Maximum‐likelihood maximum‐entropy constrained probability density function estimation for prediction of rare events

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This paper presents a novel efficient method of estimating the joint probability distribution of continuous random variables with arbitrary (nonmonotonic or monotonic) relationships. As the backbone of the method is a set of monotonization transformations that "roll out" the relationships, the method is named the rolling pin method. The method allows one to estimate joint probability distributions when the actual causal structure of the attributes is unknown or extremely intricate to be determined accurately. Once the relationships are monotonized by the transformations, an appropriate parametric copula function is used to describe the joint distribution of the transformed variables. The copula function allows modeling the joint distribution of the transformed variables with a few parameters. The monotonization transformations empower standard parametric copulas to (i) capture complicated unknown dependence structures, (ii) model multivariate joint probability distributions with different pairwise dependence structures using the same parametric copula, and (iii) model nonmonotonicity. The application and performance of the method are shown using two examples.

Section: Introductionmentioning

confidence: 99%

Rolling Pin Method: Efficient General Method of Joint Probability Modeling

Ahooyi

Arbogast

Soroush

2014

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“…Ordinary BNs mostly rely on the relative-frequency-based techniques (e.g., the maximum likelihood-based methods) to learn the conditional probability values, so they are susceptible to the cases for which there are no data available for certain regions or ranges (rare states), because of the scarcity of data. Although some methods, such as the MLME method 35 and the ME method, 36 have been proposed to estimate the conditional probabilities over the unobserved regions, using BNs for performing inference for rare events is still limited. At the same time, the rolling pin method presents a natural interpretation of rare states that have their near-zero probabilities, which can be predicted by probability density functions of continuous random variables.…”

Section: Inference Over Certain Variablesmentioning

confidence: 99%

Applications of the Rolling Pin Method. 1. An Efficient Alternative to Bayesian Network Modeling and Inference

Ahooyi

Arbogast

Soroush

2014

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This paper presents a novel probabilistic modeling and inference method that is computationally more efficient than Bayesian networks (BNs). This method is applicable to systems with continuous variables. It is also applicable to systems with discrete variables that have an adequately high number of states. This work, indeed, represents an application of the rolling pin method introduced in our earlier paper [Mohseni Ahooyi et al., Ind. Eng. Chem. Res,

“…Seider and co-workers − applied Cuadras-Augé copula and multivariate Gaussian copula to describe a nonlinear relationship between variables and behavior-based factors involving human operators in chemical process risk analysis. Ahooyi et al proposed a moment-based method for estimating the probabilities of a rare event, and a copula-based model was illustrated as a comparison study. However, the copula theory is scarcely applied in the field of process monitoring at present, which is mainly due to its cumbersome and inefficient optimization procedure of traditional multivariate copulas, known as “curse of dimensionality”.…”

Section: Introductionmentioning

confidence: 99%

Vine Copula-Based Dependence Description for Multivariate Multimode Process Monitoring

Ren

Tian

2015

A novel vine copula-based dependence description (VCDD) process monitoring approach is proposed. The main contribution is to extract the complex dependence among process variables rather than perform dimensionality reduction or other decoupling processes. For a multimode chemical process, the C-vine copula model of each mode is initially created, in which a multivariate optimization problem is simplified as coping with a series of bivariate copulas listed in a sparse matrix. To measure the distance of the process data from each non-Gaussian mode, a generalized local probability (GLP) index is defined. Consequently, the generalized Bayesian inference-based probability (GBIP) index under a given control limit can be further calculated in real time via searching the density quantile table created offline. The validity and effectiveness of the proposed approach are illustrated using a numerical example and the Tennessee Eastman benchmark process. The results show that the proposed VCDD approach achieves good performance in both monitoring results and computation load.