AR Model Prediction: Determining Model Order and Computing Error Analysis

In the field of machine learning and time series analysis, AR (AutoRegressive) modeling serves as a fundamental technique for making predictions based on historical data patterns. When implementing AR models, a critical step involves determining the optimal model order through criteria like Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC), typically computed using functions such as ar() in MATLAB or ARIMA in Python's statsmodels library. This process selects the appropriate number of lagged observations (p) to incorporate into predictive calculations.

Consider a practical example: predicting stock prices using AR modeling. The implementation begins by preprocessing historical price data (e.g., differencing for stationarity) and testing multiple model orders. For instance, in Python, one might use ARIMA(order=(p,0,0)) to fit AR models while evaluating partial autocorrelation functions (PACF) to identify significant lags. Underfitting (too few lags) reduces predictive power, whereas overfitting (excessive lags) captures noise—a balance achieved through cross-validation or information criteria minimization.

Post-model construction, error analysis becomes essential. Residual diagnostics involve computing metrics like Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) via mean_absolute_error(y_true, y_pred) in scikit-learn, or analyzing residual autocorrelation with Ljung-Box tests. Visualization tools such as residual plots (plt.plot(residuals)) help detect patterns in prediction errors. Iterative refinement—adjusting model order based on error metrics—ensures robust performance for financial forecasting or general time-series applications.