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()
from scipy.stats import levy_stable
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
# Calculate a few first moments:
alpha, beta = 1.8, -0.5
mean, var, skew, kurt = levy_stable.stats(alpha, beta, moments='mvsk')
# Display the probability density function (``pdf``):
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
gamma = np.random.uniform(-np.pi / 2, np.pi / 2)
X = np.sin(alpha * (gamma - gamma0)) / np.cos(gamma)**(1 / alpha) * (
np.cos(gamma - alpha * (gamma - gamma0)) / W)**((1 - alpha) / alpha)
result.append(X)
result = np.array(result)
# x = np.arange(levy_stable.ppf(0.01,alpha,beta),levy_stable.ppf(0.99,alpha,beta),0.01)
x = np.arange(-10, 10, 0.1)
plt.hist(result,
alpha=0.7,
edgecolor="black",
bins=20,
range=(-10, 10),
normed=True)
plt.plot(x, levy_stable.pdf(x, alpha, beta), color="red")
plt.ylabel("Frequency", fontsize=12)
plt.title("Histogram of alpha=0.5 standard stable r.v.")
plt.show()
# alpha = 1 beta = 0
alpha = 1
beta = 0.75
beta2 = beta
result = []
for i in range(N):
W = np.random.exponential(1)
gamma = np.random.uniform(-np.pi / 2, np.pi / 2)
X = (np.pi / 2 + beta2 * gamma) * np.tan(gamma) - beta2 * np.log(
W * np.cos(gamma) / (np.pi / 2 + beta2 * gamma))
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 pdf(x, alpha, beta):
levy_stable.pdf_default_method = "quadrature"
if alpha != 1:
x += beta * tan(pi * alpha / 2)
return levy_stable.pdf(x, alpha, beta)
def pdf(x, alpha, beta):
levy_stable.pdf_default_method = "zolotarev"
if alpha != 1:
x += beta * tan(pi * alpha / 2)
return levy_stable.pdf(x, alpha, beta)
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)
pdf = pd.Series(y, x)
x = np.linspace(levy_stable.ppf(0.005, al, be, de, ga),
levy_stable.ppf(0.99, al, be, de, ga), 1000)
y = levy_stable.pdf(x, al, be, de, ga)
pdflevy = pd.Series(y, x)
#plot of histogram and fit probability density function
plt.figure(figsize=(10, 6), dpi=100)
ax = pdf.plot(lw=2,
label="normal fit: μ=" + str("%.8f" % mean) + " σ^2=" +
str("%.8f" % var),
legend=True)
pdflevy.plot(lw=2,
label="levy stable fit: a=" + str("%.2f" % al) + " b=" +
str("%.2f" % be),
legend=True,
ax=ax)
data.plot(kind='hist',
def alpha_stable_pdf(x, alpha=1, beta=1, c=1, mu=0):
# Returns probability from a pdf
return levy_stable.pdf((x - mu)/c, alpha, beta)
from scipy.stats import levy_stable
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
# Calculate a few first moments:
alpha, beta = 0.357, -0.675
mean, var, skew, kurt = levy_stable.stats(alpha, beta, moments='mvsk')
# Display the probability density function (``pdf``):
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``: