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Smoothing Problem
The smoothing problem (not to be confused with smoothing in statistics, image processing and other contexts) is the problem of estimating an unknown probability density function recursively over time using incremental incoming measurements. It is one of the main problems defined by Norbert Wiener.1942, ''Extrapolation, Interpolation and Smoothing of Stationary Time Series''. A war-time classified report nicknamed "the yellow peril" because of the color of the cover and the difficulty of the subject. Published postwar 1949 MIT Press. http://www.isss.org/lumwiener.htm A smoother is an algorithm that implements a solution to this problem, typically based on recursive Bayesian estimation. The smoothing problem is closely related to the filtering problem, both of which are studied in Bayesian smoothing theory. A smoother is often a two-pass process, composed of forward and backward passes. Consider doing estimation (prediction/retrodiction) about an ongoing process (e.g. tracking a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. In smoothing, the data points of a signal are modified so individual points higher than the adjacent points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased leading to a smoother signal. Smoothing may be used in two important ways that can aid in data analysis (1) by being able to extract more information from the data as long as the assumption of smoothing is reasonable and (2) by being able to provide analyses that are both flexible and robust. Many different algorithms are used in smoothing. Compared to curve fitting Smoothing may be distinguished from the related and partially overlapping concept of curve fitting in the following ways: * curve fitting often involves the use of an explicit functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filter Design
Filter design is the process of designing a signal processing filter that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to an acceptable degree. The filter design process can be described as an optimization problem. Certain parts of the design process can be automated, but an experienced designer may be needed to get a good result. The design of digital filters is a complex topic. Although filters are easily understood and calculated, the practical challenges of their design and implementation are significant and are the subject of advanced research. Typical design requirements Typical requirements which may be considered in the design process are: * Frequency response * Phase shift or group delay * impulse response * Causal filter required? * Stable filter required? * Finite (in duration) impulse response required? * Computational complexity * Technology The f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Filters
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Estimation
In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior probability, posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation. Definition Suppose an unknown parameter \theta is known to have a prior distribution \pi. Let \widehat = \widehat(x) be an estimator of \theta (based on some measurements ''x''), and let L(\theta,\widehat) be a loss function, such as squared error. The Bayes risk of \widehat is defined as E_\pi(L(\theta, \widehat)), where the Expected value, expectation is taken over the probability distribution of \theta: this defines the risk function as a function of \widehat. An estimator \widehat is said to be a ''Bayes estimator'' if it minimizes the Bayes risk among all estimat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. In smoothing, the data points of a signal are modified so individual points higher than the adjacent points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased leading to a smoother signal. Smoothing may be used in two important ways that can aid in data analysis (1) by being able to extract more information from the data as long as the assumption of smoothing is reasonable and (2) by being able to provide analyses that are both flexible and robust. Many different algorithms are used in smoothing. Compared to curve fitting Smoothing may be distinguished from the related and partially overlapping concept of curve fitting in the following ways: * curve fitting often involves the use of an explicit functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Filtering
Generalized filtering is a generic Bayesian filtering scheme for nonlinear state-space models. It is based on a variational principle of least action, formulated in generalized coordinates of motion. Note that "generalized coordinates of motion" are related to—but distinct from—generalized coordinates as used in (multibody) dynamical systems analysis. Generalized filtering furnishes posterior densities over hidden states (and parameters) generating observed data using a generalized gradient descent on variational free energy, under the Laplace assumption. Unlike classical (e.g. Kalman-Bucy or particle) filtering, generalized filtering eschews Markovian assumptions about random fluctuations. Furthermore, it operates online, assimilating data to approximate the posterior density over unknown quantities, without the need for a backward pass. Special cases include variational filtering, dynamic expectation maximization and generalized predictive coding. Definition Definition: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kalman Filter
In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán. Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships Dynamic positioning, positioned dynamically. Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics. K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filter (signal Processing)
In signal processing, a filter is a device or process that removes some unwanted components or features from a Signal (electronics), signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequency, frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist. Correlations can be removed for certain frequency components and not for others without having to act in the frequency domain. Filters are widely used in electronics and telecommunication, in radio, television, audio recording, radar, control systems, music synthesis, image processing, computer graphics, and structural dynamics. There are many different bases of classifying filters and these overlap in many different ways; there is no simple hierarchical classification. Fil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtering Problem
In the theory of stochastic processes, filtering describes the problem of determining the state of a system from an incomplete and potentially noisy set of observations. For example, in GPS navigation, filtering helps estimate a car’s true position (the state) from noisy satellite signals (the observations). While originally motivated by problems in engineering, filtering found applications in many fields from signal processing to finance. The problem of optimal non-linear filtering (even for the non-stationary case) was solved by Ruslan L. Stratonovich (1959, 1960), see also Harold J. Kushner's work and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter known as the Zakai equation. The solution, however, is infinite-dimensional in the general case.Mireille Chaleyat-Maurel and Dominique Michel. Des resultats de non existence de filtre de dimension finie. Stochastics, 13(1+2):83-102, 1984. Certain approximations and special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Filter
In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant ( LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and additive noise. The Wiener filter minimizes the mean square error between the estimated random process and the desired process. Description The goal of the wiener filter is to compute a statistical estimate of an unknown signal using a related signal as an input and filtering it to produce the estimate. For example, the known signal might consist of an unknown signal of interest that has been corrupted by additive noise. The Wiener filter can be used to filter out the noise from the corrupted signal to provide an estimate of the underlying signal of interest. The Wiener filter is based on a statistical approach, and a more statistical account of the theory is given in the minimum mean square error (MMSE) estimator article. Typical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Retrodiction
Retrodiction is the act of making a prediction about the past. It is also known as postdiction (but this should not be confused with the use of the term in criticisms of parapsychological research). Activity The activity of retrodiction (or postdiction) involves moving backwards in time, step-by-step, in as many stages as are considered necessary, from the present into the speculated past to establish the ultimate cause of a specific event (for instance, in the case of reverse engineering, forensics, etc.). Given that retrodiction is a process in which "past observations, events and data are used as evidence to infer the process(es) that produced them" and that diagnosis "involve going from visible effects such as symptoms, signs and the like to their prior causes",Einhorn & Hogarth (1982), p.24. the essential balance between prediction and retrodiction could be characterized as: ::retrodiction : diagnosis :: prediction : prognosis regardless of whether the prognosis is of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
