AIC stands for Akaike Information Criterion. It is a metric used to compare the goodness-of-fit of statistical models, taking into account both accuracy and complexity. For SARIMA (or any ARIMA-type model), the formula is:
Where:
- = number of estimated parameters in the model (including AR, MA, seasonal AR/MA, and variance terms).
- = maximum likelihood of the model (how probable the observed data is given the model).
Key points:
Lower AIC is better. It balances model fit and complexity.
- A model with too many parameters may fit the training data well but overfit; AIC penalizes extra parameters.
Relative metric: AIC itself has no absolute meaning; it is only useful for comparing models on the same dataset.
Use in SARIMA: When tuning , we often compute AIC for each combination. The model with the lowest AIC is considered the most efficient balance of fit and parsimony.
Related
BIC
# **Bayesian Information Criterion (BIC)**
#
# - Similar to AIC, the BIC is another criterion for model selection, but it introduces a stronger penalty for models with more parameters.
# - $BIC = \ln(n)k - 2\ln(\hat{L})$, where $n$ is the number of observations, $k$ is the number of parameters, and $\hat{L}$ is the maximized likelihood.
# - A lower BIC value indicates a better model, preferring simpler models to complex ones, especially as the sample size $n$ increases.