Kais Zaman

A probabilistic approach for representation of interval uncertainty

In this paper, we propose a probabilistic approach to represent interval data for input variables in reliability and uncertainty analysis problems, using flexible families of continuous Johnson distributions. Such a probabilistic representation of interval data facilitates a unified framework for handling aleatory and epistemic uncertainty. For fitting probability distributions, methods such as moment matching are commonly used in the literature. However, unlike point data where single estimates for the moments of data can be calculated, moments of interval data can only be computed in terms of upper and lower bounds. Finding bounds on the moments of interval data has been generally considered an NP-hard problem because it includes a search among the combinations of multiple values of the variables, including interval endpoints. In this paper, we present efficient algorithms based on continuous optimization to find the bounds on second and higher moments of interval data. With numerical examples, we show that the proposed bounding algorithms are scalable in polynomial time with respect to increasing number of intervals. Using the bounds on moments computed using the proposed approach, we fit a family of Johnson distributions to interval data. Furthermore, using an optimization approach based on percentiles, we find the bounding envelopes of the family of distributions, termed as a Johnson p-box. The idea of bounding envelopes for the family of Johnson distributions is analogous to the notion of empirical p-box in the literature. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the computationally expensive nested analysis that is typically required in the presence of interval variables, the proposed probabilistic representation enables inexpensive optimization-based strategies to estimate bounds on an output quantity of interest.

Robustness-Based Design Optimization of Multidisciplinary System Under Epistemic Uncertainty

This paper proposes formulations and algorithms for design optimization of multidisciplinary systems under both aleatory uncertainty (i.e., natural or physical variability) and epistemic uncertainty (due to sparse or imprecise information) from the perspective of system robustness. The availability of sparse and interval data regarding input or design random variables introduces uncertainty about their probability distribution type and distribution parameters. A single-loop approach is developed for the design optimization, which does not require any coupled multidisciplinary uncertainty propagation analysis. Thus, the computational complexity and cost involved in estimating the mean and variation of the objective and constraints are greatly reduced. A decoupled approach is used to unnest the robustness-based design from the analysis of nondesign epistemic variables to achieve further computational efficiency. The proposed methods are illustrated for a mathematical problem and a practical engineering prob...

Representation and First-Order Approximations for Propagation of Aleatory and Distribution Parameter Uncertainty

Many probabilistic uncertainty propagation methods have been developed for many single discipline problems involving expensive computational codes in order to propagate physical variability in the input, typically expressed through random variables and random processes and/or fields. Most of these techniques have only been studied with respect to physical variability represented by probability distributions, but are not able to include both aleatory and epistemic uncertainties. Epistemic uncertainty can come from many different sources, including incertitude resulting from model form errors, solution approximation errors, and scarcity of data about the underlying variables subject to physical variability. Therefore, the contribution of this paper is to develop and illustrate approaches for the propagation aleatory and epistemic uncertainty due to sparse data, using probabilistic methods of uncertainty propagation. We reformulate the problem of deriving confidence bounds on cumulative probability estimates as an optimization problem and use the results of these optimization formulations to derive absolute and/or confidence bounds on the cumulative probability distribution of system response.

Probabilistic Analysis with Sparse Data

In this paper, the problem of reliability analysis under both aleatory uncertainty (natural variability) and epistemic uncertainty (arising when the only knowledge about the random variables is sparse-point data) is addressed. First considered is epistemic uncertainty arising from a lack of knowledge of the distribution type of the random variables. To address this uncertainty in distribution type, the use of a flexible family of distributions is proposed. The Johnson family of distributions has the ability to reproduce the shape of many named continuous probability distributions and therefore alleviate the difficulty of determining an appropriate named distribution type for the random variable. Next considered is uncertainty in the distribution parameters themselves, and methods to determine probability distributions for the distribution parameters are proposed. As a result, the uncertainty in reliability estimates for limit-state functions having random variables with imprecise probability distributions...

Representation and Propagation of both Probabilistic and Interval Uncertainty

This paper develops and illustrates a probabilistic approach for uncertainty representation and propagation in system analysis, when the information on the uncertain input variables and/or their distribution parameters may be available as either probability distributions or simply intervals (single or multiple). The uncertainty described by interval data is represented through a flexible family of probability distributions. Conversion of interval data to a probabilistic format enables the use of computationally efficient methods for probabilistic uncertainty propagation. Two methods are explored for the implementation of the proposed approach, based on: (1) sampling and (2) optimization. The sampling based strategy is more expensive and tends to underestimate the output bounds. The optimization based methodology improves both aspects.

An Optimization-Based Approach to Calculate Confidence Interval on Mean Value with Interval Data

In this paper, we propose a methodology for construction of confidence interval on mean values with interval data for input variable in uncertainty analysis and design optimization problems. The construction of confidence interval with interval data is known as a combinatorial optimization problem. Finding confidence bounds on the mean with interval data has been generally considered an NP hard problem, because it includes a search among the combinations of multiple values of the variables, including interval endpoints. In this paper, we present efficient algorithms based on continuous optimization to find the confidence interval on mean values with interval data. With numerical experimentation, we show that the proposed confidence bound algorithms are scalable in polynomial time with respect to increasing number of intervals. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the current practice for the design optimization with interval data that typically implements the constraints on interval variables through the computation of bounds on mean values from the sampled data, the proposed approach of construction of confidence interval enables more complete implementation of design optimization under interval uncertainty.

Construction of confidence interval on mean value with interval data

In this paper, we propose a methodology for construction of confidence interval on mean values with interval data for input variable in uncertainty analysis problems. Confidence interval on mean values depends on the values of moments of the sampled data. For construction of confidence interval with point data, well-established methods are available in the literature. However, unlike point data where single estimates for the moments of data can be calculated, moments of interval data can only be computed in terms of upper and lower bounds implying that with interval data every moment will be an interval itself. This suggests that the construction of confidence interval with interval data is an optimization problem. In this paper, we present efficient algorithms based on continuous optimization to find the confidence interval on mean values with interval data. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the current practice for the design optimization with interval data that typically implements the constraints on interval variables through the computation of bounds on mean values from the sampled data, the proposed approach of construction of confidence interval enables more complete implementation of design optimization under interval uncertainty.

Robustness-based Design Optimization with Sparse Point and Interval Data

This paper proposes formulations and algorithms for design optimization under both aleatory (i.e., natural or physical variability) and epistemic uncertainty (i.e., imprecise probabilistic information), from the perspective of system robustness. The proposed formulations deal with epistemic uncertainty arising from both sparse and interval data without any assumption about the probability distributions of the random variables. A decoupled approach is proposed in this paper to un-nest the robustness-based design from the analysis of non-design epistemic variables to achieve computational efficiency. The proposed methods are illustrated for the upper stage design problem of a two-stage-to-orbit (TSTO) vehicle, where the information on the random design inputs are only available as sparse point data and/or interval data. As collecting more data reduces uncertainty but increases cost, the effect of sample size on the optimality and robustness of the solution is also studied.