Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Levy alpha-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Levy alpha-stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.Comment: 7 pages, 5 figures, 1 table. Presented at the Conference on Computing in Economics and Finance in Montreal, 14-16 June 2007; at the conference "Modelling anomalous diffusion and relaxation" in Jerusalem, 23-28 March 2008; et

Closed-Form Bayesian Inferences for the Logit Model via Polynomial Expansions

Articles in Marketing and choice literatures have demonstrated the need for incorporating person-level heterogeneity into behavioral models (e.g., logit models for multiple binary outcomes as studied here). However, the logit likelihood extended with a population distribution of heterogeneity doesn't yield closed-form inferences, and therefore numerical integration techniques are relied upon (e.g., MCMC methods). We present here an alternative, closed-form Bayesian inferences for the logit model, which we obtain by approximating the logit likelihood via a polynomial expansion, and then positing a distribution of heterogeneity from a flexible family that is now conjugate and integrable. For problems where the response coefficients are independent, choosing the Gamma distribution leads to rapidly convergent closed-form expansions; if there are correlations among the coefficients one can still obtain rapidly convergent closed-form expansions by positing a distribution of heterogeneity from a Multivariate Gamma distribution. The solution then comes from the moment generating function of the Multivariate Gamma distribution or in general from the multivariate heterogeneity distribution assumed. Closed-form Bayesian inferences, derivatives (useful for elasticity calculations), population distribution parameter estimates (useful for summarization) and starting values (useful for complicated algorithms) are hence directly available. Two simulation studies demonstrate the efficacy of our approach.Comment: 30 pages, 2 figures, corrected some typos. Appears in Quantitative Marketing and Economics vol 4 (2006), no. 2, 173--20

Use of Amulet in behavioral change for geriatric obesity management

Background: Obesity in older adults is a significant public health concern. Weight-loss interventions are known to improve physical function but risk the development of sarcopenia. Mobile health devices have the potential to augment existing interventions and, if designed accordingly, could improve one’s physical activity and strength in routine physical activity interventions. Methods and results: We present Amulet, a mobile health device that has the capability of engaging patients in physical activity. The purpose of this article is to discuss the development of applications that are tailored to older adults with obesity, with the intention to engage and improve their health. Conclusions: Using a team-science approach, Amulet has the potential, as an open-source mobile health device, to tailor activity interventions to older adults

Distribution and asymptotics under beta random scaling

Let X,Y,B be three independent random variables such that $X$ has the same distribution function as Y B. Assume that B is a Beta random variable with positive parameters a,b and Y has distribution function H. Pakes and Navarro (2007) show under some mild conditions that the distribution function H_{a,b} of X determines H. Based on that result we derive in this paper a recursive formula for calculation of H, if H_{a,b} is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H_{a,b}, respectively, and the conditional limiting distribution of bivariate elliptical distributions.Comment: 12 page

Statistical mixing and aggregation in Feller diffusion

We consider Feller mean-reverting square-root diffusion, which has been applied to model a wide variety of processes with linearly state-dependent diffusion, such as stochastic volatility and interest rates in finance, and neuronal and populations dynamics in natural sciences. We focus on the statistical mixing (or superstatistical) process in which the parameter related to the mean value can fluctuate - a plausible mechanism for the emergence of heavy-tailed distributions. We obtain analytical results for the associated probability density function (both stationary and time dependent), its correlation structure and aggregation properties. Our results are applied to explain the statistics of stock traded volume at different aggregation scales.Comment: 16 pages, 3 figures. To be published in Journal of Statistical Mechanics: Theory and Experimen

Fractional oscillator process with two indices

We introduce a new fractional oscillator process which can be obtained as solution of a stochastic differential equation with two fractional orders. Basic properties such as fractal dimension and short range dependence of the process are studied by considering the asymptotic properties of its covariance function. The fluctuation--dissipation relation of the process is investigated. The fractional oscillator process can be regarded as one-dimensional fractional Euclidean Klein-Gordon field, which can be obtained by applying the Parisi-Wu stochastic quantization method to a nonlocal Euclidean action. The Casimir energy associated with the fractional field at positive temperature is calculated by using the zeta function regularization technique.Comment: 32 page

PolyEDA: Combining Estimation of Distribution Algorithms and Linear Inequality Constraints

Algorithmus , Linearität , Ungleichhei

Left Motor delta Oscillations Reflect Asynchrony Detection in Multisensory Speech Perception

During multisensory speech perception, slow delta oscillations (∼1 - 3 Hz) in the listener's brain synchronize with the speech signal, likely engaging in speech signal decomposition. Notable fluctuations in the speech amplitude envelope, resounding speaker prosody, temporally align with articulatory and body gestures and both provide complementary sensations that temporally structure speech. Further, delta oscillations in the left motor cortex seem to align with speech and musical beats, suggesting their possible role in the temporal structuring of (quasi)-rhythmic stimulation. We extended the role of delta oscillations to audio-visual asynchrony detection as a test case of the temporal analysis of multisensory prosody fluctuations in speech. We recorded EEG responses in an audio-visual asynchrony detection task while participants watched videos of a speaker. We filtered the speech signal to remove verbal content and examined how visual and auditory prosodic features temporally (mis-)align. Results confirm (i) that participants accurately detected audio-visual asynchrony, and (ii) increased delta power in the left motor cortex in response to audio-visual asynchrony. The difference of delta power between asynchronous and synchronous conditions predicted behavioural performance, and (iii) decreased delta-beta coupling in the left motor cortex when listeners could not accurately map visual and auditory prosodies. Finally, both behavioural and neurophysiological evidence was altered when a speaker's face was degraded by a visual mask. Together, these findings suggest that motor delta oscillations support asynchrony detection of multisensory prosodic fluctuation in speech.SIGNIFICANCE STATEMENTSpeech perception is facilitated by regular prosodic fluctuations that temporally structure the auditory signal. Auditory speech processing involves the left motor cortex and associated delta oscillations. However, visual prosody (i.e., a speaker's body movements) complements auditory prosody, and it is unclear how the brain temporally analyses different prosodic features in multisensory speech perception. We combined an audio-visual asynchrony detection task with electroencephalographic recordings to investigate how delta oscillations support the temporal analysis of multisensory speech. Results confirmed that asynchrony detection of visual and auditory prosodies leads to increased delta power in left motor cortex and correlates with performance. We conclude that delta oscillations are invoked in an effort to resolve denoted temporal asynchrony in multisensory speech perception