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Unformatted text preview: 1412©2008 Raj JainCSE567MWashington University in St. LouisDerivation of Regression ParametersDerivation of Regression Parameters!The error in the ith observation is:!For a sample of n observations, the mean error is:!Setting mean error to zero, we obtain:!Substituting b0 in the error expression, we get:1413©2008 Raj JainCSE567MWashington University in St. LouisDerivation of Regression Parameters (Cont)Derivation of Regression Parameters (Cont)!The sum of squared errors SSE is:1414©2008 Raj JainCSE567MWashington University in St. LouisDerivation (Cont)Derivation (Cont)!Differentiating this equation with respect to b1and equating the result to zero:!That is,1415©2008 Raj JainCSE567MWashington University in St. LouisAllocation of VariationAllocation of Variation!Error variance without Regression = Variance of the responseand1416©2008 Raj JainCSE567MWashington University in St. LouisAllocation of Variation (Cont)Allocation of Variation (Cont)!The sum of squared errors without regression would be:!This is called total sum of squaresor (SST). It is a measure of y's variability and is called variationof y. SST can be computed as follows:!Where, SSY is the sum of squares of y(or Σy2). SS0 is the sum of squares of and is equal to .1417©2008 Raj JainCSE567MWashington University in St. LouisAllocation of Variation (Cont)Allocation of Variation (Cont)!The difference between SST and SSE is the sum of squares explained by the regression. It is called SSR:or!The fraction of the variation that is explained determines the goodness of the regression and is called the coefficient of determination, R2:1418©2008 Raj JainCSE567MWashington University in St. LouisAllocation of Variation (Cont)Allocation of Variation (Cont)!The higher the value of R2, the better the regression. R2=1 ⇒Perfect fit R2=0 ⇒No fit!Coefficient of Determination = {Correlation Coefficient (x,y)}2!Shortcut formula for SSE:1419©2008 Raj JainCSE567MWashington University in St. LouisExample 14.2Example 14.2!For the disk I/OCPU time data of Example 14.1:!The regression explains 97% of CPU time's variation. 1420©2008 Raj JainCSE567MWashington University in St. LouisStandard Deviation of ErrorsStandard Deviation of Errors!Since errors are obtained after calculating two regression parameters from the data, errors have n2degrees of freedom!SSE/(n2) is called mean squared errorsor (MSE). !Standard deviation of errors = square root of MSE. !SSY has ndegrees of freedom since it is obtained from nindependent observations without estimating any parameters.!SS0 has just one degree of freedom since it can be computed simply from !SST has n1degrees of freedom, since one parameter must be calculated from the data before SST can be computed. 1421©2008 Raj JainCSE567MWashington University in St. LouisStandard Deviation of Errors (Cont)Standard Deviation of Errors (Cont)!SSR, which is the difference between SST and SSE, has the remaining one degree of freedom....
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 Spring '13
 MRR
 Math, Linear Regression, Regression Analysis, Washington University, Raj Jain

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