ARIMA & prophet

1. Principle of ARIMA (AutoRegressive Integrated Moving Average)

ARIMA is a popular time series forecasting model that combines three components:

• AR (AutoRegressive): This component represents the relationship between an observation and a number of lagged observations (previous time points).
• I (Integrated): This part refers to differencing the time series data to make it stationary, i.e., removing trends and seasonality to make the data more predictable.
• MA (Moving Average): This component models the relationship between the observation and the residual error from a moving average model applied to lagged observations.

Key Assumptions for ARIMA:

• Stationarity: The time series should be stationary (i.e., its statistical properties like mean and variance do not change over time). If the data is non-stationary, differencing is used to make it stationary.
• No Seasonality: ARIMA is generally designed for non-seasonal time series. For seasonal data, SARIMA (Seasonal ARIMA) can be used, which extends ARIMA to handle seasonality.
• Linearity: ARIMA assumes that the relationship between past observations and future predictions is linear.

Model Selection:

ARIMA model parameters are typically chosen through a process called model identification, which involves:

• ACF (AutoCorrelation Function) and PACF (Partial AutoCorrelation Function) plots to identify the lag values for AR and MA.
• Differencing to make the series stationary (if needed).
• Use of AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) to select the best model.

2. Principle of Prophet

Prophet is a forecasting tool developed by Facebook that is designed for forecasting time series with multiple seasonalities, holidays, and outliers. It is designed to be highly flexible, making it easy to use even with relatively little knowledge of time series modeling.

Key Components of Prophet:

• Trend Component: A piecewise linear or logistic growth trend that adjusts automatically to the data. It is flexible and can change at different points in time, based on the data's patterns.
• Seasonality Component: Prophet automatically detects and models both yearly and weekly seasonalities in the data (or custom seasonalities can be specified). It also includes the ability to capture daily seasonality if the data allows.
• Holiday Effects: Users can provide additional information about holidays or special events that might affect the time series. Prophet includes this information in the forecasting model.

Model Structure:

Prophet’s forecasting equation is typically represented as:

y(t)=g(t)+s(t)+h(t)+ϵty(t)=g(t)+s(t)+h(t)+ϵt​

Where:

• g(t)g(t) is the trend.
• s(t)s(t) is the seasonality.
• h(t)h(t) is the holiday effect.
• ϵtϵt​ is the error term.

Prophet uses Bayesian methods to estimate parameters, and it is capable of dealing with missing data and large outliers.

3. When to Use ARIMA vs. Prophet

When to Use ARIMA:

• Stationary Data: ARIMA is best for stationary data, where the mean, variance, and autocorrelation structure do not change over time. If the data is non-stationary, differencing can be applied to make it stationary.
• Linear Relationships: ARIMA is a linear model, so it performs well when there are linear relationships between the observations. If the relationships in the data are highly non-linear, ARIMA might not perform well unless transformed appropriately.
• No Complex Seasonality or Holidays: ARIMA (without the seasonal extension SARIMA) doesn't handle complex seasonal patterns or external events like holidays very well. If the data has strong seasonality, you should consider using SARIMA or switching to Prophet.
• Short-Term Forecasting: ARIMA tends to work better for short-term forecasts, especially when data is well-behaved and does not have many external drivers or unexpected changes.

When to Use Prophet:

• Non-Stationary Data with Trend and Seasonality: Prophet works well when the data has trends (which may change over time) and multiple seasonalities (e.g., yearly, weekly, or daily). It automatically handles these patterns, even when the seasonality is not explicitly modeled in the data.
• Complex Seasonality: Prophet can handle multiple seasonalities (e.g., yearly, weekly, daily) and can be easily adapted to model special holidays and events that might affect the time series.
• Non-Linear Trends: Prophet can handle non-linear trends, unlike ARIMA, which assumes linear relationships. If you expect sudden changes or non-linear patterns in the trend, Prophet may be a better choice.
• Missing Data & Outliers: Prophet is robust to missing data and can also handle outliers without requiring data preprocessing or cleaning. If your data has many missing values or outliers, Prophet can still produce reliable forecasts.
• Flexibility & Ease of Use: Prophet is often chosen for its simplicity and flexibility in real-world scenarios. It does not require deep expertise in time series modeling, and users can specify holiday effects, seasonalities, and even adjust trends easily.

4. Conditions for Using ARIMA and Prophet

Conditions for ARIMA:

• Stationarity: Data must be stationary. If it’s not, transformations such as differencing should be applied.
• Linear Relationships: ARIMA assumes linearity, so the underlying relationship in the time series must be linear or approximately linear for the model to work well.
• Seasonality (Optional): If the data has strong seasonal components, use SARIMA (Seasonal ARIMA). Otherwise, ARIMA can handle non-seasonal data well.
• No Complex Trend: ARIMA assumes a relatively simple trend structure. If the trend is highly non-linear or exhibits sudden changes, ARIMA might not be ideal.

Conditions for Prophet:

• Non-Stationary Data: Prophet works well with non-stationary data, as it automatically models both trend and seasonality components.
• Multiple Seasonalities: If your data exhibits multiple types of seasonality (e.g., yearly and weekly), Prophet is a great choice.
• Custom Holidays or Events: Prophet is suitable when you want to incorporate external factors such as holidays, special events, or any external regressors.
• Non-Linear Trends: Prophet is robust to non-linear trends and can adjust the model to handle abrupt changes in the trend, which ARIMA cannot do.
• Outliers and Missing Data: Prophet handles missing data and outliers well, making it useful for noisy or incomplete time series data.

5. Summary Table:

Aspect	ARIMA	Prophet
Data Type	Stationary, linear trends	Non-stationary, non-linear trends
Seasonality	Limited handling of seasonality (SARIMA for seasonal data)	Handles multiple seasonalities (weekly, yearly, daily)
Trend	Linear trends only	Non-linear trends
Missing Data	Requires interpolation or imputation	Can handle missing data natively
Outliers	Sensitive to outliers	Robust to outliers
Ease of Use	Requires model selection (AR, MA, differencing)	User-friendly, automatic seasonalities and trend adjustments
External Regressors	Can include external regressors	Easily incorporates holidays and events
Forecast Horizon	Best for short-term forecasts	Suitable for both short and long-term forecasts
Stationarity Requirement	Data must be stationary (or made stationary)	No need for stationarity