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TOPOLOGY FOR TIME
SERIES
Colleen M. Farrelly
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
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
COMPARING TIME SERIES DATA
• Stationarity assumption
• Measurements at same time
points
• Autocorrelation assumptions
• No outliers
• Fréchet distance
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
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
HOMOLOGY: CLASSIFYING HOLES
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
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
DIVING INTO R CODE
Overview of datasets
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

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Topology for Time Series.pptx

  • 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
  • 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