---
description: "Institutional alpha combination: how to merge multiple weak signals into a single high-conviction output using the 11-step procedure and the Fundamental Law of Active Management."
tags: [signals, alpha, portfolio, kelly, statistics]
---

# Signal Combination — Alpha Stacking

## Core Law

Do not search for one perfect signal. Combine many weak, independent signals.

**Fundamental Law of Active Management:**
```
IR = IC × √N
```
- `IR` = Information Ratio of the combined system (risk-adjusted edge)
- `IC` = average Information Coefficient per signal (correlation of prediction to outcome)
- `N` = number of *genuinely independent* signals

Real institutional signals have IC = 0.05–0.15. A single signal at IC=0.10 is outperformed by 50 signals at IC=0.05 (IR = 0.05 × √50 = 0.354, over 3× better).

**Critical:** N counts *effective independent signals*, not raw signal count. Fifty correlated signals may yield only 10–15 effective ones. The 11-step procedure below forces honest accounting.

---

## Five Signal Categories

| Category | What it measures | Why it persists |
|---|---|---|
| **Momentum / Price** | Direction/rate of price movement over lookback `d` | Underreaction causes short-term trend persistence |
| **Mean Reversion** | Deviation from cross-sectional fair value | Related instruments maintain consistent relative pricing |
| **Volatility** | Implied vs. realized volatility gap | Vol risk premium: sellers demand compensation |
| **Factor** | Value, momentum, carry, quality, low-vol premiums | Persistent behavioral/structural inefficiencies |
| **Microstructure** | Order book imbalance, bid-ask spread, VPIN | Informed order flow leads price movement |

> **Dexorder scope**: Only crypto OHLCV data is available. Factor signals (value, carry, quality) require TradFi data not available here. Momentum, mean reversion, volatility, and microstructure signals are all applicable.

---

## 11-Step Combination Engine

Given N signals with historical returns R(i,s) over M periods:

**Step 1.** Collect realized return series R(i,s) for each signal i, each period s.

**Step 2.** Remove drift — serially demean:
```
X(i,s) = R(i,s) − mean(R(i,·))
```

**Step 3.** Compute variance per signal:
```
σ(i)² = (1/M) × Σ X(i,s)²
```

**Step 4.** Normalize to common scale:
```
Y(i,s) = X(i,s) / σ(i)
```
Makes signals with different magnitudes directly comparable.

**Step 5.** Drop the most recent observation from Y — use only out-of-sample history.

**Step 6.** Cross-sectionally demean at each time period:
```
Λ(i,s) = Y(i,s) − avg_j(Y(j,s))
```
Removes any market-wide effect driving all signals simultaneously at that moment.

**Step 7.** Drop the final period from Λ to eliminate residual look-ahead.

**Step 8.** Estimate expected forward return per signal using d-day moving average, normalize:
```
E(i) = (1/d) × Σ R(i,s) over recent d periods
E_normalized(i) = E(i) / σ(i)
```

**Step 9. (Critical)** Regress E_normalized over Λ(i,s) without intercept, unit weights. Residuals `ε(i)` are each signal's *independent* forward-looking contribution — the component not explained by any other signal in the stack.

**Step 10.** Set weights:
```
w(i) = η × ε(i) / σ(i)
```
High independent edge + low noise → high weight. No subjective judgment.

**Step 11.** Normalize: scale η so `Σ|w(i)| = 1`. No unintended leverage.

**Combined output = Σ w(i) × signal_i_current_value**

---

## Empirical Kelly Sizing

```
f_empirical = f_kelly × (1 − CV_edge)

f_kelly = (p × b − q) / b
```
- `CV_edge` = coefficient of variation of edge estimates across 10,000 Monte Carlo path simulations of historical returns
- Higher uncertainty → smaller fraction. The formula automatically scales confidence to what is warranted.

---

## Key Failure Mode: Correlation Blindness

Believing you have 3 independent reasons for a trade when you have 1 reason expressed 3 times, sized as if for 3. This is the mechanism behind most systematic blowups where the trader was directionally correct but over-sized.

The cross-sectional demeaning (Step 6) and regression residualization (Step 9) structurally prevent this by exposing shared variance before weights are assigned.

---

## Dexorder Application Note

When combining multiple indicators into a single entry/exit signal:

1. Each indicator (momentum, RSI divergence, volume profile, spread, etc.) is a signal producing a score or directional estimate.
2. Run the 11-step engine over backtested signal histories to derive weights.
3. Combined score = weighted sum of current signal outputs.
4. Size the resulting position using empirical Kelly with CV_edge from simulation.

If computing probability estimates (e.g. probability of upward breakout), substitute probability estimates for return estimates at each step — the math is identical.