def cross_wavelet_transform(y1,
y2,
dt,
dj=1 / 12,
s0=-1,
J=-1,
wavelet='morlet',
normalize=True):
wavelet = pycwt.wavelet._check_parameter_wavelet(wavelet)
# Makes sure input signal 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)
# Calculates the cross CWT of y1 and y2.
W12 = W1 * W2.conj()
return W12, coi, freq
def get_lf0_le_cwt(lf0, le):
mother = wavelet.MexicanHat()
dt = 0.005
dj = 1
s0 = dt * 2
J = 9
C_delta = 3.541
Wavelet_lf0, scales, _, _, _, _ = wavelet.cwt(np.squeeze(lf0), dt, dj, s0, J, mother)
Wavelet_le, scales, _, _, _, _ = wavelet.cwt(np.squeeze(le), dt, dj, s0, J, mother)
Wavelet_lf0 = np.real(Wavelet_lf0).T
Wavelet_le = np.real(Wavelet_le).T # (T, D=10)
lf0_le_cwt = np.concatenate((Wavelet_lf0, Wavelet_le), -1)
return lf0_le_cwt
def cwt(self):
"""
Compute the continuous wavelet transform
Returns
-------
bool
True if successful, False otherwise
Atributes
---------
wt : ndarray
Real part of the wavelet coefficients
wtN : ndarray
Real part of the wavelet coefficients normalized
(wt-<wt>)/\sigma
"""
try:
wt, sc, freqs, coi, fft, fftfreqs = wav.cwt(
self.sig, self.dt, 0.25, self.scale, 0, self.mother)
self.wt = np.real(np.squeeze(wt))
self.wtN = (self.wt - self.wt.mean()) / self.wt.std()
return True
except BaseException:
return False
def calculate_power_wavelet(rr_intervals, heart_rate=4, mother_wave='morlet'):
"""
Method to calculate the spectral power using wavelet method.
Parameters
----------
rr_intervals: array-like
list of RR interval (in ms)
heart_rate: int
values = The range of heart rate frequency * 2
mother_wave: str
The main waveform to transform data.
Available waves are:
'gaussian':
'paul': apply lomb method to compute PSD
'mexican_hat':
Returns
-------
freq : list
Frequency of the corresponding psd points.
psd : list
Power Spectral Density of the signal.
"""
dt = 1 / heart_rate
if mother_wave in mother_wave_dict.keys():
mother_morlet = mother_wave_dict[mother_wave]
else:
mother_morlet = wavelet.Morlet()
wave, scales, freqs, coi, fft, fftfreqs = \
wavelet.cwt(rr_intervals, dt, wavelet=mother_morlet)
powers = (np.abs(wave))**2
return freqs, powers
def plot_tfr(times, freqs, data, mother=None):
'''
Plots time frequency representations of analog signal with PSD estimation
Parameters
----------
times : array
freqs : array
power : array
mother : wavelet
'''
import pycwt
if mother is None:
mother = pycwt.Morlet()
sampling_period = times[1] - times[0]
wave, scales, freqs, coi, fft, fftfreqs = pycwt.cwt(data, sampling_period, freqs=freqs, wavelet=mother)
power = (numpy.abs(wave)) ** 2
power /= scales[:, None] #rectify the power spectrum according to the suggestions proposed by Liu et al. (2007)
fft_power = numpy.abs(fft) ** 2
gs = gridspec.GridSpec(3, 3)
ax_pow = plt.subplot(gs[:2, 1:3])
ax_pow.set_xlim(*times[[0,-1]])
ax_pow.set_ylim(*freqs[[0,-1]])
ax_fft = plt.subplot(gs[:2, 0], sharey=ax_pow)
ax_sig = plt.subplot(gs[2, 1:3], sharex=ax_pow)
ax_pow.contourf(times, freqs, power, levels=100)
ax_sig.plot(times, data)
ax_fft.plot(fft_power, fftfreqs)
def calculate_cwt(self, memory_buffer, return_plot_info=False):
coefs_array = []
if not return_plot_info:
row_count = self.scale_count
col_count = memory_buffer.length
coefs_array = np.ndarray(
[row_count, col_count, memory_buffer.channel_count])
channel_index = 0
buffer = memory_buffer.get_buffer()
for idx, x in enumerate(buffer):
channel_name = memory_buffer.channels[idx]
x = self.__normalize(x)
coefs, scales, freqs, coi, fft, fftfreqs = pycwt.cwt(
x, self.delta_t, wavelet=self.wavelet_type, freqs=self.freqs)
coefs = self.__normalize(coefs.real)
# "clean" values away from the max/min
#coefs = np.power(coefs, 3)
if not return_plot_info:
for i in range(row_count):
for j in range(col_count):
coefs_array[i][j][channel_index] = coefs[i][j]
channel_index += 1
else:
coefs_array.append((channel_name, coefs, x, coi))
return coefs_array
def wavel(signal, cadence):
mother = 'morlet'
sig_level = 0.95
#/ signal.std()
t1 = np.linspace(0, cadence * signal.size, signal.size)
wave, scales, freqs, coi = wavelet.cwt((signal - signal.mean()),
cadence,
wavelet=mother,
dj=1 / 100.)
power = (np.abs(wave))**2 # / scales[:,None]
period = 1 / freqs
alpha = 0.0
## (variance=1 for the normalized SST)
signif, fft_theor = wavelet.significance(signal,
period,
scales,
0,
alpha,
significance_level=sig_level,
wavelet=mother)
sig95 = np.ones([1, signal.size]) * signif[:, None]
sig95 = power / sig95
## indices for stuff
idx = find_closest(period, coi.max())
## Into minutes
t1 /= 60
period /= 60
coi /= 60
return wave, scales, sig95, idx, t1, coi, period, power
def remove_frequencies_and_save_to_csv(dtfrm, band_pass, high_pass,
low_freq_limit, high_freq_limit, width,
delta_t):
[row, column] = dtfrm.shape
frame = pd.DataFrame()
i = 0
while i < column:
y = dtfrm.iloc[:, i].tolist()
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
y, delta_t, J=width - 1, wavelet=u'mexicanhat')
wave = select_frequencies(wave, width, band_pass, high_pass, delta_t,
low_freq_limit, high_freq_limit)
xrec = wavelet.icwt(wave, scales, delta_t, wavelet=u'mexicanhat')
xrec = normalize(xrec)
xrec = pd.Series(xrec)
frame = pd.concat([frame, xrec], axis=1, ignore_index=True)
i = i + 1
print("Select a folder")
path = th.ui.getdir('Select a directory to save the csv file'
) # prompts user to select folder
frame.to_csv(path + "/" + "new_dtfrm.csv")
def cwt(self, rectify=False):
"""
Operating cwt.
Parameter
---------
None
Return
------
wave : numpy.ndarray
Wavelet transform according to the selected mother wavelet.
Has (J+1) x N dimensions.
scales : numpy.ndarray
Vector of scale indices given by sj = s0 * 2**(j * dj),
j={0, 1, ..., J}.
freqs : array like
Vector of Fourier frequencies (in 1 / time units) that
corresponds to the wavelet scales.
coi : numpy.ndarray
Returns the cone of influence, which is a vector of N
points containing the maximum Fourier period of useful
information at that particular time. Periods greater than
those are subject to edge effects.
fft : numpy.ndarray
Normalized fast Fourier transform of the input signal.
fft_freqs : numpy.ndarray
Fourier frequencies (in 1/time units) for the calculated
FFT spectrum.
Example
-------
wave, scales, coi, fft, fft_freqs = cwt.cwt()
"""
wave, \
scales, \
freqs, \
coi, \
fft, \
fft_freqs \
= wavelet.cwt(
self.dat_norm,
self.feature.dt,
self.feature.dj,
self.feature.minimum_scale,
self.feature.J,
self.feature.mother,
self.feature.freqs
)
wavelet_spectra = WaveletSpectora(wave, scales, freqs, coi, rectify,
self.feature)
fourier_spectra = FourierSpectora(fft, fft_freqs)
self.wavelet_spectra = wavelet_spectra
self.fourier_spectra = fourier_spectra
return (wavelet_spectra, fourier_spectra)
def calculateCWT(t,s,steps=32):
mother = wavelet.Morlet(6)
deltaT = t[1] - t[0]
dj = 1 / steps # sub-octaves per octaves
s0 = 2 * deltaT # Starting scale, here 2 months
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(s, deltaT, dj, s0, -1, mother)
# Normalized wavelet power spectra
power = (np.abs(wave)) ** 2
return power,scales,coi,freqs
def get_lf0_cwt(lf0):
mother = wavelet.MexicanHat()
dt = 0.005
dj = 0.015
s0 = dt*2
J = 513 - 1
Wavelet_lf0, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(np.squeeze(lf0), dt, dj, s0, J, mother)
Wavelet_lf0 = np.real(Wavelet_lf0).T
return Wavelet_lf0, scales
def getCWT_auxInfoPlus(signal):
[_, sj, freqs, coi, _, _] = pycwt.cwt(signal,
TR,
s0=s0,
dj=dj,
J=J,
wavelet=mother)
return sj, freqs, coi, mother, s0, dj, J, s1, TR
def getCWT_coeff(signal):
[coefficients, _, _, _, _, _] = pycwt.cwt(signal,
TR,
s0=s0,
dj=dj,
J=J,
wavelet=mother)
return coefficients
def xwt(trace_ref, trace_current, fs, ns, nt, vpo, freqmin, freqmax, nptsfreq):
# Choosing a Morlet wavelet with a central frequency w0 = 6
mother = wavelet.Morlet(6.)
