NONLINEAR STATISTICAL MODELLING USING CLUSTERWISE LINEAR REGRESSION APPROACH

Abstract

Regression analysis is a statistical technique for analysing data. It can be used to build a model or obtain a relationship between variables. This regression model can then be used for several purposes, including prediction and estimation. As an example of a problem in which regression analysis may be helpful, suppose that an industrial engineer employed by a soft drink beverage bottler is analysing the product delivery and service operations for vending machines, He suspects that the time required by a route deliveryman to load and service a machine is related to the number of cases of product delivered. The engineer visits m randomly chosen retail outlets having vending machines, and the in-outlet delivery time (in minutes) and the volume of product delivered (in cases) are observed for each. This m observations can then be plotted and regression analysis can be done on these data. If the scatter plot shows that a straight line is sufficient to fit the model, then linear regression can be used to obtain the model and this will be discussed in Chapter 1. But if on the other hand, a straight line is not good enough then other means of regression has to be looked into. The objective of this project is to consider clusterwise linear regression in which no assumptions on the relationship between the variables is required. We will partition the data into different clusters and in each clusters a linear model is obtained. This model can then be used to predict delivery time for a specified number of cases of soft drinks to be delivered. This prediction may be helpful in planning delivery activities such as routing and scheduling, or in evaluating the productivity of delivery operations. This project consists of six chapters. A general review on linear and nonlinear regression will be given in Chapter 1. A brief discussion on the least squares method to obtain the estimates for the parameters in both the linear and nonlinear case will be discussed in this chapter. Regression analysis by using binary tree is introduced in Chapter 2. In this chapter we shall give a rule to partition a set of data to form an optimal binary tree and also a method to obtain a piecewise linear model for the binary tree. Chapters 3, 4 and 5 will be a discussion on clusterwise linear regression. As predictio11 is our main purpose, the most natural objective function will be one that minimizes the residual sum of squares and at the same time maximizes the volume available for prediction. The exact form of the objective function will be given in Chapter 3. In this chapter we also give the main algorithm - the exchange method, that is used in clusterwise linear regression. Chapter 4 will consist of all necessary concept needed to obtain the volume of the predictor's space. We shall present the FORTRAN subroutines that are needed in the main program in Chapter 5. Chapter 6 consists of some simulated examples. Appendix A consists of the main program for clusterwise linear regression while Appendix B and C give the computer output for the examples discuss in Chapter 6.