arXiv subjects

Tyrus Berry

Publications and source records attributed to Tyrus Berry.

35 records · Page 2Linked to original sources

arXiv2019

Consistent Manifold Representation for Topological Data Analysis

For data sampled from an arbitrary density on a manifold embedded in Euclidean space, the Continuous k-Nearest Neighbors (CkNN) graph construction is introduced. It is shown that CkNN is geometrically consistent in the sense that under certain conditions, the unnormalized graph Laplacian converges to the Laplace-Beltrami operator, spectrally as well as pointwise. It is proved for compact (and conjectured for noncompact) manifolds that CkNN is the unique unweighted construction that yields a geometry consistent with the connected components of the underlying manifold in the limit of large data. Thus CkNN produces a single graph that captures all topological features simultaneously, in contrast to persistent homology, which represents each homology generator at a separate scale. As applications we derive a new fast clustering algorithm and a method to identify patterns in natural images topologically. Finally, we conjecture that CkNN is topologically consistent, meaning that the homology of the Vietoris-Rips complex (implied by the graph Laplacian) converges to the homology of the underlying manifold (implied by the Laplace-de Rham operators) in the limit of large data.

math.CA↗

arXiv2018

Applications of topological graph theory to $2$-manifold learning

We show how, given a sufficiently large point cloud sampled from an embedded 2-manifold in $\mathbb{R}^n$, we may obtain a global representation as a cell complex with vertices given by a representative subset of the point cloud. The vertex spacing is based on obtaining an approximation of the tangent plane which insures that the vertex accurately summarizes the local data. Using results from topological graph theory, we couple our cell complex representation with the known Classification of Surfaces in order to classify the manifold. The algorithm developed gives a meaningful description of the embedding as a piecewise linear structure, which is obtained from combinatorial data by projecting points in the point cloud into estimates of tangent planes.

math.NA↗

arXiv2018

Limits on reconstruction of dynamical networks

An observability condition number is defined for physical systems modeled by network dynamics. Assuming the dynamical equations of the network are known and a noisy trajectory is observed at a subset of the nodes, we calculate the expected distance to the nearest correct trajectory as a function of the observation noise level, and discuss how it varies over the unobserved nodes of the network. When the condition number is sufficiently large, reconstructing the trajectory from observations from the subset will be infeasible. This knowledge can be used to choose an optimal subset from which to observe a network.

math.DS↗

arXiv2018

Correcting Observation Model Error in Data Assimilation

Standard methods of data assimilation assume prior knowledge of a model that describes the system dynamics and an observation function that maps the model state to a predicted output. An accurate mapping from model state to observation space is crucial in filtering schemes when adjusting the estimate of the system state during the filter's analysis step. However, in many applications the true observation function may be unknown and the available observation model may have significant errors, resulting in a suboptimal state estimate. We propose a method for observation model error correction within the filtering framework. The procedure involves an alternating minimization algorithm used to iteratively update a given observation function to increase consistency with the model and prior observations, using ideas from attractor reconstruction. The method is demonstrated on the Lorenz 1963 and Lorenz 1996 models, and on a single-column radiative transfer model with multicloud parameterization.

math.DS↗

arXiv2016

Correcting biased observation model error in data assimilation

While the formulation of most data assimilation schemes assumes an unbiased observation model error, in real applications, model error with nontrivial biases is unavoidable. A practical example is the error in the radiative transfer model (which is used to assimilate satellite measurements) in the presence of clouds. As a consequence, many (in fact 99\%) of the cloudy observed measurements are not being used although they may contain useful information. This paper presents a novel nonparametric Bayesian scheme which is able to learn the observation model error distribution and correct the bias in incoming observations. This scheme can be used in tandem with any data assimilation forecasting system. The proposed model error estimator uses nonparametric likelihood functions constructed with data-driven basis functions based on the theory of kernel embeddings of conditional distributions developed in the machine learning community. Numerically, we show positive results with two examples. The first example is designed to produce a bimodality in the observation model error (typical of "cloudy" observations) by introducing obstructions to the observations which occur randomly in space and time. The second example, which is physically more realistic, is to assimilate cloudy satellite brightness temperature-like quantities, generated from a stochastic cloud model for tropical convection and a simple radiative transfer model.

