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class_09_26

# class_09_26 - Statistical Data Mining ORIE 474 Spring 2007...

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Statistical Data Mining ORIE 474 Spring 2007 Tatiyana Apanasovich 09/26/07 Review: SLR

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Simple Linear Regression Least Squares Estimation The case of simple linear regression considers a single regressor or predictor x and a dependent or response variable Y . The observation of Y at each level of x is a random variable with mean: We assume that each observation, Y , can be described by the model x x X Y E 1 0 ) ( β β + = =
Simple Linear Regression Least Squares Estimation Suppose that we have n pairs of observations( x 1 , y 1 ), ( x 2 , y 2 ), …, ( x n , y n ).

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Simple Linear Regression Least Squares Estimation Assumptions: 1. The model can be written as 2. X’s are nonstochastic (fixed, or measured without error) 3. are iid , therefore are independently distributed as 4. X and are uncorrelated. i i i x x X Y E 1 0 ) ( β β + = = 2 ) ( σ = i i x Y V ε ) , 0 ( ~ 2 σ ε N i i i i x Y ε β β + + = 1 0 i Y ) , ( 2 1 0 σ β β i x N +
Simple Linear Regression Least Squares Estimation The method of least squares is used to estimate the parameters, β 0 and β 1 by minimizing the sum of the squares of the vertical deviations in Figure 6-6. i i x y 1 0 ˆ ˆ ˆ β β + = i.e., = - n i i i y y 1 2 ) ˆ (

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Simple Linear Regression Least Squares Estimation t he n observations in the sample can be expressed as The sum of the squares of the deviations of the observations from the true regression line is
Simple Linear Regression Least Squares Estimation

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Simple Linear Regression Least Squares Estimation

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Simple Linear Regression Least Squares Estimation can be treated as an estimate of i e i ε 0 1 = = n i i e
Simple Linear Regression Least Squares Estimation Notation

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Simple Linear Regression Least Squares Estimation
Simple Linear Regression Least Squares Estimation Do not interpret the intercept as the “salt of conc. when roadway area is

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