# nx represent the number of element in the trace_current array
nx = np.size(trace_current)
x_reference = np.transpose(trace_ref)
x_current = np.transpose(trace_current)
# Sampling interval
dt = 1 / fs
# Spacing between discrete scales, the default value is 1/12
dj = 1 / vpo
# Number of scales less one, -1 refers to the default value which is J = (log2(N * dt / so)) / dj.
J = -1
# Smallest scale of the wavelet, default value is 2*dt
s0 = 2 * dt # Smallest scale of the wavelet, default value is 2*dt
# Creation of the frequency vector that we will use in the continuous wavelet transform
freqlim = np.linspace(freqmax, freqmin, num=nptsfreq, endpoint=True, retstep=False, dtype=None, axis=0)
# Calculation of the two wavelet transform independently
# scales are calculated using the wavelet Fourier wavelength
# fft : Normalized fast Fourier transform of the input trace
# fftfreqs : Fourier frequencies for the calculated FFT spectrum.
cwt_reference, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(x_reference, dt, dj, s0, J, mother, freqs=freqlim)
cwt_current, _, _, _, _, _ = wavelet.cwt(x_current, dt, dj, s0, J, mother, freqs=freqlim)
scales = np.array([[kk] for kk in scales])
invscales = np.kron(np.ones((1, nx)), 1 / scales)
cfs1 = smoothCFS(invscales * abs(cwt_reference) ** 2, scales, dt, ns, nt)
cfs2 = smoothCFS(invscales * abs(cwt_current) ** 2, scales, dt, ns, nt)
crossCFS = cwt_reference * np.conj(cwt_current)
WXamp = abs(crossCFS)
# cross-wavelet transform operation with smoothing
crossCFS = smoothCFS(invscales * crossCFS, scales, dt, ns, nt)
WXspec = crossCFS / (np.sqrt(cfs1) * np.sqrt(cfs2))
WXangle = np.angle(WXspec)
Wcoh = abs(crossCFS) ** 2 / (cfs1 * cfs2)
pp = 2 * np.pi * freqs
pp2 = np.array([[kk] for kk in pp])
WXdt = WXangle / np.kron(np.ones((1, nx)), pp2)
return WXamp, WXspec, WXangle, Wcoh, WXdt, freqs, coi
def getCWT(signal):
[coefficients, _, _, _, _, _] = pycwt.cwt(signal,
TR,
s0=s0,
dj=dj,
J=J,
wavelet=mother)
power = np.absolute(coefficients)
return power
def get_lf0_cwt(lf0):
mother = wavelet.MexicanHat()
# dt = 0.005
dt = 0.005
dj = 1
s0 = dt * 2
J = 9
Wavelet_lf0, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
np.squeeze(lf0), dt, dj, s0, J, mother)
Wavelet_lf0 = np.real(Wavelet_lf0).T # 取实数 并进行转置
return Wavelet_lf0, scales
def noise_estimate(indata, f_h, f_l):
nt = indata.size
tt = np.arange(0, nt) / Fs
pf = np.polyfit(tt, indata, 1)
indata_norm = indata - np.polyval(pf, tt)
i_wave, i_scales, i_freqs, i_coi, i_fft, i_fftfreqs = wavelet.cwt(indata_norm, 1 / Fs, dj, s0, J, mother)
i_power = (np.abs(i_wave)) ** 2
i_period = 1 / i_freqs
i_sel = find((i_period >= 1 / f_h) & (i_period < 1 / f_l)) # select frequency band for averaging
i_Cdelta = mother.cdelta
i_scale_avg = (i_scales * np.ones((nt, 1))).transpose()
i_scale_avg = i_power / i_scale_avg # As in Torrence and Compo (1998) equation 24
i_scale_avg = dj / Fs * i_Cdelta * i_scale_avg[i_sel, :].sum(axis=0)
i_max = max(i_scale_avg)
return i_max
def wavelet_decompose_power_spectrum(signal, wl=None,
resample=None,
resample_freq=None,
sampling_frequency=None,
filter_frequency=40,
dt=1):
"""
:param signal: The signal, a numpy array or PyTorch Tensor of shape (N,)
:param wl: Provided Wavelet (see pycwt documentation for available wavelets)
:param resample: Downsample factor for signal time series.
:param resample_freq: Downsample factor for wavelet frequency plane.
:param sampling_frequency: Sampling frequency to be used by the butterworth filter, if provided.
:param filter_frequency: Filter frequency for the butterworth filter
:param dt: Sampling interval Sampling interval for the continuous wavelet transform.
:return: Resampled time series, Resamples frequency series, power spectrum of shape (Frequencies, Time),
Original signal.
"""
if resample is not None:
signal = sp.resample(signal, signal.shape[0] // resample)
if isinstance(signal, torch.Tensor):
signal = signal.numpy()
# Butterworth filter
if sampling_frequency is not None:
sos = sp.butter(5, filter_frequency, 'low', fs=sampling_frequency, output='sos')
signal = sp.sosfilt(sos, signal)
time = np.arange(signal.shape[0])
# p = np.polyfit(time, signal, 1)
# dat_notrend = signal - np.polyval(p, time)
# std = dat_notrend.std() # Standard deviation
# dat_norm = dat_notrend / std # Normalized dataset
if wl is None:
wl = wavelet.Morlet(6)
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(signal, dt, wavelet=wl)
power = (np.abs(wave)) ** 2
power /= scales[:, None]
if resample_freq is not None:
power = sp.resample(power, num=resample_freq, axis=0)
freqs = sp.resample(freqs, num=resample_freq)
return time, np.array(freqs), power, signal
def continuous_wavelet_transform(data):
""" Written using the tutorial at https://pycwt.readthedocs.io/en/latest/tutorial.html"""
dt = 0.25
dj = 1 / 12
dat = (data - data.mean()) / data.std()
s0 = 2 * dt
J = 7 / dj
mother = wavelet.Morlet(6)
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
dat, dt, dj, s0, J, mother)
power = (np.abs(wave))**2
power = np.log2(power)
return power
def wavelet(data, sampling_rate, f_start, f_stop, f_step=1, morlet=20):
mother = pycwt.Morlet(morlet) # Morlet size
freqs = np.arange(f_start, f_stop + f_step, f_step) # Frequency range
wave, scales, freqs, coi, fft, fftfreqs = pycwt.cwt(data,
1. / sampling_rate,
freqs=freqs,
wavelet=mother)
power = (np.abs(wave))**2
power /= scales[:,
None] #rectify the power spectrum according to suggestions proposed by Liu et al. (2007)
mask_coi(power, freqs, coi)
return freqs, power
def get_lf0_cwt(lf0):
mother = wavelet.MexicanHat()
#dt = 0.005
dt = 0.005
dj = 1
s0 = dt * 2
J = 9
#C_delta = 3.541
#Wavelet_lf0, scales, _, _, _, _ = wavelet.cwt(np.squeeze(lf0), dt, dj, s0, J, mother)
Wavelet_lf0, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
np.squeeze(lf0), dt, dj, s0, J, mother)
#Wavelet_le, scales, _, _, _, _ = wavelet.cwt(np.squeeze(le), dt, dj, s0, J, mother)
Wavelet_lf0 = np.real(Wavelet_lf0).T
#Wavelet_le = np.real(Wavelet_le).T # (T, D=10)
#0lf0_le_cwt = np.concatenate((Wavelet_lf0, Wavelet_le), -1)
# iwave = wavelet.icwt(np.squeeze(lf0), scales, dt, dj, mother) * std
return Wavelet_lf0, scales
def continuous_wavelet_transform(data,
octave_exponent=10,
sub_octaves=25,
starting_scale=2,
dt=1):
""" Generate a continuous wavelet transform using pycwt
"""
std = data.std() # Standard deviation
dat = (data - data.mean()) / std # Calculating anomaly and normalizing
dj = 1 / sub_octaves # x sub-octaves per octaves
s0 = starting_scale # Starting scale
J = octave_exponent / dj # x powers of two with dj sub-octaves
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, dt, dj, s0, J, mother)
#print('wave.shape', wave.shape)
# Normalized wavelet and Fourier power spectra
wave = (np.abs(wave))**2
# Resize:
averaging_window = 160
#print('before reshape', wave.shape[1])
height = wave.shape[0]
rounded_shape = wave.shape[1] - wave.shape[1] % averaging_window
#print(rounded_shape)
wave = wave[:, :rounded_shape].reshape(-1, averaging_window).mean(
axis=1).reshape(height, int(rounded_shape / averaging_window))
#print('after reshape', wave.shape)
# Normalize to (0, 1)
# wave = (wave - np.min(wave)) / (np.max(wave) - np.min(wave))
# Vertical Flip
# wave = np.flipud(wave)
# Logarithm
wave = np.log2(wave)
return wave
def _padded_cwt(params, dt, dj, s0, J, mother, padding_len):
"""Private function to compute a wavelet transform on padded data
Parameters
----------
params: arraylike
The prosodic parameters.
dt: ?
?
dj: ?
?
s0: ?
?
J: ?
?
mother: ?
The mother wavelet.
padding_len: int
The padding length
Returns
-------
wavelet_matrix: ndarray
The wavelet data resulting from the analysis
scales: arraylike
The scale indices corresponding to the wavelet data
freqs: ?
?
coi: array
The cone of influence values
fft: ?
?
fftfreqs: ?
?