stat.ME↗

arXiv2016

Kalman-Takens filtering in the presence of dynamical noise

The use of data assimilation for the merging of observed data with dynamical models is becoming standard in modern physics. If a parametric model is known, methods such as Kalman filtering have been developed for this purpose. If no model is known, a hybrid Kalman-Takens method has been recently introduced, in order to exploit the advantages of optimal filtering in a nonparametric setting. This procedure replaces the parametric model with dynamics reconstructed from delay coordinates, while using the Kalman update formulation to assimilate new observations. We find that this hybrid approach results in comparable efficiency to parametric methods in identifying underlying dynamics, even in the presence of dynamical noise. By combining the Kalman-Takens method with an adaptive filtering procedure we are able to estimate the statistics of the observational and dynamical noise. This solves a long standing problem of separating dynamical and observational noise in time series data, which is especially challenging when no dynamical model is specified.

physics.data-an↗

arXiv2016

Forecasting Turbulent Modes with Nonparametric Diffusion Models: Learning from noisy data

In this paper, we apply a recently developed nonparametric modeling approach, the "diffusion forecast", to predict the time-evolution of Fourier modes of turbulent dynamical systems. While the diffusion forecasting method assumes the availability of a noise-free training data set observing the full state space of the dynamics, in real applications we often have only partial observations which are corrupted by noise. To alleviate these practical issues, following the theory of embedology, the diffusion model is built using the delay-embedding coordinates of the data. We show that this delay embedding biases the geometry of the data in a way which extracts the most stable component of the dynamics and reduces the influence of independent additive observation noise. The resulting diffusion forecast model approximates the semigroup solutions of the generator of the underlying dynamics in the limit of large data and when the observation noise vanishes. As in any standard forecasting problem, the forecasting skill depends crucially on the accuracy of the initial conditions. We introduce a novel Bayesian method for filtering the discrete-time noisy observations which works with the diffusion forecast to determine the forecast initial densities. Numerically, we compare this nonparametric approach with standard stochastic parametric models on a wide-range of well-studied turbulent modes, including the Lorenz-96 model in weakly chaotic to fully turbulent regimes and the barotropic modes of a quasi-geostrophic model with baroclinic instabilities. We show that when the only available data is the low-dimensional set of noisy modes that are being modeled, the diffusion forecast is indeed competitive to the perfect model.

nlin.CD↗

arXiv2016

Density Estimation on Manifolds with Boundary

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 where the location of the boundary is assumed to be known. We overcome this limitation by developing a density estimation method for manifolds with boundary that does not require any prior knowledge of the location of the boundary. To accomplish this we introduce statistics that provably estimate the distance and direction of the boundary, which allows us to apply a cut-and-normalize boundary correction. By combining multiple cut-and-normalize estimators we introduce a consistent kernel density estimator that has uniform bias, at interior and boundary points, on manifolds with boundary.

math.CA↗

arXiv2015

Iterated Diffusion Maps for Feature Identification

Recently, the theory of diffusion maps was extended to a large class of local kernels with exponential decay which were shown to represent various Riemannian geometries on a data set sampled from a manifold embedded in Euclidean space. Moreover, local kernels were used to represent a diffeomorphism, H, between a data set and a feature of interest using an anisotropic kernel function, defined by a covariance matrix based on the local derivatives, DH. In this paper, we generalize the theory of local kernels to represent degenerate mappings where the intrinsic dimension of the data set is higher than the intrinsic dimension of the feature space. First, we present a rigorous method with asymptotic error bounds for estimating DH from the training data set and feature values. We then derive scaling laws for the singular values of the local linear structure of the data, which allows the identification the tangent space and improved estimation of the intrinsic dimension of the manifold and the bandwidth parameter of the diffusion maps algorithm. Using these numerical tools, our approach to feature identification is to iterate the diffusion map with appropriately chosen local kernels that emphasize the features of interest. We interpret the iterated diffusion map (IDM) as a discrete approximation to an intrinsic geometric flow which smoothly changes the geometry of the data space to emphasize the feature of interest. When the data lies on a product manifold of the feature manifold with an irrelevant manifold, we show that the IDM converges to the quotient manifold which is isometric to the feature manifold, thereby eliminating the irrelevant dimensions. We will also demonstrate empirically that if we apply the IDM to features that are not a quotient of the data space, the algorithm identifies an intrinsically lower-dimensional set embedding of the data which better represents the features.

