# Foundations I: Parameter¶

A parameter is one of the main building blocks in Effective Quadratures. Let $$s$$ be a parameter defined on a domain $$\mathcal{D} \in \mathbb{R}$$. The support of the domain $$\mathcal{D}$$ may be:

• closed $$[a,b]$$

• semi-infinite $$(-\infty, b)$$ or $$[a, \infty)$$

• infinite $$(-\infty, \infty)$$

Further, let us assume that this parameter is characterized by a positive weight function $$\rho(s)$$, which may be interpreted as the probability density function (PDF) of $$s$$, which readily implies that

$\int_{\mathcal{D}}\rho\left(s\right)ds=1.$

We now demonstrate some basic functionality of this parameter. First consider the case where $$\rho(s) = \mathcal{N} (0, 1)$$ is a standard Gaussian distribution with a mean of 0.0 and a variance of 1.0. We then plot its PDF and cumulative density function (CDF) and demonstrate how we can generate random samples from this distribution.

In :

import equadratures as eq
s = eq.Parameter(distribution='normal', shape_parameter_A = 0.0, \
shape_parameter_B = 1.0, order=3)


Now for some plots; first let us plot the PDF. We can call s.get_pdf() to get a numpy array containing the pdf values, but instead, lets use plot_pdf() here.

In :

s.plot_pdf() and similarly, lets plot the CDF.

In :

s.plot_cdf() Now, lets use the get_samples() functionality to sample from the parameter distribution. These samples can be passed to plot_pdf to create a histogram.

In :

s_samples = s.get_samples(1000)
s.plot_pdf(data=s_samples) One can repeat the above for a range of distributions. We provide a few additional definitions below. First, consider the example of a Gaussian distribution $$\mathcal{N}(0,1)$$, truncanted between $$[-1,2]$$.

In :

s = eq.Parameter(distribution='truncated-gaussian', lower=-1.0, upper=2., \
shape_parameter_A = 0.0, shape_parameter_B = 1.0, order=3)


followed by that of a custom distribution—based on user supplied data.

In :

# Create some data
import numpy as np
param1 = np.random.rand(500)
param2 = np.random.randn(600)
param3 = np.random.randn(650)*0.5 - 0.2
param4 = np.random.randn(150)*0.1 + 3
data = np.hstack([param1, param2, param3, param4])

# Fit a Weight function to this data
input_dist = eq.Weight(data, support=[0, 4], pdf=False)

# Use the weight function to define a bespoke data-driven Parameter.
# We can also can truncate the data to a tighter support.
s = eq.Parameter(distribution='data', weight_function=input_dist, order=3)

# Plot the cdf
s.plot_pdf(data=s.get_samples(2000)) 