"""
#padded = concatenate([params,params,params])
padded = pad(params, padding_len, mode='edge') #edge
wavelet_matrix, scales, freqs, coi, fft, fftfreqs = cwt.cwt(
padded, dt, dj, s0, J, mother)
wavelet_matrix = _unpad(wavelet_matrix, padding_len)
#wavelet_matrix = _unpad(wavelet_matrix, len(params))
return (wavelet_matrix, scales, freqs, coi, fft, fftfreqs)
def plot_spectrum(sig, wavename, casename, t1, t2, fs=1 / 60):
"""
Plot the spectrogram from a given signal, timeframes and frequency
:param sig: The processed signal to plot
:param wavename: The name of the wavelet
:param casename: The name of the analysed column
:param t1: The starting timeframe
:param t2: The ending timeframe
:param fs: The frequency, defaults to 1/60
:return: Plots the spectrogram saving the results to a file
"""
T = np.array(range(t1, t2))
dat = sig[T]
dt = 1 / fs
t = T / fs / 60 / 60
dat_norm = dat / dat.std() # Normalized dataset
mother = wavelet.Morlet(6)
s0 = 2 * dt
dj = 1 / 12
J = 7 / dj
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
dat_norm, dt, dj, s0, J, mother)
power = (np.abs(wave))**2
period = 1 / freqs / 60 / 60
plt.figure(1, figsize=(6.4, 4.8))
bx = plt.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,
power,
np.log2(levels),
extend='both',
cmap=plt.cm.prism)
bx.set_ylabel('Period (hours)')
bx.set_xlabel('Time (hours)')
plt.savefig('./output/{}_2.png'.format(casename), dpi=300)
plt.close('all')
def get_cwt(self, signal):
scales = self.get_scales()
# """get frequencies range of interest 0.67 ~~ 4 Hz"""
MorletFourierFactor = 4 * math.pi / (6 + math.sqrt(2 + 6**2))
freqs = 1 / (scales * MorletFourierFactor)
coef, scales, _, coi, fft, fftfreqs = wavelet.cwt(signal,
1 /
self.samplingRate,
wavelet='morlet',
freqs=freqs)
firstScaleIndex = np.where(freqs < self.maxFreq)[0][0]
lastScaleIndex = np.where(freqs > self.minFreq)[0][-1]
energyProfile = np.abs(coef)
max_index = np.argmax(energyProfile[firstScaleIndex:lastScaleIndex, :],
axis=0)
instantPulseRate = 60 * freqs[firstScaleIndex + max_index]
return coef, instantPulseRate
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()
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,
alpha,
significance_level=0.95,
wavelet=mother)
def wavelet_analysis(z,
tm,
lon=None,
lat=None,
mother='Morlet',
alpha=0.0,
siglvl=0.95,
loc=None,
onlyloc=False,
periods=None,
sel_periods=[],
show=False,
save='',
dsave='',
prefix='',
labels=dict(),
title=None,
name=None,
fpath='',
fpattern='',
std=dict(),
crange=None,
levels=None,
cmap=cm.GMT_no_green,
debug=False):
"""Continuous wavelet transform and significance analysis.
The analysis is made using the methodology and statistical approach
suggested by Torrence and Compo (1998).
Depending on the dimensions of the input array, three different
kinds of approaches are taken. If the input array is one-dimensional
then only a simple analysis is performed. If the array is
bi- or three-dimensional then spectral Hovmoller diagrams are drawn
for each Fourier period given within a range of +/-25%.
PARAMETERS
z (array like) :
Input data. The data array should have one of these forms,
z[tm], z[tm, lat] or z[tm, lat, lon].
tm (array like) :
Time axis. It should contain values in matplotlib date
format (i.e. number of days since 0001-01-01 UTC).
lon (array like, optional) :
Longitude.
lat (array like, optional) :
Latitude.
mother (string, optional) :
Gives the name of the mother wavelet to be used. Possible
values are 'Morlet' (default), 'Paul' or 'Mexican hat'.
alpha (float or dictionary, optional) :
Lag-1 autocorrelation for background noise. Default value
is 0.0 (white noise). If different autocorrelation
coefficients should be used for different locations, then
the input should contain a dictionary with 'lon', 'lat',
'map' keys as for the std parameter.
siglvl (float, optional) :
Significance level. Default value is 0.95.
loc (array like, optional) :
Special locations of interest. If the input array is of
higher dimenstions, the output of the simple wavelet
analysis of each of the locations is output. The list
should contain the pairs of (lon, lat) for each locations
of interest.
onlyloc (boolean, optional) :
If set to true then only the specified locations are
analysed. The default is false.
periods (array like, optional) :
Special Fourier periods of interest in case of analysis of
higher dimensions (in years).
sel_periods (array like, optional) :
Select which Fourier periods spectral power are averaged.
show (boolean, optional) :
If set to true the the resulting maps are shown on screen.
save (string, optional) :
The path in which the resulting plots are to be saved. If
not set, then no images will be saved.
dsave (string, optional) :
If set, saves the scale averaged power spectrum series to
this path. This is especially useful if memory is an issue.
prefix (string, optional) :
Prefix to retain naming conventions such as basin.
labels (dictionary, optional) :
Sets the labels for the plot axis.
title (string, array like, optional) :
Title of each of the selected periods.
name (string, array like, optional) :
Name of each of the selected periods. Used when saving the
results to files.
fpath (string, optional) :
Path for the source files to be loaded when memory issues
are a concern.
fpattern (string, optional) :
Regular expression pattern to match file names.
std (dictionary, optional) :
A dictionary containing a map of the standard deviation of
the analysed time series. To set the longitude and latitude
coordinates of the map, they should be included as
separate 'lon' and 'lat' key items. If they are omitted,
then the regular input parameters are assumed. Accepted
standard deviation error is set in key 'err' (default value
is 1e-2).
crange (array like, optional) :
Array of power levels to be used in average Hovmoler colour bar.
levels (array like, optional) :
Array of power levels to be used in spectrogram colour bar.
cmap (colormap, optional) :
Sets the colour map to be used in the plots. The default is
the Generic Mapping Tools (GMT) no green.
debug (boolean, optional) :
If set to True then warnings are shown.
OUTPUT
If show or save are set, plots either on screen and or on file
according to the specified parameters.
If dsave parameter is set, also saves the scale averaged power
series to files.
RETURNS
wave (dictionary) :
Dictionary containing the resulting calculations from the
wavelet analysis according to the input parameters. The
output items might be:
scale --
Wavelet scales.
period --
Equivalent Fourier periods (in days).
power_spectrum --
Wavelet power spectrum (in units**2).
power_significance --
Relative significance of the power spectrum.
global_power --
Global wavelet power spectrum (in units**2).
scale_spectrum --
Scale averaged wavelet spectra (in units**2)
according to selected periods.
scale_significance --
Relative significance of the scale averaged wavelet
spectra.
fft --
Fourier spectrum.
fft_first --
Fourier spectrum of the first half of the
time-series.
fft_second --
Fourier spectrum of the second half of the
time-series.
fft_period --
Fourier periods (in days).
trend --
Signal trend (in units/yr).
wavelet_trend --
Wavelet spectrum trends (in units**2/yr).
"""
t1 = time()
result = {}
# Resseting unit labels for hovmoller plots
hlabels = dict(labels)
hlabels['units'] = ''
# Setting some titles and paths
if name == None:
name = title
# Working with the std parameter and setting its properties:
if 'val' in std.keys():
if 'lon' not in std.keys():
std['lon'] = lon
std['lon180'] = common.lon180(std['lon'])
if 'lat' not in std.keys():
std['lat'] = lat
if 'err' not in std.keys():
std['err'] = 1e-2
std['map'] = True
else:
std['map'] = False
# Lag-1 autocorrelation parameter
if type(alpha).__name__ == 'dict':
if 'lon' not in alpha.keys():
alpha['lon'] = lon
alpha['lon180'] = common.lon180(alpha['lon'])
if 'lat' not in alpha.keys():
alpha['lat'] = lat
alpha['mean'] = alpha['val'].mean()
alpha['map'] = True
alpha['calc'] = False
else:
if alpha == -1:
alpha = {'mean': -1, 'calc': True}
else:
alpha = {'val': alpha, 'mean': alpha, 'map': False, 'calc': False}
# Shows some of the options on screen.
print('Average Lag-1 autocorrelation for background noise: %.2f' %
(alpha['mean']))
if save:
print 'Saving result figures in \'%s\'.' % (save)
if dsave:
print 'Saving result data in \'%s\'.' % (dsave)
if fpath:
# Gets the list of files to be loaded individually extracts all the
# latitudes and loads the first file to get the main parameters.
flist = os.listdir(fpath)
flist, match = common.reglist(flist, fpattern)
if len(flist) == 0:
raise Warning, 'No files matched search pattern.'
flist = numpy.asarray(flist)
lst_lat = []
for item in match:
y = string.atof(item[-2])
if item[-1].upper() == 'S': y *= -1
lst_lat.append(y)
# Detect file type from file name
ftype = fm.detect_ftype(flist[0])
x, y, tm, z = fm.load_map('%s/%s' % (fpath, flist[0]),
ftype=ftype,
masked=True)
if lon == None:
lon = x
lat = numpy.unique(lst_lat)
dim = 2
else:
# Transforms input arrays in numpy arrays and numpy masked arrays.
tm = numpy.asarray(tm)
z = numpy.ma.asarray(z)
z.mask = numpy.isnan(z)
# Determines the number of dimensions of the variable to be plotted and
# the sizes of each dimension.
a = b = c = None
dim = len(z.shape)
if dim == 3:
c, b, a = z.shape
elif dim == 2:
c, a = z.shape
b = 1
z = z.reshape(c, b, a)
else:
c = z.shape[0]
a = b = 1
z = z.reshape(c, b, a)
if tm.size != c:
raise Warning, 'Time and data lengths do not match.'
