This is the paper for Facebook Prophet. It considers time series \(y(t)\) as a composition of trend, seasonality, and holidays under generalized additive model (GAM):

\[y(t) = g(t) + s(t) + h(t) + \epsilon_t\]

which the trend \(g(t)\) is non-periodic, the seasonality \(s(t)\) is periodic, and holiday \(h(t)\) is the effect of holidays which occur irregularly. The error term \(\epsilon_t\) is assumed Gaussian. Fitting new components in GAM can be done using L-BFGS. Prophet is a curve-fitting model; in contrary to ARIMA which is a generative one. Hence data needd not to be regularly spaced and we do not need interpolation for missing data.

Prophet allows trend \(g(t)\) to be nonlinear and satuating. The basic form is

\[g(t) = \frac{C}{1+\exp(-k(t-m))}\]

which the growth rate is \(k\), time offset is \(m\), and growth ceiling is \(C\) (i.e., the capacity). The extension is a piecewise logistic growth model

\[g(t) = \frac{C(t)}{1+\exp(-(k+\mathbf{a}(t)^T\mathbf{\delta})(t-(m+\mathbf{a}(t)^T\mathbf{\gamma})))}\]

where \(C(t)\) is to model the non-constant capacity, and \(\mathbf{a}(t)\) is to model the change points at which the growth rate updated. In detail, it is a vector that

\[a_j(t) = \begin{cases}1 & \text{if }t\ge s_j\\ 0 & \text{otherwise}\end{cases}\]

and at time \(s_j\) the growth rate change by \(\delta_j\), hence at any time, the growth rate is given by

\[k + \sum_{j: t> s_j} \delta_j\]

and the time offset should be changed accordingly, which

\[\gamma_j = \big(s_j - m - \sum_{i<j} \gamma_i\big)\big(1-\frac{k+\sum_{i<j}\delta_i}{k+\sum_{i\le j}\delta_i}\big)\]

For a linear trend, it can be simplified into (with \(\gamma_j = -s_j\delta_j\)):

\[g(t) = (k+\mathbf{a}(t)^T\mathbf{\delta})t + (m+\mathbf{a}(t)^T\mathbf{\gamma})\]

The model of rate change has an implication in forecasting. The paper suggested a prior of \(\delta_j \sim \text{Laplace}(0,\tau)\) and the parameter \(\tau\) is fitted with data. Then the change and change point will be forecasted as a simulated stochastic process.

The seasonality is approximated using Fourier series model:

\[s(t) = \sum_{n=1}^N \big( a_n \cos \frac{2\pi nt}{P} + b_n \sin \frac{2\pi nt}{P}\big)\]

in the paper, it is proposed to fit the Fourier model by using a seasonality vector:

\[\begin{aligned} X(t) &= \big[\cos\frac{2\pi(1)t}{365.25}, \cdots, \sin\frac{2\pi(10)t}{365.25}\big] \\ \mathbf{\beta} &= [a_1, b_1, \cdots, a_10, b_10] \\ s(t) &= X(t)^T \mathbf{\beta} \end{aligned}\]

The paper claimed that \(N=10\) as above performs well for yearly seasonality while \(N=3\) is good for weekly. This design choice can be confirmed using AIC, for example.

Holiday terms are simple, just a binary function that injects impulse \(\kappa\) to \(y(t)\) whenever \(t\) is a holiday (as defined by a custom list). The impulse \(\kappa\sim\text{Normal}(0,\sigma^2)\).

With the model defined, then Prophet will run L-BFGS to fit the paramters.

Bibliographic data

   title = "Forecasting at Scale",
   author = "Sean J. Taylor and Benjamin Letham",
   journal = "The American Statistician",
   volume = "72",
   number = "1",
   year = "2018",
   pages = "37--45",
   doi = "10.1080/00031305.2017.1380080",