def approx_real(x): """ approx_real(x) : returns x.real if |x.imag| < |x.real| * _eps_approx. This function is needed by sqrtm and allows further functions. """ if MLab.max(MLab.max(numerix.absolute(x.imag))) <= MLab.max(MLab.max(numerix.absolute(x.real))) * _eps_approx: return x.real else: return x
def norm(a, p=2):
"""norm(a,p=2) -> l-p norm of a.flat
Return the l-p norm of a, considered as a flat array. This is NOT a true
matrix norm, since arrays of arbitrary rank are always flattened.
p can be a number or the string 'Infinity' to get the L-infinity norm."""
if p == 'Infinity':
return max(absolute(a).flat)
else:
return (sum_flat(absolute(a)**p))**(1.0 / p)
def norm(a,p=2):
"""norm(a,p=2) -> l-p norm of a.flat
Return the l-p norm of a, considered as a flat array. This is NOT a true
matrix norm, since arrays of arbitrary rank are always flattened.
p can be a number or the string 'Infinity' to get the L-infinity norm."""
if p=='Infinity':
return max(absolute(a).flat)
else:
return (sum_flat(absolute(a)**p))**(1.0/p)
def _set_offset(self, range):
# offset of 20,001 is 20,000, for example
locs = self.locs
if locs is None or not len(locs):
self.offset = 0
ave_loc = mean(locs)
if ave_loc: # dont want to take log10(0)
ave_oom = math.floor(math.log10(mean(absolute(locs))))
range_oom = math.floor(math.log10(range))
if absolute(ave_oom-range_oom) >= 3: # four sig-figs
if ave_loc < 0:
self.offset = math.ceil(amax(locs)/10**range_oom)*10**range_oom
else:
self.offset = math.floor(amin(locs)/10**(range_oom))*10**(range_oom)
else: self.offset = 0
def cohere(x,
y,
NFFT=256,
Fs=2,
detrend=detrend_none,
window=window_hanning,
noverlap=0):
"""
cohere the coherence between x and y. Coherence is the normalized
cross spectral density
Cxy = |Pxy|^2/(Pxx*Pyy)
The return value is (Cxy, f), where f are the frequencies of the
coherence vector. See the docs for psd and csd for information
about the function arguments NFFT, detrend, windowm noverlap, as
well as the methods used to compute Pxy, Pxx and Pyy.
Returns the tuple Cxy, freqs
"""
if len(x) < 2 * NFFT:
raise RuntimeError(
'Coherence is calculated by averaging over NFFT length segments. Your signal is too short for your choice of NFFT'
)
Pxx, f = psd(x, NFFT, Fs, detrend, window, noverlap)
Pyy, f = psd(y, NFFT, Fs, detrend, window, noverlap)
Pxy, f = csd(x, y, NFFT, Fs, detrend, window, noverlap)
Cxy = divide(absolute(Pxy)**2, Pxx * Pyy)
Cxy.shape = len(f),
return Cxy, f
def psd(x, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=0):
"""
The power spectral density by Welches average periodogram method.
The vector x is divided into NFFT length segments. Each segment
is detrended by function detrend and windowed by function window.
noperlap gives the length of the overlap between segments. The
absolute(fft(segment))**2 of each segment are averaged to compute Pxx,
with a scaling to correct for power loss due to windowing. Fs is
the sampling frequency.
-- NFFT must be a power of 2
-- detrend and window are functions, unlike in matlab where they are
vectors.
