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

Confidence Region (Confidence ellipse) plot for Bivariate Data

Part 1: Calculate Rotation Angle. Calculate Chi-square critical value based on required confidence (eg. alpha of 0.05, is 95% confidence ellipse).  Using CHISQ.INV() function in Excel. 2 argument in function is degree of freedom, which is number of variables measured as part of sample data, for bivariate data dof=2. Calculate Variance Covariance matrix of sample data.  Using COV() function of real stats. Calculate eigen value of covariance matrix.  Using eVALUES() function of real stats or using formula considering VarCov matrix. sqrt(λ1) is the radius of the major axis (the longer radius)  sqrt(λ1) is the radius of the major axis (the longer radius)  Calculate angle from positive X-axis to the ellipse's major axis, in counterclockwise direction Using ATAN2() excel function. atan2()'s first parameter is y and second is x. atan2(covar, λ1-Var.X).  θ is the angle in radian. Part 2: Get the points on Confidence Ellipse Calculate rotation matrix, Q, using angle...
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