Adaptive, locally-linear models of complex dynamics

The dynamics of complex systems generally include high-dimensional, non-stationary and non-linear behavior, all of which pose fundamental challenges to quantitative understanding. To address these difficulties we detail a new approach based on local linear models within windows determined adaptively from the data. While the dynamics within each window are simple, consisting of exponential decay, growth and oscillations, the collection of local parameters across all windows provides a principled characterization of the full time series. To explore the resulting model space, we develop a novel likelihood-based hierarchical clustering and we examine the eigenvalues of the linear dynamics. We demonstrate our analysis with the Lorenz system undergoing stable spiral dynamics and in the standard chaotic regime. Applied to the posture dynamics of the nematode $C. elegans$ our approach identifies fine-grained behavioral states and model dynamics which fluctuate close to an instability boundary, and we detail a bifurcation in a transition from forward to backward crawling. Finally, we analyze whole-brain imaging in $C. elegans$ and show that the stability of global brain states changes with oxygen concentration.Comment: 25 pages, 16 figure

Adaptive inferential sensors based on evolving fuzzy models

A new technique to the design and use of inferential sensors in the process industry is proposed in this paper, which is based on the recently introduced concept of evolving fuzzy models (EFMs). They address the challenge that the modern process industry faces today, namely, to develop such adaptive and self-calibrating online inferential sensors that reduce the maintenance costs while keeping the high precision and interpretability/transparency. The proposed new methodology makes possible inferential sensors to recalibrate automatically, which reduces significantly the life-cycle efforts for their maintenance. This is achieved by the adaptive and flexible open-structure EFM used. The novelty of this paper lies in the following: (1) the overall concept of inferential sensors with evolving and self-developing structure from the data streams; (2) the new methodology for online automatic selection of input variables that are most relevant for the prediction; (3) the technique to detect automatically a shift in the data pattern using the age of the clusters (and fuzzy rules); (4) the online standardization technique used by the learning procedure of the evolving model; and (5) the application of this innovative approach to several real-life industrial processes from the chemical industry (evolving inferential sensors, namely, eSensors, were used for predicting the chemical properties of different products in The Dow Chemical Company, Freeport, TX). It should be noted, however, that the methodology and conclusions of this paper are valid for the broader area of chemical and process industries in general. The results demonstrate that well-interpretable and with-simple-structure inferential sensors can automatically be designed from the data stream in real time, which predict various process variables of interest. The proposed approach can be used as a basis for the development of a new generation of adaptive and evolving inferential sensors that can a- ddress the challenges of the modern advanced process industry

Estimating Local Function Complexity via Mixture of Gaussian Processes

Real world data often exhibit inhomogeneity, e.g., the noise level, the sampling distribution or the complexity of the target function may change over the input space. In this paper, we try to isolate local function complexity in a practical, robust way. This is achieved by first estimating the locally optimal kernel bandwidth as a functional relationship. Specifically, we propose Spatially Adaptive Bandwidth Estimation in Regression (SABER), which employs the mixture of experts consisting of multinomial kernel logistic regression as a gate and Gaussian process regression models as experts. Using the locally optimal kernel bandwidths, we deduce an estimate to the local function complexity by drawing parallels to the theory of locally linear smoothing. We demonstrate the usefulness of local function complexity for model interpretation and active learning in quantum chemistry experiments and fluid dynamics simulations.Comment: 19 pages, 16 figure

Local-Aggregate Modeling for Big-Data via Distributed Optimization: Applications to Neuroimaging

Technological advances have led to a proliferation of structured big data that have matrix-valued covariates. We are specifically motivated to build predictive models for multi-subject neuroimaging data based on each subject's brain imaging scans. This is an ultra-high-dimensional problem that consists of a matrix of covariates (brain locations by time points) for each subject; few methods currently exist to fit supervised models directly to this tensor data. We propose a novel modeling and algorithmic strategy to apply generalized linear models (GLMs) to this massive tensor data in which one set of variables is associated with locations. Our method begins by fitting GLMs to each location separately, and then builds an ensemble by blending information across locations through regularization with what we term an aggregating penalty. Our so called, Local-Aggregate Model, can be fit in a completely distributed manner over the locations using an Alternating Direction Method of Multipliers (ADMM) strategy, and thus greatly reduces the computational burden. Furthermore, we propose to select the appropriate model through a novel sequence of faster algorithmic solutions that is similar to regularization paths. We will demonstrate both the computational and predictive modeling advantages of our methods via simulations and an EEG classification problem.Comment: 41 pages, 5 figures and 3 table