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lecture05 - 16.322 Stochastic Estimation and Control, Fall...

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Lecture 5 Last time: Characterizing groups of random variables Names for groups of random variables n S = X i i = 1 n n 2 S = ∑∑ X X j i i = 1 j = 1 Characterize by pairs to compute E XY ] = XY = dx xyf xy (, ) x y dy [ ∫∫ , −∞ −∞ which we define as the correlation . Often we do not know the complete distribution, but only simple statistics. The most common of the moments of higher ordered distribution functions is the covariance , )( )( µ xy = E ( X XY Y ) ⎤ = ( X Y ) = XY XY XY + XY = XY XY XY + XY = XY XY = (correlation) (product of means) Even more significant is the normalized covariance , or correlation coefficient : xy ρ = 2 xy 2 = σσ , 1 1 x y x y This correlation coefficient may be thought of as measuring the degree of linear dependence between the random variables: = 0 if the two are independent and 1 if one is a linear function of the other. First note = 0 if X and Y are independent. Calculate for Ya =+ b . xy If linearly related: Page 1 of 9
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x 2 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde a Y =+ bX XY = a X + b X 2 Y 2 2 2 = a + 2 abX + b X 2 aX + b X 2 X ( a + bX ) ρ = 2 2 2 2 2 2 2 ) σ ( a + 2 abX + b X 2 abX b X a bX 2 X 2 ) 2 = ( = b x = ± 1s g n ( b ) = 2 22 ( 2 b x X 2 ) x Degree of Linear Dependence At every observation, or trial or the experiment, we observe a pair x,y . We ask: how well can we approximate Y as a linear function of X ? a Y b X approx . 2 Choose a and b to minimize the mean squared error, ε , in the approximation. a = Y Y = + b Y approx . 2 2 2 2 = a + b X + + 2 abX 2 bXY 2 aY Y 2 2 Y 2 = a + b X 2 + + 2 abX 2 b XY 2 aY Page 2 of 9
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 2 ε = 2 a + 2 bX 2 Y = 0 a 2 2 = 2 bX + 2 aX 2 XY = 0 b 2 2 2 X = (2 b X 2 2 X Y ) b 2 X ) = 0 b a b = XY XY µ xy σ y = = ρ 2 X X 2 x 2 x aY xy y Y =− 2 X X x x Y xy X + xy X approx .
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lecture05 - 16.322 Stochastic Estimation and Control, Fall...

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