Minimum mean square error estimator on arvau

Wikipedia / Minimum mean square error estimator

In statistics and signal processing, a minimum mean square error estimator (MMSE estimator) is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In the Bayesian setting, MMSE more specifically refers to estimation with quadratic loss function. In such case, the MMSE estimator is given by the posterior mean of the parameter to be estimated. Since the posterior mean is cumbersome to calculate, the form of the MMSE estimator is usually constrained to be within a certain class of functions. Linear MMSE estimators are a popular choice since they are easy to use, easy to calculate, and very versatile. It has given rise to many popular estimators such as the Wiener–Kolmogorov filter and Kalman filter.

Motivation

The term MMSE more specifically refers to estimation in a Bayesian setting with quadratic cost function. The basic idea behind the Bayesian approach to estimation stems from practical situations where we often have some prior information about the parameter to be estimated. For instance, we may have prior information about the range that the parameter can assume; or we may have an old estimate of the parameter that we want to modify when a new observation is made available; or the statistics of an actual random signal such as speech. This is in contrast to the non-Bayesian approach like minimum-variance unbiased estimator (MVUE) where absolutely nothing is assumed to be known about the parameter in advance and which does not account for such situations. In the Bayesian approach, such prior information is captured by the prior probability density function of the parameters; and based directly on Bayes' theorem, it allows us to make better posterior estimates as more observations become available. Thus unlike non-Bayesian approach where parameters of interest are assumed to be deterministic, but unknown constants, the Bayesian estimator seeks to estimate a parameter that is itself a random variable. Furthermore, Bayesian estimation can also deal with situations where the sequence of observations are not necessarily independent. Thus Bayesian estimation provides yet another alternative to the MVUE. This is useful when the MVUE does not exist or cannot be found.

Definition

known random vector variable (the measurement or observation), both of them not necessarily of the same dimension. An estimator and its mean squared error (MSE) is given by the trace of error covariance matrix

where the expectation. Note that MSE can equivalently be defined in other ways, since

The MMSE estimator is then defined as the estimator achieving minimal MSE:

Properties

When the means and variances are finite, the MMSE estimator is uniquely defined and is given by:
matrix,
1. .
The MMSE estimator is unbiased (under the regularity assumptions mentioned above):
The MMSE estimator is asymptotically unbiased and it converges in distribution to the normal distribution:
is the Fisher information. Thus, the MMSE estimator is asymptotically efficient.
The orthogonality principle if and only if
of the measurements. For random vectors, since the MSE for estimation of a random vector is the sum of the MSEs of the coordinates, finding the MMSE estimator of a random vector decomposes into finding the MMSE estimators of the coordinates of X separately:
for all i and j. More succinctly put, the cross-correlation should be zero,
are jointly Gaussian. This can be directly shown using the Bayes' theorem. As a consequence, to find the MMSE estimator, it is sufficient to find the linear MMSE estimator.

Linear MMSE estimator

or finding the minima of MSE. Direct numerical evaluation of the conditional expectation is computationally expensive since it often requires multidimensional integration usually done via Monte Carlo methods. Another computational approach is to directly seek the minima of the MSE using techniques such as the stochastic gradient descent methods; but this method still requires the evaluation of expectation. While these numerical methods have been fruitful, a closed form expression for the MMSE estimator is nevertheless possible if we are willing to make some compromises.

. The linear MMSE estimator is the estimator achieving minimum MSE among all estimators of such form. That is, it solves the following optimization problem:

One advantage of such linear MMSE estimator is that it is not necessary to explicitly calculate the posterior probability are jointly Gaussian, it is not necessary to make this assumption, so long as the assumed distribution has well defined first and second moments. The form of the linear estimator does not depend on the type of the assumed underlying distribution.

is given by:

Thus, the expression for linear MMSE estimator, its mean, and its auto-covariance is given by

Lastly, the error covariance and minimum mean square error achievable by such estimator is

. It is required that the MMSE estimator be unbiased. This means,

in above, we get

. Thus we can re-write the estimator as

and the expression for estimation error becomes

. Here the left-hand-side term is

Thus the full expression for the linear MMSE estimator is

, we can also obtain its auto-covariance as

, we get

Lastly, the covariance of linear MMSE estimation error will then be given by

in terms of covariance matrices as

Thus the minimum mean square error achievable by such a linear estimator is

Univariate case

are scalars, the above relations simplify to

is the Pearson's correlation coefficient.

The above two equations allows us to interpret the correlation coefficient either as normalized slope of linear regression

or as square root of the ratio of two variances

, as given by the equation of straight line.

