What are the level and trend components for double exponential smoothing?

Double exponential smoothing employs a level component and a trend component at each period. It uses two weights, or smoothing parameters, to update the components at each period. The double exponential smoothing equations are:

L_t = α Y_t + (1 - α) [L_t-1 + T_t-1]
T_t = γ[L_t - L_t-1] + (1 - γ) T_t-1
= L_t-1 + T_t-1

where L_t is the level at time t, α is the weight for the level, T_t is the trend at time t, γ is the weight for the trend, Y_t is the data value at time t, and is the fitted value, or one-step-ahead forecast, at time t.

The first observation is numbered one, then level and trend estimates at time zero must be initialized in order to continue. The initialization method used to determine how the smoothed values are obtained in one of two ways: with Minitab generated weights or with specified weights.

Optimal ARIMA weights	Specified weights
Minitab fits an ARIMA (0,2,2) model to the data, in order to minimize the sum of squared errors. The trend and level components are then initialized by backward forecasting.	Minitab fits a linear regression model to time series data (y variable) versus time (x variable). The constant from this regression is the initial estimate of the level component, the slope coefficient is the initial estimate of the trend component.

When you specify weights that correspond to an equal-root ARIMA (0, 2, 2) model, Holt's method specializes to Brown's method.