def vector_to_speeddir(uv, dtype='from'):
"""
Converts complex vector to polar vector, assuming geographical angle
origin convention at north.
Parameters
----------
uv : array like
Array of cartesian vector components in complex notation.
dtype : string, optional
If `from`, uses convention that vector indicates direction from
which it is comming (e.g. wind from direction). If `to`
indicates direction to which it is pointing (e.g. currents).
Returns
-------
m, a : array like
Magnitude and angle.
"""
m = (uv.real**2 + uv.imag**2)**0.5
a = rad2deg(arctan2(uv.imag, uv.real))
# Converts angle from mathematical direction to meteorological direction.
# Check <http://wx.gmu.edu/dev/clim301/lectures/wind/wind-uv.html> for
# further explanations.
if dtype == 'from':
a = common.lon360(270 - a)
elif dtype == 'to':
a = common.lon360(90 - a)
else:
raise ValueError('Invalid data type `{}`.'.format(dtype))
#
return m, a
def vector_to_speeddir(uv, dtype='from'):
"""
Converts complex vector to polar vector, assuming geographical angle
origin convention at north.
Parameters
----------
uv : array like
Array of cartesian vector components in complex notation.
dtype : string, optional
If `from`, uses convention that vector indicates direction from
which it is comming (e.g. wind from direction). If `to`
indicates direction to which it is pointing (e.g. currents).
Returns
-------
m, a : array like
Magnitude and angle.
"""
m = (uv.real**2 + uv.imag**2)**0.5
a = rad2deg(arctan2(uv.imag, uv.real))
# Converts angle from mathematical direction to meteorological direction.
# Check <http://wx.gmu.edu/dev/clim301/lectures/wind/wind-uv.html> for
# further explanations.
if dtype == 'from':
a = common.lon360(270 - a)
elif dtype == 'to':
a = common.lon360(90 - a)
else:
raise ValueError('Invalid data type `{}`.'.format(dtype))
#
return m, a
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_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