Round 2: reverting power law and cycle-on-powerlaw tests; one-time holdout run.

Add TrendReversionVol: deviations from the power-law trend follow a daily
AR(1), so uncertainty levels off, optionally plus trend-parameter
uncertainty with an autocorrelation-adjusted effective sample size.

Two experiments, run under the unchanged verdict rule:
- powerlaw-ou: +21% to +45% vs powerlaw at 2-4 years, but slightly negative
  point estimates at 1 month make it inconclusive.
- cycle-on-powerlaw: inconclusive (+18% at 2 years, negative elsewhere).

The holdout (outcomes after 2024-11-26) was scored once, for the four
candidates fixed beforehand. powerlaw is the best long-horizon forecast
(+45% and +58% vs the random walk at 2 and 3 years); nothing beats the
random walk inside a year; cycle fails badly. Results are in the README.

README.md

@@ -96,8 +96,32 @@ expect a few flukes either way.
 | tails | Student-t, or empirical horizon-level shape | Student-t worse; empirical +5% at 1 month only |
 | cycle-phase | align cycles by fraction elapsed, not days | inconclusive |
 
-Next: a power law whose deviations revert with bounded variance, and a
+Round 2 tested the power law against two refinements (the verdict rule was
-registered test of whether the cycle's timing adds anything on top of it.
+unchanged, and the holdout candidates were fixed before it ran):
+
+| Experiment | Idea | Verdict |
+|---|---|---|
+| powerlaw-ou | deviations from the trend revert, so uncertainty levels off; optionally plus trend-parameter uncertainty | inconclusive: +21% to +45% at 2-4 years (intervals above zero), but -0.3% and -1% at 1 month fail the "never negative" clause. Bands too narrow without parameter uncertainty, too wide with it |
+| cycle-on-powerlaw | the cycle's timing, rescaled to the power-law level | inconclusive: +18% at 2 years, slightly negative at short horizons and 4 years |
+
+### Holdout (run once, 2026-09-24)
+
+Scored on outcomes after 2024-11-26 for the four candidates fixed in advance
+(`random_walk`, `drift_rw`, `cycle`, `powerlaw`). The holdout is ~22 months
+long, so at 2-4 years it is essentially one outcome period seen from several
+origins (1.4-1.9 windows), and the bootstrap intervals there mean little.
+
+| Skill vs random walk | 1mo | 3mo | 6mo | 1y | 2y | 3y | 4y |
+|---|---|---|---|---|---|---|---|
+| drift_rw | -0% | +1% | +5% | +14% | +31% | +53% | -65% |
+| cycle | -14% | -8% | -25% | -104% | -46% | -21% | -179% |
+| powerlaw | -1% | -7% | -5% | +3% | +45% | +58% | +15% |
+
+Consistent with development: nothing beats the random walk inside a year,
+`cycle` fails badly, and `powerlaw` is the best long-horizon forecast, unbiased
+at 2-3 years (mean PIT 0.53-0.56) but with intervals too wide (its 80% interval
+held every 2- and 3-year outcome). The holdout is now spent for these models;
+a new model needs new data to be tested honestly.
 
 ## Usage
 

btcmodel/experiments.py

@@ -24,7 +24,7 @@ from .models.drift import (
     TrailingMeanDrift,
 )
 from .models.shape import Empirical, StudentT
-from .models.volatility import CycleVol, EwmaVol, ReversionVol, TrailingVol
+from .models.volatility import CycleVol, EwmaVol, ReversionVol, TrailingVol, TrendReversionVol
 
 
 @dataclass(frozen=True)
@@ -64,6 +64,7 @@ def verdict(variant_summary: pd.DataFrame) -> str:
 # Idea labels (D1, V2, ...) refer to docs/2024-ideas.md.
 CYCLE = MODELS["cycle"]
 DRIFT_RW = MODELS["drift_rw"]
+POWERLAW = MODELS["powerlaw"]
 # Volatility and shape experiments use drift_rw as the control: the zero-drift
 # random walk is biased low at long horizons, so anything that merely widened
 # its intervals would look like an improvement.
@@ -138,5 +139,31 @@ EXPERIMENTS: dict[str, Experiment] = {
             CYCLE,
             (Composite("cycle_fraction", CycleDrift(phase="fraction"), TrailingVol()),),
         ),
+        # Round 2. Before running these, the holdout candidates were fixed as:
+        # the three original models, powerlaw, and any variant below that is
+        # "better" than its control.
+        Experiment(
+            "powerlaw-ou",
+            "deviations from the power-law trend fade, so long-horizon uncertainty is bounded",
+            "follow-up to diminishing-returns: powerlaw_revert beat drift_rw, but its bands"
+            " were too wide",
+            POWERLAW,
+            (
+                Composite("powerlaw_ou", PowerLawDrift(revert=True), TrendReversionVol()),
+                Composite(
+                    "powerlaw_ou_param",
+                    PowerLawDrift(revert=True),
+                    TrendReversionVol(parameter_uncertainty=True),
+                ),
+            ),
+        ),
+        Experiment(
+            "cycle-on-powerlaw",
+            "with the level set by the power law, the cycle's timing adds information",
+            "D1 on D3. This pair was already compared informally on the same data after"
+            " round 1, so only the holdout can really settle it",
+            POWERLAW,
+            (Composite("cycle_on_powerlaw", PowerLawScaledCycleDrift(), TrailingVol()),),
+        ),
     )
 }

btcmodel/models/volatility.py

@@ -6,8 +6,8 @@ import numpy as np
 import pandas as pd
 
 from ..data import log_returns
-from ..halving import cycle_position
+from ..halving import GENESIS, cycle_position
-from .drift import mean_by_cycle_day
+from .drift import PowerLawDrift, mean_by_cycle_day
 
