def test(): y = sio.loadmat(here(__file__) + '/demo/ma1.mat')['y'] # The right results are: # "biased": [-0.12250513 0.35963613 1.00586945 0.35963613 -0.12250513] # "unbiaed": [-0.12444965 0.36246791 1.00586945 0.36246791 -0.12444965] print cum2est(y, 2, 128, 0, 'unbiased') print cum2est(y, 2, 128, 0, 'biased') # For the 3rd cumulant: # "biased": [-0.18203039 0.07751503 0.67113035 0.729953 0.07751503] # "unbiased": [-0.18639911 0.07874543 0.67641484 0.74153955 0.07937539] print cum3est(y, 2, 128, 0, 'biased', 1) print cum3est(y, 2, 128, 0, 'unbiased', 1) # For testing the 4th-order cumulant # "biased": [-0.03642083 0.4755026 0.6352588 1.38975232 0.83791117 0.41641134 -0.97386322] # "unbiased": [-0.04011388 0.48736793 0.64948927 1.40734633 0.8445089 0.42303979 -0.99724968] print cum4est(y, 3, 128, 0, 'biased', 1, 1) print cum4est(y, 3, 128, 0, 'unbiased', 1, 1)
def cumest(y, norder=2, maxlag=0 ,nsamp=None, overlap=0, flag='biased' ,k1=0, k2=0):
"""
Second-, third- or fourth-order cumulants.
Parameters:
y - time-series - should be a vector
norder - cumulant order: 2, 3 or 4 [default = 2]
maxlag - maximum cumulant lag to compute [default = 0]
samp_seg - samples per segment [default = data_length]
overlap - percentage overlap of segments [default = 0]
overlap is clipped to the allowed range of [0,99].
flag - 'biased' or 'unbiased' [default = 'biased']
k1,k2 - specify the slice of 3rd or 4th order cumulants
Output:
y_cum - C2(m) or C3(m,k1) or C4(m,k1,k2), -maxlag <= m <= maxlag
depending upon the cumulant order selected
"""
(ksamp, nrecs) = y.shape
if ksamp == 1:
ksamp = nrecs
nrecs = 1
if norder < 2 or norder > 4:
raise ValueError('cumulant order must be 2, 3 or 4')
if maxlag < 0:
raise ValueError('"maxlag" must be non-negative')
if nrecs > 1: nsamp = ksamp
if nsamp <= 0 or nsamp > ksamp: nsamp = ksamp
if nrecs > 1: overlap = 0
overlap = max(0,min(overlap,99))
# estimate the cumulants
if norder == 2:
y_cum = cum2est(y, maxlag, nsamp, overlap, flag)
elif norder == 3:
y_cum = cum3est(y, maxlag, nsamp, overlap, flag, k1)
elif norder == 4:
y_cum = cum3est(y, maxlag, nsamp, overlap, flag, k1, k2)
return y_cum
def cum4est(y, maxlag=0, nsamp=0, overlap=0, flag='biased', k1=0, k2=0):
"""
CUM4EST Fourth-order cumulants.
Parameters:
Should be invoked via CUMEST for proper parameter checks
y_cum = cum4est (y, maxlag, samp_seg, overlap, flag, k1, k2)
Computes sample estimates of fourth-order cumulants
via the overlapped segment method.
y_cum = cum4est (y, maxlag, samp_seg, overlap, flag, k1, k2)
y: input data vector (column)
maxlag: maximum lag
samp_seg: samples per segment
overlap: percentage overlap of segments
flag : 'biased', biased estimates are computed
: 'unbiased', unbiased estimates are computed.
k1,k2 : the fixed lags in C3(m,k1) or C4(m,k1,k2)
Output:
y_cum : estimated fourth-order cumulant slice
C4(m,k1,k2) -maxlag <= m <= maxlag
"""
(n1, n2) = shape(y, 2)
N = n1*n2
overlap0 = overlap
overlap = np.fix(overlap/100 * nsamp)
nrecord = np.fix((N - overlap)/(nsamp - overlap))
nadvance = nsamp - overlap
# scale factors for unbiased estimates
nlags = 2 * maxlag + 1
zlag = maxlag
tmp = np.zeros([nlags,1])
if flag == 'biased':
scale = np.ones([nlags,1])/nsamp
else:
ind = np.arange(-maxlag, maxlag+1).T
kmin = min(0, min(k1, k2))
kmax = max(0,max(k1, k2))
scale = nsamp - np.maximum(ind, kmax) + np.minimum(ind, kmin)
scale = np.ones(nlags) / scale
scale = scale.reshape(-1,1)
mlag = maxlag + max(abs(k1), abs(k2))
mlag = max(mlag, abs(k1-k2) )
mlag1 = mlag + 1
nlag = maxlag
m2k2 = np.zeros([2*maxlag+1,1])
if np.any(np.any(np.imag(y) != 0)): complex_flag = 1
else: complex_flag = 0
# estimate second- and fourth-order moments combine
y_cum = np.zeros([2*maxlag+1, 1])
R_yy = np.zeros([2*mlag+1, 1])
ind = np.arange(nsamp)
for i in xrange(nrecord):
tmp = np.zeros([2*maxlag+1, 1])
x = y[ind]
x = x.ravel(order='F') - np.mean(x)
z = x * 0
cx = np.conj(x)
# create the "IV" matrix: offset for second lag
if k1 >= 0:
z[0:nsamp-k1] = x[0:nsamp-k1] * cx[k1:nsamp]
else:
z[-k1:nsamp] = x[-k1:nsamp] * cx[0:nsamp+k1]
# create the "IV" matrix: offset for third lag
if k2 >= 0:
z[0:nsamp-k2] = z[0:nsamp-k2] * x[k2:nsamp]
z[nsamp-k2:nsamp] = np.zeros([k2, 1])
else:
z[-k2:nsamp] = z[-k2:nsamp] * x[0:nsamp+k2]
z[0:-k2] = np.zeros([-k2, 1])
tmp[zlag] = tmp[zlag] + np.dot(z.T, x)
for k in xrange(1, maxlag+1):
tmp[zlag-k] = tmp[zlag-k] + np.dot(z[k:nsamp].T, x[0:nsamp-k])
tmp[zlag+k] = tmp[zlag+k] + np.dot(z[0:nsamp-k].T, x[k:nsamp])
y_cum = y_cum + tmp * scale
R_yy = cum2est(x, mlag, nsamp, overlap0, flag)
# We need E x(t)x(t+tau) stuff also:
if complex_flag:
M_yy = cum2x(np.conj(x), x, mlag, nsamp, overlap0, flag)
else:
M_yy = R_yy
y_cum = y_cum - \
R_yy[mlag1+k1-1] * R_yy[mlag1-k2-nlag-1:mlag1-k2+nlag] - \
R_yy[k1-k2+mlag1-1] * R_yy[mlag1-nlag-1:mlag1+nlag] - \
M_yy[mlag1+k2-1].T * M_yy[mlag1-k1-nlag-1:mlag1-k1+nlag]
ind = ind + int(nadvance)
y_cum = y_cum / nrecord
return y_cum