Financial Anomaly Detection for the Canadian Market

This paper evaluates topological data analysis, principal component analysis, and neural network-based methods on TSX-60 data to detect Canadian financial anomalies, finding that neural networks and TDA offer the strongest performance by leveraging global topological properties to identify major stress events.

The Gist

Imagine the Canadian stock market (the TSX-60) as a massive, bustling orchestra. Every musician (stock) plays their own instrument, but they are also listening to each other. Usually, they play in harmony. But sometimes, the music gets chaotic, dissonant, or the whole band starts playing out of sync. These moments of chaos are financial crises—like the 2008 crash or the pandemic panic.

The goal of this paper is to build a "super-listener" that can hear these moments of chaos before the music completely falls apart, warning investors to get ready.

The authors, Luigi and Nicholas, tested three different types of "super-listeners" to see which one is best at spotting these warning signs in the Canadian market.

The Three "Super-Listeners"

1. The "Shape Detective" (Topological Data Analysis - TDA)

Imagine you have a giant, tangled ball of yarn representing the relationships between all the stocks.

• How it works: Instead of looking at individual knots, this method looks at the overall shape of the ball. Is it a smooth sphere? Does it have a hole in the middle? Is it twisted into a weird knot?
• The Analogy: When the market is calm, the yarn ball looks like a nice, round beach ball. When a crisis is coming, the ball suddenly twists into a weird, spiky shape with holes.
• The Result: This "Shape Detective" was very good. It realized that the global shape of the market changes before a crash happens.

2. The "AI Mimic" (Neural Networks)

Think of this as a student trying to learn a song by listening to a master teacher.

• How it works: The AI listens to the "normal" music of the stock market for years. It builds a mental model of what "normal" sounds like. Then, it listens to the current market. If the current music sounds too different from what it learned, it raises an alarm.
• The Twist: They used two specific types of AI. One tried to memorize the whole song perfectly (One-Shot), and the other tried to copy a "teacher" AI (Knowledge Distillation).
• The Result: These were the champions. They were the most accurate at spotting both the big crashes and the smaller, sneaky stress events (like when oil prices crashed in 2015).

3. The "Flattener" (Principal Component Analysis - PCA)

Imagine trying to describe a complex 3D sculpture by squashing it flat onto a 2D piece of paper.

• How it works: This method takes all the complex data and tries to simplify it into a few main lines (like summarizing a 500-page book into a 3-sentence summary).
• The Result: It was okay at spotting the huge disasters (like 2008), but it missed the smaller, subtle warnings. It was too busy looking at the "big picture" summary and missed the fine details.

The Big Discovery

The authors tested these methods on the Canadian market from 2005 to 2021. They looked for famous "bad days" like:

• The 2008 Financial Crisis.
• The Greek Debt Crisis.
• The 2020 Pandemic crash.
• The 2015-2016 Oil Price crash (which hit Canada hard).

Here is what they found:

• The Winners: The AI Mimics and the Shape Detectives were the best. They didn't just spot the massive earthquakes; they also felt the small tremors that happened before the big ones.
• The Loser: The Flattener (PCA) missed a lot of the smaller, important stress events.
• Why it matters: The fact that the "Shape Detective" worked so well suggests that financial crises aren't just about one or two stocks going down. They are about the entire structure of the market changing shape. It's like the whole orchestra suddenly deciding to play in a different key, not just one violinist playing a wrong note.

The Takeaway

If you want to predict when the Canadian stock market is about to have a meltdown, don't just look at the individual numbers. You need to look at the shape of the connections between companies or use smart AI that understands the complex "music" of the market.

The authors even tested this on the US market (the Dow Jones), and the same rule applied: Smart AI and Shape Analysis beat simple summaries every time.

In short: To find the storm before it hits, you need to understand the shape of the clouds, not just count the raindrops.

Technical Summary

1. Problem Statement

The paper addresses the challenge of detecting financial anomalies and extreme stress events (e.g., market crashes, liquidity crises) within the Canadian stock market. The authors frame financial crashes not merely as outliers in price data, but as structural shifts in the relationships between assets.

• Objective: To evaluate and compare the effectiveness of three distinct classes of anomaly detection methods: Topological Data Analysis (TDA), Principal Component Analysis (PCA), and Graph Neural Networks (GNN).
• Target: The study focuses on the TSX-60 index (Canada's 60 largest companies) over the period 2005–2021.
• Ground Truth: The models are validated against the Canadian Financial Stress Index (CFSI), specifically targeting seven major historical stress events, including the 2008 financial crisis, the Greek debt crisis, oil price shocks (2015–2016), and the COVID-19 pandemic.

2. Methodology

The authors propose a unified pipeline that transforms time-series stock data into weighted graphs, which are then analyzed using different feature extraction and detection techniques.

