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There are numerous fields that require the use of estimation theory. |
There are numerous fields that require the use of estimation theory. |
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Some of these fields include (but by no means limited to): |
Some of these fields include (but by no means limited to): |
||
+ | * Interpretation of scientific [[experiments]] |
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− | * [[Medicine]] |
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− | + | * [[Signal processing]] |
|
+ | * [[Clinical trial]]s |
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− | ** Imaging: |
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− | *** [[Computed axial tomography|CAT]] |
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− | *** [[Electroencephalography|EEG]] |
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− | *** [[Electrocardiogram|EKG/ECG]] |
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− | *** [[MRI]] |
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− | *** [[Medical ultrasonography]] |
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* [[Opinion poll]]s |
* [[Opinion poll]]s |
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* [[Quality control]] |
* [[Quality control]] |
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− | * [[Radar]], [[sonar]] |
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− | ** Localization of objects |
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− | * [[Telecommunication]]s |
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− | ** [[Channel (communications)|Channel]] parameters |
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− | ** Noise [[variance]] |
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− | ** DC gain (see example below) |
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− | * [[Seismology]] |
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− | ** Localization of underground oil deposits |
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* [[Control theory]] |
* [[Control theory]] |
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+ | |||
− | ** [[Actuator]] changes with time |
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⚫ | |||
− | ** Position of objects in images (''see [[computer vision]]'') |
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− | * [[Digital signal processing]] |
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− | ** Parametric (e.g., [[periodogram]], [[correlogram]]) spectral analysis |
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− | ** nonparametric (e.g., [[MUSIC]], [[Root-MUSIC]], [[ESPRIT]]) spectral analysis |
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⚫ | |||
− | ** [[Wiener filter]] |
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− | ** [[Particle filter]] |
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− | * [[Network intrusion detection system]] |
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The measured data is likely to be subject to [[noise (physics)|noise]] or uncertainty and it is through statistical [[probability]] that [[optimization (mathematics)|optimal]] solutions are sought to extract as much [[Fisher information|information]] from the data. |
The measured data is likely to be subject to [[noise (physics)|noise]] or uncertainty and it is through statistical [[probability]] that [[optimization (mathematics)|optimal]] solutions are sought to extract as much [[Fisher information|information]] from the data. |
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These are the general steps to arrive at an estimator: |
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. |
* 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 |
+ | * 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). |
* 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 |
+ | * 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. |
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. |
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== Basics == |
== Basics == |
||
To build a model, several statistical "ingredients" need to be known. |
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 " |
+ | 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 sample]]s taken from a [[random vector]] (RV) of size <math>N</math> |
+ | The first is a set of [[statistical sample]]s taken from a [[random vector]] (RV) of size <math>N</math>. Put into a [[vector (spatial)|vector]], |
− | : <math>\mathbf{x} = \begin{bmatrix} x[0] \\ x[1] \\ \vdots \\ x[N-1] \end{bmatrix}</math> |
