correlation-analysis Checked

def ewma_correlation(x: pd.Series, y: pd.Series, span: int = 60) -> pd.Series:
    """Compute EWMA correlation between two return series."""
    cov_xy = x.mul(y).ewm(span=span).mean() - x.ewm(span=span).mean() * y.ewm(span=span).mean()
    std_x = x.ewm(span=span).std()
    std_y = y.ewm(span=span).std()
    return cov_xy / (std_x * std_y)

### Typical Windows

| Window | Days | Use Case |

|--------|------|----------|

| Short | 20 | Tactical trading, pairs entry/exit |

| Medium | 60 | Strategy allocation, regime detection |

| Long | 120 | Portfolio construction, strategic allocation |

Correlation Matrix Analysis

### Computing the Full Matrix

# Build return matrix for multiple assets
returns = pd.DataFrame({
    'BTC': btc_returns,
    'ETH': eth_returns,
    'SOL': sol_returns,
    'AVAX': avax_returns,
})

# Correlation matrix (Pearson)
corr_matrix = returns.corr()

# Spearman (better for crypto)
spearman_matrix = returns.corr(method='spearman')

### Eigenvalue Decomposition

Decompose the correlation matrix to identify driving factors.

eigenvalues, eigenvectors = np.linalg.eigh(corr_matrix.values)

# Sort descending
idx = eigenvalues.argsort()[::-1]
eigenvalues = eigenvalues[idx]
eigenvectors = eigenvectors[:, idx]

# First eigenvalue = market factor (explains most variance)
# Subsequent eigenvalues = sector/style factors
market_factor_pct = eigenvalues[0] / eigenvalues.sum() * 100

• **First eigenvector**: the market factor — when this dominates (>60% variance), everything moves together
• **Subsequent eigenvectors**: sector or style factors
• **Small eigenvalues**: noise / idiosyncratic risk

### Minimum Variance Portfolio

from numpy.linalg import inv

cov_matrix = returns.cov()
ones = np.ones(len(cov_matrix))
inv_cov = inv(cov_matrix.values)

# Minimum variance weights
weights = inv_cov @ ones / (ones @ inv_cov @ ones)

Hierarchical Clustering

Group assets by correlation similarity to identify natural clusters.

from scipy.cluster.hierarchy import linkage, fcluster
from scipy.spatial.distance import squareform

# Convert correlation to distance
dist_matrix = np.sqrt(2 * (1 - corr_matrix.values))
np.fill_diagonal(dist_matrix, 0)

# Hierarchical clustering
condensed = squareform(dist_matrix)
linkage_matrix = linkage(condensed, method='ward')

# Cut at threshold to get clusters
clusters = fcluster(linkage_matrix, t=1.0, criterion='distance')

**Applications**:

• **Sector detection**: assets in the same cluster behave similarly
• **Diversification**: select one asset per cluster for maximum diversification
• **Risk allocation**: allocate risk budget across clusters, not individual assets

Tail Dependence

Normal correlation understates co-movement during crashes. Tail dependence measures how often assets experience extreme returns simultaneously.

### Lower Tail Dependence

def tail_dependence(x: pd.Series, y: pd.Series, quantile: float = 0.05) -> float:
    """Estimate lower tail dependence coefficient.

    Measures P(Y < q | X < q) for quantile q.
    Higher values mean assets crash together more often.
    """
    threshold_x = x.quantile(quantile)
    threshold_y = y.quantile(quantile)
    joint_extreme = ((x < threshold_x) & (y < threshold_y)).sum()
    marginal_extreme = (x < threshold_x).sum()
    return joint_extreme / marginal_extreme if marginal_extreme > 0 else 0.0

### Crypto-Specific Tail Behavior

In crypto markets, tail dependence typically exceeds normal correlation:

• **Normal correlation** of 0.6 between two altcoins might have **tail dependence** of 0.8
• During market panics, correlations spike toward 1.0 across all risk assets
• This means diversification benefits disappear exactly when needed most

Regime-Dependent Correlation

Correlation is not constant — it changes with market regime.

| Regime | Typical Correlation | Implication |

|--------|-------------------|-------------|

| Bull (trending up) | 0.4–0.7 | Moderate — some diversification works |

| Range-bound | 0.2–0.5 | Lower — best diversification environment |

| Bear (crash) | 0.8–0.95 | Very high — diversification fails |

| Recovery | 0.5–0.7 | Declining from crash highs |

### Detecting Correlation Regime Shifts

def correlation_zscore(rolling_corr: pd.Series, lookback: int = 252) -> pd.Series:
    """Z-score of rolling correlation vs its own history."""
    mean = rolling_corr.rolling(lookback).mean()
    std = rolling_corr.rolling(lookback).std()
    return (rolling_corr - mean) / std

# Flag regime shift when z-score exceeds threshold
zscore = correlation_zscore(rolling_corr_60d)
regime_shift = zscore.abs() > 2.0

Crypto-Specific Correlation Patterns

### Typical Correlation Ranges

| Pair | Normal Range | Notes |

|------|-------------|-------|

| BTC / ETH | 0.7–0.9 | Highest among majors |

| BTC / SOL | 0.6–0.85 | SOL more volatile, slightly less correlated |

| BTC / Altcoin | 0.5–0.8 | Varies by market cap and sector |

| Meme / BTC | 0.2–0.5 | Lower normal correlation |

| Meme / Meme | 0.1–0.4 | Low normal but high tail dependence |

| Stablecoin / BTC | -0.1–0.1 | Should be near zero |

### Key Observations

• Most altcoins are hig