def cdf(self, strike, spot, texp, cp=1):
fwd, df, _ = self._fwd_factor(spot, texp)
sig2_inv = np.exp(self.sigma**2 * texp) - 1
ig = spst.invgauss(mu=sig2_inv, scale=1 / sig2_inv)
x = strike / fwd
cdf = np.where(cp > 0, ig.sf(x), ig.cdf(x))
return cdf
def estimate_tweeide_logcdf_series(x, mu, phi, p):
"""Estimate the logcdf of a given set of x, mu, phi, and p
Parameters
----------
x : array
The observed values. Must be non-negative.
mu : array
The fitted values. Must be positive.
phi : array
The scale paramter. Must be positive.
p : array
The Tweedie variance power. Must equal 0 or must be greater than or
equal to 1.
Returns
-------
estiate_tweedie_loglike_series : float
"""
x = np.array(x, ndmin=1)
mu = np.array(mu, ndmin=1)
phi = np.array(phi, ndmin=1)
p = np.array(p, ndmin=1)
logcdf = np.zeros_like(x)
# Gaussian (Normal)
mask = p == 0
if np.sum(mask) > 0:
logcdf[mask] = norm(loc=mu[mask],
scale=np.sqrt(phi[mask])).logcdf(x[mask])
# Poisson
mask = p == 1.
if np.sum(mask) > 0:
logcdf[mask] = np.log(poisson(mu=mu[mask] / phi[mask]).cdf(x[mask]))
# 1 < p < 2
mask = (1 < p) & (p < 2)
if np.sum(mask) > 0:
cond1 = mask
cond2 = x > 0
mask = cond1 & cond2
logcdf[mask] = logcdf_1to2(x[mask], mu[mask], phi[mask], p[mask])
mask = cond1 & ~cond2
logcdf[mask] = -(mu[mask]**(2 - p[mask]) / (phi[mask] * (2 - p[mask])))
# Gamma
mask = p == 2
if np.sum(mask) > 0:
logcdf[mask] = gamma(a=1 / phi[mask],
scale=phi[mask] * mu[mask]).logcdf(x[mask])
# Inverse Gaussian (Normal)
mask = p == 3
if np.sum(mask) > 0:
logcdf[mask] = invgauss(mu=mu[mask] * phi[mask],
scale=1 / phi[mask]).logcdf(x[mask])
return logcdf
def generate_values(line, length_mean, length_std):
"""
Generate values for a given kmer,
according to description provided in line.
The number of values is randomised;
it is drawn from normal distribution with mean length_mean
and standard deviation length_std
(length_std = 0 means no randomisation).
line - line of a template (row from pandas.DataFrame)
length_mean - mean of a Gauss distribution from which length of a sequence will be drawn
length_std - standard deviation of see above.
Description of columns in original template:
kmer The kmer being modelled.
level_mean The mean of a Gaussian distribution representing the current observed for this kmer.
level_stdv The standard deviation of the above Gaussian distribution of observed currents for this kmer.
sd_mean The mean of an inverse Gaussian distribution representing the noise observed for this kmer.
sd_stdv The standard deviation of the above inverse Gaussian distribution of noise observed for this kmer.
ig_lambda The lambda parameter for the above inverse Gaussian distribution of noise observed for this kmer.
(so sd_stdv == sqrt(sd_mean ^ 3 / ig_lambda). See Wikipedia for more info.
weight Used internally for model training purposes.
"""
signal_var = norm(line["level_mean"], line["level_stdv"])
mu, lmbda = line["sd_mean"], line["ig_lambda"]
noise_var = invgauss(mu / lmbda, scale=lmbda)
n = norm.rvs(loc=length_mean, scale=length_std)
#n = max(1, int(n))
n = max(0, int(n))
values = signal_var.rvs(n) + noise_var.rvs(n)
return values
def log_event_sd_inv_gaussian_probability_match(self, event_sd, kmer):
"""Get the probability of the event_sd coming from the model's kmer inv-gaussian distribution
:param event_sd: sd of event
:param kmer: kmer for model distribution selection
"""
inv_gauss_mean, inv_gauss_lambda = self.get_event_sd_inv_gaussian_parameters(
kmer)
return invgauss(inv_gauss_mean / inv_gauss_lambda,
scale=inv_gauss_lambda).logpdf(event_sd)
def price(self, strike, spot, texp, cp=1):
fwd, df, _ = self._fwd_factor(spot, texp)
sig2_inv = np.exp(self.sigma**2 * texp) - 1
ig = spst.invgauss(mu=sig2_inv, scale=1 / sig2_inv)
kk = strike / fwd
price = np.where(
cp > 0,
ig.cdf(1 / kk) - kk * ig.sf(kk),
kk * ig.cdf(kk) - ig.sf(1 / kk),
)
return df * fwd * price
def test_invgauss_pdf():
for i in range(10):
a = np.random.rand()*10.
