signal-processingmachine-learningstrategy-designposition-sizing

Nonlinear Signal Mapping (非线性信号映射)

Mapping a raw alpha signal s_t into a portfolio position f(s_t) via a nonlinear function. The core design choice behind every modern trend-following / momentum strategy.

Why nonlinear?

A linear mapping f(s) = s (or a binary f(s) = \text{sign}(s)) treats all signal magnitudes identically — a noise blip and a regime shift get the same weight. A theory-consistent nonlinear mapping (Ferson-Siegel Optimal Weighting) mutes extreme signals because variance grows quadratically while expected return grows linearly, so the unconditional Sharpe is maximized by saturating in the tails.

Design axes

When designing f(s_t), there are five choices to make:

  1. Functional form: parametric (FS S-curve, sign, threshold), non-parametric (Nadaraya-Watson, splines), or learned (ANN, GBDT).
  2. Training objective: MSE / cross-entropy vs. economic objective (Sharpe ratio, Sortino, Calmar, turnover-penalized). Moskowitz et al. (2025) use negative-Sharpe loss and find that the recovered f matches FS theory.
  3. Constraints:
    • Symmetry f(-s) = -f(s) — keeps the strategy unbiased.
    • Beta-neutrality f(0) = 0 — strips out buy-and-hold exposure.
    • Variance preservation \text{var}(f(s)) \approx \text{var}(s) — keeps risk scaling comparable across designs.
  4. Estimation window: in-sample vs. expanding window. Moskowitz et al. (2025) note that fixing the empirical f over 1994–2024 leads to some degradation in 2015–2024; periodic re-estimation would help.
  5. Re-scaling after f: the outer Volatility Targeting layer multiplies f(s_t) by \sigma_{target} / \hat\sigma_t to risk-balance across contracts.

Practical recipes

Recipe 1 — Ferson-Siegel (theory)

f_{FS}(s_t) = \lambda \cdot \frac{s_t}{\sigma_\varepsilon^2}, \quad \lambda \approx 0.394 \text{ for unit-variance normalized signal}

Pros: closed-form, robust, well-motivated. Cons: assumes risk-adjusted return is linear in s_t at small magnitudes.

Recipe 2 — Empirical ANN

Train a small MLP (1–2 layers, 2–16 nodes, tanh activation, L_2 reg, dropout 25–50%) with Keras + Adam to minimize the negative Sharpe ratio of f(s_t) \cdot r_{t+1} / \hat\sigma_{t+1} on symmetrized training data. Symmetrize by including (-s, -r) pairs. (Moskowitz et al., 2025, §3.3.)

Recipe 3 — Non-parametric Nadaraya–Watson

\hat f(s) = \frac{\sum_i K_b(s - s_i) \cdot y_i}{\sum_i K_b(s - s_i)}

with Epanechnikov kernel and bandwidth chosen to avoid instability. Moskowitz et al. (2025) find that simple cross-validation tends to pick too-small bandwidths; recommend b = 1 on volatility-normalized inputs.

Recipe 4 — Thresholded / piecewise linear

A simpler practitioner alternative: f(s) = \text{clip}(\alpha s, -1, 1) or f(s) = \text{sign}(s) \cdot \min(\alpha |s|, 1). Captures saturation without ML.

What the empirical ANN recovers

Moskowitz et al. (2025) report the recovered f has these properties:

  • Approximately linear for |s| \in [0, 1].
  • Rollback (slope decreases toward zero or even reverses) for |s| > 1.5 — consistent with FS.
  • Shape depends on observation frequency more than lookback period. At daily frequency, rollback is more pronounced for short lookbacks (1M). At monthly frequency, rollback is more pronounced for long lookbacks (12M).
  • Robust to loss function: MSE-trained and Sharpe-trained networks converge to similar shapes.

When NOT to use nonlinear mapping

  • If the signal is already well-calibrated and bounded, the linear map may suffice.
  • If transaction costs are very high and the signal flips sign frequently, the smooth nonlinear map helps (binary rule causes flip-flopping); but a steeper saturation can hurt in slow-trending regimes.
  • If you have very few assets, the empirical ANN may overfit; the FS closed-form is safer.

References

  • Moskowitz, T.J., Sabbatucci, R., Tamoni, A., & Uhl, B. (2025). "Nonlinear Time Series Momentum." Working paper, §3.3.
  • Ferson, W.E. & Siegel, A.F. (2001). "The Efficient Use of Conditioning Information in Portfolios." Journal of Finance, 56(3).
  • Levine, A. & Pedersen, L.H. (2016). "Which Trend Is Your Friend?" Financial Analysts Journal, 72(3).