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

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

Exam 2

Probability Ellipsoids

Example Bivariate Normal Distributions

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

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

The Probability Ellipsoid

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:

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

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. ???

The Probability Ellipsoid

The Probability Ellipsoid

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:

2D Example

2D Example

The Probability Ellipsoid The Euler angles are defined by:

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

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

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

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

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

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

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

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

Why is the ellipsoid useful?