AIC and BICThe challenge of selecting a model from a group of potential models is known as model selection. Selecting the bestperforming model on a holdout test dataset or estimating model performance using a resampling technique like kfold crossvalidation are frequent approaches. Using probabilistic statistical measurements to characterize the model's complexity and performance on the training dataset is an alternate method of selecting models. The Minimum Description Length and the Akaike and Bayesian Information Criterion are two examples. These information criteria statistics have the advantage of not requiring a holdout test set; nevertheless, they have the drawback of not accounting for model uncertainty, which might lead to the selection of too simplistic models. Probabilistic Model SelectionThe process of scoring and selecting candidate models may be done analytically using probabilistic model selection, often known as "information criteria." Models are graded according to their complexity as well as how well they perform on the training dataset.
A probabilistic framework, such as loglikelihood within the context of maximum likelihood estimation, can be used to assess the performance of the model. The number of parameters or degrees of freedom in the model may be used to assess model complexity. Statistical metrics like the Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) are frequently applied in the evaluation and selection of models. They both seek to balance model complexity with goodness of fit, and they are founded on the ideas of information theory. Although they accomplish comparable goals, their underlying presumptions and penalties for model complexity are different. AIC: Akaike Information CriterionA statistical metric called the Akaike Information Criterion (AIC) is employed in the selection, assessment, and comparison of models. It was created in 1973 by Hirotugu Akaike, a statistician from Japan, and is extensively utilized in a number of disciplines, including econometrics, machine learning, and statistics. AIC measures how well a model fits its data while balancing its complexity. In order to prevent overfitting, it penalizes models that are too complicated and instead selects a model that appropriately describes the data. Finding the model that minimizes information loss while skillfully striking a balance between model complexity and explanatory power is the foundation of the AIC approach. The formula for calculating AIC is: where k represents the number of parameters in the model and L denotes the maximum value of the likelihood function for the model. As seen by the formula's 2k term, AIC penalizes models with more parameters. The model's loglikelihood serves as the basis for the penalty term 2ln(L). The objective is to strike a compromise between the model's complexity and goodness of fit, as indicated by the likelihood. In real life, the model with the lowest AIC score is favored when evaluating several models fitted to the same set of data. A model that better explains the data without being too complicated is shown by lower AIC values, which also show a better balance between model fit and complexity. AIC is a useful tool for evaluating and choosing models since it offers a methodical approach to comparing several models and choosing the best fit for a particular dataset. It's crucial to remember that AIC has drawbacks and that, in order to achieve robust model selection, it should be used in combination with other assessment approaches. BIC: Bayesian Information CriterionA criterion for choosing a model from a limited number of options is the Bayesian Information criteria (BIC), often referred to as the Schwarz Criterion or Schwarz Information Criterion (SIC). The BIC punishes models for their complexity in a manner akin to that of the Akaike Information Criterion (AIC), but it does so more severely by penalizing models with more parameters. The formula for calculating BIC is: Where k represents the number of parameters in the model and n denotes the sample size whereas L denotes the maximum value of the likelihood function for the model. A penalty term proportional to the sample size's logarithm is included in the BIC. In comparison to AIC, this penalty becomes more pronounced with increasing sample size, indicating a higher cost for model complexity. This means that BIC favors simpler models more strongly than AIC. In actuality, the model with the lowest BIC value is favored when comparing different models fitted to the same set of data. A model that better explains the data without being too complicated is shown by lower BIC values, which also show a better balance between model fit and complexity. Because BIC favors models with fewer parameters than AIC, it is especially helpful when the main objective is to choose the most parsimonious model. To guarantee robust model selection, BIC should be used in conjunction with other assessment approaches as it has constraints similar to those of AIC. Code: Now we will try to select the model using AIC and BIC Importing Libraries Now we will try to perform an Augmented DickeyFuller (ADF) test and fit an ARIMA model to a time series dataset. Output: We will now plot the series as such to find any key information. Output: Output: Output: Now we will fit ARIMA models with different autoregressive (AR) orders as AR(1), AR(4), AR(6), and AR(10). Output: We will now compare through AIC Output: Based on AIC criteria, we can pick AR(6) We will now compare through BIC Output: Based on BIC criteria, we can pick AR(6)
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