Squeeze Kernel Covariance Estimator

A streaming covariance estimator for panels of financial returns whose entire public surface is one number — the decay lam of the anchor correlation timescale. Every other quantity is derived from it, fixed by a structural argument, or computed online from the estimator's own state. An O(Kn²) state update per day, positive semi-definite by construction, missing values handled natively, no tuning, no refits. Only dependency: NumPy.

from squeeze_kernel import SqueezeKernel

sk = SqueezeKernel(lam=0.996)        # the entire public surface
for r_t in returns:                  # NaN marks missing assets
    sk.update(r_t)
cov = sk.covariance()

Reference: "The Squeeze Kernel Covariance Estimator: Dual-Timescale Tracking with Adaptive Shrinkage" (Kende, 2026) — SSRN abstract 6455918; the 2.0 estimator is described in the paper's current revision.

Why

Markets do not keep calendar time. Following Mandelbrot, the estimator treats a panel as a collection of partially coupled markets, each advancing on its own activity-driven clock — and reads those clocks from the panel's own correlation structure, so a hot cluster (say precious metals and FX) advances its correlation state while an idle one (agriculture) does not, without anyone identifying a cluster. On those clocks it runs a single recursion that:

• is PSD at every step, structurally — the correlation state evolves by a diagonal-congruence flow (a congruence plus a rank-one term); no eigenvalue clipping, no nearest-PSD repair, and no factorisation anywhere in the state update. (The adaptive timescale weights are the one exception: they read each timescale's predictive likelihood, which costs one Cholesky per timescale per day. Turn them off and the estimator is pure O(Kn²).)
• learns in market time and forgets in calendar time — observations enter with a saturating, self-studentising weight (no day counts more than one unit of trading time); memory decays at fixed per-day rates on a geometric ladder of three timescales (lam⁴, lam, lam^¼). Pairs accrue covariance at the geometric mean of their two clock increments, which is the most positive semi-definiteness allows and exactly the Cauchy–Schwarz bound on how far two assets' clocks can overlap;
• regularises itself — each timescale's shrinkage intensity is computed from two online statistics, the concentration n/ν (dimension per unit trading time) and the de-noised fraction of correlation dispersion the target explains; the target is the Hadamard square of the running correlation, which is exactly the correlation matrix of the squared returns (cluster-respecting, PSD by the Schur product theorem, and sign-blind by construction — the signs are carried by the unshrunk term);
• adapts its memory to regime breaks, in both stages — the timescale mix moves by an exponentiated-gradient step on the blend's own log score (not on any single timescale's, which would select rather than blend), at a temperature calibrated so that uninformative evidence leaves the mix within a factor e of its prior; the marginal variance is tracked on its own three-rung ladder, two octaves below the correlation ladder, and pooled panel-wide by the same rule on saturated evidence, so one spike day cannot hand a stale rung weeks of weight. Nothing in either mixture is fitted: the ladders are derived from lam, the evidence memory is the fastest rung's, and the temperature is the null's;
• ingests missing values natively — listings, delistings, halts enter as NaN;
• is fast — one Cholesky and one triangular solve per day; a thirty-year daily pass at n=300 runs in about two minutes single-threaded, well under daily rolling-window refits.

Evidence. On thirty years of S&P 500 constituents against an eleven-method field (EWMA, DCC, Ledoit–Wolf, OAS, nonlinear shrinkage, RMT filtering, Gerber, IEWMA, CM-IEWMA, and the published v1 estimator) it leads at every universe size from 50 to 300 and is the sole member of the 90% model confidence set at every size. Carried zero-shot to a diversified panel of 121 futures across eight asset classes it beats the same field calibrated on that panel's own history — matched-backbone IEWMA by 6.9 NLL/day (p = 4·10⁻⁴), calibrated DCC by 17.9 — out-of-time.

See the difference

A passive strategy any allocator would recognize: long-only minimum-variance over 300 liquid US stocks, scaled to a 15% volatility target, rebalanced monthly, 5 bps costs. Two runs on identical data; the only difference is the covariance matrix. The Squeeze Kernel arm runs SqueezeKernel(lam=0.996) — nothing tuned on this panel.

Reproduce from the repo alone (the 300-stock panel ships as a parquet; survivorship and provenance are documented in the script):

pip install squeeze-kernel pandas pyarrow scikit-learn matplotlib
python examples/vol_targeted_portfolio.py    # ~2 minutes

Installation

pip install squeeze-kernel          # NumPy only
pip install "squeeze-kernel[full]"  # + SciPy (faster detector factorisations)

Quickstart

import numpy as np
from squeeze_kernel import SqueezeKernel, estimate_squeeze_cov

returns = np.random.default_rng(42).normal(0.0, 0.01, size=(500, 30))

sk = SqueezeKernel()                 # lam=0.996 (anchor half-life ~173 days)
for r_t in returns:
    w = sk.update(r_t)               # returns the day's kernel weight
cov, corr = sk.covariance(), sk.correlation()
sk.state()                           # kernel scale, per-timescale effective sizes, mixture tilt

# batch mode: full panel in, covariance path out
cov_path, corr_path, weights = estimate_squeeze_cov(returns, with_weights=True)

Missing values: pass NaN (or mask= on update). Newly listed, delisted or halted assets need no imputation and no complete-case subsetting.

What derives from lam

from squeeze_kernel import CONSTANTS exposes the structural constants for research. The published v1 estimator (all its knobs) remains available as SqueezeKernelEstimator / SqueezeKernel.v1(...); every 2.0 mechanism is also an estimator-level switch for ablation. See MIGRATION.md.

How it works

One daily update: variance on a three-rung ladder per asset, pooled by panel-wide self-adapting weights → standardised surprise → per-asset clock increments from the Schur-square-weighted neighbourhood mean of squared surprises → diagonal-congruence update of each timescale's correlation state on those clocks → per-timescale shrinkage as a learned pool of the raw correlation, its equicorrelation level and its Hadamard square, with the self-tuning intensity rule as the prior and the blend gradient as the correction → blend across timescales, moved by the gradient of the blend's own log score → covariance. The paper gives the derivations, guarantees (PSD, conditioning floor, exact reductions to the published special cases), and the full evaluation.

Development

uv sync --extra full --extra dev
uv run python -m pytest        # test suite
uv run python -m ruff check .  # lint
uv run mypy                      # strict type check (src/squeeze_kernel)
uv build                       # build sdist + wheel

Citation

@article{kende2026squeeze,
  title  = {The Squeeze Kernel Covariance Estimator: Dual-Timescale Tracking with Adaptive Shrinkage},
  author = {Kende, Robert},
  year   = {2026},
  note   = {Available at SSRN: \url{https://ssrn.com/abstract=6455918}}
}

See also CITATION.cff.

License

MIT