# Transforms coordinate arrays into numpy arrays
s = type(lat).__name__
if s in ['int', 'float', 'float64']:
lat = numpy.asarray([lat])
elif s != 'NoneType':
lat = numpy.asarray(lat)
s = type(lon).__name__
if s in ['int', 'float', 'float64']:
lon = numpy.asarray([lon])
elif s != 'NoneType':
lon = numpy.asarray(lon)
# Starts the mother wavelet class instance and determines important
# analysis parameters
mother = mother.lower()
if mother == 'morlet':
mother = wavelet.Morlet()
elif mother == 'paul':
mother = wavelet.Paul()
elif mother in ['mexican hat', 'mexicanhat', 'mexican_hat']:
mother = wavelet.Mexican_hat()
else:
raise Warning, 'Mother wavelet unknown.'
t = tm / common.daysinyear # Time array in years
dt = tm[1] - tm[0] # Temporal sampling interval
try: # Zonal sampling interval
dx = lon[1] - lon[0]
except:
dx = 1
try: # Meridional sampling interval
dy = lat[1] - lat[0]
except:
dy = dx
if numpy.isnan(dt): dt = 1
if numpy.isnan(dx): dx = 1
if numpy.isnan(dy): dy = dx
dj = 0.25 # Four sub-octaves per octave
s0 = 2 * dt # Smallest scale
J = 7 / dj - 1 # Seven powers of two with dj sub-octaves
scales = period = None
if type(crange).__name__ == 'NoneType':
crange = numpy.arange(0, 1.1, 0.1)
if type(levels).__name__ == 'NoneType':
levels = 2.**numpy.arange(-3, 6)
if fpath:
N = lat.size
# TODO: refactoring # lon = numpy.arange(-81. - dx / 2., 290. + dx / 2, dx)
# TODO: refactoring # lat = numpy.unique(numpy.asarray(lst_lat))
c, b, a = tm.size, lat.size, lon.size
else:
N = a * b
# Making sure that the longitudes range from -180 to 180 degrees and
# setting the squared search radius R2.
try:
lon180 = common.lon180(lon)
except:
lon180 = None
R2 = dx**2 + dy**2
if numpy.isnan(R2):
R2 = 65535.
if loc != None:
loc = numpy.asarray([[common.lon180(item[0]), item[1]]
for item in loc])
# Initializes important result variables such as the global wavelet power
# spectrum map, scale avaraged spectrum time-series and their significance,
# wavelet power trend map.
global_power = numpy.ma.empty([J + 1, b, a]) * numpy.nan
try:
C = len(periods) + 1
dT = numpy.diff(periods)
pmin = numpy.concatenate([[periods[0] - dT[0] / 2],
0.5 * (periods[:-1] + periods[1:])])
pmax = numpy.concatenate(
[0.5 * (periods[:-1] + periods[1:]), [periods[-1] + dT[-1] / 2]])
except:
# Sets the lowest period to null and the highest to half the time
# series length.
C = 1
pmin = numpy.array([0])
pmax = numpy.array([(tm[-1] - tm[0]) / 2])
if type(sel_periods).__name__ in ['int', 'float']:
sel_periods = [sel_periods]
elif len(sel_periods) == 0:
sel_periods = [-1.]
try:
if fpath:
raise Warning, 'Process files individually'
avg_spectrum = numpy.ma.empty([C, c, b, a]) * numpy.nan
mem_error = False
except:
avg_spectrum = numpy.ma.empty([C, c, a]) * numpy.nan
mem_error = True
avg_spectrum_signif = numpy.ma.empty([C, b, a]) * numpy.nan
trend = numpy.ma.empty([b, a]) * numpy.nan
wavelet_trend = numpy.ma.empty([C, b, a]) * numpy.nan
fft_trend = numpy.ma.empty([C, b, a]) * numpy.nan
std_map = numpy.ma.empty([b, a]) * numpy.nan
zero = numpy.ma.empty([c, a])
fft_spectrum = None
fft_spectrum1 = None
fft_spectrum2 = None
# Walks through each latitude and then through each longitude to perform
# the temporal wavelet analysis.
if N == 1:
plural = ''
else:
plural = 's'
s = 'Spectral analysis of %d location%s... ' % (N, plural)
stdout.write(s)
stdout.flush()
for j in range(b):
t2 = time()
isloc = False # Ressets 'is special location' flag
hloc = [] # Cleans location list for Hovmoller plots
zero *= numpy.nan
if mem_error:
# Clears average spectrum for next step.
avg_spectrum *= numpy.nan
avg_spectrum.mask = False
if fpath:
findex = pylab.find(lst_lat == lat[j])
if len(findex) == 0:
continue
ftype = fm.detect_ftype(flist[findex[0]])
try:
x, y, tm, z = fm.load_dataset(fpath,
flist=flist[findex],
ftype=ftype,
masked=True,
lon=lon,
lat=lat[j:j + 1],
verbose=True)
except:
continue
z = z[:, 0, :]
x180 = common.lon180(x)
# Determines the first and second halves of the time-series and some
# constants for the FFT
fft_ta = numpy.ceil(t.min())
fft_tb = numpy.floor(t.max())
fft_tc = numpy.round(fft_ta + fft_tb) / 2
fft_ia = pylab.find((t >= fft_ta) & (t <= fft_tc))
fft_ib = pylab.find((t >= fft_tc) & (t <= fft_tb))
fft_N = int(2**numpy.ceil(numpy.log2(max([len(fft_ia), len(fft_ib)]))))
fft_N2 = fft_N / 2 - 1
fft_dt = t[fft_ib].mean() - t[fft_ia].mean()
for i in range(a):
# Some string output.
try:
Y, X = common.num2latlon(lon[i],
lat[j],
mode='each',
padding=False)
except:
Y = X = '?'
# Extracts individual time-series from the whole dataset and
# sets or calculates its standard deviation, squared standard
# deviation and finally the normalized time-series.
if fpath:
try:
ilon = pylab.find(x == lon[i])[0]
fz = z[:, ilon]
except:
continue
else:
fz = z[:, j, i]
if fz.mask.all():
continue
if std['map']:
try:
u = pylab.find(std['lon180'] == lon180[i])[0]
v = pylab.find(std['lat'] == lat[j])[0]
except:
if debug:
warnings.warn(
'Unable to locate standard deviation '
'for (%s, %s)' % (X, Y), Warning)
continue
fstd = std['val'][v, u]
estd = fstd - fz.std()
if (estd < 0) & (abs(estd) > std['err']):
if debug:
warnings.warn('Discrepant input standard deviation '
'(%f) location (%.3f, %.3f) will be '
'disregarded.' %
(estd, lon180[i], lat[j]))
continue
else:
fstd = fz.std()
fstd2 = fstd**2
std_map[j, i] = fstd
zero[:, i] = fz
fz = (fz - fz.mean()) / fstd
# Calculates the distance of the current point to any special
# location set in the 'loc' parameter. If only special locations
# are to be analysed, then skips all other ones. If the input
# array is one dimensional, then do the analysis anyway.
if dim == 1:
dist = numpy.asarray([0.])
else:
try:
dist = numpy.asarray([
((item[0] - (lon180[i]))**2 + (item[1] - lat[j])**2)
for item in loc
])
except:
dist = []
if (dist > R2).all() & (loc != 'all') & onlyloc:
continue
# Determines the lag-1 autocorrelation coefficient to be used in
# the significance test from the input parameter
if alpha['calc']:
ac = acorr(fz)
alpha_ij = (ac[c + 1] + ac[c + 2]**0.5) / 2
elif alpha['map']:
try:
u = pylab.find(alpha['lon180'] == lon180[i])[0]
v = pylab.find(alpha['lat'] == lat[j])[0]
alpha_ij = alpha['val'][v, u]
except:
if debug:
warnings.warn(
'Unable to locate standard deviation '
'for (%s, %s) using mean value instead' % (X, Y),
Warning)
alpha_ij = alpha['mean']
else:
alpha_ij = alpha['mean']
# Calculates the continuous wavelet transform using the wavelet
# Python module. Calculates the wavelet and Fourier power spectrum
# and the periods in days. Also calculates the Fourier power
# spectrum for the first and second halves of the timeseries.
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(
fz, dt, dj, s0, J, mother)
power = abs(wave * wave.conj())
fft_power = abs(fft * fft.conj())
period = 1. / freqs
fftperiod = 1. / fftfreqs
psel = pylab.find(period <= pmax.max())
# Calculates the Fourier transform for the first and the second
# halves ot the time-series for later trend analysis.
fft_1 = numpy.fft.fft(fz[fft_ia], fft_N)[1:fft_N / 2] / fft_N**0.5
fft_2 = numpy.fft.fft(fz[fft_ib], fft_N)[1:fft_N / 2] / fft_N**0.5
fft_p1 = abs(fft_1 * fft_1.conj())
fft_p2 = abs(fft_2 * fft_2.conj())
# Creates FFT return array and stores the spectrum accordingly
try:
fft_spectrum[:, j, i] = fft_power * fstd2
fft_spectrum1[:, j, i] = fft_p1 * fstd2
fft_spectrum2[:, j, i] = fft_p2 * fstd2
except:
fft_spectrum = (numpy.ma.empty([len(fft_power), b, a]) *
numpy.nan)
fft_spectrum1 = (numpy.ma.empty([fft_N2, b, a]) * numpy.nan)
fft_spectrum2 = (numpy.ma.empty([fft_N2, b, a]) * numpy.nan)
#
fft_spectrum[:, j, i] = fft_power * fstd2
fft_spectrum1[:, j, i] = fft_p1 * fstd2
fft_spectrum2[:, j, i] = fft_p2 * fstd2
# Performs the significance test according to the article by
# Torrence and Compo (1998). The wavelet power is significant
# if the ratio power/sig95 is > 1.
signif, fft_theor = wavelet.significance(1.,
dt,
scales,
0,
alpha_ij,
significance_level=siglvl,
wavelet=mother)
sig95 = (signif * numpy.ones((c, 1))).transpose()
sig95 = power / sig95
# Calculates the global wavelet power spectrum and its
# significance. The global wavelet spectrum is the average of the
# wavelet power spectrum over time. The degrees of freedom (dof)
# have to be corrected for padding at the edges.
glbl_power = power.mean(axis=1)
dof = c - scales
glbl_signif, tmp = wavelet.significance(1.,
dt,
scales,
1,
alpha_ij,
significance_level=siglvl,
dof=dof,
wavelet=mother)
