Prophet is an open-source forecasting library created by Facebook’s data science team (now Meta). It’s designed to make time-series forecasting simple, scalable, and accurate, especially for business applications where data shows trends + seasonality + holidays.

1) Intuition

Prophet views time series as a sum of separable components:

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

  • Trend $g(t)$: long-term growth or decline (linear or logistic).
  • Seasonality $s(t)$: periodic patterns (daily, weekly, yearly).
  • Holiday effects $h(t)$: one-off or recurring special events.
  • Error $\varepsilon_t$​: random noise.

This decomposition makes it interpretable and flexible.

2) Trend component

Prophet supports two main trend models:

  • Linear growth: $g(t) = k t + m$ (slope $k$, offset $m$)
  • Logistic growth (saturation): $g(t) = \frac{C}{1 + \exp(-k(t – m))}$​ (capacity $C$, slope $k$, offset $m$)
    → Useful when growth has a natural ceiling (e.g., user adoption, product sales).

Prophet allows automatic change points: places where trend slope can shift (e.g., sudden business changes, policy changes).

3) Seasonality component

  • Modeled with Fourier series (sine/cosine functions).
  • Flexible enough to capture weekly, yearly, and custom seasonal cycles.
    Example:
    • Weekly pattern (weekdays vs weekends).
    • Yearly cycle (summer vs winter sales).

4) Holiday effects

  • You can feed Prophet a list of holidays or events (e.g., Black Friday, Christmas).
  • It assigns extra parameters to model their impact on the time series.
  • This is critical in retail, finance, and tourism forecasting.

5) Error term

  • Captures irregular fluctuations Prophet doesn’t explain via trend/seasonality/holidays.
  • Treated as noise (assumed independent, not autocorrelated).

6) Advantages

  • User-friendly: Only requires a DataFrame with ds (date) and y (value).
  • Interpretable: You can separately view trend, seasonality, holiday effects.
  • Handles missing data/outliers: Robust to imperfect business data.
  • Automatic change-point detection: Adapts when trends shift.
  • Scales to many time series: Efficient to run in parallel.

7) Limitations

  • Assumes additive structure (trend + seasonality + holidays). Complex dynamics (like autoregressive correlations) aren’t captured.
  • Works best for daily, weekly, monthly business data (sales, demand, traffic).
  • Not ideal for high-frequency data (seconds, milliseconds).
  • Forecast intervals can be too optimistic if uncertainty isn’t well estimated.
  • Less powerful than ML/DL approaches (e.g., LSTMs, Transformers) for complex patterns.

8) Typical use cases

  • Business forecasting: sales, revenue, website traffic, call volumes.
  • Operations: demand forecasting, staffing needs.
  • Finance: transaction volumes, risk modeling.
  • Public data: forecasting COVID-19 cases, energy consumption, or tourism.

9) Workflow in Python

from prophet import Prophet
import pandas as pd

# DataFrame: must have 'ds' (date) and 'y' (value)
df = pd.DataFrame({
    'ds': pd.date_range(start="2020-01-01", periods=100),
    'y': [i + (i%7)*5 for i in range(100)]
})

# Initialize and fit model
model = Prophet()
model.fit(df)

# Make future dataframe
future = model.make_future_dataframe(periods=30)
forecast = model.predict(future)

# Plot forecast
model.plot(forecast)
model.plot_components(forecast)

Bottom line:
Prophet is a decomposable time-series forecasting model combining trend + seasonality + holidays in an interpretable way. It’s powerful for business-oriented, daily/weekly data with strong recurring patterns but less suitable for high-frequency or highly autoregressive processes.