## Problem definition

In speech processing and elsewhere, a frequently appearing task is to make a prediction of an unknown vector *y* from available observation vectors *x*. Specifically, we want to have an estimate
\( \hat y = f(x) \)
such that
\( \hat y \approx y. \)
In particular, we will focus on *linear estimates* where
\( \hat y=f(x):=A^T x, \)
and where *A* is a matrix of parameters.

## The minimum mean square estimate (MMSE)

Suppose we want to minimise the squared error of our estimate on average. The estimation error is* *
\( e=y-\hat y \)
and the squared error is the *L _{2}*-norm of the error, that is,
\( \left\|e\right\|^2 = e^T e \)
and its mean can be written as the expectation
\( E\left[\left\|e\right\|^2\right] = E\left[\left\|y-\hat y\right\|^2\right] = E\left[\left\|y-A^T x\right\|^2\right]. \)
Formally, the minimum mean square problem can then be written as

This can in generally not be directly implemented because we have the abstract expectation-operation in the middle.

*(Advanced derivation begins)* To get a computational model, first note that the error expectation can be written in terms of the mean of a sample of vector *e _{k}* as

where
\( E=\left[e_1,\,e_2,\dotsc,e_N\right] \)
and *tr()* is the matrix trace. To minimize the error energy expectation, we can then set its derivative to zero

where the observation matrix is
\( X=\left[x_1,\,x_2,\dotsc,x_N\right] \)
and the desired output matrix is
\( Y=\left[y_1,\,y_2,\dotsc,y_N\right] \)
. *(End of advanced derivation)*

It follows that the optimal weight matrix *A* can be solved as

where the superscript \( \dagger \) denotes the Moore-Penrose pseudo-inverse.

### Estimates with a mean parameter

Suppose that instead of an estimate \( \hat y=A^T x \) , we want to include a mean vector in the estimate as \( \hat y=A^T x + \mu \) . While it is possible to derive all of the above equations for this modified model, it is easier to rewrite the model into a similar form as above with

\[ \hat y=A^T x + \mu = \begin{bmatrix} \mu^T \\ A^T \end{bmatrix} \begin{bmatrix} 1 \\ x \end{bmatrix} := A'^T x'. \]That is, we can extend *x* by a single 1, (the observation *X* similarly with a row of constant 1s), and extend *A* to include the mean vector. With this modifications, the above Moore-Penrose pseudo-inverse can again be used to solve the modified model.

### Estimates with linear equality constraints

*(Advanced derivation begins)*

In practical situations we often have also linear constraints, such as \( A^T C = B \) , which is equivalent with \( A^T C - B = 0. \) The modified programming task is then

\[ \min_A\, E\left[\left\|y-A^T x\right\|^2\right]\quad\text{such that}\quad A^T C - B = 0. \]Such constraints can be included into the objective function using the method of Lagrange multipliers such that the modified objective function is

\[ \eta(A,G) = E\, \left[\left(Y - A^T X\right)^T\left(Y - A^T X\right) - 2 \,{\mathrm{tr}}\left[ G^T \left(A^T C - B\right)\right]\left]. \]A heuristic explanation of this objective function is based on the fact the *G* is a free parameter. Since its value can be anything, then
\( A^T C - B \)
must be zero, because otherwise the output value of the objective function could be anything. That is, when optimizing with respect to *A*, we find the minimum of the mean square error, while simultaneously satisfying the constraint.

The objective function can further be rewritten as

\[ \eta(A,G) = E\, \left[\left(Y - A^T X\right)^T\left(Y - A^T X\right) - 2 \,{\mathrm{tr}}\left[ G^T \left(A^T C - B\right)\right]\left]. \]\( \int_{-\infty}^\infty \mbox{e}^{-x^2} \mbox{d}x = \sqrt{\pi} \) TBC

*(Advanced derivation ends)*