Variables

Variables#

In this section, the variables used by UQPCE will be documented.

Continuous#

Multiple continuous variables are supported in UQPCE, and these variables will be discussed below.

All bounds \(a\) and \(b\) are such that \(a \ \epsilon \ \mathbb{R}\), \(b \ \epsilon \ \mathbb{R}\), and both are finite.

Normal Variable#

The equation UQPCE uses for a normal variable is shown below where \(\mu \ \epsilon \ \mathbb{R}\) and \(\sigma^{2} > 0\). The support range of this variable is [\(-\infty\), \(\infty\)].

\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}} \]

Uniform Variable#

The equation UQPCE uses for a uniform variable is shown below. The support range of this variable is [a, b].

\[ f(x) = \frac{1}{b - a} \]

Beta Variable#

The equation UQPCE uses for a beta variable is shown below where \(\alpha > 0\), \(\beta > 0\), and the gamma function \(\Gamma(n) = (n - 1)!\). The support range of this variable is [a, b].

\[ f(x) = \frac{1}{b - a} \frac{\Gamma(\alpha + \beta)} {\Gamma(\alpha) \Gamma(\beta)} \Bigg(\frac{x - a}{b - a}\Bigg)^{\alpha - 1} \Bigg(\frac{b - x}{b - a}\Bigg)^{\beta - 1} \]

Note

The implementation of the density function used for this program is the more-common equation for the beta distribution, while common orthogonal polynomial conventions use a different equation. The user must treat their data according to the above equation.

Exponential Variable#

The equation UQPCE uses for an exponential variable is shown below where \(\lambda > 0\). The support range of this variable is [a, \(\infty\)].

\[ f(x) = \lambda e^{-\lambda (x\ -\ a)} \]

Gamma Variable#

The equation UQPCE uses for a gamma variable is shown below where \(\alpha > 0\), \(\theta > 0\), and the gamma function \(\Gamma(n) = (n - 1)!\). The support range of this variable is [a, \(\infty\)].

\[ f(x) = \frac{1}{\Gamma(\alpha)\theta^{\alpha}} (x\ -\ a)^{\alpha - 1} e^{-\frac{(x\ -\ a)}{\theta}} \]

Lognormal Variable#

The equation UQPCE uses for a lognormal variable is shown below where \(\mu \ \epsilon \ \mathbb{R}\) and \(\sigma > 0\). Parameter \(\mu\) is the mean of the variable’s natural logarithm, and \(\sigma\) is the standard deviation of the variable’s natural logarithm. The support range of this variable is [a, \(\infty\)].

\[ f(x) = \frac{1}{(x - a) \sigma \sqrt{2 \pi}} \cdot e^{-\frac{\bigg(ln \big(x-a \big) - \mu \bigg)^{2}}{2 \sigma}} \]

User-Input Variable#

This is an option for the user to input a variable that has an arbitrary continuous distribution. While this gives the user flexibility, there are some requirements the distribution must adhere to:

  • The distribution must have all values included in the PDF. For example, a user-input variable for a normal distribution would need to input the mean and standard deviation values into the equation explicitly.

  • The distribution and samples need to be standardized as inputs, as UQPCE currently has no way of standardizing the ContinuousVariable class distribution and samples.

  • The distribution must be continuous.

  • The distribution must have a finite integral over its support range.

Note

Since every continuous distribution is impossible to test and this variable type relies on the user’s understanding of the inputs, it is recommended that users first test a UQPCE case using their user-input variables against a Monte Carlo to ensure that the variable works correctly for their purposes.

For many distributions, this user-input variable works well. It is, however, possible that the orthogonal polynomials and/or norm squared values for an equation take significant computational time to converge.

Discrete#

Multiple discrete variables are supported in UQPCE, and these variables will be discussed below.

All bounds \(a\) and \(b\) are such that \(a \ \epsilon \ \mathbb{R}\), \(b \ \epsilon \ \mathbb{R}\), and both are finite.

Poisson#

The equation UQPCE uses for a Poisson variable is shown below where \(\lambda \geq 0\). The support range of this variable is [a, \(\infty\)].

\[ f(x) = e^{-\lambda} \frac{\lambda^{(x-a)}}{(x-a)!} \]

Uniform#

The equation UQPCE uses for a discrete uniform variable is shown below for \(x = \{a, a+1, ..., b-1, b\}\). The support range of this variable is [a, b].

\[ f(x) = \frac{1}{b-a} \]

Negative Binomial#

The equation UQPCE uses for a negative binomial variable is shown below where \(0 \leq p \leq 1\) and \(r > 0\). The support range of this variable is [a, \(\infty\)].

\[ f(x) = \frac{((x-a)+r-1)!}{(x-a)!(r-1)!} \ (1-p)^{(x-a)} p^r \]

Hypergeometric#

The equation UQPCE uses for a hypergeometric variable is shown below where \(M \geq 0\), \(n \geq 0\), and \(1 \leq N \leq M+n\). The support range of this variable is [a, \(N\) + a].

\[ f(x) = \frac{M! \ n! \ N! \ (M+n-N)!}{(x-a)! \ (n-(x-a))! \ (M+(x-a)-N)! \ (N-(x-a))! \ (M+n)!} \]

User-Input Variable#

This is an option for the user to input a variable that has an arbitrary discrete distribution. While this gives the user flexibility, there are some requirements the distribution must adhere to:

  • The distribution must have numbers plugged in for any parameters.

  • The distribution and samples need to be standardized as inputs, as UQPCE currently has no way of standardizing the DiscreteVariable class distribution and samples.

  • The distribution must be discrete.

  • The distribution must have a finite integral over its support range.

Note

Since every discrete distribution is impossible to test and this variable type relies on the user’s understanding of the inputs, it is recommended that users first test a UQPCE case using their user-input variables against a Monte Carlo to ensure that the variable works correctly for their purposes.

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