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mean-reversion

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Mean Reversion

均值回归

Mean reversion is the statistical tendency for prices, spreads, or other financial variables to return toward a long-run average after deviating from it. A mean-reverting series overshoots its mean, then corrects back -- creating predictable oscillations that can be traded.
均值回归是指价格、价差或其他金融变量在偏离长期均值后，回归至该均值的统计趋势。均值回归序列会先偏离均值，随后修正回归——形成可被交易的可预测波动。

When Mean Reversion Works

均值回归策略适用场景

  • Ranging markets: Sideways price action with clear support/resistance
  • Pairs spreads: Spread between cointegrated assets reverts to equilibrium
  • Oversold/overbought extremes: RSI, Bollinger Band, or z-score extremes in stationary series
  • Funding rate arbitrage: Perpetual funding rates revert to baseline
  • Stablecoin depegs: Classic mean-reversion opportunity (peg = known mean)
  • Post-dump recovery: Brief mean-reversion windows after initial PumpFun dumps
  • 区间震荡市场：有明确支撑/阻力位的横盘走势
  • 配对价差：协整资产之间的价差会回归至均衡水平
  • 超卖/超买极值：平稳序列中RSI、布林带或z-score达到极值
  • 资金费率套利：永续合约资金费率回归至基准水平
  • 稳定币脱钩：经典的均值回归机会（挂钩价=已知均值）
  • 暴跌后修复：PumpFun初始拉盘后的短暂均值回归窗口

When Mean Reversion Fails

均值回归策略失效场景

  • Strong trending markets (most crypto most of the time)
  • Regime changes: what was stationary becomes non-stationary
  • Structural breaks: token migration, protocol upgrade, delistings
  • Low liquidity: wide spreads consume mean-reversion profits
  • 强势趋势市场（加密货币多数时间处于该状态）
  • 市场结构转变：原本平稳的序列变为非平稳序列
  • 结构性断裂：代币迁移、协议升级、退市
  • 低流动性：点差过大吞噬均值回归利润

Testing for Mean Reversion

均值回归检验

Before trading mean reversion, you must statistically confirm the series is mean-reverting. Three complementary tests:
在进行均值回归交易前，必须通过统计方法确认序列具备均值回归特性。三种互补检验方法：

1. Augmented Dickey-Fuller (ADF) Test

1. 增广迪基-富勒（ADF）检验

Tests the null hypothesis that a series has a unit root (non-stationary).
python
from scipy import stats
import numpy as np

def adf_test(series: np.ndarray, max_lag: int = 0) -> dict:
    """Run ADF test. Reject null (p < 0.05) → stationary → mean-reverting."""
    # See references/statistical_tests.md for full implementation
    # Use statsmodels.tsa.stattools.adfuller for production
    pass
  • p < 0.01: Strong evidence of stationarity
  • p < 0.05: Evidence of stationarity
  • p > 0.10: Cannot reject unit root -- likely non-stationary
检验序列存在单位根（非平稳）的原假设。
python
from scipy import stats
import numpy as np

def adf_test(series: np.ndarray, max_lag: int = 0) -> dict:
    """Run ADF test. Reject null (p < 0.05) → stationary → mean-reverting."""
    # See references/statistical_tests.md for full implementation
    # Use statsmodels.tsa.stattools.adfuller for production
    pass
  • p < 0.01：强烈表明序列平稳
  • p < 0.05：表明序列平稳
  • p > 0.10：无法拒绝单位根假设——序列大概率非平稳

2. Hurst Exponent

2. Hurst指数

Measures the long-range dependence of a time series.
Hurst ValueInterpretationTrading Implication
H < 0.5Mean-revertingTrade mean reversion
H = 0.5Random walkNo edge
H > 0.5TrendingTrade momentum
python
def hurst_exponent(series: np.ndarray) -> float:
    """Compute Hurst exponent via R/S method. H < 0.5 → mean-reverting."""
    # See references/statistical_tests.md for full R/S algorithm
    pass
衡量时间序列的长期依赖性。
Hurst值解释交易启示
H < 0.5均值回归型采用均值回归策略
H = 0.5随机游走无交易优势
H > 0.5趋势型采用动量策略
python
def hurst_exponent(series: np.ndarray) -> float:
    """Compute Hurst exponent via R/S method. H < 0.5 → mean-reverting."""
    # See references/statistical_tests.md for full R/S algorithm
    pass

