Parameter estimation and model selection for stochastic differential equations for biological growth
In this paper, we consider stochastic versions of three classical growth models given by ordinary differential equations (ODEs). Indeed we use stochastic versions of Von Bertalanffy, Gompertz, and Logistic differential equations as models. We assume that each stochastic differential equation (SDE) has some crucial parameters in the drift to be estimated and we use the Maximum Likelihood Estimator (MLE) to estimate them. For estimating the diffusion parameter, we use the MLE for two cases and the quadratic variation of the data for one of the SDEs. We apply the Akaike information criterion (AIC) to choose the best model for the simulated data. We consider that the AIC is a function of the drift parameter. We present a simulation study to validate our selection method. The proposed methodology could be applied to datasets with continuous and discrete observations, but also with highly sparse data. Indeed, we can use this method even in the extreme case where we have observed only one point for each path, under the condition that we observed a sufficient number of trajectories. For the last two cases, the data can be viewed as incomplete observations of a model with a tractable likelihood function; then, we propose a version of the Expectation Maximization (EM) algorithm to estimate these parameters. This type of datasets typically appears in fishery, for instance.
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There are multiple cluster randomised trial designs that vary in when the clusters cross between control and intervention states, when observations are made within clusters, and how many observations are made at that time point. Identifying the most efficient study design is complex though, owing to the correlation between observations within clusters and over time. In this article, we present a review of statistical and computational methods for identifying optimal cluster randomised trial designs. We also adapt methods from the experimental design literature for experimental designs with correlated observations to the cluster trial context. We identify three broad classes of methods: using exact formulae for the treatment effect estimator variance for specific models to derive algorithms or weights for cluster sequences; generalised methods for estimating weights for experimental units; and, combinatorial optimisation algorithms to select an optimal subset of experimental units. We also discuss methods for rounding weights to whole numbers of clusters and extensions to non-Gaussian models. We present results from multiple cluster trial examples that compare the different methods, including problems involving determining optimal allocation of clusters across a set of cluster sequences, and selecting the optimal number of single observations to make in each cluster-period for both Gaussian and non-Gaussian models, and including exchangeable and exponential decay covariance structures.
Determining causal relationship between high dimensional observations are among the most important tasks in scientific discoveries. In this paper, we revisited the \emph{linear trace method}, a technique proposed in~\citep{janzing2009telling,zscheischler2011testing} to infer the causal direction between two random variables of high dimensions. We strengthen the existing results significantly by providing an improved tail analysis in addition to extending the results to nonlinear trace functionals with sharper confidence bounds under certain distributional assumptions. We obtain our results by interpreting the trace estimator in the causal regime as a function over random orthogonal matrices, where the concentration of Lipschitz functions over such space could be applied. We additionally propose a novel ridge-regularized variant of the estimator in \cite{zscheischler2011testing}, and give provable bounds relating the ridge-estimated terms to their ground-truth counterparts. We support our theoretical results with encouraging experiments on synthetic datasets, more prominently, under high-dimension low sample size regime.
Large-scale multiple testing under static factor models is commonly used to select skilled funds in financial market. However, static factor models are arguably too stringent as it ignores the serial correlation, which severely distorts error rate control in large-scale inference. In this manuscript, we propose a new multiple testing procedure under dynamic factor models that is robust against both heavy-tailed distributions and the serial dependence. The idea is to integrate a new sample-splitting strategy based on chronological order and a two-pass Fama-Macbeth regression to form a series of statistics with marginal symmetry properties and then to utilize the symmetry properties to obtain a data-driven threshold. We show that our procedure is able to control the false discovery rate (FDR) asymptotically under high-dimensional dynamic factor models. As a byproduct that is of independent interest, we establish a new exponential-type deviation inequality for the sum of random variables on a variety of functionals of linear and non-linear processes. Numerical results including a case study on hedge fund selection demonstrate the advantage of the proposed method over several state-of-the-art methods.
