Glossary

10.4 Model selection criteria (AIC, BIC, adjusted R-squared)

3 min read•august 9, 2024

Model selection criteria help us choose the best forecasting model by balancing accuracy and complexity. AIC, BIC, and are key tools for comparing models, each with its own strengths in evaluating fit and penalizing unnecessary complexity.

These criteria are crucial for avoiding and selecting models that will perform well on new data. By using them together, we can make informed decisions about which forecasting model to use, ensuring our predictions are as accurate and reliable as possible.

Information Criteria

Understanding AIC and BIC

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  • quantifies the relative quality of statistical models for a given dataset
  • AIC balances against model complexity penalizes overly complex models
  • functions similarly to AIC but imposes a stricter penalty for model complexity
  • BIC tends to select simpler models compared to AIC, especially with large sample sizes
  • Both AIC and BIC use the likelihood function measures how well a model fits the observed data
  • Lower AIC or BIC values indicate better models (Toyota Corolla vs. Ferrari)

Calculating Information Criteria

  • AIC formula: AIC=2k−2ln⁡(L)AIC = 2k - 2\ln(L)
    • k represents the number of parameters in the model
    • L denotes the maximum value of the likelihood function
  • BIC formula: BIC=kln⁡(n)−2ln⁡(L)BIC = k\ln(n) - 2\ln(L)
    • n represents the number of observations in the dataset
  • Model complexity penalty increases with the number of parameters (k) in both AIC and BIC
  • BIC penalizes complexity more severely due to the inclusion of sample size (n) in its formula

Applying Information Criteria in Practice

  • Use AIC and BIC to compare multiple models fitted to the same dataset
  • Select the model with the lowest AIC or BIC value as the preferred model
  • AIC often preferred for predictive modeling (weather forecasting)
  • BIC often preferred for explanatory modeling (identifying key economic indicators)
  • Consider using both criteria to gain a comprehensive understanding of model performance
  • Information criteria help avoid overfitting by balancing model fit and complexity

Goodness-of-Fit Measures

Understanding Adjusted R-squared

  • Adjusted R-squared measures the proportion of variance in the dependent variable explained by the independent variables
  • Regular R-squared increases with the addition of any variable, even if irrelevant
  • Adjusted R-squared penalizes the inclusion of unnecessary variables
  • Formula for Adjusted R-squared: Radj2=1−(1−R2)(n−1)n−k−1R^2_{adj} = 1 - \frac{(1-R^2)(n-1)}{n-k-1}
    • n represents the number of observations
    • k denotes the number of predictor variables
  • Values range from 0 to 1, with higher values indicating better model fit
  • Useful for comparing models with different numbers of predictors (comparing 3-variable vs. 5-variable economic growth models)

Balancing Model Fit and Complexity

  • Trade-off between fit and complexity fundamental concept in model selection
  • Overly complex models may fit training data well but perform poorly on new data (overfitting)
  • Overly simple models may fail to capture important relationships in the data ()
  • Adjusted R-squared helps identify the optimal balance between fit and complexity
  • Increasing model complexity improves fit up to a point, after which it leads to overfitting
  • Use adjusted R-squared in conjunction with other criteria (AIC, BIC) for comprehensive model evaluation
  • Consider the practical implications of model complexity in terms of interpretability and computational resources

Model Selection Principles

Applying the Parsimony Principle

  • principle states that simpler explanations should be preferred over complex ones, all else being equal
  • In modeling, parsimony favors simpler models with fewer parameters
  • philosophical basis for the parsimony principle
  • Simpler models often more generalizable and less prone to overfitting
  • Parsimonious models easier to interpret and explain (simple linear regression vs. complex neural network)
  • Apply parsimony principle by selecting models with fewer parameters when performance similar

Implementing Model Selection Strategies

  • Use a combination of information criteria, goodness-of-fit measures, and parsimony principle for robust model selection
  • Stepwise regression automated approach to model selection based on these principles
    • Forward selection starts with no variables and adds them one by one
    • Backward elimination starts with all variables and removes them one by one
    • Bidirectional elimination combines both approaches
  • technique to assess model performance on unseen data
  • Consider domain knowledge and theoretical foundations when selecting models
  • Balance statistical criteria with practical considerations (cost, interpretability, implementation feasibility)
  • Regularly reassess and update models as new data becomes available or business needs change

Key Terms to Review (15)

