ARIMA vs Prophet vs Gradient Boosting

Comparing linear statistical autoregression against additive generalized additive models and tree-based gradient boosting for time series.

🔴 advanced5 min readtime-series

ARIMA (AutoRegressive Integrated Moving Average) is a linear statistical model combining autoregressive lags (p), differencing (d), and moving average error lags (q). Prophet (Meta) is a Generalized Additive Model (GAM) fitting non-linear trend, multi-period seasonality (weekly, yearly), and holiday effects via curve fitting. Gradient Boosting (XGBoost / LightGBM) treats forecasting as tabular regression over lag and rolling window features.

Comparison Matrix

       ARIMA (p, d, q)                        PROPHET (Meta)                  XGBoost / LightGBM
Autoregressive Linear Lags              Generalized Additive Model        Tabular Supervised Regression
  y_t = c + φ_1 y_{t-1} + θ_1 ε_{t-1}     y(t) = g(t) + s(t) + h(t) + ε_t   y_{t+1} = GBDT(Lags, Rolling, Features)

Dimension	ARIMA / SARIMAX	Prophet (Meta)	XGBoost / LightGBM
Underlying Model	Linear Autoregressive (p,d,q)	Non-Linear GAM Curve-Fitting	Ensembles of Decision Trees
Stationarity Needed?	Yes (Requires differencing `d`)	No (Handles trend internally)	No (Handles non-linear trends)
Missing Data Handling	Fails (Requires continuous time step)	Robust (Curve fitting handles gaps)	Robust (Native missing values)
Multiple Seasonality	Complex (SARIMAX gets heavy)	Native (Daily, Weekly, Yearly)	Native (Engineered calendar features)
Exogenous Features	Supported (SARIMAX)	Supported (Extra regressors)	Extremely Strong (Hundreds of features)
Multi-Item Scaling	1 model per item (Slow)	1 model per item (Parallelizable)	1 global model for 100k SKUs

Mathematical Formulations

1. ARIMA(p, d, q)

For stationary series $y'_t = \Delta^d y_t$ :

y'_t = c + \sum_{i=1}^p \phi_i y'_{t-i} + \sum_{j=1}^q \theta_j \epsilon_{t-j} + \epsilon_t

$p$ (AR): Partial Autocorrelation Function (PACF) determines $p$ .
$d$ (I): Order of differencing needed for stationarity.
$q$ (MA): Autocorrelation Function (ACF) determines $q$ .

2. Prophet Decomposition Model

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

$g(t)$ : Piecewise linear or logistic growth trend.
$s(t)$ : Fourier series modeling seasonality (Weekly $s_W(t)$ , Yearly $s_Y(t)$ ).
$h(t)$ : Holiday effects matrix.

Say this out loud

"ARIMA is a linear statistical model combining autoregressive lags p, differencing d, and moving average error terms q, requiring strict stationarity. Prophet is an additive curve-fitting model robust to missing data and multiple seasonalities. For large-scale enterprise e-commerce with thousands of SKUs and exogenous features (prices, promotions), we use LightGBM/XGBoost to train a single global model across all items."

Follow-ups to expect

What is DeepAR (Amazon)? An autoregressive recurrent neural network model that learns probabilistic forecasters across thousands of related time series simultaneously, predicting full predictive probability distributions rather than point estimates.
How do you choose between ARIMA and SARIMAX? SARIMAX adds Seasonal terms $(P, D, Q)_s$ and exogenous regressors $X_t$ , essential when modeling data with fixed seasonal periods (e.g. quarterly or monthly cycles).

Check yourself

Question 1 of 3

What do the three parameters (p, d, q) represent in an ARIMA(p, d, q) model?

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