global_power[:, j, i] = glbl_power * fstd2
# Calculates the average wavelet spectrum along the scales and its
# significance according to Torrence and Compo (1998) eq. 24. The
# scale_avg_full variable is used multiple times according to the
# selected periods range.
#
# Also calculates the average Fourier power spectrum.
Cdelta = mother.cdelta
scale_avg_full = (scales * numpy.ones((c, 1))).transpose()
scale_avg_full = power / scale_avg_full
for k in range(C):
if k == 0:
sel = pylab.find((period >= pmin[0])
& (period <= pmax[-1]))
pminmax = [period[sel[0]], period[sel[-1]]]
les = pylab.find((fftperiod >= pmin[0])
& (fftperiod <= pmax[-1]))
fminmax = [fftperiod[les[0]], fftperiod[les[-1]]]
else:
sel = pylab.find((period >= pmin[k - 1])
& (period < pmax[k - 1]))
pminmax = [pmin[k - 1], pmax[k - 1]]
les = pylab.find((fftperiod >= pmin[k - 1])
& (fftperiod <= pmax[k - 1]))
fminmax = [fftperiod[les[0]], fftperiod[les[-1]]]
scale_avg = numpy.ma.array(
(dj * dt / Cdelta * scale_avg_full[sel, :].sum(axis=0)))
scale_avg_signif, tmp = wavelet.significance(
1.,
dt,
scales,
2,
alpha_ij,
significance_level=siglvl,
dof=[scales[sel[0]], scales[sel[-1]]],
wavelet=mother)
scale_avg.mask = (scale_avg < scale_avg_signif)
if mem_error:
avg_spectrum[k, :, i] = scale_avg
else:
avg_spectrum[k, :, j, i] = scale_avg
avg_spectrum_signif[k, j, i] = scale_avg_signif
# Trend analysis using least square polynomial fit of one
# degree of the original input data and scale averaged
# wavelet power. The wavelet power trend is calculated only
# where the cone of influence spans the highest analyzed
# period. In the end, the returned value for the trend is in
# units**2.
#
# Also calculates the trends in the Fourier power spectrum.
# Note that the FFT power spectrum is already multiplied by
# the signal's standard deviation.
incoi = pylab.find(coi >= pmax[-1])
if len(incoi) == 0:
incoi = numpy.arange(c)
polyw = numpy.polyfit(t[incoi], scale_avg[incoi].data, 1)
wavelet_trend[k, j, i] = polyw[0] * fstd2
fft_trend[k, j, i] = (
fft_spectrum2[les[les < fft_N2], j, i] -
fft_spectrum1[les[les < fft_N2], j, i]).mean() / fft_dt
if k == 0:
polyz = numpy.polyfit(t, fz * fstd, 1)
trend[j, i] = polyz[0]
# Plots the wavelet analysis results for the individual
# series. The plot is only generated if the dimension of the
# input variable z is one, if a special location is within a
# range of the search radius R and if the show or save
# parameters are set.
if (show | (save != '')) & ((k in sel_periods)):
if (dist < R2).any() | (loc == 'all') | (dim == 1):
# There is an interesting spot within the search
# radius of location (%s, %s).' % (Y, X)
isloc = True
if (dist < R2).any():
try:
hloc.append(loc[(dist < R2)][0, 0])
except:
pass
if save:
try:
sv = '%s/tz_%s_%s_%d' % (
save, prefix,
common.num2latlon(lon[i], lat[j]), k)
except:
sv = '%s' % (save)
else:
sv = ''
graphics.wavelet_plot(tm,
period[psel],
fz,
power[psel, :],
coi,
glbl_power[psel],
scale_avg.data,
fft=fft,
fft_period=fftperiod,
power_signif=sig95[psel, :],
glbl_signif=glbl_signif[psel],
scale_signif=scale_avg_signif,
pminmax=pminmax,
labels=labels,
normalized=True,
std=fstd,
ztrend=polyz,
wtrend=polyw,
show=show,
save=sv,
levels=levels,
cmap=cmap)
# Saves and/or plots the intermediate results as zonal temporal
# diagrams.
if dsave:
for k in range(C):
if k == 0:
sv = '%s/%s/%s_%s.xt.gz' % (
dsave, 'global', prefix,
common.num2latlon(lon[i], lat[j], mode='each')[0])
else:
sv = '%s/%s/%s_%s.xt.gz' % (
dsave, name[k - 1].lower(), prefix,
common.num2latlon(lon[i], lat[j], mode='each')[0])
if mem_error:
fm.save_map(lon, tm, avg_spectrum[k, :, :].data, sv,
lat[j])
else:
fm.save_map(lon, tm, avg_spectrum[k, :, j, :].data, sv,
lat[j])
if ((dim > 1) and (show or (save != '')) & (not onlyloc)
and len(hloc) > 0):
hloc = common.lon360(numpy.unique(hloc))
if save:
sv = '%s/xt_%s_%s' % (save, prefix,
common.num2latlon(
lon[i], lat[j], mode='each')[0])
else:
sv = ''
if mem_error:
# To include overlapping original signal, use zz=zero
gis.hovmoller(lon,
tm,
avg_spectrum[1:, :, :],
zo=avg_spectrum_signif[1:, j, :],
title=title,
crange=crange,
show=show,
save=sv,
labels=hlabels,
loc=hloc,
cmap=cmap,
bottom='avg',
right='avg',
std=std_map[j, :])
else:
gis.hovmoller(lon,
tm,
avg_spectrum[1:, :, j, :],
zo=avg_spectrum_signif[1:, j, :],
title=title,
crange=crange,
show=show,
save=sv,
labels=hlabels,
loc=hloc,
cmap=cmap,
bottom='avg',
right='avg',
std=std_map[j, :])
# Flushing profiling text.
stdout.write(len(s) * '\b')
s = 'Spectral analysis of %d location%s (%s)... %s ' % (
N, plural, Y, common.profiler(b, j + 1, 0, t1, t2))
stdout.write(s)
stdout.flush()
stdout.write('\n')
result['scale'] = scales
result['period'] = period
if dim == 1:
result['power_spectrum'] = power * fstd2
result['power_significance'] = sig95
result['cwt'] = wave
result['fft'] = fft
result['global_power'] = global_power
result['scale_spectrum'] = avg_spectrum
if fpath:
result['lon'] = lon
result['lat'] = lat
result['scale_significance'] = avg_spectrum_signif
result['trend'] = trend
result['wavelet_trend'] = wavelet_trend
result['fft_power'] = fft_spectrum
result['fft_first'] = fft_spectrum1
result['fft_second'] = fft_spectrum2
result['fft_period'] = fftperiod
result['fft_trend'] = fft_trend
return result
def wavelet(self, signal, mother='morlet', plot=True):
"""
Takes a 1D signal and perfroms a continous wavelet transform.
Parameters
----------
time: ndarray
The 1D time series for the data
data: ndarray
The actual 1D data
mother: string
The name of the family. Acceptable values are Paul, Morlet, DOG, Mexican_hat
plot: bool
If True, will return a plot of the result.
