Second order exponential smoothing

If first order exponential smoothing does not perform well, then there is a trend in the time series data. The trend is commonly observed in many domains such as when marketing campaigns are run by e-commerce companies, the sales rise or any good annual performance by a company will have a bullish effect on its stock prices. The linear trend can occur due to linear trend between time and response:

x_t = constant + ωt + ε_t

Here, ω is the coefficient that leads to trend. The second order exponential smoothing helps capture the trend in time series data by including another term to the first order exponential smoothing as follows:

Here, T_t captures the trend component of the exponential smoothing and is represented as follows:

Here, α is the data smoothing factor and β is the trend smoothing factor with values between [0, 1]. The next stage forecast can be generated as follows:

In second order smoothing, the initial value for the trend component can be assigned...