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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)

- Timothy Sauer
- 2013

Stochastic differential equations (SDEs) provide accessible mathematical models that combine deterministic and probabilistic components of dynamic behavior. This article is an overview of numerical solution methods for SDEs. The solutions are stochastic processes that represent diffusive dynamics, a common modeling assumption in many application areas. We… (More)

- Timothy Sauer
- 2009

Consider a difference equation whose evolution rule is defined as the maximum of several first-order equations. It is shown that if the first-order equations are individually contractive, then the aggregated max-type equation converges to a fixed point. A corresponding result holds for local convergence.

- Timothy Sauer
- 2009

This chapter is an introduction and survey of numerical solution methods for stochastic differential equations. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial systems. We include a review of fundamental concepts, a description of elementary numerical methods and the… (More)

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)

Density estimation is a crucial component of many machine learning methods, and manifold learning in particular, where geometry is to be constructed from data alone. A significant practical limitation of the current density estimation literature is that methods have not been developed for manifolds with boundary, except in simple cases of linear manifolds… (More)

- Tyrus Berry, Timothy Sauer
- 2015

Spectral methods have received attention as powerful theoretical and practical approaches to a number of machine learning problems. The methods are based on the solution of the eigenproblem of a similarity matrix formed from distance kernels. In this article we discuss three problems that are endemic in current implementations of spectral clustering: (1)… (More)

- Tyrus Berry, Timothy Sauer
- 2015

We consider practical density estimation from large data sets sampled on manifolds embedded in Euclidean space. Existing density estimators on manifolds typically require prior knowledge of the geometry of the manifold, and all density estimation on embedded manifolds is restricted to compact manifolds without boundary. First, motivated by recent… (More)

- Tyrus Berry, Timothy Sauer
- 2010

Consider a difference equation which takes the k-th largest output of m functions of the previous m terms of the sequence. If the functions are also allowed to change periodically as the difference equation evolves, this is analogous to a differential equation with periodic forcing. A large class of such non-autonomous difference equations are shown to… (More)