A. Data Preprocessing & Network Construction

1. Data Source: Daily adjusted closing prices for 39 stocks (from the original 60, filtered for continuous trading 2005–2021) from Yahoo Finance.
2. Log-Returns: Calculated as $q^i_t = \log(p^i_t / p^i_{t-1})$.
3. Graph Construction:
   • A sliding window of $W=25$ days is applied.
   • Convergent Cross Mapping (CCM) is used to compute causal correlations between pairs of stock time series within the window, generating a correlation matrix $C_t$.
   • Negative entries in $C_t$ are set to 0.
   • The matrix serves as the adjacency matrix for a weighted directed graph representing the market state on day $t$.

B. Feature Extraction & Detection Approaches

The paper compares three distinct pathways for processing these graphs:

1. Principal Component Analysis (PCA):

   • Correlation matrices are flattened into vectors.
   • PCA is applied for dimensionality reduction (tested with dimensions 10 and 100).
   • Anomaly scores are derived using Mahalanobis Distance and Local Outlier Factor (LOF).

2. Topological Data Analysis (TDA):

   • The weighted directed graph is converted into a Directed Flag Complex.
   • Persistent Homology is computed to generate persistence diagrams (capturing topological features like connected components and loops).
   • Features are extracted as $L_1$ and $L_2$ norms of the persistent barcodes (measuring the "lifetime" of topological features).
   • Anomaly detection is performed using Mahalanobis Distance and LOF on these topological features.

3. Graph Neural Networks (GNN):

   • One-Shot GIN(E): A One-Class GNN (OCGIN) architecture where the network learns a representation centered around a vector $c$. Anomalies are identified by the distance from this center. The authors use GINE (Graph Isomorphism Network with Edge attributes) to handle edge weights.
   • Knowledge Distillation (GlocalKD): A "student" network is trained to mimic a frozen "teacher" network. The anomaly score is based on the reconstruction error (difference between student and teacher outputs), assuming anomalous graphs will deviate significantly from the learned global patterns.

C. Anomaly Scoring

• A graph (representing a specific day) is classified as anomalous if its score exceeds the 97.5th percentile of the empirical distribution.
• A "financial stress event" is considered successfully signaled if an anomalous graph is detected within a 50-business-day window preceding the event.

3. Key Contributions

• Comparative Framework: The first systematic comparison of TDA, PCA, and GNN specifically for the Canadian market using a unified graph-based preprocessing pipeline.
• Methodological Adaptation:
  • Adaptation of the Gid17 TDA pipeline (using CCM correlations and Directed Flag Complexes) for the TSX-60.
  • Implementation of GINE (handling edge weights) for both One-Shot learning and Knowledge Distillation, moving beyond standard GIN.
• Empirical Validation: Demonstration that methods capturing global structural properties (TDA and GNN) significantly outperform linear methods (PCA) in detecting both major crises and subtle market stress (e.g., oil shocks).
• Robustness Check: Replication of experiments on the Dow Jones Industrial Average (DJIA), confirming that the superiority of TDA and GNN over PCA is not unique to the Canadian market.

4. Experimental Results

The study evaluates performance using Precision, Recall, and F1-Score.

• Top Performers (GNN):

  • GlocalKD (GINE): Achieved the highest F1-score of 0.68 (Precision: 0.86, Recall: 0.56).
  • One-Shot GIN(E): Achieved an F1-score of 0.60 (Precision: 0.86, Recall: 0.47).
  • Observation: GNNs demonstrated high precision, effectively filtering out false positives while capturing major and minor stress events.

• Intermediate Performers (TDA):

  • TDA methods (using $L_1/L_2$ norms of persistence diagrams) achieved F1-scores between 0.55 and 0.59.
  • They successfully identified major crises and showed improved recall over PCA, particularly in detecting the 2015–2016 oil price shocks.

• Underperformers (PCA):

  • PCA-based methods (both raw and reduced dimensions) performed substantially worse, with F1-scores ranging from 0.28 to 0.45.
  • While some PCA variants showed high precision (e.g., 0.86 for Mahalanobis on raw data), they suffered from very low recall (0.20), meaning they missed many actual stress events.

• Key Insight: The TDA and GNN approaches achieved approximately 10% higher recall than PCA methods. Crucially, they detected smaller-scale stress events (like the 2015 oil crash) that PCA methods missed, suggesting that global topological and non-linear structural features are more informative than linear correlations for anomaly detection.

5. Significance and Conclusion

• Theoretical Implication: The success of TDA suggests that global topological properties (homotopy types, connectivity structures) of the market network are meaningful indicators of financial stress. This supports the hypothesis that financial crashes are structural phase transitions rather than simple statistical outliers.
• Practical Application: The proposed pipeline offers a robust early-warning system for Canadian investors and regulators. The ability to detect "smaller-scale" stress events (like oil shocks) provides a competitive advantage over traditional linear models.
• Generalizability: The replication of results on the US DJIA confirms that the superiority of non-linear, structure-aware methods (TDA and GNN) over linear dimensionality reduction (PCA) is a general phenomenon in financial anomaly detection.

Code Availability: The authors have made the implementation (using PyTorch Geometric, Giotto-AI, and Scikit-learn) publicly available on GitHub.