Summary. We propose a new method for estimating parameters in models that are defined by a system of non-linear differential equations. Such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself. Current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation. The paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non-linear differential equations. The approach is based on a modification of data smoothing methods along with a generalization of profiled estimation. We derive estimates and confidence intervals, and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology. The performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus.Keywords: Differential equation; Dynamic system; Estimating equation; Functional data analysis; Gauss-Newton method; Parameter cascade; Profiled estimation Challenges in dynamic systems estimation Basic properties of dynamic systemsWe have in mind a process that transforms a set of m input functions u.t/ into a set of d output functions x.t/. Dynamic systems model output change directly by linking the output derivativeṡ x.t/ to x.t/ itself, as well as to inputs u:x.t/ = f.x, u, t|θ/, t ∈ [0, T ]:. 1/ Vector θ contains any parameters defining the system whose values are not known from experimental data, theoretical considerations or other sources of information. Systems involving derivatives of x of order n > 1 are reducible to expression (1) by defining new variables, x 1 = x and x 2 =ẋ 1 , . . . , x n =ẋ n−1 : Further generalizations of expression (1) are also candidates for the approach that is developed in this paper but will not be considered. Dependences of f on t other than through x and u arise when, for example, certain quantities defining the system are themselves time varying. Differential equations as a rule do not define their solutions uniquely, but rather as a manifold of solutions of typical dimension d. For example, d2 x=dt 2 = −ω 2 x.t/, reduced toẋ 1 = x 2 anḋ x 2 = −ω 2 x 1 , implies solutions of the form x 1 .t/ = c 1 sin.ωt/ + c 2 cos.ωt/, where coefficients c 1 and c 2 are arbitrary; and at least d = 2 observations are required to identify the solution that Address for correspondence: J. O. Ramsay, 2748 Howe Street, Ottawa, Ontario, K2B 6W9, Canada. E-mail: ramsay@psych.mcgill.ca 742 J. O. Ramsay, G. Hooker, D. Campbell and J. Cao best fits the data. Initial value problems supply x.0/, whereas boundary value problems require d values selected from x(0) and x.T/.However, we assume more generally that only a subset I of the d output variables x may be measured at time point...

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We present a maximum likelihood argument for the Bennett acceptance ratio method, and derive a simple formula for the variance of free energy estimates generated using this method. This derivation of the acceptance ratio method, using a form of logistic regression, a common statistical technique, allows us to shed additional light on the underlying physical and statistical properties of the method. For example, we demonstrate that the acceptance ratio method yields the lowest variance for any estimator of the free energy which is unbiased in the limit of large numbers of measurements.

Understanding long‐term coexistence of numerous competing species is a longstanding challenge in ecology. Progress requires determining which processes and species differences are most important for coexistence when multiple processes operate and species differ in many ways. Modern coexistence theory (MCT), formalised by Chesson, holds out the promise of doing that, but empirical applications remain scarce. We argue that MCT's mathematical complexity and subtlety have obscured the simplicity and power of its underlying ideas and hindered applications. We present a general computational approach that extends our previous solution for the storage effect to all of standard MCT's spatial and temporal coexistence mechanisms, and also process‐defined mechanisms amenable to direct study such as resource partitioning, indirect competition, and life history trade‐offs. The main components are a method to partition population growth rates into contributions from different mechanisms and their interactions, and numerical calculations in which some mechanisms are removed and others retained. We illustrate how our approach handles features that have not been analysed in the standard framework through several case studies: competing diatom species under fluctuating temperature, plant–soil feedbacks in grasslands, facilitation in a beach grass community, and niche differences with independent effects on recruitment, survival and growth in sagebrush steppe.

Standard generalized additive models (GAMs) usually model the dependent variable as a sum of univariate models. Although previous studies have shown that standard GAMs can be interpreted by users, their accuracy is significantly less than more complex models that permit interactions.In this paper, we suggest adding selected terms of interacting pairs of features to standard GAMs. The resulting models, which we call GA 2 M-models, for Generalized Additive Models plus Interactions, consist of univariate terms and a small number of pairwise interaction terms. Since these models only include one-and two-dimensional components, the components of GA 2 M-models can be visualized and interpreted by users. To explore the huge (quadratic) number of pairs of features, we develop a novel, computationally efficient method called FAST for ranking all possible pairs of features as candidates for inclusion into the model.In a large-scale empirical study, we show the effectiveness of FAST in ranking candidate pairs of features. In addition, we show the surprising result that GA 2 M-models have almost the same performance as the best full-complexity models on a number of real datasets. Thus this paper postulates that for many problems, GA 2 M-models can yield models that are both intelligible and accurate.