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Ordinary differential equation (ODE) models are widely used to describe chemical or biological processes. This article considers the estimation and assessment of such models on the basis of time-course data. Due to experimental limitations, time-course data are often noisy and some components of the system may not be observed. Furthermore, the computational demands of numerical integration have hindered the widespread adoption of time-course analysis using ODEs. To address these challenges, we explore the efficacy of the recently developed MAGI (MAnifold-constrained Gaussian process Inference) method for ODE inference. First, via a range of examples we show that MAGI is capable of inferring the parameters and system trajectories, including unobserved components, with appropriate uncertainty quantification. Second, we illustrate how MAGI can be used to assess and select different ODE models with time-course data based on MAGI's efficient computation of model predictions. Overall, we believe MAGI is a useful method for the analysis of time-course data in the context of ODE models, which bypasses the need for any numerical integration., Comment: 26 pages, 8 figures, with code supplement

Catalytic prior distributions provide general, easy-to-use, and interpretable specifications of prior distributions for Bayesian analysis. They are particularly beneficial when the observed data are inadequate to stably estimate a complex target model. A catalytic prior distribution is constructed by augmenting the observed data with synthetic data that are sampled from the predictive distribution of a simpler model estimated from the observed data. We illustrate the usefulness of the catalytic prior approach using an example from labor economics. In the example, the resulting Bayesian inference reflects many important aspects of the observed data, and the estimation accuracy and predictive performance of the inference based on the catalytic prior are superior to, or comparable to, that of other commonly used prior distributions. We further explore the connection between the catalytic prior approach and a few popular regularization methods. We expect the catalytic prior approach to be useful in many applications.

This article presents the MAGI software package for the inference of dynamic systems. The focus of MAGI is on dynamics modeled by nonlinear ordinary differential equations with unknown parameters. While such models are widely used in science and engineering, the available experimental data for parameter estimation may be noisy and sparse. Furthermore, some system components may be entirely unobserved. MAGI solves this inference problem with the help of manifold-constrained Gaussian processes within a Bayesian statistical framework, whereas unobserved components have posed a significant challenge for existing software. We use several realistic examples to illustrate the functionality of MAGI. The user may choose to use the package in any of the R, MATLAB, and Python environments., Comment: 47 pages, 10 figures

Exceptional points are interesting physical phenomena in non-Hermitian physics at which the eigenvalues are degenerate and the eigenvectors coalesce. In this paper, we find that the universal feature of arbitrary non-Hermitian two level systems with singularities is basis defectiveness rather than energy degeneracy or state coalescence. This leads to the discovery of general exceptional points (GEPs). For GEPs, more subtle structures (e.g., Bloch peach), additional classification, and' 'hidden" quantum phase transitions are explored. By using the topologically protected subspace from two edge states in the non-Hermitian SSH model as an example, we illustrate the physical properties of different types of GEPs.

Big data generated from the Internet offer great potential for predictive analysis. Here we focus on using online users' Internet search data to forecast unemployment initial claims weeks into the future, which provides timely insights into the direction of the economy. To this end, we present a novel method PRISM (Penalized Regression with Inferred Seasonality Module), which uses publicly available online search data from Google. PRISM is a semi-parametric method, motivated by a general state-space formulation, and employs nonparametric seasonal decomposition and penalized regression. For forecasting unemployment initial claims, PRISM outperforms all previously available methods, including forecasting during the 2008-2009 financial crisis period and near-future forecasting during the COVID-19 pandemic period, when unemployment initial claims both rose rapidly. The timely and accurate unemployment forecasts by PRISM could aid government agencies and financial institutions to assess the economic trend and make well-informed decisions, especially in the face of economic turbulence.

Parameter estimation for nonlinear dynamic system models, represented by ordinary differential equations (ODEs), using noisy and sparse data is a vital task in many fields. We propose a fast and accurate method, MAGI (MAnifold-constrained Gaussian process Inference), for this task. MAGI uses a Gaussian process model over time-series data, explicitly conditioned on the manifold constraint that derivatives of the Gaussian process must satisfy the ODE system. By doing so, we completely bypass the need for numerical integration and achieve substantial savings in computational time. MAGI is also suitable for inference with unobserved system components, which often occur in real experiments. MAGI is distinct from existing approaches as we provide a principled statistical construction under a Bayesian framework, which incorporates the ODE system through the manifold constraint. We demonstrate the accuracy and speed of MAGI using realistic examples based on physical experiments.

