APME_3 - Click to edit Master subtitle style Applied...

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Unformatted text preview: Click to edit Master subtitle style Applied Probability Methods for Engineers Slide Set 3 Click to edit Master subtitle style Chapter 12 Simple Linear Regression and Correlation Simple Linear Regression n Used when we believe some random variable is dependent on some other factor and is a linear function of this factor n We consider a random variable Yi that depends on the value of an independent variable xi n Assume that an observation yi is the sum of a linear function of xi and an error term i, i.e., yi = 0 + 1xi + i n Generally assumed that the error terms are normally distributed (and iid) with zero mean and variance 2 n Observations y1, , yn are therefore observations of independent random variables Yi ~ N(0 + 1xi, 2) n Expected value of Yi equals 0 + 1xi Simple Linear Regression Simple Linear Regression n y is the dependent variable and x is the explanatory variable n 0 and 1 are the slope and intercept parameters n 1 slope parameter determines how expected value of y changes as a function of x n 1 = 0 implies y and x are unrelated n 0, 1, and 2 typically estimated from a data set n Should first look at a graph of the data to make sure linearity is a reasonable assumption Example Data Set Factory electricity usage as a function of production Example Data Set Fitting a Regression Line n How do we take a data set and fit the best line to it? n Consider vertical deviations Fitting a Regression Line n Given the data set of xi, yi values, we want the 0 and 1 values that minimize the sum of squared deviations from the line n The vertical error is given by i = yi (0 + 1xi) n Why not minimize the sum of the errors? n Min ii = Min i{yi (0 + 1xi)} = -Max i{0 + 1xi} n Since we dont have sign restrictions on 0 and 1, optimal solution is 0 = 1 = & & (assuming xis nonnegative) n This is a meaningless result that essentially interprets negative errors as a good thing Fitting a Regression Line n We need to consider negative and positive errors as equally bad n Min ii2 = Min Q = i{yi (0 + 1xi)}2 n Setting these to zero gives ( 29 ( 29 1 1 2 n i i i Q y x = = -- + ( 29 ( 29 1 1 1 2 n i i i i Q x y x = = -- + 1 1 1 n n i i i i y n x = = = + 2 1 1 1 1 n n n i i i i i i i x y x x = = = = + Side Note n Is Q a convex function of 0 and 1? n Jointly convex in 0 and 1 if a 0, c 0, and ac b2 0 2 2 2 Q a n = = 2 2 2 1 1 2 n i i Q c x = = = 2 1 1 2 n i i Q b x = = = 2 1 1 2 n i i Q b x = = = ( 29 ( 29 2 2 2 2 1 1 1 4 4 4 n n n i i i i i i ac b n x x n x x = = =- =- =- Back to the Fitted Line n Solve the two equations simultaneously: 1 1 1 n n i i i i y n x = = = + 2 1 1 1 1 n n n i i i i i i i x y x x = = = = + 1 1 1 1 n n i i i i y x y x n n = =...
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APME_3 - Click to edit Master subtitle style Applied...

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