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Axis-aligned confidence ellipses In general, the equation of an axis-aligned ellipse with a major axis of length and a minor axis of length , centered at the origin, is defined by the following equation: In our case, the length of the axes are defined by the standard deviations and of the data such that the equation of the error ellipse becomes: where s defines the scale of the ellipse and could be any arbitrary number (e.g. s=1). The question is now how to choose , such that the scale of the resulting ellipse represents a chosen confidence level (e.g. a 95% confidence level corresponds to s=5.991). The left hand side of equation ( 2 ) actually represents the sum of squares of independent normally distributed data samples, cov is zero. The sum of squared Gaussian data points is known to be distributed according to a so called Chi-Square distribution . A Chi-Square distribution is defined in terms of ‘degrees of fre...

Visualizing Ellipse rotation and associated parametric equations

June 05, 2023

Diagonal matrix Ellipse axes are parallel to coordinate axes (no rotation). Zeros in off-diagonals means zero correlation. Same values in diagonals Ellipse is rotated 45 degrees if correlation (off-diagonal) is positive regardless of its magnitude. Similarly -45 degrees if correlation is negative. Correlation matrix Correlation matrix has 1 in diagonals and values between -1 and 1 inclusive in off-diagonals. Ellipse size remains the same (always touches square of side 2 units). Whether the correlation is positive or negative can be observed by the orientation of the ellipse. The amount of correlation can be interpreted by how thin the ellipse is. How It Works Below is a list of parametric equations starting from that of a general ellipse and modifying it step by step into a prediction ellipse, showing how different parts contribute at each step. Reference How to Draw Ellipse of Covariance Matrix (cookierobotics.com)

Confidence Region (Confidence ellipse) plot for Bivariate Data

June 05, 2023

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...

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