Using extremal events to characterize noisy time series.

Area of Science:

• Dynamical Systems and Time Series Analysis
• Computational Mathematics
• Data Science

Background:

• Experimental time series offer insights into dynamical systems.
• Extremal timing in time series reveals structural information.
• Measurement errors often complicate time series analysis.

Purpose of the Study:

• To develop a robust method for time series characterization despite measurement errors.
• To introduce a new object, the normalized branch decomposition, for interval computation.
• To establish ordering principles for intervals across datasets.

Main Methods:

• Characterizing time series using intervals guaranteed to contain extrema for a given error level.
• Utilizing the merge tree of a continuous function to define normalized branch decomposition.
• Defining total and partial orders on computed intervals.

Main Results:

• The normalized branch decomposition allows interval computation for any error level [Formula: see text].
• A well-defined total order exists for intervals of a single time series.
• This order extends to a partial order across datasets.

Conclusions:

• The method enables pattern matching against switching models for network model rejection.
• Graph distances of partial orders can quantify similarity between biological replicates.
• This approach provides a robust framework for analyzing noisy time series data.