v = np.random.rand()*10.
sigma = np.random.rand()*10.
mu = a / v
lam = a**2 / sigma**2
x = np.linspace(0.1, 100, 1000)
y1 = np.exp(likelihoods.invgauss_logpdf(x, 0, a, v, sigma=sigma, p_outlier=0.))
# needs transform: https://github.com/scipy/scipy/issues/2367#issuecomment-17028905
y2 = stats.invgauss(mu/lam, loc=0, scale=lam).pdf(x)
np.testing.assert_array_almost_equal(y1, y2)
def _distDivergence(self, _model1, _model2):
"""
Compute the divergence between the two probability models (Variation of Jensen Shannon)
_model1: The basic model.
_model2: The test models.
RETURN: The divergence score [0,1]
"""
POINT_DENSITY = 1000 # how many points to fit
# basis fit
model = stats.invgauss(mu=_model1[0], loc=_model1[1], scale=_model1[2])
x = np.linspace(model.ppf(0.010), model.ppf(0.999), POINT_DENSITY)
p1 = model.pdf(x)
avgP = p1
# test fit
p2 = np.zeros((_model2.shape[0], POINT_DENSITY))
for i in range(0, _model2.shape[0]):
model = stats.invgauss(mu=_model2[i, 0],
loc=_model2[i, 1],
scale=_model2[i, 2])
x = np.linspace(model.ppf(0.001), model.ppf(0.999), POINT_DENSITY)
p2[i, :] = model.pdf(x)
avgP = avgP + p2[i, :]
# compute average
avgP = avgP / (1 + _model2.shape[0])
gJS = np.sum(p1 * np.log(np.divide(p1, avgP)))
for i in range(0, _model2.shape[0]):
gJS = gJS + np.sum(p2 * np.log(np.divide(p2[i, :], avgP)))
gJS = gJS / (1 + _model2.shape[0])
return (math.exp(-gJS / 1000))
def test_invgauss_pdf():
for i in range(10):
a = np.random.rand() * 10.
v = np.random.rand() * 10.
sigma = np.random.rand() * 10.
mu = a / v
lam = a**2 / sigma**2
x = np.linspace(0.1, 100, 1000)
y1 = np.exp(
likelihoods.invgauss_logpdf(x, 0, a, v, sigma=sigma, p_outlier=0.))
# needs transform: https://github.com/scipy/scipy/issues/2367#issuecomment-17028905
y2 = stats.invgauss(mu / lam, loc=0, scale=lam).pdf(x)
np.testing.assert_array_almost_equal(y1, y2)
def _ppf(self, q, p, mu, phi):
p = np.broadcast_to(p, q.shape)
mu = np.broadcast_to(mu, q.shape)
phi = np.broadcast_to(phi, q.shape)
single1to2v = np.vectorize(self._ppf_single1to2, otypes='d')
ppf = np.zeros(q.shape, dtype=float)
# Gaussian
mask = p == 0
if np.sum(mask) > 0:
ppf[mask] = norm(loc=mu[mask],
scale=np.sqrt(phi[mask])).ppf(q[mask])
# Poisson
mask = p == 1
if np.sum(mask) > 0:
ppf[mask] = poisson(mu=mu[mask] / phi[mask]).ppf(q[mask])
# 1 < p < 2
mask = (1 < p) & (p < 2)
if np.sum(mask) > 0:
zero_mass = np.zeros_like(ppf)
zeros = np.zeros_like(ppf)
zero_mass[mask] = self._cdf(zeros[mask], p[mask], mu[mask],
phi[mask])
right = 10 * mu * phi**p
cond1 = mask
cond2 = q > zero_mass
if np.sum(cond1 & ~cond2) > 0:
ppf[cond1 & ~cond2] = zeros[cond1 & ~cond2]
if np.sum(cond1 & cond2) > 0:
single1to2v = np.vectorize(self._ppf_single1to2, otypes='d')
mask = cond1 & cond2
ppf[mask] = single1to2v(q[mask], p[mask], mu[mask], phi[mask],
zero_mass[mask], right[mask])
# Gamma
mask = p == 2
if np.sum(mask) > 0:
ppf[mask] = gamma(a=1 / phi[mask],
scale=phi[mask] * mu[mask]).ppf(q[mask])
# Inverse Gamma
mask = p == 3
if np.sum(mask) > 0:
ppf[mask] = invgauss(mu=mu[mask] * phi[mask],
scale=1 / phi[mask]).ppf(q[mask])
return ppf
def test_invgauss_sampling():
samples = 5000
likelihoods.init_rands((samples, 3), seed=101)
for i in range(10):
a = np.random.rand()*10.