A study is presented in which a contrastive learning approach is used to extract low-dimensional representations of the acoustic environment from single-channel, reverberant speech signals. Convolution of room impulse responses (RIRs) with anechoic source signals is leveraged as a data augmentation technique that offers considerable flexibility in the design of the upstream task. We evaluate the embeddings across three different downstream tasks, which include the regression of acoustic parameters reverberation time RT60 and clarity index C50, and the classification into small and large rooms. We demonstrate that the learned representations generalize well to unseen data and perform similarly to a fully-supervised baseline.
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is necessary to resort to accurate numerical methods to derive the corresponding behaviours in the time domain. In this regard, we develop a computational procedure to solve variable-order fractional differential equations of this novel class. Furthermore, we provide some numerical experiments to show the effectiveness of the proposed techniques.
This paper presents a new approach for the estimation and inference of the regression parameters in a panel data model with interactive fixed effects. It relies on the assumption that the factor loadings can be expressed as an unknown smooth function of the time average of covariates plus an idiosyncratic error term. Compared to existing approaches, our estimator has a simple partial least squares form and does neither require iterative procedures nor the previous estimation of factors. We derive its asymptotic properties by finding out that the limiting distribution has a discontinuity, depending on the explanatory power of our basis functions which is expressed by the variance of the error of the factor loadings. As a result, the usual ``plug-in" methods based on estimates of the asymptotic covariance are only valid pointwise and may produce either over- or under-coverage probabilities. We show that uniformly valid inference can be achieved by using the cross-sectional bootstrap. A Monte Carlo study indicates good performance in terms of mean squared error. We apply our methodology to analyze the determinants of growth rates in OECD countries.
We consider identification and inference for the average treatment effect and heterogeneous treatment effect conditional on observable covariates in the presence of unmeasured confounding. Since point identification of average treatment effect and heterogeneous treatment effect is not achievable without strong assumptions, we obtain bounds on both average and heterogeneous treatment effects by leveraging differential effects, a tool that allows for using a second treatment to learn the effect of the first treatment. The differential effect is the effect of using one treatment in lieu of the other, and it could be identified in some observational studies in which treatments are not randomly assigned to units, where differences in outcomes may be due to biased assignments rather than treatment effects. With differential effects, we develop a flexible and easy-to-implement semi-parametric framework to estimate bounds and establish asymptotic properties over the support for conducting statistical inference. We provide conditions under which causal estimands are point identifiable as well in the proposed framework. The proposed method is examined by a simulation study and two case studies using datasets from National Health and Nutrition Examination Survey and Youth Risk Behavior Surveillance System.
Rather than refining individual candidate solutions for a general non-convex optimization problem, by analogy to evolution, we consider minimizing the average loss for a parametric distribution over hypotheses. In this setting, we prove that Fisher-Rao natural gradient descent (FR-NGD) optimally approximates the continuous-time replicator equation (an essential model of evolutionary dynamics) by minimizing the mean-squared error for the relative fitness of competing hypotheses. We term this finding "conjugate natural selection" and demonstrate its utility by numerically solving an example non-convex optimization problem over a continuous strategy space. Next, by developing known connections between discrete-time replicator dynamics and Bayes's rule, we show that when absolute fitness corresponds to the negative KL-divergence of a hypothesis's predictions from actual observations, FR-NGD provides the optimal approximation of continuous Bayesian inference. We use this result to demonstrate a novel method for estimating the parameters of stochastic processes.
Causal discovery aims to uncover causal structure among a set of variables. Score-based approaches mainly focus on searching for the best Directed Acyclic Graph (DAG) based on a predefined score function. However, most of them are not applicable on a large scale due to the limited searchability. Inspired by the active learning in generative flow networks, we propose a novel approach to learning a DAG from observational data called GFlowCausal. It converts the graph search problem to a generation problem, in which direct edges are added gradually. GFlowCausal aims to learn the best policy to generate high-reward DAGs by sequential actions with probabilities proportional to predefined rewards. We propose a plug-and-play module based on transitive closure to ensure efficient sampling. Theoretical analysis shows that this module could guarantee acyclicity properties effectively and the consistency between final states and fully-connected graphs. We conduct extensive experiments on both synthetic and real datasets, and results show the proposed approach to be superior and also performs well in a large-scale setting.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
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