Stats 443.3 & 851.3

1. Stats 443.3 & 851.3 Linear Models

2. Instructor: W.H.Laverty Office: 235 McLean Hall Phone: 966-6096 Lectures: M W F 9:30am - 10:20am Geol 269 Lab 2:30pm – 3:30 pm Tuesday Evaluation: Assignments, Term tests - 40%Final Examination - 60%

3. The lectures will be given in Power Point

4. Course Outline

5. Introduction

11. Independent variables Dependent Variables Categorical Continuous Continuous & Categorical Categorical Multiway frequency Analysis (Log Linear Model) Discriminant Analysis Discriminant Analysis Continuous ANOVA (single dep var) MANOVA (Mult dep var) MULTIPLE REGRESSION (single dep variable) MULTIVARIATE MULTIPLE REGRESSION (multiple dependent variable) ANACOVA (single dep var) MANACOVA (Mult dep var) Continuous & Categorical ?? ?? ?? A chart illustrating Statistical Procedures

12. A Review of Linear Algebra With some Additions

13. Matrix Algebra Definition An n × m matrix, A, is a rectangular array of elements n = # of columns m = # of rows dimensions = n × m

14. Definition A vector, v,of dimension n is an n × 1matrix rectangular array of elements vectors will be column vectors (they may also be row vectors)

15. A vector, v,of dimension n can be thought a point in n dimensional space

16. v3 v2 v1

17. Matrix Operations Addition Let A = (aij) and B = (bij) denote two n × m matrices Then the sum, A + B, is the matrix The dimensions of A and B are required to be bothn × m.

18. Scalar Multiplication Let A = (aij) denote an n × m matrix and let c be any scalar. Then cA is the matrix

19. Addition for vectors v3 v2 v1

20. Scalar Multiplication for vectors v3 v2 v1

21. Matrix multiplication Let A = (aij) denote an n × m matrix and B = (bjl) denote an m × k matrix Then the n × k matrixC = (cil) where is called the product of A and B and is denoted by A∙B

22. In the case that A = (aij) is an n × m matrix and B = v = (vj) is an m × 1vector Then w =A∙v = (wi) where is an n × 1vector w3 v3 w2 v2 w1 v1

23. Definition An n × n identity matrix, I, is the square matrix Note: • AI = A • IA = A.

24. Definition (The inverse of an n × n matrix) Let Adenote the n × n matrix Let Bdenote an n × n matrix such that AB = BA = I, If the matrix B exists then A is called invertibleAlso B is called the inverse of A and is denoted by A-1

25. Note: Let A and B be two matrices whose inverse exists. Let C = AB. Then the inverse of the matrix C exists and C-1 = B-1A-1. Proof C[B-1A-1] = [AB][B-1A-1] = A[BB-1]A-1 = A[I]A-1 = AA-1=I

26. The Woodbury Theorem where the inverses

27. Proof: Let Then all we need to show is that H(A + BCD) = (A + BCD) H = I.

29. The Woodbury theorem can be used to find the inverse of some pattern matrices: Example: Find the inverse of the n × n matrix

30. where hence and

31. Thus Now using the Woodbury theorem

32. Thus

33. where

34. Note: for n = 2

35. Also

36. Now

37. and This verifies that we have calculated the inverse

38. Block Matrices Let the n × m matrix be partitioned into sub-matrices A11, A12, A21, A22, Similarly partition the m × k matrix

39. Product of Blocked Matrices Then

40. The Inverse of Blocked Matrices Let the n × n matrix be partitioned into sub-matrices A11, A12, A21, A22, Similarly partition the n × n matrix Suppose that B = A-1

41. Product of Blocked Matrices Then

42. Hence From (1) From (3)

43. Hence or using the Woodbury Theorem Similarly

44. From and similarly

45. Summarizing Let Suppose that A-1 = B then

46. Example Let Find A-1 = B

48. The transpose of a matrix Consider the n × m matrix, A then the m × n matrix, (also denoted by AT) is called the transpose of A

49. Symmetric Matrices • An n × n matrix, A, is said to be symmetric if Note:

50. The trace and the determinant of a square matrix Let A denote then n × n matrix Then