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) of the fitted values of a dependent variable, which is a common measure of estimator quality.
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. 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.
Let be a unknown (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, then MSE expression simplifies to . Note that MSE could equivalently be defined in other ways, since
The MMSE estimator is then defined as the estimator achieving minimal MSE.
- Under some weak regularity assumptions, the MMSE estimator is uniquely defined, 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
- 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,
- If and are jointly Gaussian, then the MMSE estimator is linear, i.e., it has the form for matrix and constant . As a consequence, to find the MMSE estimator, it is sufficient to find the linear MMSE estimator. Such a situation occurs in the example presented in the next section.
Linear MMSE estimator
In many cases, it is not possible to determine a closed form expression for the conditional expectation required to obtain the MMSE estimator. Direct numerical evaluation of the conditional expectation is computationally expensive, since they often require multidimensional integration. In such cases, 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. The linear MMSE estimator is the estimator achieving minimum MSE among all estimators of such form. Such linear estimator only depends on the first two moments of the probability density function. So although it may be convenient to assume that and are jointly Gaussian, it is not necessarily to make this assumption, so long as the assumed distribution has well defined first and second moments.
The expression for optimal and is given by
Thus the expression for linear MMSE estimator, its mean, and its auto-covariance is given by
where , the is cross-covariance matrix between and , the is auto-covariance matrix of , and the is cross-covariance matrix between and . Lastly, the error covariance and minimum mean square error achievable by such estimator is
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. Levinson recursion is a fast method when is also a Toeplitz matrix. This can happen when is a wide sense stationary process.
These estimators are sometimes referred to as Wiener-Kolmogorov filters.
Linear MMSE estimator for linear observation process
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.
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
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 such that where is an identity matrix, then which has the same expression as the ordinary least square estimate.
Sequential linear MMSE estimation
For stationary process
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 and thus . The new estimate based on additional data is now
which works out to be
and the new error covariance is
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
The matrix is often referred to as the gain factor.
For example, an easy to use recursive expression can be derived when at each m-th time instant the underlying linear observation process yields a scalar such that , where is 1-by-n known row 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 (m+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 .
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 section, 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 . 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.
Consider a vector formed by taking observations of a random 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. Let the aprior distribution of be uniform 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
The last step is due to a special case of the matrix binomial inverse theorem (also known as Woodbury matrix identity). The matrix thus obtained in the last step, , will have as diagonal terms and as off-diagonal terms. Taking the product with respect to , we get the required estimator
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 unknown random variable 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.
- Lehmann and Casella, Corollary 4.1.2.
- Moon and Stirling.
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