Response language: match the user's input language — Simplified Chinese / Traditional Chinese / English.

• User supplies 2–10 symbols and asks for correlation analysis, whether two stocks move together, rolling correlation trends, or pairs-trading pre-screening.
• Triggers: "AAPL MSFT GOOGL 相关矩阵", "TSLA 和 NVDA 滚动相关", "correlation matrix for my watchlist", "协整检验 700.HK 5.HK".

1. For each symbol, fetch 252 daily candles:

   longbridge kline <SYMBOL> --period day --count 252 --format json

2. Align all series on

   time

   ; drop dates missing in any series.
3. Compute daily log-returns for each symbol.
4. Pearson correlation matrix: pairwise Pearson correlation of returns; flag pairs with |ρ| > 0.8 (high) or < 0.2 (low).
5. Spearman correlation (rank-based, robust to outliers): compute alongside Pearson for comparison.
6. Rolling 60-day correlation for the highest-correlated pair: show trend over time.
7. Cointegration screen (for pairs only):
   • OLS spread residuals → ADF test → report p-value and verdict
   • Half-life = −ln(2) / OLS slope of Δspread ~ spread_{t-1}
8. Output correlation matrix heatmap description (text-based) and a summary of key relationships.

longbridge kline --help

• longbridge-kline

  — raw OHLCV data

• longbridge-pairs-trading

  — execute a pairs trade after correlation pre-screen

• longbridge-multifactor

  — factor correlation and collinearity check

• longbridge-performance-attribution

  — covariance matrix for factor decomposition