טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentShamsonov Mark
SubjectEstimating Uncertain Probability Distributions Using
Info-Gap Theory
DepartmentDepartment of Mechanical Engineering
Supervisors Professor Miriam Zacksenhouse
Professor Yakov Ben-Haim
Full Thesis textFull thesis text - English Version


Abstract

Probability density estimation has an important role in many science based applications, from reliability estimation, failure rate and damage models to econometrics, probabilistic forecasting, and climatic change modeling. The two main alternatives for probability density estimation are parametric and nonparametric estimation. The first (and more common one) assumes the data is drawn from a specific density model and the model parameters are then fitted to the data. In most practical applications, pdfs are usually estimated based on past records and available data, so the real shape of the pdf that generated the sample of observations is not known for sure. A thorough investigation of the real pdf shape is impossible, or costly and laborious in many cases. Nonparametric estimation doesn’t assume a pdf shape a priori, but this comes at a cost of less efficient estimation and other limitations such as necessity of selecting a smoothing parameter, or the requirement of a relatively large data set.

Info-gap decision theory provides a quantitative tool for decision-making under severe uncertainty. A system model, an info-gap model of uncertainty, a performance demand, and a robustness function are the main components constituting the theory. The robustness function provides a quantitative tool for robustness assessment. In addition, it enables one to explore the desirability of different performance demands.

 In this work an approach to estimate an uncertain probability density based on info-gap theory is provided. The approach is applied to the exponential and Gaussian distributions, and different potential cases of pdf shape uncertainty are examined. For the various cases of uncertainty the phenomenon of robustness curves crossing was examined (curve crossing describes the change in preference between alternatives). Effect of parameters such as sample size, distribution parameters size, and type of uncertainty in the distribution form on the location of robustness curves crossing was investigated. The results show that for exponential distribution, choice of the robust parameter depends only on the info-gap model, while for the normal distribution the behavior of the expected value is sample dependent, and the standard deviation depends only on the info-gap model. In addition, in all cases examined, smaller sample size entails smaller range where the Maximum Likelihood parameter is the most robust choice. The info-gap method and probability theory give a similar result: higher confidence in a parameter computed from a large sample and vice versa.