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# lec29 - Lecture 29 Simple linear regression 29.1 Method of...

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Lecture 29 Simple linear regression. 29.1 Method of least squares. Suppose that we are given a sequence of observations ( X 1 , Y 1 ) , . . . , ( X n , Y n ) where each observation is a pair of numbers X , Y i . Suppose that we want to predict variable Y as a function of X because we believe that there is some underlying relationship between Y and X and, for example, Y can be approximated by a function of X, i.e. Y f ( X ). We will consider the simplest case when f ( x ) is a linear function of x : f ( x ) = β 0 + β 1 x. x x x x x X Y Figure 29.1: The least-squares line. Of course, we want to find the line that fits our data best and one can define the measure of the quality of the fit in many different ways. The most common approach 116

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LECTURE 29. SIMPLE LINEAR REGRESSION. 117 is to measure how Y i is approximated by β 0 + β 1 X i in terms of the squared difference ( Y i - ( β 0 + β 1 X i )) 2 which means that we measure the quality of approximation globally by the loss function L = n X i =1 ( Y i |{z} actual - ( β 0 + β 1 X i | {z } estimate )) 2 minimize over β 0 , β 1 and we want to minimize it over all choices of parameters β 0 , β 1 . The line that mini- mizes this loss is called the least-squares line . To find the critical points we write: ∂L ∂β 0 = - n X i =1 2( Y i - ( β 0 + β 1 X i )) = 0 ∂L ∂β 1 = - n X i =1 2( Y i - ( β 0 + β 1 X i )) X i
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