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- Franz Hamilton, Tyrus Berry, Timothy Sauer
- Physical review. E, Statistical, nonlinear, and…
- 2015

Methods for forecasting time series are a critical aspect of the understanding and control of complex networks. When the model of the network is unknown, nonparametric methods for prediction have been developed, based on concepts of attractor reconstruction pioneered by Takens and others. In this Rapid Communication we consider how to make use of a subset… (More)

- Tyrus Berry, John Robert Cressman, Zrinka Greguric-Ferencek, Timothy Sauer
- SIAM J. Applied Dynamical Systems
- 2013

It has long been known that the method of time-delay embedding can be used to reconstruct non-linear dynamics from time series data. A less-appreciated fact is that the induced geometry of time-delay coordinates increasingly biases the reconstruction toward the stable directions as delays are added. This bias can be exploited, using the diffusion maps… (More)

- Tyrus Berry, Timothy Sauer
- 2014

a r t i c l e i n f o a b s t r a c t We introduce a theory of local kernels, which generalize the kernels used in the standard diffusion maps construction of nonparametric modeling. We prove that evaluating a local kernel on a data set gives a discrete representation of the generator of a continuous Markov process, which converges in the limit of large… (More)

- Tyrus Berry, Franz Hamilton, Nathalia Peixoto, Timothy Sauer
- Journal of neuroscience methods
- 2012

We develop a method from semiparametric statistics (Cox, 1972) for the purpose of tracking links and connection strengths over time in a neuronal network from spike train data. We consider application of the method as implemented in Masud and Borisyuk (2011), and evaluate its use on data generated independently of the Cox model hypothesis, in particular… (More)

- Domenico Napoletani, Timothy D Sauer
- Physical review. E, Statistical, nonlinear, and…
- 2008

Given a general physical network and measurements of node dynamics, methods are proposed for reconstructing the network topology. We focus on networks whose connections are sparse and where data are limited. Under these conditions, common in many biological networks, constrained optimization techniques based on the L1 vector norm are found to be superior… (More)

Methods of data assimilation are established in physical sciences and engineering for the merging of observed data with dynamical models. When the model is nonlinear, methods such as the ensemble Kalman filter have been developed for this purpose. At the other end of the spectrum, when a model is not known, the delay coordinate method introduced by Takens… (More)

- Franz Hamilton, Tyrus Berry, Nathalia Peixoto, Timothy Sauer
- Physical review. E, Statistical, nonlinear, and…
- 2013

A nonlinear data assimilation technique is applied to determine and track effective connections between ensembles of cultured spinal cord neurons measured with multielectrode arrays. The method is statistical, depending only on confidence intervals, and requiring no form of arbitrary thresholding. In addition, the method updates connection strengths… (More)

- Timothy Sauer
- Applied Mathematics and Computation
- 2011

Convergence results are presented for rank-type difference equations , whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univari-ate… (More)

Differential equation modeling is central to applications of mathematics to science and engineering. When a particular system of equations is used in an application, it is often important to determine unknown parameters. We compare the traditional shooting method to versions of multiple shooting methods in chaotic systems with noise added and conduct… (More)

– Assimilation of data with models of physical processes is a critical component of modern scientific analysis. In recent years, nonlinear versions of Kalman filtering have been developed, in addition to methods that estimate model parameters in parallel with the system state. We propose a substantial extension of these tools to deal with the specific case… (More)