-- if length x < NFFT, it will be zero padded to NFFT
Returns the tuple Pxx, freqs
Refs:
Bendat & Piersol -- Random Data: Analysis and Measurement
Procedures, John Wiley & Sons (1986)
"""
if NFFT % 2:
raise ValueError, 'NFFT must be a power of 2'
# zero pad x up to NFFT if it is shorter than NFFT
if len(x)<NFFT:
n = len(x)
x = resize(x, (NFFT,))
x[n:] = 0
# for real x, ignore the negative frequencies
if x.typecode()==Complex: numFreqs = NFFT
else: numFreqs = NFFT//2+1
windowVals = window(ones((NFFT,),x.typecode()))
step = NFFT-noverlap
ind = range(0,len(x)-NFFT+1,step)
n = len(ind)
Pxx = zeros((numFreqs,n), Float)
# do the ffts of the slices
for i in range(n):
thisX = x[ind[i]:ind[i]+NFFT]
thisX = windowVals*detrend(thisX)
fx = absolute(fft(thisX))**2
Pxx[:,i] = divide(fx[:numFreqs], norm(windowVals)**2)
# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2
if n>1:
Pxx = mean(Pxx,1)
freqs = Fs/NFFT*arange(numFreqs)
Pxx.shape = len(freqs),
return Pxx, freqs
def cohere(x, y, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=0):
"""
cohere the coherence between x and y. Coherence is the normalized
cross spectral density
Cxy = |Pxy|^2/(Pxx*Pyy)
The return value is (Cxy, f), where f are the frequencies of the
coherence vector. See the docs for psd and csd for information
about the function arguments NFFT, detrend, windowm noverlap, as
well as the methods used to compute Pxy, Pxx and Pyy.
Returns the tuple Cxy, freqs
"""
if len(x)<2*NFFT:
raise RuntimeError('Coherence is calculated by averaging over NFFT length segments. Your signal is too short for your choice of NFFT')
Pxx, f = psd(x, NFFT, Fs, detrend, window, noverlap)
Pyy, f = psd(y, NFFT, Fs, detrend, window, noverlap)
Pxy, f = csd(x, y, NFFT, Fs, detrend, window, noverlap)
Cxy = divide(absolute(Pxy)**2, Pxx*Pyy)
Cxy.shape = len(f),
return Cxy, f
def psd(x, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=0):
"""
The power spectral density by Welches average periodogram method.
The vector x is divided into NFFT length segments. Each segment
is detrended by function detrend and windowed by function window.
noperlap gives the length of the overlap between segments. The
absolute(fft(segment))**2 of each segment are averaged to compute Pxx,
with a scaling to correct for power loss due to windowing. Fs is
the sampling frequency.
-- NFFT must be a power of 2
-- detrend and window are functions, unlike in matlab where they are
vectors.
-- if length x < NFFT, it will be zero padded to NFFT
Returns the tuple Pxx, freqs
Refs:
Bendat & Piersol -- Random Data: Analysis and Measurement
Procedures, John Wiley & Sons (1986)
"""
if NFFT % 2:
raise ValueError, 'NFFT must be a power of 2'
# zero pad x up to NFFT if it is shorter than NFFT
if len(x)<NFFT:
n = len(x)
x = resize(x, (NFFT,))
x[n:] = 0
# for real x, ignore the negative frequencies
if x.typecode()==Complex: numFreqs = NFFT
else: numFreqs = NFFT//2+1
windowVals = window(ones((NFFT,),x.typecode()))
step = NFFT-noverlap
ind = range(0,len(x)-NFFT+1,step)
n = len(ind)
Pxx = zeros((numFreqs,n), Float)
# do the ffts of the slices
for i in range(n):
thisX = x[ind[i]:ind[i]+NFFT]
thisX = windowVals*detrend(thisX)
fx = absolute(fft(thisX))**2
Pxx[:,i] = divide(fx[:numFreqs], norm(windowVals)**2)
# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2
if n>1:
Pxx = mean(Pxx,1)
freqs = Fs/NFFT*arange(numFreqs)
Pxx.shape = len(freqs),
return Pxx, freqs
def norm(x, y=2):
"""
Norm of a matrix or a vector according to Matlab.
The description is taken from Matlab:
For matrices...
NORM(X) is the largest singular value of X, max(svd(X)).
NORM(X,2) is the same as NORM(X).
NORM(X,1) is the 1-norm of X, the largest column sum,
= max(sum(abs((X)))).
NORM(X,inf) is the infinity norm of X, the largest row sum,
= max(sum(abs((X')))).
NORM(X,'fro') is the Frobenius norm, sqrt(sum(diag(X'*X))).