For epidemics control and prevention, timely insights of potential hot spots are invaluable. Alternative to traditional epidemic surveillance, which often lags behind real time by weeks, big data from the Internet provide important information of the current epidemic trends. Here we present a methodology, ARGOX (Augmented Regression with GOogle data CROSS space), for accurate real-time tracking of state-level influenza epidemics in the United States. ARGOX combines Internet search data at the national, regional and state levels with traditional influenza surveillance data from the Centers for Disease Control and Prevention, and accounts for both the spatial correlation structure of state-level influenza activities and the evolution of people's Internet search pattern. ARGOX achieves on average 28\% error reduction over the best alternative for real-time state-level influenza estimation for 2014 to 2020. ARGOX is robust and reliable and can be potentially applied to track county- and city-level influenza activity and other infectious diseases.

Dengue is a mosquito-borne disease that threatens more than half of the world's population. Despite being endemic to over 100 countries, government-led efforts and mechanisms to timely identify and track the emergence of new infections are still lacking in many affected areas. Multiple methodologies that leverage the use of Internet-based data sources have been proposed as a way to complement dengue surveillance efforts. Among these, the trends in dengue-related Google searches have been shown to correlate with dengue activity. We extend a methodological framework, initially proposed and validated for flu surveillance, to produce near real-time estimates of dengue cases in five countries/regions: Mexico, Brazil, Thailand, Singapore and Taiwan. Our result shows that our modeling framework can be used to improve the tracking of dengue activity in multiple locations around the world.

This paper discusses the simultaneous inference of mean parameters in a family of distributions with quadratic variance function. We first introduce a class of semiparametric/parametric shrinkage estimators and establish their asymptotic optimality properties. Two specific cases, the location-scale family and the natural exponential family with quadratic variance function, are then studied in detail. We conduct a comprehensive simulation study to compare the performance of the proposed methods with existing shrinkage estimators. We also apply the method to real data and obtain encouraging results., Comment: Published at http://dx.doi.org/10.1214/15-AOS1377 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Accurate real-time tracking of influenza outbreaks helps public health officials make timely and meaningful decisions that could save lives. We propose an influenza tracking model, ARGO (AutoRegression with GOogle search data), that uses publicly available online search data. In addition to having a rigorous statistical foundation, ARGO outperforms all previously available Google-search-based tracking models, including the latest version of Google Flu Trends, even though it uses only low-quality search data as input from publicly available Google Trends and Google Correlate websites. ARGO not only incorporates the seasonality in influenza epidemics but also captures changes in people's online search behavior over time. ARGO is also flexible, self-correcting, robust, and scalable, making it a potentially powerful tool that can be used for real-time tracking of other social events at multiple temporal and spatial resolutions., Comment: 23 pages, 2 figures, Proceedings of the National Academy of Sciences (2015)

