Covariance matrix as product of rotation matrix and scaling matrix/factor

 






Reference

  1. A geometric interpretation of the covariance matrix (visiondummy.com)

Comments

Post a Comment

Popular posts from this blog

Robust covariance matrix estimation in SPSS

   IN SPSS,  Produces the parameter estimates along with robust or heteroskedasticity-consistent (HC) standard errors. When  Parameter estimates with robust standard errors  is selected, the following methods are available for the robust covariance matrix estimation. HC0 Based on the original asymptotic or large sample robust, empirical, or "sandwich" estimator of the covariance matrix of the parameter estimates. The middle part of the sandwich contains squared OLS (ordinary least squares) or squared weighted WLS (weighted least squares) residuals. HC1 A finite-sample modification of HC0, multiplying it by N/(N-p), where N is the sample size and p is the number of non-redundant parameters in the model. HC2 A modification of HC0 that involves dividing the squared residual by 1-h, where h is the leverage for the case. HC3 A modification of HC0 that approximates a jackknife estimator. Squared residuals are divided by the square of 1-h. HC4 A modification of HC0 tha...
Read more

Clear Understanding on Mahalanobis Distance

Image
  Clear Understanding on Mahalanobis Distance In multivariate/ multicharacteristics data, a measure of divergence or distance between groups in terms of  multiple characteristics is required. Lets consider, you are interested in measuring the difference (distance) between groups G1 and G2 (each of p-dimensional). A common assumption is to take the p-dimensional random vector X  , from each group, as having same variation about its mean within either group. The difference between the groups can be considered in terms of difference between mean vectors of X, in each group relative to the common within-group variation  (using common (pooled) covariance matrix). The most often used measure for multiple characteristics data is, Mahalanobis distance (Mahalanobis  Δ , where  Δ  is Uppercase Delta). The square of Mahalanobis distance is given by,  Δ 2 =  (µ 1 -µ 2 ) T Σ -1 (µ 1 -µ 2 ) or          Δ 2 =  (µ 1 -µ 2 ) ′ ...
Read more

Vignettes for Matrix concepts, related operations

  Geometric Transformations, compute the matrix of a rotation transformation and visualize it. Wolfram|Alpha Examples: Geometric Transformations (wolframalpha.com) Reference Change of Basis – GeoGebra Eigenvalues of a Matrix Calculator - Online Eigen Values λ Finder (dcode.fr) Eigenvectors of a Matrix Calculator (with Eigenvalues) - Online (dcode.fr) Matrix Visualization | Sarah Greer (sygreer.com)   Matrix visualizer (utexas.edu)  gives rotation by angle. Eigenvalue Calculator: Wolfram|Alpha (wolframalpha.com) Determinant Calculator: Wolfram|Alpha (wolframalpha.com)
Read more