Transfer Entropy as a Log-likelihood Ratio

Transfer entropy, an information-theoretic measure of time-directed information transfer between joint processes, has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields, including the neurosciences, ecology, climatology and econometrics. We show that for a broad class of predictive models, the log-likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself. For finite Markov chains, furthermore, no explicit model is required. In the general case, an asymptotic chi-squared distribution is established for the transfer entropy estimator. The result generalises the equivalence in the Gaussian case of transfer entropy and Granger causality, a statistical notion of causal influence based on prediction via vector autoregression, and establishes a fundamental connection between directed information transfer and causality in the Wiener-Granger sense

Spectral Analysis of Multi-dimensional Self-similar Markov Processes

In this paper we consider a discrete scale invariant (DSI) process $\{X(t), t\in {\bf R^+}\}$ with scale $l>1$. We consider to have some fix number of observations in every scale, say $T$, and to get our samples at discrete points $\alpha^k, k\in {\bf W}$ where $\alpha$ is obtained by the equality $l=\alpha^T$ and ${\bf W}=\{0, 1,...\}$. So we provide a discrete time scale invariant (DT-SI) process $X(\cdot)$ with parameter space $\{\alpha^k, k\in {\bf W}\}$. We find the spectral representation of the covariance function of such DT-SI process. By providing harmonic like representation of multi-dimensional self-similar processes, spectral density function of them are presented. We assume that the process $\{X(t), t\in {\bf R^+}\}$ is also Markov in the wide sense and provide a discrete time scale invariant Markov (DT-SIM) process with the above scheme of sampling. We present an example of DT-SIM process, simple Brownian motion, by the above sampling scheme and verify our results. Finally we find the spectral density matrix of such DT-SIM process and show that its associated $T$-dimensional self-similar Markov process is fully specified by $\{R_{j}^H(1),R_{j}^H(0),j=0, 1,..., T-1\}$ where $R_j^H(\tau)$ is the covariance function of $j$th and $(j+\tau)$th observations of the process.Comment: 16 page

Multivariate Granger Causality and Generalized Variance

Granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables. A shortcoming of the standard framework for Granger causality is that it only allows for examination of interactions between single (univariate) variables within a system, perhaps conditioned on other variables. However, interactions do not necessarily take place between single variables, but may occur among groups, or "ensembles", of variables. In this study we establish a principled framework for Granger causality in the context of causal interactions among two or more multivariate sets of variables. Building on Geweke's seminal 1982 work, we offer new justifications for one particular form of multivariate Granger causality based on the generalized variances of residual errors. Taken together, our results support a comprehensive and theoretically consistent extension of Granger causality to the multivariate case. Treated individually, they highlight several specific advantages of the generalized variance measure, which we illustrate using applications in neuroscience as an example. We further show how the measure can be used to define "partial" Granger causality in the multivariate context and we also motivate reformulations of "causal density" and "Granger autonomy". Our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems, neural and otherwise.Comment: added 1 reference, minor change to discussion, typos corrected; 28 pages, 3 figures, 1 table, LaTe

A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Levy-noise

We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by strictly stable Levy-processes with stability index bigger than one. The limit process turns out to be a strictly stable Levy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity.Comment: Accepted to Journal of Theoretical Probabilit

Measured quantum probability distribution functions for Brownian motion

The quantum analog of the joint probability distributions describing a classical stochastic process is introduced. A prescription is given for constructing the quantum distribution associated with a sequence of measurements. For the case of quantum Brownian motion this prescription is illustrated with a number of explicit examples. In particular it is shown how the prescription can be extended in the form of a general formula for the Wigner function of a Brownian particle entangled with a heat bath.Comment: Phys. Rev. A, in pres

Sentencing Outcomes of Convicted Child Sex Offenders

This research examines the sentencing outcomes of convicted child sexual offenders from data collected over an eight year period. Multiple regression and nominal log linear regression are used to examine length of prison sentence, length of probation sentence, and whether or not the convicted offender is actually sent to prison or to probation. While many independent variables appear to be related to sentence outcome, they fall into three categories: characteristics of the offender, characteristics of the victim, and characteristics of the crime. Additionally, while many variables appear related at the bivariate level, when multivariate analysis is applied, fewer variables remain significant and these are mostly from the characteristics of the offense

Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. III: potential function in local stochastic dynamics and in steady state of Boltzmann-Gibbs type distribution function

From a logic point of view this is the third in the series to solve the problem of absence of detailed balance. This paper will be denoted as SDS III. The existence of a dynamical potential with both local and global meanings in general nonequilibrium processes has been controversial. Following an earlier explicit construction by one of us (Ao, J. Phys. {\bf A37}, L25 '04, arXiv:0803.4356, referred to as SDS II), in the present paper we show rigorously its existence for a generic class of situations in physical and biological sciences. The local dynamical meaning of this potential function is demonstrated via a special stochastic differential equation and its global steady-state meaning via a novel and explicit form of Fokker-Planck equation, the zero mass limit. We also give a procedure to obtain the special stochastic differential equation for any given Fokker-Planck equation. No detailed balance condition is required in our demonstration. For the first time we obtain here a formula to describe the noise induced shift in drift force comparing to the steady state distribution, a phenomenon extensively observed in numerical studies. The comparison to two well known stochastic integration methods, Ito and Stratonovich, are made ready. Such comparison was made elsewhere (Ao, Phys. Life Rev. {\bf 2} (2005) 117. q-bio/0605020).Comment: latex. 13 page

A Pathwise Ergodic Theorem for Quantum Trajectories

If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravelings the time averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states the quantum trajectory converges in the mean to a random choice from these states.Comment: 8 page

Scaling Limits for Internal Aggregation Models with Multiple Sources

We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.Comment: 74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in v2: added "least action principle" (Lemma 3.2); small corrections in section 4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version); expanded section 6.

Eyewitness descriptions without memory: The (f)utility of describing faces

Eyewitness descriptions provide critical information for the police and other agencies to use during investigations. While researchers have typically considered the impact of memory, little consideration has been given to the utility of facial descriptions themselves, without the additional memory demands. In Experiment 1, participants described face images to their partners, who were then required to select these faces from photographic lineups. Performance was error‐prone when the same image appeared in the lineup (73% correct), and decreased further when a different image of the same face was presented (22% correct). We found some evidence to suggest this was due, in part, to difficulties with recognizing that two different images depicted the same person. In Experiment 2, we demonstrated that descriptions of the same face given by different people showed only moderate agreement. Taken together, these results highlight the problematic nature of facial descriptions, even without memory, and their limited utility