1. ASEN 5070: Statistical Orbit Determination I Fall 2015
Professor Brandon A. Jones
Lecture 33: Exam 2 and Probability Ellipsoids

2. Announcements Homework 10 due on Friday 11/20
Guest Lecture on Wednesday

3. Exam 2

4. Probability Ellipsoids

5. Example Bivariate Normal Distributions

6. Probability Ellipsoids
An “ellipsoid” is an n-dimensional ellipse or, more generally, a hyperellipsoid.
P, the variance-covariance matrix, represents the uncertainty in the state estimate.
The truth is that there is a relationship between the variances in each component, and visualization of such relationships provides information on quality of the state

7. Probability Ellipsoids
Generally the best to represent the probability ellipsoid using the covariance matrix’s principal axes.

8. The Probability Ellipsoid

9. Probability Ellipsoid
For a filter estimated Cartesian state, it is easier to use the principal axes of P to construct the probability ellipsoid
For this we use the eigenvector/value decomposition:

10. Probability Ellipsoid
The matrix U may be used to diagonalize P
Called a principal axis transformation
Any realization of the random vector x may be rotated into the principal axes via

11. The Probability Ellipsoid
This is really useful, because if P is oriented in Cartesian coordinates, we don’t really know what the size of a probability ellipsoid is.
???

12. The Probability Ellipsoid

13. The Probability Ellipsoid

14. The Probability Ellipsoid
The axis sizes of the lσ ellipsoid are then
The orientation of the ellipsoids is determined by the eigenvector matrix U.
We may interpret this matrix as an Euler angle sequence of rotations:

15. 2D Example

16. 2D Example

17. The Probability Ellipsoid
The Euler angles are defined by:

18. Implementation
Although the Euler-rotation interpretation provides some understanding of the probability ellipsoid, we do not have to compute the angles
The eigenvector matrix U defines the transformation from the frame used to define P and the principal axis frame

19. The Probability Ellipsoid
Consider the case:
We will plot this case in MATLAB using:

20. The Probability Ellipsoid
Views of Error Ellipsoid
view (0,0)*
azimuth =0, elevation =0
view down the negative y-axis
*view(azimuth, elevation), azimuth is a clockwise rotation about the positive z-axis

21. The Probability Ellipsoid
Views of Error Ellipsoid
view (90°,0)
view down the positive x-axis

22. The Probability Ellipsoid
Views of Error Ellipsoid
view (0,90°)
view down the positive z-axis

23. The Probability Ellipsoid
Views of Error Ellipsoid
view (-37.5°,0)
standard matlab view

24. Trivariate Probabilities
Probability of being within 1σ, 2σ, and 3σ differs based on dimension
Univariate:
1σ – 0.683
2σ – 0.954
3σ – 0.997
Trivariate
1σ – 0.200
2σ – 0.739
3σ – 0.971

25. Beware issues with eig() Functions!
Output of eig() (in almost any language) is not always consistent with the presented formulation
Always inspect the covariance matrix and compare it to the plotted ellipsoid
For example, does the standard deviation in the x- component in P match the plot

26. Why is the ellipsoid useful?