def scipy_modified_levy(x, alpha, beta, mu=0.0, sigma=1.0, a=0, b=np.inf, cdf=False):
"""
Levy distribution with the tail replaced by the analytical (power law) approximation.
`alpha` in (0, 2] is the index of stability, or characteristic exponent.
`beta` in [-1, 1] is the skewness. `mu` in the reals and `sigma` > 0 are the
location and scale of the distribution (corresponding to `delta` and `gamma`
in Nolan's notation; note that sigma in levy corresponds to sqrt(2) sigma
in the Normal distribution).
'a' and 'b' are the cut off limits for the distribution.
Both a and b are > 0 and the distribution is built on the assumption.
It uses parametrization 0 (to get it from another parametrization, convert).
Example:
>>> x = np.array([1, 2, 3])
>>> modified_levy(x, 1.5, 0, a=0.1, b=3.1)
array([0.23897864, 0.09999616, 0.0372704])
:param x: values where the function is evaluated
:type x: :class:`~numpy.ndarray`
:param alpha: alpha
:type alpha: float
:param beta: beta
:type beta: float
:param mu: mu
:type mu: float
:param sigma: sigma
:type sigma: float
:param a: a
:type a: float
:param b: b
:type b: float
:return: values of the pdf (or cdf if parameter 'cdf' is set to True) at 'x'
:rtype: :class:`~numpy.ndarray`
"""
if x == 0:
raise ValueError
else:
if cdf == False:
if b == np.inf:
return levy_stable.pdf(x, alpha, beta, loc=mu, scale=sigma) / 2 / (1 - levy_stable.cdf(x, alpha, beta, loc=mu, scale=sigma))
else:
return levy_stable.pdf(x, alpha, beta, loc=mu, scale=sigma) / 2 / (levy_stable.cdf(b, alpha, beta, loc=mu, scale=sigma) - levy_stable.cdf(a, alpha, beta, loc=mu, scale=sigma))
else:
if x < 0:
return (levy_stable.cdf(x, alpha, beta, loc=mu, scale=sigma) - levy_stable.cdf(-b, alpha, beta, loc=mu, scale=sigma)) / 2 / (levy_stable.cdf(b, alpha, beta, loc=mu, scale=sigma) - levy_stable.cdf(a, alpha, beta, loc=mu, scale=sigma))
else:
return (levy_stable.cdf(x, alpha, beta, loc=mu, scale=sigma) - levy_stable.cdf(a, alpha, beta, loc=mu, scale=sigma)) / 2 / (levy_stable.cdf(b, alpha, beta, loc=mu, scale=sigma) - levy_stable.cdf(a, alpha, beta, loc=mu, scale=sigma))
def plot_marginal(self, bounds=(-.5, .5), bins=50, show=False):
""" Plot a histogram of the marginal distribution of increments, with the expected PDF overlayed on top of it
:param bounds: largest increments to be included in histogram
:param bins: number of bins in histogram of empirical distribution
:param show: show the plot when done
:type bounds: tuple
:type bins: int
:type show: bool
"""
x = np.linspace(bounds[0], bounds[1], 1000)
hist, bin_edges = np.histogram(self.noise.flatten(),
bins=bins,
range=bounds,
density=True)
# account for part of PDF that is chopped off. Using density=True makes hist sum to 1
area_covered = levy_stable.cdf(bounds[1], self.alpha, 0, loc=0, scale=self.scale) - \
levy_stable.cdf(bounds[0], self.alpha, 0, loc=0, scale=self.scale)
hist *= area_covered
# plot bars. Can't use plt.hist since I needed to modify the bin heights
bin_width = bin_edges[1] - bin_edges[0]
bin_centers = [i + bin_width / 2 for i in bin_edges[:-1]]
plt.bar(bin_centers, hist, width=bin_width)
plt.plot(x,
levy_stable.pdf(x, self.alpha, 0, loc=0, scale=self.scale),
'--',
color='black',
lw=2)
# print(np.abs(x).sum(), levy_stable.pdf(x, self.alpha, 0, loc=0, scale=self.scale).sum())
# exit()
# formatting
plt.xlabel('Step Size', fontsize=14)
plt.ylabel('Frequency', fontsize=14)
plt.gcf().get_axes()[0].tick_params(labelsize=14)
plt.tight_layout()
if show:
plt.show()
def cdf(x):
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always", category=IntegrationWarning)
result = levy_stable.cdf(x, alpha, beta)
# Scipy has only an experimental .cdf() function for alpha=1, beta!=0.
# It sometimes passes and sometimes xfails.
if w and alpha == 1 and beta != 0:
pytest.xfail(reason="scipy.stats.levy_stable.cdf is unstable")
return result
x = np.linspace(levy_stable.ppf(0.01, alpha, beta),
levy_stable.ppf(0.99, alpha, beta), 100)
ax.plot(x, levy_stable.pdf(x, alpha, beta),
'r-', lw=5, alpha=0.6, label='levy_stable pdf')
# Alternatively, the distribution object can be called (as a function)
# to fix the shape, location and scale parameters. This returns a "frozen"
# RV object holding the given parameters fixed.
# Freeze the distribution and display the frozen ``pdf``:
rv = levy_stable(alpha, beta)
ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
# Check accuracy of ``cdf`` and ``ppf``:
vals = levy_stable.ppf([0.001, 0.5, 0.999], alpha, beta)
np.allclose([0.001, 0.5, 0.999], levy_stable.cdf(vals, alpha, beta))
# True
# Generate random numbers:
r = levy_stable.rvs(alpha, beta, size=1000)
# And compare the histogram:
ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
ax.legend(loc='best', frameon=False)
plt.show()
def scipy_levy(x, alpha, beta, mu=0.0, sigma=1.0, cdf=False):
if cdf == False:
return levy_stable.pdf(x, alpha, beta, loc=mu, scale=sigma)
else:
return levy_stable.cdf(x, alpha, beta, loc=mu, scale=sigma)
def cdf(x, alpha, beta):
if alpha != 1:
x += beta * tan(pi * alpha / 2)
return levy_stable.cdf(x, alpha, beta)
#which is the number of distribution parameters
chisqnorm, pvalnorm = st.chisquare(observed, expected, ddof=2)
print("Chi-square for normal =", chisqnorm)
print("Normal skew =", skew(z), ";\nexcess kurtosis =", kurtosis(z))
#Levy-stable fit
[al, be, de, ga] = levy_stable._fitstart(z)
print("Levy stable fit parameters:", al, be, de, ga)
#Calculate chi-square test statistic:
k = 34 #number of bins in histogram
#use numpy histogram for the expected value
observed, hist_binedges = np.histogram(z, bins=k)
#use the cumulative density function (c.d.f.)
#of the distribution for the expected value:
cdf = levy_stable.cdf(hist_binedges, al, be, de, ga)
expected = len(z) * np.diff(cdf)
#use scipy.stats chisquare function, where
#ddof is the adjustment to the k-1 degrees of freedom,
#which is the number of distribution parameters
chisqlevy, pvallevy = st.chisquare(observed, expected, ddof=4)
print("Chi-square for levy stable =", chisqlevy)
#Plotting
matplotlib.style.use('ggplot')
data = pd.Series(z)
#determines left and right limits of fit displayed (adjust numbers for more/less)
x = np.linspace(norm.ppf(0.001, mean, var), norm.ppf(0.999, mean, var), 1000)
y = norm.pdf(x, mean, var)