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 [2]:
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 [3]:
s.plot_pdf()
../../_images/_documentation_tutorials_1_Defining_a_Parameter_4_0.png

and similarly, lets plot the CDF.

In [4]:
s.plot_cdf()
../../_images/_documentation_tutorials_1_Defining_a_Parameter_6_0.png

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 [5]:
s_samples = s.get_samples(1000)
s.plot_pdf(data=s_samples)
../../_images/_documentation_tutorials_1_Defining_a_Parameter_8_0.png

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 [6]:
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 [7]:
# 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))
../../_images/_documentation_tutorials_1_Defining_a_Parameter_12_0.png
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