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.

Contents 1 Fields that use estimation theory 2 Estimation process 3 Basics 4 Estimators 5 Example: DC gain in white Gaussian noise 5.1 Maximum likelihood 5.2 Cramér-Rao lower bounds 6 References 7 See also

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 as possible.

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,

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

Secondly, we have the corresponding $M$ parameters

$\backslash mathbf\{\backslash theta\}\; =\; \backslash begin\{bmatrix\}\; \backslash theta\_1\; \backslash \backslash \; \backslash theta\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash theta\_M\; \backslash end\{bmatrix\}$,

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

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

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

$\backslash pi(\; \backslash mathbf\{\backslash theta\})$.

After the model is formed, the goal is to estimate the parameters, commonly denoted $\backslash hat\{\backslash mathbf\{\backslash theta\}\}$, 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

$\backslash mathbf\{e\}\; =\; \backslash hat\{\backslash mathbf\{\backslash theta\}\}\; -\; \backslash mathbf\{\backslash theta\}$

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

Estimators

Commonly-used estimators, and 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 — 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 $\backslash sigma^2$ (i.e., $\backslash mathcal\{N\}(0,\; \backslash sigma^2)$). Since the variance is known then the only unknown parameter is $A$.

The model for the signal is then

$x[n]\; =\; A\; +\; w[n]\; \backslash quad\; n=0,\; 1,\; \backslash dots,\; N-1$

Two possible (of many) estimators are:

• $\backslash hat\{A\}\_1\; =\; x[0]$
• $\backslash hat\{A\}\_2\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{n=0\}^\{N-1\}\; x[n]$ 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

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

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.

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

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 $N\; \backslash to\; \backslash infty$, 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

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

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

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

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

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

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 $-\backslash mathrm\{E\}\; \backslash left[$

\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.

References

• 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

See also

• Best linear unbiased estimator (BLUE)
• Completeness (statistics)
• Cramér-Rao bound
• Detection theory
• Efficiency (statistics)
• Estimator, Estimator bias
• Expectation-maximization algorithm (EM algorithm)
• Information theory
• Kalman filter
• Least-squares spectral analysis
• Markov chain Monte Carlo (MCMC)
• Matched filter
• Maximum a posteriori (MAP)
• Maximum likelihood
• Maximum entropy spectral estimation
• Method of moments, generalized method of moments
• Minimum mean squared error (MMSE)
• Minimum variance unbiased estimator (MVUE)
• Nuisance variable
• Parametric equation
• Particle filter
• Rao-Blackwell theorem
• Spectral density, Spectral density estimation
• Statistical signal processing
• Sufficiency (statistics)
• Wiener filter

Digital signal processing
Theory — Discrete frequency | Nyquist–Shannon sampling theorem | estimation theory | detection theory
Sub-fields — audio signal processing | control engineering | digital image processing | speech processing | statistical signal processing
Techniques — Discrete Fourier transform (DFT) | Discrete-time Fourier transform (DTFT) | Impulse invariance | bilinear transform | Z-transform, advanced Z-transform
Sampling — oversampling | undersampling | downsampling | upsampling | aliasing | anti-aliasing filter | sampling rate | Nyquist rate/frequency
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