Multi-signal combination

One signal becomes one alpha. Real research has several — a trend read, a volume read, a mean-reversion read — and they are correlated. src/alphas/combine.py combines them into one alpha while avoiding the two classic mistakes:

1. Naive weighting double-counts. Weighting by raw IC over-weights redundant signals (three flavors of trend look like three bets but are one) and under-weights a weak-but-independent signal that adds the most.
2. Estimated ICs are uncertain, and that uncertainty should shrink a signal's contribution toward zero — more for short histories and low ICs.

Optimal combination via the signal correlation matrix

Given each signal's information coefficient IC and the signal correlation matrix Ω, the weights that maximize the combined IC are the GLS solution:

w = Ω⁻¹ · IC            combined IC = √(ICᵀ Ω⁻¹ IC)

Because Ω⁻¹ accounts for correlation, two redundant signals split a weight rather than each getting full credit. For two signals this is exactly the closed form:

IC₁' = (IC₁ − ρ·IC₂)/(1 − ρ²)        IC_comb² = (IC₁² + IC₂² − 2ρ·IC₁·IC₂)/(1 − ρ²)

So adding a weak independent signal raises the combined IC; adding a strong redundant one barely does. A small ridge regularizes Ω⁻¹ so a near-duplicate pair (ρ → 1) stays finite.

Bayesian IC-uncertainty shrinkage

Each measured IC is shrunk toward zero by its estimation confidence before combining:

IC' = IC · g/(g+1)        g = n · IC²

IC' → 0 when the history n is short or the IC is small; IC' → IC as n·IC² → ∞. This is what stops a noisy, short-history IC from being trusted at face value — a primary reason backtested alphas disappoint live.

Measured, not assumed

The ICs and Ω are measured over a trailing window, never assumed. At each of several rebalance dates, every signal is scored on the cross-section (using only bars ≤ t) and correlated with the subsequent realized residual return (return minus β·benchmark — so it rewards skill, not beta). The mean over rebalances is the IC; the mean cross-sectional correlation between signals is Ω. Measuring on out-of-sample data is what keeps the combination from over-fitting its own weights.

The combined score then flows through the same refine_alpha pipeline, scaled by the combined IC. So the single-signal assumed IC scalar is replaced by one measured, shrunk, redundancy-aware number — applied once, never twice.

Who owns the level shrink

The per-signal Bayesian shrink here and the IC-uncertainty level shrink are the same g/(g+1) math, so applying both would double-shrink and undertrade forever. The rule: the level shrink owns "is the IC real"; the combination owns "how correlated signals share credit." On this combined path the combination discharges the level; the level shrink is not re-applied. The result echoes a shrink_chain so the single application is auditable.

Where it runs

services/analysis.py::compute_combined_alphas measures the signals, combines the current cross-section, refines it, and returns the ranked alphas plus the measured ICs, shrunk ICs, GLS weights, and correlation matrix. The CLI (python main.py alphas --combine volume_spike,ma_crossover,mean_reversion) and the read-only MCP tool combine_alphas route through it.

On the bundled synthetic data this is its own honesty check: the two trend strategies measure as highly correlated (ρ ≈ 0.9) while mean-reversion is the contrarian foil (ρ < 0), and the shrinkage drives the (genuinely skill-less) random-walk ICs toward zero — so the combined alpha is near-flat, as it should be.