In order to analyse large complex stochastic dynamical models such as those studied in systems biology there is currently a great need for both analytical tools and also algorithms for accurate and fast simulation and estimation. We present a new stochastic approximation of biological oscillators that addresses these needs. Our method, called phase-corrected LNA (pcLNA) overcomes the main limitations of the standard Linear Noise Approximation (LNA) to remain uniformly accurate for long times, still maintaining the speed and analytically tractability of the LNA. As part of this, we develop analytical expressions for key probability distributions and associated quantities, such as the Fisher Information Matrix and Kullback-Leibler divergence and we introduce a new approach to system-global sensitivity analysis. We also present algorithms for statistical inference and for long-term simulation of oscillating systems that are shown to be as accurate but much faster than leaping algorithms and algorithms for integration of diffusion equations. Stochastic versions of published models of the circadian clock and NF-κB system are used to illustrate our results.
In order to analyse large complex stochastic dynamical models such as those studied in systems biology there is currently a great need for both analytical tools and also algorithms for accurate and fast simulation and estimation. We present a new stochastic approximation of biological oscillators that addresses these needs. Our method, called phase-corrected LNA (pcLNA) overcomes the main limitations of the standard Linear Noise Approximation (LNA) to remain uniformly accurate for long times, still maintaining the speed and analytically tractability of the LNA. As part of this, we develop analytical expressions for key probability distributions and associated quantities, such as the Fisher Information Matrix and KullbackLeibler divergence and we introduce a new approach to system-global sensitivity analysis. We also present algorithms for statistical inference and for long-term simulation of oscillating systems that are shown to be as accurate but much faster than leaping algorithms and algorithms for integration of diffusion equations. Stochastic versions of published models of the circadian clock and NF-κB system are used to illustrate our results. Author summaryMany cellular and molecular systems such as the circadian clock and the cell cycle are oscillators that are modelled using nonlinear dynamical systems. Moreover, oscillatory systems are ubiquitous elsewhere in science. There is an extensive theory for perfectly noise-free dynamical systems and very effective algorithms for simulating their temporal behaviour. On the other hand, biological systems are inherently stochastic and the presence of stochastic noise can play a crucial role. Unfortunately, there are far fewer analytical tools and much less understanding for stochastic models especially when they are nonlinear and have lots of state variables and parameters. Moreover simulation is not so effective and can be very slow if the system is large. In this article we describe how to accurately approximate such systems in a way that facilitates fast simulation, parameter estimation and new approaches to analysis, such as calculating probability distributions that describe the system's stochastic behaviour and describing how these distributions change when the parameters of the system are varied. PLOS Computational Biology | https://doi
In multivariate clinical trials, a key research endpoint is ascertaining whether a candidate treatment is more efficacious than an established alternative. This global endpoint is clearly of high practical value for studies, such as those arising from neuroimaging, where the outcome dimensions are not only numerous but they are also highly correlated and the available sample sizes are typically small. In this paper, we develop a two-stage procedure testing the null hypothesis of global equivalence between treatments effects and demonstrate its application to analysing phase II neuroimaging trials. Prior information such as suitable statistics of historical data or suitably elicited expert clinical opinions are combined with data collected from the first stage of the trial to learn a set of optimal weights. We apply these weights to the outcome dimensions of the second-stage responses to form the linear combination z and t tests statistics while controlling the test's false positive rate. We show that the proposed tests hold desirable asymptotic properties and characterise their power functions under wide conditions. In particular, by comparing the power of the proposed tests with that of Hotelling's T(2), we demonstrate their advantages when sample sizes are close to the dimension of the multivariate outcome. We apply our methods to fMRI studies, where we find that, for sufficiently precise first stage estimates of the treatment effect, standard single-stage testing procedures are outperformed.
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