tailieunhanh - Lecture Signals, systems & inference – Lecture 15: Normal equations random processes

The following will be discussed in this chapter: Zestimates, LMMSE for multivariate case, Geometric picture, applying orthogonality gives the “normal equations”, estimating mean vector and covariance matrix from data, random variable, random process,. | Lecture Signals, systems & inference – Lecture 15: Normal equations random processes Normal equations Random processes , Spring 2018 Lec 15 1 Zillow (founded 2006) 2 © Zillow. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Zestimates 3 © Zillow. All rights reserved. This content is excluded from our Creative Commons license. For more information, see LMMSE for multivariate case min E[(Y { a0 + ⌃ L a X j=1 j j }) 2 ] a0 ,.,aL | {z } b` Y First min ) a0 = µY – ⌃j =1 aj µXj L a0 This ensures unbiasedness of the estimator. Now min E[(Ye 4 ⌃L e j=1 j j ] a X ) 2 a1 ,.,aL Geometric picture 5 Applying orthogonality gives the “normal equations” h⇣ ⌘ i E Ye - ⌃L e e j=1 aj Xj Xi = 0 2 32 3 2 3 C X1 X1 CX1 X2 ··· CX 1 X L a1 C X1 Y 6 C X2 X1 CX2 X2 ··· CX 2 X L 76 a2 7 6 CX2 Y 7 6 76 7 6 7 6 76 7=6 7 4 . . . . 54 . 5 4 . 5 C XL X1 CXL X 2 ··· CX L X L aL CXL Y (CXX ) a = cXY MMSE: CY2 - cY X (CXX )-1 cXY = CY2 - cY X .a 6 Estimating mean vector and covariance matrix from data Given N independent measurements: Xi , i = 1, · · · , N N X 1 Estimate of mean: bX µ = Xi N 1 N X b XX = 1 Estimate of covariance: C (Xi � µ bX )T bX )(Xi � µ N �1 1 7 Random variable ° Real line X(c) c 8 Random process ° Amplitude X(t; c) c t1 t 9 Signal ensemble for outcomes a,b,c,d; & determination of RXX(t1,t2) X(t) = Xa(t) t X(t) = Xb(t) t X(t) = Xc(t) t X(t) = Xd(t) t t1 10 t2 MIT OpenCourseWare Signals, Systems and Inference Spring 2018 For information about citing these materials or our Terms of Use, visit: . 11