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Regression Analysis

Regression analysis is a part of statistics and includes the techniques which are used for the modeling and analyses of many variables and focuses on the relationship between a dependent variable with one or more independent variables. By the help of regression analysis, the change in the typical values of the dependent variable is noted, when any one of the independent variable is varied while keeping the other independent variables constant. The regression analysis is often used for the estimation of conditional execution of any dependent variable when the independent variable is fixed. Sometimes the focus is on the quintile or the location parameter of the dependent variable’s conditional distribution. But whatever be the focus, in all the cases the target of estimation is a function of independent variable which is called regression function.

Regression analysis is often used to infer the casual relationship which occurs between a dependent and independent variable and its various forms.

Regression analysis often overlaps with machine learning and is used for prediction and for forecasting in various realms of engineering.

There are various techniques used for regression analysis. The basic three models of it are:-

1. The unknown parameters (can be both scalar or vector and are denoted by β)

2. The dependent variables, Y

3. The independent variable X

The terminologies which are used differ for dependent and independent variables in the various fields of application.

To a function of X and β, Y is related by the regression model as:- y ~= f(X, β)

This approximation is usually given a formal equation in the form of

E(Y | X) = f(X, β)

The form of the function F has to be specified for carrying out the regression analysis. The form of this function is many a times base on the knowledge of the relationship which occurs between X and Y, and which does not relies on the data. When no such knowledge is available, more convenient and flexible form of f is chosen.

Assumptions for regression analysis

1. The sample is a representative for the whole population and is used for inference prediction.

2. The error is random variable and has a mean of zero which is conditional on the variables which are explanatory.

3. The measurement of independent variables has no errors.

4. The variance- covariance matrix of the error is diagonal and each of the non zero elements are the variance of the errors which are uncorrelated.

5. The error variances are constant across observations. This is also called homscedasticity.

Interpolation and extrapolation

The regression models predict a value for the Y variable when the values of X variables are known. The prediction, which is made within this range of value in the dataset used for the process of model fitting, is called as interpolation. When the predictions are made outside this range of data, the process is called extrapolation. Regression assumptions are strongly relied upon while performing extrapolation. When the extrapolation goes outside the data, the chances of model failure increase as there are differences between the assumptions and the sample data (the true values). It is advised that while the extrapolations are performed the estimated value of the dependent variable is accompanied by a predication interval which represents the uncertainty. These intervals tend to expand rapidly when the values of the independent variables move outside the range which is covered by observe data.

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