I assume you want to do a best-fit ellipse to the wind data. Here is a link to my notes on how to find the best-fit ellipse to a set of x,y points. The notes refer to center-of-presssure data , but it could be any set of x,y points.
I apologize for the poor quality of the file. This old file online is the only copy of my notes that I can locate at the moment. Key points from the notes:
The following procedure finds the “best fit” ellipse.
- Start with N pairs (x1, y1), (x2, y2), …, (xN, yN).
- Compute the means (xm, ym).
- Compute the covariance matrix Cov= (cxx , cxy ; cyx , cyy) where
- cxx= [ ∑(xi-xm)² ] / (N-1)
- cxy = cyx = [ ∑(xi-xm)(yi-ym) ] / (N-1)
- cyy = [ ∑(yi-ym)² ] / (N-1)
- Find the eigenvalues (λ1, λ2) of the covariance matrix (λ1 > λ2).
- Find the unit eigenvectors (v11 ; v21) and (v12 ; v22) of the covariance matrix.
- The best fit ellipse has its center at (xm, ym).
- The semimajor axis of the 95% ellipse has direction (v11; v21) and length sqrt(5.99 λ1).
- The semiminor axis of the 95% ellipse has direction (v12; v22) and length sqrt(5.99 λ2).
- The area of the 95% ellipse is A = 5.99 π sqrt(λ1 λ2)
- The angle between the semimajor axis and the y-axis is θ = sin-1(v11).
Plot your wind data:
% Example data for time series of u and v velocities
u = [-3.45 -2.35 0.85 -1.35 0.20 0.75 3.85 5.00 4.90 8.00 6.20 14.65 21.80 14.85 13.05];
v = [13.10 12.30 8.00 3.80 2.90 1.85 3.75 1.00 3.30 4.85 -3.90 9.90 11.40 16.55 20.30];
figure; plot(u,v,'k*');
grid on; xlabel('u (m/s)'); ylabel('v (m/s)'); hold on
Calculate best fit ellipse. I notice that you do not take the square root of the eigenvalues, in your calculations. You should. And you should scale them by a suitable factor, such as sqrt(5.99) to catch 95% of the points, or sqrt(4.61) to catch 90% of the points. (The scaling factors come from the chi squared distribution, and are based on the assumption that the noise in the data is Gaussian and independent for u and v.)
