A Central Limit Theorem for non-overlapping return times

Define the non-overlapping return time of a random process to be the number of blocks that we wait before a particular block reappears. We prove a Central Limit Theorem based on these return times. This result has applications to entropy estimation, and to the problem of determining if digits have come from an independent equidistribted sequence. In the case of an equidistributed sequence, we use an argument based on negative association to prove convergence under weaker conditions

Шары в пространствах последовательностей

We introduce a new metric on a space of right-sided infinite sequences drawn from a finite alphabet. Emerging from a problem of entropy estimation of a discrete stationary ergodic process, the metric is important on its own part and exhibits some interesting properties. For example, the measure of a ball is discontinuous at every binary rational value of log r, where r is the radius.Предлагается новая метрика на пространстве правосторонних бесконечных последовательностей над конечным алфавитом. Введенная в задаче оценивания энтропии дискретных стационарных процессов, эта метрика обладает рядом интересных свойств. Например, мера шара является разрывной при любом двоично-рациональном значении log r, где r – радиус шара

Несмещенная оценка энтропии для бинарных потоков

A new class of metrics on a space of right-sided infinite sequences drawn from a binary alphabet was introduced.Предлагается новый класс метрик на пространстве правосторонних бесконечных последовательностей над бинарным алфавитом. Показано, что параметры, определяющие этот класс метрик, можно выбрать так, что смещение оценки энтропии будет O(n¯с ), где n – число заданных последовательностей, c – некоторая константа

Асимптотика дисперсии оценки энтропии для симметричных мер Бернулли

We consider Bernoulli measures with equally probable symbols and obtain a closed-form expression of the Grassberger entropy estimator variance.Найдена точная асимптотика для дисперсии оценки энтропии П. Грассбер-гера [1] в случае симметричной бернуллиевской меры (с равными вероятностями символов)

Gibbs distribution analysis of temporal correlations structure in retina ganglion cells

We present a method to estimate Gibbs distributions with \textit{spatio-temporal} constraints on spike trains statistics. We apply this method to spike trains recorded from ganglion cells of the salamander retina, in response to natural movies. Our analysis, restricted to a few neurons, performs more accurately than pairwise synchronization models (Ising) or the 1-time step Markov models (\cite{marre-boustani-etal:09}) to describe the statistics of spatio-temporal spike patterns and emphasizes the role of higher order spatio-temporal interactions.Comment: To appear in J. Physiol. Pari

Detecting and Reducing Biases in Cellular-Based Mobility Data Sets

Correctly estimating the features characterizing human mobility from mobile phone traces is a key factor to improve the performance of mobile networks, as well as for mobility model design and urban planning. Most related works found their conclusions on location data based on the cells where each user sends or receives calls or messages, data known as Call Detail Records (CDRs). In this work, we test if such data sets provide enough detail on users' movements so as to accurately estimate some of the most studied mobility features. We perform the analysis using two different data sets, comparing CDRs with respect to an alternative data collection approach. Furthermore, we propose three filtering techniques to reduce the biases detected in the fraction of visits per cell, entropy and entropy rate distributions, and predictability. The analysis highlights the need for contextualizing mobility results with respect to the data used, since the conclusions are biased by the mobile phone traces collection approach.This research was partially funded by the Spanish Ministry of Economy, Industry and Competitiveness through TEC2017-84197-C4-1-R (Inteligencia de fuentes abiertas para redes electricas inteligentes seguras), TEC2014-54335-C4-2-R (INRISCO: INcident monitoRing In Smart COmmunities), and IPT-2011-1272-430000 (MONOLOC) projects

Существование несмещенной оценки энтропии для специальной меры Бернулли

Let $\Omega = A^{N}$  be a space of right-sided infinite sequences drawn from a finite alphabet $A = \{0,1\}$,  $N = \{1,2,\dots \}$, \label{rho}   \rho(\boldsymbol{x},\boldsymbol{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} a metric on $\Omega = A^{N}$, and $\mu$ is a probability measure on $\Omega$. Let $\boldsymbol{\xi_0}, \boldsymbol{\xi_1}, \dots, \boldsymbol{\xi_n}$ be independent identically distributed points on $\Omega$. We study the estimator $\eta_n^{(k)}(\gamma)$ of the reciprocal of the entropy $1/h$ that are defined as \label{etan} \eta_n^{(k)}(\gamma) = k \left(r_{n}^{(k)}(\gamma) - r_{n}^{(k+1)}(\gamma)\right), where \label{def_r} r_n^{(k)}(\gamma) = \frac{1}{n+1}\sum_{j=0}^{n} \gamma\left(\min_{i:i \neq j} {^{(k)}} \rho(\boldsymbol{\xi_{i}}, \boldsymbol{\xi_{j}})\right),  $\min ^{(k)}\{X_1,\dots,X_N\}=  X_k$, if  $X_1\leq X_2\leq \dots\leq X_N$. The number $k$ and the function $\gamma(t)$ are auxiliary parameters.The main result of this paper isTheorem. Let $\mu$ be the Bernoulli measure  with probabilities p_0,p_1>0, $p_0+p_1=1$, $p_0=p_1^2$. There exists a function $\gamma(t)$ such that $$E\eta_n^{(k)}(\gamma) =  \frac1h.$$ Пусть $\Omega = A^{N}$  - пространство правосторонних бесконечных последовательностей символов из алфавита $A = \{0,1\}$,  $N = \{1,2,\dots \}$, \label{rho}   \rho(\boldsymbol{x},\boldsymbol{y}) = \sum_{k=1}^{\infty}|x_{k} - y_{k}|2^{-k} - метрика на $\Omega$ и $\mu$ - вероятностная мера на $\Omega$. Пусть $\boldsymbol{\xi_0}, \boldsymbol{\xi_1}, \dots, \boldsymbol{\xi_n}$ - независимые случайные точки на $\Omega$, распределенные по мере $\mu$. Будем изучать оценку $\eta_n^{(k)}(\gamma)$ величины обратной к энтропии $1/h$, которая определяется следующим образом: \label{etan} \eta_n^{(k)}(\gamma) = k \left(r_{n}^{(k)}(\gamma) - r_{n}^{(k+1)}(\gamma)\right), где \label{def_r} r_n^{(k)}(\gamma) = \frac{1}{n+1}\sum_{j=0}^{n} \gamma\left(\min_{i:i \neq j} {^{(k)}} \rho(\boldsymbol{\xi_{i}}, \boldsymbol{\xi_{j}})\right), $\min ^{(k)}\{X_1,\dots,X_N\}=  X_k$, if  $X_1\leq X_2\leq \dots\leq X_N$. Число $k$ и функция $\gamma(t)$  - вспомогательные параметры. Основной результат работы: Теорема. Пусть $\mu$ -  мера Бернулли с вероятностями p_0,p_1>0, $p_0+p_1=1$, $p_0=p_1^2$, тогда существует функция $\gamma(t)$ такая, что \[E\eta_n^{(k)}(\gamma) =  \frac1h.\