Solutions to Homework Set #6 (Prepared by Lele Wang)

Transcription

1 Solutions to Homework Set #6 (Prepared by Lele Wang) Gaussian random vector Given a Gaussian random vector X N (µ, Σ), where µ ( 5 ) T and 0 Σ (a) Find the pdfs of i X, ii X + X 3, iii X + X + X 3, iv X 3 given (X, X ), and v (X, X 3 ) given X (b) What is P{X + X X 3 < 0}? Express your answer using the Q function (c) Find the joint pdf on AX, where A [ ] Solution: (a) i The marginal pdfs of a jointly Gaussian pdf are Gaussian Therefore X N (, ) ii Since X and X 3 are independent (σ 3 0), the variance of the sum is the sum of the variances Also the sum of two jointly Gaussian random variables is also Gaussian Therefore X + X 3 N (7, 3) iii Since X + X + X 3 is a linear transformation of a Gaussian random vector, X + X + X 3 [ ] X X, X 3 it is a Gaussian random vector with mean and variance µ [ ] 5 9 and σ [ ] Thus X + X + X 3 N (9, ) iv Since σ 3 0, X 3 and X are uncorrelated and hence independent since they are jointly Gaussian; similarly, since σ 3 0, X 3 and X are independent Therefore the conditional pdf of X 3 given (X, X ) is the same as the pdf of X 3, which is N (, 9)

2 v We use the general formula for the conditional Gaussian pdf: X {X x } N ( Σ Σ (x µ ) + µ, Σ Σ Σ Σ ) In the case of (X, X 3 ) X, Σ [ ], Σ [ ], Σ 0 [ ] Therefore the mean and variance of (X, X 3 ) given X x are [ ] [ ] [ µ (X,X 3 ) X x ] [ ] [ ] 5 x + 4 +, 0 [ ] [ ] [ ] [ ] [ ] 4 0 [ ] Σ (X,X 3 ) X Thus X and X 3 are conditionally independent given X The conditional densities are X {X x } N (x + 4, 3) and X 3 {X x} N (, 9) (b) Let X +X X 3 Similarly as part (a)iii, X +X X 3 is a linear transformation of a Gaussian random vector, X + X X 3 [ ] X X, X 3 it is a Gaussian random vector with mean and variance µ [ ] 5 5 and σ [ ] Thus X + X X 3 N (5, ), ie, N (5, ) Thus { } ( ) ( 5) (0 5) 5 P{ < 0} P < Q (c) In general, AX N (Aµ X, AΣ X A T ) For this problem, Thus N µ Aµ X Σ AΣ X A T [ ([ ] [ ]) 9 6, 6 [ ] 5 [ ] 9, ] [ ] 6 6

3 Gaussian Markov chain Let X,, and Z be jointly Gaussian random variables with zero mean and unit variance, ie, E(X) E( ) E(Z) 0 and E(X ) E( ) E(Z ) Let ρ X, denote the correlation coefficient between X and, and let ρ,z denote the correlation coefficient between and Z Suppose that X and Z are conditionally independent given (a) Find ρ X,Z in terms of ρ X, and ρ,z (b) Find the MMSE estimate of Z given (X, ) and the corresponding MSE Solution: (a) From the definition of ρ X,Z, we have where, ρ X,Z Cov(X, Z) σ X σ Z, Cov(X, Z) E(XZ) E(X)E(Z) E(XZ) 0 E(XZ), σ X E(X ) E(X) 0, σ E( ) E( ) 0 Thus, ρ X,Z E(XZ) Moreover, since X and Z are conditionally independent given, E(XZ) E(E(XZ )) E[E(X )E(Z )] Now E(X ) can be easily calculated from the bivariate Gaussian conditional density Similarly, we have Therefore, combining the above, E(X ) E(X) + ρ X, σ X σ ( E( )) ρ X, E(Z ) ρ,z ρ X,Z E(XZ) E[E(X )E(Z )] E(ρ X, ρ,z ) ρ X, ρ,z E( ) ρ X, ρ,z (b) X, and Z are jointly Gaussian random variables Thus, the minimum MSE estimate of Z given (X, ) is linear [ ] ρx, Σ (X, ) T, ρ X, [ ] [ ] E(XZ) ρx,z Σ (X, ) T Z, E( Z) Σ Z(X, ) T [ ρ X,Z ρ,z ] ρ,z 3

