Update a small state whenever new data arrives.

What does smoothing do?

A time series mixes useful structure with short-term randomness, measurement error, and unusual events. Smoothing reduces the influence of those temporary movements so that the current level, trend, and seasonal pattern become easier to estimate.

Smoothing is controlled memory: when a new observation arrives, the model decides how much to revise its previous belief. It should react to genuine change without chasing every noisy point.

Begin with a moving average

A three-period moving average gives equal weight to the latest three observations:

If the observations are 100, 110, and 105, then \(M_t=105\). This reduces noise, but an observation has full weight inside the window and suddenly receives zero weight after leaving it.

Exponential smoothing replaces that hard cutoff with gradually declining weights. Every past observation can contribute, but recent evidence matters more.

Simple exponential smoothing

Think of \(\ell_{t-1}\) as yesterday's best belief about the current level and \(y_t\) as fresh evidence. The update takes a weighted average: the first term accepts a fraction \(\alpha\) of the new evidence, while the second retains a fraction \(1-\alpha\) of the old belief.

Walk toward the observation: start at the old level, measure how wrong it was, and move \(\alpha\) of that distance. With \(\alpha=0\), ignore the new point. With \(\alpha=1\), jump fully to it. Intermediate values trade stability for responsiveness.

Repeated substitution reveals why the method is called exponential smoothing:

Why every observation still matters: an observation \(j\) steps old receives weight \(\alpha(1-\alpha)^j\). The weights shrink geometrically rather than suddenly becoming zero at a fixed window boundary.

A two-step SES example

Every future forecast from this origin is 106 because SES has only a level state:

If the next observation is \(y_2=90\), the forecast error is \(e_2=90-106=-16\), and the new level becomes:

Error-correction view: the same update can be written \(\ell_t=\ell_{t-1}+\alpha(y_t-\ell_{t-1})\). Move the old level by a fraction \(\alpha\) of its latest error.

Add a trend with Holt's method

The level update compares the observation with the previously projected level. The trend update blends the newest estimated change with the previous trend. Parameter \(\beta\) controls how quickly the trend changes.

Worked Holt update

Suppose yesterday's level was \(\ell_{t-1}=100\), the estimated increase was \(b_{t-1}=5\), today's observation is \(y_t=112\), and \(\alpha=0.4,\ \beta=0.3\).

1. Project the old state into today: \(100+5=105\).
2. Blend the observation with that projection: \(\ell_t=0.4(112)+0.6(105)=107.8\).
3. The newly observed level change is \(107.8-100=7.8\). Smooth it with the old trend: \(b_t=0.3(7.8)+0.7(5)=5.84\).
4. Forecast: \(\widehat y_{t+1\mid t}=107.8+5.84=113.64\), while \(\widehat y_{t+2\mid t}=107.8+2(5.84)=119.48\).

What changed? SES would forecast the same level at every horizon. Holt carries the estimated slope forward, so the path rises by \(5.84\) per step.

Damped trend: long straight-line extrapolation can become implausible. A damping parameter \(\phi\), usually between 0 and 1, gradually reduces the trend contribution at longer horizons.

Additive Holt-Winters

For daily data with weekly seasonality, \(m=7\). The final term retrieves the latest estimated seasonal effect for the forecasted weekday. Parameter \(\gamma\) determines how quickly that seasonal pattern changes.

Worked additive Holt-Winters update

Assume weekly seasonality with \(m=7\). Before observing today, \(\ell_{t-1}=100\), \(b_{t-1}=2\), and today's weekday usually contributes \(s_{t-7}=15\) rentals. We observe \(y_t=120\), with \(\alpha=0.4,\ \beta=0.2,\ \gamma=0.3\).

1. Remove the old seasonal effect: \(y_t-s_{t-7}=120-15=105\).
2. Update the nonseasonal level: \(\ell_t=0.4(105)+0.6(100+2)=103.2\).
3. Update trend: \(b_t=0.2(103.2-100)+0.8(2)=2.24\).
4. Today's new seasonal evidence is \(120-100-2=18\). Blend it with the old weekday effect: \(s_t=0.3(18)+0.7(15)=15.9\).
5. For the same weekday one week ahead, \(\widehat y_{t+7\mid t}=103.2+7(2.24)+15.9=134.78\).

Three memories cooperate: level says where the series is, trend says how fast it is moving, and the seasonal state says how this weekday normally differs from the underlying level.

Fitting: software estimates smoothing parameters and initial states by minimizing in-sample forecast errors. Model choice still belongs to rolling-origin validation; the smallest training SSE is not enough.

One family, three forecast shapes

The forecast shape follows directly from the states the model stores. A model cannot extrapolate a component it does not represent.

Additive or multiplicative seasonality?

Practical distinction: use additive seasonality when seasonal amplitude is stable in units; consider multiplicative seasonality when the swings grow with the level. Multiplicative models require positive values. A log transformation can often turn multiplicative structure into additive structure.

How smoothing models are fitted and chosen

1. Select the state structure. Level only, level plus trend, or level plus trend and seasonality.
2. Choose the seasonal period. Daily data may use \(m=7\); hourly demand may need \(m=24\) or \(m=168\).
3. Initialize the states. Estimate the starting level, trend, and one seasonal effect for each position in a cycle.
4. Estimate smoothing parameters. Software usually selects \(\alpha\), \(\beta\), \(\gamma\), and possibly damping \(\phi\) by minimizing a likelihood or one-step training errors.
5. Validate the whole model. Compare SES, Holt, damped Holt, and Holt-Winters on identical rolling forecast origins and horizons.
6. Inspect residuals. Remaining trend, seasonality, or autocorrelation means the state structure has not captured all predictable information.

Final intuition: exponential smoothing is a small adaptive memory. Each observation asks whether the level moved, whether the slope changed, and whether this seasonal position behaved differently from usual.