ENMA 420-520 Lecture 9 Slides

# ENMA 420-520 Lecture 9 Slides - Statistical Concepts for...

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Click to edit Master subtitle style 10/17/09 Statistical Concepts for Engineering Management ENMA 420 / 520 Lecture #9 Multiple Regression Analysis 11

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10/17/09 Multiple Regression Models More than one independent variable: Has a deterministic component and a random error Can contain higher-order functions of independent variables 22 l = + 0 + + 1 l 1 + + 2 l 2 + ⋯+ + + + + + +
10/17/09 Model Assumptions The quantities x1, x2, … xk can be measured without error when a value of y is observed and they do not involve any unknown parameters. The mean of the random error component is 0. The variance of the random error component is constant for all x. 33

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10/17/09 Matrix Algebra 44 Matric es ( ) ( ): arecomposedofelementsinrows horizontal andcolumns vertical + = + + 11 l 12 l 13 l 21 l 22 l 23 l 31 l 32 l 33 l Avector isa3 x1 ( ) column . matrix . Additional andsubtractionrequireidenticallysizedmatricesandisdoneelementbyelement
10/17/09 Matrix Algebra (Cont’d) 55 Multiplicationisaccomplish . edbyrowandcolumnelements Thenumber ofcolumnsinthefirst ( . . ): matrixmustequal thenumber ofrowsinthesecond eg itisNOT commutative + = + + 11 l 12 l 13 l 21 l 22 l 23 l 31 l 32 l 33 l l l 11 l 12 l 13 l 21 l 22 l 23 l 31 l 32 l 33 l = + + 11 l 11 + + 12 l 21 + + 13 l 31 l 11 l 12 + + 12 l 22 + + 13 l 32 l 11 l 13 + + 12 l 23 + + 13 l 33 l 21 l 11 + + 22 l 21 + + 23 l 31 l 21 l 12 + + 22 l 22 + + 23 l 32 l 21 l 13 + + 22 l 23 + + 23 l 33 l 31 l 11 + + 32 l 21 + + 33 l 31 l 31 l 12 + + 32 l 22 + + 33 l 32 l 31 l 13 + + 32 l 23 + + 33 l 33 l : Noticethattheij elementoftheproductwill alwaystaketheform ++ + + = + + 1 l 1l + + + 2 l 2l + + + 3 l 3l : Theidentitymatrixis += + 1 0 0 0 1 0 0 0 1 l so that ++ = + 1 0 0 0 1 0 0 0 1 l l l 11 l 12 l 13 l 21 l 22 l 23 l 31 l 32 l 33 l = + + 11 l 12 l 13 l 21 l 22 l 23 l 31 l 32 l 33 l = + ( Divisionisaccomplishedbymultiplicationoftheinverse seeAppendixfor techniquestoinvert ). matrices Thetransposeofthematri ( x denotedby T ) ; swapseachelementbycolumnandrow m ij becomesm ji

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10/17/09 Matrix Algebra (Cont’d) 66 : Sidenote Acrossproductisformedbytheproductofaskew - : symmetricmatrixformofavector + = + + + + +; + = + 0 −+ + + 0 −+ −+ + 0 l S o : that + ×+ = ++ = + 0 −+ + + 0 −+ −+ + 0 l l l l l l = + + − + + − + + − + +
10/17/09 Estimating Individual Parameters 77 : Inmatrixformthegeneral linear model is + = + + + w : here + = + ێێێێێ± ێێێێێ± ێێێێێ± l l 1 l 2 l 3 + + + + + + + + = + ێێێێێ± ێێێێێ± ێێێێێ± l 1 l 11 l 12 ⋯ + 1l 1 l 21 l 22 ⋮ + 2l 1 l 31 l 32 ⋮ + 3l ⋮ ⋮ ⋮ ⋱ ⋮ 1 l l 1 l l 2 ⋮ + ++ + + + + + + = + ێێێێێ± ێێێێێ± ێێێێێ± ێێێێێ± l l l 0 l l 1 l l 2 + + + + + + + + + = + ێێێێێ± ێێێێێ± ێێێێێ± l l 1 l 2 l 3 + + + + + + + Check : (n x 1) = (n x k) (k x 1) + (n x 1) = (n x 1) + (n x 1)

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10/17/09 Estimating Individual Parameters (Cont’d) 88 LeastSquaresMatrixEquation ++ + +++ + = + + + LeastSquaresSolution + + = + + + + + 1 l l l
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ENMA 420-520 Lecture 9 Slides - Statistical Concepts for...

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