Overview of mathematical and machine learning models related to climate risk modeling, climate change simulations, and change point detection. Includes a hands-on session with geometry-based systems analysis of food prices related to climate change and geopolitical factors.

1. MODELING
CLIMATE
CHANGE
C O L L E E N M . FAR R E L LY

2. ABOUT ME
• Author of The Shape of Data
• Mathematician at Post Urban
• Advisor at Candlesticks
• Topology/geometry/network
science researcher
• Colleagues:
• Yae, Franck, Louis, and Peter

3. WHY TACKLE CLIMATE CHANGE?
• Mass extinctions
• Human displacement
• Famines
• Natural disasters

4. CLIMATE CHANGE DATA
• Spatial datasets
• Temporal datasets
• Spatiotemporal datasets

5. MODELING
APPROACHES Traditional
• Spatial regression
• ARIMA models
• SSA models
More advanced approaches
• Network-based models
• Geometry-based models
• Dynamic systems models

6. CLIMATE
RISK
MODELING

7. ACTUARIAL
APPROACH Sum of different risk scores
• Fire hazard + cyclone hazard +
drought hazard…
• Can be weighted
• Easy in scipy and numpy
Probability tables and event
frequency estimates
• 1 in 10000-year floods
• 1 in 10-year earthquakes

8. PARTITION MODELS
• By geography
• By multivariate clustering of risk
• Common models
• Partitioned regression models
• Tree models
• Many options can be created with scikit-
learn

9. MORSE FUNCTIONS
• Akin to a height function over a data
manifold (univariate or multivariate)
• Slices manifold into sections
• Morse-Smale regression
• Partitions manifold topologically
• Based on critical points of function
• Fits partitioned regression to each manifold
partition
• Topopy package as starting point

10. NETWORKS,
CURVATURE, AND
CHANGE POINTS

11. TYPES OF
CURVATURE:
INTRINSIC
• Curvature at a point (local)
• Does not depend on the surface
embedding
• Can be measured by:
• Defining a metric
• Measuring a geodesic
• Example:
• Gaussian curvature
• (Product of principal curvatures at a point)

12. TYPES OF CURVATURE: EXTRINSIC
• Curvature of a surface
• Depends on the surface embedding
• Can be measured by:
• Examining rate of change from unit normal along surface
• Involves lifting manifold to higher-dimensional space,
measuring, and comparing to manifold measurements
• Example:
• Geodesic curvature
• Can have extrinsic curvature without intrinsic
curvature

13. RICCI CURVATURE
• Tensor defined locally to measure
how a metric tensor differs locally
from Euclidean space
• Measures how a shape deforms as you
move along a geodesic line on the manifold
• Relates to the Laplacian tensor
• Symmetric properties
• Flow of Ricci curvature directly related to
solutions of partial differential equations
defined geometrically

14. RICCI FLOW
• Partial differential equation
for Riemannian metric on a
manifold
• Analogous to heat diffusion

15. RICCI CURVATURE: FROM
CONTINUOUS TO DISCRETE
• Connection to Laplacian (and Bochner
formula)
• Allows for extensions of Ricci curvature
and flow to discrete objects like graphs
• Includes Forman-Ricci curvature and
Forman-Ricci flow

16. FORMAN-RICCI
CURVATURE FORMULA
• Forman-Ricci curvature formula for weighted
networks:
• 𝑅𝑖𝑐 𝑒 = 𝑤(𝑒)(
𝑤(𝑣1)
𝑤(𝑒)
+
𝑤(𝑣2)
𝑤(𝑒)
-
𝑒 𝑣2 ~𝑣
𝑒 𝑣1 ~𝑣
𝑤(𝑣1)
𝑤(𝑒)𝑤(𝑒𝑣1)
+

17. FORMAN-RICCI
FLOW
• To calculate flow:
• Subtract edge curvature between
time points
• Differences in Forman-Ricci
centrality of vertices related to
importance of vertex in network
over time

18. CHANGE POINT DETECTION
• Changes in Forman-Ricci
curvature over time and/or space
• Large Forman-Ricci flow over
networks

19. DYNAMIC
SYSTEMS

20. DYNAMIC SYSTEMS MODELS
• Landau-Ginzburg model
• Free energy function
• Initial conditions
• Natural and human conditions
• Bifurcation points to plot
• Bernardini, A. E., Bertolami, O., &
Francisco, F. (2022). Chaotic Behaviour of
the Earth System in the
Anthropocene. arXiv preprint
arXiv:2204.08955.

21. CONNECTION TO TOPOLOGY
• Dynamic systems underlying
topology
• Useful tool to:
• Classify simulations
• Find bifurcation points (change points)
• Partition systems
• Some math:
• Morse theory again
• Traditional dynamic systems

22. CHANGE POINT
EXAMPLE
PYTHON CODE