NORM(X,P) is available for matrix X only if P is 1, 2, inf or 'fro'.
For vectors...
NORM(V,P) = sum(abs(V).^P)^(1/P).
NORM(V) = norm(V,2).
NORM(V,inf) = max(abs(V)).
NORM(V,-inf) = min(abs(V)).
"""
x = numerix.asarray(x)
if MLab.rank(x) == 2:
if y == 2:
return MLab.max(MLab.svd(x)[1])
elif y == 1:
return MLab.max(MLab.sum(numerix.absolute((x))))
elif y == 'inf':
return MLab.max(MLab.sum(numerix.absolute((MLab.transpose(x)))))
elif y == 'fro':
return MLab.sqrt(
MLab.sum(
MLab.diag(numerix.matrixmultiply(MLab.transpose(x), x))))
else:
verbose.report_error('Second argument not permitted for matrices')
return None
else:
if y == 'inf':
return MLab.max(numerix.absolute(x))
elif y == '-inf':
return MLab.min(numerix.absolute(x))
else:
return numerix.power(
MLab.sum(numerix.power(numerix.absolute(x), y)), 1 / float(y))
def norm(x,y=2):
"""
Norm of a matrix or a vector according to Matlab.
The description is taken from Matlab:
For matrices...
NORM(X) is the largest singular value of X, max(svd(X)).
NORM(X,2) is the same as NORM(X).
NORM(X,1) is the 1-norm of X, the largest column sum,
= max(sum(abs((X)))).
NORM(X,inf) is the infinity norm of X, the largest row sum,
= max(sum(abs((X')))).
NORM(X,'fro') is the Frobenius norm, sqrt(sum(diag(X'*X))).
NORM(X,P) is available for matrix X only if P is 1, 2, inf or 'fro'.
For vectors...
NORM(V,P) = sum(abs(V).^P)^(1/P).
NORM(V) = norm(V,2).
NORM(V,inf) = max(abs(V)).
NORM(V,-inf) = min(abs(V)).
"""
x = numerix.asarray(x)
if MLab.rank(x)==2:
if y==2:
return MLab.max(MLab.svd(x)[1])
elif y==1:
return MLab.max(MLab.sum(numerix.absolute((x))))
elif y=='inf':
return MLab.max(MLab.sum(numerix.absolute((MLab.transpose(x)))))
elif y=='fro':
return MLab.sqrt(MLab.sum(MLab.diag(numerix.matrixmultiply(MLab.transpose(x),x))))
else:
verbose.report_error('Second argument not permitted for matrices')
return None
else:
if y == 'inf':
return MLab.max(numerix.absolute(x))
elif y == '-inf':
return MLab.min(numerix.absolute(x))
else:
return numerix.power(MLab.sum(numerix.power(numerix.absolute(x),y)),1/float(y))
def specgram(x,
NFFT=256,
Fs=2,
detrend=detrend_none,
window=window_hanning,
noverlap=128):
"""
Compute a spectrogram of data in x. Data are split into NFFT
length segements and the PSD of each section is computed. The
windowing function window is applied to each segment, and the
amount of overlap of each segment is specified with noverlap
See pdf for more info.
