MMSE Minimum mean squared error

Last updated on May 4, 2023

Minimum mean squared error (MMSE) is a widely used criterion in signal processing, communication systems, and statistical estimation theory. MMSE is used to determine the best estimate of a random variable based on a noisy observation of the variable. The goal of MMSE is to minimize the expected value of the squared difference between the true value of the random variable and its estimate.

In this article, we will explain the concept of MMSE in detail, including its mathematical derivation, its applications, and its limitations.

Derivation of MMSE

Suppose we have a random variable X and we want to estimate its value based on an observation Y. However, the observation Y is corrupted by noise, so we cannot directly observe the value of X. Instead, we can use a function f(Y) to estimate the value of X. Our goal is to find the best possible function f(Y) that minimizes the expected value of the squared difference between the true value of X and its estimate.

To formalize this problem, we define the following variables:

X: the true value of the random variable we want to estimate
Y: the noisy observation of X
Z: the estimate of X based on Y

The goal is to find the function f(Y) that minimizes the expected value of the squared difference between X and Z:cssCopy codeE[(X-Z)^2]

We can expand this equation using the definition of Z:cssCopy codeE[(X-Z)^2] = E[(X-f(Y))^2]

We can simplify this expression using the following identity:scssCopy codeE[(X-f(Y))^2] = E[(X-E[Z]+E[Z]-f(Y))^2]

We can further simplify this expression using the fact that E[Z] is the expected value of Z:mathematicaCopy codeE[(X-f(Y))^2] = E[(X-E[Z]+E[Z]-f(Y))^2] = E[((X-E[Z])+(E[Z]-f(Y)))^2] = E[(X-E[Z])^2] + E[(E[Z]-f(Y))^2] + 2E[(X-E[Z])(E[Z]-f(Y))]

The first term in this expression is the variance of X:cssCopy codeE[(X-E[Z])^2] = var(X)

The second term is the variance of the estimation error:scssCopy codeE[(E[Z]-f(Y))^2] = var(Z-Y)

The third term is the covariance between X and the estimation error:scssCopy codeE[(X-E[Z])(E[Z]-f(Y))] = cov(X,Z-Y)

Substituting these expressions into the previous equation, we get:scssCopy codeE[(X-Z)^2] = var(X) + var(Z-Y) + 2cov(X,Z-Y)

This equation is known as the mean squared error (MSE). Our goal is to minimize the MSE by finding the best function f(Y). To do this, we can take the derivative of the MSE with respect to f(Y) and set it to zero:scssCopy coded/d(f(Y)) E[(X-Z)^2] = -2E[(X-Z)d/d(f(Y))Z] = 0

Solving for f(Y), we get:scssCopy codef(Y) = E[X|Y]

This equation states that the optimal estimate of X given Y is the conditional expectation of X given Y. In other words, we estimate X by taking the average of X over all possible values of Y.

Applications of MMSE

MMSE has many applications in signal processing, communication systems, and statistical estimation theory. Here are a few examples:

Denoising

One of the most common applications of MMSE is in denoising. Suppose we have a noisy signal Y that we want to recover the underlying clean signal X. We can use MMSE to estimate the value of X based on Y. To do this, we first estimate the conditional probability distribution of X given Y using the training data. Then, we use this conditional distribution to compute the optimal estimate of X for each observation of Y. This method is known as the MMSE estimator, and it has been shown to be effective in reducing the noise in signals.

Channel equalization

In communication systems, MMSE is used for channel equalization. A communication channel is typically characterized by its impulse response, which describes how the channel responds to a signal. The impulse response can be used to model the distortion introduced by the channel. MMSE can be used to estimate the clean signal based on the received signal and the channel impulse response. The MMSE estimator can be derived using the same method as for denoising, and it has been shown to be effective in mitigating the effects of channel distortion.

Estimation theory

MMSE is a fundamental concept in statistical estimation theory. It is used to derive the optimal estimator for a wide range of estimation problems, including parameter estimation, state estimation, and signal estimation. MMSE provides a benchmark for the performance of estimators and can be used to compare different estimation methods. In addition, MMSE can be used to derive the Cramer-Rao bound, which provides a lower bound on the variance of any unbiased estimator.

Limitations of MMSE

Although MMSE is a powerful and widely used criterion, it has several limitations. One of the main limitations is that it assumes that the conditional probability distribution of X given Y is known. In practice, this distribution may be unknown or difficult to estimate, especially if the dimensionality of X is high. In addition, MMSE is sensitive to the choice of the loss function. If the loss function is not well-suited to the problem at hand, the MMSE estimator may not perform well.

Another limitation of MMSE is that it assumes that the noise is Gaussian. In practice, the noise may be non-Gaussian, which can lead to suboptimal performance. In addition, MMSE assumes that the noise is uncorrelated and has a constant variance. If the noise is correlated or has a non-constant variance, the MMSE estimator may not perform well.

Finally, MMSE assumes that the observation Y is a linear function of X. If the relationship between Y and X is nonlinear, the MMSE estimator may not be optimal. In these cases, other estimation methods, such as maximum likelihood estimation or Bayesian estimation, may be more appropriate.

Conclusion

MMSE is a widely used criterion in signal processing, communication systems, and statistical estimation theory. MMSE is used to derive the optimal estimator for a wide range of estimation problems, including denoising, channel equalization, and parameter estimation. Although MMSE has several limitations, it provides a powerful tool for estimating random variables in noisy environments.