16. Mean Square Estimation

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16. Mean Square Estimation

Given some information that is related to an unknown quantity of interest, the problem is to obtain a good estimate for the unknown in terms of the observed data.

Suppose represent a sequence of random variables about whom one set of observations are available, and Y represents an unknown random variable. The problem is to obtain a good estimate for Y in terms of the observations Let

represent such an estimate for Y.

Note that can be a linear or a nonlinear function of the observation Clearly

represents the error in the above estimate, and the square of

n X X X₁, ₂,", . , , , ₂ 1 X Xn X " ) ( ) , , , ( ˆ _{X1 X2 X X} Y =ϕ " _n =ϕ _(16-1) ) (⋅ ϕ . , , , ₂ 1 X Xn X " ) ( ˆ ) (X Y Y Y ϕ X ε = − = − (16-2) 2 | |ε

2

the error. Since is a random variable, represents the mean square error. One strategy to obtain a good estimator would be to

minimize the mean square error by varying over all possible forms of and this procedure gives rise to the Minimization of the Mean Square Error (MMSE) criterion for estimation. Thus under MMSE criterion,the estimator is chosen such that the mean square error is at its minimum.

Next we show that the conditional mean of Y given X is the best estimator in the above sense.

Theorem1: Under MMSE criterion, the best estimator for the unknown

Y in terms of is given by the conditional mean of Y gives X. Thus

Proof : Let represent an estimate of Y in terms of

Then the error and the mean square error is given by ε E{| ε |2} ), (⋅ ϕ ) (⋅ ϕ n X X X₁, ₂,", }. | { ) ( ˆ _{X E Y X} Y = ϕ = _(16-3) ) ( ˆ _X Y =ϕ ). , , , (X₁ X₂ X_n X = " _{ε = Y −Y}ˆ_, ˆ } | | { ε 2 E

3

Since

we can rewrite (16-4) as

where the inner expectation is with respect to Y, and the outer one is with respect to

Thus

To obtain the best estimator we need to minimize in (16-6) with respect to In (16-6), since and the variable appears only in the integrand term, minimization of the mean square error in (16-6) with respect to is equivalent to minimization of with respect to

}] | { [ ] [z E E z X E = _{X z} }] | ) ( | { [ } | ) ( | { z 2 z 2 2 _{E Y X E E Y X X} Y X      ϕ ϕ σ_ε = − = − . X

∫

−+∞∞ − = − = . ) ( } | ) ( | { }] | ) ( | { [ 2 2 2 dx X f X X Y E X X Y E E X ϕ ϕ σ_ε (16-6) (16-5) , ϕ 2 ε σ . ϕ _{fX (X ) ≥ 0, E{| Y −ϕ(X ) |}2 _{X}≥ 0,} ϕ ϕ 2 ε σ } | ) ( | { Y X 2 X E −ϕ ϕ.

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Since X is fixed at some value, is no longer random, and hence minimization of is equivalent to

This gives

or

But

since when is a fixed number Using (16-9) ) (X ϕ } | ) ( | { Y X 2 X E −ϕ . 0 } | ) ( | { − 2 = ∂ ∂ X X Y E ϕ ϕ (16-7) 0 } | ) ( {|Y − X X = E ϕ (16-8) ), ( } | ) ( { X X X E ϕ =ϕ (16-9) ) ( , X x X = ϕ ϕ(x). . 0 } | ) ( { } | {Y X − E X X = E ϕ

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in (16-8) we get the desired estimator to be

Thus the conditional mean of Y given represents the best estimator for Y that minimizes the mean square error.

The minimum value of the mean square error is given by

As an example, suppose is the unknown. Then the best MMSE estimator is given by

Clearly if then indeed is the best estimator for Y }. , , , | { } | { ) ( ˆ _{X E Y X E Y X1 X2 Xn} Y = ϕ = = " n X X X₁, ₂,", . 0 )} | {var( ] } | ) | ( | { [ } | ) | ( | { ) var( 2 2 2 min ≥ = − = − = X Y E X X Y E Y E E X Y E Y E X Y          σ (16-11) (16-10) 3 X Y = , 3 X Y = Yˆ = X 3 . } | { } | { ˆ _{E Y X E X} 3 _{X X} 3 Y = = = _(16-12)

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in terms of X. Thus the best estimator can be nonlinear. Next, we will consider a less trivial example.

