OtherRegressionModels - CPE 619 Other Regression Models...

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CPE 619 Other Regression Models Aleksandar Milenković The LaCASA Laboratory Electrical and Computer Engineering Department The University of Alabama in Huntsville http://www.ece.uah.edu/~milenka http://www.ece.uah.edu/~lacasa
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2 Overview Multiple Linear Regression : More than one predictor variables Categorical Predictors : Predictor variables are categories such as CPU type, disk type, and so on Curvilinear Regression : Relationship is nonlinear Transformations : Errors are not normally distributed or the variance is not homogeneous Outliers Common Mistakes in Regression
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3 Multiple Linear Regression Models Given a sample of n observations with k predictors n kn k n n k k k k e x b x b x b b y e x b x b x b b y e x b x b x b b y + + + + + = + + + + + = + + + + + = ... ... ... ... 2 2 1 1 0 1 2 2 22 2 12 1 0 2 1 1 21 2 11 1 0 1
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4 Vector Notation In vector notation, we have: or All elements in the first column of X are 1. See Box 15.1 for regression formulas.
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5 Multiple Linear Regression Where, y – a column vector of n observed values X – an n row by (k+1) column matrix b – a column vector with (k+1) elements e – a column vector of n error terms Parameter estimation ) ( ) ( 1 y X X X b T T - = e Xb y + =
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6 Multiple Linear Regression (cont’d) Variations Coefficient of determination, multiple correlation SSE SST SSR y X b y y SSE SS SSY SST y n SS y SSY T T T n i i - = - = - = = = = } { 0 0 , 2 1 2 SST SSR R SST SSR R = = , 2
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7 Multiple Linear Regression (cont’d) Degrees of freedom Analysis of variance Regression is significant is MSR/MSE is greater than F [1- α ,k,n-k-1] ( 29 1 1 1 0 - - + = - = - + = - = k n k n n SSE SSR SS SSY SST 1 ; - - = = k n SSE MSE k SSR MSR
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8 Multiple Linear Regression (cont’d) Standard deviation Standard deviation of parameters Regression is significant is MSR/MSE is greater than F [1- α ,k,n-k-1] MSE s e = 1 ) ( , - = = X X C of term diagonal j is c where c s s T th jj jj e b j
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9 Multiple Linear Regression (cont’d) Prediction Standard deviation Correlations among predictors kp k p p p x b x b x b b y + + + + = ... ˆ 2 2 1 1 0 } ) ( 1 { 1 ˆ p T T p e y x X X x m s S p - + = = = = - - - = n i i n i i n i i i x x x n x x n x x x n x x R 1 2 2 2 2 1 2 1 2 1 2 1 1 2 1 , 2 1
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10 Model Assumptions Errors are independent and identically distributed normal variates with zero mean Errors have the same variance for all values of the predictors Errors are additive Xi’s and y are linearly related Xi’s are nonstochastic and are measured without error
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11 Example 15.1 Seven programs were monitored to observe their resource demands. In particular, the number of disk I/O's, memory size (in kBytes), and CPU time (in milliseconds) were observed
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12 Example 15.1 (cont’d) In this case:
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13 Example 15.1 (cont’d) The regression parameters are: The regression equation is:
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14 Example 15.1 (cont’d) From the table we see that SSE is:
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This note was uploaded on 12/13/2011 for the course CPE 619 taught by Professor Milenkovic during the Fall '09 term at University of Alabama - Huntsville.

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OtherRegressionModels - CPE 619 Other Regression Models...

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