+ | : <math>\mathbf{x} = \begin{bmatrix} x[0] \\ x[1] \\ \vdots \\ x[N-1] \end{bmatrix}</math>. |
− | + | Secondly, we have the corresponding <math>M</math> parameters |
|
− | : <math>\mathbf{\theta} = \begin{bmatrix} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_M \end{bmatrix}</math> |
+ | : <math>\mathbf{\theta} = \begin{bmatrix} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_M \end{bmatrix}</math>, |
− | need to be established with their [[probability density function]] (pdf) or [[probability mass function]] (pmf) |
+ | which need to be established with their [[probability density function]] (pdf) or [[probability mass function]] (pmf) |
− | : <math>p(\mathbf{x} | \mathbf{\theta})</math> |
+ | : <math>p(\mathbf{x} | \mathbf{\theta})</math>. |
− | It is also possible for the parameters themselves to have a probability distribution (e.g., [[Bayesian statistics]]) |
+ | 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]] |
− | : <math>\pi( \mathbf{\theta})</math> |
+ | : <math>\pi( \mathbf{\theta})</math>. |
− | After the model is formed, the goal is to estimate the parameters, |
+ | After the model is formed, the goal is to estimate the parameters, commonly denoted <math>\hat{\mathbf{\theta}}</math>, where the "hat" indicates the estimate. |
+ | |||
− | One common estimator is the [[minimum mean squared error]] (MMSE) estimator |
+ | 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 |
: <math>\mathbf{e} = \hat{\mathbf{\theta}} - \mathbf{\theta}</math> |
: <math>\mathbf{e} = \hat{\mathbf{\theta}} - \mathbf{\theta}</math> |
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This list is some of the more common [[estimator]]s used, and some topics related to them: |
This list is some of the more common [[estimator]]s used, and some topics related to them: |
||
*[[Maximum likelihood]] estimators |
*[[Maximum likelihood]] estimators |
||
+ | *[[Bayes estimator]]s |
||
− | *[[Method of moments]] estimators |
+ | *[[method of moments (statistics)|Method of moments]] estimators |
− | *[[Cramér-Rao |
+ | *[[Cramér-Rao bound]] |
− | *[[Minimum mean squared error]] (MMSE) |
+ | *[[Minimum mean squared error]] (MMSE), also known as Bayes least squared error (BLSE) |
*[[Maximum a posteriori]] (MAP) |
*[[Maximum a posteriori]] (MAP) |
||
*[[Minimum variance unbiased estimator]] (MVUE) |
*[[Minimum variance unbiased estimator]] (MVUE) |
||
*[[Best linear unbiased estimator]] (BLUE) |
*[[Best linear unbiased estimator]] (BLUE) |
||
− | *Unbiased estimators — see [[ |
+ | *Unbiased estimators — see [[estimator bias]]. |
*[[Particle filter]] |
*[[Particle filter]] |
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*[[Markov chain Monte Carlo]] (MCMC) |
*[[Markov chain Monte Carlo]] (MCMC) |
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*[[Kalman filter]] |
*[[Kalman filter]] |
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⚫ | |||
*[[Wiener filter]] |
*[[Wiener filter]] |
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= |
= |
||
\mathrm{var} \left( \frac{1}{N} \sum_{n=0}^{N-1} x[n] \right) |
\mathrm{var} \left( \frac{1}{N} \sum_{n=0}^{N-1} x[n] \right) |
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+ | \overset{independence}{=} |
||
− | = |
||
\frac{1}{N^2} \left[ \sum_{n=0}^{N-1} \mathrm{var} (x[n]) \right] |
\frac{1}{N^2} \left[ \sum_{n=0}^{N-1} \mathrm{var} (x[n]) \right] |
||
= |
= |
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This example of DC gain + WGN is a recurring example in Kay's ''Fundamentals of Statistical Signal Processing''. |
This example of DC gain + WGN is a recurring example in Kay's ''Fundamentals of Statistical Signal Processing''. |
||
+ | == History == |
||
+ | Frank Frieman invented parametric cost modeling for estimating acquisition and development of hardware systems in 1969. |
||
− | ==Books== |
||
+ | |||
+ | In 1975 [[PRICE Systems]] business launched within RCA Corporation with introduction of first parametric Hardware Acquisition Cost Model (Frank Frieman, founder) |
||
+ | |||
+ | ==See also== |
||
+ | *[[Parametric equation]] |
||
+ | |||
+ | |||
+ | |||
+ | ==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) |
* ''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- |
+ | * ''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- |
+ | * ''Detection, Estimation, and Modulation Theory, Part 1'' by Harry L. Van Trees (ISBN 0-471-09517-6; [http://gunston.gmu.edu/demt/demtp1/ website]) |
+ | * ''Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches'' by Dan Simon [http://academic.csuohio.edu/simond/estimation/ website] |
||
==See also== |
==See also== |
||
⚫ | |||
⚫ | |||
+ | * [[Estimator bias]] |
||
* [[Completeness (statistics)]] |
