Constructing A Covariance-Operator-Using-Matrix-Algebra_ Problem Set
Developing a Multi-Variate Pattern Covariance Matrix Utilizing Matrix Operations Suppose that now we have an n x okay multivariate pattern: (1) 1 2, ....,, kY y y y =   the place jy for j=1,…okay is an n x 1 vector containing the pattern values for variable j. Recall that pattern imply for variable j can then be computed as ( )1 1j jny y′= the place 1 is an n-vector of "ones". An nx1 vector with every component equal to the pattern common jy may be constructed as: ( )11 11j jnjy y y′= = . The deviation or error vector for variable j can then be constructed as: (2) ( ) ( )1 1ˆ ( ) ( 11 ) ( 11 )j jn nj j j j je y y y y I y M y′ ′= − = − = − = . Word that M is each idempotent and symmetric. Word additionally that the matrix 1 1' is a rank 1 sq. matrix with every component equal to unity or 1. The pattern variance may be computed as: (three) ( ) ( ) ( ) ( ) ( ) ( ) 22 21 1 1 1 , ,1 1 1 1 1 1 1 1 ˆ ˆ ˆ ˆ ( ) n n j i j j i j j j j jn n n n i i j j j jn y y e e e y M y y M y y y σ − − − − = = − ′ ′= − = = = ′ ′= = ∑ ∑  The pattern covariance is outlined and may be computed as: (four) ( ) ( )( ) ( ) ( ) ( ) ( ) 2 1 1 1 1 , , , , ,1 1 1 1 1 1 1 1 ˆ ˆ ˆ ˆ ˆ ( ) n n i j i t i j t j i t j t i j i jn n n n t i i j i jn y y y y e e e e y M y y M y y y σ − − − − = = − ′ ′= − − = = = ′ ′= = ∑ ∑  the place ( ) ( ) ( )1 1 11 1[ 1 1 ]nn nI M− −′= − = is a “covariance operator”. It's simply proven (displaying this might be one in all your issues in an issue set) that the kxk estimated covariance matrix 2 1 12 1 2 12 2 2 2 1 2 ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ okay okay okay okay okay σ σ σ σ σ σ σ σ σ      Σ =               may be instantly computed as: (5) ( ) ( )1 11ˆ ( [ 1 1 ])nnY Y Y I Y−′ ′ ′Σ = = − -research paper writing service