def main():
fig, ax = plt.subplots(figsize=gfs)
xs = np.linspace(0., 10., 500)
rvs = stats.lomax(c).pdf(xs)
ax.plot(xs, rvs, label='$\lambda(x) = c/x$')
rvs = stats.expon(scale=1 / c).pdf(xs)
ax.plot(xs, rvs, label='$\lambda(x) = c$')
rvs = stats.rayleigh(scale=np.sqrt(1 / c)).pdf(xs)
ax.plot(xs, rvs, label='$\lambda(x) = cx$')
ax.set_xlim(0, 10)
ax.set_ylim(0, 2)
ax.set_xlabel('$X$ (failure rv)', fontsize=fs)
ax.set_ylabel('$p(x)$', fontsize=fs)
ax.tick_params(labelsize=fs)
ax.legend(fontsize=fs)
plt.tight_layout()
plt.savefig('../hf2pdf-examples.png', bbox_inches='tight')
plt.close()
# logspace now
fig, ax = plt.subplots(figsize=gfs)
xs = np.linspace(0.1, 100., 1000)
rvs = stats.lomax(c).pdf(xs)
ax.plot(xs, rvs, label='$\lambda(x) = c/x$')
rvs = stats.expon(scale=1 / c).pdf(xs)
ax.plot(xs, rvs, label='$\lambda(x) = c$')
rvs = stats.rayleigh(scale=np.sqrt(1 / c)).pdf(xs)
ax.plot(xs, rvs, label='$\lambda(x) = cx$')
ax.set_xlabel('$X$ (failure rv)', fontsize=fs)
ax.set_ylabel('$p(x)$', fontsize=fs)
ax.tick_params(labelsize=fs)
ax.legend(fontsize=fs)
ax.set_ylim(10**-9, )
ax.set_yscale('log')
ax.set_xscale('log')
plt.tight_layout()
plt.savefig('../hf2pdf-examples-loglog.png', bbox_inches='tight')
plt.close()
g1 = gamma(alpha_, loc=0, scale=1. / float(beta_))
plt.figure(figsize=(8, 5))
plt.hist(theta_samples, bins=55, density=True, alpha=.5)
plt.plot(theta_range, g1.pdf(theta_range), 'b-')
plt.xlim(np.min(theta_samples), np.max(theta_samples))
plt.title('Theta posterior: samples versus pdf', fontsize=15)
plt.xlabel('theta (-)', fontsize=15)
# Compute predictive posterior directly from analytical formula: the Lomax distribution
# See https://en.wikipedia.org/wiki/Lomax_distribution
# and https://en.wikipedia.org/wiki/Conjugate_prior
X = np.linspace(0, 100, 10000) # fixed sample locations
PredPost = lomax_manual(X, alpha_, beta_)
PredPost = PredPost / np.sum(PredPost)
ll = lomax(c=alpha_, scale=float(beta_))
PredPost2 = ll.pdf(X)
PredPost2 /= np.sum(PredPost2)
print(np.allclose(PredPost2, PredPost))
true_pdf = true_theta * np.exp(-true_theta * X)
true_pdf /= np.sum(true_pdf)
print('True mean of exponential: %1.3f' % (1. / true_theta))
print('Estimated mean from pred.post.: %1.3f' %
(np.dot(X, PredPost / np.sum(PredPost))))
# Now let's try to replicate this posterior predictive from the (discrete) prob. mass function of
# the theta parameter, namely by constructing a mixture of exponentials, not using
# a 'meta-pdf' structure as outlined in <<Think Bayes>> (essentially based on the essential construct
# of a nested dictionary and bypassing all classes etc., see MakeMixture() function)), but instead with a nested for-loop.
def interval(self, alpha):
return st.lomax(scale=self.alpha, c=self.beta).interval(alpha)
def log_marginal_likelihood(self, data):
return st.lomax(scale=self.alpha, c=self.beta).logpdf(data)
print('max. observed delay: %i days' % (df.delay_obs.max()))
print('\n')
# True delay distribution
True_delay = expon(loc=testgroups['A'].min_delay,
scale=testgroups['A'].lambda_)
# Compute observed delay dist. via Bayesian conj. priors (see script <exponential_post_predictive.py>)
alpha_, beta_ = Bayesian_conjugate_inference(observed_delays -
testgroups['A'].min_delay)
# (marginal) posterior
lambd_dist_uncorrected = gamma(alpha_, loc=0,
scale=1. / float(beta_)).pdf(lambda_range)
# posterior predictive
ll = lomax(c=alpha_, loc=testgroups['A'].min_delay, scale=float(beta_))
Observed_delay_dist_Bayesian = ll.pdf(x)
# Compute corrected delay dist. by adjusting likelihood function (see script <decay_problem_MacKay.py>)
today = df.event0.values[-1]
df['days_ago'] = (today - df.event0) / pd.Timedelta(1, 'D')
completed_events = df[['delay_obs', 'days_ago']].dropna().copy()
#lambda_likelh = likelihood_over_all_observations(completed_events, min_delay=testgroups['A'].min_delay, lambda_range=lambda_range)
lambda_likelh = pd.Series(data=np.ones((len(lambda_range), )))
for _, obs in completed_events.iterrows():
Z = expon_integral(
lambda_range, 0, obs.days_ago
) # integration constants, dependent on observation window but independent of observation
likelihood_of_lambda_for_this_observation = expon.pdf(
c = 1.88 mean, var, skew, kurt = lomax.stats(c, moments='mvsk') # Display the probability density function (``pdf``): x = np.linspace(lomax.ppf(0.01, c), lomax.ppf(0.99, c), 100) ax.plot(x, lomax.pdf(x, c), 'r-', lw=5, alpha=0.6, label='lomax 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 = lomax(c) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') # Check accuracy of ``cdf`` and ``ppf``: vals = lomax.ppf([0.001, 0.5, 0.999], c) np.allclose([0.001, 0.5, 0.999], lomax.cdf(vals, c)) # True # Generate random numbers: r = lomax.rvs(c, size=1000) # And compare the histogram: ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
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),
}
def _reset_distribution(self):
self._distribution = lomax(c=self.alpha_prime, scale=self.beta_prime)
def _reset_distribution(self):
self._distribution: rv_continuous = lomax(c=self._alpha,
scale=self._lambda)