Minimum mean square error

In statistics and signal processing, a minimum mean square error (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, the term 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.

MotivationEdit

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.

DefinitionEdit

Let   be a   hidden random vector variable, and let   be a   known random vector variable (the measurement or observation), both of them not necessarily of the same dimension. An estimator   of   is any function of the measurement  . The estimation error vector is given by   and its mean squared error (MSE) is given by the trace of error covariance matrix

 

where the expectation   is taken over both   and  . When   is a scalar variable, the MSE expression simplifies to  . Note that MSE can equivalently be defined in other ways, since

 

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

 

PropertiesEdit

  • When the means and variances are finite, the MMSE estimator is uniquely defined[1] and is given by:
 
In other words, the MMSE estimator is the conditional expectation of   given the known observed value of the measurements.
  • The MMSE estimator is unbiased (under the regularity assumptions mentioned above):
 
 
where   is the Fisher information of  . Thus, the MMSE estimator is asymptotically efficient.
  • The orthogonality principle: When   is a scalar, an estimator constrained to be of certain form   is an optimal estimator, i.e.   if and only if
 
for all   in closed, linear subspace   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 between the minimum estimation error   and the estimator   should be zero,
 
  • If   and   are jointly Gaussian, then the MMSE estimator is linear, i.e., it has the form   for matrix   and constant  . 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 estimatorEdit

In many cases, it is not possible to determine the analytical expression of the MMSE estimator. Two basic numerical approaches to obtain the MMSE estimate depends on either finding the conditional expectation   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.

One possibility is to abandon the full optimality requirements and seek a technique minimizing the MSE within a particular class of estimators, such as the class of linear estimators. Thus, we postulate that the conditional expectation of   given   is a simple linear function of  ,  , where the measurement   is a random vector,   is a matrix and   is a vector. This can be seen as the first order Taylor approximation of  . The linear MMSE estimator is the estimator achieving minimum MSE among all estimators of such form. That is, it solves the following the optimization problem:

 

One advantage of such linear MMSE estimator is that it is not necessary to explicitly calculate the posterior probability density function of  . Such linear estimator only depends on the first two moments of   and  . So although it may be convenient to assume that   and   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.

The expression for optimal   and   is given by:

 
 

where  ,   the   is cross-covariance matrix between   and  , the   is auto-covariance matrix of  .

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

 
 
 

where the   is cross-covariance matrix between   and  .

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

 
 
Derivation using orthogonality principle

Let us have the optimal linear MMSE estimator given as  , where we are required to find the expression for   and  . It is required that the MMSE estimator be unbiased. This means,

 

Plugging the expression for   in above, we get

 

where   and  . Thus we can re-write the estimator as

 

and the expression for estimation error becomes

 

From the orthogonality principle, we can have  , where we take  . Here the left-hand-side term is

 

When equated to zero, we obtain the desired expression for   as

 

The   is cross-covariance matrix between X and Y, and   is auto-covariance matrix of Y. Since  , the expression can also be re-written in terms of   as

 

Thus the full expression for the linear MMSE estimator is

 

Since the estimate   is itself a random variable with  , we can also obtain its auto-covariance as

 

Putting the expression for   and  , we get

 

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

 

The first term in the third line is zero due to the orthogonality principle. Since  , we can re-write   in terms of covariance matrices as

 

This we can recognize to be the same as   Thus the minimum mean square error achievable by such a linear estimator is

 .

Univariate caseEdit

For the special case when both   and   are scalars, the above relations simplify to

 
 

where   is the Pearson's correlation coefficient between   and  .

ComputationEdit

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

Linear MMSE estimator for linear observation processEdit

Let us further model the underlying process of observation as a linear process:  , where   is a known matrix and   is random noise vector with the mean   and cross-covariance  . Here the required mean and the covariance matrices will be

 
 
 

Thus the expression for the linear MMSE estimator matrix   further modifies to

 

Putting everything into the expression for  , 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, (i.e. the dimension of  ) need not be at least as large as the number of unknowns, n, (i.e. the dimension of  ). The estimate for the linear observation process exists so long as the m-by-m matrix   exists; this is the case for any m if, for instance,   is positive definite. Physically the reason for this property is that since   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, there need be no measurement error. Thus, we may have  , because as long as   is positive definite, the estimate still exists. Lastly, this technique can handle cases where the noise is correlated.

Alternative formEdit

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

 

which can be established by post-multiplying by   and pre-multiplying by   to obtain

 

and

 

Since   can now be written in terms of   as  , we get a simplified expression for   as

 

In this form the above expression can be easily compared with weighed least square and Gauss–Markov estimate. In particular, when  , corresponding to infinite variance of the apriori information concerning  , the result   is identical to the weighed linear least square estimate with   as the weight matrix. Moreover, if the components of   are uncorrelated and have equal variance such that   where   is an identity matrix, then   is identical to the ordinary least square estimate.

Sequential linear MMSE estimationEdit

In many real-time application, observational data is not available in a single batch. Instead the observations are made in a sequence. A naive application of previous formulas would have us discard an old estimate and recompute a new estimate as fresh data is made available. But then we lose all information provided by the old observation. When the observations are scalar quantities, one possible way of avoiding such re-computation is to first concatenate the entire sequence of observations and then apply the standard estimation formula as done in Example 2. But this can be very tedious because as the number of observation increases so does the size of the matrices that need to be inverted and multiplied grow. Also, this method is difficult to extend to the case of vector observations. Another approach to estimation from sequential observations is to simply update an old estimate as additional data becomes available, leading to finer estimates. Thus a recursive method is desired where the new measurements can modify the old estimates. Implicit in these discussions is the assumption that the statistical properties of   does not change with time. In other words,   is stationary.