Adjusted r-squared: Adjusted r-squared is a statistical measure that provides an adjusted version of the traditional r-squared value, which indicates the proportion of variance in the dependent variable that can be explained by the independent variables in a regression model. Unlike r-squared, adjusted r-squared accounts for the number of predictors in the model, penalizing excessive use of variables that do not contribute significantly to explaining variability. This adjustment helps in evaluating the model's performance, especially when comparing models with different numbers of predictors.
Akaike information criterion (AIC): The Akaike Information Criterion (AIC) is a statistical tool used for model selection that estimates the quality of a model relative to other models. It helps in balancing the complexity of the model with its goodness of fit, providing a means to choose between competing models by considering both the likelihood of the model and the number of parameters it uses. This concept is crucial when assessing autoregressive and moving average processes, as well as in addressing non-linear relationships.
Bayesian Information Criterion (BIC): The Bayesian Information Criterion (BIC) is a statistical measure used for model selection that balances the goodness of fit of a model against its complexity. It helps in determining which model among a set of candidates is more likely to predict future observations accurately, particularly by penalizing models with more parameters to avoid overfitting. This criterion is closely associated with other model selection criteria and is particularly useful when evaluating autoregressive and moving average processes, as well as in addressing non-linear relationships.
Cross-validation: Cross-validation is a statistical method used to assess the performance and generalizability of a forecasting model by partitioning the data into subsets, training the model on some subsets, and validating it on others. This technique helps ensure that the model is not overfitting to the training data, allowing for better predictions on unseen data. It plays a crucial role in refining model specifications, selecting appropriate variables, and choosing between different forecasting models based on their predictive accuracy.
Holdout method: The holdout method is a technique used in model validation where a subset of data is reserved for testing the performance of a model after it has been trained on the remaining data. This approach helps ensure that the model's predictions are not overly fitted to the training data and provides a more realistic assessment of how well the model can generalize to new, unseen data. It connects to various model selection criteria that evaluate the effectiveness of different models based on their ability to predict outcomes accurately.
Law of parsimony: The law of parsimony, also known as Occam's Razor, is a principle that suggests when faced with competing hypotheses or models, the simplest one is usually preferred. This concept is especially relevant in model selection, where it emphasizes the importance of choosing a model that explains the data with the fewest parameters while still providing a good fit, thereby avoiding overfitting.
Linear regression models: Linear regression models are statistical methods used to predict the value of a dependent variable based on one or more independent variables by fitting a linear equation to observed data. These models help in understanding the relationship between variables and are crucial for evaluating how changes in predictors affect the outcome, which ties into assessing model performance through various selection criteria.
Mean absolute error (MAE): Mean Absolute Error (MAE) is a measure used to evaluate the accuracy of a forecasting model by calculating the average absolute differences between predicted values and actual outcomes. This metric provides insights into how close the forecasts are to the actual values, making it essential for model selection, assessing service level accuracy, and understanding the performance of integrated processes.
Model fit: Model fit refers to how well a statistical model represents the data it is intended to explain. It assesses the accuracy of predictions made by the model and is essential for understanding whether the chosen model is appropriate for the underlying data. Good model fit ensures that the relationships identified in the model accurately reflect real-world patterns, which is critical in methods like regression analysis, variable selection, and evaluating model performance using criteria.
Occam's Razor: Occam's Razor is a philosophical principle that suggests that the simplest explanation is usually the correct one. This principle plays a vital role in model selection, where it emphasizes choosing models that make fewer assumptions while still adequately explaining the data. In the context of evaluating models, it encourages analysts to prefer simpler models over more complex ones, as they are often more generalizable and easier to interpret.
Overfitting: Overfitting occurs when a statistical model captures noise or random fluctuations in the training data instead of the underlying pattern, leading to poor generalization to new, unseen data. This issue is particularly important in model development as it can hinder the model's predictive performance and mislead interpretation.
Parsimony: Parsimony refers to the principle that suggests choosing the simplest model among competing models that adequately explain the data. This idea is important in statistical modeling as it emphasizes avoiding unnecessary complexity, which can lead to overfitting and make models less generalizable. In model selection, criteria such as AIC, BIC, and adjusted R-squared help assess how well a model balances simplicity and explanatory power.
Root mean square error (RMSE): Root Mean Square Error (RMSE) is a widely used measure of the differences between values predicted by a model and the actual values observed. It provides a way to quantify the accuracy of a forecasting model by calculating the square root of the average of the squares of these errors, giving more weight to larger errors. This metric is crucial for evaluating model performance, especially when dealing with various forecasting contexts such as economic indicators, model selection criteria, service level forecasting, integrated processes, and non-linear relationships.
Time series models: Time series models are statistical methods used to analyze data points collected or recorded at specific time intervals, allowing for the identification of trends, seasonal patterns, and cyclical behaviors. These models help in forecasting future values based on historical data, which is crucial for decision-making in various fields such as finance, economics, and business. Understanding the challenges and limitations of these models is essential for effective forecasting, as well as employing appropriate model selection criteria to achieve the best predictive performance.
Underfitting: Underfitting occurs when a statistical model is too simplistic to capture the underlying patterns in the data, resulting in poor performance on both the training and test datasets. This situation arises when the model does not have enough complexity or flexibility to represent the relationships present in the data, often leading to high bias and low variance.
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