多时间序列符号回归：寻求多变量对目标时序变量的影响分析方法

Got it, let's break down how to analyze the impact of columns a and b on your target time series c. You're right that pandas' corr() only gives pairwise correlations, and ARMA/ARIMA are univariate—so here are the best methods to solve your problem, including symbolic regression and other multi-variable time-series tools:

1. Symbolic Regression (Your Exact Request) #

Symbolic regression is perfect here because it doesn't force you to predefine a model structure (like linear regression does). Instead, it evolves mathematical expressions (using operators like +, -, *, /, sqrt, log) that map your input variables (a, b, and their lagged values) to your target c.

• Tool to use: The gplearn library in Python is a great open-source option. Here's a quick snippet to get you started:

  from gplearn.genetic import SymbolicRegressor
  import pandas as pd
  
  # Create lag features for a and b (e.g., lag 1 to 3)
  df = pd.DataFrame({'a': your_a_data, 'b': your_b_data, 'c': your_c_data})
  for lag in range(1, 4):
      df[f'a_lag_{lag}'] = df['a'].shift(lag)
      df[f'b_lag_{lag}'] = df['b'].shift(lag)
  df = df.dropna()
  
  # Define features (a, b, their lags) and target (c)
  X = df[['a', 'b', 'a_lag_1', 'b_lag_1', 'a_lag_2', 'b_lag_2', 'a_lag_3', 'b_lag_3']]
  y = df['c']
  
  # Train symbolic regressor
  sr = SymbolicRegressor(population_size=5000, generations=20, verbose=1)
  sr.fit(X, y)
  
  # Print the best found expression linking a/b to c
  print(sr._program)
  

• Pros: Discovers non-obvious, non-linear relationships between a/b and c that you might miss with traditional models.
• Cons: Can be computationally expensive; results might be hard to interpret if the evolved expression gets too complex.

2. Multivariate Autoregressive Models (VAR/VARMA/VARIMA) #

Quick correction: while ARMA/ARIMA are univariate, their multivariate counterparts are built explicitly to model interactions between multiple time series. These models assume that your target c depends on its own past values and the past values of a and b.

• Tool to use: The statsmodels library has solid support for VAR/VARMA. Example workflow:

  from statsmodels.tsa.vector_ar.var_model import VAR
  
  # Prepare multivariate time series (a, b, c)
  data = df[['a', 'b', 'c']].dropna()
  
  # Fit VAR model (choose lag order based on AIC/BIC scores)
  model = VAR(data)
  results = model.fit(maxlags=3)
  
  # View coefficients for c's equation (shows how lags of a/b impact c)
  print("Coefficients for c's equation:")
  print(results.coefs[:, 2])  # Column 2 corresponds to the target c
  

• Pros: Highly interpretable; ideal for linear or weakly non-linear relationships.
• Cons: Struggles with complex non-linear patterns that symbolic regression might catch.

3. Non-Linear Time Series Models (ML-Based) #

If your data has strong non-linear dependencies, machine learning models can capture them while letting you quantify the impact of a and b on c:

• Tree-Based Models (XGBoost/LightGBM): Convert your time series into supervised learning data by adding lag features for a, b, and c. Train a regressor, then use feature importance scores to compare how much a (and its lags) vs. b (and its lags) contribute to predicting c.
• Deep Learning (LSTM/GRU): Use recurrent neural networks to model sequential relationships. You can add attention mechanisms or calculate permutation importance to infer the relative impact of a and b on c.

4. Granger Causality (For Causal Insight) #

While not a regression method, Granger causality tests complement your analysis by checking whether past values of a or b help predict c (beyond using past values of c alone). This is a quick way to confirm if a/b have a meaningful predictive relationship with c before diving into regression.

• Example with statsmodels:

  from statsmodels.tsa.stattools import grangercausalitytests
  
  # Test if a Granger-causes c
  grangercausalitytests(df[['c', 'a']], maxlag=3)
  # Test if b Granger-causes c
  grangercausalitytests(df[['c', 'b']], maxlag=3)
  

内容的提问来源于stack exchange，提问作者tardis