Question

Elaborate what are the least square lines and Regression?

Answer 1

A method of curve fitting is Least Square Method. Curves or lines fitted in this method is called least square lines.  Method of least Squares is a device for finding the equation of a specific type of curve, which best fits a given set of observations. The method depends upon the principle of least squares, which suggests that for the "best fitting" curve, the sum of the squares of differences between the observed and the corresponding estimated values should be the minimum possible.

Suppose we are given n pairs of observations (x₁, y1), (x₂,y₂),....................,(x_n,y_n) and it is required to fit a straight line to these data. The general equation of a straight line y=a+bx is taken, where a and b are constants. Any values for a and b would give a straight line,and once these values are obtained, an estimate of y can be had by substituting value of x. That is to say, the estimated value of y when x=x₁,x₂,.........,x_n would be more a+bx₁,a+bx₂,.................,a+bx_n respectively. In order that the equation y=a+bx gives a good representation of the relationship between x and y, it is desirable that the estimated values a+bx₁,a+bx₂,.................,a+bx_n are on the whole ,close enough to the corresponding observed values y₁,y₂,................................,y_n. For the best fitting straight line, therefore our problem is only to choose such values of a and b for the equation y=a+bx which will provide estimates of y as close as possible to the observed values. This can be done in different ways. However according to the principle of least squares, the best fitting equation is interpreted as that which minimises the sum of squares of differences

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The word regression is used to denote estimation or prediction of the average value of one variable for a specific value of the other variable. The estimations is done by means of suitable equations, derived on the basis of available bivariate data. Such an equation is known as a regression equation and its geometrical representation is called a regression curve.

In linear regression the relationship between the variables is assumed to be linear. The estimates of y (say, y') is obtained from an equation of the form  

.........................................................(1)

and the estimate of x (say,x') from another equation of the form

.........................................................(2)

equation (1) is called regression equation of y on x .

equation (2) is called regression equation of x on y.

b_yx is regression coeeficien of y an x.

b_xy is regression coeeficien of x an y.

. geometrical representaion of (1) and (2) are called regression lines.

where r is correlation coefficient between x and y

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