ENMA 420-520 Lecture 9 Slides

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

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Click to edit Master subtitle style 10/17/09 Statistical Concepts for Engineering Management ENMA 420 / 520 Lecture #9 Multiple Regression Analysis 11
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 + ⋯+ + + + + + +
Background image of page 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 = + + − + + − + + − + +
Background image of page 6
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)
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
10/17/09 Estimating Individual Parameters (Cont’d) 88 LeastSquaresMatrixEquation ++ + +++ + = + + + LeastSquaresSolution + + = + + + + + 1 l l l
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/17/2009 for the course MET 387 taught by Professor Dean during the Spring '09 term at Old Dominion.

Page1 / 59

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

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online