Approximating differentiable relationships between delay embedded dynamical systems with radical basis functions

This thesis is about the study of relationships between experimental dynamical systems. The basic approach is to fit radial basis function maps between time delay embeddings of manifolds. We have shown that under certain conditions these maps are generically diffeomorphisms, and can be analysed to determine whether or not the manifolds in question are diffeomorphically related to each other. If not, a study of the distribution of errors may provide information about the lack of equivalence between the two. The method has applications wherever two or more sensors are used to measure a single system, or where a single sensor can respond on more than one time scale: their respective time series can be tested to determine whether or not they are coupled, and to what degree. One application which we have explored is the determination of a minimum embedding dimension for dynamical system reconstruction. In this special case the diffeomorphism in question is closely related to the predictor for the time series itself. Linear transformations of delay embedded manifolds can also be shown to have nonlinear inverses under the right conditions, and we have used radial basis functions to approximate these inverse maps in a variety of contexts. This method is particularly useful when the linear transformation corresponds to the delay embedding of a finite impulse response filtered time series. One application of fitting an inverse to this linear map is the detection of periodic orbits in chaotic attractors, using suitably tuned filters. This method has also been used to separate signals with known bandwidths from deterministic noise, by tuning a filter to stop the signal and then recovering the chaos with the nonlinear inverse. The method may have applications to the cancellation of noise generated by mechanical or electrical systems. In the course of this research a sophisticated piece of software has been developed. The program allows the construction of a hierarchy of delay embeddings from scalar and multi-valued time series. The embedded objects can be analysed graphically, and radial basis function maps can be fitted between them asynchronously, in parallel, on a multi-processor machine. In addition to a graphical user interface, the program can be driven by a batch mode command language, incorporating the concept of parallel and sequential instruction groups and enabling complex sequences of experiments to be performed in parallel in a resource-efficient manner.

Improving energy efficiency in a wireless sensor network by combining cooperative MIMO with data aggregation

In wireless sensor networks where nodes are powered by batteries, it is critical to prolong the network lifetime by minimizing the energy consumption of each node. In this paper, the cooperative multiple-input-multiple-output (MIMO) and data-aggregation techniques are jointly adopted to reduce the energy consumption per bit in wireless sensor networks by reducing the amount of data for transmission and better using network resources through cooperative communication. For this purpose, we derive a new energy model that considers the correlation between data generated by nodes and the distance between them for a cluster-based sensor network by employing the combined techniques. Using this model, the effect of the cluster size on the average energy consumption per node can be analyzed. It is shown that the energy efficiency of the network can significantly be enhanced in cooperative MIMO systems with data aggregation, compared with either cooperative MIMO systems without data aggregation or data-aggregation systems without cooperative MIMO, if sensor nodes are properly clusterized. Both centralized and distributed data-aggregation schemes for the cooperating nodes to exchange and compress their data are also proposed and appraised, which lead to diverse impacts of data correlation on the energy performance of the integrated cooperative MIMO and data-aggregation systems.