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 Value	Interpretation	Trading Implication
H < 0.5	Mean-reverting	Trade mean reversion
H = 0.5	Random walk	No edge
H > 0.5	Trending	Trade 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

Parameter	Rule of Thumb
Lookback window	2x half-life
Holding period	1x half-life
Maximum hold	3x half-life (stop)
Signal recalc	0.5x 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

Condition	Signal	Action
z < -2.0	Buy	Enter long (price below mean)
z > +2.0	Sell	Enter short (price above mean)
z crosses 0	Exit	Close position (returned to mean)
abs(z) > 3.0	Stop	Close 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

Parameter	Meaning	Estimation
theta	Speed of mean reversion	From AR(1) beta: theta = -ln(1+beta)/dt
mu	Long-run mean	From AR(1) intercept: mu = -alpha/beta
sigma	Volatility of innovations	Residual 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:

Confirm cointegration (see
```
cointegration-analysis
```
skill)
Compute spread:
```
S = Y - beta * X
```
Apply z-score framework to the spread
Go long spread (buy Y, sell X) when z < -2
Go short spread (sell Y, buy X) when z > +2

交易两个协整资产之间的价差：

确认协整关系（参考
```
cointegration-analysis
```
技能）
计算价差：
```
S = Y - beta * X
```
对价差应用z-score框架
当z < -2时，做多价差（买入Y，卖出X）
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

Most crypto trends: Hurst exponent for BTC, ETH, SOL is typically 0.55-0.70. Raw price mean reversion is rare.
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
Short lookbacks: Crypto mean reversion has short half-lives (hours to days, not weeks)
Transaction costs: DEX swap fees (0.25-1%) can eat mean-reversion profits. Factor in slippage.
Regime awareness: Use
```
regime-detection
```
skill to only trade mean reversion in ranging regimes.

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。
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
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）
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。
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

Skill	Integration
`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

Scan for mean-reverting pairs

扫描均值回归型资产对

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。*