PATTERN ANALYSIS AND RECOGNITION IN NATURAL SYSTEMS: A CASE FOR PERSISTENT HOMOLOGY

Recognition and detection - leading to prediction - within natural systems is an incredibly complex, but necessary, area of active, growing, and continued research. Researchers in many science/engineering fields strive for predicting phenomena in natural systems motivated by the ability to optimize understanding, preparation, and planning, so lives can be saved, property damages can be prevented, and/or the quality of life can be improved. Examples of predicting phenomena in the geosciences include the location of natural resources, and the occurrence of landslides and earthquakes. Natural systems phenomena are all spatiotemporal data types in that the information they convey are strongly linked to geospatial (latitude, longitude, and elevation) and (often) temporal dimensions (t). While humans are capable of complex pattern recognition and prediction of spatiotemporal data, we are limited by subjective biases, accuracy, scale (big data), and efficiency. Computers can surpass the human capacity for complicated numerical operations with reduced biases introduced by the cognitive process, but require robust mathematical methods. The vast majority of research in spatiotemporal pattern analysis methods consider only statistical methods, but these methods are unable to take into consideration the structure of spatiotemporal data that may cluster around geometrical shapes. Successful pattern recognition and detection requires novel mathematical methods that can comprehensively and accurately recognize and detect spatiotemporal patterns. Topological data science is a relatively recent field of study that has seen a surge in development over the past ten years, and algebraic topology (i.e., homology) can formally detect topological features, such as the number of components and holes, and types of holes in each dimension. As a means to overcome shortcomings of traditional mathematical approaches, persistent homology is described within the context of geoscientific data.