def fftsurr(x):
"""
Compute an FFT phase randomized surrogate of x
"""
z = fft(x)
a = 2.*pi*1j
phase = a*rand(len(x))
z = z*exp(phase)
return inverse_fft(z).real
def fftsurr(x, detrend=detrend_none, window=window_none):
"""
Compute an FFT phase randomized surrogate of x
"""
x = window(detrend(x))
z = fft(x)
a = 2.*pi*1j
phase = a*rand(len(x))
z = z*exp(phase)
return inverse_fft(z).real
def fftsurr(x, detrend=detrend_none, window=window_none):
"""
Compute an FFT phase randomized surrogate of x
"""
x = window(detrend(x))
z = fft(x)
a = 2. * pi * 1j
phase = a * rand(len(x))
z = z * exp(phase)
return inverse_fft(z).real
def exp_safe(x):
"""Compute exponentials which safely underflow to zero.
Slow but convenient to use. Note that NumArray will introduce proper
floating point exception handling with access to the underlying
hardware."""
if type(x) is ArrayType:
return exp(clip(x,exp_safe_MIN,exp_safe_MAX))
else:
return math.exp(x)
def exp_safe(x):
"""Compute exponentials which safely underflow to zero.
Slow but convenient to use. Note that NumArray will introduce proper
floating point exception handling with access to the underlying
hardware."""
if type(x) is ArrayType:
return exp(clip(x, exp_safe_MIN, exp_safe_MAX))
else:
return math.exp(x)
def bivariate_normal(X, Y, sigmax=1.0, sigmay=1.0,
mux=0.0, muy=0.0, sigmaxy=0.0):
"""
Bivariate gaussan distribution for equal shape X, Y
http://mathworld.wolfram.com/BivariateNormalDistribution.html
"""
Xmu = X-mux
Ymu = Y-muy
rho = sigmaxy/(sigmax*sigmay)
z = Xmu**2/sigmax**2 + Ymu**2/sigmay - 2*rho*Xmu*Ymu/(sigmax*sigmay)
return 1.0/(2*pi*sigmax*sigmay*(1-rho**2)) * exp( -z/(2*(1-rho**2)))
def bivariate_normal(X, Y, sigmax=1.0, sigmay=1.0,
mux=0.0, muy=0.0, sigmaxy=0.0):
"""
Bivariate gaussan distribution for equal shape X, Y
http://mathworld.wolfram.com/BivariateNormalDistribution.html
"""
Xmu = X-mux
Ymu = Y-muy
rho = sigmaxy/(sigmax*sigmay)
z = Xmu**2/sigmax**2 + Ymu**2/sigmay - 2*rho*Xmu*Ymu/(sigmax*sigmay)
return 1.0/(2*pi*sigmax*sigmay*(1-rho**2)) * exp( -z/(2*(1-rho**2)))
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 pyxplothist(graph, x_sample, Nbins=80, bin_range=None, norm=False, bars=0,
lw=0, lt=0, color=(0,0,0), xlogscale=False, ylogscale=False,
y_given=None, title=False):
""" Plots a histogram of 'x_sample'.
Arguments:
'x_sampe' - data
'Nbins' - number of bins
'bin_range' - intervall to be divided into 'Nbins' equidistant parts
'norm' - normalization of histogram
(comparison with density function)
'bars' - style parameter: bars (bars=1) or steps (bars=0)
'lw', 'lt' - linewidth, linetype
'title' - title
'color' - color of histogram (r,g,b)
'xlogscale' - takes bins with logarithmically constant width,
makes only sense, when 'xaxistype' of 'graph'
is set to 'log'
'ylogscale' - activates necessary changes, if y-axis is logarithmic,
makes only sense, when 'yaxistype' of 'graph'
is set to 'log'
'ygiven' - if you already know the y-values
"""
steps = 1-bars
# determine max. x-value for the bins
x_max = max(x_sample)
if bin_range != None:
x_max = bin_range[1]
# changes due to logarithmic x-axis:
# take log of the data, check if data or range is <= zero
if xlogscale:
if min(x_sample) > 0:
x_sample = log(x_sample)
if bin_range != None:
if bin_range[0] > 0 and bin_range[1] > 0:
bin_range = (log(bin_range[0]),log(bin_range[1]))
else:
print "Given range includes values <= 0!"
print "Ignoring given range..."
bin_range = None
else:
print "Data is smaller than zero, no logarithmic x-axis possible!"
print "Continuing linearly..."
xlogscale = False
if y_given != None:
x_l = x_sample
x_hist = y_given
Nbins = len(x_l)
else:
## histogram of 'x_sample': gives number 'x_hist' of sample points within
## 'Nbins' different of bins within 'bin_range', where the bins are
## specified by their left edge coordinate 'x_l'
x_hist, x_l = histogram(x_sample, bins=Nbins, range=bin_range, normed=norm)
# if logarithmic x-axis: exponentiate bin positions and data
if xlogscale:
x_l = exp(x_l)
x_sample = exp(x_sample)
y_min = 0
# offset for logarithmic y-axis
if ylogscale:
x_hist = x_hist+1e-16
y_min = 1e-16
# plot histogram manually
for i in xrange(Nbins):
# horzontal lines
if i < Nbins-1:
graph.pyxplot(([x_l[i],x_l[i+1]],[x_hist[i],x_hist[i]]),
style="l", color=color, lt=lt, lw=lw, title=False)
else:
graph.pyxplot(([x_l[i],x_max],[x_hist[i],x_hist[i]]),
style="l", color=color, lt=lt, lw=lw, title=False)
# vertical lines
if i != Nbins-1:
graph.pyxplot(([x_l[i+1],x_l[i+1]],[x_hist[i],x_hist[i+1]]),
style="l", color=color, lt=lt, lw=lw, title=False)
if i == 0:
graph.pyxplot(([x_l[i],x_l[i]],[x_hist[i],y_min]),
style="l", color=color, lt=lt, lw=lw, title=title)
if i == Nbins-1:
graph.pyxplot(([x_max,x_max],[x_hist[i],y_min]),
style="l", color=color, lt=lt, lw=lw, title=False)
def normpdf(x, *args): "Return the normal pdf evaluated at x; args provides mu, sigma" mu, sigma = args return 1/(numerix.mlab.sqrt(2*pi)*sigma)*exp(-0.5 * (1/sigma*(x - mu))**2)
def normpdf(x, *args):
"Return the normal pdf evaluated at x; args provides mu, sigma"
mu, sigma = args
return 1 / (numerix.mlab.sqrt(2 * pi) * sigma) * exp(-0.5 * (1 / sigma *
(x - mu))**2)