16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 22 ast time N、O3 ken+I Ok o keN,+I Or The estimate of x based on n data points can then be made without reprocessing the first N, points. Their effect can be included simply by starting with a pseudo observation which is equal to the estimate based on the first N points having a variance equal to the variance of the estimate based on N The same is true of the variance of the estimate based on 1=1+y1 A priori information about x can be included in exactly this way whether or not it was derived from previous measurements. Whatever the source of the prior information, it can be expressed as an a priori distribution f(x), or at least as an expected value and a variance. Take the expected value as a pseudo observation, do, and accumulate this data with the actual data using the standard formulae With the prior information included as a pseudo observation, the least squares estimate is formed just as if there were no prior information. The result, for normal variables at least, is identical to the estimators based on the conditional distribution of x Bayes rule can be used to form the distribution f(xl:, 2, Ev)starting from the original a priori distribution f(x) f(x) f(x|=1) f(x)f(-1|x) ∫f(ao(-l)lh f(x|=12=2)= f(x|=1)f(=2|x) f(u1=)f(=2|)dh if the measurements are conditionally independent. Two disadvantages relative to the previous method Page 1 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 22 Last time: 1 1 1 2 2 ˆ 1 2 2 ˆ 1 ˆ ˆ 1 1 N k x k k N N x k k N x z x σ σ σ σ = + = + + = + ∑ ∑ The estimate of x based on N data points can then be made without reprocessing the first N1 points. Their effect can be included simply by starting with a pseudo observation which is equal to the estimate based on the first N1 points having a variance equal to the variance of the estimate based on N1 . The same is true of the variance of the estimate based on N . 1 1 22 2 ˆ ˆ 1 11 1 N σ xx k σ σ k N= + = + ∑ A priori information about x can be included in exactly this way whether or not it was derived from previous measurements. Whatever the source of the prior information, it can be expressed as an a priori distribution f ( ) x , or at least as an expected value and a variance. Take the expected value as a pseudo observation, 2 σ 0 , and accumulate this data with the actual data using the standard formulae. With the prior information included as a pseudo observation, the least squares estimate is formed just as if there were no prior information. The result, for normal variables at least, is identical to the estimators based on the conditional distribution of x . Bayes’ rule can be used to form the distribution ( ) 1 2 | , ,..., N f xzz z starting from the original a priori distribution f ( ) x ( ) ( ) ( )( ) ( )( ) 1 1 1 1 2 1 2 1 2 ( ) () | (| ) () | | | (| , ) | | etc. f x fxf z x fxz f u f z u du f xz f z x fxzz f u z f z u du = = ∫ ∫ if the measurements are conditionally independent. Two disadvantages relative to the previous method: