A Möbius transformation-induced distribution on the torus

Kato, Shogo; Pewsey, Arthur

doi:10.1093/biomet/asv003

Cited by 33 publications

(53 citation statements)

References 27 publications

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“…Data simulation from a w C (m,£) is already implemented in the "Circular" R package [21]. Kato and Pewsey [10] proposed a five-parameter bivariate wrapped Cauchy distribution, which is unimodal, pointwise symmetric around the mean and has a closed-form expression for the mode. A dependence parameter controls the cor relation from total independence to perfect correlation.…”

Section: Univariate and Bivariate Wrapped Cauchymentioning

confidence: 99%

“…(2), there is no closed-form expression for the maximum likelihood estimates, and numerical optimization methods must be used to find them. Although maximum likelihood estimation is the most common parameter estimation method, Kato and Pewsey [10] showed that method of moments is more efficient for our purpose. We use the method of moments, where all formulas for the estimates can be expressed in a closed form, as it is easier to implement and is computationally very fast.…”

Section: Parameter Estimationmentioning

confidence: 99%

“…1a). We use this algorithm for several reasons: (a) It uses only second-order sta tistics, which are easily and reliably measured from data, (b) we avoid the use of conditional independence tests, which are often computationally expensive and only available by simulation [10] from wrapped Cauchy and (c) is a way of ensuring a tree-structure.…”

Section: Parameter Estimationmentioning

confidence: 99%

“…Further, it was shown that it could be obtained by mapping Cauchy distributions [19] onto the circle. Recently Kato and Pewsey [10] pro posed a five-parameter bivariate wrapped Cauchy distribution for toroidal data, whose marginals and conditionals follow univariate wrapped Cauchy distribu tions. Their family is therefore said to be closed.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Tree-Structured Bayesian Networks for Wrapped Cauchy Directional Distributions

Leguey

Bielza

Larrañaga

2016

Advances in Artificial Intelligence

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Section: Univariate and Bivariate Wrapped Cauchymentioning

confidence: 99%

Section: Parameter Estimationmentioning

confidence: 99%

Section: Parameter Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tree-Structured Bayesian Networks for Wrapped Cauchy Directional Distributions

Leguey

Bielza

Larrañaga

2016

Advances in Artificial Intelligence

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“…Even though the simplest of our proposals (i.e., wCNB) is a straightforward naive Bayes classifier extension to using wrapped Cauchy distributions, this has never been attempted before to the best of our knowledge.The remainder of this paper is organized as follows. Section 2 reviews the bivariate wrapped Cauchy distribution of Kato and Pewsey [35]. Section 3 describes the four novel wrapped Cauchy classifiers proposed here.…”

mentioning

confidence: 99%

Circular Bayesian classifiers using wrapped Cauchy distributions

Leguey

Bielza

Larrañaga

2019

Data & Knowledge Engineering

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A B S T R A C TCapturing the dependences among circular variables within supervised classification models is a challenging task. In this paper, we propose four different supervised Bayesian classification algorithms where the predictor variables follow all circular wrapped Cauchy distributions. For this purpose, we introduce four wrapped Cauchy classifiers. The bivariate wrapped Cauchy distribution is the only bivariate circular distribution whose marginals and conditionals are also wrapped Cauchy distributions, a property that makes it possible to define these models easily. Furthermore, the wrapped Cauchy tree-augmented naive Bayes classifier requires the definition of a conditional circular mutual information measure between variables that follow wrapped Cauchy distributions. Synthetic data is used to illustrate, compare and evaluate the classification algorithms (including a comparison with the Gaussian TAN classifier, decision tree, random forest, multinomial logistic regression, support vector machine and simple neural network), leading to satisfactory predictive results. We also use a real neuromorphological dataset obtained from juvenile rat somatosensory cortex cells, where we measure the bifurcation angles of the dendritic basal arbors.characteristic that each variable is conditionally independent of those that are non-descendants in the graph given the value of their parents. Therefore the joint probability distribution is expressed as the product of the local distributions conditioned to their parents. For these reasons, Bayesian networks can deal efficiently with supervised classification i.e., the Bayesian network classifiers [13] and offer an explicit, graphical and interpretable representation of uncertain knowledge, which has made it possible to successfully apply them to real-world problems.Supervised classification [19] deals with the problem of assigning a label to an instance, based on a set of variables that characterize it. Yet circular data has been commonly treated as linear data in supervised classification tasks. Only a few circular classifiers exist, and almost none of them are based on the principles of Bayesian networks, capable of capturing multivariate relationships among variables. Most of them focus on discriminant analysis and assume several circular distributions such as the von Mises distribution [20], later extended to the von Mises-Fisher distribution [21]. There are also circular discriminant analysis studies for the Watson, Selby and Arnold distributions on the sphere [22,23]. SenGupta and Roy [24] used a classification discriminant rule based on the mean chord-length to classify a new observation into one of two different circular populations that are von Mises, when training samples are available for each of them. Also a likelihood ratio test based on a bootstrapping approach for classifying into two populations was proposed for linear and circular data [25]. Kirby and Miranda [26] proposed a variation of a neural network, including a circular node, which was able to ke...

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Circular Data Models

Pewsey¹

2018

Wiley StatsRef: Statistics Reference Online

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In contrast to linear data, for which the support is the real line, circular data are those for which the natural support is the unit circle. Examples include angles measured using a compass or weather vane expressed in degrees or radians and measured either clockwise or anticlockwise from some previously specified origin. Times of the day, week, month, or year can also be treated as circular data after conversion to angles. Cyber attacks recorded on a 24‐h clock provide an example. Circular data arise in animal movement and wind‐power generation studies, geology, astronomy, meteorology, oceanography, psychology, medicine, and the health sciences. As extensions of circular data, toroidal (circular–circular) data are obtained when the values of two circular random variables are observed simultaneously, and cylindrical (circular–linear) data when those of one circular and one linear random variable are jointly observed. In this article, we review probability models for circular data as well as their toroidal and cylindrical extensions.

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A Möbius transformation-induced distribution on the torus

Cited by 33 publications

References 27 publications

Tree-Structured Bayesian Networks for Wrapped Cauchy Directional Distributions

Tree-Structured Bayesian Networks for Wrapped Cauchy Directional Distributions

Circular Bayesian classifiers using wrapped Cauchy distributions

Circular Data Models

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