Returns
-------
Examples
--------
"""
sig_level = 0.95
std2 = signal.std() ** 2
signal_orig = signal[:]
signal = (signal - signal.mean())/ signal.std()
t1 = np.linspace(0,self.period*signal.size,signal.size)
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(signal,
self.period,
wavelet=mother, dj=1/100)
power = (np.abs(wave)) ** 2
period = 1/freqs
# alpha, _, _ = wavelet.ar1(signal)
alpha = 0.0
## (variance=1 for the normalized SST)
signif, fft_theor = wavelet.significance(1.0, self.period, scales, 0, alpha,
significance_level=sig_level, wavelet=mother)
sig95 = np.ones([1, signal.size]) * signif[:, None]
sig95 = power / sig95
glbl_power = std2 * power.mean(axis=1)
dof = signal.size - scales
glbl_signif, tmp = wavelet.significance(std2, self.period, scales, 1, alpha,
significance_level=sig_level, dof=dof, wavelet=mother)
## indices for stuff
idx = self.find_closest(period,coi.max())
## Into minutes
t1 /= 60
period /= 60
coi /= 60
if plot:
plt.figure(figsize=(12,12))
ax = plt.axes([0.1, 0.75, 0.65, 0.2])
ax.plot(t1, signal_orig-signal_orig.mean(), 'k', linewidth=1.5)
extent = [t1.min(),t1.max(),0,max(period)]
bx = plt.axes([0.1, 0.1, 0.65, 0.55], sharex=ax)
im = NonUniformImage(bx, interpolation='nearest', extent=extent)
im.set_cmap('cubehelix')
im.set_data(t1, period[:idx], power[:idx,:])
bx.images.append(im)
bx.contour(t1, period[:idx], sig95[:idx,:], [-99,1], colors='w', linewidths=2, extent=extent)
bx.fill(np.concatenate([t1, t1[-1:]+self.period, t1[-1:]+self.period,t1[:1]-self.period, t1[:1]-self.period]),
(np.concatenate([coi,[1e-9], period[-1:], period[-1:], [1e-9]])),
'k', alpha=0.3,hatch='x', zorder=100)
bx.set_xlim(t1.min(),t1.max())
cx = plt.axes([0.77, 0.1, 0.2, 0.55], sharey=bx)
cx.plot(glbl_signif[:idx], period[:idx], 'k--')
cx.plot(glbl_power[:idx], period[:idx], 'k-', linewidth=1.5)
cx.set_ylim(([min(period), period[idx]]))
plt.setp(cx.get_yticklabels(), visible=False)
plt.show()
return wave, scales, freqs, coi, power
import pycwt import numpy as np from pylab import * filename = 'sst_nino3.dat' data = loadtxt(filename) # remove mean data = (data - np.nansum(data) / len(data)) data[np.isnan(data)] = 0 t = pycwt.cwt(data, pycwt.Morlet(), octaves=8, dscale=0.1) b = pycwt.bootstrap_signif(t, 200) imshow(t.power(), aspect='auto') contour(b, levels=[0.05], colors='w') figure() plot(pycwt.time_avg(t), t.scales)
# 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)
power1 = (np.abs(W1)) ** 2 # Normalized wavelet power spectrum
power2 = (np.abs(W2)) ** 2 # Normalized wavelet power spectrum
period1 = 1/freqs1
period2 = 1/freqs2
sig95_1 = np.ones([1, n1]) * signif1[:, None]
sig95_1 = power1 / sig95_1 # Where ratio > 1, power is significant
sig95_2 = np.ones([1, n2]) * signif2[:, None]
def wavelet_analysis(z, tm, lon=None, lat=None, mother='Morlet', alpha=0.0,
siglvl=0.95, loc=None, onlyloc=False, periods=None,
sel_periods=[], show=False, save='', dsave='', prefix='',
labels=dict(), title=None, name=None, fpath='',
fpattern='', std=dict(), crange=None, levels=None,
cmap=cm.GMT_no_green, debug=False):
"""Continuous wavelet transform and significance analysis.
The analysis is made using the methodology and statistical approach
suggested by Torrence and Compo (1998).
Depending on the dimensions of the input array, three different
kinds of approaches are taken. If the input array is one-dimensional
then only a simple analysis is performed. If the array is
bi- or three-dimensional then spectral Hovmoller diagrams are drawn
for each Fourier period given within a range of +/-25%.
PARAMETERS
z (array like) :
Input data. The data array should have one of these forms,
z[tm], z[tm, lat] or z[tm, lat, lon].
tm (array like) :
Time axis. It should contain values in matplotlib date
format (i.e. number of days since 0001-01-01 UTC).
lon (array like, optional) :
Longitude.
lat (array like, optional) :
Latitude.
mother (string, optional) :
Gives the name of the mother wavelet to be used. Possible
values are 'Morlet' (default), 'Paul' or 'Mexican hat'.
alpha (float or dictionary, optional) :
Lag-1 autocorrelation for background noise. Default value
is 0.0 (white noise). If different autocorrelation
coefficients should be used for different locations, then
the input should contain a dictionary with 'lon', 'lat',
'map' keys as for the std parameter.
siglvl (float, optional) :
Significance level. Default value is 0.95.
loc (array like, optional) :
Special locations of interest. If the input array is of
higher dimenstions, the output of the simple wavelet
analysis of each of the locations is output. The list
should contain the pairs of (lon, lat) for each locations
of interest.
onlyloc (boolean, optional) :
If set to true then only the specified locations are
analysed. The default is false.
periods (array like, optional) :
Special Fourier periods of interest in case of analysis of
higher dimensions (in years).
sel_periods (array like, optional) :
Select which Fourier periods spectral power are averaged.
show (boolean, optional) :
If set to true the the resulting maps are shown on screen.
save (string, optional) :
The path in which the resulting plots are to be saved. If
not set, then no images will be saved.
dsave (string, optional) :
If set, saves the scale averaged power spectrum series to
this path. This is especially useful if memory is an issue.
prefix (string, optional) :
Prefix to retain naming conventions such as basin.
labels (dictionary, optional) :
Sets the labels for the plot axis.
title (string, array like, optional) :
Title of each of the selected periods.
name (string, array like, optional) :
Name of each of the selected periods. Used when saving the
results to files.
fpath (string, optional) :
Path for the source files to be loaded when memory issues
are a concern.
fpattern (string, optional) :
Regular expression pattern to match file names.
std (dictionary, optional) :
A dictionary containing a map of the standard deviation of
the analysed time series. To set the longitude and latitude
coordinates of the map, they should be included as
separate 'lon' and 'lat' key items. If they are omitted,
then the regular input parameters are assumed. Accepted
standard deviation error is set in key 'err' (default value
is 1e-2).
crange (array like, optional) :
Array of power levels to be used in average Hovmoler colour bar.
levels (array like, optional) :
Array of power levels to be used in spectrogram colour bar.
cmap (colormap, optional) :
Sets the colour map to be used in the plots. The default is
the Generic Mapping Tools (GMT) no green.
debug (boolean, optional) :
If set to True then warnings are shown.
OUTPUT
If show or save are set, plots either on screen and or on file
according to the specified parameters.
If dsave parameter is set, also saves the scale averaged power
series to files.
RETURNS
wave (dictionary) :
Dictionary containing the resulting calculations from the
wavelet analysis according to the input parameters. The
output items might be:
scale --
Wavelet scales.
period --
Equivalent Fourier periods (in days).
power_spectrum --
Wavelet power spectrum (in units**2).
power_significance --
Relative significance of the power spectrum.
global_power --
Global wavelet power spectrum (in units**2).
scale_spectrum --
Scale averaged wavelet spectra (in units**2)
according to selected periods.
scale_significance --
Relative significance of the scale averaged wavelet
spectra.
fft --
Fourier spectrum.
fft_first --
Fourier spectrum of the first half of the
time-series.
fft_second --
Fourier spectrum of the second half of the
time-series.
fft_period --
Fourier periods (in days).
trend --
Signal trend (in units/yr).
wavelet_trend --
Wavelet spectrum trends (in units**2/yr).
"""
t1 = time()
result = {}
# Resseting unit labels for hovmoller plots
hlabels = dict(labels)
hlabels['units'] = ''
# Setting some titles and paths
if name == None:
name = title
# Working with the std parameter and setting its properties:
if 'val' in std.keys():
if 'lon' not in std.keys():
std['lon'] = lon
std['lon180'] = common.lon180(std['lon'])
if 'lat' not in std.keys():
std['lat'] = lat
if 'err' not in std.keys():
std['err'] = 1e-2
std['map'] = True
else:
std['map'] = False
# Lag-1 autocorrelation parameter
if type(alpha).__name__ == 'dict':
if 'lon' not in alpha.keys():
alpha['lon'] = lon
alpha['lon180'] = common.lon180(alpha['lon'])
if 'lat' not in alpha.keys():
alpha['lat'] = lat
alpha['mean'] = alpha['val'].mean()
alpha['map'] = True
alpha['calc'] = False
else:
if alpha == -1:
alpha = {'mean': -1, 'calc': True}
else:
alpha = {'val': alpha, 'mean': alpha, 'map': False, 'calc': False}
# Shows some of the options on screen.
print ('Average Lag-1 autocorrelation for background noise: %.2f' %
(alpha['mean']))
if save:
print 'Saving result figures in \'%s\'.' % (save)
if dsave:
print 'Saving result data in \'%s\'.' % (dsave)
if fpath:
# Gets the list of files to be loaded individually extracts all the
# latitudes and loads the first file to get the main parameters.
flist = os.listdir(fpath)
flist, match = common.reglist(flist, fpattern)
if len(flist) == 0:
raise Warning, 'No files matched search pattern.'
flist = numpy.asarray(flist)
lst_lat = []
for item in match:
y = string.atof(item[-2])
if item[-1].upper() == 'S': y *= -1
lst_lat.append(y)
# Detect file type from file name
ftype = fm.detect_ftype(flist[0])
x, y, tm, z = fm.load_map('%s/%s' % (fpath, flist[0]),
ftype=ftype, masked=True)
if lon == None:
lon = x
lat = numpy.unique(lst_lat)
dim = 2
else:
# Transforms input arrays in numpy arrays and numpy masked arrays.
tm = numpy.asarray(tm)
z = numpy.ma.asarray(z)
z.mask = numpy.isnan(z)
# Determines the number of dimensions of the variable to be plotted and
# the sizes of each dimension.
a = b = c = None
dim = len(z.shape)
if dim == 3:
c, b, a = z.shape
elif dim == 2:
c, a = z.shape
b = 1
z = z.reshape(c, b, a)
else:
c = z.shape[0]
a = b = 1
z = z.reshape(c, b, a)
if tm.size != c:
raise Warning, 'Time and data lengths do not match.'
