10PDFEst - ensemble averaging over N statistically...

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MAE 591 RANDOM DATA Probability Density Function Estimate and Errors Consider N data values {} from a transformed record xn N n , , , ,. .., , = 1 2 3 x t () that is stationary with x = 0 . The probability density function of x t () can be estimated by ± () px N Nx x = with x s 02 . where x is a narrow interval centered at x , is the number of data values that fall within the range of N x x x ± 2 , and the standard deviation of the sample data is given by s N x n n N = = 1 1 2 1 The bias error in the estimate is given by ± () px ε b px xpx px [ ± () ] () " ( ) () 2 24 The normalized random error in the estimate is approximated by, when it is estimated by ensemble averaging over N
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Unformatted text preview: ensemble averaging over N statistically independent time history records, 1 r ˆ [ p( x )] N x p( x ) ε ≈ ∆ and, when it is computed by time-averaging a single sample record of length T with energy distributed uniformly over a bandwidth B , 1 2 r ˆ [ p( x )] BT x p( x ) ≈ ∆ Note #1: Decrease in ∆ x tends to decrease the bias error but increase the random error. Note #2: Normalized bias and random errors of are defined as ± φ [ ] ( ) ε φ φ φ b E = 1 ± − , ( ) [ ] ε φ φ φ r E E = − 1 2 1 2 ± [ ± ]...
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This note was uploaded on 11/21/2009 for the course ME . taught by Professor . during the Spring '09 term at Korea University.

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