Estimation theory

Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data. The parameters describe the physical scenario or object that answers a question posed by the estimator.

For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the unobservable parameter; the estimate is based on a small random sample of voters.

Or, for example, in radar the goal is to estimate the location of objects (airplanes, boats, etc.) by analyzing the received echo and a possible question to be posed is "where are the airplanes?" To answer where the airplanes are, it is necessary to estimate the distance the airplanes are at from the radar station, which can provide an absolute location if the absolute location of the radar station is known.

In estimation theory, it is assumed that the desired information is embedded into a noisy signal. Noise adds uncertainty and if there was no uncertainty then there would be no need for estimation.

Fields that use estimation theory
There are numerous fields that require the use of estimation theory. Some of these fields include (but by no means limited to):
 * Interpretation of scientific experiments
 * Signal processing
 * Clinical trials
 * Opinion polls
 * Quality control
 * Telecommunications
 * Control theory
 * Kalman filter
 * Actuator changes with time
 * Network intrusion detection system

The measured data is likely to be subject to noise or uncertainty and it is through statistical probability that optimal solutions are sought to extract as much information from the data.

Estimation process
The entire purpose of estimation theory is to arrive at an estimator, and preferably an implementable one that could actually be used. The estimator takes the measured data as input and produces an estimate of the parameters.

It is also preferable to derive an estimator that exhibits optimality. An optimal estimator would indicate that all available information in the measured data has been extracted, for if there was unused information in the data then the estimator would not be optimal.

These are the general steps to arrive at an estimator:
 * In order to arrive at a desired estimator for estimating a single or multiple parameters, it is first necessary to determine a model for the system. This model should incorporate the process being modeled as well as points of uncertainty and noise. The model describes the physical scenario in which the parameters apply.
 * After deciding upon a model, it is helpful to find the limitations placed upon an estimator. This limitation, for example, can be found through the Cramér-Rao bound.
 * Next, an estimator needs to be developed or applied if an already known estimator is valid for the model. The estimator needs to be tested against the limitations to determine if it is an optimal estimator (if so, then no other estimator will perform better).
 * Finally, experiments or simulations can be run using the estimator to test its performance.

After arriving at an estimator, real data might show that the model used to derive the estimator is incorrect, which may require repeating these steps to find a new estimator. A non-implementable or infeasible estimator may need to be scrapped and the process start anew.

In summary, the estimator estimates the parameters of a physical model based on measured data.

Basics
To build a model, several statistical "ingredients" need to be known. These are needed to ensure the estimator has some mathematical tractability instead of being based on "good feel".

The first is a set of statistical samples taken from a random vector (RV) of size $$N$$. Put into a vector,


 * $$\mathbf{x} = \begin{bmatrix} x[0] \\ x[1] \\ \vdots \\ x[N-1] \end{bmatrix}$$.

Secondly, we have the corresponding $$M$$ parameters


 * $$\mathbf{\theta} = \begin{bmatrix} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_M \end{bmatrix}$$,

which need to be established with their probability density function (pdf) or probability mass function (pmf)


 * $$p(\mathbf{x} | \mathbf{\theta})$$.

It is also possible for the parameters themselves to have a probability distribution (e.g., Bayesian statistics). It is then necessary to define the epistemic probability


 * $$\pi( \mathbf{\theta})$$.

After the model is formed, the goal is to estimate the parameters, commonly denoted $$\hat{\mathbf{\theta}}$$, where the "hat" indicates the estimate.

One common estimator is the minimum mean squared error (MMSE) estimator, which utilizes the error between the estimated parameters and the actual value of the parameters


 * $$\mathbf{e} = \hat{\mathbf{\theta}} - \mathbf{\theta}$$

as the basis for optimality. This error term is then squared and minimized for the MMSE estimator.

Estimators
This list is some of the more common estimators used, and some topics related to them:
 * Maximum likelihood estimators
 * Bayes estimators
 * Method of moments estimators
 * Cramér-Rao bound
 * Minimum mean squared error (MMSE), also known as Bayes least squared error (BLSE)
 * Maximum a posteriori (MAP)
 * Minimum variance unbiased estimator (MVUE)
 * Best linear unbiased estimator (BLUE)
 * Unbiased estimators &mdash; see estimator bias.
 * Particle filter
 * Markov chain Monte Carlo (MCMC)
 * Kalman filter
 * Ensemble Kalman filter (EnKF)
 * Wiener filter

Example: DC gain in white Gaussian noise
Consider a received discrete signal, $$x[n]$$, of $$N$$ independent samples that consists of a DC gain $$A$$ with Additive white Gaussian noise $$w[n]$$ with known variance $$\sigma^2$$ (i.e., $$\mathcal{N}(0, \sigma^2)$$). Since the variance is known then the only unknown parameter is $$A$$.

