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Encyclopedia > 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. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Signal processing is the processing, amplification and interpretation of signals and deals with the analysis and manipulation of signals. ... It has been suggested that this article or section be merged with estimation theory. ...


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. This long range RADAR antenna, known as ALTAIR, is used to detect and track space objects in conjunction with ABM testing at the Ronald Reagan Test Site on the Kwajalein atoll[1]. RADAR is a system that uses radio waves to detect, determine the direction and distance and/or speed...


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. In science, and especially in physics and telecommunication, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ...

Contents

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):

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. From Latin ex- + -periri (akin to periculum attempt). ... Signal processing is the processing, amplification and interpretation of signals and deals with the analysis and manipulation of signals. ... In medicine, a clinical trial (synonyms: clinical studies, research protocols, medical research) is a research study. ... Opinion polls are surveys of opinion using sampling. ... In engineering and manufacturing, quality control and quality engineering are involved in developing systems which ensure that products or services are designed and produced to meet or exceed customer requirements and expectations. ... Copy of the original phone of Graham Bell at the Musée des Arts et Métiers in Paris Telecommunication is the transmission of signals over a distance for the purpose of communication. ... In engineering and mathematics, control theory deals with the behavior of dynamical systems. ... The Kalman filter is an efficient recursive filter which estimates the state of a dynamic system from a series of incomplete and noisy measurements. ... An actuator is the mechanism by which an agent acts upon an environment. ... A network intrusion detection system (NIDS) is a system that tries to detect malicious activity such as denial of service attacks, port-scans or even attempts to crack into computers by monitoring network traffic. ... In science, and especially in physics and telecommunication, noise is fluctuations in and the addition of external factors to the stream of target information (signal) being received at a detector. ... Informally, probable is one of several words applied to uncertain events or knowledge, being closely related in meaning to likely, risky, hazardous, and doubtful. ... In mathematics, the term optimization refers to the study of problems that have the form Given: a function f : A R from some set A to the real numbers Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A (minimization) or such that... In statistics and information theory, the Fisher information (denoted ) is the variance of the score. ...


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. In mathematics, the term optimization refers to the study of problems that have the form Given: a function f : A R from some set A to the real numbers Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A (minimization) or such that...


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 inequality.
  • 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 statistics, the Cramér-Rao inequality, named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, expresses an upper bound on the precision of a statistical estimator, based on Fisher information. ...


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, A sample is that part of a population which is actually observed. ... A multivariate random variable or random vector is a vector X=(X1,...,Xn) whose components are scalar-valued random variables on the same probability space (Ω, P). ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...

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) In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ...

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 Bayesian inference is statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ... Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements; when used with Bayes theorem, it then becomes Bayesian inference. ...

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 Minimum mean-square error (MMSE) relates to an estimator having estimates with the minimum mean squared error possible. ...

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: It has been suggested that this article or section be merged with estimation theory. ...

Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. ... In decision theory and estimation theory, a Bayes estimator is an estimator or decision rule which is optimal given a prior probability of the estimated parameters. ... In statistics, the method of moments is a method of estimation of population parameters such as mean, variance, median, etc. ... In statistics, the Cramér-Rao inequality, named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, expresses an upper bound on the precision of a statistical estimator, based on Fisher information. ... Minimum mean-square error (MMSE) relates to an estimator having estimates with the minimum mean squared error possible. ... In statistics, the method of maximum a posteriori (MAP, or posterior mode) estimation can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. ... In statistics, and more specifically in estimation theory, a minimum-variance unbiased estimator (MVUE or MVU estimator) is an unbiased estimator of parameters, whose variance is minimized for all values of the parameters. ... This article is not about Gauss-Markov processes. ... In statistics, the term bias is used for two different concepts. ... Result of particle filtering (red line) based on observed data generated from the blue line ( Much larger image) Particle filter methods, also known as Sequential Monte Carlo (SMC), are sophisticated model estimation techniques based on simulation. ... Markov chain Monte Carlo (MCMC) methods (which include random walk Monte Carlo methods) are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. ... The Kalman filter is an efficient recursive filter which estimates the state of a dynamic system from a series of incomplete and noisy measurements. ... The Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published [1]. // Description Unlike the typical filtering theory of designing a filter for a desired frequency response the Wiener filter approaches filtering from a different angle. ...

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 σ2 (i.e., mathcal{N}(0, sigma^2)). Since the variance is known then the only unknown parameter is A. A discrete signal is a signal that has been sampled from a continuous signal. ... A sample is that part of a population which is actually observed. ... Direct current (DC or continuous current) is the continuous flow of electricity through a conductor such as a wire from high to low potential. ... In communications, the additive white Gaussian noise (AWGN) channel model is one in which the only impairment is the linear addition of wideband or white noise with a constant spectral density (expressed as watts per hertz of bandwidth) and a Gaussian distribution of amplitude. ... In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...


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 In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ... In statistics, mean has two related meanings: the average in ordinary English, which is also called the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

mathrm{E}left[hat{A}_1right] = 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) = 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 Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

p(w[n]) = frac{1}{sigma sqrt{2 pi}} expleft(- 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}} expleft(- 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{2pi}right)^N} expleft(- frac{1}{2 sigma^2} sum_{n=0}^{N-1}(x[n] - A)^2 right)

Taking the natural logarithm of the pdf The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...

ln p(mathbf{x}; A) = -N ln left(sigma sqrt{2pi}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 In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...

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 In statistics, the Cramér-Rao inequality, named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, states that the reciprocal of the Fisher information, , of a parameter , is a lower bound on the variance of an unbiased estimator of the parameter (denoted ). In some cases, no unbiased estimator... In statistics and information theory, the Fisher information (denoted ) is the variance of the score. ...

mathcal{I}(A) = mathrm{E} left( left[ frac{partial}{partialtheta} ln p(mathbf{x}; A) right]^2 right) = -mathrm{E} left[ frac{partial^2}{partialtheta^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. In statistics, and more specifically in estimation theory, a minimum-variance unbiased estimator (MVUE or MVU estimator) is an unbiased estimator of parameters, whose variance is minimized for all values of the parameters. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. ...


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


Books

  • 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 0-38-794173-8)
  • Detection, Estimation, and Modulation Theory, Part 1 by Harry L. Van Trees (ISBN 0-47-109517-6; website)
  • Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches by Dan Simon website

See also


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