3. Variance Ratio Test

3. 方差比检验

Compares variance of multi-period returns to single-period variance.
  • VR < 1: Negative autocorrelation (mean-reverting)
  • VR = 1: Random walk
  • VR > 1: Positive autocorrelation (trending)
python
def variance_ratio(series: np.ndarray, q: int = 5) -> float:
    """Compute variance ratio at horizon q. VR < 1 → mean-reverting."""
    returns = np.diff(np.log(series))
    var_1 = np.var(returns)
    returns_q = np.diff(np.log(series[::q]))
    var_q = np.var(returns_q)
    return var_q / (q * var_1)
See 
references/statistical_tests.md
for complete implementations and interpretation guides.
比较多期收益率方差与单期收益率方差的比值。
  • VR < 1：负自相关（均值回归型）
  • VR = 1：随机游走
  • VR > 1：正自相关（趋势型）
python
def variance_ratio(series: np.ndarray, q: int = 5) -> float:
    """Compute variance ratio at horizon q. VR < 1 → mean-reverting."""
    returns = np.diff(np.log(series))
    var_1 = np.var(returns)
    returns_q = np.diff(np.log(series[::q]))
    var_q = np.var(returns_q)
    return var_q / (q * var_1)
完整实现及解读指南请参考
references/statistical_tests.md
。

Half-Life Estimation

半衰期估计

The half-life tells you how many periods it takes for a deviation to decay to half its size. This is the single most important parameter for mean-reversion trading.
半衰期指偏差衰减至初始值一半所需的周期数，是均值回归交易中最重要的参数。

AR(1) Regression Method

AR(1)回归法

Fit the autoregressive model: 
delta_X_t = alpha + beta * X_{t-1} + epsilon
python
def half_life(series: np.ndarray) -> float:
    """Estimate mean-reversion half-life from AR(1) regression.

    Returns:
        Half-life in periods. Negative means non-mean-reverting.
    """
    y = np.diff(series)
    x = series[:-1]
    x = np.column_stack([np.ones(len(x)), x])
    beta = np.linalg.lstsq(x, y, rcond=None)[0][1]
    if beta >= 0:
        return -1.0  # Not mean-reverting
    return -np.log(2) / np.log(1 + beta)
拟合自回归模型：
delta_X_t = alpha + beta * X_{t-1} + epsilon
python
def half_life(series: np.ndarray) -> float:
    """Estimate mean-reversion half-life from AR(1) regression.

    Returns:
        Half-life in periods. Negative means non-mean-reverting.
    """
    y = np.diff(series)
    x = series[:-1]
    x = np.column_stack([np.ones(len(x)), x])
    beta = np.linalg.lstsq(x, y, rcond=None)[0][1]
    if beta >= 0:
        return -1.0  # Not mean-reverting
    return -np.log(2) / np.log(1 + beta)

Using Half-Life

半衰期应用准则

ParameterRule of Thumb
Lookback window2x half-life
Holding period1x half-life
Maximum hold3x half-life (stop)
Signal recalc0.5x half-life
参数经验法则
回溯窗口2倍半衰期
持有周期1倍半衰期
最大持有时间3倍半衰期（止损）
信号重算周期0.5倍半衰期

Z-Score Signal Framework

Z-Score信号框架

The z-score normalizes the deviation from the mean, providing standardized entry/exit signals.
z = (price - rolling_mean) / rolling_std
Z-score将偏离均值的幅度标准化，提供标准化的进场/离场信号。
z = (price - rolling_mean) / rolling_std

Signal Rules

信号规则

ConditionSignalAction
z < -2.0BuyEnter long (price below mean)
z > +2.0SellEnter short (price above mean)
z crosses 0ExitClose position (returned to mean)
abs(z) > 3.0StopClose position (reversion failed)
条件信号操作
z < -2.0买入做多（价格低于均值）
z > +2.0卖出做空（价格高于均值）
z穿越0离场平仓（回归至均值）
abs(z) > 3.0止损平仓（回归失败）

Lookback Window

回溯窗口

Set the rolling window to approximately 2x the half-life:
python
def z_score_signals(
    prices: np.ndarray,
    lookback: int,
    entry_z: float = 2.0,
    exit_z: float = 0.0,
    stop_z: float = 3.0,
) -> np.ndarray:
    """Generate z-score-based mean-reversion signals.

    Returns:
        Array of signals: 1 (long), -1 (short), 0 (flat).
    """
    rolling_mean = pd.Series(prices).rolling(lookback).mean().values
    rolling_std = pd.Series(prices).rolling(lookback).std().values
    z = (prices - rolling_mean) / rolling_std
    # See scripts/mean_reversion_test.py for full signal generation
    ...
将滚动窗口设置为约2倍半衰期：
python
def z_score_signals(
    prices: np.ndarray,
    lookback: int,
    entry_z: float = 2.0,
    exit_z: float = 0.0,
    stop_z: float = 3.0,
) -> np.ndarray:
    """Generate z-score-based mean-reversion signals.

