- Orthogonality principle
In statistics and signal processing, the orthogonality principle is a necessary and sufficient condition for the optimality of a Bayesian estimator. Loosely stated, the orthogonality principle says that the error vector of the optimal estimator (in a mean square error sense) is orthogonal to any possible estimator. The orthogonality principle is most commonly stated for linear estimators, but more general formulations are possible. Since the principle is a necessary and sufficient condition for optimality, it can be used to find the minimum mean square error estimator.
Orthogonality principle for linear estimators
The orthogonality principle is most commonly used in the setting of linear estimation. In this context, let x be an unknown random vector which is to be estimated based on the observation vector y. One wishes to construct a linear estimator for some matrix H and vector c. Then, the orthogonality principle states that an estimator achieves minimum mean square error if and only if
If x and y have zero mean, then it suffices to require the first condition.
Suppose x is a Gaussian random variable with mean m and variance Also suppose we observe a value y = x + w, where w is Gaussian noise which is independent of x and has mean 0 and variance We wish to find a linear estimator minimizing the MSE. Substituting the expression into the two requirements of the orthogonality principle, we obtain
Solving these two linear equations for h and c results in
so that the linear minimum mean square error estimator is given by
This estimator can be interpreted as a weighted average between the noisy measurements y and the prior expected value m. If the noise variance is low compared with the variance of the prior (corresponding to a high SNR), then most of the weight is given to the measurements y, which are deemed more reliable than the prior information. Conversely, if the noise variance is high relative to the prior variance, then the estimate will be close to m, as the measurements are not reliable enough to outweigh the prior information.
Finally, note that because the variables x and y are jointly Gaussian, the minimum MSE estimator is linear. Therefore, in this case, the estimator above minimizes the MSE among all estimators, not only linear estimators.
Let V be a Hilbert space of random variables with an inner product defined by . Suppose W is a closed subspace of V, representing the space of all possible estimators. One wishes to find a vector which will approximate a vector . More accurately, one would like to minimize the mean squared error (MSE) between and x.
In the special case of linear estimators described above, the space V is the set of all functions of x and y, while W is the set of linear estimators, i.e., linear functions of y only. Other settings which can be formulated in this way include the subspace of causal linear filters and the subspace of all (possibly nonlinear) estimators.
Geometrically, we can see this problem by the following simple case where W is a one-dimensional subspace:
We want to find the closest approximation to the vector x by a vector in the space W. From the geometric interpretation, it is intuitive that the best approximation, or smallest error, occurs when the error vector, e, is orthogonal to vectors in the space W.
More accurately, the general orthogonality principle states the following: Given a closed subspace W of estimators within a Hilbert space V and an element x in V, an element achieves minimum MSE among all elements in W if and only if for all
Stated in such a manner, this principle is simply a statement of the Hilbert projection theorem. Nevertheless, the extensive use of this result in signal processing has resulted in the name "orthogonality principle."
A solution to error minimization problems
The following is one way to find the minimum mean square error estimator by using the orthogonality principle.
We want to be able to approximate a vector x by
is the approximation of x as a linear combination of vectors in the subspace W spanned by Therefore, we want to be able to solve for the coefficients, ci, so that we may write our approximation in known terms.
By the orthogonality theorem, the square norm of the error vector, , is minimized when, for all j,
Developing this equation, we obtain
If there is a finite number n of vectors pi, one can write this equation in matrix form as
Assuming the pi are linearly independent, the Gramian matrix can be inverted to obtain
thus providing an expression for the coefficients ci of the minimum mean square error estimator.
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