def harmonic(data, k=3):
data = data.reorder('t...')
cdutil.setAxisTimeBoundsDaily(data.getTime())
axislist = data.getAxisList()
dataid = data.id
daily = True
monthly = False
timeAxis = axislist[0]
N = 365. #len(timeAxis)
# P = 10. # 10 year, yearly harmonic oscilation
# P = 10*12 # 10 year, monthly harmonic oscilation
# P = 10*365 # 10 year, daily harmonic oscilation
# if P > N:
# raise ValueError("P('%d') value should not exceed N(%d)" % (P,N))
if k > N/2:
raise ValueError("k value should not exceed (%d) i.e. N/2 value" % (N/2))
if len(timeAxis) > 366:
print 'found more than 1 year data.'
# y_t = dailyClimatology(data, action='sum')
else:
y_t = data
# end of if len(timeAxis) > 366:
Y_0 = cdutil.averager(data, axis='t', action='average', weights='equal')
# make memory free
del data
t = numpy.arange(1, N+1, dtype='float')
otheraxis = list(Y_0.shape)
ax_product = 1
for ax in otheraxis:
ax_product *= ax
otheraxis.insert(0,N)
t = t.repeat(ax_product).reshape(otheraxis)
angle = 2 * math.pi * t/N
Y_k = 0.
for i in range(1,k+1):
kangle = angle*i
A_k = (2./N) * cdutil.averager(y_t * numpy.cos(kangle), axis='t', action='sum')
B_k = (2./N) * cdutil.averager(y_t * numpy.sin(kangle), axis='t', action='sum')
C_k = MV2.sqrt((A_k*A_k) + (B_k*B_k))
# if A_k is positiv, then retain this phase_angle as it is.
# phase_angle should be in degrees
phase_angle = phase_arc_angle = MV2.arctan(B_k/A_k)
# if A_k is zero, then replace phase_angle with pi/2 else retain same
phase_angle = MV2.where(MV2.equal(A_k, 0.), math.pi/2.0, phase_arc_angle)
# if A_k is negative, then add pi with phase_angle (if it is <= pi )
condition1 = MV2.logical_and(MV2.less(A_k, 0.), MV2.less_equal(phase_arc_angle, math.pi))
phase_angle = MV2.where(condition1, phase_arc_angle+math.pi, phase_arc_angle)
# if A_k is negative, then subtract pi from phase_angle (if it is > pi )
condition2 = MV2.logical_and(MV2.less(A_k, 0.), MV2.greater(phase_arc_angle, math.pi))
condition3 = MV2.logical_or(condition1, condition2)
phase_angle = MV2.where(condition3, phase_arc_angle-math.pi, phase_arc_angle)
# make memory free
del phase_arc_angle
if daily and not monthly:
# subtract 15 days lag to adjust phase_angle w.r.t daily
print "Daily Subtraction"
phase_angle -= (15.*2*math.pi)/N
# end of if daily and not monthly:
phase_angle = numpy.array(phase_angle)
# phase_angle = numpy.tile(phase_angle, N).reshape(kangle.shape)
kangle = numpy.array(kangle)
Y_k += C_k * MV2.cos(kangle - phase_angle)
# end of for i in range(1,k+1):
# add mean to the sum of first k-th harmonic of data
Y_k += Y_0
# make memory free
del y_t, Y_0
sumOfMean_and_first_k_harmonic = cdms2.createVariable(Y_k, id=dataid)
sumOfMean_and_first_k_harmonic.setAxisList(axislist)
sumOfMean_and_first_k_harmonic.comments = 'sumOfMean_and_first_%d_harmonic' % k
# make memory free
del Y_k
# return result
return sumOfMean_and_first_k_harmonic
def harmonic(data, k=3, time_type='daily', phase_shift=15):
"""
Inputs :
data : climatology data
k : Integer no to compute K th harmonic. By default it takes 3.
time_type : daily | monthly | full (time type of input climatology)
'daily' -> it returns 365 days harmonic,
'monthly' -> it returns 12 month harmonic,
'full' -> it retuns harmonic for full length of
input data.
phase_shift : Used to subtract 'phase_shift' days lag to adjust
phase_angle w.r.t daily or monthly. By default it takes
15 days lag to adjust phase_angle w.r.t daily data.
User can pass None disable this option.
Returns :
Returns "sum mean of mean and first K th harmonic" of input
climatology data.
