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ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 14: Probability and Statistics (Part 4) University of Colorado Boulder Lecture Quiz Due by 5pm Homework #5 Due 10/2 Exam 1 – Oct. 9 ◦ More details to come University of Colorado Boulder 2 Variance-Covariance Matrix Multivariate Gaussian Distribution Central Limit Theorem Bayes’ Theorem Statistical Least Squares University of Colorado Boulder 3 Variance-Covariance Matrix University of Colorado Boulder 4 Covariance provides a measure of correlation between variables University of Colorado Boulder 5 Indicates the degree of linear correlations between variables University of Colorado Boulder 6 When we have a linear relationship between random variables, then we have an extreme value of the correlation coefficient, and vice versa In other words, See pages 458-459 of the textbook University of Colorado Boulder 7 University of Colorado Boulder 8 Symmetric Is it non-singular? University of Colorado Boulder 9 Multivariate Normal Distribution University of Colorado Boulder 10 Univariate: Multivariate: University of Colorado Boulder 11 University of Colorado Boulder 12 It may be shown that: Although the above assumes a bivariate normal distribution, the idea extends to higher dimensions with minor changes University of Colorado Boulder 13 The conditional density function is also a normal distribution (anyone seeing a trend here?) What happens if ρ = 0? What happens if ρ = ±1? University of Colorado Boulder 14 The conditional PDF from the previous slide is a special case of the general conditional PDF University of Colorado Boulder 15 University of Colorado Boulder 16 Central Limit Theorem University of Colorado Boulder 17 University of Colorado Boulder 18 University of Colorado Boulder 19 The CLT implies that we can treat ε as a Gaussian random variable What about when we have a small number of observations from different sensors? University of Colorado Boulder 20 Bayes’ Theorem University of Colorado Boulder 21 University of Colorado Boulder 22 Allows for updating a hypothesis’ probability when given additional information ◦ Known as Bayesian Inference Modern estimation research is rooted in Bayesian Inference! University of Colorado Boulder 23 University of Colorado Boulder 24 University of Colorado Boulder 25
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