v = np.random.rand()*10.
sigma = np.random.rand()*10.
mu = a / v
lam = a**2 / sigma**2
samples = likelihoods.invgauss(0, a, v, sigma=sigma)
# needs transform: https://github.com/scipy/scipy/issues/2367#issuecomment-17028905
D, p_value = stats.kstest(samples, stats.invgauss(mu/lam, loc=0, scale=lam).cdf)
assert p_value > .05, "Not from the same distribution. Params: a=%f, v=%f, mu=%f, lam=%f" % (a, v, mu, lam)
def test_invgauss_sampling():
samples = 5000
likelihoods.init_rands((samples, 3), seed=101)
for i in range(10):
a = np.random.rand() * 10.
v = np.random.rand() * 10.
sigma = np.random.rand() * 10.
mu = a / v
lam = a**2 / sigma**2
samples = likelihoods.invgauss(0, a, v, sigma=sigma)
# needs transform: https://github.com/scipy/scipy/issues/2367#issuecomment-17028905
D, p_value = stats.kstest(
samples,
stats.invgauss(mu / lam, loc=0, scale=lam).cdf)
assert p_value > .05, "Not from the same distribution. Params: a=%f, v=%f, mu=%f, lam=%f" % (
a, v, mu, lam)
def all_dists():
# dists param were taken from scipy.stats official
# documentaion examples
# Total - 89
return {
"alpha":
stats.alpha(a=3.57, loc=0.0, scale=1.0),
"anglit":
stats.anglit(loc=0.0, scale=1.0),
"arcsine":
stats.arcsine(loc=0.0, scale=1.0),
"beta":
stats.beta(a=2.31, b=0.627, loc=0.0, scale=1.0),
"betaprime":
stats.betaprime(a=5, b=6, loc=0.0, scale=1.0),
"bradford":
stats.bradford(c=0.299, loc=0.0, scale=1.0),
"burr":
stats.burr(c=10.5, d=4.3, loc=0.0, scale=1.0),
"cauchy":
stats.cauchy(loc=0.0, scale=1.0),
"chi":
stats.chi(df=78, loc=0.0, scale=1.0),
"chi2":
stats.chi2(df=55, loc=0.0, scale=1.0),
"cosine":
stats.cosine(loc=0.0, scale=1.0),
"dgamma":
stats.dgamma(a=1.1, loc=0.0, scale=1.0),
"dweibull":
stats.dweibull(c=2.07, loc=0.0, scale=1.0),
"erlang":
stats.erlang(a=2, loc=0.0, scale=1.0),
"expon":
stats.expon(loc=0.0, scale=1.0),
"exponnorm":
stats.exponnorm(K=1.5, loc=0.0, scale=1.0),
"exponweib":
stats.exponweib(a=2.89, c=1.95, loc=0.0, scale=1.0),
"exponpow":
stats.exponpow(b=2.7, loc=0.0, scale=1.0),
"f":
stats.f(dfn=29, dfd=18, loc=0.0, scale=1.0),
"fatiguelife":
stats.fatiguelife(c=29, loc=0.0, scale=1.0),
"fisk":
stats.fisk(c=3.09, loc=0.0, scale=1.0),
"foldcauchy":
stats.foldcauchy(c=4.72, loc=0.0, scale=1.0),
"foldnorm":
stats.foldnorm(c=1.95, loc=0.0, scale=1.0),
# "frechet_r": stats.frechet_r(c=1.89, loc=0.0, scale=1.0),
# "frechet_l": stats.frechet_l(c=3.63, loc=0.0, scale=1.0),
"genlogistic":
stats.genlogistic(c=0.412, loc=0.0, scale=1.0),
"genpareto":
stats.genpareto(c=0.1, loc=0.0, scale=1.0),
"gennorm":
stats.gennorm(beta=1.3, loc=0.0, scale=1.0),
"genexpon":
stats.genexpon(a=9.13, b=16.2, c=3.28, loc=0.0, scale=1.0),
"genextreme":
stats.genextreme(c=-0.1, loc=0.0, scale=1.0),
"gausshyper":
stats.gausshyper(a=13.8, b=3.12, c=2.51, z=5.18, loc=0.0, scale=1.0),
"gamma":
stats.gamma(a=1.99, loc=0.0, scale=1.0),
