An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning

Manifold-learning techniques are routinely used in mining complex spatiotemporal data to extract useful, parsimonious data representations/parametrizations; these are, in turn, useful in nonlinear model identification tasks. We focus here on the case of time series data that can ultimately be modelled as a spatially distributed system (e.g. a partial differential equation, PDE), but where we do not know the space in which this PDE should be formulated. Hence, even the spatial coordinates for the distributed system themselves need to be identified - to emerge from - the data mining process. We will first validate this emergent space reconstruction for time series sampled without space labels in known PDEs; this brings up the issue of observability of physical space from temporal observation data, and the transition from spatially resolved to lumped (order-parameter-based) representations by tuning the scale of the data mining kernels. We will then present actual emergent space discovery illustrations. Our illustrative examples include chimera states (states of coexisting coherent and incoherent dynamics), and chaotic as well as quasiperiodic spatiotemporal dynamics, arising in partial differential equations and/or in heterogeneous networks. We also discuss how data-driven spatial coordinates can be extracted in ways invariant to the nature of the measuring instrument. Such gauge-invariant data mining can go beyond the fusion of heterogeneous observations of the same system, to the possible matching of apparently different systems

Gaussian Nonlinear Line Attractor for Learning Multidimensional Data

The human brain’s ability to extract information from multidimensional data modeled by the Nonlinear Line Attractor (NLA), where nodes are connected by polynomial weight sets. Neuron connections in this architecture assumes complete connectivity with all other neurons, thus creating a huge web of connections. We envision that each neuron should be connected to a group of surrounding neurons with weighted connection strengths that reduces with proximity to the neuron. To develop the weighted NLA architecture, we use a Gaussian weighting strategy to model the proximity, which will also reduce the computation times significantly. Once all data has been trained in the NLA network, the weight set can be reduced using a locality preserving nonlinear dimensionality reduction technique. By reducing the weight sets using this technique, we can reduce the amount of outputs for recognition tasks. An appropriate distance measure can then be used for comparing testing data and the trained data when processed through the NLA architecture. It is observed that the proposed GNLA algorithm reduces training time significantly and is able to provide even better recognition using fewer dimensions than the original NLA algorithm. We have tested this algorithm and showed that it works well in different datasets, including the EO Synthetic Vehicle database and the Sheffield face database

Computational neural learning formalisms for manipulator inverse kinematics

An efficient, adaptive neural learning paradigm for addressing the inverse kinematics of redundant manipulators is presented. The proposed methodology exploits the infinite local stability of terminal attractors - a new class of mathematical constructs which provide unique information processing capabilities to artificial neural systems. For robotic applications, synaptic elements of such networks can rapidly acquire the kinematic invariances embedded within the presented samples. Subsequently, joint-space configurations, required to follow arbitrary end-effector trajectories, can readily be computed. In a significant departure from prior neuromorphic learning algorithms, this methodology provides mechanisms for incorporating an in-training skew to handle kinematics and environmental constraints

Multiple Steady States, Limit Cycles and Chaotic Attractors in Evolutionary Games with Logit Dynamics

This paper investigates, by means of simple, three and four strategy games, the occurrence of periodic and chaotic behaviour in a smooth version of the Best Response Dynamics, the Logit Dynamics. The main finding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles, do occur under the Logit Dynamics, even for three strategy games. Moreover, we show that the Logit Dynamics displays another bifurcation which cannot to occur under the Replicator Dynamics: the fold catastrophe. Finally, we find, in a four strategy game, a period-doubling route to chaotic dynamics under a 'weighted' version of the Logit Dynamics.

A Robust Rational Route to in a Simple Asset Pricing Model (revised March 2004)

We investigate asset pricing dynamics in an adaptive evolutionary asset pricing model with fundamentalists, trend followers and a market maker. Agents can choose between a fundamentalist strategy at positive information cost or choose a trend following strategy for free. Price adjustment is proportional to the excess demand in the asset market. Agents asynchronously update their strategy according to realized net profits in the recent past. As agents become more sensitive to differences in strategy performance, the fundamental steady state becomes unstable and multiple steady states may arise. As the traders' sensitivity to differences in fitness increases, a bifurcation route to chaos sets in due to homoclinic bifurcations of stable and unstable manifolds of the fundamental steady state.

Heterogeneous Agents Models: two simple examples, forthcoming In: Lines, M. (ed.) Nonlinear Dynamical Systems in Economics, CISM Courses and Lectures, Springer, 2005, pp.131-164.

These notes review two simple heterogeneous agent models in economics and finance. The first is a cobweb model with rational versus naive agents introduced in Brock and Hommes (1997). The second is an asset pricing model with fundamentalists versus technical traders introduced in Brock and Hommes (1998). Agents are boundedly rational and switch between different trading strategies, based upon an evolutionary fitness measure given by realized past profits. Evolutionary switching creates a nonlinearity in the dynamics. Rational routes to randomness, that is, bifurcation routes to complicated dynamical behaviour occur when agents become more sensitive to differences in evolutionary fitness.