 
 @dataclass(frozen=True)
@@ -104,3 +104,43 @@ class CycleVol:
         future = history.index[-1] + pd.to_timedelta(np.arange(1, horizons.max() + 1), unit="D")
         _, future_day = cycle_position(future)
         return base * np.sqrt(np.cumsum(ratio[future_day] ** 2)[horizons - 1])
+
+
+@dataclass(frozen=True)
+class TrendReversionVol:
+    """
+    Uncertainty for a price that reverts to the power-law trend.
+
+    Deviations from the trend follow a daily AR(1) with coefficient φ (fitted
+    by PowerLawDrift), so their variance levels off: after h days it is
+    σ²(1 − φ^2h) / (1 − φ²), with σ the trailing `window`-day volatility.
+
+    With `parameter_uncertainty`, the uncertainty of the fitted trend line is
+    added. The residuals are so autocorrelated that ~5000 days carry the
+    information of only n(1 − φ)/(1 + φ) independent points, and the
+    coefficient covariance is inflated to match.
+    """
+
+    window: int = 365
+    parameter_uncertainty: bool = False
+
+    def sd(self, history: pd.DataFrame, horizons: np.ndarray) -> np.ndarray:
+        intercept, slope, phi = PowerLawDrift().fit(history)
+        phi = min(phi, 0.9999)
+        sigma = log_returns(history).iloc[-self.window :].std()
+        variance = sigma**2 * (1 - phi ** (2 * horizons)) / (1 - phi**2)
+        if self.parameter_uncertainty:
+            variance = variance + self._trend_variance(history, horizons, intercept, slope, phi)
+        return np.sqrt(variance)
+
+    @staticmethod
+    def _trend_variance(history, horizons, intercept, slope, phi) -> np.ndarray:
+        t = (history.index - GENESIS).days.to_numpy()
+        x = np.column_stack([np.ones(len(t)), np.log(t)])
+        resid = np.log(history["close"].to_numpy()) - x @ [intercept, slope]
+        n_eff = len(t) * (1 - phi) / (1 + phi)
+        cov = resid.var() * np.linalg.inv(x.T @ x) * len(t) / n_eff
+        # The forecast mean is a(1 − φ^h) + b(ln t_h − φ^h ln t_0) + φ^h ln P_0.
+        decay = phi**horizons
+        g = np.column_stack([1 - decay, np.log(t[-1] + horizons) - decay * np.log(t[-1])])
+        return np.einsum("hi,ij,hj->h", g, cov, g)

tests/test_components.py

@@ -6,7 +6,7 @@ from btcmodel.experiments import EXPERIMENTS
 from btcmodel.halving import GENESIS
 from btcmodel.models.drift import CycleDrift, PowerLawDrift, ShrunkDrift
 from btcmodel.models.shape import StudentT
-from btcmodel.models.volatility import ReversionVol, TrailingVol
+from btcmodel.models.volatility import ReversionVol, TrailingVol, TrendReversionVol
 
 from .test_models import synthetic_prices
 
@@ -64,3 +64,22 @@ def test_student_t_shape_has_unit_variance_and_fatter_tails():
     q = StudentT(4).standard_quantiles(None, np.array([1]))[0]
     assert q[-1] > 2.576  # beyond the normal 99.5% quantile
     assert np.interp(0.8413, np.linspace(0.005, 0.995, 100), q) < 1.0  # thinner shoulders
+
+
+def test_trend_reversion_vol_levels_off():
+    # A power-law trend plus AR(1) deviations with a ~70-day half-life.
+    rng = np.random.default_rng(0)
+    dates = pd.date_range("2011-01-01", "2024-11-26", freq="D", name="date")
+    t = (dates - GENESIS).days.to_numpy()
+    gap = np.zeros(len(t))
+    for i in range(1, len(t)):
+        gap[i] = 0.99 * gap[i - 1] + 0.03 * rng.standard_normal()
+    prices = pd.DataFrame({"close": np.exp(-40 + 5.8 * np.log(t) + gap)}, index=dates)
+
+    horizons = np.array([30, 365, 1460, 5000])
+    plain = TrendReversionVol().sd(prices, horizons)
+    with_params = TrendReversionVol(parameter_uncertainty=True).sd(prices, horizons)
+    assert np.all(np.diff(plain) >= 0)
+    stationary = 0.03 / np.sqrt(1 - 0.99**2)
+    assert plain[-1] == pytest.approx(stationary, rel=0.15)
+    assert np.all(with_params >= plain)