"gengamma":
stats.gengamma(a=4.42, c=-3.12, loc=0.0, scale=1.0),
"genhalflogistic":
stats.genhalflogistic(c=0.773, loc=0.0, scale=1.0),
"gilbrat":
stats.gilbrat(loc=0.0, scale=1.0),
"gompertz":
stats.gompertz(c=0.947, loc=0.0, scale=1.0),
"gumbel_r":
stats.gumbel_r(loc=0.0, scale=1.0),
"gumbel_l":
stats.gumbel_l(loc=0.0, scale=1.0),
"halfcauchy":
stats.halfcauchy(loc=0.0, scale=1.0),
"halflogistic":
stats.halflogistic(loc=0.0, scale=1.0),
"halfnorm":
stats.halfnorm(loc=0.0, scale=1.0),
"halfgennorm":
stats.halfgennorm(beta=0.675, loc=0.0, scale=1.0),
"hypsecant":
stats.hypsecant(loc=0.0, scale=1.0),
"invgamma":
stats.invgamma(a=4.07, loc=0.0, scale=1.0),
"invgauss":
stats.invgauss(mu=0.145, loc=0.0, scale=1.0),
"invweibull":
stats.invweibull(c=10.6, loc=0.0, scale=1.0),
"johnsonsb":
stats.johnsonsb(a=4.32, b=3.18, loc=0.0, scale=1.0),
"johnsonsu":
stats.johnsonsu(a=2.55, b=2.25, loc=0.0, scale=1.0),
"ksone":
stats.ksone(n=1e03, loc=0.0, scale=1.0),
"kstwobign":
stats.kstwobign(loc=0.0, scale=1.0),
"laplace":
stats.laplace(loc=0.0, scale=1.0),
"levy":
stats.levy(loc=0.0, scale=1.0),
"levy_l":
stats.levy_l(loc=0.0, scale=1.0),
"levy_stable":
stats.levy_stable(alpha=0.357, beta=-0.675, loc=0.0, scale=1.0),
"logistic":
stats.logistic(loc=0.0, scale=1.0),
"loggamma":
stats.loggamma(c=0.414, loc=0.0, scale=1.0),
"loglaplace":
stats.loglaplace(c=3.25, loc=0.0, scale=1.0),
"lognorm":
stats.lognorm(s=0.954, loc=0.0, scale=1.0),
"lomax":
stats.lomax(c=1.88, loc=0.0, scale=1.0),
"maxwell":
stats.maxwell(loc=0.0, scale=1.0),
"mielke":
stats.mielke(k=10.4, s=3.6, loc=0.0, scale=1.0),
"nakagami":
stats.nakagami(nu=4.97, loc=0.0, scale=1.0),
"ncx2":
stats.ncx2(df=21, nc=1.06, loc=0.0, scale=1.0),
"ncf":
stats.ncf(dfn=27, dfd=27, nc=0.416, loc=0.0, scale=1.0),
"nct":
stats.nct(df=14, nc=0.24, loc=0.0, scale=1.0),
"norm":
stats.norm(loc=0.0, scale=1.0),
"pareto":
stats.pareto(b=2.62, loc=0.0, scale=1.0),
"pearson3":
stats.pearson3(skew=0.1, loc=0.0, scale=1.0),
"powerlaw":
stats.powerlaw(a=1.66, loc=0.0, scale=1.0),
"powerlognorm":
stats.powerlognorm(c=2.14, s=0.446, loc=0.0, scale=1.0),
"powernorm":
stats.powernorm(c=4.45, loc=0.0, scale=1.0),
"rdist":
stats.rdist(c=0.9, loc=0.0, scale=1.0),
"reciprocal":
stats.reciprocal(a=0.00623, b=1.01, loc=0.0, scale=1.0),
"rayleigh":
stats.rayleigh(loc=0.0, scale=1.0),
"rice":
stats.rice(b=0.775, loc=0.0, scale=1.0),
"recipinvgauss":
stats.recipinvgauss(mu=0.63, loc=0.0, scale=1.0),
"semicircular":
stats.semicircular(loc=0.0, scale=1.0),
"t":
stats.t(df=2.74, loc=0.0, scale=1.0),
"triang":
stats.triang(c=0.158, loc=0.0, scale=1.0),
"truncexpon":
stats.truncexpon(b=4.69, loc=0.0, scale=1.0),
"truncnorm":
stats.truncnorm(a=0.1, b=2, loc=0.0, scale=1.0),
"tukeylambda":
stats.tukeylambda(lam=3.13, loc=0.0, scale=1.0),
"uniform":
stats.uniform(loc=0.0, scale=1.0),
"vonmises":
