Theory of Maximum Entropy Image Reconstruction

Skilling, John; Justice, James H.

doi:10.1017/cbo9780511569678.011

Cited by 23 publications

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“…The MaxEnt inference procedure has been applied in many fields, including image reconstruction in medicine and forensics (Frieden, 1972; Skilling, 1984; Gull & Newton, 1986; Roussev, 2010), neural net firing patterns (Meshulam, 2017), protein folding (Steinbach et. al, 2002; Mora et al, 2010) and reconstruction of incomplete input–output data and other applications in economics (Golan et al .…”

Section: Introductionmentioning

confidence: 99%

“…The maximum entropy form of a probability distribution, p(n), is obtained by maximising its Shannon information entropy (Shannon 1948), À∑ n p n ð Þlog p n ð Þ ð Þ, under imposed constraints. The MaxEnt inference procedure has been applied in many fields, including image reconstruction in medicine and forensics (Frieden, 1972;Skilling, 1984;Gull & Newton, 1986;Roussev, 2010), neural net firing patterns (Meshulam, 2017), protein folding (Steinbach et. al, 2002;Mora et al, 2010) and reconstruction of incomplete input-output data and other applications in economics (Golan et al 1996;Golan 2018).…”

Section: Introductionmentioning

confidence: 99%

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DynaMETE: a hybrid MaxEnt‐plus‐mechanism theory of dynamic macroecology

2021

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The Maximum Entropy Theory of Ecology (METE) predicts the shapes of macroecological metrics in relatively static ecosystems, across spatial scales, taxonomic categories and habitats, using constraints imposed by static state variables. In disturbed ecosystems, however, with time‐varying state variables, its predictions often fail. We extend macroecological theory from static to dynamic by combining the MaxEnt inference procedure with explicit mechanisms governing disturbance. In the static limit, the resulting theory, DynaMETE, reduces to METE but also predicts a new scaling relationship among static state variables. Under disturbances, expressed as shifts in demographic, ontogenic growth or migration rates, DynaMETE predicts the time trajectories of the state variables as well as the time‐varying shapes of macroecological metrics such as the species abundance distribution and the distribution of metabolic rates over individuals. An iterative procedure for solving the dynamic theory is presented. Characteristic signatures of the deviation from static predictions of macroecological patterns are shown to result from different kinds of disturbance. By combining MaxEnt inference with explicit dynamical mechanisms of disturbance, DynaMETE is a candidate theory of macroecology for ecosystems responding to anthropogenic or natural disturbances.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

DynaMETE: a hybrid MaxEnt‐plus‐mechanism theory of dynamic macroecology

2021

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“…For more details, see Geman and Geman (1984), Daniell and Gull (1980), and Skilling (1986). For more details, see Geman and Geman (1984), Daniell and Gull (1980), and Skilling (1986).…”

