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Improving Density Estimation by Incorporating Spatial Information
, 2009
"... Given discrete event data, we wish to produce a probability density that can model the relative probability of events occurring in a spatial region. Common methods of density estimation, such as Kernel Density Estimation, do not incorporate geographical information. Using these methods could result ..."
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Given discrete event data, we wish to produce a probability density that can model the relative probability of events occurring in a spatial region. Common methods of density estimation, such as Kernel Density Estimation, do not incorporate geographical information. Using these methods could result in nonnegligible portions of the support of the density in unrealistic geographic locations. For example, crime density estimation models that do not take geographic information into account may predict events in unlikely places such as oceans, mountains, etc. We propose a set of Maximum Penalized Likelihood Estimation methods based on Total Variation and H1 Sobolev norm regularizers in conjunction with a priori high resolution spatial data to obtain more geographically accurate density estimates. We apply this method to a residential burglary data set of the San Fernando Valley using geographic features obtained from satellite images of the region and housing density information. 1
The alter egos of the regularized maximum likelihood density estimators: deregularized maximumentropy,
"... Abstract: Various properties of maximum likelihood density estimators penalizing the total variation of some derivative of the logarithm of the estimated density are discussed, in particular the properties of their dual formulations and connections to stretched (taut) string methodology. ..."
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Abstract: Various properties of maximum likelihood density estimators penalizing the total variation of some derivative of the logarithm of the estimated density are discussed, in particular the properties of their dual formulations and connections to stretched (taut) string methodology.
Mathematical Publications PRIMAL AND DUAL FORMULATIONS RELEVANT FOR THE NUMERICAL ESTIMATION OF A PROBABILITY DENSITY VIA REGULARIZATION
"... ABSTRACT. General schemes relevant for the estimation of a probability density via regularization—primal and dual versions in the discretized setting—are investigated. Conditions for the dual solution to be a probability density are given, and a strong duality theorem is proved. We study various ins ..."
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ABSTRACT. General schemes relevant for the estimation of a probability density via regularization—primal and dual versions in the discretized setting—are investigated. Conditions for the dual solution to be a probability density are given, and a strong duality theorem is proved. We study various instances of the problem −w T Lh + s T Ψ(g) + J(−Ph) = min! subject to h ≼ g, g,h where L and w are evaluation operator and averaging functional described later in the text; Ψ(g) indicates the application of a real convex function ψ to the components of g, while J(h) is a general convex function applied to the whole vector −Ph, the negative of the result of a linear operator P applied on h. We assume that vectors w and s have nonnegative elements; hereafter, ≽ and ≼ denote componentwise inequalities. If ψ is nondecreasing, the primal formulation (P) can be simplified—it is equivalent to the unconstrained problem (U) −w T Lg + s T Ψ(g) + J(−Pg) = min! g Convex functions are allowed to attain + ∞ as a value; the domain, dom Φ, is the set where Φ is finite. We assume that all convex functions in (P) and (U) have domains with nonempty interiors. Concave functions are handled in an analogous manner, only the rôle of + ∞ is played by −∞.
Noname manuscript No. (will be inserted by the editor) Statistical Density Estimation using Threshold Dynamics for Geometric Motion
"... Abstract Our goal is to estimate a probability density based on discrete point data via segmentation techniques. Since point data may represent certain activities, such as crime, our method can be successfully used for detecting regions of high activity. In this work we design a binary segmentation ..."
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Abstract Our goal is to estimate a probability density based on discrete point data via segmentation techniques. Since point data may represent certain activities, such as crime, our method can be successfully used for detecting regions of high activity. In this work we design a binary segmentation version of the wellknown Maximum Penalized Likelihood Estimation (MPLE) model, as well as a minimization algorithm based on thresholding dynamics originally proposed by Merriman, Bence and Osher [20]. We also present some computational examples, including one with actual residential burglary data from the San Fernando Valley.
and
, 2009
"... Copula density estimation by total variation penalized likelihood with linear equality constraints ..."
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Copula density estimation by total variation penalized likelihood with linear equality constraints
Research Article Improving Density Estimation by Incorporating Spatial Information
, 2010
"... License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Given discrete event data, we wish to produce a probability density that can model the relative probability of events occurring in a spatial region. Common methods of ..."
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License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Given discrete event data, we wish to produce a probability density that can model the relative probability of events occurring in a spatial region. Common methods of density estimation, such as Kernel Density Estimation, do not incorporate geographical information. Using these methods could result in nonnegligible portions of the support of the density in unrealistic geographic locations. For example, crime density estimation models that do not take geographic information into account may predict events in unlikely places such as oceans, mountains, and so forth. We propose a set of Maximum Penalized Likelihood Estimation methods based on Total Variation and H1 Sobolev norm regularizers in conjunction with a priori high resolution spatial data to obtain more geographically accurate density estimates. We apply this method to a residential burglary data set of the San Fernando Valley using geographic features obtained from satellite images of the region and housing density information. 1.
By SYLVAIN SARDY ∗
"... Density estimation by total variation penalized likelihood driven by the sparsity ℓ1 information criterion ..."
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Density estimation by total variation penalized likelihood driven by the sparsity ℓ1 information criterion
Computational Statistics and Data Analysis ( ) – Contents lists available at SciVerse ScienceDirect Computational Statistics and Data Analysis
"... journal homepage: www.elsevier.com/locate/csda Copula density estimation by total variation penalized likelihood with linear equality constraints ..."
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journal homepage: www.elsevier.com/locate/csda Copula density estimation by total variation penalized likelihood with linear equality constraints
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"... ABSTRACT. We investigate general schemes relevant for the estimation of a probability density via regularization—their primal and dual versions in the discretized setting. In particular, conditions for the dual solution to be a probability density are given, and a strong duality theorem is proved. ..."
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ABSTRACT. We investigate general schemes relevant for the estimation of a probability density via regularization—their primal and dual versions in the discretized setting. In particular, conditions for the dual solution to be a probability density are given, and a strong duality theorem is proved. We study various instances of the problem (P) −wTLh+ sTΨ(g) + J(−Ph) = min g,h! subject to h g, where Ψ(g) indicates the application of a real convex function ψ to the components of g, while J(h) is rather a general convex function applied to the whole vector −Ph, the negative of the result of a linear operator P applied on h. We assume that vectors w and s have positive nonzero elements; hereafter, and stand for componentwise inequalities. In some cases, the primal formulation (P) can be simplified. If the function ψ is nondecreasing, then it immediately follows that (P) is equivalent to the unconstrained problem (U) −wTLg + sTΨ(g) + J(−Pg) = min g As is customary in convex analysis, we consider convex functions that may attain + ∞ as a value; the set where such a function Φ is finite is called its domain, domΦ. We assume that all convex functions appearing in (P) have domains with