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