We'll focus on two methods to analyze and forecast a single time series: exponential smoothing and Autoregressive Integrated Moving Average (ARIMA) models. We'll start by looking at exponential smoothing models.
Like moving average models, exponential smoothing models use weights for past observations. But unlike moving average models, the more recent the observation, the more weight it's given relative to the later ones. There are three possible smoothing parameters to estimate: the overall smoothing parameter, a trend parameter, and the seasonal smoothing parameter. If no trend or seasonality is present, then these parameters become null.
The smoothing parameter produces a forecast with the following equation:
In this equation, Yt is the value at the time, T, and alpha (α) is the smoothing parameter. Algorithms optimize the alpha (and other parameters) by minimizing the errors, Sum of Squared Error (SSE) or maximum likelihood.
The forecast equation along with...