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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 secondmoment 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 autocovariance and crosscovariance 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 12 and covariance matrix 12 given by m 12 = m 1 + 12 22 1 x 2 m 2 ( ) 12 = 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
 DonHill

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