For sequential estimation, if we have an estimate   based on measurements generating space  , 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.

Suppose an optimal estimate   has been formed on the basis of past measurements and that error covariance matrix is  . For linear observation processes the best estimate of   based on past observation, and hence old estimate  , is  . Subtracting   from  , we obtain the prediction error

 .

The new estimate based on additional data is now

 

where   is the cross-covariance between   and   and   is the auto-covariance of  

Using the fact that   and  , we can obtain the covariance matrices in terms of error covariance as

 
 

Putting everything together, we have the new estimate as

 

and the new error covariance as

 

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

  1.  
  2.  
  3.  

The matrix   is often referred to as the gain factor. 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.

Special case: scalar observationsEdit

As an important special case, an easy to use recursive expression can be derived when at each t-th time instant the underlying linear observation process yields a scalar such that  , where   is n-by-1 known column vector whose values can change with time,   is n-by-1 random column vector to be estimated, and   is scalar noise term with variance  . After (t+1)-th observation, the direct use of above recursive equations give the expression for the estimate   as:

 

where   is the new scalar observation and the gain factor   is n-by-1 column vector given by

 

The   is n-by-n error covariance matrix given by

 

Here, no matrix inversion is required. Also, the gain factor,  , depends on our confidence in the new data sample, as measured by the noise variance, versus that in the previous data. The initial values of   and   are taken to be the mean and covariance of the aprior probability density function of  .

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. However, since the estimation error   cannot be directly observed, these methods try to minimize the mean squared prediction error  . For instance, in the case of scalar observations, we have the gradient   Thus, the update equation for the least mean square filter is given by

 

where   is the scalar step size and the expectation is approximated by the instantaneous value  . As we can see, these methods bypass the need for covariance matrices.

ExamplesEdit

Example 1Edit

We shall take a linear prediction problem as an example. Let a linear combination of observed scalar random variables   and   be used to estimate another future scalar random variable   such that  . If the random variables   are real Gaussian random variables with zero mean and its covariance matrix given by

 

then our task is to find the coefficients   such that it will yield an optimal linear estimate  .

In terms of the terminology developed in the previous sections, for this problem we have the observation vector  , the estimator matrix   as a row vector, and the estimated variable   as a scalar quantity. The autocorrelation matrix   is defined as

 

The cross correlation matrix   is defined as

 

We now solve the equation   by inverting   and pre-multiplying to get

 

So we have     and   as the optimal coefficients for  . Computing the minimum mean square error then gives  .[2] Note that it is not necessary to obtain an explicit matrix inverse of   to compute the value of  . 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 2Edit

Consider a vector   formed by taking   observations of a fixed but unknown scalar parameter   disturbed by white Gaussian noise. We can describe the process by a linear equation  , where  . Depending on context it will be clear if   represents a scalar or a vector. Suppose that we know   to be the range within which the value of   is going to fall in. We can model our uncertainty of   by an aprior uniform distribution over an interval  , and thus   will have variance of  . Let the noise vector   be normally distributed as   where   is an identity matrix. Also   and   are independent and  . It is easy to see that

 

Thus, the linear MMSE estimator is given by

 

We can simplify the expression by using the alternative form for   as

 

where for   we have  

Similarly, the variance of the estimator is

 

Thus the MMSE of this linear estimator is

 

For very large  , 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

 

while the variance will be unaffected by data   and the LMMSE of the estimate will tend to zero.

However, the estimator is suboptimal since it is constrained to be linear. Had the random variable   also been Gaussian, then the estimator would have been optimal. Notice, that the form of the estimator will remain unchanged, regardless of the apriori distribution of  , so long as the mean and variance of these distributions are the same.

Example 3Edit

Consider a variation of the above example: Two candidates are standing for an election. Let the fraction of votes that a candidate will receive on an election day be   Thus the fraction of votes the other candidate will receive will be   We shall take   as a random variable with a uniform prior distribution over   so that its mean is   and variance is   A few weeks before the election, two independent public opinion polls were conducted by two different pollsters. The first poll revealed that the candidate is likely to get   fraction of votes. Since some error is always present due to finite sampling and the particular polling methodology adopted, the first pollster declares their estimate to have an error   with zero mean and variance   Similarly, the second pollster declares their estimate to be   with an error   with zero mean and variance   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

 

Here, both the  . Thus, we can obtain the LMMSE estimate as the linear combination of   and   as

 

where the weights are given by

 

Here, since the denominator term is constant, the poll with lower error is given higher weight in order to predict the election outcome. Lastly, the variance of the prediction is given by

 

which makes   smaller than  

In general, if we have   pollsters, then   where the weight for i-th pollster is given by  

Example 4Edit

Suppose that a musician is playing an instrument and that the sound is received by two microphones, each of them located at two different places. Let the attenuation of sound due to distance at each microphone be   and  , which are assumed to be known constants. Similarly, let the noise at each microphone be   and  , each with zero mean and variances   and   respectively. Let   denote the sound produced by the musician, which is a random variable with zero mean and variance   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

 

Here both the  . Thus, we can combine the two sounds as

 

where the i-th weight is given as

 

See alsoEdit

NotesEdit

  1. ^ "Mean Squared Error (MSE)". www.probabilitycourse.com. Retrieved 9 May 2017.
  2. ^ Moon and Stirling.

Further readingEdit