Prophet Forecasting

Overview

Prophet (Meta) decomposes time series into trend + seasonality + holidays + error. Uses an additive (or multiplicative) model fitted with Stan. Handles missing data, outliers, and holiday effects natively. Designed for business time series at daily/weekly granularity.

When to Use

Trigger conditions:

• Forecasting business metrics (sales, traffic, engagement) at daily/weekly frequency
• Data with strong seasonal patterns and known holiday effects
• Need quick, reasonable forecasts without deep time series expertise

When NOT to use:

• For high-frequency data (sub-hourly) — Prophet is designed for daily+
• When you need causal/explanatory models (Prophet is descriptive)
• For very short time series (< 2 seasonal cycles)

Algorithm

IRON LAW: Prophet Is an Additive Regression Model, NOT Classical Time Series
y(t) = g(t) + s(t) + h(t) + ε(t)
- g(t): piecewise linear or logistic trend with automatic changepoints
- s(t): Fourier series for yearly/weekly/daily seasonality
- h(t): user-specified holiday effects
Prophet does NOT model autocorrelation in residuals. If residuals are
autocorrelated, the uncertainty intervals will be too narrow.

Phase 1: Input Validation

Prepare DataFrame with columns: ds (datestamp), y (metric). Add regressor columns if available. Specify: country holidays, custom holidays, growth type. Gate: Data formatted, minimum 2 full seasonal cycles.

Phase 2: Core Algorithm

1. Choose growth model: 'linear' (default) or 'logistic' (with cap and floor)
2. Set seasonality: yearly (default), weekly (default), custom (e.g., monthly)
3. Add holidays: country built-ins + custom events (promotions, launches)
4. Fit model: m = Prophet(); m.fit(df)
5. Generate future DataFrame and predict: m.predict(future)

Phase 3: Verification

Check: forecast components (trend, seasonality, holidays) are intuitive. Cross-validate: use Prophet's built-in cross_validation() with rolling windows. Evaluate MAPE, RMSE. Gate: MAPE acceptable for use case, components pass visual inspection.

Phase 4: Output

Return forecast with decomposed components.

Output Format

{
  "forecasts": [{"ds": "2025-04-15", "yhat": 1200, "yhat_lower": 1050, "yhat_upper": 1350}],
  "components": {"trend": "upward_3pct", "yearly_seasonality": "peak_in_december", "weekly_seasonality": "low_on_weekends"},
  "metadata": {"mape": 0.08, "training_days": 730, "forecast_days": 90}
}

Examples

Sample I/O

Input: 2 years of daily website traffic with Christmas spike and summer dip Expected: Forecast captures: upward trend, weekly pattern (weekday > weekend), annual pattern (Christmas spike, summer dip).

Edge Cases

Input	Expected	Why
Many missing days	Prophet handles natively	Unlike ARIMA, no imputation needed
Sudden trend change	Changepoint detected automatically	Prophet's key feature vs ARIMA
Multiplicative seasonality	Set seasonality_mode='multiplicative'	When seasonal amplitude grows with trend

Gotchas

• Default changepoint sensitivity: Prophet may over/under-detect trend changes. Tune changepoint_prior_scale (default 0.05): higher = more flexible, lower = smoother.
• Flat forecasts: If trend changepoints are too conservative, long-range forecasts can be unrealistically flat. Increase flexibility or specify growth cap.
• Holiday effects require specification: Prophet doesn't discover holidays automatically. You must provide a holiday DataFrame — missing holidays will not be modeled.
• Not for causal inference: Prophet finds patterns but doesn't explain why. Adding a regressor shows correlation, not causation.
• Uncertainty intervals: Based on historical trend change variance, not residual autocorrelation. May be too narrow if residuals are structured.

References

• For Prophet hyperparameter tuning guide, see references/prophet-tuning.md
• For cross-validation best practices, see references/prophet-cv.md