Confidence Region (Confidence ellipse) plot for Bivariate Data

Part 1: Calculate Rotation Angle.
  1. 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.
  2. Calculate Variance Covariance matrix of sample data. 
    • Using COV() function of real stats.
  3. 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) 
  4. 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
  1. Calculate rotation matrix, Q, using angle in radian.
    • Diagonal elements are, Cos(θ)
    • Off diagonal elements are, Sin(θ) and -Sin(θ)
  2. Create an input column for angle with values 0 to 2π in increments of π/10.
    • First cell, angle=0.
    • 2nd cell, angel =PI()/10
    • 3rd onward, cumulative sum, get values from 0 to 6.28 (21 values)
  3. Calculate the lengths of the two axes of the ellipse.
    • Length of major axis, a, a=sqrt(λ1)*sqrt(Chi-square Critical value).
    • Length of minor axis, b, b=sqrt(λ2)*sqrt(Chi-square Critical value).
  4. Calculate the points on confidence ellipse using formula.
    • X-value is, Xmean+(Cos(angle)*dig.of Q*mj_axis)+sin(angle)*offdig.neg.value*mn_axis)
    • Y-value is, Ymean+(Cos(angle)*offdigQ*mj_axis)+sin(angle)*dig.value*mn_axis)
    • Using Matrix operations, to be demonstrated.
  5. Get pair of points on confidence ellipse.
Part 3: Plot the graphs of confidence ellipse and include individual data and its mean in plot.
Part 4: Determine if point is inside or outside ellipse using inverse of covariance matrix.

Reference

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