# Transforms coordinate arrays into numpy arrays
s = type(lat).__name__
if s in ['int', 'float', 'float64']:
lat = numpy.asarray([lat])
elif s != 'NoneType':
lat = numpy.asarray(lat)
s = type(lon).__name__
if s in ['int', 'float', 'float64']:
lon = numpy.asarray([lon])
elif s != 'NoneType':
lon = numpy.asarray(lon)
# Starts the mother wavelet class instance and determines important
# analysis parameters
mother = mother.lower()
if mother == 'morlet':
mother = wavelet.Morlet()
elif mother == 'paul':
mother = wavelet.Paul()
elif mother in ['mexican hat', 'mexicanhat', 'mexican_hat']:
mother = wavelet.Mexican_hat()
else:
raise Warning, 'Mother wavelet unknown.'
t = tm / common.daysinyear # Time array in years
dt = tm[1] - tm[0] # Temporal sampling interval
try: # Zonal sampling interval
dx = lon[1] - lon[0]
except:
dx = 1
try: # Meridional sampling interval
dy = lat[1] - lat[0]
except:
dy = dx
if numpy.isnan(dt): dt = 1
if numpy.isnan(dx): dx = 1
if numpy.isnan(dy): dy = dx
dj = 0.25 # Four sub-octaves per octave
s0 = 2 * dt # Smallest scale
J = 7 / dj - 1 # Seven powers of two with dj sub-octaves
scales = period = None
if type(crange).__name__ == 'NoneType':
crange = numpy.arange(0, 1.1, 0.1)
if type(levels).__name__ == 'NoneType':
levels = 2. ** numpy.arange(-3, 6)
if fpath:
N = lat.size
# TODO: refactoring # lon = numpy.arange(-81. - dx / 2., 290. + dx / 2, dx)
# TODO: refactoring # lat = numpy.unique(numpy.asarray(lst_lat))
c, b, a = tm.size, lat.size, lon.size
else:
N = a * b
# Making sure that the longitudes range from -180 to 180 degrees and
# setting the squared search radius R2.
try:
lon180 = common.lon180(lon)
except:
lon180 = None
R2 = dx ** 2 + dy ** 2
if numpy.isnan(R2):
R2 = 65535.
if loc != None:
loc = numpy.asarray([[common.lon180(item[0]), item[1]] for item in
loc])
# Initializes important result variables such as the global wavelet power
# spectrum map, scale avaraged spectrum time-series and their significance,
# wavelet power trend map.
global_power = numpy.ma.empty([J + 1, b, a]) * numpy.nan
try:
C = len(periods) + 1
dT = numpy.diff(periods)
pmin = numpy.concatenate([[periods[0] - dT[0] / 2],
0.5 * (periods[:-1] + periods[1:])])
pmax = numpy.concatenate([0.5 * (periods[:-1] + periods[1:]),
[periods[-1] + dT[-1] / 2]])
except:
# Sets the lowest period to null and the highest to half the time
# series length.
C = 1
pmin = numpy.array([0])
pmax = numpy.array([(tm[-1] - tm[0]) / 2])
if type(sel_periods).__name__ in ['int', 'float']:
sel_periods = [sel_periods]
elif len(sel_periods) == 0:
sel_periods = [-1.]
try:
if fpath:
raise Warning, 'Process files individually'
avg_spectrum = numpy.ma.empty([C, c, b, a]) * numpy.nan
mem_error = False
except:
avg_spectrum = numpy.ma.empty([C, c, a]) * numpy.nan
mem_error = True
avg_spectrum_signif = numpy.ma.empty([C, b, a]) * numpy.nan
trend = numpy.ma.empty([b, a]) * numpy.nan
wavelet_trend = numpy.ma.empty([C, b, a]) * numpy.nan
fft_trend = numpy.ma.empty([C, b, a]) * numpy.nan
std_map = numpy.ma.empty([b, a]) * numpy.nan
zero = numpy.ma.empty([c, a])
fft_spectrum = None
fft_spectrum1 = None
fft_spectrum2 = None
# Walks through each latitude and then through each longitude to perform
# the temporal wavelet analysis.
if N == 1:
plural = ''
else:
plural = 's'
s = 'Spectral analysis of %d location%s... ' % (N, plural)
stdout.write(s)
stdout.flush()
for j in range(b):
t2 = time()
isloc = False # Ressets 'is special location' flag
hloc = [] # Cleans location list for Hovmoller plots
zero *= numpy.nan
if mem_error:
# Clears average spectrum for next step.
avg_spectrum *= numpy.nan
avg_spectrum.mask = False
if fpath:
findex = pylab.find(lst_lat == lat[j])
if len(findex) == 0:
continue
ftype = fm.detect_ftype(flist[findex[0]])
try:
x, y, tm, z = fm.load_dataset(fpath, flist=flist[findex],
ftype=ftype, masked=True, lon=lon, lat=lat[j:j+1],
verbose=True)
except:
continue
z = z[:, 0, :]
x180 = common.lon180(x)
# Determines the first and second halves of the time-series and some
# constants for the FFT
fft_ta = numpy.ceil(t.min())
fft_tb = numpy.floor(t.max())
fft_tc = numpy.round(fft_ta + fft_tb) / 2
fft_ia = pylab.find((t >= fft_ta) & (t <= fft_tc))
fft_ib = pylab.find((t >= fft_tc) & (t <= fft_tb))
fft_N = int(2 ** numpy.ceil(numpy.log2(max([len(fft_ia),
len(fft_ib)]))))
fft_N2 = fft_N / 2 - 1
fft_dt = t[fft_ib].mean() - t[fft_ia].mean()
for i in range(a):
# Some string output.
try:
Y, X = common.num2latlon(lon[i], lat[j], mode='each',
padding=False)
except:
Y = X = '?'
# Extracts individual time-series from the whole dataset and
# sets or calculates its standard deviation, squared standard
# deviation and finally the normalized time-series.
if fpath:
try:
ilon = pylab.find(x == lon[i])[0]
fz = z[:, ilon]
except:
continue
else:
fz = z[:, j, i]
if fz.mask.all():
continue
if std['map']:
try:
u = pylab.find(std['lon180'] == lon180[i])[0]
v = pylab.find(std['lat'] == lat[j])[0]
except:
if debug:
warnings.warn('Unable to locate standard deviation '
'for (%s, %s)' % (X, Y), Warning)
continue
fstd = std['val'][v, u]
estd = fstd - fz.std()
if (estd < 0) & (abs(estd) > std['err']):
if debug:
warnings.warn('Discrepant input standard deviation '
'(%f) location (%.3f, %.3f) will be '
'disregarded.' % (estd, lon180[i], lat[j]))
continue
else:
fstd = fz.std()
fstd2 = fstd ** 2
std_map[j, i] = fstd
zero[:, i] = fz
fz = (fz - fz.mean()) / fstd
# Calculates the distance of the current point to any special
# location set in the 'loc' parameter. If only special locations
# are to be analysed, then skips all other ones. If the input
# array is one dimensional, then do the analysis anyway.
if dim == 1:
dist = numpy.asarray([0.])
else:
try:
dist = numpy.asarray([((item[0] - (lon180[i])) **
2 + (item[1] - lat[j]) ** 2) for item in loc])
except:
dist = []
if (dist > R2).all() & (loc != 'all') & onlyloc:
continue
# Determines the lag-1 autocorrelation coefficient to be used in
# the significance test from the input parameter
if alpha['calc']:
ac = acorr(fz)
alpha_ij = (ac[c + 1] + ac[c + 2] ** 0.5) / 2
elif alpha['map']:
try:
u = pylab.find(alpha['lon180'] == lon180[i])[0]
v = pylab.find(alpha['lat'] == lat[j])[0]
alpha_ij = alpha['val'][v, u]
except:
if debug:
warnings.warn('Unable to locate standard deviation '
'for (%s, %s) using mean value instead' %
(X, Y), Warning)
alpha_ij = alpha['mean']
else:
alpha_ij = alpha['mean']
# Calculates the continuous wavelet transform using the wavelet
# Python module. Calculates the wavelet and Fourier power spectrum
# and the periods in days. Also calculates the Fourier power
# spectrum for the first and second halves of the timeseries.
wave, scales, freqs, coi, fft, fftfreqs = wavelet.cwt(fz, dt, dj,
s0, J, mother)
power = abs(wave * wave.conj())
fft_power = abs(fft * fft.conj())
period = 1. / freqs
fftperiod = 1. / fftfreqs
psel = pylab.find(period <= pmax.max())
# Calculates the Fourier transform for the first and the second
# halves ot the time-series for later trend analysis.
fft_1 = numpy.fft.fft(fz[fft_ia], fft_N)[1:fft_N/2] / fft_N ** 0.5
fft_2 = numpy.fft.fft(fz[fft_ib], fft_N)[1:fft_N/2] / fft_N ** 0.5
fft_p1 = abs(fft_1 * fft_1.conj())
fft_p2 = abs(fft_2 * fft_2.conj())
# Creates FFT return array and stores the spectrum accordingly
try:
fft_spectrum[:, j, i] = fft_power * fstd2
fft_spectrum1[:, j, i] = fft_p1 * fstd2
fft_spectrum2[:, j, i] = fft_p2 * fstd2
except:
fft_spectrum = (numpy.ma.empty([len(fft_power), b, a]) *
numpy.nan)
fft_spectrum1 = (numpy.ma.empty([fft_N2, b, a]) *
numpy.nan)
fft_spectrum2 = (numpy.ma.empty([fft_N2, b, a]) *
numpy.nan)
#
fft_spectrum[:, j, i] = fft_power * fstd2
fft_spectrum1[:, j, i] = fft_p1 * fstd2
fft_spectrum2[:, j, i] = fft_p2 * fstd2
# Performs the significance test according to the article by
# Torrence and Compo (1998). The wavelet power is significant
# if the ratio power/sig95 is > 1.
signif, fft_theor = wavelet.significance(1., dt, scales, 0,
alpha_ij, significance_level=siglvl, wavelet=mother)
sig95 = (signif * numpy.ones((c, 1))).transpose()
sig95 = power / sig95
# Calculates the global wavelet power spectrum and its
# significance. The global wavelet spectrum is the average of the
# wavelet power spectrum over time. The degrees of freedom (dof)
# have to be corrected for padding at the edges.
glbl_power = power.mean(axis=1)
dof = c - scales
glbl_signif, tmp = wavelet.significance(1., dt, scales, 1,
alpha_ij, significance_level=siglvl, dof=dof, wavelet=mother)
