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Stats 443.3 & 851.3 Summary
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The Woodbury Theorem where the inverses
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Block Matrices Let the n × m matrix be partitioned into sub-matrices A 11, A 12, A 21, A 22, Similarly partition the m × k matrix
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Product of Blocked Matrices Then
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The Inverse of Blocked Matrices Let the n × n matrix be partitioned into sub-matrices A 11, A 12, A 21, A 22, Similarly partition the n × n matrix Suppose that B = A -1
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Summarizing Let Suppose that A -1 = B then
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Symmetric Matrices An n × n matrix, A, is said to be symmetric if Note:
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The trace and the determinant of a square matrix Let A denote then n × n matrix Then
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also where
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Some properties
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Special Types of Matrices 1.Orthogonal matrices –A matrix is orthogonal if P ˊ P = PP ˊ = I –In this cases P -1 =P ˊ. –Also the rows (columns) of P have length 1 and are orthogonal to each other
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Special Types of Matrices (continued) 2.Positive definite matrices –A symmetric matrix, A, is called positive definite if: –A symmetric matrix, A, is called positive semi definite if:
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Theorem The matrix A is positive definite if
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Special Types of Matrices (continued) 3.Idempotent matrices –A symmetric matrix, E, is called idempotent if: –Idempotent matrices project vectors onto a linear subspace
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Eigenvectors, Eigenvalues of a matrix
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Definition Let A be an n × n matrix Let then is called an eigenvalue of A and and is called an eigenvector of A and
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Note:
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= polynomial of degree n in. Hence there are n possible eigenvalues 1, …, n
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Thereom If the matrix A is symmetric with distinct eigenvalues, 1, …, n, with corresponding eigenvectors Assume
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The Generalized Inverse of a matrix
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Definition B (denoted by A - ) is called the generalized inverse (Moore – Penrose inverse) of A if 1. ABA = A 2. BAB = B 3. (AB)' = AB 4. (BA)' = BA Note: A - is unique
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Hence B 1 = B 1 AB 1 = B 1 AB 2 AB 1 = B 1 (AB 2 ) ' (AB 1 ) ' = B 1 B 2 ' A ' B 1 ' A ' = B 1 B 2 ' A ' = B 1 AB 2 = B 1 AB 2 AB 2 = (B 1 A)(B 2 A)B 2 = (B 1 A) ' (B 2 A) ' B 2 = A ' B 1 ' A ' B 2 ' B 2 = A ' B 2 ' B 2 = (B 2 A) ' B 2 = B 2 AB 2 = B 2 The general solution of a system of Equations The general solution where is arbitrary
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Let C be a p×q matrix of rank k < min(p,q), then C = AB where A is a p×k matrix of rank k and B is a k×q matrix of rank k
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The General Linear Model
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Geometrical interpretation of the General Linear Model
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Estimation The General Linear Model
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the Normal Equations
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The Normal Equations
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Solution to the normal equations
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Estimate of 2
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Properties of The Maximum Likelihood Estimates Unbiasedness, Minimum Variance
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s 2 is an unbiased estimator of 2. Unbiasedness
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Distributional Properties Least square Estimates (Maximum Likelidood estimates)
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The General Linear Model and The Estimates
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Theorem
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The General Linear Model with an intercept
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The matrix formulation (intercept included) Then the model becomes Thus to include an intercept add an extra column of 1’s in the design matrix X and include the intercept in the parameter vector
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The Gauss-Markov Theorem An important result in the theory of Linear models Proves optimality of Least squares estimates in a more general setting
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The Gauss-Markov Theorem Assume Consider the least squares estimate of, an unbiased linear estimator of and Let denote any other unbiased linear estimator of
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Hypothesis testing for the GLM The General Linear Hypothesis
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Testing the General Linear Hypotheses The General Linear Hypothesis H 0 :h 11 1 + h 12 2 + h 13 3 +... + h 1p p = h 1 h 21 1 + h 22 2 + h 23 3 +... + h 2p p = h 2... h q1 1 + h q2 2 + h q3 3 +... + h qp p = h q where h 11 h 12, h 13,..., h qp and h 1 h 2, h 3,..., h q are known coefficients. In matrix notation
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Testing
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An Alternative form of the F statistic
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Confidence intervals, Prediction intervals, Confidence Regions General Linear Model
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One at a time (1 – )100 % confidence interval for (1 – )100 % confidence interval for 2 and .
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Multiple Confidence Intervals associated with the test Theorem: Let H be a q × p matrix of rank q. then form a set of (1 – )100 % simultaneous confidence interval for
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