* [[Completeness (statistics)]] |
||
⚫ | |||
* [[Detection theory]] |
* [[Detection theory]] |
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* [[Efficiency (statistics)]] |
* [[Efficiency (statistics)]] |
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+ | * [[Estimator]] |
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* [[Expectation-maximization algorithm]] (EM algorithm) |
* [[Expectation-maximization algorithm]] (EM algorithm) |
||
* [[Information theory]] |
* [[Information theory]] |
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− | * [[ |
+ | * [[Kalman filter]] |
⚫ | |||
− | * [[Sufficiency (statistics)]] |
||
⚫ | |||
⚫ | |||
* [[Maximum likelihood]] |
* [[Maximum likelihood]] |
||
+ | * [[Maximum entropy spectral estimation]] |
||
− | * [[Method of moments]], [[generalized method of moments]] |
+ | * [[Method of moments (statistics)|Method of moments]], [[generalized method of moments]] |
⚫ | |||
* [[Minimum mean squared error]] (MMSE) |
* [[Minimum mean squared error]] (MMSE) |
||
⚫ | |||
* [[Minimum variance unbiased estimator]] (MVUE) |
* [[Minimum variance unbiased estimator]] (MVUE) |
||
⚫ | |||
− | * Unbiased estimators — see [[bias (statistics)]]. |
||
* [[Particle filter]] |
* [[Particle filter]] |
||
+ | * [[Rao-Blackwell theorem]] |
||
⚫ | |||
⚫ | |||
⚫ | |||
⚫ | |||
* [[Wiener filter]] |
* [[Wiener filter]] |
||
+ | |||
⚫ | |||
+ | |||
− | [[Category:Estimation]] |
||
⚫ | |||
[[Category:Signal processing]] |
[[Category:Signal processing]] |
||
− | [[Category:Statistics]] |
||
+ | :es:Estimación estadística |
||
:he:אמידה |
:he:אמידה |
||
+ | :zh:估计理论 |
||
{{enWP|Estimation_theory}} |
{{enWP|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 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
- Control theory
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 . Put into a vector,
- .
Secondly, we have the corresponding parameters
- ,
which need to be established with their probability density function (pdf) or probability mass function (pmf)
- .
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
- .
After the model is formed, the goal is to estimate the parameters, commonly denoted , 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
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 — 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, , of independent samples that consists of a DC gain with Additive white Gaussian noise with known variance (i.e., ). Since the variance is known then the only unknown parameter is .
The model for the signal is then
Two possible (of many) estimators are:
- which is the sample mean
Both of these estimators have a mean of , which can be shown through taking the expected value of each estimator
and
At this point, these two estimators would appear to perform the same. However, the difference between them becomes apparent when comparing the variances.
and
It would seem that the sample mean is a better estimator since, as , 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 is
and the probability of becomes ( can be thought of a )
By independence, the probability of becomes
Taking the natural logarithm of the pdf
and the maximum likelihood estimator is
Taking the first derivative of the log-likelihood function
and setting it to zero
This results in the maximum likelihood estimator
which is simply the sample mean. From this example, it was found that the sample mean is the maximum likelihood estimator for 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
and copying from above
Taking the second derivative
and finding the negative expected value is trivial since it is now a deterministic constant
Finally, putting the Fisher information into
results in
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 and . 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.
History
Frank Frieman invented parametric cost modeling for estimating acquisition and development of hardware systems in 1969.
In 1975 PRICE Systems business launched within RCA Corporation with introduction of first parametric Hardware Acquisition Cost Model (Frank Frieman, founder)
See also
- Parametric equation
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)
- Estimator bias
- Completeness (statistics)
- Cramér-Rao bound
- Detection theory
- Efficiency (statistics)
- Estimator
- Expectation-maximization algorithm (EM algorithm)
- Information theory
- Kalman filter
- 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)
- Particle filter
- Rao-Blackwell theorem
- Statistical signal processing
- Sufficiency (statistics)
- Wiener filter
- es:Estimación estadística
- he:אמידה
- zh:估计理论
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