statistical-arbitragebasket-tradinggraph-clusteringsigned-grapheigendecompositionus-equity

SPONGEsym

SPONGE 对称版：一类带符号股票图聚类算法，把股票嵌入到正/负 Laplacian 联合特征空间后用 k-means 形成多空对冲篮子。是目前 forward-looking 统计套利策略的事实标准聚类后端。

Definition

SPONGE = Signed Positive Over Negative Generalized Eigenproblem. The symmetric variant (SPONGEsym) treats the equity universe as a signed, weighted graph where positive and negative correlations are decomposed into positive and negative Laplacians, normalised, and jointly eigendecomposed. Stocks are clustered in the resulting embedding with k-means++.

SPONGEsym is the clustering back-end used by Cartea Cucuringu Jin 2023 SPONGE (the original paper), extended by Korniejczuk Slepaczuk 2024 Graph Clustering Stat Arb with ML-ensemble signal filtering, and used unchanged by Fanshawe Masih Cameron 2025 THGNN Correlation Forecasting with forward-looking predicted correlations instead of rolling-window ones.

Algorithm

Inputs: a correlation matrix \rho \in [-1, 1]^{N \times N} across N stocks. Steps:

  1. Adjacency construction. A_{ij} = \rho_{ij} (signed). Decompose A = A_+ - A_- where A_+ = \max(A, 0), A_- = \max(-A, 0).

  2. Signed Laplacians. Build the positive and negative graph Laplacians L_+ = D_+ - A_+, \qquad L_- = D_- - A_- with D_\pm the diagonal degree matrices of A_\pm.

  3. Normalised symmetric Laplacians. L^{\text{sym}}_+ = D_+^{-1/2} L_+ D_+^{-1/2}, \qquad L^{\text{sym}}_- = D_-^{-1/2} L_- D_-^{-1/2}

  4. Generalised eigenvalue problem. Solve L^{\text{sym}}_- \, v = \lambda \, (L^{\text{sym}}_+ + \tau_+ I) \, v for the k smallest eigenvalues (Cartea et al. 2023 pick k by a 90% explained-variance criterion on the symmetrised similarity matrix). This is the symetric variant; the asymmetric version uses different normalisations.

  5. k-means++ on the k-dimensional eigenvector embedding to obtain k clusters.

The explicit positive/negative Laplacian split is the key design choice: it lets the algorithm find clusters where stocks are strongly positively correlated within cluster and strongly negatively correlated across clusters, which is exactly the structure of a market-neutral long/short basket.

Why not other clustering methods

  • Spectral clustering (Ng et al. 2002): uses only positive similarity, ignores sign structure.
  • Signed Laplacian clustering (Kunegis et al. 2010): does not balance positive and negative evidence as cleanly.
  • Minimum Spanning Tree / PMFG (Mantegna 1999; Tumminello et al. 2005): produces a sparse tree, not a soft embedding for k-means.
  • Sector classifications (Fama-French 48): static, ignores cross-sector dependencies, completely breaks down in crisis regimes where correlations spike within days.

Cartea et al. (2023) benchmarked all of the above on S&P 500 and reported SPONGEsym as the best — specifically the symmetric variant. The original SPONGEsym strategy (long/short baskets every 3 days, equal contrarian weights) returned 12.2% ARC with Sharpe 1.1, before transaction costs.

The forward-looking extension — the contribution of THGNN

The static SPONGEsym strategy has two failure modes:

  1. Transaction costs erode the 3-day rebalance profits. Korniejczuk & Slepaczuk (2024) lengthen rebalance to 10 days and add an ML-ensemble filter (5 classifiers with soft voting) that only trades the top-10% highest-probability-of-profitability signals.
  2. Backward-looking correlations are slow to adapt. In COVID-19 (2020) or Russia-Ukraine (2022) correlations spike within days but rolling 30-day windows need weeks to adjust, generating losses until the cluster structure catches up.

Fanshawe Masih Cameron 2025 THGNN Correlation Forecasting keeps the Korniejczuk–Slepaczuk trading engine but swaps the rolling 30-day correlation matrix for THGNN-predicted 10-day-ahead correlations. Result:

Metric Korniejczuk–Slepaczuk static THGNN forward-looking
Sharpe (paper claims large gain; not reproduced) 1.84
Annualised return (unstable baseline) 19.20%
Max drawdown — −9.4%
Max loss duration — 82 days

Period 2019–2024, S&P 500, top-10% filtered basket of ~45 stocks.

Position construction inside each cluster

Within each cluster:

  • Compute the cluster-mean cumulative return over the last 5 days.
  • Long stocks that have underperformed the cluster mean; short stocks that have outperformed. Equal weight inside the cluster.
  • Rebalance every 10 days to control turnover / transaction costs.

The "contrarian within cluster" logic is the mean-reversion assumption: stocks that diverge from their cluster peers will revert. This only works if the cluster structure itself is correct — which is what the forward-looking correlation forecast is supposed to guarantee.

Adaptive k — handling regime change

The number of clusters k is chosen per rebalance by the 90% explained-variance criterion on the symmetrised similarity matrix. This is critical during regime shifts: in COVID-19 correlations spike and the "correct" number of clusters drops as everything becomes highly correlated. The adaptive k lets the algorithm collapse clusters gracefully rather than forcing a fixed-k structure that no longer matches market reality.

Limitations

  • No position-sizing beyond equal weight — no vol targeting, no risk parity, no Kelly.
  • No slippage / market impact in the published backtest.
  • The fan-favourite baseline (Korniejczuk–Slepaczuk ML-filtered gains) was not reproducible by the THGNN authors, who benchmark against SPX buy-and-hold instead.
  • Cluster interpretability is limited — the eigenvectors are not directly mappable to economic factors.

Practitioner checklist for SPONGEsym deployment

  • [ ] Decide: rolling vs forward-looking correlation matrix. THGNN-style forward predictions are materially better but require ML infra.
  • [ ] Adaptive k via explained-variance criterion — never hard-code.
  • [ ] ML-ensemble signal filter at top of stack to control turnover.
  • [ ] Rebalance period of ~10 trading days balances mean-reversion holding power vs transaction cost.
  • [ ] Add a vol-targeting layer (see Volatility Targeting) — equal-weight long-short ignores cluster-volatility heterogeneity.
  • [ ] Use Stationary Bootstrap for any Sharpe-ratio inference on the basket return stream.

See also