# Forecasting models

Exponential smoothing models are a popular class of time series models.

Exponential smoothing models are applicable to a single set of values that are recorded over equal time increments only. However, they support data properties that are frequently found in business applications such as trend, seasonality, and time dependence. All specified model features are estimated based on available observed data. An estimated model can then be used to forecast future values and provide upper and lower confidence bounds for the forecast values.

Each model type is suited for modeling a different combination of properties that are found in the data. The model type that can provide the best match to the observed data is selected for modeling the observed data and is used to forecast any future values.

## Model estimation algorithms

Models are specified by the smoothing equations that include the model parameters and initial smoothing states. Model parameters are estimated with values that minimize the model error.

## Smoothing equations

Exponential smoothing models derive their name from the smoothing equations that specify the model. They provide formulas for computing smoothing states for each observed point by using the current observed value and the previous smoothing states. Smoothing equations provide weighted averages of the current value and the previous states in the time series. Weight for the current value or state is given by a model parameter between 0 and 1, while the weights for previous values are exponentially decreasing.

## Level smoothing equations

All model types compute a level state for each time series point by using the corresponding level smoothing equation. Level states for the model without trend and seasonal components are computed as the weighted average of the time series value at the current point and the level state at the previous point. The weight that is associated with the current value is a parameter, alpha, with its value restricted between 0 and 1. For other models, previous trend and seasonal states are also included in the level smoothing equation.

## Trend smoothing equations

Model types with additive or damped additive trend compute a trend state for each time series point by using the corresponding trend smoothing equation. The trend state for the current point is based on the difference of level states at the current and the previous point, and on the trend state at the previous point. The weight that is associated with the difference of level states at the current and previous point is a parameter that is named beta with its value restricted between 0 and 1. An extra parameter, phi, is added to the damped trend smoothing equations. Phi multiplies the trend state contribution from the preceding point and its value is also restricted between 0 and 1. The purpose of this parameter is to estimate the degree of trend damping from one point to the next.

## Seasonal smoothing equations

Model types that support additive or multiplicative seasonality compute a seasonal state for each time series point. The seasonal states are computed by using seasonal smoothing equations. The seasonal state for the current point includes the difference of the time series value and the current level state for additive seasonality or ratio of the two same values for multiplicative seasonality. The weight that is associated with this term is a parameter, gamma, with its value restricted between 0 and 1. The rest of the contribution comes from the corresponding seasonal state in the previous seasonal period. Notice that the seasonal period has a fixed length, and while the seasonal state can change for each point, only matching seasonal indices from different periods are considered together in the seasonal smoothing equations.

## Initial smoothing states

Values must be specified for level, trend, and seasonality states for points that precede the time series. The values are needed for the smoothing equations. To compute the various states at the first point of the time series requires state values at the corresponding previous points.

## Model parameters

Each smoothing equation uses corresponding model parameters:

- alpha
- Controls the level states.
- beta
- Controls the trend states.
- gamma
- Controls seasonal indices across seasonal periods.
- phi
- An extra parameter that is used for specifying the damped trend.

All four parameters have values between 0 and 1. Higher values of alpha, beta, and gamma mean that more recent observations have higher weight, while lower values mean higher weights for older observations. A higher value of phi corresponds to a higher degree of dampening the forecast trend.

## Model estimation

Model parameters in the smoothing equations are estimated based on the time series data. Parameters cannot be estimated directly by using a formula. They are estimated by an iterative process that searches for parameter values that minimize the model error. The model error is computed as Mean Absolute Scaled Error (MASE). The iterations stop when no further reduction in the model error can be achieved. Corresponding parameter values together with the initial smoothing states fully specify the estimated model. They are used to compute the model states for all other data points and generate the model forecasts by using a corresponding forecast equation.