Exponential smoothing ( $S_{t} = K X_{t} + (1 - K) S_{t - 1}$ )

With a fixed gain

0 < K < 1

, exponential smoothing recursively combines the newest observation with the preceding estimate. Iteration assigns geometric weights

K (1 - K)^{j}

to older observations, together with a diminishing weight on the initial estimate. The steady-state local-level Kalman gain gives a probabilistic interpretation of this recursion.

Double exponential smoothing

Double exponential smoothing supplements a smoothed level with a smoothed slope so that forecasts can extrapolate a local linear trend. Two common versions are Holt's linear trend method, with separate level and slope gains, and Brown's double exponential smoothing, which applies the same smoothing gain twice. They are different algorithms, not two names for one recursion. Neither version by itself models seasonal variation.

For

0 < α < 1

s_{t}^{(1)} = α X_{t} + (1 - α) s_{t - 1}^{(1)}

and

s_{t}^{(2)} = α s_{t}^{(1)} + (1 - α) s_{t - 1}^{(2)}

. Use the displayed corrected level and slope, then forecast

a_{t} + h b_{t}

. For an exact linear trend after initialization transients, the first and second smooths have lags

(1 - α) / α

and twice that lag; their difference recovers the slope and the corrected level removes the lag. This is distinct from Holt's linear trend method.

Update the level and slope by

ℓ_{t} = α X_{t} + (1 - α) (ℓ_{t - 1} + b_{t - 1})

and

b_{t} = β (ℓ_{t} - ℓ_{t - 1}) + (1 - β) b_{t - 1}

, with gains in

(0, 1)

. The first update compares the new observation with the previous level extrapolated one step; the second smooths the latest level increment. The displayed forecast extrapolates that local slope. This is a double exponential smoothing method for a nonseasonal trending time series.

For

0 < α < 1

, update a local level by the displayed recursion and forecast every future horizon by

ℓ_{t}

. Iteration gives geometrically decreasing weights

α (1 - α)^{j}

on past observations and a vanishing contribution from the initial level. It is useful for a locally stable level without a persistent trend or seasonality. A trend causes lag, motivating double exponential smoothing.

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Exponential smoothing is a statistical technique used for forecasting time series data. It involves using weighted averages of past observations, with the weights decaying exponentially. This means that more recent observations have a greater influence on the forecast than older observations. Exponential smoothing is particularly useful for data with trends and seasonal patterns. There are several types of exponential smoothing methods, including: 1. **Simple Exponential Smoothing**: This method is used for time series data without trends or seasonal patterns.