A topology-based approach for nonlinear time series with applications in computer performance analysis

We present a topology-based methodology for the analysis of experimental data generated by a discrete-time, nonlinear dynamical system. This methodology has significant applications in the field of computer performance analysis. Our approach consists of two parts. In the first part, we propose a novel signal separation algorithm that exploits the continuity of the dynamical system being studied. We use established tools from computational topology to test the connectedness of various regions of state space. In particular, a connected region of space that has a disconnected image under the experimental dynamics suggests the presence of multiple signals in the data. Using this as a guideline, we are able to model experimental data as an Iterated Function System (IFS). We demonstrate the success of our algorithm on several synthetic examples—including a Henon-like IFS. Additionally, we successfully model experimental computer performance data as an IFS. In the second part of the analysis, we represent an experimental dynamical system with an algebraic structure that allows for the computation of algebraic topological invariants. Previous work has shown that a cubical grid and the associated cubical complex are effective tools that can be used to identify isolating neighborhoods and compute the corresponding Conley Index—thereby rigorously verifying the existence of periodic orbits and/or chaotic dynamics. Our contribution is to adapt this technique by altering the underlying data structure--improving flexibility and efficiency. We represent the state space of the dynamical system with a simplicial complex and its induced simplicial multivalued map. This contains information about both geometry and dynamics, whereas the cubical complex is restricted by the geometry of the experimental data. This representation has several advantages; most notably, the complexity of the algorithm that generates the associated simplicial multivalued map is linear in the number of data points—as opposed to exponential in dimension for the cubical multivalued map. The synthesis of the two parts of our methodology results in a nonlinear time-series analysis framework that is particularly well suited for computer performance analysis. Complex computer programs naturally switch between `regimes' and are appropriately modeled as IFSs by part one of our program. Part two of our methodology provides the correct tools for analyzing each regime independently.