Holt-Winters Exponential Smoothing

Background

Holt's linear smoothing (the level + trend variant of exponential smoothing) is the canonical lightweight forecaster for series with a roughly-linear trend and no seasonality. It maintains two state variables — level $L_t$ and trend $T_t$ — each as a weighted average of the new observation and the previous extrapolation. Used as a baseline forecaster in production capacity planning, ETA prediction, and time-series anomaly detection.

Problem statement

Implement holt(series, alpha, beta, horizon). Initialise $L = y_0$, $T = y_1 - y_0$. For $t = 1, \dots, T-1$:

$$L_t = \alpha y_t + (1 - \alpha)(L_{t-1} + T_{t-1}),\qquad T_t = \beta (L_t - L_{t-1}) + (1 - \beta) T_{t-1}$$

Forecast $h$ steps ahead is $L_T + h \cdot T_T$.

Input

• series — array-like of floats, the observed time series ($\ge 2$ values).
• alpha — smoothing factor for the level, in $(0, 1)$.
• beta — smoothing factor for the trend, in $(0, 1)$.
• horizon — int >= 1, number of future steps to forecast.

Output

Returns a tuple (level, trend, forecast):

• level — final $L_T$ (Python float).
• trend — final $T_T$ (Python float).
• forecast — np.ndarray of length horizon.

Examples

Example 1 — constant series

Input:  series=[5.0, 5.0, ..., 5.0], alpha=0.5, beta=0.3, horizon=3
Output: level=5.0, trend≈0.0, forecast=[5.0, 5.0, 5.0]

Example 2 — linear series recovers the slope

Input:  series=[0, 1, 2, ..., 19], alpha=0.5, beta=0.5, horizon=3
Output: trend ≈ 1.0

Constraints

• series must have $\ge 2$ observations.
• alpha and beta must lie in $(0, 1)$.
• horizon >= 1.

Notes

• Adding seasonality. Holt-Winters' triple-exponential variant adds a per-season component $S_t$; this implementation is the level+trend version.
• Picking $\alpha, \beta$. Production setups grid-search over a held-out window or use AIC/BIC; defaults of $\alpha=0.5, \beta=0.3$ are reasonable for monthly data.