16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Assuming zero mean, which is often the case f(x)= exp For zero mean variables: contours of constant probability density are given by: M-x=c X Not expressed in principal coordinates if the Xi are correlated Need to know the rudimentary properties of eigenvalues and eigenvectors Mis symmetric and full rank. Mv=2y VV=0= 0,i≠ This probability density function can be better visualized in terms of its principal coordinates. These coordinates are defined by the directions of the eigenvectors of the covariance matrix. The appropriate transformation is Vx Thus yi is the component of x in the direction y (In terms of the new variable y, the contours of constant probability density are) 9/30/2004955AM Page 8 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 8 of 10 Assuming zero mean, which is often the case: ( ) 1 2 1 1 ( ) exp 2 2 n f x xM x π M ⎡ ⎤ Τ − = −⎢ ⎥ ⎣ ⎦ For zero mean variables: contours of constant probability density are given by: T 1 2 x Mx c − = Not expressed in principal coordinates if the Xi are correlated. Need to know the rudimentary properties of eigenvalues and eigenvectors. M is symmetric and full rank. 1, 0, i ii T i j ij Mv v i j v v i j λ δ = ⎧ = = = ⎨ ⎩ ≠ This probability density function can be better visualized in terms of its principal coordinates. These coordinates are defined by the directions of the eigenvectors of the covariance matrix. The appropriate transformation is 1 T T n y Vx v V v = ⎡ ⎤ ← → ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ← → M Thus yi is the component of x in the direction i v . (In terms of the new variable y , the contours of constant probability density are)