math.CA↗

arXiv2015

Nonparametric Uncertainty Quantification for Stochastic Gradient Flows

This paper presents a nonparametric statistical modeling method for quantifying uncertainty in stochastic gradient systems with isotropic diffusion. The central idea is to apply the diffusion maps algorithm to a training data set to produce a stochastic matrix whose generator is a discrete approximation to the backward Kolmogorov operator of the underlying dynamics. The eigenvectors of this stochastic matrix, which we will refer to as the diffusion coordinates, are discrete approximations to the eigenfunctions of the Kolmogorov operator and form an orthonormal basis for functions defined on the data set. Using this basis, we consider the projection of three uncertainty quantification (UQ) problems (prediction, filtering, and response) into the diffusion coordinates. In these coordinates, the nonlinear prediction and response problems reduce to solving systems of infinite-dimensional linear ordinary differential equations. Similarly, the continuous-time nonlinear filtering problem reduces to solving a system of infinite-dimensional linear stochastic differential equations. Solving the UQ problems then reduces to solving the corresponding truncated linear systems in finitely many diffusion coordinates. By solving these systems we give a model-free algorithm for UQ on gradient flow systems with isotropic diffusion. We numerically verify these algorithms on a 1-dimensional linear gradient flow system where the analytic solutions of the UQ problems are known. We also apply the algorithm to a chaotically forced nonlinear gradient flow system which is known to be well approximated as a stochastically forced gradient flow.

math.DS↗

arXiv2015

Semiparametric forecasting and filtering: correcting low-dimensional model error in parametric models

Semiparametric forecasting and filtering are introduced as a method of addressing model errors arising from unresolved physical phenomena. While traditional parametric models are able to learn high-dimensional systems from small data sets, their rigid parametric structure makes them vulnerable to model error. On the other hand, nonparametric models have a very flexible structure, but they suffer from the curse-of-dimensionality and are not practical for high-dimensional systems. The semiparametric approach loosens the structure of a parametric model by fitting a data-driven nonparametric model for the parameters. Given a parametric dynamical model and a noisy data set of historical observations, an adaptive Kalman filter is used to extract a time-series of the parameter values. A nonparametric forecasting model for the parameters is built by projecting the discrete shift map onto a data-driven basis of smooth functions. Existing techniques for filtering and forecasting algorithms extend naturally to the semiparametric model which can effectively compensate for model error, with forecasting skill approaching that of the perfect model. Semiparametric forecasting and filtering are a generalization of statistical semiparametric models to time-dependent distributions evolving under dynamical systems.

stat.ME↗

arXiv2015

Variable Bandwidth Diffusion Kernels

Practical applications of kernel methods often use variable bandwidth kernels, also known as self-tuning kernels, however much of the current theory of kernel based techniques is only applicable to fixed bandwidth kernels. In this paper, we derive the asymptotic expansion of these variable bandwidth kernels for arbitrary bandwidth functions; generalizing the theory of Diffusion Maps and Laplacian Eigenmaps. We also derive pointwise error estimates for the corresponding discrete operators which are based on finite data sets; generalizing a result of Singer which was restricted to fixed bandwidth kernels. Our analysis reveals how areas of small sampling density lead to large errors, particularly for fixed bandwidth kernels. We explain the limitation of the existing theory to data sampled from compact manifolds by showing that when the sampling density is not bounded away from zero (which implies that the data lies on an open set) the error estimates for fixed bandwidth kernels will be unbounded. We show that this limitation can be overcome by choosing a bandwidth function inversely proportional to the sampling density (which can be estimated from data) which allows us to control the error estimates uniformly over a non-compact manifold. We numerically verify these results on non-compact manifolds by constructing the generator of the Ornstein-Uhlenbeck process on a real line and a two-dimensional plane using data sampled independently from the respective invariant measures. We also verify our results on compact manifolds by constructing the Laplacian on the unit circle and the unit sphere and we show that the variable bandwidth kernels exhibit reduced sensitivity to bandwidth selection and give better results for an automatic bandwidth selection algorithm.

