7.1 Information criteria (AIC, BIC) for model selection

Time series analysis requires careful model selection to balance fit and complexity. Choosing the right model ensures accurate forecasting and captures essential data patterns without overfitting. This process is crucial for understanding complex time-dependent data.

The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are key tools for model selection. These methods compare models based on their fit and complexity, helping analysts choose the most appropriate model for their time series data.

Model Selection in Time Series Analysis

Importance of model selection

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  MR - Bootstrap aggregation for model selection in the model-free formalism View original

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• Process of choosing the best model from a set of candidate models balances goodness of fit and model complexity to avoid overfitting (selecting an overly complex model that fits noise) and underfitting (selecting a model that is too simple to capture the underlying patterns)
• Crucial for accurate forecasting and inference in time series analysis as data often exhibit complex patterns and dependencies (autocorrelation, seasonality, trend)
• Ensures the selected model captures the essential features of the data without being overly complex, leading to better generalization and predictive performance

Role of Akaike Information Criterion

• Widely used model selection criterion developed by Hirotugu Akaike in 1974 assesses the relative quality of a model based on its likelihood (measure of how well the model fits the data) and complexity (number of parameters)
• Calculated using the formula: $AIC = 2k - 2\ln(L)$, where $k$ is the number of parameters in the model and $L$ is the maximum likelihood estimate of the model
• Lower AIC values indicate better-fitting models as it penalizes models with a larger number of parameters to discourage overfitting and allows for the comparison of non-nested models (models that cannot be obtained by imposing restrictions on another model)

Bayesian Information Criterion vs AIC

• BIC is another commonly used model selection criterion developed by Gideon Schwarz in 1978 based on Bayesian principles (incorporating prior knowledge) and sample size
• Calculated using the formula: $BIC = k\ln(n) - 2\ln(L)$, where $k$ is the number of parameters, $n$ is the sample size, and $L$ is the maximum likelihood estimate
• BIC penalizes model complexity more heavily than AIC, especially for large sample sizes, favoring simpler models
• BIC is consistent, meaning it selects the true model with probability approaching 1 as sample size increases, while AIC is not consistent and may select an overly complex model even with large sample sizes

Application of AIC and BIC

1. Fit the candidate models (ARIMA, SARIMA, exponential smoothing) to the time series data
2. Calculate the likelihood and the number of parameters for each model
3. Compute AIC and BIC values for each model using the respective formulas
4. Select the model with the lowest AIC or BIC value as the best-fitting model

• AIC and BIC provide a relative comparison of models, not an absolute measure of model quality, so the selected model should also be assessed for interpretability and practical relevance
• Consider the context and purpose of the analysis (short-term forecasting, long-term forecasting, identifying underlying patterns) when making the final model choice