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lecture22 - 16.322 Stochastic Estimation and Control, Fall...

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 22 Last time: 1 1 1 22 1 ˆ 1 ˆ ˆ ˆ 11 N k kN xk N x z x σ =+ + = + The estimate of x based on N data points can then be made without reprocessing the first 1 N points. Their effect can be included simply by starting with a pseudo observation which is equal to the estimate based on the first 1 N points having a variance equal to the variance of the estimate based on 1 N . The same is true of the variance of the estimate based on N . 1 1 2 1 ˆˆ 1 N xx k σσ 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 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 ( ) 12 | , ,..., N fxzz z starting from the original a priori distribution f x 1 1 1 | (| ) | || (| , ) etc. fx fxfz x fxz fuf z ud u f xz f z x fuz f z ud u = = if the measurements are conditionally independent. Two disadvantages relative to the previous method:
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 More computation unless you know each conditional density is going to be normal Must provide () f x - the a priori distribution. This is both the advantage and disadvantage of this method. Other estimators include the effect of a priori information directly. Several estimators are based on the conditional probability distribution of x given the values of the observations . In this approach, we think of x as a random variable having some distribution. This troubles some people since we know x is in fact fixed at some value throughout all the experiment. However, the fact that we do not know what the value is is expressed in terms of a distribution of possible values for x . The extent of our a priori knowledge is reflected in the variance of the a priori distribution we assign. Having an a priori distribution for x , and the values of the observations, we can in principle – and often in fact – calculate the conditional distribution of x given the observations. This is in fact the a posteriori distribution, ( ) 1 | ,..., N f xz z . This distribution expresses the probability density for various values of x given the values of the observations and the a priori distribution. Having this distribution, one can define a number of reasonable estimates . One is the minimum variance estimate – that value ˆ x which minimizes the error variance.
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lecture22 - 16.322 Stochastic Estimation and Control, Fall...

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