Uncorrelatedness and Independence

Transcription

1 Uncorrelatedness and Independence Uncorrelatedness:Two r.v. x and y are uncorrelated if C xy = E[(x m x )(y m y ) T ] = 0 or equivalently R xy = E[xy T ] = E[x]E[y T ] = m x m T y White random vector:this is defined to be a r.v. with zero mean and unit covariance (correlation) matrix. m x = 0,R x = C x = I Example: What will be the mean and covariance of the white r.v. under the orthogonal transform? Let T denote an orthogonal matrix (i.e. T T T = TT T = I).Such a matrix defines an orthogonal transform (i.e. a rotation of an coordinate system - preserves distances in the space). Thus, define y = Tx. 1

2 Hence, m y =... = 0 and C y =... = I Therefore the orthogonal transformation preserves whiteness. Example: Calculate the R x for x = As + n, where s is a random signal with correlation matrix R s and the noise vector n has zero mean and is uncorrelated with the signal. Independence:Two random variables x, y are statistically independent if p x,y (x,y) = p x (x)p y (y) i.e. if the joint pdf of (x,y) factors out into the product of their marginal probability distributions p x and p y. 2

3 From the definition of statistical independence it follows that E[g(x)h(y)] = E[g(x)]E[h(y)] where g,h are any absolutely integrable functions. Similarly for random vectors the definition of statistical independence reads and the property reads Properties: p x,y (x,y) = p x (x)p y (y) E[g(x)h(y)] = E[g(x)]E[h(y)] the statistical independence of two r.v. s implies their uncorrelatedness Independence is a stronger property than uncorrelatedness. Only for gaussian variables uncorrelatedness and independence coincide. 3

4 Example: Consider the discrete random vector from our example X \ Y /18 1/9 1/6 6/18 1 1/9 1/18 1/9 5/18 2 1/6 1/6 1/18 7/18 6/18 6/18 6/18 Are X,Y independent? To check, Let s construct a table which entries are products of the corresponding marginal probabilities of X, Y. X \ Y /18*6/18 6/18*6/18 6/18*6/18 1 6/18*5/18 6/18*5/18 6/18*5/18 2 6/18*7/18 6/18*7/18 6/18*7/18 Hence, X, Y are not independent. Are they uncorrelated? 4

5 E(XY) = XYp(X,Y) X X = 0 0 1/ / / / / / / / /18 = 15/18 However, E(X)E(Y) = 19/18 hence X, Y are correlated. Central limit theorem (CLT) Classical probability is concerned with random variables and sequences of independent identically distributed (iid) r.v. s. A very important case - sequence of partial sums of iid r.v. s x k = k z i i=1 5

6 Consider the normalised variables y k = x k m xk σ xk where m xk and σ xk are the mean and variance of X k. Central limit theorem asserts that the distribution of y k converges to a normal distribution with k. Analogous formulation of the CLT holds in the case of random vectors. CLT justifies use of the gaussian variables for modelling random phenomena in practice sums of a relatively small number of r.v. s will show gaussianity even if individual components are not necessarily identical. 6

7 Conditional probability Conditional density:consider random vectors x,y with marginal pdf s p x (x) and p y (y), respectively and a joint pdf p x,y (x,y). Conditional density of x given y is defined as p x y (x y) = p x,y(x,y) p y (y) Similarly, conditional density of y given x is defined as p y x (y x) = p x,y(x,y) p x (x) The conditional probability distributions allow to address questions like, what is the probability density of a r.v. x given that a random vector y has a fixed value y 0. For statistically independent r.v. s the conditional densities equal the respective marginal densities. 7

8 Example: Consider the bivariate discrete random vector X \ Y /18 1/9 1/6 6/18 1 1/9 1/18 1/9 5/18 2 1/6 1/6 1/18 7/18 6/18 6/18 6/18 The conditional probability function of Y given X = 1 is Y X= /9 5/18 1/18 5/18 1/9 5/18 Bayes Rule:From definitions of the conditional densities we can obtain the following alternative formulas for calculating the joint pdf p x,y (x,y) = p y x (y x)p x (x) = p x y (x y)p y (y) From the above follows so called Bayes rule for calculating the conditional density of y given x: 8

9 p y x (y x) = p x y (x y)p y(y) p x (x) where the denominator can be calculated by integration p x (x) = p x y (x η)p y(η)dη Bayes rule allows to compute the posterior density p y x (y x) given the observed vector x and either knowing or assuming the prior distribution p y (y). Conditional expectations E[g(x, y) y] = g(ξ,y)p x y (ξ y)dξ The conditional expectation is a random variable - it depends on the r.v. y. The following relationship holds E[g(x, y)] = E[E[g(x, y) y]] 9

10 The family of multivariate gaussian densities p x (x) = exp 1 (2π) n/2 (detc x ) 1/2 ( 1 ) 2 (x m x) T C 1 x (x m x ) where n is the dimension of x, m x is the mean and C x is the covariance matrix of x and is assumed to be strictly positive definite. Properties: m x and C x define uniquely the Gaussian pdf. closed under linear transforms - if x is a random gaussian vector then y = Ax is also gaussian with m y = Am x and C y = AC x A T marginal and conditionals are gaussian 10

11 uncorrelatedness and geometric structure: If the covariance matrix C x of the multidimensional gaussian density is not diagonal, then the components of x are not independent. C x is symmetric and positive definite matrix, hence it can be represented as C x = EDE T = n i=1 λ i e i e T i where E is an orthogonal matrix containing eigenvectors of C x as its columns and D = diag(λ 1,λ 2,...,λ n ) is a diagonal matrix containing the corresponding eigenvalues of C x. Transform u = E T (x m x ) rotates the data so that the components of u are uncorrelated and hence independent. 11

12 The cross-section of gaussian pdf with constant value of the density is a hyper-ellipsoid (x m x ) T C 1 x (x m x) = c centered at the mean, with axis parallel to the eigenvectors of C x and the eigenvalues being the corresponding variances. Higher-order Statistics Consider a scalar r.v. x with a probability density function p x (x). The j th moment of x is α j = E[x j ] = ξj p x (ξ)dξ and the j th central moment of x µ j = E[(x α 1 ) j ] = (ξ m x) j p x (ξ)dξ 12

13 Skewness and Kurtosis: The third central moment called skewness provides a measure of asymmetricity of the pdf. The fourth order statistics called kurtosis indicates nongaussianity of r.v. It is defined for zero-mean r.v. as kurt(x) = E[x 4 ] 3(E[x 2 ]) 2 Distribution with negative kurtosis are called subgaussian (usually flatter than Gaussian or multimodal). Distribution with positive kurtosis are called supergaussian (usually sharper peaked than Gaussian with longer tails). Properties of kurtosis: for 2 statistically independent r.v. x, y, kurt(x+y) = kurt(x)+kurt(y) for any scalar a: kurt(ax) = a 4 kurt(x) 13

14 Example: Laplacian density has a pdf p x (x) = λ 2 exp( λ x ) Example: Exponential family of pdf s (with zero mean) contains Gaussian, Laplacian and uniform pdf s as special cases: p x (x) = Cexp ( x ν νe[ x ν ] i.e. for ν = 2 the above pdf is equivalent to the Gaussian pdf ) ( ) p x (x) = Cexp ( x 2 2E[ x 2 = Cexp x2 ] 2σx 2 ν = 1 gives Laplacian pdf and ν yields uniform pdf. ) 14