Filtering problem (stochastic processes) (original) (raw)
在随机过程理論中的濾波問題（Filtering problem）是指針對信号处理及相關領域中，許多狀態估測問題的數學模型。大致概念是從不完整的、可能包括雜訊的觀測值中，建立有關系統真實值的「最佳估測」。最佳非線性濾波問題（甚至也包括非平稳过程問題）由（1959年、1960年）找到解答，在的研究及的研究中也有提到，Zakai建立了濾波器在條件機率未歸一情況下的簡化動態模型，稱為。不過一般情形下的解是無限維的。 目前已針對一些近似以及一些特定條件有深入的研究。例如在高斯隨機變數的假設下，最佳解是線性濾波器，也稱為维纳滤波及卡尔曼滤波。更一般的情形下，其解為無限維度，為了在有限記憶體的電腦中計算，需要進行有限維度的近似，有限維的近似型比較會以启发為基礎，例如或是假定密度濾波器（Assumed Density Filters），也有更方法論導向的作法，例如Projection Filters，其中有些子系列恰好和假定密度濾波器相同。 一般來說，若可以適用分離原理，這些濾波器也可以成為最优控制問題解的一部份。例如在LQG控制最佳控制問題中，其估測部份的解就是卡爾曼濾波。
| Property | Value |
|---|---|
| dbo:abstract | 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. 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 Zakai equation. The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the extended Kalman filter or the assumed density filters, or more methodologically oriented such as for example the Projection Filters, some sub-families of which are shown to coincide with the Assumed Density Filters. In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the linear-quadratic-Gaussian control problem. (en) 在随机过程理論中的濾波問題（Filtering problem）是指針對信号处理及相關領域中，許多狀態估測問題的數學模型。大致概念是從不完整的、可能包括雜訊的觀測值中，建立有關系統真實值的「最佳估測」。最佳非線性濾波問題（甚至也包括非平稳过程問題）由（1959年、1960年）找到解答，在的研究及的研究中也有提到，Zakai建立了濾波器在條件機率未歸一情況下的簡化動態模型，稱為。不過一般情形下的解是無限維的。 目前已針對一些近似以及一些特定條件有深入的研究。例如在高斯隨機變數的假設下，最佳解是線性濾波器，也稱為维纳滤波及卡尔曼滤波。更一般的情形下，其解為無限維度，為了在有限記憶體的電腦中計算，需要進行有限維度的近似，有限維的近似型比較會以启发為基礎，例如或是假定密度濾波器（Assumed Density Filters），也有更方法論導向的作法，例如Projection Filters，其中有些子系列恰好和假定密度濾波器相同。 一般來說，若可以適用分離原理，這些濾波器也可以成為最优控制問題解的一部份。例如在LQG控制最佳控制問題中，其估測部份的解就是卡爾曼濾波。 (zh) |
| dbo:wikiPageID | 12433418 (xsd:integer) |
| dbo:wikiPageLength | 8529 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1121590071 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Probability_space dbr:Sigma_algebra dbr:Kalman-Bucy_filter dbr:Moshe_Zakai dbr:Conditional_expectation dbr:Stochastic_processes dbr:Harold_J._Kushner dbr:State_(controls) dbr:Measurable_function dbr:Linear_subspace dbc:Control_theory dbc:Signal_estimation dbr:Euclidean_space dbr:Noise_(signal_processing) dbr:Kalman_filter dbr:Random_variable dbr:Hilbert_space dbr:Wiener_filter dbr:Zakai_equation dbr:Dimension dbr:Filter_(signal_processing) dbr:Brownian_motion dbr:Optimal_control dbr:Orthogonal_projection dbc:Stochastic_differential_equations dbr:Stochastic_differential_equation dbr:Separation_principle dbr:Smoothing dbr:Smoothing_problem dbr:Extended_Kalman_filter dbr:Linear-quadratic-Gaussian_control dbr:Nonlinear_filter dbr:Kiyoshi_Itō dbr:Ruslan_L._Stratonovich |
| dbp:wikiPageUsesTemplate | dbt:Cite_book dbt:Reflist dbt:Short_description |
