notes11 - Stat 430/Math 468 Notes #11 Chapter 5: Inference...

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Stat 430/Math 468 – Notes #11 Chapters 5, 8 Chapter 5: Inference about a Mean Vector (Continued) Simultaneous Confidence Interval for ' a μ (i.e. for 1 p ii i a μ = ) Result: Let be random sample from a normal population . The sample mean and sample variance-covariance matrix are, respectively, 1 ,..., n XX (, ) p N μΣ 1 12 ( ... ) n n =+ + + X X and 1 1 1 () n jj n j = =− SX X X ' X . 1. ( -confidence interval) Simultaneously for all 2 T 1 ( ,. .., )' p aa = a , the 100(1 )% α confidence interval for ' a μ is , (1 ) ' '( pn p pn F np n ± aSa aX ) . Example: For choices 1 1, 0, ,0 = a K , 2 0,1, ,0 = K a , K , and , we have 0,0, ,1 p = a K 11 ' = a μ , 22 ' = μ a , K , and ' pp = a μ . The - intervals for the above choices are 2 T 11 11 1, 1 ss xF n n αμ −− −≤ + 22 22 2, 2 n n + M M M ,2 , pp pp n p n p n n + 2. ( Bonferroni confidence interval ) The )% simultaneous confidence interval for m linear combinations of the mean vector is ' , ' ,..., ' m a μ a μ a μ 1 ' ' , 1,2,. .., 2 in ti mn ⎛⎞ ±= ⎜⎟ ⎝⎠ aS a m . Example: For the restricted set 1 = a K , 2 ,0 = K ) a , K , and , where m=p, the Bonferroni confidence intervals are ( 0,0, ,1 m = a K 11 11 1 nn xt p αα + n 22 22 21 2 p + n 1
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M M M 12 1 22 pp pp pn ss xt p np αα μ −− ⎛⎞ −≤ + ⎜⎟ ⎝⎠ n Chapter 8: Principal Component Analysis A principal component analysis (PCA) is concerned with explaining the variance- covariance structure of a set of variables (e.g. p) through a few linear combinations (e.g. k with k < p) of these variables. Objectives: 1. Data reduction 2. Interpretation Analyses of principle components (PCs) are often served as intermediate steps in a much larger investigations. For example, the PCs may be inputs to a multiple regression, or cluster analysis or used in factor analysis. We will first discuss the properties of population PCs and then see how to use sample PCs to summarize sample variation. Population Principal Components Algebraically, PCs are particular linear combinations of the p random variables 1 ,..., p X X . Geometrically, these linear combinations represent the selection of a new coordinate system obtained by rotating the original system with 1 ,..., p X X as the coordinate axes. PCs depend only on the variance-covariance matrix Σ (or correlation matrix ρ ). Suppose the random vector 1 ( ,..., )' p X X = X have variance-covariance matrix . Let be p unit-length constant vectors. Consider the linear combinations: .
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notes11 - Stat 430/Math 468 Notes #11 Chapter 5: Inference...

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