The returned times are the midpoints of the intervals over which
the ffts are calculated
"""
x = asarray(x)
assert (NFFT > noverlap)
if log(NFFT) / log(2) != int(log(NFFT) / log(2)):
raise ValueError, 'NFFT must be a power of 2'
# zero pad x up to NFFT if it is shorter than NFFT
if len(x) < NFFT:
n = len(x)
x = resize(x, (NFFT, ))
x[n:] = 0
# for real x, ignore the negative frequencies
if typecode(x) == Complex: numFreqs = NFFT
else: numFreqs = NFFT // 2 + 1
windowVals = window(ones((NFFT, ), typecode(x)))
step = NFFT - noverlap
ind = arange(0, len(x) - NFFT + 1, step)
n = len(ind)
Pxx = zeros((numFreqs, n), Float)
# do the ffts of the slices
for i in range(n):
thisX = x[ind[i]:ind[i] + NFFT]
thisX = windowVals * detrend(thisX)
fx = absolute(fft(thisX))**2
# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2
Pxx[:, i] = divide(fx[:numFreqs], norm(windowVals)**2)
t = 1 / Fs * (ind + NFFT / 2)
freqs = Fs / NFFT * arange(numFreqs)
return Pxx, freqs, t
def _set_orderOfMagnitude(self,range):
# if scientific notation is to be used, find the appropriate exponent
# if using an numerical offset, find the exponent after applying the offset
locs = absolute(self.locs)
if self.offset: oom = math.floor(math.log10(range))
else:
if locs[0] > locs[-1]: val = locs[0]
else: val = locs[-1]
if val == 0: oom = 0
else: oom = math.floor(math.log10(val))
if oom <= -3:
self.orderOfMagnitude = oom
elif oom >= 4:
self.orderOfMagnitude = oom
else:
self.orderOfMagnitude = 0
def _set_orderOfMagnitude(self,range):
# if scientific notation is to be used, find the appropriate exponent
# if using an numerical offset, find the exponent after applying the offset
locs = absolute(self.locs)
if self.offset: oom = math.floor(math.log10(range))
else:
if locs[0] > locs[-1]: val = locs[0]
else: val = locs[-1]
if val == 0: oom = 0
else: oom = math.floor(math.log10(val))
if oom <= -3:
self.orderOfMagnitude = oom
elif oom >= 4:
self.orderOfMagnitude = oom
else:
self.orderOfMagnitude = 0
def specgram(x, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=128):
"""
Compute a spectrogram of data in x. Data are split into NFFT
length segements and the PSD of each section is computed. The
windowing function window is applied to each segment, and the
amount of overlap of each segment is specified with noverlap
See pdf for more info.
The returned times are the midpoints of the intervals over which
the ffts are calculated
"""
assert(NFFT>noverlap)
if log(NFFT)/log(2) != int(log(NFFT)/log(2)):
raise ValueError, 'NFFT must be a power of 2'
# zero pad x up to NFFT if it is shorter than NFFT
if len(x)<NFFT:
n = len(x)
x = resize(x, (NFFT,))
x[n:] = 0
# for real x, ignore the negative frequencies
if x.typecode()==Complex: numFreqs = NFFT
else: numFreqs = NFFT//2+1
windowVals = window(ones((NFFT,),x.typecode()))
step = NFFT-noverlap
ind = arange(0,len(x)-NFFT+1,step)
n = len(ind)
Pxx = zeros((numFreqs,n), Float)
# do the ffts of the slices
for i in range(n):
thisX = x[ind[i]:ind[i]+NFFT]
thisX = windowVals*detrend(thisX)
fx = absolute(fft(thisX))**2
# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2
Pxx[:,i] = divide(fx[:numFreqs], norm(windowVals)**2)
t = 1/Fs*(ind+NFFT/2)
freqs = Fs/NFFT*arange(numFreqs)
return Pxx, freqs, t
def levypdf(x, gamma, alpha):
"Returm the levy pdf evaluated at x for params gamma, alpha"
N = len(x)
if N % 2 != 0:
raise ValueError, 'x must be an event length array; try\n' + \
'x = linspace(minx, maxx, N), where N is even'
dx = x[1] - x[0]
f = 1 / (N * dx) * arange(-N / 2, N / 2, Float)
ind = concatenate([arange(N / 2, N, Int), arange(N / 2, Int)])
df = f[1] - f[0]
cfl = exp(-gamma * absolute(2 * pi * f)**alpha)
px = fft(take(cfl, ind) * df).astype(Float)
return take(px, ind)
def levypdf(x, gamma, alpha):
"Returm the levy pdf evaluated at x for params gamma, alpha"
N = len(x)
if N%2 != 0:
raise ValueError, 'x must be an event length array; try\n' + \
'x = linspace(minx, maxx, N), where N is even'
dx = x[1]-x[0]
f = 1/(N*dx)*arange(-N/2, N/2, Float)
ind = concatenate([arange(N/2, N, Int),
arange(N/2,Int)])
df = f[1]-f[0]
cfl = exp(-gamma*absolute(2*pi*f)**alpha)
px = fft(take(cfl,ind)*df).astype(Float)
return take(px, ind)
def cohere_pairs( X, ij, NFFT=256, Fs=2, detrend=detrend_none,
window=window_hanning, noverlap=0,
preferSpeedOverMemory=True,
progressCallback=donothing_callback,
returnPxx=False):
"""
Cxy, Phase, freqs = cohere_pairs( X, ij, ...)
