Analytic implementation of Facebook Prophet.
Prophet's a time series forecasting algorithm published by Facebook Core Data Science Team back in 2017. It gained popularity since then and several companies reported using it.
A successor - NeuralProphet - wad released in 2021, but it's out of the scope of this repo :)
Prophet model is a Generalized Additive Model (GAM) composed of 3 components, is defined as follows
$ \displaystyle y(t) = g(t) + s(t) + h(t) + \epsilon_t \ g(t) = (k + a(t)^T\delta)t + (m+a(t)^T\gamma) \ s(t) = \sum_n a_n \cos(\frac{2\pi tn}{P}) + b_n \sin(\frac{2\pi tn}{P}) = X(t)^T\beta \ h(t) = \sum_i \kappa_i \mathcal{1}_{t \in D_i} = Z(t)^T\kappa $
where
Prophet's trained udner Maximum A Posteriori - meaning priors are declared on the parameters, likelihood is declared on the data, and a function - posterior distribution - is now well-defined, using Bayes formula.
The original version of Prophet is no more than the L-BFGS optimization algorithm used on the loss function. The gradient being estimated using Automatic Differentiation - the L-BFGS optimizer as well as the Automatic Differentiaiton feature are defined within Stan, which itself is written upon C++ and relies on the eigen library.
In this implementaiton, the same optimizer is used on the same loss, however, no gradient estimation part, instead, the gradient is provided by an analytical expression. The backend is written in C++, using dlib for L-BFGS and eigen for numerical computations, An interface similar to the original Prophet is provided.