# Automatic Probabilistic Modelling of Dynamical Systems Based on Global Geometry & Topology of Data

Max Snijders,

December 7th, 2018

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## The Team

 Prof. Dr. Ulrich Bauer Prof. Dr. Oliver Junge Prof. Dr. Konstantin Mischaikow Max Snijders Technical University of Munich Technical University of Munich Rutgers University Technical University of Munich & Ludwig-Maximilians University Munich
Introduction

Introduction

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## Homology Intuition

 Example complex Cycles Boundaries Reps. for Homology
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## Cohomology Intuition

1 By the Universal Coefficient Theorem
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## First Integral Cohomology and Coordinate Maps

$H^1(X; \mathbb{Z}) \simeq \lbrack X, S^1 \rbrack$

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Example

## The Lorenz Attractor

$\frac{\partial}{\partial t} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \sigma (y - x) \\ x (\rho - z) - y \\ x y - \beta z \end{pmatrix}$

We use: $$\beta = \nicefrac{8}{3}, \sigma = 10, \rho = 28$$

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## Previous Work

This work relies on a large body of pre-existing work, including:

• Fast Rips-complex cohomology computations enabled by Ripser [Bauer, 2018]
• Integral cohomology computations enabled by Polymake [Assarf et al., 2017]
• Circular coordinates computation [de Silva et al., 2009]
• GAIO dynamical systems algorithms [Dellnitz, Junge et al. 2002]
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## Outlook

 Paper Persistent Homology Comparison against competition Real-world data
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# Automatic Probabilistic Modelling of Dynamical Systems Based on Global Geometry & Topology of Data

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## Backup Material

 Thesis refs.: Univ. Coef. Thm. Cohom. Repr. Cor. Backup slides: Normal slides:
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## Rips Complexes

 Def. For a given point metric space $$(X, d)$$ we denote by $$\mathrm{Rips}_r(X,d)$$ the Rips complex of radius $$r \in \mathbb{R}$$ with vertex set $$X$$ and as simplices all subsets of $$X$$ whose constituent vertices are pairwise within distance $$r$$.
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## Coordinate Smoothing (1/2)

Starting from a cocycle $$\eta$$ which represents a cohomology class, which can be seen as a map on the complex' edges, we perform the following optimisation programme: \begin{align} \bar{f} &= \mathrm{argmin}_{f \in C^0(\bar{X}, \Phi; \mathbb{R})} ||\eta + d_0 f||^2_2 \\ ||\eta||_2^2 &= \sum_{ab \in (\bar{X}, \Phi)} || \eta(ab) ||^2 \end{align}

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## Coordinate Smoothing (2/2)

 $$\eta_{\alpha}, || \eta_{\alpha} ||^2_2 = 1$$ $$\eta_{\beta}, || \eta_{\beta} ||^2_2 = \nicefrac{1}{2}$$ $$\eta_{\gamma}, || \eta_{\gamma} ||^2_2 = \nicefrac{1}{7}$$ Coordinate $$\alpha$$ s.t. $$\eta_{\gamma} = \eta_{\alpha} + d_0 \alpha$$
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## Cohomology

Cohomology is defined by dualizing the chains and the boundary maps between them by applying the $$\mathrm{Hom}\left( \_, G \right)$$ functor as follows:

• We dualize the chains by replacing them with maps from the chains to our coefficient group $$G$$ giving us the cochains $$C^n(K, \Phi; G) = \mathrm{Hom}\left( C_n(K, \Phi \right), G)$$.
• Furthermore, we dualize the boundary operator to get the coboundary operator , which behaves as follows: \begin{align} d_n : C^n(K, \Phi; G) &\rightarrow C^{n+1}(K, \Phi; G) \\ c &\mapsto c \circ \partial_{n+1} \end{align}

Intuitively, whereas an $$n$$-chain is a sum of $$n$$-simplices, an $$n$$-cochain is a function taking values on each $$n$$-simplex.

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## Simplicial Complexes

 Def. A simplicial complex $$(K, \Phi)$$ is a totally ordered vertex set $$K$$ together with a family of nonempty subsets $$\Psi$$ of $$K$$ called the simplices such that the following hold: $$k \in K \implies \{ k \} \in \Phi$$ $$a, \in \Phi, b \subseteq a \implies b \in \Phi$$ We call a simplex $$a \in \Phi$$ an $$n$$-simplex if $$|a| = n+1$$. Def. We define the $$n$$-chains of a simplicial complex $$(K, \Phi)$$, $$C_n(K,\Phi)$$ to be the group generated by the $$n$$-simplices of $$(K,\Phi)$$ under formal summation. Def. For a given $$n \in \mathbb{Z}$$ we define the $$n$$th boundary operator $$\partial_n: C_n(K, \Phi) \rightarrow C_{n-1}(K,\Phi)$$ on individual $$n$$-simplices: $\partial_n: \{ v_0, ..., v_N \} \mapsto \sum_{i=0}^n (-1)^i \{ v_0, ..., v_{i-1}, v_{i+1}, ..., v_N \}$ and extend it linearly w.r.t. the summation on simplices.