stats.vonmises(kappa=3.99, loc=0.0, scale=1.0),
"vonmises_line":
stats.vonmises_line(kappa=3.99, loc=0.0, scale=1.0),
"wald":
stats.wald(loc=0.0, scale=1.0),
"weibull_min":
stats.weibull_min(c=1.79, loc=0.0, scale=1.0),
"weibull_max":
stats.weibull_max(c=2.87, loc=0.0, scale=1.0),
"wrapcauchy":
stats.wrapcauchy(c=0.0311, loc=0.0, scale=1.0),
}
mu = 0.145 mean, var, skew, kurt = invgauss.stats(mu, moments='mvsk') # Display the probability density function (``pdf``): x = np.linspace(invgauss.ppf(0.01, mu), invgauss.ppf(0.99, mu), 100) ax.plot(x, invgauss.pdf(x, mu), 'r-', lw=5, alpha=0.6, label='invgauss 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 = invgauss(mu) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') # Check accuracy of ``cdf`` and ``ppf``: vals = invgauss.ppf([0.001, 0.5, 0.999], mu) np.allclose([0.001, 0.5, 0.999], invgauss.cdf(vals, mu)) # True # Generate random numbers: r = invgauss.rvs(mu, size=1000) # And compare the histogram: ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
print(abs(logabsderiv - np.log(-back(1))).max())
if False:
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
from scipy import stats
# y = np.linspace(-5,EER2logdprime(0.000001),200)
# pdf_x = stats.beta(1,1,scale=0.5).pdf #uniform between 0 and 0.5
# y2x = logdprime2EER
# pdf_y = reparam_pdf(pdf_x,y2x,y)
# ax.plot(y,pdf_y,label="log(d')")
# y = np.linspace(np.exp(-5),np.exp(EER2logdprime(0.000001)),200)
# pdf_x = stats.beta(1,4,scale=0.5).pdf #uniform between 0 and 0.5
# y2x = dprime2EER
# pdf_y = reparam_pdf(pdf_x,y2x,y)
# ax.plot(y,pdf_y,label="d'")
y = np.linspace(0, 0.49, 200)
pdf_x = stats.invgauss(mu=1, scale=1).pdf
y2x = EER2dprime
pdf_y = reparam_pdf(pdf_x, y2x, y)
ax.plot(y, pdf_y, label="EER")
ax.legend(loc='best', frameon=False)
plt.xlabel("EER")
plt.grid()
plt.show()
from scipy.stats import invgauss import matplotlib.pyplot as plt import numpy as np #invgauss.pdf(x, mu) = 1 / sqrt(2*pi*x**3) * exp(-(x-mu)**2/(2*x*mu**2)) fig, ax = plt.subplots(1, 1) mu = 1 x = np.linspace(invgauss.pdf(0.01, mu),invgauss.pdf(0.99, mu), 100) ax.plot(x, invgauss.pdf(x, mu),'r-', lw=5, alpha=0.6, label='invgauss pdf') rv = invgauss(mu) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') r = invgauss.rvs(mu, size=1000) ax.hist(r, normed=True, histtype='stepfilled', alpha=0.2) ax.legend(loc='best', frameon=False) plt.show()
def _scipy_invgauss(loc, concentration): # Wrapper of scipy's invgauss function, which is used to generate expected # output. # scipy uses a different parameterization. # See https://github.com/scipy/scipy/issues/4654. return stats.invgauss(mu=loc/concentration, scale=concentration)
def estimate_tweedie_loglike_series(x, mu, phi, p):
"""Estimate the loglikihood of a given set of x, mu, phi, and p
Parameters
----------
x : array
The observed values. Must be non-negative.