math.SP↗

arXiv2015

Nonparametric forecasting of low-dimensional dynamical systems

This letter presents a non-parametric modeling approach for forecasting stochastic dynamical systems on low-dimensional manifolds. The key idea is to represent the discrete shift maps on a smooth basis which can be obtained by the diffusion maps algorithm. In the limit of large data, this approach converges to a Galerkin projection of the semigroup solution to the underlying dynamics on a basis adapted to the invariant measure. This approach allows one to quantify uncertainties (in fact, evolve the probability distribution) for non-trivial dynamical systems with equation-free modeling. We verify our approach on various examples, ranging from an inhomogeneous anisotropic stochastic differential equation on a torus, the chaotic Lorenz three-dimensional model, and the Niño-3.4 data set which is used as a proxy of the El-Niño Southern Oscillation.

math.DS↗

arXiv2015

Local Kernels and the Geometric Structure of Data

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 data. We explicitly connect the drift and diffusion coefficients of the process to the moments of the kernel. Moreover, when the kernel is symmetric, the generator is the Laplace-Beltrami operator with respect to a geometry which is influenced by the embedding geometry and the properties of the kernel. In particular, this allows us to generate any Riemannian geometry by an appropriate choice of local kernel. In this way, we continue a program of Belkin, Niyogi, Coifman and others to reinterpret the current diverse collection of kernel-based data analysis methods and place them in a geometric framework. We show how to use this framework to design local kernels invariant to various features of data. These data-driven local kernels can be used to construct conformally invariant embeddings and reconstruct global diffeomorphisms.

math.CA↗

arXiv2014

Linear theory for filtering nonlinear multiscale systems with model error

We study filtering of multiscale dynamical systems with model error arising from unresolved smaller scale processes. The analysis assumes continuous-time noisy observations of all components of the slow variables alone. For a linear model with Gaussian noise, we prove existence of a unique choice of parameters in a linear reduced model for the slow variables. The linear theory extends to to a non-Gaussian, nonlinear test problem, where we assume we know the optimal stochastic parameterization and the correct observation model. We show that when the parameterization is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa. Given the correct parameterization, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that parameters estimated online, as part of a filtering procedure, produce accurate filtering and equilibrium statistical prediction. In contrast, a linear regression based offline method, which fits the parameters to a given training data set independently from the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed.

math.DS↗

arXiv2010

Convergence of Periodically-Forced Rank-Type Equations

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 converge to a periodic limit which is independent of the initial condition. The period of the limit does not depend on how far back each term is allowed to look back in the sequence, and is in fact equal to the period of the forcing.

math.DS↗

arXiv2009

Outer Approximation of the Spectrum of a Fractal Laplacian

We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside. The method allows a numerical approximation of eigenvalues and eigenfunctions for lower portions of the spectrum. We present experimental evidence that the method works by looking at examples where the spectrum of the fractal Laplacian is known (the unit interval and the Sierpinski Gasket (SG)). We also present a speculative description of the spectrum on the standard Sierpinski carpet (SC), where existence of a self-similar Laplacian is known, and also on nonsymmetric and random carpets and the octagasket, where existence of a self-similar Laplacian is not known. At present we have no explanation as to why the method should work. Nevertheless, we are able to prove some new results about the structure of the spectrum involving "miniaturization" of eigenfunctions that we discovered by examining the experimental results obtained using our method.

math.AP↗