The model for the signal is then
 * $$x[n] = A + w[n] \quad n=0, 1, \dots, N-1$$

Two possible (of many) estimators are:
 * $$\hat{A}_1 = x[0]$$
 * $$\hat{A}_2 = \frac{1}{N} \sum_{n=0}^{N-1} x[n]$$ which is the sample mean

Both of these estimators have a mean of $$A$$, which can be shown through taking the expected value of each estimator


 * $$\mathrm{E}\left[\hat{A}_1\right] = \mathrm{E}\left[ x[0] \right] = A$$

and

\mathrm{E}\left[ \hat{A}_2 \right] = \mathrm{E}\left[ \frac{1}{N} \sum_{n=0}^{N-1} x[n] \right] = \frac{1}{N} \left[ \sum_{n=0}^{N-1} \mathrm{E}\left[ x[n] \right] \right] = \frac{1}{N} \left[ N A \right] = A $$

At this point, these two estimators would appear to perform the same. However, the difference between them becomes apparent when comparing the variances.


 * $$\mathrm{var} \left( \hat{A}_1 \right) = \mathrm{var} \left( x[0] \right) = \sigma^2$$

and

\mathrm{var} \left( \hat{A}_2 \right) = \mathrm{var} \left( \frac{1}{N} \sum_{n=0}^{N-1} x[n] \right) \overset{independence}{=} \frac{1}{N^2} \left[ \sum_{n=0}^{N-1} \mathrm{var} (x[n]) \right] = \frac{1}{N^2} \left[ N \sigma^2 \right] = \frac{\sigma^2}{N} $$

It would seem that the sample mean is a better estimator since, as $$N \to \infty$$, the variance goes to zero.

Maximum likelihood
Continuing the example using the maximum likelihood estimator, the probability density function (pdf) of the noise for one sample $$w[n]$$ is


 * $$p(w[n]) = \frac{1}{\sigma \sqrt{2 \pi}} \exp\left(- \frac{1}{2 \sigma^2} w[n]^2 \right)$$

and the probability of $$x[n]$$ becomes ($$x[n]$$ can be thought of a $$\mathcal{N}(A, \sigma^2)$$)


 * $$p(x[n]; A) = \frac{1}{\sigma \sqrt{2 \pi}} \exp\left(- \frac{1}{2 \sigma^2} (x[n] - A)^2 \right)$$

By independence, the probability of $$\mathbf{x}$$ becomes



p(\mathbf{x}; A) = \prod_{n=0}^{N-1} p(x[n]; A) = \frac{1}{\left(\sigma \sqrt{2\pi}\right)^N} \exp\left(- \frac{1}{2 \sigma^2} \sum_{n=0}^{N-1}(x[n] - A)^2 \right) $$

Taking the natural logarithm of the pdf



\ln p(\mathbf{x}; A) = -N \ln \left(\sigma \sqrt{2\pi}\right) - \frac{1}{2 \sigma^2} \sum_{n=0}^{N-1}(x[n] - A)^2 $$

and the maximum likelihood estimator is


 * $$\hat{A} = \arg \max \ln p(\mathbf{x}; A)$$

Taking the first derivative of the log-likelihood function



\frac{\partial}{\partial A} \ln p(\mathbf{x}; A) = \frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}(x[n] - A) \right] = \frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}x[n] - N A \right] $$

and setting it to zero



0 = \frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}x[n] - N A \right] = \sum_{n=0}^{N-1}x[n] - N A $$

This results in the maximum likelihood estimator



\hat{A} = \frac{1}{N} \sum_{n=0}^{N-1}x[n] $$

which is simply the sample mean. From this example, it was found that the sample mean is the maximum likelihood estimator for $$N$$ samples of AWGN with a fixed, unknown DC gain.

Cramér-Rao lower bounds
To find the Cramér-Rao lower bounds (CRLB) of the sample mean estimator, it is first necessary to find the Fisher information number



\mathcal{I}(A) = \mathrm{E} \left( \left[ \frac{\partial}{\partial\theta} \ln p(\mathbf{x}; A) \right]^2 \right) = -\mathrm{E} \left[ \frac{\partial^2}{\partial\theta^2} \ln p(\mathbf{x}; A) \right] $$

and copying from above



\frac{\partial}{\partial A} \ln p(\mathbf{x}; A) = \frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}x[n] - N A \right] $$

Taking the second derivative

\frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A) = \frac{1}{\sigma^2} (- N) = \frac{-N}{\sigma^2} $$

and finding the negative expected value is trivial since it is now a deterministic constant $$ -\mathrm{E} \left[ \frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A) \right] = \frac{N}{\sigma^2} $$

Finally, putting the Fisher information into



\mathrm{var}\left( \hat{A} \right) \geq \frac{1}{\mathcal{I}} $$

results in



\mathrm{var}\left( \hat{A} \right) \geq \frac{\sigma^2}{N} $$

Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is equal to the Cramér-Rao lower bounds for all values of $$N$$ and $$A$$. The sample mean is the minimum variance unbiased estimator (MVUE) in addition to being the maximum likelihood estimator.

This example of DC gain + WGN is a recurring example in Kay's Fundamentals of Statistical Signal Processing.