4 Therefore, [ ] Ẑ Σ Z(X, ) T Σ X (X, ) T [ ] [ ] [ ] ρ ρ X,Z ρ X, X,Z ρ X, [ [ ] ρ ρ X,Z ρ X,,Z ρ ρ X, X, ρ X, [ 0 ρ X, ρ,z + ρ,z ] [ X ], ][ ] X where the last equality follows from the result of (a) Thus, Ẑ [ ] [ ] X 0 ρ,z ρ,z The corresponding MSE is MSE Σ Z Σ Z(X, ) T Σ Σ (X, ) T (X, ) T Z [ ] [ ] [ ] ρ ρ X,Z ρ X, ρx,z,z ρ X, ρ,z [ [ ] [ ] ] ρ ρ X,Z ρ X, ρx,z,z ρ ρ X, X, ρ,z [ ] [ ] ρ 0 ρ X,Z,Z ρ,z ρ,z 3 Prediction of an autoregressive process Let X be a random vector with zero mean and covariance matrix α α α n α α Σ X α α α n for α < X, X,, X n are observed, find the best linear MSE estimate (predictor) of X n Compute its MSE Solution: We have α n α n Σ X α n α α n α 4

5 By defining [ X ] T X n, we have α n Σ, α n Therefore, and Σ X [ α n α ] T, Σ X [ α n α ], σ x ˆX n Σ X Σ [ α n α ] α n α n h T (where h T Σ X Σ ) [ 0 0 α ] (since h T Σ Σ X ) αx n ; MSE σ x Σ X Σ Σ X h T Σ X [ 0 0 α ] α α n 4 Noise cancellation A classical problem in statistical signal processing involves estimating a weak signal (eg, the heart beat of a fetus) in the presence of a strong interference (the heart beat of its mother) by making two observations; one with the weak signal present and one without (by placing one microphone on the mother s belly and another close to her heart) The observations can then be combined to estimate the weak signal by cancelling out the interference The following is a simple version of this application Let the weak signal X be a random variable with mean µ and variance P, and the observations be X + Z (Z being the strong interference), and Z + Z (Z is a measurement noise), where Z and Z are zero mean with variances N and N, respectively Assume that X, Z and Z are uncorrelated Find the best linear MSE estimate of X given and and its MSE Interprete the results Solution: This is a vector linear MSE problem Since Z and Z are zero mean, µ X µ µ and µ 0 We first normalize the random variables by subtracting off their means to get α 5

6 X X µ, and [ ] µ Now using the orthogonality principle we can find the best linear MSE estimate ˆX of X To do so we first find [ ] [ ] P + N N Σ P and Σ N N + N X 0 Thus, ˆX Σ T XΣ [ P 0 ] [ ] N + N N P (N + N ) + N N N P + N P [ ] (N + N ) N P (N + N ) + N N The best linear MSE estimate is ˆX ˆX + µ Thus, P ˆX ((N + N )( µ) N ) + µ P (N + N ) + N N (P ((N + N ) N )) + N N µ) P (N + N ) + N N The MSE can be calculated by MSE σx Σ T XΣ Σ X P P [ ] (N + N ) N P (N + N ) + N N P (N + N ) P P (N + N ) + N N P N N P (N + N ) + N N The equation for the MSE makes perfect sense First, note that if N and N are held constant but P goes to infinity, the MSE tends to N N N +N Next, note that if both N and N go to infinity, the MSE goes to σx, ie, the estimate becomes worthless Finally, note that if either N or N goes to 0, the MSE also goes to 0 This is because the estimator will then use the measurement with zero noise variance (that is, the one with no noise) and ignore the other measurement [ ] P 0 6

7 Solutions to Additional Exercises Markov chain Suppose X and X 3 are independent given X Show that f(x, x, x 3 ) f(x )f(x x )f(x 3 x ) f(x 3 )f(x x 3 )f(x x ) In other words, if X X X 3 forms a Markov Chain, then so does X 3 X X Solution: By definition of conditional independence, f(x, x 3 x ) f(x x )f(x 3 x ) Therefore, using the definition of conditional density, f(x 3 x, x ) f(x, x, x 3 ) f(x, x ) f(x, x 3 x )f(x ) f(x x )f(x ) f(x x )f(x 3 x ) f(x x ) f(x 3 x ) We are given that X and X 3 are independent given X Then f(x, x, x 3 ) f(x )f(x x )f(x 3 x, x ) f(x )f(x x )f(x 3 x ), In this case X X X 3 is said to form a Markov chain Similarly, f(x, x, x 3 ) f(x 3 )f(x x 3 )f(x x, x 3 ) f(x 3 )f(x x 3 )f(x x ), This shows that if X X X 3 is a Markov chain, then X 3 X X is also a Markov chain Proof of Property 4 In Lecture Notes #6 it was stated that conditionals of a Gaussian random vector are Gaussian In this problem you will prove that fact If [ ] is a zero-mean GRV then X { y} N ( Σ X X Σ y, σ X Σ XΣ Σ X) Justify each of the following steps of the proof (a) Let ˆX be the best MSE linear estimate of X given Then ˆX and X ˆX are individually zero-mean Gaussians Find their variances (b) ˆX and X ˆX are independent (c) Now write X ˆX + (X ˆX) If y then X Σ X Σ y + (X ˆX) (d) Now complete the proof Remark: This proof can be extended to vector X Solution: (a) Let ˆX be the best MSE linear estimate of X given In the MSE vector case section of Lecture Notes #6 it was shown that ˆX and X ˆX are individually zero-mean Gaussian random variables with variances Σ X Σ Σ X and σx Σ XΣ Σ X, respectively 7