Example : Let

where k > 0 is a suitable normalization constant. To determine the best estimate for Y in terms of X, we need

Thus

Hence the best MMSE estimator is given by    < < < = otherwise, 0 1 0 , ) , ( , y x kxy y x f_{X Y} ). | ( | y x f_{Y X} 1. x 0 , 2 ) 1 ( 2 ) , ( ) ( 2 1 2 1 1 , < < − = = = =

_∫

_∫

x kx kxy kxydy dy y x f x f x x X Y x X y x 1 1 . 1 0 ; 1 2 2 / ) 1 ( ) ( ) , ( ) | ( , _{2 2} < < < − = − = = x y x y x kx kxy x f y x f x y f X Y X X Y (16-13)

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Once again the best estimator is nonlinear. In general the best estimator is difficult to evaluate, and hence next we will examine the special subclass of best linear estimators.

Best Linear Estimator

In this case the estimator is a linear function of the observations Thus

where are unknown quantities to be determined. The mean square error is given by

. 1 ) 1 ( 3 2 1 1 3 2 1 3 2 ) | ( } | { ) ( ˆ 2 2 2 3 1 2 3 1 ₂ 1 2 1 1 2 1 2 2 | x x x x x x y dy y dy y dy x y f y X Y E X Y x x x x x y x Y X − + + = − − = − = = = = = =

∫

∫

∫

− − ϕ } | {Y X E (16-14) Yˆ . , , , ₂ 1 X Xn X " n a a a₁, ₂,",

∑

= = + + + = n i i i n n l a X a X a X a X Y 1 2 2 1 1 . ˆ _" (16-15) ) ˆ (ε = Y −Y_l

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and under the MMSE criterion should be chosen so that the mean square error is at its minimum possible value. Let represent that minimum possible value. Then

To minimize (16-16), we can equate

This gives But n a a a₁, ₂,", } | | { } | ˆ {| } | | { 2 = E Y −Y_l 2 = E Y −

∑

a_iX_i 2 E ε } | | { ε 2 E 2 n σ

∑

= − = n i i i a a a n E Y a X n ₁ 2 , , , 2 _{min {| | }.} 2 1 " σ (16-17) ,n. , , k E a_k {| | } 0, 1 2 " 2 _{= =} ∂ ∂ _ε (16-18) . 0 2 | | } | {| * 2 2 ₌               ∂ ∂ =       ∂ ∂ = ∂ ∂ k k k a E a E E a ε ε ε ε _(16-19) (16-16)

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Substituting (16-19) in to (16-18), we get

or the best linear estimator must satisfy

Notice that in (16-21), represents the estimation error and represents the data. Thus from (16-21), the error is orthogonal to the data for the best linear estimator. This is the orthogonality principle.

In other words, in the linear estimator (16-15), the unknown constants must be selected such that the error

. ) ( ) ( 1 1 k k n i i i k k n i i i k X a X a a Y a X a Y a ∂ = − ∂ − ∂ ∂ = ∂ − ∂ = ∂ ∂ε

∑

₌

∑

₌ , 0 } { 2 } | {| 2 _* = − = ∂ ∂ k k X E a E ε _ε . , , 2 , 1 , 0 } { X * k n E ε _k = = " (16-20) (16-21)

∑

= − n i aiXi Y ₁ ), ( ε ε Xk, k = 1→ n X_k, k = 1→ n n a a a₁, ₂,",

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is orthogonal to every data for the

best linear estimator that minimizes the mean square error.

Interestingly a general form of the orthogonality principle holds good in the case of nonlinear estimators also.

