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Unformatted text preview: Application Example 16 (Conditional second moment analysis with vectors) PREDICTION OF DAILY TEMPERATURES USING SEVERAL PAST OBSERVATIONS In Application Example 15, we have seen how one can use second-moment results for normally distributed variables to update uncertainty on a scalar quantity X 1 given a scalar observation X 2 . For many applications, one needs to extend those results to the case when the predicted quantity and/or the observed quantity is a vector. We first review these extended results and then make an application to the prediction of temperature at different time lags. Conditional Distribution Results for Jointly Normal Vectors Consider two random vectors X 1 and X 2 with joint normal distribution, mean value vectors m 1 and m 2 , and auto-covariance and cross-covariance matrices 11 , 22 and 12 = 21 T . These matrices are defined as ij = E[(X i- m i )(X j- m j ) T ], i, j = 1, 2, where the superscript T denotes transposition. This means that the vector X = X 1 X 2 has joint normal distribution X = X 1 X 2 N m 1 m 2 , 11 12 21 22 (1) Now suppose that X 2 is measured and found to be equal to x 2 . What is the conditional distribution of (X 1 |X 2 = x 2 )? One can show that this conditional distribution is also normal, with mean value vector m 1|2 and covariance matrix 1|2 given by m 1|2 = m 1 + 12 22 1 x 2 m 2 ( ) 1|2 = 11 12 22 1 12 T (2) 1 In the special case when X 1 and X 2 are scalar quantities, 11 =...
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This note was uploaded on 02/27/2008 for the course PTE 461 taught by Professor Donhill during the Fall '07 term at USC.
- Fall '07