Section: Resultsmentioning

confidence: 99%

A Simple Statistical Project: Image Reconstruction

Yu¹

1994

The American Statistician

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 128.235.251.160 on Sun, 22 Mar 2015 15:29:22 UTC All use subject to JSTOR Terms and Conditionsrestricted to satisfy the conditions pij > q (i = 1, . . ., k -1) with respective observed values xij (j = 1, . . ., k -1) and xik +y. The dimension of the new parameter space now decreases 1 unit to k. If Xlk 1 /Mik , > (Xik +Y)/(mik +n), then 9=Q51,. . **Pk,4),wherePik = 4 = (Xik +Y)/(Mik +n) and ij =Xi /mi for the otherjs. If (xik+y)/(mlk+n) > Xikl/mikl, use Lemma 2.1 again and reduce the dimension of the parameter space 1 more unit by setting pik_, = Pik = q and restricting pij > q (j = 1, . . ., k -2). If necessary, repeat the procedure to reduce the parameter space until the MLE 9 of 0 restricted to 0R is obtained. The maximum number of reductions will occur if reductions must be made to a one-dimensional space. In this case,This method for finding the restricted MLE of 0 can be considered a method of pooling the violating treatment sample of the smallest nonrestricted MLE with the control sample.The following algorithm can be used to find 9.Algorithm 2.1.Step 1. Compute Pi = ki/mi (i = 1, . . ., k) and q = y/n.Step 2. Compare q with the smallest of Pi. If q is less than or equal to this smallest Pi, the restricted MLE is obtained and is equal to the nonrestricted MLE.Step 3. Otherwise, pool the treatment sample with the smallest Pi with the control, and consider this treatment to be as effective as the control. Compute the new nonrestricted MLE q of q. Then go back to Step 2. Example 2.1. Consider five groups of patients, with each group containing 10 patients. The patients in the first four groups are under four different new treatments, respectively, and the patients in the last group are under the control treatment. Suppose that the numbers of patients having positive response are 3, 5, 4, 7, and 6, respectively. Then, the MLE's of the probabilities of positive response PI, P2, P3, P4, and q of these treatments under the restriction pi > q (i = 1, *. ., 4) through Algorithm 2.1 are given by Pi = 13/30, P2 = 5/10, p3 = 13/30, P4 = 7/10, and q= 13/30.The detailed steps in computing these are as follows.Step 1. j5 = 3/10, P2= 5/10, P3 = 4/10, P4= 7/10, and q = 6/10.Step 2. min=^3 =3/10< q=6/10. PoolingSample 1 to the control (p = q), q=9/20.Step 3. ml =nP = 4/10 < q= 9/20. Pooling Sample 3 to the control (pi =P3 = q), q = 13/30.Step 4. min =P2= 5/10 > q = 13/30. Then, pii=P3 = q = 13/30, Pj2= 5/10, andjP4 = 7/10. APPENDIX: PROOF OF LEMMA 2.1The proof of this lemma is based on the fact that the likelihood function L is strictly increasing in each pi for pi < x1/mi (i = 1, . . ., k) and in q for q < y/n and is strictly decreasing in each pi for pi > xi/mi...

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“…One special application of MaxEnt familiar to many ecologists is its use in machine learning and species distribution modelling (Phillips, Anderson & Schapire ), though there are a variety of applications in ecology (Shipley, Vile & Garnier ; Williams ) and other sciences (e.g. Skilling ; Jaynes ; Banavar, Maritan & Volkov ). The conceptual goal of the MaxEnt approach in macroecology is to build biodiversity predictions that are not sensitive to potentially arbitrary model parameter choices (e.g.…”

Section: Background On the Maximum Entropy Theory Of Ecologymentioning

confidence: 99%

meteR: an r package for testing the maximum entropy theory of ecology

Rominger

Merow

2016

Methods Ecol Evol

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Summary1. Macroecological patterns appear to follow consistent forms across a range of natural systems; however, the origin of their regularity remains obscured. The maximum entropy theory of ecology (METE) predicts macroecological patterns of abundance, metabolic rates and their distribution within communities and across space using an information theoretic approach. METE's success in predicting empirical patterns demands that we further press the theory's predictions to determine how (or whether) predictability depends on attributes of the system and the temporal, spatial and biological scales at which we study it. 2. Maximum entropy theory of ecology predicts multiple macroecological metrics using statistical idealizations from information theory; thus, confronting METE with data represents a strong test of the underlying biological mechanisms that could drive real communities away from statistical idealizations. METE has remained somewhat inaccessible due to its highly mathematical nature and a lack of software for model construction/evaluation. To remedy this, we have developed an R package implementation of METE.3. Our open-source (GNU General Public License v2) R package, meteR (version 1.2; https://cran.r-project.org/package=meteR), (i) directly calculates all of METE's predictions from a variety of data formats; (ii) automatically handles approximations and other technical details; and (iii) provides high-level plotting and model comparison functions to explore and interrogate models. 4. With these tools in hand, ecologists can more readily test the predictions of METE for their data sets. By facilitating tests of METE, we expect that a better understanding of its strengths and limitations will emerge. A better understanding of the strengths and limitations of METE will offer insight into how biological mechanisms and statistical constraints combine to drive macroecological patterns.

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Theory of Maximum Entropy Image Reconstruction

Cited by 23 publications

References 0 publications

DynaMETE: a hybrid MaxEnt‐plus‐mechanism theory of dynamic macroecology

DynaMETE: a hybrid MaxEnt‐plus‐mechanism theory of dynamic macroecology

A Simple Statistical Project: Image Reconstruction

meteR: an r package for testing the maximum entropy theory of ecology

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