Principal component analysis (PCA) for binary data, known as logistic PCA, has become a popular alternative to dimensionality reduction of binary data. It is motivated as an extension of ordinary PCA by means of a matrix factorization, akin to the singular value decomposition, that maximizes the Bernoulli log-likelihood. We propose a new formulation of logistic PCA which extends Pearson’s formu...

Dear Editor, The topic addressed by this brief communication is the quantification of diagnostic information and, in particular, a method of illustrating graphically the quantification of diagnostic information for binary tests. To begin, consider the following outline procedure for development of such a test. An appropriate indicator variable that will serve as a proxy for the actual variable ...

Probabilistic models for binary spike patterns provide a powerful tool for understanding the statistical dependencies in large-scale neural recordings. Maximum entropy (or “maxent”) models, which seek to explain dependencies in terms of low-order interactions between neurons, have enjoyed remarkable success in modeling such patterns, particularly for small groups of neurons. However, these mode...

Monitoring Markov Dependent Observations with a Log-Likelihood Based CUSUM Shabnam Mousavi and Marion R. Reynolds, Jr. Department of Statistics, The Pennsylvania State University, University Park, PA 16802-2111, U.S.A., Max Planck Institute for Human Development, Lentzeallee 94, 14195 Berlin, Germany, Departments of Statistics and Forestry, Virginia Polytechnic Institute and State University, B...

We prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn-Minkowski inequality and an Urysohn’s type inequality for the Bernoulli Constant and we study the behaviour of the free boundary with respect to the given boundary data. Moreover we prove a uniqueness result regarding the interior non-linear Bernoulli problem.

and Applied Analysis 3 we derive some interesting identities and relations on the modified q-Bernoulli numbers and polynomials. 2. The Modified q-Bernoulli Numbers and Polynomials with Weight α In this section, we assume α ∈ Q. Now, we define the modified q-Bernoulli numbers with weight α B̃ α n,q as follows:

We investigate some algorithms that produce Bernoulli, Euler and Genocchi polynomials. We also give closed formulas for Bernoulli, Euler and Genocchi polynomials in terms of weighted Stirling numbers of the second kind, which are extensions of known formulas for Bernoulli, Euler and Genocchi numbers involving Stirling numbers of the second kind.

We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the β-numeration. A matrix decomposition of these measures is obtained in the case when β is a PV number. We also determine their Gibbs properties for β being a multi-nacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.