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Estimation theory

 
Sci-Tech Dictionary: estimation theory
 
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(′es·tə′mā·shən ′thē·ə·rē)

(statistics) A branch of probability and statistics concerned with deriving information about properties of random variables, stochastic processes, and systems based on observed samples.

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Sci-Tech Encyclopedia: Estimation theory
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A branch of probability and statistics concerned with deriving information about properties of random variables, stochastic processes, and systems based on observed samples. Some of the important applications of estimation theory are found in control and communication systems, where it is used to estimate the unknown states and parameters of the system.

The estimation problem for dynamic systems may be divided into two parts: parameter estimation and state estimation. The basic difference between a parameter and the state is that the former either does not change at all or changes slowly in time, whereas the latter continuously evolves in time. For example, the state of a satellite is a six-dimensional vector consisting of three position variables and three velocity variables along the axes of an orthogonal coordinate system. The parameters of the satellite are its mass, inertia, and so on. In many control and communication problems, some of the system parameters are not known with desired accuracy. The problem of estimating these parameters from observed data is called parameter identification, though it is basically a problem of estimation. The more general problem of developing a mathematical model of the system from observed data is called system identification. On the other hand, the problem of state estimation is described by names such as signal processing, filtering, and smoothing. The problem belongs to the theory of stochastic processes and is also commonly known as time series analysis. See also Stochastic process.

Three basic approaches used for estimation are least-squares, maximum-likelihood, and Bayesian. An estimator is defined as a function of the observations possessing certain desirable properties such as unbiasedness, consistency, and minimum variance. A Kalman filter provides estimates that are optimal in the least-squares, maximum-likelihood and Bayesian sense for a Gauss-Markov model. (A stochastic process is Markov if, given its present state, its future is independent of its past.)

Since their introduction in 1960, Kalman filters and their extensions have found numerous applications. Initially, these filters were developed for space applications such as satellite orbit determination, inertial navigation, Apollo lunar landing module guidance, and so on. The applications to power systems and industrial processes were developed shortly thereafter. Kalman filters have been used for forecasting, water quality prediction, hurricane tracking, aircraft landing systems, and stochastic control. See also Control systems; Flight controls; Guidance systems; Process control; Statistics; Systems engineering.

 
Wikipedia: Estimation theory
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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 an underlying physical setting in such a way that the value of the parameters affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.

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 in a noisy signal. Noise adds uncertainty, without which the problem would be deterministic and estimation would not be needed.

Contents

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. Estimator optimality usually refers to achieving minimum average error over some class of estimators, for example, a minimum variance unbiased estimator. In this case, the class is the set of unbiased estimators, and the average error measure is variance (average squared error between the value of the estimate and the parameter). However, optimal estimators do not always exist.

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 started 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

Commonly-used estimators and estimation methods, and topics related to them:

Examples

Unknown constant in additive white Gaussian noise

Consider a received discrete signal, x[n], of N independent samples that consists of an unknown constant A with additive white Gaussian noise w[n] with known variance σ2 (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:

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

Main article: 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 a fixed, unknown parameter corrupted by AWGN.

Cramér–Rao lower bound

For more details on this topic, see Cramér–Rao bound.

To find the Cramér-Rao lower bound (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 bound for all values of N and A. In other words, the sample mean is the (necessarily unique) efficient estimator, and thus also the minimum variance unbiased estimator (MVUE), in addition to being the maximum likelihood estimator.

Maximum of a uniform distribution

Main article: German tank problem

One of the simplest non-trivial examples of estimation is the estimation of the maximum of a uniform distribution. It is used as a hands-on classroom exercise and to illustrate basic principles of estimation theory. Further, in the case of estimation based on a single sample, it demonstrates philosophical issues and possible misunderstandings in the use of maximum likelihood estimators and likelihood functions.

Given a discrete uniform distribution 1,2,\dots,N with unknown maximum, the UMVU estimator for the maximum is given by

\frac{k+1}{k} m - 1 = m + \frac{m}{k} - 1

where m is the sample maximum and k is the sample size, sampling without replacement.[1][2] This problem is commonly known as the German tank problem, due to application of maximum estimation to estimates of German tank production during World War II.

The formula may be understood intuitively as:

"The sample maximum plus the average gap between observations in the sample",

the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum.[note 1]

This has a variance of[1]

\frac{1}{k}\frac{(N-k)(N+1)}{(k+2)} \approx \frac{N^2}{k^2} \text{ for small samples } k \ll N

so a standard deviation of approximately N / k, the (population) average size of a gap between samples; compare \frac{m}{k} above. This can be seen as a very simple case of maximum spacing estimation.

The sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased.

Applications

Numerous fields require the use of estimation theory. Some of these fields include (but are by no means limited to):

Measured data are 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 as possible.

See also

Notes

  1. ^ The sample maximum is never more than the population maximum, but can be less, hence it is a biased estimator: it will tend to underestimate the population maximum.

References

  1. ^ a b Johnson, Roger (1994), "Estimating the Size of a Population", Teaching Statistics 16 (2 (Summer)), doi:10.1111/j.1467-9639.1994.tb00688.x 
  2. ^ Johnson, Roger (2006), "Estimating the Size of a Population", Getting the Best from Teaching Statistics, http://www.rsscse.org.uk/ts/gtb/johnson.pdf 

Reference list

  • Mathematical Statistics and Data Analysis by John Rice. (ISBN 0-534-209343)
  • Fundamentals of Statistical Signal Processing: Estimation Theory by Steven M. Kay (ISBN 0-13-345711-7)
  • An Introduction to Signal Detection and Estimation by H. Vincent Poor (ISBN 0-387-94173-8)
  • Detection, Estimation, and Modulation Theory, Part 1 by Harry L. Van Trees (ISBN 0-471-09517-6; website)
  • Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches by Dan Simon website
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