TSEUQLib: Taylor Series Expansion Uncertainty Quantification Library

Construct Taylor series expansions for surrogate modelling. Compute UQ and sensitivity analysis metrics of the Taylor series expansion.

Features

Function	Description	Symbol
tse	n-th-order Taylor Series	$T_{n}(\boldsymbol{x})$
expectation	Expected Value	$E\left[T_{n}(\boldsymbol{x})\right]$
central_moment	Central Moment	$\mu_{k}\left[T_{n}(\boldsymbol{x})\right]$
sobol_indices	Partial Variances	$V_{ij\ldots}\left[T_{n}(\boldsymbol{x})\right]$
sobol_indices	Sobol’ Indices	$S_{ij\ldots}\left[T_{n}(\boldsymbol{x})\right]$
total_sobol_indices	Total Sobol’ Indices	$S_{i}^{T}\left[T_{n}(\boldsymbol{x})\right]$
shapley_values	Shapley effects	$\phi_{i}\left[T_{n}(\boldsymbol{x})\right]$
remainder	Error of $\xi\left[T_{n}(\boldsymbol{x})\right]$	$\varepsilon\left[\xi\left[T_{n}(\boldsymbol{x})\right]\right]$
—	Sensitivities, $\frac{\partial \xi[T_n(\boldsymbol{x})]}{\partial \theta_i}$	$S^{\xi}_{\theta_i}$

Module Overview

Main functions and classes in tseuqlib/:

File	Function / Class	Description
oti_moments.py	tse_uq (class)	Core UQ engine that operates on OTI numbers to propagate Taylor-series-expansion (TSE) statistics.
	.expectation(y)	Computes the expectation of the TSE represented by OTI number y.
	.central_moment(y, k, Ey=None)	Computes the k-th central moment of the TSE expansion.
	.conditional_expectation(y, basis)	Computes the conditional expectation of the TSE w.r.t. a given basis (subset of variables).
	.sobol_indices(y, max_order=None)	Computes Sobol sensitivity indices up to a given interaction order.
	.build_hdmr(y, max_order=None, Ey=None)	Builds the High-Dimensional Model Representation (HDMR) decomposition of y.
	.shapley_values_v1/v2(y, max_order=None)	Two implementations for computing Shapley effects from variance decomposition.
	.tse_remainder / .expectation_remainder / .central_moment_remainder / .sobol_indices_remainder	Estimate truncation-error remainder terms for the TSE and its statistics.
oti_moments_sens.py	tse_uq (class)	Extended/sensitivity-focused version of tse_uq (adds HDMR-aware variants of expectation, central moment, conditional expectation).
	get_extra_bases(abases, n_dim)	Helper to build extra OTI bases used when constructing the class.
	.central_moment_hdmr, .conditional_expectation_hdmr	HDMR-consistent versions of the central moment / conditional expectation calculations.
oti_util.py	build_rv_joint_moments(mu_ind)	Builds the joint-distribution moment structure from independent input random variables.
	gen_OTI_basis_vector(nvars, order)	Generates the list of OTI bases for a given number of variables/order.
	gen_OTI_indices(nvars, order)	Generates index lists identifying each basis term in an OTI number.
	convert_index_to_exponent_form(lst)	Converts basis-index lists into exponent form.
	tse_remainder(y, n, k)	Estimates the remainder term of an n-th order Taylor series expansion.
rv_moments.py	rv_central_moments (class)	Computes central moments of input random variables from their PDFs/parameters.
	get_pdf_params(rv_pdf_name, rv_mean, rv_stdev)	Derives distribution parameters (e.g., shape/scale) from mean & stdev for a named PDF.
	gamma_oti(z) / beta_oti(a, b)	OTI-compatible (Lanczos-approximation-based) gamma/beta functions.
	convert_raw_moments_to_central_moments(mu_r, mean)	Converts raw moments to central moments.
	joint_central_moments(rv_mu, rv_moments_order)	Computes joint central moments across independent random variables.
rv_moments_v2.py	get_pdf_params(...), build_joint(mu_ind), generate_moments(...)	Newer/alternate versions of the PDF-parameter and joint-moment-building utilities above.
tse_moments.py	tse_central_moment (class)	Large (auto-generated) class with explicit closed-form expressions for TSE statistics.
	.tseEvFirst … tseEvFifth	Expected value of the TSE at 1st–5th order.
	.tseVarFirst … tseVarFourth	Variance of the TSE at 1st–4th order.
	.tseTcmFirst … tseTcmThird	Third central moment (skewness component) at 1st–3rd order.
	.tseFcmFirst … tseFcmThird	Fourth central moment (kurtosis component) at 1st–3rd order.
util.py	create_symbolic_variables(nVar)	Creates a list of symbolic variables (for symbolic derivative work).
	extract_derivatives(oti_number, nVar, tse_order_max)	Extracts derivatives (up to 5th order) from an OTI number.

The two tse_uq classes (in oti_moments.py and oti_moments_sens.py) are the primary public API — they take an OTI-represented model output and compute expectation, variance/central moments, Sobol indices, HDMR decompositions, and Shapley values via Taylor-series propagation. The rv_moments*.py and oti_util.py files supply supporting machinery (input-distribution moments, OTI basis/index generation), and tse_moments.py contains large auto-generated closed-form moment expressions used internally.

Prerequisites

Install OTILib from https://github.com/mauriaristi/otilib. This will create a new conda environment with OTILib installed.

Activate the otilib conda environment, for example:

conda activate pyoti

Add the TSEUQLib dependencies in one of two ways (TESTING IN PROGRESS).

1. Update the pyoti environment with:

conda env update -f environment.yml

Or, 2) Update the pyoti environment with:

conda install line_profiler sympy scipy pandas

Usage

Run the examples in examples/ with python. For example, run the ishigami example by:

cd TSEUQLib/examples/
python ishigami.py

Copyright

LANL O5031

© 2025. Triad National Security, LLC. All rights reserved.

This program was produced under U.S. Government contract 89233218CNA000001 for Los Alamos National Laboratory (LANL), which is operated by Triad National Security, LLC for the U.S. Department of Energy/National Nuclear Security Administration. All rights in the program are reserved by Triad National Security, LLC, and the U.S. Department of Energy/National Nuclear Security Administration. The Government is granted for itself and others acting on its behalf a nonexclusive, paid-up, irrevocable worldwide license in this material to reproduce, prepare. derivative works, distribute copies to the public, perform publicly and display publicly, and to permit others to do so.