Computation

Standard method like Gauss elimination. A more numerically stable method is provided by QR decomposition can be solved twice as fast with the Cholesky decomposition, while for large sparse systems conjugate gradient method is more effective. Levinson recursion is also a Toeplitz matrix is a wide sense stationary process. In such stationary cases, these estimators are also referred to as Wiener–Kolmogorov filterss.

Linear MMSE estimator for linear observation process

. Here the required mean and the covariance matrices will be

further modifies to

, we get

Lastly, the error covariance is

The significant difference between the estimation problem treated above and those of least squares and Gauss–Markov estimate is that the number of observations m) need not be at least as large as the number of unknowns, n). The estimate for the linear observation process exists so long as the m-by-m exists; this is the case for any m is now a random variable, it is possible to form a meaningful estimate (namely its mean) even with no measurements. Every new measurement simply provides additional information which may modify our original estimate. Another feature of this estimate is that for m < n is positive definite, the estimate still exists. Lastly, this technique can handle cases where the noise is correlated.

Alternative form

An alternative form of expression can be obtained by using the matrix identity

to obtain

and

In this form the above expression can be easily compared with weighed least square, Gauss–Markov estimate, and ridge regression, which is identical to ridge regression solution.

Sequential linear MMSE estimation

In many real-time applications, observational data is not available in a single batch. Instead the observations are made in a sequence. One possible approach is to use the sequential observations to update an old estimate as additional data becomes available, leading to finer estimates. One crucial difference between batch estimation and sequential estimation is that sequential estimation requires an additional Markov assumption.

is the prior density of k as

This is the Markov assumption.

given the k changes with time, we will make a further stationarity assumption about the prior:

Thus, the prior density for k-th time step is the posterior density of (k-1)-th time step. This structure allows us to formulate a recursive approach to estimation.

, respectively.

, its mean is given by the previous MMSE estimate,

and its covariance matrix is given by the previous error covariance matrix,

as per by the properties of MMSE estimators and the stationarity assumption.

and the covariance matrix is as before

arrives as

and the new error covariance as

, then after receiving another set of measurements, we should subtract out from these measurements that part that could be anticipated from the result of the first measurements. In other words, the updating must be based on that part of the new data which is orthogonal to the old data.

The repeated use of the above two equations as more observations become available lead to recursive estimation techniques. The expressions can be more compactly written as

is often referred to as the Kalman gain factor. The alternative formulation of the above algorithm will give

The repetition of these three steps as more data becomes available leads to an iterative estimation algorithm. The generalization of this idea to non-stationary cases gives rise to the Kalman filter. The three update steps outlined above indeed form the update step of the Kalman filter.

Special case: scalar observations

As an important special case, an easy-to-use recursive expression can be derived when at each k is n is n. After (k as:

is n-by-1 column vector given by

is n-by-n error covariance matrix given by

Alternative approaches: This important special case has also given rise to many other iterative methods (or adaptive filters), such as the least mean squares filter and recursive least squares filter, that directly solves the original MSE optimization problem using stochastic gradient descents Thus, the update equation for the least mean square filter is given by

. As we can see, these methods bypass the need for covariance matrices.

Special case: scalar unknown

Another important and an easy-to-use recursive expression can also be derived when at each k is m is m. After (k is given by

is now a 1-by-m row vector given by

Special Case: vector observation with uncorrelated noise

scalar measurements. The use of scalar update formula avoids matrix inversion in the implementation of the covariance update equations, thus improving the numerical robustness against roundoff errors. The update can be implemented iteratively as:

Examples

Example 1

We shall take a linear prediction are real Gaussian random variables with zero mean and its covariance matrix given by

is defined as

and pre-multiplying to get

.. The matrix equation can be solved by well known methods such as Gauss elimination method. A shorter, non-numerical example can be found in orthogonality principle.

Example 2

represents a scalar by an aprior uniform distribution. It is easy to see that

Thus, the linear MMSE estimator is given by

Similarly, the variance of the estimator is

Thus the MMSE of this linear estimator is

, we see that the MMSE estimator of a scalar with uniform aprior distribution can be approximated by the arithmetic average of all the observed data

and the LMMSE of the estimate will tend to zero.

, so long as the mean and variance of these distributions are the same.

Example 3

Note that except for the mean and variance of the error, the error distribution is unspecified. How should the two polls be combined to obtain the voting prediction for the given candidate?

As with previous example, we have

where the weights are given by

is given by

Thus, the LMMSE is given by

where the weight for i

Example 4

How should the recorded music from these two microphones be combined, after being synced with each other?

We can model the sound received by each microphone as

. Thus, we can combine the two sounds as

where the i-th weight is given as