The Generalized Linear Models

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02 Nov 2017

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4.1 Introduction

4.2 Regression

A model that has both deterministic as well as probabilistic components is called a regression model [12]. In deterministic model, with the help of one variable, value of other variable can be predicted and represented by y=f(x) which means value of y depends upon x, this is the reason why this model is known as deterministic model. The prediction generated by this model is a hypothetical, "what–if" statement and does not necessarily occur in past, future or even in the present but in real scenario, the chances of y being fully dependent upon x are very slim hence we use probabilistic model.

Probabilistic model [12] or probability model are used to predict the value of a variable on the basis of previous information and represented by Y~p(y) where Y is randomly generated from probability distribution p(y). As per the value of y this model makes "what-if" prediction. However the model does not say precisely what the value of y will be and also the prediction generated by the model is need not compulsorily occur in the past, future or even in the present. When large number of values for y occurs the probability model allows us to predict aggregate outcomes. Probability model does not exactly tell what the value of Y will be, hence for increasing the prediction accuracy, we combine the feature of both the models (deterministic and probability) that builds up regression model.

Like deterministic model [12], Regression model [13] also predicts the value of one variable based on other variable, represented by Y ~ p(y|x), where Y is generated at random from the probability distribution for known x. The regression model has proven to be a powerful tool that makes prediction about past, present or future events with the help of information about past or present events. For constructing a regression model, value of x and y is taken from the sample of object and comparing with other model regression model takes less time and/or resources for retrieving the information for computing the prediction.

4.3 Types of Regression

Regression is used to predict the value of variable, which helps in recommendation.

There are two basic types of regression which we consider are simple regression and multiple regression.

4.3.1 Simple Regression

Simple linear regression is used to represent the relationship between a scalar dependent variable  and one explanatory variables denoted. In case when there is only one explanatory variable is present called Simple Regression. For predicting the outcome simple regression uses one independent variable. The model in which data are modelled using linear predictor functions, and unknown model parameters are estimated from the data, called linear models.

4.3.2 Multiple Regression

With the help of one or more explanatory variables multiple regression determined the value of scalar dependent variable. For more than one explanatory variable, it is called multiple regression. Multiple regression uses two or more independent variables to predict the outcome.

The general form of Simple and Multiple type of Regression is:

Simple Linear Regression: Y = a + bX

Multiple Linear Regression: Y = a + b1X1 + b2X2 + b3X3 + ... + btXt

Where

Y= predicted variable or the variable whose value we want to find out. regression are used to predict the dependent variable, that always start with a set of known y values and use these values to build a regression model. The known value y also referred to as observed values.

X= Explanatory variable or the variable that we are using to predict Y. In regression dependent variable is the function of Explanatory variable.

a=the regression intercept. It represent the exact value for dependent variable, if all the dependent variables are zero.

b= the slope b is the regression coefficient which are computed by the regression tool. For each explanatory variable there is regression coefficient that represent the strength and type of relationship of explanatory variable has to the dependent variable.

Linear Regression:   An approach to modeling the relationship between a scalar dependent variable and one or more explanatory variables is called a Linear Regression. With the help of standard estimation techniques linear regression models make a number of assumptions about the predictor variables, the response variables and their relationship, where assumption may be erogeneity, Linearity, Constant variance, independence etc. Relaxation is provided to these assumptions by number of extension and in some cases these assumption can be eliminated entirely. Some method provides the relaxation in multiple assumptions at once or by combining different extension it can be achieved. Because of these extensions, estimating procedure become more complex, time consuming and required more data in order to get a accurate model.

4.3.1.3 General Linear Models

The general linear model considers the situation when the response variable Y is not a scalar but a vector. Conditional linearity of E(y|x) = Bx is still assumed, with a matrix B replacing the vector b of the classical linear regression model.

4.3.1.4 Generalized Linear Models

Generalized linear models are a framework for modelling a response variable y that is bounded or discrete. Generalised linear model is the extensions of fixed effects linear models and used where standard assumptions are violated. Poisson regression for count data, Logistic regression and probit regression for binary data, Multinomial logistic regression and multinomial probit regression for categorical data, ordered probit regression for ordinal data, are some of the example of Generalised linear models.

4.3.1.5 Hierarchical Linear Models

Hierarchical linear models (or multilevel regression) organizes the data into a hierarchy of regressions, for example where A is regressed on B, and B is regressed on C. It is often used where the data have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms are nested in schools, and schools are nested in some administrative grouping such as a school district. The response variable might be a measure of student achievement such as a test score, and different covariates would be collected at the classroom, school, and school district levels.

4.3.1.6 Heteroscedastic Models

Various models have been created that allow for heteroscedasticity, i.e. the errors for different response variables may have different variances. For example, weighted least squares is a method for estimating linear regression models when the response variables may have different error variances, possibly with correlated errors.

4.3.2 Non Linear Model

Non linear model[15] is one in which at least one of the parameter is occur nonlinearly, such model plays a very important role in understanding the complex interrelationship among variables. In non linear model at least one derivative with respect to parameter should involve that parameter. Example of non linear model is Logistic model, monomolecular model, and Richard model etc. In generalised form non linear model is represented as:

Y (t) =exp (at+bt2) or

Y (t) = at + exp (-bt)

4.3.2.1 Malthus Model

In this model rate of growth of population size is given by

After integration we get

N(t)=No exp(rt)

Where N(t) denote the population size at time t and r is intrinsic growth rate. No denote the population size at t=0.

4.3.2.2 Logistic Model

This type of model is symmetric and represented by differential equation

= rN(1-N/K)

After integration

N(t)=

Gompertz Model

This type of model have a sigmoid type behaviour and quite useful in biological work. This type of model is not symmetric unlike logistic model. This model is represented by differential equation

e(K/N)

After integration

N(t)=K exp[loge(No/K)exp(-rt)]

4.4 Application of Linear Regression

Linear regression is helps in prediction, and widely used in biological, behavioural and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines. Linear regression used in business as trend line which show changes in data over time. It tells whether a particular data set (say GDP, oil prices or stock prices) have increased or decreased over the period of time. A trend line could simply be drawn by eye through a set of data points, but more properly their position and slope is calculated using statistical techniques like linear regression. This is a simple technique, and does not require a control group, experimental design, or a sophisticated analysis technique. In economics linear regression is the predominant empirical tool. for example, it is used to predict consumption spending, fixed investment spending, inventory investment, purchases of a country's exports, spending on imports, the demand to hold liquid assets, labor demand, and labor supply. In finance capital asset pricing model uses linear regression as well as the concept of Beta for analyzing and quantifying the systematic risk of an investment. This comes directly from the Beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets. Linear regression finds application in a wide range of environmental science applications. The Environmental Effects Monitoring Program uses statistical analyses on fish and benthic surveys to measure the effects of pulp mill or metal mine effluent on the aquatic ecosystem.

[12] http://courses.ttu.edu/isqs5349-westfall/images/5349/deterministic_stochastic.htm

[13] http://www.psychstat.missouristate.edu/introbook/sbk16.htm.

[15]http://www.iasri.res.in/ebook/EB_SMAR/e-book_pdf%20files/Manual%20IV/1-Nonlinear%20Regression.pdf

http://en.wikipedia.org/wiki/Linear_regression



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