Symbols and Definitions

Symbols

In the next sections, concepts that UQPCE relies on will be covered. Below are the symbols and their meanings.

\(X\) - the variable basis matrix

\(x\) - the set of variable values from the run_matrix.dat file

\(y\) - the responses of the system from the results.dat file

\(\hat{y}\) - the vector of predicted responses

\(x_v\) - the set of variable values from the verification_run_matrix.dat file

\(y_v\) - the responses of the system from the verification_results.dat file

\(\beta\) - the vector of matrix coefficients

\(\langle\bf{\Psi}^{2}\rangle\) - the multivariate norm squared matrix

\(\langle \Psi^{2} \rangle\) - the univariate norm squared matrix

\(\psi_i\) - the \(i^{th}\) order orthogonal polynomial basis component

\(p\) - the number of terms, including the intercept, in the model

\(k\) - the number of terms, not including the intercept, in the model

\(n\) - the number of responses used to create the model

\(n_v\) - the number of responses used to verify the model

\(m\) - the number of variables

\(M\) - the model interaction matrix

\(\alpha\) - the significance level from confidence interval

\(t\) - a Student’s t-distributed random variable

\(f\) - an f distributed random variable

\(\mathcal{R}\) - residual

Abbreviations

PDF : Probability Density Function

CDF : Cumulative Distribution Function

Definitions

There are a few definitions that the user will need to be familiar with in order to correctly use UQPCE and analyze its results.

Aleatory : This is a type of uncertainty due to inherent variation of a parameter, physical system or environment. This type of uncertainty is often referred to as irreducible uncertainty.

Epistemic : This is a type of uncertainty due to imperfect knowledge or ignorance of a variable.

Sobol : Sobol indices are global non-linear estimates of sensitivity. In general, the larger a Sobol index, the more sensitive the model is to that term.