Non asymptotic variance bounds and deviation inequalities by optimal transport

This book relies on maximum likelihood (ML) estimation of parameters. Asymptotic theory assumes regularity conditions hold when the ML estimator is consistent. Typically an additional third derivative condition is assumed to ensure that the ML estimator is also asymptotically normally distributed. Standard asymptotic results that then hold are summarized in this chapter; for example, the asymptotic variance of the ML estimator is then given by the Fisher information formula, and the log-likelihood ratio, the Wald and the score statistics for testing the statistical significance of parameter estimates are all asymptotically equivalent. Also, the useful profile log-likelihood then behaves exactly as a standard log-likelihood only in a parameter space of just one dimension. Further, the model can be reparametrized to make it locally orthogonal in the neighbourhood of the true parameter value. The large exponential family of models is briefly reviewed where a unified set of regular conditions can be obtained.

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On the Markov Transition Kernels for First Passage Percolation on the Ladder

Journal of Applied Probability ◽

10.1017/s0021900200007932 ◽

2011 ◽

Vol 48 (02) ◽

pp. 366-388 ◽

Cited By ~ 1

Author(s):

Eckhard Schlemm

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Asymptotic Variance ◽

Almost Sure Convergence ◽

First Passage Percolation ◽

First Passage ◽

Percolation Problem ◽

Vertex Set ◽

Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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Optimal transport and unitary orbits in C⁎-algebras

Journal of Functional Analysis ◽

10.1016/j.jfa.2021.109068 ◽

2021 ◽

Vol 281 (5) ◽

pp. 109068

Author(s):

Bhishan Jacelon ◽

Karen R. Strung ◽

Alessandro Vignati

Keyword(s):

Optimal Transport ◽

C Algebras

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Modeling Epidemic Spread among a Commuting Population Using Transport Schemes

Mathematics ◽

10.3390/math9161861 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1861

Author(s):

Daniela Calvetti ◽

Alexander P. Hoover ◽

Johnie Rose ◽

Erkki Somersalo

Keyword(s):

Network Model ◽

Transport Theory ◽

Optimal Transport ◽

State Level ◽

Compartment Model ◽

Census Bureau ◽

Mitigation Strategies ◽

Mitigation Measures ◽

Metapopulation Model ◽

Infection Dynamics

Understanding the dynamics of the spread of COVID-19 between connected communities is fundamental in planning appropriate mitigation measures. To that end, we propose and analyze a novel metapopulation network model, particularly suitable for modeling commuter traffic patterns, that takes into account the connectivity between a heterogeneous set of communities, each with its own infection dynamics. In the novel metapopulation model that we propose here, transport schemes developed in optimal transport theory provide an efficient and easily implementable way of describing the temporary population redistribution due to traffic, such as the daily commuter traffic between work and residence. Locally, infection dynamics in individual communities are described in terms of a susceptible-exposed-infected-recovered (SEIR) compartment model, modified to account for the specific features of COVID-19, most notably its spread by asymptomatic and presymptomatic infected individuals. The mathematical foundation of our metapopulation network model is akin to a transport scheme between two population distributions, namely the residential distribution and the workplace distribution, whose interface can be inferred from commuter mobility data made available by the US Census Bureau. We use the proposed metapopulation model to test the dynamics of the spread of COVID-19 on two networks, a smaller one comprising 7 counties in the Greater Cleveland area in Ohio, and a larger one consisting of 74 counties in the Pittsburgh–Cleveland–Detroit corridor following the Lake Erie’s American coastline. The model simulations indicate that densely populated regions effectively act as amplifiers of the infection for the surrounding, less densely populated areas, in agreement with the pattern of infections observed in the course of the COVID-19 pandemic. Computed examples show that the model can be used also to test different mitigation strategies, including one based on state-level travel restrictions, another on county level triggered social distancing, as well as a combination of the two.

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Quantum Optimal Transport with Quantum Channels

Annales Henri Poincaré ◽

10.1007/s00023-021-01042-3 ◽

2021 ◽

Author(s):

Giacomo De Palma ◽

Dario Trevisan

Keyword(s):

Optimal Transport ◽

Quantum Channels

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279 WACSAW: A Statistical and Individually Adaptive Method to Determine Sleep and Wakefulness from Activity

SLEEP ◽

10.1093/sleep/zsab072.278 ◽

2021 ◽

Vol 44 (Supplement_2) ◽

pp. A111-A112

Author(s):

Austin Vandegriffe ◽

V A Samaranayake ◽

Matthew Thimgan

Keyword(s):

Time Series ◽

Optimal Transport ◽

Activity Patterns ◽

Adaptive Method ◽

Wasserstein Distance ◽

Activity Data ◽

Commercial Grade ◽

Movement Data ◽

The Difference ◽

Levene Test

Abstract Introduction Technological innovations have broadened the type and amount of activity data that can be captured in the home and under normal living conditions. Yet, converting naturalistic activity patterns into sleep and wakefulness states has remained a challenge. Despite the successes of current algorithms, they do not fill all actigraphy needs. We have developed a novel statistical approach to determine sleep and wakefulness times, called the Wasserstein Algorithm for Classifying Sleep and Wakefulness (WACSAW), and validated the algorithm in a small cohort of healthy participants. Methods WACSAW functional routines: 1) Conversion of the triaxial movement data into a univariate time series; 2) Construction of a Wasserstein weighted sum (WSS) time series by measuring the Wasserstein distance between equidistant distributions of movement data before and after the time-point of interest; 3) Segmenting the time series by identifying changepoints based on the behavior of the WSS series; 4) Merging segments deemed similar by the Levene test; 5) Comparing segments by optimal transport methodology to determine the difference from a flat, invariant distribution at zero. The resulting histogram can be used to determine sleep and wakefulness parameters around a threshold determined for each individual based on histogram properties. To validate the algorithm, participants wore the GENEActiv and a commercial grade actigraphy watch for 48 hours. The accuracy of WACSAW was compared to a detailed activity log and benchmarked against the results of the output from commercial wrist actigraph. Results WACSAW performed with an average accuracy, sensitivity, and specificity of >95% compared to detailed activity logs in 10 healthy-sleeping individuals of mixed sexes and ages. We then compared WACSAW’s performance against a common wrist-worn, commercial sleep monitor. WACSAW outperformed the commercial grade system in each participant compared to activity logs and the variability between subjects was cut substantially. Conclusion The performance of WACSAW demonstrates good results in a small test cohort. In addition, WACSAW is 1) open-source, 2) individually adaptive, 3) indicates individual reliability, 4) based on the activity data stream, and 5) requires little human intervention. WACSAW is worthy of validating against polysomnography and in patients with sleep disorders to determine its overall effectiveness. Support (if any):

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