The Statistical Estimation of Entropy in the Non-Parametric Case

Harris, Bernard

doi:10.21236/ada020217

Cited by 40 publications

(50 citation statements)

References 3 publications

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“…(MacArthur, 1955;Goodman, 1975;McCann, 2000;Ives and Carpenter, 2007). Other important research questions include 120 the sources of diversity (Tsimring et al, 1996;De'ath, 2012), extensions of diversity to include a concept of species distance Chao et al (2010); Leinster and Cobbold (2012);Feoli (2012);Scheiner (2012), and techniques for measuring diversity (Chao and Shen, 2003;Schmera and Podani, 2013;125 Gotelli and Chao, 2013), despite the well-known difficulties in estimating entropy in an unbiased fashion (Harris, 1975;Paninski, 2003). Beyond ecology, the study of diversity is also important in economics ( D i of order q for varying q, shown for three different particles (inset square plots).…”

Section: N Riemer and M West: Quantifying Aerosol Mixing Statementioning

confidence: 99%

“…Other important research questions include the sources of diversity (Tsimring et al, 1996;De'ath, 2012), extensions of diversity to include a concept of species distance Chao et al (2010); Leinster and Cobbold (2012);Feoli (2012);Scheiner (2012), and techniques for measuring diversity (Chao and Shen, 2003;Schmera and Podani, 2013;Gotelli and Chao, 2013), despite the well-known difficulties in estimating entropy in an unbiased fashion (Harris, 1975;Paninski, 2003). Beyond ecology, the study of diversity is also important in economics (Garrison and Paulson, 1973;Hannah and Kay, 1977;Attaran and Zwick, 1989;Malizia and Ke, 1993;Drucker, 2013), immunology (Tsimring et al, 1996), neuroscience (Panzeri and Treves, 1996;Strong et al, 1998), and genetics (Innan et al, 1999;Rosenberg et al, 2002;Falush et al, 2007).…”

Section: Introductionmentioning

confidence: 99%

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Quantifying aerosol mixing state with entropy and diversity measures

2013

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Abstract. This paper presents the first quantitative metric for aerosol population mixing state, defined as the distribution of per-particle chemical species composition. This new metric, the mixing state index χ, is an affine ratio of the average per-particle species diversity D α and the bulk population species diversity D γ , both of which are based on information-theoretic entropy measures. The mixing state index χ enables the first rigorous definition of the spectrum of mixing states from so-called external mixture to internal mixture, which is significant for aerosol climate impacts, including aerosol optical properties and cloud condensation nuclei activity. We illustrate the usefulness of this new mixing state framework with model results from the stochastic particle-resolved model PartMC-MOSAIC. These results demonstrate how the mixing state metrics evolve with time for several archetypal cases, each of which isolates a specific process such as coagulation, emission, or condensation. Further, we present an analysis of the mixing state evolution for a complex urban plume case, for which these processes occur simultaneously. We additionally derive theoretical properties of the mixing state index and present a family of generalized mixing state indexes that vary in the importance assigned to low-mass-fraction species.

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Section: N Riemer and M West: Quantifying Aerosol Mixing Statementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantifying aerosol mixing state with entropy and diversity measures

2013

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“…The standard deviation of the direct estimate H Dir can be expressed as (Harris, 1975): where again, the Ο(N −1 ) term is independent of the distribution. Thus we focus on distributions that reduce the Ο(N −½ ) term.…”

Section: Theoretical Considerationsmentioning

confidence: 99%

“…This term is smaller for higher entropy distributions, reaching zero for the uniform distribution. When comparing two distributions with equal N and K, the † A straight forward expansion of the bias to higher order: (Harris, 1975) reveals that the Ο(N −2 ) term does depend on the distribution of p k , the probability associated with the k th bin. The Ο(N −2 ) term is always negative, and minimal in the absolute sense for the uniform distribution: Ο(N −2 ) uniform = (1-K 2 )/(N 2 12 ln(2)).…”

Section: Theoretical Considerationsmentioning

confidence: 99%

Probability distributions of the logarithm of inter-spike intervals yield accurate entropy estimates from small datasets

Dorval

2008

Journal of Neuroscience Methods

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The maximal information that the spike train of any neuron can pass on to subsequent neurons can be quantified as the neuronal firing pattern entropy. Difficulties associated with estimating entropy from small datasets have proven an obstacle to the widespread reporting of firing pattern entropies and more generally, the use of information theory within the neuroscience community. In the most accessible class of entropy estimation techniques, spike trains are partitioned linearly in time and entropy is estimated from the probability distribution of firing patterns within a partition. Ample previous work has focused on various techniques to minimize the finite dataset bias and standard deviation of entropy estimates from under-sampled probability distributions on spike timing events partitioned linearly in time. In this manuscript we present evidence that all distribution-based techniques would benefit from inter-spike intervals being partitioned in logarithmic time. We show that with logarithmic partitioning, firing rate changes become independent of firing pattern entropy. We delineate the entire entropy estimation process with two example neuronal models, demonstrating the robust improvements in bias and standard deviation that the logarithmic time method yields over two widely used linearly partitioned time approaches.

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“…-Percent of count -Estimated Shannon entropy [35][36][37] To analyze the precision of the estimated Shannon entropy values, the standard deviation was also calculated [38]. This enabled us to assess the spread of the estimated entropy over the number of possible responses to a question:…”

Section: Analysis Methodsmentioning

confidence: 99%

Users’ Assessment of Orthoimage Photometric Quality for Visual Interpretation of Agricultural Fields

et al. 2015

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Land cover identification and area quantification are key aspects of implementing the European Common Agriculture Policy. Legitimacy of support provided to farmers is monitored using the Land Parcel Identification System (LPIS), with land cover identification performed by visual image interpretation. While the geometric orthoimage quality required for correct interpretation is well understood, little is known about the photometric quality needed for LPIS applications. This paper analyzes the orthoimage quality characteristics chosen by authors as being most suitable for visual identification of agricultural fields. We designed a survey to assess users' preferred brightness and contrast ranges for orthoimages used for LPIS purposes. Survey questions also tested the influence of a background color on the preferred orthoimage brightness and contrast, the preferred orthoimage format and color composite, assessments of orthoimages with shadowed areas, appreciation of image enhancements and, finally, consistency of individuals' preferred brightness and contrast settings across multiple sample images. We find that image appreciation is stable at the individual level, but preferences vary across respondents. We therefore recommend that LPIS operators be enabled to personalize photometric settings, such as brightness and contrast values, and to choose the displayed band combination from at least four spectral bands. OPEN ACCESSRemote Sens. 2015, 7 4920

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The Statistical Estimation of Entropy in the Non-Parametric Case

Cited by 40 publications

References 3 publications

Quantifying aerosol mixing state with entropy and diversity measures

Quantifying aerosol mixing state with entropy and diversity measures

Probability distributions of the logarithm of inter-spike intervals yield accurate entropy estimates from small datasets

Users’ Assessment of Orthoimage Photometric Quality for Visual Interpretation of Agricultural Fields

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