Timeseries Engineering

Two unbiased forecasts with equal error variance and an error correlation of 0.3. Each alone has a mean squared error of 1.00; their simple average has 0.65 — matching the theoretical (1 + ρ)/2. The less the forecasters’ errors are correlated, the more averaging helps, even when neither is better than the other.

• Estimated optimal weights usually do worse than a simple average out of sample — the “forecast combination puzzle” (Smith & Wallis 2009).
• Averaging biased forecasts averages the biases; it does not remove them.
• Highly correlated models add cost and little else; diversity is what an ensemble buys.

rng = np.random.default_rng(15)
rho = 0.3
cov = [[1, rho], [rho, 1]]
e = rng.multivariate_normal([0, 0], cov, 100_000)     # errors of two unbiased forecasts
mse_single = (e[:, 0] ** 2).mean()
mse_avg = (e.mean(axis=1) ** 2).mean()
theory = (1 + rho) / 2

Assumes import numpy as np and import pandas as pd. Each snippet simulates or makes up its own series, as the comments say.

# Python — forecast ensemble
import numpy as np

# Simple average ensemble
forecasts = [arima_pred, ets_pred, lstm_pred]
ensemble = np.mean(forecasts, axis=0)

# Inverse-error weighted
errors = [mae_arima, mae_ets, mae_lstm]
weights = [1/e for e in errors]
w_sum = sum(weights)
ensemble_w = sum(f*w/w_sum for f,w in zip(forecasts, weights))