8 (b) The random variables ˆX and X ˆX are jointly Gaussian since they are obtained by a linear transformation of the GRV [ X ] T By orthogonality, ˆX and X ˆX are uncorrelated, so they are also independent By the same reasoning, X ˆX and are independent (c) Now write X ˆX + (X ˆX) Then given y since X ˆX is independent of X Σ X Σ y + (X ˆX), (d) Thus X { y} is Gaussian with mean Σ X Σ y and variance σ X Σ XΣ Σ X 3 Additive nonwhite Gaussian noise channel Let i X + Z i for i,,, n be n observations of a signal X N(0, P ) The additive noise random variables Z, Z,, Z n are zero mean jointly Gaussian random variables that are independent of X and have correlation E(Z i Z j ) N i j for i, j n (a) Find the best MSE estimate of X given,,, n (b) Find the MSE of the estimate in part (a) Hint: the coefficients for the best estimate are of the form h T [ a b b b b a ] Solution: (a) The best estimate of X is of the form ˆX n h i i i We apply the orthogonality condition E(X j ) E( ˆX j ) for j n: P n h i E( i j ) i n h i E((X + Z i )(X + Z j )) i n h i (P + N i j ) i There are n equations with n unknowns: P P + N P + N/ P + N/ n P + N/ n P P + N/ P + N P + N/ n 3 P + N/ n P P + N/ n P + N/ n 3 P + N P + N/ P P + N/ n P + N/ n P + N/ P + N h h h n h n 8

9 By the hint, there are only degrees of freedom given, a and b Solving this equation using the first rows of the matrix, we obtain h h P 3N + (n + )P h n h n (b) The minimum mean square error is MSE E(X ˆX)X n P P h i i P ( i ) (n + )P 3N + (n + )P 3P N 3N + (n + )P 4 Sufficient statistic The bias of a coin is a random variable P U[0, ] Let Z, Z,, Z 0 be the outcomes of 0 coin flips Thus Z i B(P ) and {Z, Z,, Z 0 } are conditionally independent given P If X is the total number of heads, then X {P p} Binom(0, p) Assuming that the total number of heads is 9, show that is independent of the order of the outcomes Solution: f P Z,Z,,Z 0 (p z, z,, z 0 ) f P X (p 9) The tosses of the coin are conditionally independent given the bias, that is, p Z,,Z 0 P (z,, z 0 p) p Z P (z p)p Z P (z p) p Z0 P (z 0 p) Suppose that the order of the outcomes is nine heads followed by one tail Then p Z,,Z 0 P (H, H,, H, T p) p 9 ( p) We use Bayes rule to find the conditional pdf of P f P Z,,Z 0 (p H,, H, T ) p Z,Z,,Z 0 P (H,, H, T p) p Z,Z,,Z 0 (H, H,, H, T ) f P (p) This expression is zero when p < 0 or p > since the a priori pdf f P (p) is zero For 0 p : f P Z,,Z 0 (p H,, H, T ) p 9 ( p) 0 f Z,Z,,Z 0 P (H,, H, T p)f P (p) dp p 9 ( p) 0 p9 ( p) dp p9 ( p) /0 0p 9 ( p) 9

10 Note that the result is independent of the order of heads and tails This is due to the fact that the tosses are conditionally independent, and therefore the conditional pmf of (Z,, Z 0 ) given P is a function only of the number of heads 5 Gambling Let X, X, X 3, be independent random variables with the same mean µ > 0 and the same variance σ Find the limit of P{ n n i X i < µ/} as n Solution: By the weak law of large numbers, the sample mean n n i X i converges to the mean E(X) in probability, so P( S n µ < ɛ) as n But if S n µ > µ/ then P (S n < µ/) 0 0