var * fft_power, np.log2(1./fftfreqs), '-', color='#cccccc',
linewidth=1.)
return_data['global_power'] = var * glbl_power
return_data['fourier_spectra'] = var * fft_power
return_data['per'] = np.log2(period)
return_data['amp'] = np.log2(1./fftfreqs)
cx.plot(var * glbl_power, np.log2(period), 'k-', linewidth=1.5)
cx.set_title('c) Power Spectrum')
cx.set_xlabel(r'Power [({})^2]'.format(units))
if b_lims != None:
cx.set_xlim([0,b_lims])
#cx.set_xlim([0,max(glbl_power.max(), var*fft_power.max())])
#print(max(glbl_power.max(), var*fft_power.max()))
cx.set_ylim(np.log2([period.min(), period.max()]))
cx.set_yticks(np.log2(Yticks))
cx.set_yticklabels(Yticks)
return_data['yticks'] = Yticks
plt.setp(cx.get_yticklabels(), visible=False)
# Fourth sub-plot, the scale averaged wavelet spectrum.
dx = plt.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) {}--{} cm scale-averaged power'.format(per_min, per_max))
dx.set_xlabel('Displacement (cm)')
dx.set_ylabel(r'Average variance [{}]'.format(units))
ax.set_xlim([t.min(), t.max()])
if d_lims != None:
dx.set_ylim([0,d_lims])
plt.savefig("C:\pyscripts\wavelet_analysis\Calibrated Images\{}".format(title))
return fig, return_data
def simple_sample(sls):
# Then, we load the dataset and define some data related parameters. In this
# case, the first 19 lines of the data file contain meta-data, that we ignore,
# since we set them manually (*i.e.* title, units).
# url = 'http://paos.colorado.edu/research/wavelets/wave_idl/nino3sst.txt'
# dat = numpy.genfromtxt(url, skip_header=19)
title = 'Sentence Length'
label = 'Zhufu Sentence Length'
units = 'Characters'
t0 = 1
dt = 1 # In years
dat = numpy.array(sls)
# We also create a time array in years.
N = dat.size
t = numpy.arange(0, N) * dt + t0
# We write the following code to detrend and normalize the input data by its
# standard deviation. Sometimes detrending is not necessary and simply
# removing the mean value is good enough. However, if your dataset has a well
# defined trend, such as the Mauna Loa CO\ :sub:`2` dataset available in the
# above mentioned website, it is strongly advised to perform detrending.
# Here, we fit a one-degree polynomial function and then subtract it from the
# original data.
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
# The next step is to define some parameters of our wavelet analysis. We
# select the mother wavelet, in this case the Morlet wavelet with
# :math:`\omega_0=6`.
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
# The following routines perform the wavelet transform and inverse wavelet
# transform using the parameters defined above. Since we have normalized our
# input time-series, we multiply the inverse transform by the standard
# deviation.
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
dat_norm, dt, dj, s0, J, mother)
iwave = wavelet.icwt(wave, scales, dt, dj, mother) * std
# We calculate the normalized wavelet and Fourier power spectra, as well as
# the Fourier equivalent periods for each wavelet scale.
power = (numpy.abs(wave))**2
fft_power = numpy.abs(fft)**2
period = 1 / freqs
# We could stop at this point and plot our results. However we are also
# interested in the power spectra significance test. The power is significant
# where the ratio ``power / sig95 > 1``.
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
# Then, we calculate the global wavelet spectrum and determine its
# significance level.
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)
# We also calculate the scale average between 2 years and 8 years, and its
# significance level.
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)
# Finally, we plot our results in four different subplots containing the
# (i) original series anomaly and the inverse wavelet transform; (ii) the
# wavelet power spectrum (iii) the global wavelet and Fourier spectra ; and
# (iv) the range averaged wavelet spectrum. In all sub-plots the significance
# levels are either included as dotted lines or as filled contour lines.
# 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)
# Third sub-plot, the global wavelet and Fourier power spectra and theoretical
# noise spectra. Note that period scale is logarithmic.
dx = pyplot.axes([0.1, 0.07, 0.65, 0.2])
dx.plot(numpy.log2(fftfreqs), numpy.log2(fft_power), 'k')
dx.plot(numpy.log2(freqs), var * fft_theor, '--', color='#cccccc')
dx.plot(numpy.log2(1. / fftfreqs),
var * fft_power,
'-',
color='#cccccc',
linewidth=1.)
dx.plot(fftfreqs, fft_power, 'k-', linewidth=1.5)
dx.set_title('d) Global Wavelet Spectrum')
dx.set_ylabel(r'Power [({})^2]'.format(units))
dx.set_xlim([0, 2 * fftfreqs.max()])
Yticks = 2**numpy.arange(numpy.ceil(numpy.log2(fft_power.min())),
numpy.ceil(numpy.log2(fft_power.max())))
dx.set_ylim(numpy.log2([fft_power.min(), fft_power.max()]))
dx.set_yticks(numpy.log2(Yticks))
dx.set_yticklabels(Yticks)
pyplot.setp(dx.get_yticklabels(), visible=False)
pyplot.show()