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## Homology

 Def. For a given simplicial complex $$K, \Phi$$ and integer $$n \in \mathbb{Z}$$ we denote two special subgroups of $$C_n(K,\Phi)$$, namely the $$n$$-cycles $$Z_n(K, \Phi) = \mathrm{ker}\left( \partial_n: C_n(K, \Phi) \rightarrow C_{n-1}(K, \Phi)\right)$$ and the $$n$$-boundaries $$B_n(K, \Phi) = \mathrm{im}\left( \partial_{n+1}: C_{n+1}(K, \Phi) \rightarrow C_{n}(K, \Phi)\right)$$. We note that $$B_n(K, \Phi) \subseteq Z_n(K, \Phi)$$ as $$\partial_{n} \circ \partial_{n+1} = 0$$ and so the following group is well-defined: $H_n(K, \Phi) = \nicefrac{Z_n(K, \Phi)}{B_n(K, \Phi)}$ we call this group the $$n$$th homology group for $$(K, \Phi)$$.

We will only need the first two homology groups, $$H_0$$ and $$H_1$$.

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## Monotonic, Cycling, Transient Definitions

 Def. A lifted coordinate function $$\theta: \{ t_1, ..., t_N \} \rightarrow \mathbb{R}$$ is said to be $$\epsilon$$-increasing on $$J \subseteq \{ t_1, ..., t_N \}$$ iff $$\frac{\theta(t') - \theta(t)}{t' - t} \ge \epsilon$$ for all $$t, t' \in J$$ whenever $$t' > t$$. Def. A lifted coordinate function $$\theta: \{ t_1, ..., t_N \} \rightarrow \mathbb{R}$$ is said to be $$\epsilon$$-cycling on an interval $$(a, b) \subseteq \mathbb{R}$$ for $$a,b \in \{ t_1, ..., t_N \}$$ if $$\theta$$ is $$\epsilon$$-increasing on $$\{t_1, ..., t_N\} \big|_{(a,b)}$$ and $$\theta(b) - \theta(a) \ge 1$$. Def. A cube $$\xi$$ in the cover is said to be $$\epsilon$$-cycling w.r.t. the function $$f: \bar{X} \rightarrow \mathbb{R}$$ and the boxed trajectory $$\gamma: \{t_1, ..., t_N\} \rightarrow \bar{X}$$ if there is some interval $$(a,b) \subseteq \mathbb{R}$$ s.t. $$\gamma((a,b)) \ni \xi$$ where $$f \circ \gamma: \{ t_1, ..., t_N \} \rightarrow \mathbb{R}$$ is $$\epsilon$$-cycling on the interval $$(a, b)$$. Def. A cube $$\xi$$ in the cover is said to be $$\epsilon$$-transient w.r.t. the function $$f: \bar{X} \rightarrow \mathbb{R}$$ and the boxed trajectory $$\gamma: \{t_1, ..., t_N\} \rightarrow \bar{X}$$ if for a maximal interval $$(a,b) \subseteq \mathbb{R}$$ s.t. $$\gamma((a,b)) \ni \xi$$ and $$f \circ \gamma: \{ t_1, ..., t_N \} \rightarrow \mathbb{R}$$ is $$\epsilon$$-cycling on the interval $$(a, b)$$ we have that $$f(\xi) - f(\gamma(b)) \le 1$$, i.e. the box is in the last cycle of the interval .

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## Dynamical Systems and Decompositions

 Def. A dynamical system is some system with time-dependent state, i.e. some system $$S$$ with a transition map $$f: S \rightarrow S$$ or a continuous-time evolution map $$f: \mathbb{R} \times S \rightarrow S$$.

In our case, our state-space consists of $$\mathbb{R}^n$$ and our time-evolution map is provided by an ODE.

We want to decompose the state-space of a given dynamical system into regions where we can understand behaviour and then study how the system transitions between these regions.

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## Homotopy-equivalence

 Def. Let $$f, g: X \rightarrow Y$$ be two maps between topological spaces. $$f$$ and $$g$$ are said to be homotopy-equivalent or homotopic if there exists a continuous map $$H: X \times \lbrack 0, 1 \rbrack \rightarrow Y$$ such that $$H \big|_{X \times \{0\}} = f$$ and $$H \big|_{X \times \{1\}} = g$$. Def. Two topological spaces $$X$$ and $$Y$$ are said to be homotopy-equivalent or homotopic if there exist continuous maps $$f : X \rightarrow Y$$ and $$g: Y \rightarrow X$$ such that $$f \circ g: Y \rightarrow Y$$ is homotopic to the identity map on $$Y$$ and $$g \circ f: X \rightarrow X$$ is homotopic to the identity map on $$X$$.
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