Reference no: EM132528456

Uncertainty modelling

Background
Probability distribution fitting or simply distribution fitting is the fitting of a probability distribution to a data sample of an unknown population, that includes: 1) Preliminary data analysis; 2) Model selection; 2) Estimation of model parameters; 3) Goodness of fit statistical tests.

First, an exploratory data analysis is conducted that consists of getting descriptive statistics (mean, standard deviation, skewness, kurtosis, etc.) and using graphical techniques (e.g. histograms, empirical cumulative distribution functions) to identify the adequate probability density functions (pdfs) to use to fit the model.

The information obtained from the sample is then used to make generalizations about the populations from which the samples were obtained. This includes the choice of probabilistic models and the estimation of their parameters. Goodness of fit tests indicate whether or not it is reasonable to assume that a random sample comes from a specific distribution or are used to discriminate the relative validity or goodness of the different distributions.

Accurate estimates of parameters using classical statistical approaches require large amounts of data. When the observed data is limited there are uncertainty in the model parameters, which can be represented by random variables. This uncertainty is usually known as statistical uncertainty (a special case of epistemic uncertainty). As the sample size increases, the mean values of parameters approximate their true values and their variances decrease. Modelling parameters as random variables has two main advantages. First, it guarantees that the statistical uncertainty is properly included in any subsequent probabilistic calculations. Second, it allows the probabilistic models to be updated as new data become available using the Bayesian probabilistic model framework.

Objective
To construct probability models from data and to assess the quality of the models using different methods available in the statistical literature.

To represent the statistical uncertainty on the model parameters and to update the prior model of the distribution parameters on the basis of new data using the Bayesian probabilistic model framework.

Description
Consider the sample in the excel file "sample.xls" of 100 values of the ultimate strength (σ_u) [in MPa] of a steel square plate with initial imperfections subjected to in-plane longitudinal displacement evaluated by non-linear FE analysis.

The work to be performed should include the following:

1) Descriptive statistics of the ultimate strength of the plate calculated by nonlinear FEM.

2) Probability distribution model fitting. Fit at least 2 probability distributions to the sample using the Least Squares Fit of Transformed Data method (Probability Plot) and the Maximum Likelihood Method.

3) Test the validity of the assumed distributions using the Chi-square and the Kolomogorov- Smirnov tests.

4) Compare the two probability distributions using the Akaike Information Criterion (AIC).

5) Assume that the ultimate strength (σ_u ) of the plate is described by a lognormal distribution and that the uncertainty in its mean value is described by a normal distribution (prior distribution) with mean estimated by the Maximum Likelihood Method and coefficient of variation of 0.2.

Assume that the following 5 new observations of the ultimate strength of the plate were obtained { 285, 289, 278, 272, 299}.

Update the prior probability model of the distribution mean value and calculate the 1% and 5% percentiles of the predictive prior and posterior distributions of the ultimate strength of the plate.

Attachment:- Uncertainty modelling.rar