如何用PCA对高相关变量进行时间序列预测？含数据集实操疑问

Alright, let's walk through this problem step by step—time series data adds a layer of complexity because order matters, but once you lock that down, using PCA for forecasting is totally doable. Here's how to handle each part:

This is foundational—if your years are out of whack, everything downstream will be unreliable. Try these checks/fixes:

• Sort explicitly by year: If your dataset has a dedicated year column, sort it and reset the index to eliminate any gaps or misordering:

  import pandas as pd
  df = df.sort_values('year').reset_index(drop=True)
  

• Visualize to confirm: Plot one of your original time series against year to spot obvious out-of-order points:

  import matplotlib.pyplot as plt
  plt.plot(df['year'], df['your_first_economic_column'])
  plt.xlabel('Year')
  plt.title('Check for Continuous Time Order')
  plt.show()
  

• Set year as index: This makes all subsequent time series operations (like forecasting) use the time axis directly, reducing the chance of mix-ups:

  df = df.set_index('year')
  

Since PCA gives you orthogonal (independent) components, you can model each one separately as its own time series. Here are two common approaches:

Option A: ARIMA (Best for Non-Linear/Seasonal Time Series) #

ARIMA is designed for time series data and handles trends, stationarity, and autocorrelation well:

from statsmodels.tsa.arima.model import ARIMA

# Assume you have your two PCs stored in a DataFrame called `pcs_df` with year as index
# Fit ARIMA to PC1 (adjust order=(p,d,q) based on your data's stationarity/autocorrelation)
model_pc1 = ARIMA(pcs_df['pc1'], order=(1,1,0))
results_pc1 = model_pc1.fit()
# Forecast next 10 years
pc1_pred = results_pc1.forecast(steps=10)

# Repeat for PC2
model_pc2 = ARIMA(pcs_df['pc2'], order=(1,1,0))
results_pc2 = model_pc2.fit()
pc2_pred = results_pc2.forecast(steps=10)

Pro tip: Use the ADF test to check if your PCs are stationary, and adjust the d parameter in ARIMA accordingly (d=1 if non-stationary, d=0 if stationary).

Option B: Linear Regression (Best for Strong Linear Trends) #

If your economic trends are clearly linear, a simple linear regression with year as the predictor works:

from sklearn.linear_model import LinearRegression
import numpy as np

# Get last year from your data to generate future years
last_year = df.index[-1]
future_years = np.arange(last_year + 1, last_year + 11).reshape(-1,1)

# Fit regression for PC1
X_train = np.array(df.index).reshape(-1,1)
y_train = pcs_df['pc1']
reg_pc1 = LinearRegression().fit(X_train, y_train)
pc1_pred = reg_pc1.predict(future_years)

# Repeat for PC2
reg_pc2 = LinearRegression().fit(X_train, pcs_df['pc2'])
pc2_pred = reg_pc2.predict(future_years)

To get back to your original 5 economic columns, you need to use the inverse transform from your original PCA model—this is why you must save the PCA object you used to generate the PCs initially!

# Assume your original PCA model is stored as `pca` (e.g., pca = PCA(n_components=2))
# Combine predicted PCs into a 2D array
predicted_pcs = np.column_stack((pc1_pred, pc2_pred))
# Inverse transform to get original feature space predictions
original_pred = pca.inverse_transform(predicted_pcs)

# Wrap into a DataFrame for readability
future_predictions = pd.DataFrame(
    original_pred,
    columns=df.columns,
    index=future_years.flatten()
)

Before trusting your 10-year forecast, validate with your existing data:

• Use the first 8 years of data to fit PCA and your forecasting model, then predict the last 2 years. Compare the reconstructed values to the actual data to gauge error.
• Check the explained variance ratio of your PCs: If they explain 80%+ of the variance in your original data, your reconstructed predictions will be meaningful. If not, consider adding more PCs (but keep in mind diminishing returns).

内容的提问来源于stack exchange，提问作者liberalartsthot