Compute the coherence for all pairs in ij. X is a
numSamples,numCols Numeric array. ij is a list of tuples (i,j).
Each tuple is a pair of indexes into the columns of X for which
you want to compute coherence. For example, if X has 64 columns,
and you want to compute all nonredundant pairs, define ij as
ij = []
for i in range(64):
for j in range(i+1,64):
ij.append( (i,j) )
The other function arguments, except for 'preferSpeedOverMemory'
(see below), are explained in the help string of 'psd'.
Return value is a tuple (Cxy, Phase, freqs).
Cxy -- a dictionary of (i,j) tuples -> coherence vector for that
pair. Ie, Cxy[(i,j) = cohere(X[:,i], X[:,j]). Number of
dictionary keys is len(ij)
Phase -- a dictionary of phases of the cross spectral density at
each frequency for each pair. keys are (i,j).
freqs -- a vector of frequencies, equal in length to either the
coherence or phase vectors for any i,j key. Eg, to make a coherence
Bode plot:
subplot(211)
plot( freqs, Cxy[(12,19)])
subplot(212)
plot( freqs, Phase[(12,19)])
For a large number of pairs, cohere_pairs can be much more
efficient than just calling cohere for each pair, because it
caches most of the intensive computations. If N is the number of
pairs, this function is O(N) for most of the heavy lifting,
whereas calling cohere for each pair is O(N^2). However, because
of the caching, it is also more memory intensive, making 2
additional complex arrays with approximately the same number of
elements as X.
The parameter 'preferSpeedOverMemory', if false, limits the
caching by only making one, rather than two, complex cache arrays.
This is useful if memory becomes critical. Even when
preferSpeedOverMemory is false, cohere_pairs will still give
significant performace gains over calling cohere for each pair,
and will use subtantially less memory than if
preferSpeedOverMemory is true. In my tests with a 43000,64 array
over all nonredundant pairs, preferSpeedOverMemory=1 delivered a
33% performace boost on a 1.7GHZ Athlon with 512MB RAM compared
with preferSpeedOverMemory=0. But both solutions were more than
10x faster than naievly crunching all possible pairs through
cohere.
See test/cohere_pairs_test.py in the src tree for an example
script that shows that this cohere_pairs and cohere give the same
results for a given pair.