Concept :
Earth science data consists of a strong seasonality component as
indicated by the cycles of repeated patterns in climate variables such
as air pressure, temperature and precipitation. The seasonality forms
the strongest signals in this data and in order to find other patterns,
the seasonality is removed by subtracting the monthly mean values of the
raw data for each month. However since the raw data like air temperature,
pressure, etc. are constantly being generated with the help of satellite
observations, the climate scientists usually use a moving reference base
interval of some years of raw data to calculate the mean in order to
generate the anomaly time series and study the changes with respect to
that.
Fourier series analysis decomposes a signal into an infinite series of
harmonic components. Each of these components is comprised initially of
a sine wave and a cosine wave of equal integer frequency. These two waves
are then combined into a single cosine wave, which has characteristic
amplitude (size of the wave) and phase angle (offset of the wave).
Convergence has been established for bounded piecewise continuous
functions on a closed interval, with special conditions at points of
discontinuity. Its convergence has been established for other conditions
as well, but these are not relevant to the analysis at hand.
Reference: Daniel S Wilks, 'Statistical Methods in the Atmospheric
Sciences' second Edition, page no(372-378).
Written By : Arulalan.T
Date : 16.05.2014
"""
data = data.reorder('t...')
cdutil.setAxisTimeBoundsDaily(data.getTime())
axislist = data.getAxisList()
timeAxis = axislist[0]
dataid = data.id
if time_type in ['daily']:
N = 365.0 # must be float
elif time_type[:3] in ['mon']:
N = 12.0 # must be float
elif time_type in ['full']:
N = float(len(timeAxis))
if k > N/2:
raise ValueError("k value should not exceed (%d) i.e. N/2 value" % (N/2))
if len(timeAxis) > 366:
print 'found more than 1 year data.'
raise ValueError("Kindly pass only climatology data")
else:
y_t = data
# end of if len(timeAxis) > 366:
Y_0 = cdutil.averager(data, axis='t', action='average', weights='equal')
# make memory free
del data
t = numpy.arange(1, N+1, dtype='float')
otheraxis = list(Y_0.shape)
ax_product = 1
for ax in otheraxis:
ax_product *= ax
otheraxis.insert(0,N)
t = t.repeat(ax_product).reshape(otheraxis)
angle = 2 * math.pi * t/N
Y_k = 0.
for i in range(1,k+1):
kangle = angle*i
A_k = (2./N) * cdutil.averager(y_t * numpy.cos(kangle), axis='t', action='sum')
B_k = (2./N) * cdutil.averager(y_t * numpy.sin(kangle), axis='t', action='sum')
C_k = MV2.sqrt((A_k*A_k) + (B_k*B_k))
# if A_k is positiv, then retain this phase_angle as it is.
# phase_angle should be in degrees
phase_angle = phase_arc_angle = MV2.arctan(B_k/A_k)
# if A_k is zero, then replace phase_angle with pi/2 else retain same
phase_angle = MV2.where(MV2.equal(A_k, 0.), math.pi/2.0, phase_arc_angle)
# if A_k is negative, then add pi with phase_angle (if it is <= pi )
condition1 = MV2.logical_and(MV2.less(A_k, 0.), MV2.less_equal(phase_arc_angle, math.pi))
phase_angle = MV2.where(condition1, phase_arc_angle+math.pi, phase_arc_angle)
# if A_k is negative, then subtract pi from phase_angle (if it is > pi )
condition2 = MV2.logical_and(MV2.less(A_k, 0.), MV2.greater(phase_arc_angle, math.pi))
condition3 = MV2.logical_or(condition1, condition2)
phase_angle = MV2.where(condition3, phase_arc_angle-math.pi, phase_arc_angle)
# make memory free
del phase_arc_angle
if phase_shift:
# subtract 15 days lag to adjust phase_angle w.r.t daily
phase_angle -= (phase_shift *2 * math.pi) / N
# end of if daily and not monthly:
phase_angle = numpy.array(phase_angle)
kangle = numpy.array(kangle)
Y_k += C_k * MV2.cos(kangle - phase_angle)
# end of for i in range(1,k+1):
# add mean to the sum of first k-th harmonic of data
Y_k += Y_0
# make memory free
del y_t, Y_0
sumOfMean_and_first_k_harmonic = cdms2.createVariable(Y_k, id=dataid)
sumOfMean_and_first_k_harmonic.setAxisList(axislist)
sumOfMean_and_first_k_harmonic.comments = 'sumOfMean_and_first_%d_harmonic' % k
# make memory free
del Y_k
# return result
return sumOfMean_and_first_k_harmonic