GTU MCA III (SM): Regression (UNIT -5)

In statistics, regression analysis includes any techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables.

History

The earliest form of regression was the method of least squares (French: méthode des moindres carrés), which was published by Legendre in 1805, and by Gauss in 1809.Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun. Gauss published a further development of the theory of least squares in 1821, including a version of the Gauss–Markov theorem.
The term "regression" was coined by Francis Galton in the nineteenth century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average (a phenomenon also known as regression toward the mean). For Galton, regression had only this biological meaning, but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context.. In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925 . Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.
Regression methods continue to be an area of active research. In recent decades, new methods have been developed for robust regression, regression involving correlated responses such as time series and growth curves, regression in which the predictor or response variables are curves, images, graphs, or other complex data objects, regression methods accommodating various types of missing data, nonparametric regression, Bayesian methods for regression, regression in which the predictor variables are measured with error, regression with more predictor variables than observations, and causal inference with regression.

Regression models

Regression models involve the following variables:

The unknown parameters denoted as β; this may be a scalar or a vector.
The independent variables, X.
The dependent variable, Y.

In various fields of application, different terminologies are used in place of dependent and independent variables.
A regression model relates Y to a function of X and β.

$Y \approx f (\mathbf {X}, \boldsymbol{\beta} )$

The approximation is usually formalized as E(Y | X) = f(X, β). To carry out regression analysis, the form of the function f must be specified. Sometimes the form of this function is based on knowledge about the relationship between Y and X that does not rely on the data. If no such knowledge is available, a flexible or convenient form for f is chosen.
Assume now that the vector of unknown parameters β is of length k. In order to perform a regression analysis the user must provide information about the dependent variable Y:

If N data points of the form (Y,X) are observed, where N < k, most classical approaches to regression analysis cannot be performed: since the system of equations defining the regression model is underdetermined, there is not enough data to recover β.
If exactly N = k data points are observed, and the function f is linear, the equations Y = f(X, β) can be solved exactly rather than approximately. This reduces to solving a set of N equations with N unknowns (the elements of β), which has a unique solution as long as the X are linearly independent. If f is nonlinear, a solution may not exist, or many solutions may exist.
The most common situation is where N > k data points are observed. In this case, there is enough information in the data to estimate a unique value for β that best fits the data in some sense, and the regression model when applied to the data can be viewed as an overdetermined system in β

In the last case, the regression analysis provides the tools for:

Finding a solution for unknown parameters β that will, for example, minimize the distance between the measured and predicted values of the dependent variable Y (also known as method of least squares).
Under certain statistical assumptions, the regression analysis uses the surplus of information to provide statistical information about the unknown parameters β and predicted values of the dependent variable Y.

Necessary number of independent measurements

Consider a regression model which has three unknown parameters, β₀, β₁, and β₂. Suppose an experimenter performs 10 measurements all at exactly the same value of independent variable vector X (which contains the independent variables X₁, X₂, and X₃). In this case, regression analysis fails to give a unique set of estimated values for the three unknown parameters; the experimenter did not provide enough information. The best one can do is to estimate the average value and the standard deviation of the dependent variable Y. Similarly, measuring at two different values of X would give enough data for a regression with two unknowns, but not for three or more unknowns.

If the experimenter had performed measurements at three different values of the independent variable vector X, then regression analysis would provide a unique set of estimates for the three unknown parameters in β.
In the case of general linear regression, the above statement is equivalent to the requirement that matrix X^TX is invertible.

Statistical assumptions

When the number of measurements, N, is larger than the number of unknown parameters, k, and the measurement errors ε_i are normally distributed then the excess of information contained in (N - k) measurements is used to make statistical predictions about the unknown parameters. This excess of information is referred to as the degrees of freedom of the regression.

Underlying assumptions

Classical assumptions for regression analysis include:

The sample is representative of the population for the inference prediction.
The error is a random variable with a mean of zero conditional on the explanatory variables.
The independent variables are measured with no error. (Note: If this is not so, modeling may be done instead using errors-in-variables model techniques).
The predictors are linearly independent, i.e. it is not possible to express any predictor as a linear combination of the others. See Multicollinearity.
The errors are uncorrelated, that is, the variance-covariance matrix of the errors is diagonal and each non-zero element is the variance of the error.
The variance of the error is constant across observations (homoscedasticity). (Note: If not, weighted least squares or other methods might instead be used).

Linear regression

In linear regression, the model specification is that the dependent variable,

y i

is a linear combination of the parameters (but need not be linear in the independent variables). For example, in simple linear regression for modeling

n

data points there is one independent variable:

x i

, and two parameters,

β 0

and

β 1

straight line: $y_i=\beta_0 +\beta_1 x_i +\varepsilon_i,\quad i=1,\dots,n.\!$

In multiple linear regression, there are several independent variables or functions of independent variables. For example, adding a term in x_i² to the preceding regression gives:

parabola: $y_i=\beta_0 +\beta_1 x_i +\beta_2 x_i^2+\varepsilon_i,\ i=1,\dots,n.\!$

This is still linear regression; although the expression on the right hand side is quadratic in the independent variable

x i

, it is linear in the parameters

β 0

β 1

and

β 2 .

In both cases, $\varepsilon_i$ is an error term and the subscript

i

indexes a particular observation. Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

$\widehat{y_i} = \widehat{\beta}_0 + \widehat{\beta}_1 x_i.$

The residual, $e_i = y_i - \widehat{y}_i$ , is the difference between the value of the dependent variable predicted by the model, $\widehat{y_i}$ and the true value of the dependent variable

y i

. One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSE, also sometimes denoted RSS:

$SSE=\sum_{i=1}^N e_i^2. \,$

Minimization of this function results in a set of normal equations, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators, $\widehat{\beta}_0, \widehat{\beta}_1$ .

Illustration of linear regression on a data set.

In the case of simple regression, the formulas for the least squares estimates are

$\widehat{\beta_1}=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}\text{ and }\hat{\beta_0}=\bar{y}-\widehat{\beta_1}\bar{x}$

where $\bar{x}$ is the mean (average) of the

x

values and $\bar{y}$ is the mean of the

y

values. See simple linear regression for a derivation of these formulas and a numerical example. Under the assumption that the population error term has a constant variance, the estimate of that variance is given by:

$\hat{\sigma}^2_\varepsilon = \frac{SSE}{N-2}.\,$

This is called the mean square error (MSE) of the regression. The standard errors of the parameter estimates are given by

$\hat\sigma_{\beta_0}=\hat\sigma_{\varepsilon} \sqrt{\frac{1}{N} + \frac{\bar{x}^2}{\sum(x_i-\bar x)^2}}$

$\hat\sigma_{\beta_1}=\hat\sigma_{\varepsilon} \sqrt{\frac{1}{\sum(x_i-\bar x)^2}}.$

Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create confidence intervals and conduct hypothesis tests about the population parameters.