"""
numRows, numCols = X.shape
# zero pad if X is too short
if numRows < NFFT:
tmp = X
X = zeros( (NFFT, numCols), X.typecode())
X[:numRows,:] = tmp
del tmp
numRows, numCols = X.shape
# get all the columns of X that we are interested in by checking
# the ij tuples
seen = {}
for i,j in ij:
seen[i]=1; seen[j] = 1
allColumns = seen.keys()
Ncols = len(allColumns)
del seen
# for real X, ignore the negative frequencies
if X.typecode()==Complex: numFreqs = NFFT
else: numFreqs = NFFT//2+1
# cache the FFT of every windowed, detrended NFFT length segement
# of every channel. If preferSpeedOverMemory, cache the conjugate
# as well
windowVals = window(ones((NFFT,), X.typecode()))
ind = range(0, numRows-NFFT+1, NFFT-noverlap)
numSlices = len(ind)
FFTSlices = {}
FFTConjSlices = {}
Pxx = {}
slices = range(numSlices)
normVal = norm(windowVals)**2
for iCol in allColumns:
progressCallback(i/Ncols, 'Cacheing FFTs')
Slices = zeros( (numSlices,numFreqs), Complex)
for iSlice in slices:
thisSlice = X[ind[iSlice]:ind[iSlice]+NFFT, iCol]
thisSlice = windowVals*detrend(thisSlice)
Slices[iSlice,:] = fft(thisSlice)[:numFreqs]
FFTSlices[iCol] = Slices
if preferSpeedOverMemory:
FFTConjSlices[iCol] = conjugate(Slices)
Pxx[iCol] = divide(mean(absolute(Slices)**2), normVal)
del Slices, ind, windowVals
# compute the coherences and phases for all pairs using the
# cached FFTs
Cxy = {}
Phase = {}
count = 0
N = len(ij)
for i,j in ij:
count +=1
if count%10==0:
progressCallback(count/N, 'Computing coherences')
if preferSpeedOverMemory:
Pxy = FFTSlices[i] * FFTConjSlices[j]
else:
Pxy = FFTSlices[i] * conjugate(FFTSlices[j])
if numSlices>1: Pxy = mean(Pxy)
Pxy = divide(Pxy, normVal)
Cxy[(i,j)] = divide(absolute(Pxy)**2, Pxx[i]*Pxx[j])
Phase[(i,j)] = arctan2(Pxy.imag, Pxy.real)
freqs = Fs/NFFT*arange(numFreqs)
if returnPxx:
return Cxy, Phase, freqs, Pxx
else:
return Cxy, Phase, freqs
def rms_flat(a):
"""Return the root mean square of all the elements of a, flattened out."""
return numerix.mlab.sqrt(sum_flat(absolute(a)**2) / float(size(a)))
def l1norm(a):
"""Return the l1 norm of a, flattened out.
Implemented as a separate function (not a call to norm() for speed)."""
return sum_flat(absolute(a))
def l2norm(a):
"""Return the l2 norm of a, flattened out.
Implemented as a separate function (not a call to norm() for speed)."""
return numerix.mlab.sqrt(sum_flat(absolute(a)**2))
def l2norm(a):
"""Return the l2 norm of a, flattened out.
Implemented as a separate function (not a call to norm() for speed)."""
return numerix.mlab.sqrt(sum_flat(absolute(a)**2))
def l1norm(a):
"""Return the l1 norm of a, flattened out.
Implemented as a separate function (not a call to norm() for speed)."""
return sum_flat(absolute(a))
def rms_flat(a):
"""Return the root mean square of all the elements of a, flattened out."""
return numerix.mlab.sqrt(sum_flat(absolute(a)**2)/float(size(a)))
def cohere_pairs(X,
ij,
NFFT=256,
Fs=2,
detrend=detrend_none,
window=window_hanning,
noverlap=0,
preferSpeedOverMemory=True,
progressCallback=donothing_callback,
returnPxx=False):
"""
Cxy, Phase, freqs = cohere_pairs( X, ij, ...)
Compute the coherence for all pairs in ij. X is a
numSamples,numCols Numeric array. ij is a list of tuples (i,j).
Each tuple is a pair of indexes into the columns of X for which
you want to compute coherence. For example, if X has 64 columns,
and you want to compute all nonredundant pairs, define ij as
ij = []
for i in range(64):
for j in range(i+1,64):
ij.append( (i,j) )
The other function arguments, except for 'preferSpeedOverMemory'
(see below), are explained in the help string of 'psd'.
Return value is a tuple (Cxy, Phase, freqs).
Cxy -- a dictionary of (i,j) tuples -> coherence vector for that
pair. Ie, Cxy[(i,j) = cohere(X[:,i], X[:,j]). Number of
dictionary keys is len(ij)
Phase -- a dictionary of phases of the cross spectral density at
each frequency for each pair. keys are (i,j).
freqs -- a vector of frequencies, equal in length to either the
coherence or phase vectors for any i,j key. Eg, to make a coherence
Bode plot:
subplot(211)
plot( freqs, Cxy[(12,19)])
subplot(212)
plot( freqs, Phase[(12,19)])
For a large number of pairs, cohere_pairs can be much more
efficient than just calling cohere for each pair, because it
caches most of the intensive computations. If N is the number of
pairs, this function is O(N) for most of the heavy lifting,
whereas calling cohere for each pair is O(N^2). However, because
of the caching, it is also more memory intensive, making 2
additional complex arrays with approximately the same number of
elements as X.