Nonlinear Orthogonality Rule: Let represent any functional

form of the data and the best estimator for Y given With we shall show that

implying that

This follows since

∑

= − = n i aiXi Y ₁ ε X₁, X₂,", X_n ) (X h } | {Y X E _X. } | {Y X E Y e = − ). ( } | {Y X h X E Y e = − ⊥ , 0 )} ( {eh X = E . 0 )} ( { )} ( { ]} | ) ( [ { )} ( { } ) ( ] | [ { )} ( { )} ( ]) | [ {( )} ( { = − = − = − = − = X Yh E X Yh E X X Yh E E X Yh E X h X Y E E X Yh E X h X Y E Y E X eh E (16-22)

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Thus in the nonlinear version of the orthogonality rule the error is orthogonal to any functional form of the data.

The orthogonality principle in (16-20) can be used to obtain the unknowns in the linear case.

For example suppose n = 2, and we need to estimate Y in terms of linearly. Thus

From (16-20), the orthogonality rule gives

Thus or n a a a₁, ₂,", 2 1 and X X 2 2 1 1 ˆ _{a X a X} Y_l = + 0 } ) {( } X { 0 } ) {( } X { * 2 2 2 1 1 * 2 * 1 2 2 1 1 * 1 = − − = = − − = X X a X a Y E E X X a X a Y E E ε ε } { } | {| } { } { } { } | {| * 2 2 2 2 1 * 2 1 * 1 2 * 1 2 1 2 1 YX E a X E a X X E YX E a X X E a X E = + = +

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(16-23) can be solved to obtain in terms of the cross-correlations.

The minimum value of the mean square error in (16-17) is given by

But using (16-21), the second term in (16-24) is zero, since the error is orthogonal to the data where are chosen to be

optimum. Thus the minimum value of the mean square error is given by 2 1 and a a       =             } { } { } | {| } { } { } | {| * 2 * 1 2 1 2 2 * 2 1 * 1 2 2 1 YX E YX E a a X E X X E X X E X E (16-23) 2 n σ n a a a₁, ₂,", , i X }. { min } { min } ) ( { min } { min } | {| min * 1 , , , * , , , 1 * , , , * , , , 2 , , , 2 2 1 2 1 2 1 2 1 2 1 l n i i a a a a a a n i i i a a a a a a a a a n X E a Y E X a Y E E E n n n n n ε ε ε εε ε σ

∑

∑

= = − = − = = = " " " " " (16-24)

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where are the optimum values from (16-21).

Since the linear estimate in (16-15) is only a special case of the general estimator in (16-1), the best linear estimator that satisfies (16-20) cannot be superior to the best nonlinear estimator

Often the best linear estimator will be inferior to the best estimator in (16-3).

This raises the following question. Are there situations in which the best estimator in (16-3) also turns out to be linear ? In those situations it is enough to use (16-21) and obtain the best

linear estimators, since they also represent the best global estimators.

Such is the case if Y and are distributed as jointly Gaussian We summarize this in the next theorem and prove that result.

Theorem2: If and Y are jointly Gaussian zero

n a a a₁, ₂,", ) (X ϕ }. | {Y X E } { } | {| } ) {( } { 1 * 2 1 * * 2

∑

∑

= = − = − = = n i i i n i i i n Y X E a Y E Y X a Y E Y E ε σ (16-25) n X X X₁, ₂,", n X X X₁, ₂,",

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mean random variables, then the best estimate for Y in terms of is always linear.

Proof : Let

represent the best (possibly nonlinear) estimate of Y, and

the best linear estimate of Y. Then from (16-21)

is orthogonal to the data Thus

Also from (16-28), n X X X₁, ₂,", } | { ) , , , ( ˆ _{X1 X2 X E Y X} Y = ϕ " _n = _(16-26)

∑

= = n i i i l a X Y 1 ˆ (16-27) . 1 , k n X_k = → (16-28) . 1 , 0 } X { *_k k n E ε = = → _(16-29) . 0 } { } { } { = E Y −

∑

n a E X = E ε _(16-30) 1 _l n _{i i} i Y Y Y a X ε = =∆ − = −

∑

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Using (16-29)-(16-30), we get

From (16-31), we obtain that and are zero mean uncorrelated random variables for But itself represents a Gaussian random variable, since from (16-28) it represents a linear combination of a set of jointly Gaussian random variables. Thus and X are

jointly Gaussian and uncorrelated random variables. As a result, and

X are independent random variables. Thus from their independence

But from (16-30), and hence from (16-32)