New advances in nano sciences open the door for scientists to study biological processes on a microscopic molecule-by-molecule basis. Recent single-molecule biophysical experiments on enzyme systems, in particular, reveal that enzyme molecules behave fundamentally differently from what classical model predicts. A stochastic network model was previously proposed to explain the experimental discovery. This paper conducts detailed theoretical and data analyses of the stochastic network model, focusing on the correlation structure of the successive reaction times of a single enzyme molecule. We investigate the correlation of experimental fluorescence intensity and the correlation of enzymatic reaction times, and examine the role of substrate concentration in enzymatic reactions. Our study shows that the stochastic network model is capable of explaining the experimental data in depth., Comment: Published in at http://dx.doi.org/10.1214/12-AOAS541 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Doubly stochastic Poisson processes, also known as the Cox processes, frequently occur in various scientific fields. In this article, motivated primarily by analyzing Cox process data in biophysics, we propose a nonparametric kernel-based inference method. We conduct a detailed study, including an asymptotic analysis, of the proposed method, and provide guidelines for its practical use, introducing a fast and stable regression method for bandwidth selection. We apply our method to real photon arrival data from recent single-molecule biophysical experiments, investigating proteins' conformational dynamics. Our result shows that conformational fluctuation is widely present in protein systems, and that the fluctuation covers a broad range of time scales, highlighting the dynamic and complex nature of proteins' structure., Comment: Published in at http://dx.doi.org/10.1214/10-AOAS352 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Advances in nanotechnology have allowed scientists to study biological processes on an unprecedented nanoscale molecule-by-molecule basis, opening the door to addressing many important biological problems. A phenomenon observed in recent nanoscale single-molecule biophysics experiments is subdiffusion, which largely departs from the classical Brownian diffusion theory. In this paper, by incorporating fractional Gaussian noise into the generalized Langevin equation, we formulate a model to describe subdiffusion. We conduct a detailed analysis of the model, including (i) a spectral analysis of the stochastic integro-differential equations introduced in the model and (ii) a microscopic derivation of the model from a system of interacting particles. In addition to its analytical tractability and clear physical underpinning, the model is capable of explaining data collected in fluorescence studies on single protein molecules. Excellent agreement between the model prediction and the single-molecule experimental data is seen., Comment: Published in at http://dx.doi.org/10.1214/07-AOAS149 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Rejoinder to ``Equi-energy sampler with applications in statistical inference and statistical mechanics'' by Kou, Zhou and Wong [math.ST/0507080], Comment: Published at http://dx.doi.org/10.1214/009053606000000524 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper studies, under the setting of spline regression, the connection between finite-sample properties of selection criteria and their asymptotic counterparts, focusing on bridging the gap between the two. We introduce a bias-variance decomposition of the prediction error, using which it is shown that in the asymptotics the bias term dominates the variability term, providing an explanation of the gap. A geometric exposition is provided for intuitive understanding. The theoretical and geometric results are illustrated through a numerical example., Comment: Published at http://dx.doi.org/10.1214/009053604000000841 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We introduce a new sampling algorithm, the equi-energy sampler, for efficient statistical sampling and estimation. Complementary to the widely used temperature-domain methods, the equi-energy sampler, utilizing the temperature--energy duality, targets the energy directly. The focus on the energy function not only facilitates efficient sampling, but also provides a powerful means for statistical estimation, for example, the calculation of the density of states and microcanonical averages in statistical mechanics. The equi-energy sampler is applied to a variety of problems, including exponential regression in statistics, motif sampling in computational biology and protein folding in biophysics., Comment: This paper discussed in: [math.ST/0611217], [math.ST/0611219], [math.ST/0611221], [math.ST/0611222]. Rejoinder in [math.ST/0611224]. Published at http://dx.doi.org/10.1214/009053606000000515 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The periodic bounce configurations responsible for quantum tunneling are obtained explicitly and are extended to the finite energy case for minisuperspace models of the Universe. As a common feature of the tunneling models at finite energy considered here we observe that the period of the bounce increases with energy monotonically. The periodic bounces do not have bifurcations and make no contribution to the nucleation rate except the one with zero energy. The sharp first order phase transition from quantum tunneling to thermal activation is verified with the general criterions., Comment: 17 pages, 5 postscript figures included

In this paper we study the Macroscopic Quantum Oscillation (MQO) effect in ferromagnetic single domain magnets with a magnetic field applied along the hard anistropy axis. The level splitting for the ground state, derived with the conventional instanton method, oscillates with the external field and is quenched at some field values. A formula for quantum tunneling at excited levels is also obtained. The existence of topological phase accounts for this kind of oscillation and the corresponding thermodynamical quantities exhibit similar interference effects which resembles to some extent the electron quantum phase interference induced by gauge potential in the Aharonov-Bohm effect and the $\Theta $-vacuum in Yang-Mills field theory.., Comment: 12 pages, 4 figures, to appear in Phys. Rev. B

The crossover from thermal hopping to quantum tunneling is studied. We show that the decay rate $\Gamma$ with dissipation can accurately be determined near the crossover temperature. Besides considering the Wentzel-Kramers-Brillouin (WKB) exponent, we also calculate contribution of the fluctuation modes around the saddle point and give an extended account of a previous study of crossover region. We deal with two dangerous fluctuation modes whose contribution can't be calculated by the steepest descent method and show that higher order couplings between the two dangerous modes need to be taken into considerations. At last the crossover from thermal hopping to quantum tunneling in the molecular magnet Mn_{12}Ac is studied., Comment: 10 pages, 3 figures

The escape rates of the biaxial single domain spin particles with and without an applied magnetic field are investigated. Using the strict potential field description of spin systems developed by Ulyanov and Zaslavskii we obtain new effective Hamiltonians which are considered to be in exact spin-coordinate correspondence unlike the well studied effective Hamiltonians with the approximate correspondence. The sharp first-order transition is found in both cases. The phase diagram of the transitions depending on the anisotropy constant and the external field is also given., Comment: 15 pages, 8 figures

The quantum-classical transitions of the escape rates in a uniaxial spin model relevant to the molecular magnet Mn$_{12}$Ac and a biaxial anisotropic ferromagnetic particle are investigated by applying the periodic instanton method. The effective free energies are expanded around the top of the potential barrier in analogy to Landau theory of phase transitions. We show that the first-order transitions occur below the critical external magnetic field $h_x = 1/4$ for the uniaxial spin model and beyond the critical anisotropy constant ratio $\lambda = 1/2$ for the biaxial ferromagnetic grains, which are in good agreement with earlier works., Comment: 14 pages, revtex, 5 postscript figures