ARIMA

ARIMA (AutoRegressive Integrated Moving Average) is a widely used method for Time Series Forecasting that models the autocorrelations within the data. It is particularly effective for datasets with trends or patterns that are non-seasonal. It is more stochastic, based on past correlations rather than explicit decomposition.

When to use:

• When shocks, noise structure, or lagged dependencies drive the series more than trend/seasonality.
• Strong Autocorrelation

• When seasonality changes over time (SARIMA can capture evolving seasonal relationships better than Holt-Winters).

Requirement: Stationary Time Series

ARIMA Explained

ARIMA consists of three components:

• AutoRegressive (AR): Uses past values to predict the current value.
• Integrated (I): Applies differencing to make the series stationary.
• Moving Average (MA): Uses past forecast errors to improve predictions.

The main ARIMA parameters are:

$p$ – Autoregressive (AR) Order

• Represents the number of lagged observations included in the model.
• Purpose: AR terms capture how past values influence the current value.
• Example: $p=2$ → the model uses the previous 2 time points to predict the current one.
• How to pick $p$: Look at the Partial Autocorrelation Function (PACF Plots) plot:
  • The lag after which PACF cuts off (drops to near zero) suggests the p value.
  • Example: PACF is significant at lags 1 and 2, then drops → p = 2.

$d$ – Differencing Order

• Represents the number of times the series is differenced (Differencing in Time Series) to achieve stationarity (removing trends).
• Example: $d=1$ → model uses $y_t-y_{t-1}$.
• How to pick $d$:
  • Visual inspection: Check if the time series has a trend or changing mean.
  • ADF Test (Augmented Dickey-Fuller): If p-value > 0.05 → series is non-stationary → increase $d$.
  • Rule of thumb: Usually $d\le 2$. Most series become stationary after 1–2 differences.

$q$ – Moving Average (MA) Order

• Represents the number of lagged forecast errors included in the model.
• Purpose: MA terms capture short-term shocks (Time Series Shocks) or noise.
• Example: $q=1$ → model uses the previous time step’s error to adjust the prediction.
• How to pick $q$: Look at the Autocorrelation Function (ACF) plot:
  • The lag after which ACF cuts off suggests the q value.
  • Example: ACF is significant at lag 1 → q = 1.

Fine-Tuning and Iterative Refinement

• Test several ARIMA models around initial (p,d,q) estimates.
• Evaluate using AIC in Model Evaluation BIC, or Cross Validation (lower values indicate a better model).
• Check Residuals in Time Series: They should resemble white noise (uncorrelated and zero mean).

What ARIMA Does

1. Checks for stationarity and applies differencing ($d$ times) if needed.
2. Models the relationship between current values and:
   • Past values (AR component)
   • Past forecast errors (MA component)
3. Fits parameters by minimizing a loss function (typically log-likelihood).
4. Forecasts future values using the learned structure.

Why ARIMA Isn’t Enough for Seasonal Data

ARIMA does not model repeating patterns (e.g., quarterly or monthly seasonality). For such cases, SARIMA is used.

Related Resources

In ML_Tools see:

• ARIMA Jupyter Notebook
• Forecasting_AutoArima.py
• pmdarima
• ARIMA vs Random Forest in Time Series
• SARIMA