Using extremal events to characterize noisy time series

Abstract

Experimental time series provide an informative window into the underlying dynamical system, and the timing of the extrema of a time series (or its derivative) contains information about its structure. However, the time series often contain significant measurement errors. We describe a method for characterizing a time series for any assumed level of measurement error $\varepsilon$ by a sequence of intervals, each of which is guaranteed to contain an extremum for any function that $\varepsilon$-approximates the time series. Based on the merge tree of a continuous function, we define a new object called the normalized branch decomposition, which allows us to compute intervals for any level $\varepsilon$. We show that there is a well-defined total order on these intervals for a single time series, and that it is naturally extended to a partial order across a collection of time series comprising a dataset. We use the order of the extracted intervals in two applications. First, the partial order describing a single dataset can be used to pattern match against switching model output (Cummins et al. in SIAM J Appl Dyn Syst 17(2):1589–1616, 2018), which allows the rejection of a network model. Second, the comparison between graph distances of the partial orders of different datasets can be used to quantify similarity between biological replicates.

Yeast data analysis

Time-series transcriptomic data for one replicate of wild-type yeast Saccharomyes cerevisiae were previously published in Orlando et al. (2008). Microarray (Affymetrix Yeast Genome 2.0) expression data were normalized as previously described (Orlando et al. 2008), although for this study Affymetrix probe IDs were re-annotated using Affymetrix Yeast Genome 2.0 microarray annotation 35. Expression data were aligned to a common cell-cycle time line using the CLOCCS (Characterizing Loss of Cell Cycle Synchrony) (Orlando et al. 2007) population synchrony model, as previously described (Orlando et al. 2008). Briefly, the CLOCCS model allows multiple time-series experiments to be aligned to a common cell-cycle timeline, using experimentally-derived yeast budding data. The CLOCCS model converts time points in the series to life points, which indicate the progression through the cell cycle. Expression data for both replicates were interpolated to integer life points with an interval of one using a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) spline. Life points were then trimmed so both replicate time series were of identical location and duration in the cell cycle.