Substituting (16-28) into (16-33), we get . 1 n k = → ε X_k ε ε . 1 , 0 } { } { } { X * E E X * k n E ε _k = ε _k = = → _(16-31) ε }. { } | {ε X E ε E = _(16-32) , 0 } {ε = E . 0 } | { X = E ε _(16-33) 0 } | { } | { 1 = − =

∑

= X X a Y E X E n i i i ε

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or

From (16-26), represents the best possible estimator, and from (16-28), represents the best linear estimator.

Thus the best linear estimator is also the best possible overall estimator in the Gaussian case.

Next we turn our attention to prediction problems using linear estimators.

Linear Prediction

Suppose are known and is unknown. Thus and this represents a one-step prediction problem. If the unknown is then it represents a k-step ahead prediction problem. Returning back to the one-step predictor, let

represent the best linear predictor. Then

. } | { } | { 1 1 l n i i i n i i i Y X a X X a E X Y E =

∑

=

∑

= = = (16-34) ) ( } | {Y X x E =ϕ

∑

= n i 1aiXi n X X X₁, ₂,", X_n+1 , 1 + = X_n Y , k n X ₊ 1 ˆ _n+ X

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where the error

is orthogonal to the data, i.e.,

Using (16-36) in (16-37), we get

Suppose represents the sample of a wide sense stationary X_i

(16-35) , 1 , ˆ 1 1 1 1 2 2 1 1 1 1 1 1 = = + + + + = + = − = + + = + = + + +

∑

∑

n n i i i n n n n i i i n n n n a X a X X a X a X a X a X X X " ε (16-36) . 1 , 0 } { X * k n E ε_{n k} = = → _(16-37)

∑

+ = → = = = 1 1 * *_{} { } 0, 1 .} { n i i i k k n X a E X X k n E ε (16-38) 1 1 ˆ ₌ n _, n i i i X ₊ a X = −

∑

∆

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stochastic process so that

Thus (16-38) becomes

Expanding (16-40) for we get the following set of linear equations.

Similarly using (16-25), the minimum mean square error is given by

* *_{} ( )} {X_iX_k R i k r_{i k} r_{k i} E = − = ₋ = ₋ ) (t X (16-39) . 1 , 1 , 0 } { 1 ₁ 1 * _{a r a k n} X E n _n i i i k k n = = + = = → + = −

∑

ε (16-40) , , , 2 , 1 n k = " . 0 2 0 1 0 1 0 * 3 3 * 2 2 * 1 1 1 2 1 3 0 2 * 1 1 1 2 3 1 2 0 1 n k r r a r a r a r a k r r a r a r a r a k r r a r a r a r a n n n n n n n n n n = ← = + + + + + = ← = + + + + + = ← = + + + + + − − − − − − " # " " (16-41)

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The n equations in (16-41) together with (16-42) can be represented as

Let . } ) {( } { } { } | {| 0 1 * 2 3 * 1 2 * 1 1 1 * 1 1 1 * 1 * 1 * 2 2 r r a r a r a r a r a X X a E X E Y E E n n n n n i i n i n i i i n n n n n + + + + + = = = = = = − − + = + − + = + +

∑

∑

" ε ε ε σ (16-42) . 0 0 0 0 1 2 n 3 2 1 0 * 1 * 1 * 1 0 * 2 * 1 2 0 * 1 * 2 1 1 0 * 1 2 1 0                   =                                       − − − − − σ # # " " # " " " n n n n n n n n a a a a r r r r r r r r r r r r r r r r r r r r (16-43)

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Notice that is Hermitian Toeplitz and positive definite. Using (16-44), the unknowns in (16-43) can be represented as

Let n T . 0 * 1 * 1 * 1 1 0 * 1 2 1 0               = − − r r r r r r r r r r r r T n n n n n " # " " (16-44)             =                   =                   − − 1 2 2 n 1 3 2 1 of column Last 0 0 0 0 1 n n n n T T a a a a σ σ # # (16-45)

