The document discusses the application of topological data analysis to time series data, emphasizing the importance of comparing trends and forecasting across various domains such as stock markets and public health. It details concepts such as stationarity, autocorrelation, and persistent homology, which help in understanding the underlying structures of time series data. The use of statistical tests and R code for analyzing datasets, particularly from stock exchanges, is also covered.

1. TOPOLOGY FOR TIME
SERIES
Colleen M. Farrelly

2. ABOUT ME
• Senior data scientist
• Topological data analysis
• Time series
• Network science
• Natural language processing
• Spatial data analytics
• Author of The Shape of Data (No
Starch Press)
• Miami artist and author

3. TIME SERIES DATA
• Comparing trends over time across different regions/products
• Comparing how multiple stock markets perform across similar time periods
• Comparing product sentiment over time across different types of bread
• Forecasting future sales or population
• Forecasting population growth
• Forecasting how much impact an epidemic might have in a region
• Detecting future changes in system behavior
• Price spikes in grain
• Stock market crashes
• Public health crisis leveling out

4. COMPARING TIME SERIES DATA
• Stationarity assumption
• Measurements at same time
points
• Autocorrelation assumptions
• No outliers
• Fréchet distance

5. TIME SERIES AS DYNAMIC SYSTEMS
• Sampling of dynamic system
at different or similar
intervals of time
• Comparison of the systems
themselves useful in
comparison tasks
• Allows for the use of a field
called topology to model the
components of the system
underlying the time series
data

6. WHAT IS TOPOLOGY?
• Global properties of a space
• Many invariants (hole-
counting)
• Not concerned with
distances or curvature
• Donut and coffee mug
example
• Solid ring with a hole in the
middle

7. HOMOLOGY: CLASSIFYING HOLES

8. PERSISTENT
HOMOLOGY
• Create a series of spaces based on
distance between points in the dataset
• Filtration (technical term)
• Allows for analysis at different scales
• Analyze this series for topological
features
• Connected pieces
• Points
• Holes

9. COMPARING TIME SERIES WITH
PERSISTENT HOMOLOGY
• Create a persistence diagram
for each time series
• Any number of series
• In our example, two series
• Apply a statistical test
• Wasserstein distance
• Bonferroni correction
• Review test results
• Significance level

10. DIVING INTO R CODE
Overview of datasets

11. CLOSING PRICE STOCK DATASETS
• South African stock exchange
• JSE
• July 11, 2022-October 7, 2022
• Egypt stock exchange
• EGX 30
• July 18, 2022-October 7, 2022