The parameter 'preferSpeedOverMemory', if false, limits the
caching by only making one, rather than two, complex cache arrays.
This is useful if memory becomes critical. Even when
preferSpeedOverMemory is false, cohere_pairs will still give
significant performace gains over calling cohere for each pair,
and will use subtantially less memory than if
preferSpeedOverMemory is true. In my tests with a 43000,64 array
over all nonredundant pairs, preferSpeedOverMemory=1 delivered a
33% performace boost on a 1.7GHZ Athlon with 512MB RAM compared
with preferSpeedOverMemory=0. But both solutions were more than
10x faster than naievly crunching all possible pairs through
cohere.
See test/cohere_pairs_test.py in the src tree for an example
script that shows that this cohere_pairs and cohere give the same
results for a given pair.
"""
numRows, numCols = X.shape
# zero pad if X is too short
if numRows < NFFT:
tmp = X
X = zeros((NFFT, numCols), typecode(X))
X[:numRows, :] = tmp
del tmp
numRows, numCols = X.shape
# get all the columns of X that we are interested in by checking
# the ij tuples
seen = {}
for i, j in ij:
seen[i] = 1
seen[j] = 1
allColumns = seen.keys()
Ncols = len(allColumns)
del seen
# for real X, ignore the negative frequencies
if typecode(X) == Complex: numFreqs = NFFT
else: numFreqs = NFFT // 2 + 1
# cache the FFT of every windowed, detrended NFFT length segement
# of every channel. If preferSpeedOverMemory, cache the conjugate
# as well
windowVals = window(ones((NFFT, ), typecode(X)))
ind = range(0, numRows - NFFT + 1, NFFT - noverlap)
numSlices = len(ind)
FFTSlices = {}
FFTConjSlices = {}
Pxx = {}
slices = range(numSlices)
normVal = norm(windowVals)**2
for iCol in allColumns:
progressCallback(i / Ncols, 'Cacheing FFTs')
Slices = zeros((numSlices, numFreqs), Complex)
for iSlice in slices:
thisSlice = X[ind[iSlice]:ind[iSlice] + NFFT, iCol]
thisSlice = windowVals * detrend(thisSlice)
Slices[iSlice, :] = fft(thisSlice)[:numFreqs]
FFTSlices[iCol] = Slices
if preferSpeedOverMemory:
FFTConjSlices[iCol] = conjugate(Slices)
Pxx[iCol] = divide(mean(absolute(Slices)**2), normVal)
del Slices, ind, windowVals
# compute the coherences and phases for all pairs using the
# cached FFTs
Cxy = {}
Phase = {}
count = 0
N = len(ij)
for i, j in ij:
count += 1
if count % 10 == 0:
progressCallback(count / N, 'Computing coherences')
if preferSpeedOverMemory:
Pxy = FFTSlices[i] * FFTConjSlices[j]
else:
Pxy = FFTSlices[i] * conjugate(FFTSlices[j])
if numSlices > 1: Pxy = mean(Pxy)
Pxy = divide(Pxy, normVal)
Cxy[(i, j)] = divide(absolute(Pxy)**2, Pxx[i] * Pxx[j])
Phase[(i, j)] = arctan2(Pxy.imag, Pxy.real)
freqs = Fs / NFFT * arange(numFreqs)
if returnPxx:
return Cxy, Phase, freqs, Pxx
else:
return Cxy, Phase, freqs
def pprint_val(self, x):
xp = (x-self.offset)/10**self.orderOfMagnitude
if absolute(xp) < 1e-8: xp = 0
return self.format % xp
def pprint_val(self, x):
xp = (x-self.offset)/10**self.orderOfMagnitude
if absolute(xp) < 1e-8: xp = 0
return self.format % xp