How Adaptive Filtering Optimizes Time Series Forecasting

Adaptive Filtering Method

Adaptive filtering, like moving average and exponential smoothing, predicts a time series by weighting past observations. It searches for an optimal set of weights by first using a given set to compute a forecast, then calculating the prediction error, and finally adjusting the weights to reduce the error. This iterative process continues until the error reaches a minimum.

The basic prediction formula is omitted here. The weight‑adjustment formula updates each weight w_i as w_i(new) = w_i(old) + μ * e_t * x_{t-i}, where μ is the learning constant, e_t is the prediction error at time t, and x_{t-i} is the i‑th past observation.

The learning constant controls how quickly the weights adapt. In practice, weight adjustment can be computationally intensive and usually requires a computer.

When initializing the algorithm, the number of weights and the learning constant must be chosen. For seasonal data, the seasonal length (e.g., 12 for monthly data with yearly seasonality, 4 for quarterly data) is often used as the number of weights. If no clear seasonality exists, the lag with the highest autocorrelation can be selected.

The learning constant is typically set to 0.1, though different values can be tested to find the one that minimizes error. If no prior information is available, the initial weights can be set uniformly, for example, each weight equal to 1/N.

Adaptive filtering has two main advantages:

It is relatively simple; the number of weights and learning constant can be chosen to control the forecast, and they can also be selected automatically by a computer.

It uses all historical data to seek optimal coefficients, continuously updating the weights as new data arrive, which improves prediction accuracy. Its simple model and suitability for computer implementation make it widely used.

Reference

ThomsonRen GitHub, https://github.com/ThomsonRen/mathmodels