21 Then from (16-45), Thus . 1 , 1 2 , 1 1 , 1 1 , 2 22 21 1 , 1 12 11 1               = + + + + + + − n n n n n n n n n n n n n n n n T T T T T T T T T T " # " " . 1 1, 1 1 , 2 1 , 1 2 2 1               =                 + + + + n n n n n n n n n T T T a a a # # σ , 0 1 1 , 1 2 _{= >} + + n n n n T σ (16-46) (16-47) (16-48)

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and

Eq. (16-49) represents the best linear predictor coefficients, and they can be evaluated from the last column of in (16-45). Using these, The best one-step ahead predictor in (16-35) taken the form

and from (16-48), the minimum mean square error is given by the (n +1, n +1) entry of

From (16-36), since the one-step linear prediction error . 1 1 , 1 1 , 2 1 , 1 1 , 1 2 1               =             + + + + + + n n n n n n n n n n n T T T T a a a # # (16-49) n T . 1 − n T . ) ( 1 ˆ 1 1 , 1 , 1 1

∑

= + + + +       − = n i i n i n n n n n T X T X _(16-50) , X a X a X a X + + + + = " ε (16-51)

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we can represent (16-51) formally as follows

Thus, let

them from the above figure, we also have the representation

The filter

represents an AR(n) filter, and this shows that linear prediction leads to an auto regressive (AR) model.

n n n n n a z a z a z X ₊₁ → 1+ −1 + ₋₁ −2 +"+ ₁ − → ε , 1 ) ( _n 1 _{n 1} 2 ₁ n n z a z a z a z A = + − + ₋ − +"+ − _(16-52) . ) ( 1 → ₊₁ → _n n n X z A ε n n n n z a z a z a z A z H _{− −} − − _{+ + +} + = = 1 2 1 1 1 1 ) ( 1 ) ( " (16-53)

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The polynomial in (16-52)-(16-53) can be simplified using (16-43)-(16-44). To see this, we rewrite as

To simplify (16-54), we can make use of the following matrix identity ) (z A_n ) (z A_n                 =                 = + + + + + = − − − − − − − − − − − − − − − 2 n 1 1 ) 1 ( 2 1 1 ) 1 ( 1 2 1 ) 1 ( 2 1 0 0 0 ] 1 , , , , [ 1 ] 1 , , , , [ 1 ) ( σ # " # " " n n n n n n n n n n n T z z z a a a z z z z a z a z a z a z A (16-54) . 0 0 1       − =     −     − _B CA D C A I AB I D C B A (16-55)

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Taking determinants, we get

In particular if we get Using (16-57) in (16-54), with . ₁ B CA D A D C B A ₋ − = (16-56) , 0 ≡ D . 0 ) 1 ( 1 C B A A B CA n − = − (16-57)           = = = − − − − 2 1 ) 1 ( 0 , ], 1 , , , , [ n n n n _{z z A T B} z C σ # "

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we get

Referring back to (16-43), using Cramer’s rule to solve for we get . 1 | | 0 1 , , , 0 0 0 | | ) 1 ( ) ( 1 ) 1 ( 1 0 * 2 * 1 1 1 0 * 1 2 1 0 2 1 2 − − − − − − − − − = − = z z z r r r r r r r r r r r r T z z T T z A n n n n n n n n n n n n n n " " # " " " # _σ σ (16-58) ), 1 ( 1 = + n a 1 | | | | | | 1 2 0 1 1 0 2 1 = = − = − − + n n n n n n n n T T T r r r r a σ σ " # "

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or

Thus the polynomial (16-58) reduces to

The polynomial in (16-53) can be alternatively represented as in (16-60), and in fact represents a stable

. 0 | | | | 1 2 _{= >} − n n n T T σ . 1 1 | | 1 ) ( 1 2 1 1 1 ) 1 ( 1 0 * 2 * 1 1 1 0 * 1 2 1 0 1 n n n n n n n n n n n z a z a z a z z z r r r r r r r r r r r r T z A − − − − − − − − − − − − + + + + = = " " " # " " (16-59) (16-60) ) (z A_n ) ( ~ ) ( 1 ) ( AR n z A z H n =

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AR filter of order n, whose input error signal is white noise of constant spectral height equal to and output is

It can be shown that has all its zeros in provided thus establishing stability.