mu : array
The fitted values. Must be positive.
phi : array
The scale paramter. Must be positive.
p : array
The Tweedie variance power. Must equal 0 or must be greater than or
equal to 1.
Returns
-------
estiate_tweedie_loglike_series : float
"""
x = np.array(x, ndmin=1)
mu = np.array(mu, ndmin=1)
phi = np.array(phi, ndmin=1)
p = np.array(p, ndmin=1)
ll = np.ones_like(x) * -np.inf
# Gaussian (Normal)
gaussian_mask = p == 0.
if np.sum(gaussian_mask) > 0:
ll[gaussian_mask] = norm(loc=mu[gaussian_mask],
scale=np.sqrt(phi[gaussian_mask])).logpdf(
x[gaussian_mask])
# Poisson
poisson_mask = p == 1.
if np.sum(poisson_mask) > 0:
poisson_pdf = poisson(mu=mu[poisson_mask] / phi[poisson_mask]).pmf(
x[poisson_mask] / phi[poisson_mask]) / phi[poisson_mask]
ll[poisson_mask] = np.log(poisson_pdf)
# 1 < p < 2
ll_1to_2_mask = (1 < p) & (p < 2)
if np.sum(ll_1to_2_mask) > 0:
# Calculating logliklihood at x == 0 is pretty straightforward
zeros = x == 0
mask = zeros & ll_1to_2_mask
ll[mask] = -(mu[mask]**(2 - p[mask]) / (phi[mask] * (2 - p[mask])))
mask = ~zeros & ll_1to_2_mask
ll[mask] = ll_1to2(x[mask], mu[mask], phi[mask], p[mask])
# Gamma
gamma_mask = p == 2
if np.sum(gamma_mask) > 0:
ll[gamma_mask] = gamma(a=1 / phi, scale=phi * mu).logpdf(x[gamma_mask])
# (2 < p < 3) or (p > 3)
ll_2plus_mask = ((2 < p) & (p < 3)) | (p > 3)
if np.sum(ll_2plus_mask) > 0:
zeros = x == 0
mask = zeros & ll_2plus_mask
ll[mask] = -np.inf
mask = ~zeros & ll_2plus_mask
ll[mask] = ll_2orMore(x[mask], mu[mask], phi[mask], p[mask])
# Inverse Gaussian (Normal)
invgauss_mask = p == 3
if np.sum(invgauss_mask) > 0:
cond1 = invgauss_mask
cond2 = x > 0
mask = cond1 & cond2
ll[mask] = invgauss(mu=mu[mask] * phi[mask],
scale=1. / phi[mask]).logpdf(x[mask])
return ll
def _scipy_invgauss(loc, concentration):
# Wrapper of scipy's invgauss function, which is used to generate expected
# output.
# scipy uses a different parameterization.
# See https://github.com/scipy/scipy/issues/4654.
return stats.invgauss(mu=loc / concentration, scale=concentration)
"""Plot some basic BPT distributions
"""
import os, sys
from matplotlib import pyplot as plt
import numpy as np
from scipy.stats import expon, gamma, weibull_min, invgauss
mu = 100
alphas = [0.5, 1., 2., 5., 10.]
#alpha = 1
x_vals = np.arange(0, 4 * mu)
# Plot for a range of alpha values
for alpha in alphas:
bpt = invgauss(alpha, scale=mu)
pdf_vals = bpt.pdf(x_vals)
plt.plot(x_vals, pdf_vals, label=alpha)
# Now add exponential
exp_dist = expon(scale=mu)
pdf_vals = exp_dist.pdf(x_vals)
plt.plot(x_vals, pdf_vals, label='Exponential', color='k')
plt.legend()
plt.savefig('BPT_distribution.png')
