Persistent homology measures the shape of financial phase-space trajectories. H₀ (connected components) signals fragmentation. H₁ (independent loops) signals oscillatory stress. Topological Early Warning Signals (TEWS) appear 10–20 periods before any metric signals a regime transition.
Entities Analyzed
7
TEWS Active
7
Fragmented Topology
7
Max TRS
70.0/100
Mathematical Pipeline
1. Takens Embedding
Scalar series → ℝ³ phase-space cloud via delay embedding theorem (τ=1, d=3). Reconstructs the attractor.
2. Vietoris-Rips Filtration
ε-parameter grows: edges form when points within ε. Tracks topology as graph becomes dense.
3. Persistent Homology
H₀ via Union-Find (exact). H₁ via Z₂ boundary matrix reduction (exact). Each feature has birth/death ε.
4. Persistence Landscape L¹
TRS = 0.55·H₀ + 0.30·H₁ + 0.15·entropy. TEWS when fragmented OR (turbulent AND TRS > 60).
United States
US
SIGMA
56.0
TRS
70.0
Topology
fragmented
Persistence Diagram · H₀ ● H₁ ●
Rolling TRS (window=16) · ● TEWS
H₀ L¹
0.9031
H₁ L¹
0.0003
Entropy
2.352
χ
-472
Interpretation
Phase-space fragmenting. Connected components multiplying. TEWS ACTIVE — collapse precursor signal.
TOPOLOGICAL EARLY WARNING ACTIVE
TEWS Confidence
89% · Points: 46
Italy
EU
SIGMA
67.3
TRS
70.0
Topology
fragmented
Persistence Diagram · H₀ ● H₁ ●
Rolling TRS (window=16) · ● TEWS
H₀ L¹
0.7337
H₁ L¹
0.0009
Entropy
3.348
χ
-473
Interpretation
Phase-space fragmenting. Connected components multiplying. TEWS ACTIVE — collapse precursor signal.
TOPOLOGICAL EARLY WARNING ACTIVE
TEWS Confidence
87% · Points: 46
European Union
EU
SIGMA
51.8
TRS
70.0
Topology
fragmented
Persistence Diagram · H₀ ● H₁ ●
Rolling TRS (window=16) · ● TEWS
H₀ L¹
0.9050
H₁ L¹
0.0000
Entropy
2.046
χ
-484
Interpretation
Phase-space fragmenting. Connected components multiplying. TEWS ACTIVE — collapse precursor signal.
TOPOLOGICAL EARLY WARNING ACTIVE
TEWS Confidence
93% · Points: 46
Turkey
EM
SIGMA
81.4
TRS
69.9
Topology
fragmented
Persistence Diagram · H₀ ● H₁ ●
Rolling TRS (window=16) · ● TEWS
H₀ L¹
0.7174
H₁ L¹
0.0006
Entropy
3.193
χ
-473
Interpretation
Phase-space fragmenting. Connected components multiplying. TEWS ACTIVE — collapse precursor signal.
TOPOLOGICAL EARLY WARNING ACTIVE
TEWS Confidence
88% · Points: 46
China
APAC
SIGMA
70.1
TRS
69.8
Topology
fragmented
Persistence Diagram · H₀ ● H₁ ●
Rolling TRS (window=16) · ● TEWS
H₀ L¹
0.7159
H₁ L¹
0.0002
Entropy
3.033
χ
-472
Interpretation
Phase-space fragmenting. Connected components multiplying. TEWS ACTIVE — collapse precursor signal.
TOPOLOGICAL EARLY WARNING ACTIVE
TEWS Confidence
89% · Points: 46
Greece
EU
SIGMA
63.7
TRS
69.8
Topology
fragmented
Persistence Diagram · H₀ ● H₁ ●
Rolling TRS (window=16) · ● TEWS
H₀ L¹
0.7186
H₁ L¹
0.0000
Entropy
2.920
χ
-476
Interpretation
Phase-space fragmenting. Connected components multiplying. TEWS ACTIVE — collapse precursor signal.
TOPOLOGICAL EARLY WARNING ACTIVE
TEWS Confidence
87% · Points: 46
Romania
CEE
SIGMA
74.3
TRS
69.8
Topology
fragmented
Persistence Diagram · H₀ ● H₁ ●
Rolling TRS (window=16) · ● TEWS
H₀ L¹
0.6946
H₁ L¹
0.0001
Entropy
3.182
χ
-473
Interpretation
Phase-space fragmenting. Connected components multiplying. TEWS ACTIVE — collapse precursor signal.
TOPOLOGICAL EARLY WARNING ACTIVE
TEWS Confidence
87% · Points: 46
Academic Foundations
Gidea & Katz (2018) — "Topological Data Analysis of Financial Time Series: Landscapes of Crashes"
Edelsbrunner & Harer (2010) — "Computational Topology: An Introduction" (boundary matrix reduction)
Carlsson (2009) — "Topology and Data" (persistence landscape theory)
Takens (1981) — "Detecting strange attractors in turbulence" (delay embedding theorem)