| dct:subject | dbc:Control_theory dbc:Signal_estimation dbc:Stochastic_differential_equations |
| gold:hypernym | dbr:Model |
| rdf:type | dbo:Person yago:WikicatStochasticDifferentialEquations yago:WikicatStochasticProcesses yago:Abstraction100002137 yago:Cognition100023271 yago:Communication100033020 yago:Concept105835747 yago:Content105809192 yago:DifferentialEquation106670521 yago:Equation106669864 yago:Hypothesis105888929 yago:Idea105833840 yago:MathematicalStatement106732169 yago:Message106598915 yago:Model105890249 yago:PsychologicalFeature100023100 yago:Statement106722453 yago:StochasticProcess113561896 |
| rdfs:comment | 在随机过程理論中的濾波問題（Filtering problem）是指針對信号处理及相關領域中，許多狀態估測問題的數學模型。大致概念是從不完整的、可能包括雜訊的觀測值中，建立有關系統真實值的「最佳估測」。最佳非線性濾波問題（甚至也包括非平稳过程問題）由（1959年、1960年）找到解答，在的研究及的研究中也有提到，Zakai建立了濾波器在條件機率未歸一情況下的簡化動態模型，稱為。不過一般情形下的解是無限維的。 目前已針對一些近似以及一些特定條件有深入的研究。例如在高斯隨機變數的假設下，最佳解是線性濾波器，也稱為维纳滤波及卡尔曼滤波。更一般的情形下，其解為無限維度，為了在有限記憶體的電腦中計算，需要進行有限維度的近似，有限維的近似型比較會以启发為基礎，例如或是假定密度濾波器（Assumed Density Filters），也有更方法論導向的作法，例如Projection Filters，其中有些子系列恰好和假定密度濾波器相同。 一般來說，若可以適用分離原理，這些濾波器也可以成為最优控制問題解的一部份。例如在LQG控制最佳控制問題中，其估測部份的解就是卡爾曼濾波。 (zh) 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. While originally motivated by problems in engineering, filtering found applications in many fields from signal processing to finance. In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the linear-quadratic-Gaussian control problem. (en) |
| rdfs:label | Filtering problem (stochastic processes) (en) 濾波問題 (zh) |
| owl:sameAs | freebase:Filtering problem (stochastic processes) yago-res:Filtering problem (stochastic processes) wikidata:Filtering problem (stochastic processes) dbpedia-zh:Filtering problem (stochastic processes) https://global.dbpedia.org/id/4jc87 |
| prov:wasDerivedFrom | wikipedia-en:Filtering_problem_(stochastic_processes)?oldid=1121590071&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Filtering_problem_(stochastic_processes) |
| is dbo:wikiPageDisambiguates of | dbr:Filter |
| is dbo:wikiPageRedirects of | dbr:Filtering_problem dbr:Filtering_theory |
| is dbo:wikiPageWikiLink of | dbr:Robert_Liptser dbr:Sanjoy_K._Mitter dbr:Particle_filter dbr:Innovation_(signal_processing) dbr:Kushner_equation dbr:List_of_named_differential_equations dbr:Ruslan_Stratonovich dbr:Moshe_Zakai dbr:Maamar_Bettayeb dbr:Kalman_filter dbr:Jan_H._van_Schuppen dbr:Filter dbr:Zakai_equation dbr:Catalog_of_articles_in_probability_theory dbr:List_of_statistics_articles dbr:Moving_horizon_estimation dbr:Nonlinear_filter dbr:Switching_Kalman_filter dbr:Filtering_problem dbr:Filtering_theory |
| is foaf:primaryTopic of | wikipedia-en:Filtering_problem_(stochastic_processes) |