Prophet

The model#

Taylor and Letham (2018) specify a decomposable additive model — a generalised additive model in time, with no autoregressive component whatsoever:

$$ y(t) = g(t) + s(t) + h(t) + \varepsilon_t $$

• $g(t)$ — the trend, piecewise linear or logistic.
• $s(t)$ — periodic seasonality, represented by a truncated Fourier series.
• $h(t)$ — holiday and event effects, from a user-supplied date table.
• $\varepsilon_t \sim \mathrm{N}(0,\sigma^2)$ — assumed i.i.d.

The last assumption is the design's defining commitment and its defining weakness. Prophet models $y$ as a function of $t$, not as a function of $y_{t-1}$. Fitting is a curve-fitting problem, which is why it tolerates gaps, unequal spacing and outliers, and why it is uncompetitive when the dominant signal is short-lag autocorrelation.

Trend#

The piecewise-linear form with changepoints $s_1,\dots,s_S$ is

$$ g(t) = (k + \mathbf{a}(t)^\top \boldsymbol{\delta})\,t + (m + \mathbf{a}(t)^\top \boldsymbol{\gamma}) $$

• $k$ — the base growth rate.
• $\boldsymbol{\delta} \in \mathbb{R}^S$ — the rate adjustments at each changepoint.
• $a_j(t) = \mathbf{1}[t \ge s_j]$ — the changepoint indicator vector.
• $m$ — the offset, with $\gamma_j = -s_j\delta_j$ chosen to keep $g$ continuous.

Sparsity is imposed by a Laplace prior $\delta_j \sim \mathrm{Laplace}(0, \tau)$, where $\tau$ is changepoint_prior_scale. This is the Bayesian analogue of an $L_1$ penalty, so most $\delta_j$ shrink to near zero and a few survive. Candidate changepoints default to 25 points uniformly spaced over the first 80 percent of history.

Forecast uncertainty in the trend is generated by simulating future changepoints at the historical rate, with magnitudes drawn from a Laplace fitted to the observed $\delta_j$. This is an explicitly heuristic device, acknowledged as such in the paper: it assumes the future will bend as often as the past did.

The logistic form is

$$ g(t) = \frac{C(t)}{1 + \exp!\left(-(k + \mathbf{a}(t)^\top\boldsymbol{\delta})(t - (m + \mathbf{a}(t)^\top\boldsymbol{\gamma}))\right)} $$

with a time-varying capacity $C(t)$ that the user must supply. Prophet does not estimate the ceiling.

Seasonality and holidays#

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

• $P$ — the period in days (365.25 yearly, 7 weekly).
• $N$ — the Fourier order, defaulting to 10 for yearly and 3 for weekly.

The $2N$ coefficients get a prior $\mathrm{N}(0, \sigma_s^2)$ with $\sigma_s$ set by seasonality_prior_scale. Fourier terms rather than seasonal indices are what allow non-integer periods and multiple simultaneous seasonalities at $2N$ parameters each.

Holidays are a design matrix of indicators over the union of each event's window $[\text{lower}, \text{upper}]$, with coefficients $\kappa \sim \mathrm{N}(0, \nu^2)$. Each window day gets its own coefficient, so an event with a seven-day window costs seven parameters and needs several past occurrences to estimate them.

Estimation is MAP by L-BFGS in Stan by default, with mcmc_samples > 0 switching to full HMC and giving seasonality uncertainty that MAP omits. MAP fitting is the reason the point forecast is deterministic while the intervals are not.

The empirical record, stated plainly#

Prophet's adoption far exceeds its measured accuracy, and this is worth being precise about.

Makridakis, Spiliotis and Assimakopoulos (2018), Statistical and Machine Learning forecasting methods: Concerns and ways forward, evaluated a range of methods on M3 series and found Prophet among the weaker performers relative to statistical baselines. Later reproductions on M-competition data have consistently placed Prophet below well-tuned ETS and ARIMA on standard benchmarks, and no Prophet-based entry has featured in the top of any M competition.

The reason is structural rather than a matter of tuning. M-competition series are mostly short, monthly or quarterly, and dominated by autocorrelation rather than calendar effects. Prophet has no autoregressive term, so on those series it is fitting a smooth curve to something whose predictability lives in the lags.

Where Prophet does earn its place is the regime it was designed for: long daily business series with strong multi-scale seasonality, irregular holiday effects, missing observations, outliers, and level shifts — combined with an analyst who needs a defensible, decomposable explanation rather than the last percent of accuracy. That is a real and common regime, and it is not the one the benchmarks measure.

Extensions#

NeuralProphet (Triebe et al., 2021) keeps the decomposable structure and adds an autoregressive module (AR-Net, a sparse linear or shallow network over lags) plus lagged covariates, addressing the missing $y_{t-1}$ dependence directly while retaining interpretable components. It is a PyTorch implementation, so it also gets minibatch training over long series.

Orbit (Uber) and PyMC-based structural time series offer the same decomposition under full Bayesian inference with proper posterior intervals.

Structural time series in the classical sense (Harvey, 1989) is the closer statistical ancestor: local level plus local trend plus seasonal, estimated in state space by the Kalman filter, which does model the dependence and gives calibrated intervals. statsmodels.tsa.statespace.structural.UnobservedComponents implements it. If you like Prophet's decomposition but need honest intervals, that is where to look.

Reading#

• Taylor and Letham (2018), Forecasting at scale, The American Statistician 72(1) — the Prophet paper. Preprint at peerj.com/preprints/3190.
• Triebe, Laptev and Rajagopal (2021), NeuralProphet: Explainable Forecasting at Scale — arxiv.org/abs/2111.15397.
• Harvey (1989), Forecasting, Structural Time Series Models and the Kalman Filter, CUP.
• Makridakis, Spiliotis and Assimakopoulos (2018), Statistical and Machine Learning forecasting methods: Concerns and ways forward, PLoS ONE 13(3).
• Hastie and Tibshirani (1987), Generalized Additive Models: Some Applications, JASA 82(398) — the framework Prophet instantiates.

What to learn next#

• Feature engineering for time series — the approach that won M5, and a strong alternative to Prophet.
• Evaluating a forecast — running the baseline comparison this lesson keeps demanding.
• Multivariate forecasting — adding outside variables such as temperature or price.