Linear prediction Error

From (16-59), the mean square error using n samples is given by

Suppose one more sample from the past is available to evaluate

( i.e., are available). Proceeding as above the new coefficients and the mean square error can be determined. From (16-59)-(16-61), n ε | | / | |T_n T_n−1 X_n+1. 1 + n X . 0 | | | | 1 2 _{= >} − n n n T T σ _(16-61) 0 1 1, , , , X X X X_{n n−} " 2 1 + n σ . | | | | ₁ 2 1 n n T T ₊ + = σ _(16-62) ) (z A_{n | z |>1} 0 | |T_n >

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Using another matrix identity it is easy to show that

Since we must have or for every n. From (16-63), we have

or

since Thus the mean square error decreases as more and more samples are used from the past in the linear predictor.

In general from (16-64), the mean square errors for the one-step predictor form a monotonic nonincreasing sequence

). | | 1 ( | | | | | | 2 1 1 2 1 + − + = − n n n n s T T T (16-63) , 0 | |T_k > _{(1 | |}2_{) 0} 1 > − s_n+ | sn+1 |< 1 ) | | 1 ( | | | | | | | | 2 1 1 1 2 2 1 + − + _{= −} + n n n n n _s T T T T n n   σ σ , ) | | 1 ( ₁ 2 2 2 2 1 n n n n σ s σ σ ₊ = − ₊ < (16-64) . 1 ) | | 1 ( − s_n+1 2 <

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whose limiting value

Clearly, corresponds to the irreducible error in linear prediction using the entire past samples, and it is related to the power spectrum of the underlying process through the relation

where represents the power spectrum of For any finite power process, we have

and since Thus

2 2 2 1 2 ∞ + ≥ > → ≥ σ σ σ σ_{n n} " _k " _(16-65) . 0 2 _≥ ∞ σ ) (nT X ). (nT X ( ) 0 XX S ω ≥ 2 _exp 1 _{ln ( ) 0.} 2 SXX d π π σ ω ω π + ∞ =  ₋  ≥ 

∫

 (16-66) ( )S_XX d , π π ω ω + − < ∞

∫

0 2 _≥ ∞ σ ln S_XX ( )d S_XX ( )d . π π π ω ω π ω ω + + − ≤ − < ∞

∫

∫

(S_XX ( ) 0), lnω ≥ S_XX ( )ω ≤ S_XX ( ).ω (16-67)

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Moreover, if the power spectrum is strictly positive at every Frequency, i.e.,

then from (16-66)

and hence

i.e., For processes that satisfy the strict positivity condition in (16-68) almost everywhere in the interval the final minimum mean square error is strictly positive (see (16-70)). i.e., Such processes are not completely predictable even using their entire set of past samples, or they are inherently stochastic,

( ) 0, in - , XX S ω > π ω π< < ln S_XX ( )d . π π ω ω + − > −∞

∫

2 _exp 1 _{ln ( ) 0} 2 SXX d

e

π π σ ω ω π + _−∞ ∞ =  ₋  > = 

∫

 (16-68) (16-69) (16-70) ), , (−π π

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since the next output contains information that is not contained in the past samples. Such processes are known as regular stochastic processes, and their power spectrum is strictly positive.

) (ω XX S ω π π −

Power Spectrum of a regular stochastic Process

Conversely, if a process has the following power spectrum,

such that in then from (16-70), S_XX ( ) 0ω = ω₁ <ω < ω₂ 2 _{= 0.} ∞ σ ω 2 ω 1 ω π − π ) (ω XX S

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Such processes are completely predictable from their past data samples. In particular

is completely predictable from its past samples, since consists of line spectrum.

in (16-71) is a shape deterministic stochastic process.

∑

+ = k k k k t a nT X ( ) cos(ω φ ) (16-71) ( ) XX S ω ) (nT X " 1 ω ω_k ω ) (ω XX S