    Returns:
        Array of signals: 1 (long), -1 (short), 0 (flat).
    """
    rolling_mean = pd.Series(prices).rolling(lookback).mean().values
    rolling_std = pd.Series(prices).rolling(lookback).std().values
    z = (prices - rolling_mean) / rolling_std
    # See scripts/mean_reversion_test.py for full signal generation
    ...

Position Sizing with Z-Score

基于Z-Score的仓位管理

Scale position size with z-score magnitude for better risk-adjusted returns:
python
size = base_size * min(abs(z) / entry_threshold, max_scale)
See 
references/strategy_design.md
for complete entry/exit framework and sizing.
根据Z-score的幅度调整仓位大小，以获得更优的风险调整收益：
python
size = base_size * min(abs(z) / entry_threshold, max_scale)
完整的进场/离场框架及仓位管理请参考
references/strategy_design.md
。

Ornstein-Uhlenbeck (OU) Process

Ornstein-Uhlenbeck (OU)过程

The OU process is the continuous-time model of mean reversion:
dX = theta * (mu - X) * dt + sigma * dW
ParameterMeaningEstimation
thetaSpeed of mean reversionFrom AR(1) beta: theta = -ln(1+beta)/dt
muLong-run meanFrom AR(1) intercept: mu = -alpha/beta
sigmaVolatility of innovationsResidual std from AR(1)
OU过程是均值回归的连续时间模型：
dX = theta * (mu - X) * dt + sigma * dW
参数含义估计方法
theta均值回归速度由AR(1)的beta计算：theta = -ln(1+beta)/dt
mu长期均值由AR(1)的截距计算：mu = -alpha/beta
sigma创新项波动率AR(1)的残差标准差

Parameter Estimation

参数估计

python
def estimate_ou_params(series: np.ndarray, dt: float = 1.0) -> dict:
    """Estimate OU process parameters from observed series.

    Returns:
        Dict with keys: theta, mu, sigma, half_life.
    """
    y = np.diff(series)
    x = series[:-1]
    x_with_const = np.column_stack([np.ones(len(x)), x])
    params = np.linalg.lstsq(x_with_const, y, rcond=None)[0]
    alpha, beta = params[0], params[1]

    theta = -np.log(1 + beta) / dt
    mu = -alpha / beta if beta != 0 else np.mean(series)
    residuals = y - (alpha + beta * x)
    sigma = np.std(residuals) * np.sqrt(2 * theta / (1 - np.exp(-2 * theta * dt)))

    return {
        "theta": theta,
        "mu": mu,
        "sigma": sigma,
        "half_life": np.log(2) / theta if theta > 0 else -1,
    }
python
def estimate_ou_params(series: np.ndarray, dt: float = 1.0) -> dict:
    """Estimate OU process parameters from observed series.

    Returns:
        Dict with keys: theta, mu, sigma, half_life.
    """
    y = np.diff(series)
    x = series[:-1]
    x_with_const = np.column_stack([np.ones(len(x)), x])
    params = np.linalg.lstsq(x_with_const, y, rcond=None)[0]
    alpha, beta = params[0], params[1]

    theta = -np.log(1 + beta) / dt
    mu = -alpha / beta if beta != 0 else np.mean(series)
    residuals = y - (alpha + beta * x)
    sigma = np.std(residuals) * np.sqrt(2 * theta / (1 - np.exp(-2 * theta * dt)))

    return {
        "theta": theta,
        "mu": mu,
        "sigma": sigma,
        "half_life": np.log(2) / theta if theta > 0 else -1,
    }

Strategy Types

策略类型

Single-Asset Mean Reversion

单资产均值回归

Apply z-score framework directly to a token's price series. Works best on:
  • Stablecoins (USDC/USDT spread)
  • Tokens in established ranges
  • After confirming stationarity with ADF test
将z-score框架直接应用于代币价格序列。最适用于：
  • 稳定币（USDC/USDT价差）
  • 处于明确区间的代币
  • 经ADF检验确认平稳的序列

Pairs Trading

配对交易

Trade the spread between two cointegrated assets:
  1. Confirm cointegration (see 
    cointegration-analysis
    skill)
  2. Compute spread: 
    S = Y - beta * X
  3. Apply z-score framework to the spread
  4. Go long spread (buy Y, sell X) when z < -2
  5. Go short spread (sell Y, buy X) when z > +2
交易两个协整资产之间的价差：
  1. 确认协整关系（参考
    cointegration-analysis
    技能）
  2. 计算价差：
    S = Y - beta * X
  3. 对价差应用z-score框架
  4. 当z < -2时，做多价差（买入Y，卖出X）
  5. 当z > +2时，做空价差（卖出Y，买入X）

