Types of Regression Model: Linear Regression, Local Regression, and Logistic Regression (Especially for GATE-Geospatial 2022)

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1. Linear Regression Model

In linear regression, the model specification is that the dependent variable, is a linear combination of the parametres. For example, in simple linear regression for modeling N data points there is one independent variable: and two parametres, and :

A multiple linear regression, there are several independent variables or functions of independent variables. For example, adding a term in to the preceding regression gives:

This is still linear regression; although the expression on the right hand side is quadratic in the independent , it is linear in the parameters and .

Linear regression is a simple approach to supervised learning. It assumes that the dependence of on is linear.

True regression functions are never linear!

Figure Shows Linear Regression Function

Although it may seem overly simplistic, linear regression is extremely useful both conceptually and practically.

2. Local Regression Models

Local regression is used to model a relation between a predictor variable and response variable. To keep things simple we will consider the fixed design model. We assume a model of the form

Where f (x) is an unknown function and is an error term, representing random errors in the observations or variability form sources not included in the

We assume the errors are IID with mean 0 and finite variance var

We make no global assumptions about the function f but assume that locally it can be well approximated with a member of a simple class of parametric function, e. g. a constant or straight line. Taylor՚s theorem says that any continuous function can be approximated with polynomial.

3. Logistic Regression Model

Logistic regression, also called a logit model, is used to model dichotomous outcome variables. In the logit model the log odds of the outcome is modeled as a linear combination of the predictor variables.

We start by introducing an example that will be used to illustrate the analysis of binary data. We then discuss the stochastic structure of the data in terms of the Bernoulli and binomial distributions, and the systematic structure in terms of the logit transformation. The result is a generalized linear model with binomial response and link logit.

Logistic regression is a statistical method for analyzing a dataset in which there are one or more independent variables that determine an outcome. The outcome is measured with a dichotomous variable (in which there are only two possible outcomes) .

The goal of logistic regression is to find the best fitting model to describe the relationship between the dichotomous characteristic of interest and a set of independent variables. Logistic regression generates the coefficients of a formula to predict a logit transformation of the probability of presence of the characteristic of interest:

where p is the probability of presence of the characteristic of interest.

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