This material is compiled for the course Empirical Modelling. Sections marked with a star (∗) are not central in the courses. The main source of inspiration when writing this text has been Chapter 4 in the book ”System Identification” by Söderström and Stoica (Prentice Hall, 1989) which also may be consulted for a more thorough treatment of the material presented here. The book is available for...

Linear regression is probably the most popular model for predicting a RV Y ∈ R based on multiple RVs X1, . . . , Xd ∈ R. It predicts a numeric variable using a linear combination of variables ∑ θiXi where the combination coefficients θi are determined by minimizing the sum of squared prediction error on the training set. We use below the convention that the first variable is always one i.e., X1...

Probablistic Model: We start with the assumption that prior to starting a sequence of experiments we have a family of random variables with means that vary linearly with respect to some deterministic independent variable. That is, there exist an intercept β0 and a slope β1 such that for each value of the independent variable x, we have a random variable Y with mean β0 + β1x. We are then given p...

fuzzy linear regression models are used to obtain an appropriate linear relation between a dependent variable and several independent variables in a fuzzy environment. several methods for evaluating fuzzy coefficients in linear regression models have been proposed. the first attempts at estimating the parameters of a fuzzy regression model used mathematical programming methods. in this the...

introduction: the development of an appropriate model for the quality control of an industrial wastewater treatment system can save the time as well as the cost. this study was performed to determine an appropriate model in order to predict the cod and tkn parameters by bod5 and nh4+ in the meybod industrial estate waste water treatment plant (wwtp). methods: this descriptive – analytical study...