Statistical Arbitrage

统计套利

Multi-asset extension of pairs trading:
  • Eigenportfolios from PCA of correlated assets
  • Trade the smallest eigenvalue portfolios (most mean-reverting)
  • Requires larger universe (10+ assets)
配对交易的多资产扩展版本：
  • 从相关资产的PCA中提取特征投资组合
  • 交易最小特征值的投资组合（最具均值回归性）
  • 需要较大的资产池（10种以上资产）

Crypto-Specific Considerations

加密货币专属考量

  1. Most crypto trends: Hurst exponent for BTC, ETH, SOL is typically 0.55-0.70. Raw price mean reversion is rare.
  2. Where to find mean reversion:
    • Pairs spreads (SOL/ETH ratio, BTC dominance)
    • Funding rates on perpetuals
    • Basis between spot and futures
    • Stablecoin depegs
    • Fee tier spreads across DEXs
  3. Short lookbacks: Crypto mean reversion has short half-lives (hours to days, not weeks)
  4. Transaction costs: DEX swap fees (0.25-1%) can eat mean-reversion profits. Factor in slippage.
  5. Regime awareness: Use 
    regime-detection
    skill to only trade mean reversion in ranging regimes.
  1. 多数加密货币呈趋势性：BTC、ETH、SOL的Hurst指数通常在0.55-0.70之间，原始价格的均值回归现象罕见。
  2. 均值回归机会的来源：
    • 配对价差（SOL/ETH比值、BTC dominance）
    • 永续合约资金费率
    • 现货与期货的基差
    • 稳定币脱钩
    • DEX间的手续费层级价差
  3. 短回溯窗口：加密货币的均值回归半衰期较短（数小时至数天，而非数周）
  4. 交易成本：DEX的滑点手续费（0.25-1%）可能吞噬均值回归利润，需将滑点纳入考量。
  5. 市场结构识别：使用
    regime-detection
    技能，仅在区间震荡市场中进行均值回归交易。

Integration with Other Skills

与其他技能的集成

SkillIntegration
cointegration-analysis
Find cointegrated pairs for pairs trading
pandas-ta
RSI, Bollinger Bands as mean-reversion indicators
regime-detection
Filter: only trade MR in ranging regimes
vectorbt
Backtest mean-reversion strategies
volatility-modeling
Estimate sigma for OU model
slippage-modeling
Factor execution costs into P&L estimates
position-sizing
Size positions using Kelly + z-score scaling
技能集成方式
cointegration-analysis
寻找适合配对交易的协整资产对
pandas-ta
将RSI、布林带作为均值回归指标
regime-detection
过滤：仅在区间震荡市场中进行均值回归交易
vectorbt
回测均值回归策略
volatility-modeling
为OU模型估计sigma参数
slippage-modeling
将执行成本纳入损益估算
position-sizing
结合Kelly准则与z-score缩放进行仓位管理

Files

文件

References

参考文档

  • references/statistical_tests.md
    -- ADF, Hurst exponent, variance ratio, and half-life estimation with full implementations and interpretation
  • references/strategy_design.md
    -- Z-score framework, position sizing, pairs trading setup, risk management, and backtest considerations
  • references/statistical_tests.md
    -- ADF检验、Hurst指数、方差比检验、半衰期估计的完整实现及解读
  • references/strategy_design.md
    -- Z-score框架、仓位管理、配对交易设置、风险管理及回测考量

Scripts

脚本

  • scripts/mean_reversion_test.py
    -- Comprehensive mean-reversion analysis: ADF, Hurst, variance ratio, half-life, OU estimation, z-score signals
  • scripts/pairs_scanner.py
    -- Scan multiple assets for mean-reverting pairs: correlation, cointegration, spread analysis, ranking
  • scripts/mean_reversion_test.py
    -- 全面的均值回归分析：ADF检验、Hurst指数、方差比检验、半衰期、OU参数估计、z-score信号生成
  • scripts/pairs_scanner.py
    -- 扫描多资产以寻找均值回归型资产对：相关性分析、协整检验、价差分析、排名

Quick Start

快速开始

bash
undefined
bash
undefined

Run mean-reversion analysis on synthetic data

对合成数据进行均值回归分析

python scripts/mean_reversion_test.py --demo
python scripts/mean_reversion_test.py --demo

Scan for mean-reverting pairs

扫描均值回归型资产对

python scripts/pairs_scanner.py --demo
python scripts/pairs_scanner.py --demo

Analyze a specific token (requires BIRDEYE_API_KEY)

分析特定代币（需要BIRDEYE_API_KEY）

BIRDEYE_API_KEY=your_key TOKEN_MINT=So11...1 python scripts/mean_reversion_test.py

*This skill provides analytical tools and information only. It does not constitute financial advice or trading recommendations.*
BIRDEYE_API_KEY=your_key TOKEN_MINT=So11...1 python scripts/mean_reversion_test.py

*本技能仅提供分析工具及信息，不构成财务建议或交易推荐。*