global_power[:, j, i] = glbl_power * fstd2
# Calculates the average wavelet spectrum along the scales and its
# significance according to Torrence and Compo (1998) eq. 24. The
# scale_avg_full variable is used multiple times according to the
# selected periods range.
#
# Also calculates the average Fourier power spectrum.
Cdelta = mother.cdelta
scale_avg_full = (scales * numpy.ones((c, 1))).transpose()
scale_avg_full = power / scale_avg_full
for k in range(C):
if k == 0:
sel = pylab.find((period >= pmin[0]) &
(period <= pmax[-1]))
pminmax = [period[sel[0]], period[sel[-1]]]
les = pylab.find((fftperiod >= pmin[0]) &
(fftperiod <= pmax[-1]))
fminmax = [fftperiod[les[0]], fftperiod[les[-1]]]
else:
sel = pylab.find((period >= pmin[k - 1]) &
(period < pmax[k - 1]))
pminmax = [pmin[k-1], pmax[k-1]]
les = pylab.find((fftperiod >= pmin[k - 1]) &
(fftperiod <= pmax[k - 1]))
fminmax = [fftperiod[les[0]], fftperiod[les[-1]]]
scale_avg = numpy.ma.array((dj * dt / Cdelta *
scale_avg_full[sel, :].sum(axis=0)))
scale_avg_signif, tmp = wavelet.significance(1., dt, scales,
2, alpha_ij, significance_level=siglvl,
dof=[scales[sel[0]], scales[sel[-1]]], wavelet=mother)
scale_avg.mask = (scale_avg < scale_avg_signif)
if mem_error:
avg_spectrum[k, :, i] = scale_avg
else:
avg_spectrum[k, :, j, i] = scale_avg
avg_spectrum_signif[k, j, i] = scale_avg_signif
# Trend analysis using least square polynomial fit of one
# degree of the original input data and scale averaged
# wavelet power. The wavelet power trend is calculated only
# where the cone of influence spans the highest analyzed
# period. In the end, the returned value for the trend is in
# units**2.
#
# Also calculates the trends in the Fourier power spectrum.
# Note that the FFT power spectrum is already multiplied by
# the signal's standard deviation.
incoi = pylab.find(coi >= pmax[-1])
if len(incoi) == 0:
incoi = numpy.arange(c)
polyw = numpy.polyfit(t[incoi], scale_avg[incoi].data, 1)
wavelet_trend[k, j, i] = polyw[0] * fstd2
fft_trend[k, j, i] = (fft_spectrum2[les[les<fft_N2], j, i] -
fft_spectrum1[les[les<fft_N2], j, i]).mean() / fft_dt
if k == 0:
polyz = numpy.polyfit(t, fz * fstd, 1)
trend[j, i] = polyz[0]
# Plots the wavelet analysis results for the individual
# series. The plot is only generated if the dimension of the
# input variable z is one, if a special location is within a
# range of the search radius R and if the show or save
# parameters are set.
if (show | (save != '')) & ((k in sel_periods)):
if (dist < R2).any() | (loc == 'all') | (dim == 1):
# There is an interesting spot within the search
# radius of location (%s, %s).' % (Y, X)
isloc = True
if (dist < R2).any():
try:
hloc.append(loc[(dist < R2)][0, 0])
except:
pass
if save:
try:
sv = '%s/tz_%s_%s_%d' % (save, prefix,
common.num2latlon(lon[i], lat[j]), k)
except:
sv = '%s' % (save)
else:
sv = ''
graphics.wavelet_plot(tm, period[psel], fz,
power[psel, :], coi, glbl_power[psel],
scale_avg.data, fft=fft, fft_period=fftperiod,
power_signif=sig95[psel, :],
glbl_signif=glbl_signif[psel],
scale_signif=scale_avg_signif, pminmax=pminmax,
labels=labels, normalized=True, std=fstd,
ztrend=polyz, wtrend=polyw, show=show, save=sv,
levels=levels, cmap=cmap)
# Saves and/or plots the intermediate results as zonal temporal
# diagrams.
if dsave:
for k in range(C):
if k == 0:
sv = '%s/%s/%s_%s.xt.gz' % (dsave, 'global', prefix,
common.num2latlon(lon[i], lat[j], mode='each')[0])
else:
sv = '%s/%s/%s_%s.xt.gz' % (dsave, name[k - 1].lower(),
prefix,
common.num2latlon(lon[i], lat[j], mode='each')[0])
if mem_error:
fm.save_map(lon, tm, avg_spectrum[k, :, :].data,
sv, lat[j])
else:
fm.save_map(lon, tm, avg_spectrum[k, :, j, :].data,
sv, lat[j])
if ((dim > 1) and (show or (save != '')) & (not onlyloc) and
len(hloc) > 0):
hloc = common.lon360(numpy.unique(hloc))
if save:
sv = '%s/xt_%s_%s' % (save, prefix,
common.num2latlon(lon[i], lat[j], mode='each')[0])
else:
sv = ''
if mem_error:
# To include overlapping original signal, use zz=zero
gis.hovmoller(lon, tm, avg_spectrum[1:, :, :],
zo=avg_spectrum_signif[1:, j, :], title=title,
crange=crange, show=show, save=sv, labels=hlabels,
loc=hloc, cmap=cmap, bottom='avg', right='avg',
std=std_map[j, :])
else:
gis.hovmoller(lon, tm, avg_spectrum[1:, :, j, :],
zo=avg_spectrum_signif[1:, j, :], title=title,
crange=crange, show=show, save=sv, labels=hlabels,
loc=hloc, cmap=cmap, bottom='avg', right='avg',
std=std_map[j, :])
# Flushing profiling text.
stdout.write(len(s) * '\b')
s = 'Spectral analysis of %d location%s (%s)... %s ' % (N, plural, Y,
common.profiler(b, j + 1, 0, t1, t2))
stdout.write(s)
stdout.flush()
stdout.write('\n')
result['scale'] = scales
result['period'] = period
if dim == 1:
result['power_spectrum'] = power * fstd2
result['power_significance'] = sig95
result['cwt'] = wave
result['fft'] = fft
result['global_power'] = global_power
result['scale_spectrum'] = avg_spectrum
if fpath:
result['lon'] = lon
result['lat'] = lat
result['scale_significance'] = avg_spectrum_signif
result['trend'] = trend
result['wavelet_trend'] = wavelet_trend
result['fft_power'] = fft_spectrum
result['fft_first'] = fft_spectrum1
result['fft_second'] = fft_spectrum2
result['fft_period'] = fftperiod
result['fft_trend'] = fft_trend
return result
data = data.ravel()
# get the log spaced frequencies
numfreqs = 50
nyq = np.floor_divide(sample_rate[0], 2.0)
maxfreq = np.min([100, nyq])
minfreq = 2
freqs = np.logspace(np.log10(minfreq), np.log10(maxfreq), num=numfreqs)
# make an empty ndarray to hold the freq * electrode * timepoint data
powers_by_freq = np.zeros(shape=(len(freqs), og_shape[0], og_shape[1]))
# convolve!
for i, freq in enumerate(freqs):
wav_transform = wavelet.cwt(data,
1 / sample_rate[0],
freqs=np.full(1, freq),
wavelet=wavelet.Morlet(4))
# get the power and reshape data back into original shape
wav_transform = (np.abs(wav_transform[0])**2).reshape(og_shape)
powers_by_freq[i] = np.log(wav_transform)
# prep some variables for the robust regression done in parallel
xs = np.log(freqs).reshape(-1, 1)
midpoint = (np.log(maxfreq) - np.log(minfreq)) / 2
nworkers = int(config['nnodes'] * config['ppn'] * 0.5)
# get the indices for the chunks
chunk_indices = array_split(powers_by_freq, nworkers, axis=2)
mhq = Queue(nworkers)
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
fft_power = numpy.abs(fft) ** 2
period = 1 / freqs
# Significance test. Where ratio power/sig95 > 1, power is significant.
signif, fft_theor = wavelet.significance(1.0, ds.dt, scales, 0, alpha,
significance_level=slevel,
wavelet=mother)
sig95 = numpy.ones([1, N]) * signif[:, None]
sig95 = power / sig95
# Power rectification as of Liu et al. (2007). TODO: confirm if significance
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
var = (var - var.mean()) / std # Calculating anomaly and normalizing
N = var.size # Number of measurements
time = np.arange(0, N) * dt + 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
alpha, _, _ = wavelet.ar1(var) # Lag-1 autocorrelation for red noise
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(var, dt, dj, s0, J,
mother)
iwave = wavelet.icwt(wave, scales, dt, dj, mother)
# Normalized wavelet and Fourier power spectra
power = (np.abs(wave)) ** 2
fft_power = np.abs(fft) ** 2
period = 1/ freqs
# Significance test. Where ratio power/sig95 > 1, power is significant.
signif, fft_theor = wavelet.significance(1.0, dt, scales, 0, alpha,
significance_level=slevel, wavelet=mother)
sig95 = np.ones([1, N]) * signif[:, None]
sig95 = power / sig95
# Power rectification as of Liu et al. (2007). TODO: confirm if significance
# test ratio should be calculated first.