Distribution Continuous 1D Example

This examples shows the use of a univariate continuous distributions class. In particular:

• How to define one of the univariate distributions supported by UQpy

• How to plot the pdf and log_pdf of the distribution

• How to modify the parameters of the distribution

• How to extract the moments of the distribution

• How to draw random samples from the distribution

Initially we have to import the necessary modules.

import numpy as np
import matplotlib.pyplot as plt
from UQpy.distributions import Lognormal

Example of a univariate lognormal distribution

In order to define the lognormal distribution, the user must provide its three parameters, s, loc and scale. Printing the dist object created after the Lognormal distribution initialization, will display as output the class of the distribution as can be seen below. The provided parameters of the distribution can be retrieved using the parameters attribute available to all distributions.

dist = Lognormal(s=1., loc=0., scale=np.exp(5))
print(dist)
print(dist.parameters)

<UQpy.distributions.collection.Lognormal.Lognormal object at 0x74d188c8fa30>
{'s': 1.0, 'loc': 0.0, 'scale': 148.4131591025766}

Plot the pdf of the distribution.

The user must provide x as a ndarray of shape (nsamples, 1) or (nsamples,) - the former if preferred. The result of pdf or log_pdf will be a 1D array (nsamples, ).

x = np.linspace(0.01, 1000, 1000).reshape((-1, 1))  # Use reshape to provide a 2D array (1000, 1)
fig, ax = plt.subplots(ncols=2, figsize=(15, 4))
ax[0].plot(x, dist.pdf(x))  # Do not give params
ax[0].set_xlabel('x')
ax[0].set_ylabel('pdf(x)')
ax[0].set_title('pdf of lognormal distribution')

ax[1].plot(x, dist.log_pdf(x))
ax[1].set_xlabel('x')
ax[1].set_ylabel('log pdf(x)')
ax[1].set_title('Log pdf of lognormal distribution')
plt.show()

print('size of input x:')
print(x.shape)
print('size of dist.pdf(x):')
print(dist.pdf(x).shape)

size of input x:
(1000, 1)
size of dist.pdf(x):
(1000,)

Modify one of the parameters of the distribution.

Use the update_parameters method. The user must provide as input to the update_parameters method the name, as well as the updated value of the specified parameter. Note that in case the user provides a non-existing parameter name, the update_parameters method will raise an exception.

dist.update_parameters(loc=100.)
print(dist.parameters)

{'s': 1.0, 'loc': 100.0, 'scale': 148.4131591025766}

Plot the pdf and log_pdf functions of the lognormal distribution with the updated parameters.

x = np.linspace(0.01, 1000, 1000).reshape((-1, 1))  # Use reshape to provide a 2D array (1000, 1)
fig, ax = plt.subplots(ncols=2, figsize=(15, 4))
ax[0].plot(x, dist.pdf(x))  # Do not give params
ax[0].set_xlabel('x')
ax[0].set_ylabel('pdf(x)')
ax[0].set_title('pdf of lognormal distribution')

ax[1].plot(x, dist.log_pdf(x))
ax[1].set_xlabel('x')
ax[1].set_ylabel('log pdf(x)')
ax[1].set_title('Log pdf of lognormal distribution')
plt.show()

Print the mean, standard deviation, skewness, and kurtosis of the distribution.

Using the moments method existing in all univariate distributions, the user can retrieve the available moments. The order in which the moments are extracted can be seen in the moments_list variable.

moments_list = ['mean', 'variance', 'skewness', 'kurtosis']
m = dist.moments()
print('Moments with inherited parameters:')
for i, moment in enumerate(moments_list):
    print(moment + ' = {0:.2f}'.format(m[i]))

Moments with inherited parameters:
mean = 344.69
variance = 102880.65
skewness = 6.18
kurtosis = 110.94

Generate 5000 random samples from the lognormal distribution.

The number of samples is provided as nsamples (default 1). The user can fix the seed of the pseudo-random generator via input random_state.

Important: the output of rvs is a (nsamples, 1) ndarray.

y1 = dist.rvs(nsamples=5000)
print('Shape of output provided by rvs is (nsamples, dimension), i.e. here:')
print(y1.shape)
plt.hist(y1[:, 0], bins=50)
plt.xlabel('x')
plt.ylabel('count')
plt.show()

Shape of output provided by rvs is (nsamples, dimension), i.e. here:
(5000, 1)