After subtracting the seasonality estimated with STL (Seasonal and Trend decomposition using Loess), this model forecasts the deseasonalized data using an ETS model adjusted for trends.

This ETS model adjusted for trends involves two smoothing equations (one for the level and one for the trend) that can be summarized by the following coefficients:

• smoothing_level is the smoothing parameter for the level
• initial_level is the level value at t = 0
• smoothing_trend is the smoothing parameter for the trend
• initial_trend is the trend value at t = 0

The ETS model involves two smoothing equations (one for the level and one for the trend) that can be summarized by the following coefficients:

• smoothing_level is the smoothing parameter for the level
• initial_level is the level value at t = 0
• smoothing_trend is the smoothing parameter for the trend
• initial_trend is the trend value at t = 0

A seasonal ARIMA model (p, d, q)(P, D, Q) can be summarized by the following coefficients:

• p_i are the auto-regressive (AR) parameters
• q_i are the moving-average (MA) parameters
• P_i are the seasonal auto-regressive parameters
• Q_i are the seasonal moving-average parameters
• The other parameters are the coefficients of the external regressors

Prophet is a decomposable model with three main components (trend, seasonality, holidays) that are combined in the following equation: ŷ(t) = g(t) + s(t) + h(t)

Where g(t) describes a piecewise linear or logistic trend, s(t) describes the various seasonal patterns using Fourier series for yearly, weekly and daily seasonalities and h(t) represents the external events and are simply dummy variables.

These three components can be summarized by the following coefficients:

• k, m and the delta_i are the growth rate, offset and the changepoints rate adjustements of the piecewise trend component
• yearly_i, weekly_i and daily_i are the Fourier coefficients of the seasonality component
• The other parameters are the coefficients of the external events