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Describes random variables in terms of Hilbert spaces, defining inner products, norms, and minimum mean square error estimation.

Random variable spaces

Probability – notation primer

Definition 1 A random variable x is defined by a distribution function

P ( x ) = F X ( x ) = Prob ( X x )

The density function is given by

P ( x ) x = f X ( x ) = Prob ( X x ) x

Definition 2 The expectation of a function g ( x ) over the random variable x is

E X [ g ( x ) ] = - g ( x ) f X ( x ) d x

Definition 3 Pairs of random variables X , Y are defined by the joint distribution function

P ( x , y ) = F X Y ( x , y ) = Prob ( X x , Y y )

The joint density function is given by

2 P ( x , y ) x y = f X Y ( x , y ) = 2 Prob ( X x , Y y ) x y

The expectation of a function g ( x , y ) is given by

E X , Y [ g ( x , y ) ] = - - g ( x , y ) f X Y ( x , y ) d x d y

A hilbert space of random variables

Definition 4 Let { Y 1 , , Y n } be a collection of zero-mean ( E [ Y i ] = 0 ) random variables. The space H of all random variables that are linear combinations of those n random variables { y 1 , , y n } is a Hilbert space with inner product

X , Y = E [ x y ¯ ] = - - x y ¯ f X Y ( x , y ) d x d y .

We can easily check that this is a valid inner product:

  • x , x = E [ x x ¯ ] = - | x | 2 f x ( x ) d x = E | x | 2 0 ;
  • x , x = 0 if and only if f X ( x ) = δ ( x ) , i.e., if X is a random variable that is deterministically zero (and this random variable is the “zero” of this Hilbert space);
  • x , y = y , x ¯ ;
  • x + y , z = E [ ( x + y ) z ¯ ] = E [ x z ¯ ] + E [ y z ¯ ] = x , z + y , z ;

Note in particular that orthogonality, i.e., x , y = 0 , implies E [ x y ¯ ] = 0 , i.e., x and y are independent random variables. Additionally, the induced norm X = X , X = E [ | x | 2 ] is the standard deviation of the zero-mean random variable X .

A hilbert space of random vectors

One can define random vectors X , Y whose entries are random variables:

X = X 1 X N , Y = Y 1 Y N .

For these, the following inner product is an extension of that given above:

X , Y = E [ y H x ] = E i = 1 n y i ¯ x i = E trace [ x y H ] .

The induced norm is

X = X , X = E x H x = E i = 1 N | x i | 2 ,

the expected norm of the vector x .

Minimum mean square error estimation

In an MMSE estimation problem, we consider Y = A X + N , where X , Y are two random vectors and N is usually additive white Gaussian noise ( Y is m × 1 , A is m × n , X is n × 1 , and N N ( 0 , σ 2 I ) is m × 1 ). Due to this noise model, we want an estimate X ^ of X that minimizes E X - X ^ 2 ; such an estimate has highest likelihood under an additive white Gaussian noise model. For computational simplicity, we often want to restrict the estimator to be linear, i.e.

X ^ = K Y = K 1 H K n H Y ,

where K i H denotes the i t h row of the estimation matrix K and X ^ i = K i H Y . We use the definition of the 2 norm to simplify the equation:

min K E X - X ^ 2 2 = min K E X - K Y 2 2 = min K E i = 1 n ( X i - K i H Y ) 2

Since the terms involved in the sum are independent from each other and nonnegative, this minimization can be posed in terms of n individual minimizations: for i = 1 , 2 , . . . , n , we solve

min K i E ( X i - K i H Y ) 2 = min K i E ( X i - i = 1 n K i j ¯ Y j ) 2 = min K i X i - i = 1 n K i j ¯ Y j ,

where the norm is the induced norm for the Hilbert space of random variables. Note at this point that the set of random variables i = 1 n K i j ¯ Y j over the choices of K i can be written as span ( { Y j } j = 1 m ) . Thus, the optimal K i is given by the coefficients of the closest point in span ( { Y j } j = 1 m ) to the random variable X i according to the induced norm for the Hilbert space of random variables. Therefore, we solve for K i using results from the projection theorem with the corresponding inner product. Recall that given a basis Y i for the subspace of interest, we obtain the equation β i = G ( K i H ) T = G K i ¯ , where β i , j = X i , Y j and G is the Gramian matrix. More specifically, we have

X i , Y 1 X i , Y 2 X i , Y m β i = Y 1 , Y 1 Y 2 , Y 1 Y m , Y 1 Y 1 , Y 2 Y 2 , Y 2 Y m , Y 2 Y 1 , Y m Y 2 , Y m Y m , Y m G K i 1 ¯ K i 2 ¯ K i m ¯ K i .

Thus, one can solve for K i ¯ = G - 1 β i . In the Hilbert space of random variables, we have

G = E [ Y 1 Y 1 ] E [ Y 2 Y 1 ] E [ Y m Y 1 ] E [ Y 1 Y 2 ] E [ Y 2 Y 2 ] E [ Y m Y 2 ] E [ Y 1 Y m ] E [ Y 2 Y m ] E [ Y m Y m ] = R Y , β = E [ X i Y 1 ] E [ X i Y 2 ] E [ X i Y m ] = ρ X i Y .

Here R Y is the correlation matrix of the random vector Y and ρ X i Y is the cross-correlation vector of the random variable X i and vector Y . Thus, we have K i ¯ = G - 1 β i = R Y - 1 ρ X i Y , and so K i H = ρ X i Y T R Y - 1 . Concatenating all the rows of K together, we get K = R X , Y R Y - 1 , where R X , Y is the cross-correlation matrix for the random vectors X and Y . We therefore obtain the optimal linear estimator X ^ = K Y = R X , Y R Y - 1 Y .

At first, there may be some confusion on the difference between least squares and minimum mean-square error. To summarize:

  • Least Squares are applied when the quantities observed are deterministic (i.e., a “single draw” of data or observations).
  • Minimum Mean Square Error Estimation are applied when random variables are observed under Gaussian noise; one must know a distribution over inputs, and the error must be measured in expectation.

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Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
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