Publications with Abstracts by Alfred Hübler (1994)
* Publications in journals with a strict referee process
- A. Hübler, D. Pines, Prediction and Adaptation in an Evolving Chaotic Environment, in Complexity: From Metapher to Reality, G. Cowan, D. Pines, G. Meltzer (Eds.), Adison-Wesley, 42 pages (1994)
Abstract: We describe work in progress on computer simulations of adaptive agents responding to an evolving chaotic environment and to one another. Our simulations are designed to quantify adaptation and to explore co-adaptation for a simple calculable model of a complex adaptive system. We first consider the ability of a single agent, exposed to a chaotic environment, to model, control, and predict the future states of that environment. We then introduce a second agent which, in attempting to model and control both the chaotic environment and the first agent, modifies the extent to which that agent can identify patterns and exercise control. The competition between the two predictive agents can lead either to chaos, or to meta-stable emergent behavior, best described as a leader-follower relationship. Our results suggests a correlation between optimal adaptation, optimal complexity, and emergent behavior, and provide preliminar support for the concept of optimal coadaptation near the edge of chaos.
We consider adaptive predictors for a system specified by a logistic map dynamics in which the parameters evolve in a random fashion. The prediction is a single-step prediction of the dynamics of a single map within a non stationary network which provides the environmental dynamics. The model is a set of Km functions and their weights, which relate the actual state of the map to a future state of the map. Therefore, Km is a measure of complexity of the adaptive system. Since we keep the set of functions fixed, the weights are the only parameters of the actual model. They are extracted through a maximum likelihood estimation from the most recent history of the map.
The length of the corresponding time series Nm, as well as the number of parameters of the model Km, and the number of events Nmig which are ignored between succeeding modeling processes are adjusted by trial and error. Lmig = Nmig /( Nmig + Nm) is called level of ignorance. Nm describes certain features of the rationality of the adaptive systems, i.e., the number of events taken into account in order to predict and to optimize the quality function. The adaptive system could in principle determine all Km parameters of its internal model from Km events. Because noise is present, it has to use Nm(>>Km) events, in order to reduce statistical errors. The statistical errors could be made smaller by taking more events form the past; however, since the map dynamics is not stationary, such data would introduce large systematic errors. Balancing the reduction of the statistical error against the increase of the systematic error gives an optimal Nm, i.e., an optimal bounded rationality.
We find (i) optimal adaptive predictors have an optimal rationality and an optimal complexity, which are small in a rapidly changing environment, (ii) that the predictive power can be improved by imposing chaos or random noise onto the environment, (iii) the predictive power and the maximum level of ignorance decrease linearly with the rate of change of the environment, and (iv) the typical time scale of the adaptive process equals the rate of change of the environment if the adaptive system is capable of modeling the experimental dynamics with a small number of parameters. In this case there is a simple way to detect optimal predictors experimentally. For competing adaptive predictors, a configuration appears to be most stable when one imposes a weakly chaotic dynamics on the environment and the other predicts this controlled environment, i.e., a leader-follower relation emerges, in which the leader imposes a weakly chaotic dynamics on the environment.
From a randomly evolving network of weakly coupled logistic maps we find that models of optimal adaptive predictors for individual maps which use a Fourier series for the modeling are (i) simple, i.e., the number of significant parameters of the model equals approximately the number of significant parameters of the dynamics of the environment, (ii)reproducible, i.e., the modeling process yields a unique set of model parameters, and (iii) meaningful, i.e., there is a simple relation between the control parameters of the experiment and the parameters of the models which makes it possible to predict future settings of those parameter values. The models have predictive power in the region of interest of the adaptive system which may not necessarily overlap with the natural dynamics of the environmental system.
Reprint
- * D. Pierre, A. Hübler, A Theory for Adaptation and Competition Applied to Logistic Map Dynamics, Physica D 75, 343-360 (1994)
Abstract: We present a theory for adaptive, predictive, and competitive agents in an evolving chaotic environment. The agents are simple algorithmic agents designed to model, predict, and exert open loop control on their environment. The environment is the time iteration of the logistic map with external noise added. We find that passive agents can make accurate single step predictions even if the environmental dynamics is chaotic, while accurate multiple step predictions are possible only if the Liapunov exponent of the environmental dynamics is negative. Multiple step predictions with high precision can be made over a broad range of conditions when one agent exerts control on the emvironment. When two agents are simultaneously attempting control of the environment an agent will achieve the smallest prediction error when the second agent's goal dynamics has a stable fixed point which coincides with a stable or unstable fixed point of the goal dynamics of the first agent. When the fixed points of the goal dynamics of the two agents do not match, we find that the prediction errors of both agents approach a constant value while the amplitudes of the driving forces grow at a constant rate. Further, our studies suggest that generally the agent with the more complicated goal dynamics may achieve an extremely small prediction error by a perfect entrainment of the environmental dynamics.
Reprint