Exponential Smoothing

Exponential smoothing methods produce forecasts as weighted averages of past observations, where the weights decay exponentially. They are the most widely used forecasting family for operational time-series.

Simple Exponential Smoothing (SES)

Model

SES maintains a single level component:

$$ \hat{y}{t+1} = \ell_t = \alpha, y_t + (1 - \alpha), \ell{t-1} $$

where $\alpha \in (0, 1]$ is the smoothing parameter.

Expanding recursively:

$$ \ell_t = \alpha \sum_{j=0}^{t-1} (1-\alpha)^j, y_{t-j} + (1-\alpha)^t, \ell_0 $$

The weight on observation $y_{t-j}$ is $\alpha(1-\alpha)^j$, which decays geometrically — hence "exponential" smoothing.

Choosing α

Chronix selects α by minimizing SSE (sum of squared one-step-ahead errors) over the training window using Brent's method.

Flat Forecast

SES produces a flat forecast: $\hat{y}_{t+h} = \ell_t$ for all horizons h. This makes it unsuitable for trended series.

Holt's Linear Trend Method

Model

Holt (1957) adds a trend component $b_t$:

$$ \ell_t = \alpha, y_t + (1 - \alpha)(\ell_{t-1} + b_{t-1}) $$

$$ b_t = \beta, (\ell_t - \ell_{t-1}) + (1 - \beta), b_{t-1} $$

The h-step forecast extrapolates the trend linearly:

$$ \hat{y}_{t+h} = \ell_t + h, b_t $$

Parameters

• $\alpha$ — level smoothing (same role as SES)
• $\beta$ — trend smoothing
  • $\beta \to 0$: trend changes slowly (stiff)
  • $\beta \to 1$: trend responds immediately

Damped Trend

Gardner & McKenzie (1985) introduced a damping parameter $\phi \in (0, 1]$:

$$ \hat{y}_{t+h} = \ell_t + (\phi + \phi^2 + \cdots + \phi^h), b_t $$

As $h \to \infty$, the forecast converges to $\ell_t + \frac{\phi}{1-\phi} b_t$ instead of diverging. Damped trends are the default in Chronix because linear extrapolation is rarely realistic for infrastructure metrics.

Holt-Winters (Triple Exponential Smoothing)

Model

Holt-Winters (Winters 1960) adds a seasonal component $s_t$ with period m (e.g. m = 24 for hourly data with daily seasonality):

Additive seasonality:

$$ \ell_t = \alpha, (y_t - s_{t-m}) + (1 - \alpha)(\ell_{t-1} + b_{t-1}) $$

$$ b_t = \beta, (\ell_t - \ell_{t-1}) + (1 - \beta), b_{t-1} $$

$$ s_t = \gamma, (y_t - \ell_t) + (1 - \gamma), s_{t-m} $$

$$ \hat{y}{t+h} = \ell_t + h, b_t + s{t-m+h_m^+} $$

where $h_m^+ = ((h-1) \mod m) + 1$.

Multiplicative seasonality:

$$ \ell_t = \alpha, \frac{y_t}{s_{t-m}} + (1 - \alpha)(\ell_{t-1} + b_{t-1}) $$

$$ s_t = \gamma, \frac{y_t}{\ell_t} + (1 - \gamma), s_{t-m} $$

$$ \hat{y}{t+h} = (\ell_t + h, b_t) \cdot s{t-m+h_m^+} $$

Additive vs Multiplicative

Parameters

Initialization

Good initial values for $\ell_0$, $b_0$, and $s_{1..m}$ are critical. Chronix uses the OLS decomposition method:

1. Compute initial level: average of first complete season
2. Compute initial trend: slope of a linear regression over 2–3 seasons
3. Compute initial seasonal indices: ratios (multiplicative) or differences (additive) of the first complete season from the initial level

Computational Complexity

All exponential smoothing variants are O(n) in the training set size and O(m) in memory (storing m seasonal indices). This makes them extremely efficient for real-time retraining on sliding windows.