Model Selection Criteria

Model Selection Criteria, Akiake Selection Criteria, Bayesian information Criteria, Selection of Good Model, Schewars Information Criteria

Akaike Information Criteria: A Comprehensive Guide

by

The Akaike Information Criteria/Criterion (AIC) is a method used in statistics and machine learning to compare the relative quality of different models for a given dataset. The AIC method helps in selecting the best model out of a bunch by penalizing models that are overly complex. Akaike Information Criterion provides a means for comparing among models i.e. a tool for model selection.

  • A too-simple model leads to a large approximation error.
  • A too-complex model leads to a large estimation error.

AIC is a measure of goodness of fit of a statistical model developed by Hirotsugo Akaike under the name of “an information Criteria (AIC) and published by him in 1974 first time. It is grounded in the concept of information entropy in between bias and variance in model construction or between accuracy and complexity of the model.

The Formula of Akaike Information Criteria

Given a data set, several candidate models can be ranked according to their AIC values. From AIC values one may infer that the top two models are roughly in a tie and the rest far worse.

$$AIC = 2k-ln(L)$$

where $k$ is the number of parameters in the model, and $L$ is the maximized value of the likelihood function for the estimated model.

Akaike Information Criteria/ Criterion (AIC)

For a set of candidate models for the data, the preferred model is the one that has a minimum AIC value. AIC estimates relative support for a model, which means that AIC scores by themselves are not very meaningful

Akaike Information Criteria focuses on:

  • Balances fit and complexity: A model that perfectly fits the data might not be the best because it might be memorizing the data instead of capturing the underlying trend. AIC considers both how well a model fits the data (goodness of fit) and how complex it is (number of variables).
  • A lower score is better: Models having lower AIC scores are preferred as they achieve a good balance between fitting the data and avoiding overfitting.
  • Comparison tool: AIC scores are most meaningful when comparing models for the same dataset. The model with the lowest AIC score is considered the best relative to the other models being evaluated.

Summary

The AIC score is a single number and is used as model selection criteria. One cannot interpret the AIC score in isolation. However, one can compare the AIC scores of different model fits to the same data. The model with the lowest AIC is generally considered the best choice.

The AIC is the most useful model selection criterion when there are multiple candidate models to choose from. It works well for larger datasets. However, for smaller datasets, the corrected AIC should be preferred. AIC is not perfect, and there can be situations where it fails to choose the optimal model.

There are many other model selection criteria. For more detail read the article: Model Selection Criteria

Akaike Information Criteria

https://rfaqs.com

https://gmstat.com

https://itfeature.com

Model Selection Criteria (2019)

by

All models are wrong, but some are useful. Model selection criteria are rules used to select a (statistical) model among competing models, based on given data.

Several model selection criteria are used to choose among a set of candidate models, and/ or compare models for forecasting purposes.

All model selection criteria aim at minimizing the residual sum of squares (or increasing the coefficient of determination value). The criterion Adj-$R^2$, Akaike Information, Bayesian Information Criterion, Schwarz Information Criterion, and Mallow’s $C_p$ impose a penalty for including an increasingly large number of regressors. Therefore, there is a trade-off between the goodness of fit of the model and its complexity. The complexity refers to the number of parameters in the model.

Model Selection Criteria

Model Selection Criteria: Coefficient of Determination ($R^2$)

$$R^2=\frac{\text{Explained Sum of Square}}{\text{Total Sum of Squares}}=1-\frac{\text{Residuals Sum of Squares}}{\text{Total Sum of Squares}}$$

Adding more variables to the model may increase $R^2$ but it may also increase the variance of forecast error.
There are some problems with $R^2$

  • It measures in-sample goodness of fit (how close an estimated $Y$ value is to its actual values) in the given sample. There is no guarantee that $R^2$ will forecast well out-of-sample observations.
  • In comparing two or more $R^2$’s, the dependent variable must be the same.
  • $R^2$ cannot fall when more variables are added to the model.

Model Selection Criteria: Adjusted Coefficient of Determination ($R^2$)

$$\overline{R}^2=1-\frac{RSS/(n-k}{TSS(n-1)}$$

$\overline{R}^2 \ge R^2$ shows that the adjusted $R^2$ penalizes for adding more regressors (explanatory variables). Unlike $R^2$, the adjusted $R^2$ will increase only if the absolute $t$-value of the added variable is greater than 1. For comparative purposes, $\overline{R}^2$ is a better measure than $R^2$. The regressand (dependent variable) must be the same for the comparison of models to be valid.

Model Selection Criteria: Akaike’s Information Criterion (AIC)

$$AIC=e^{\frac{2K}{n}}\frac{\sum \hat{u}^2_i}{n}=e^{\frac{2k}{n}}\frac{RSS}{n}$$
where $k$ is the number of regressors including the intercept. The formula of AIC is

$$\ln AIC = \left(\frac{2k}{n}\right) + \ln \left(\frac{RSS}{n}\right)$$
where $\ln AIC$ is natural log of AIC and $\frac{2k}{n}$ is penalty factor.

AIC imposes a harsher penalty than the adjusted coefficient of determination for adding more regressors. In comparing two or more models, the model with the lowest value of AIC is preferred. AIC is useful for both in-sample and out-of-sample forecasting performance of a regression model. AIC is used to determine the lag length in an AR(p) model also.

Model Selection Criteria: Schwarz’s Information Criterion (SIC)

\begin{align*}
SIC &=n^{\frac{k}{n}}\frac{\sum \hat{u}_i^2}{n}=n^{\frac{k}{n}}\frac{RSS}{n}\\
\ln SIC &= \frac{k}{n} \ln n + \ln \left(\frac{RSS}{n}\right)
\end{align*}
where $\frac{k}{n}\ln\,n$ is the penalty factor. SIC imposes a harsher penalty than AIC.

Like AIC, SIC is used to compare the in-sample or out-of-sample forecasting performance of a model. The lower the values of SIC, the better the model.

Model Selection Criteria: Mallow’s $C_p$ Criterion

For Model selection the Mallow criteria is
$$C_p=\frac{RSS_p}{\hat{\sigma}^2}-(n-2p)$$
where $RSS_p$ is the residual sum of the square using the $p$ regression in the model.
\begin{align*}
E(RSS_p)&=(n-p)\sigma^2\\
E(C_p)&\approx \frac{(n-p)\sigma^2}{\sigma^2}-(n-2p)\approx p
\end{align*}
A model that has a low $C_p$ value, about equal to $p$ is preferable.

Model Selection Criteria: Bayesian Information Criteria (BIC)

The Bayesian information Criteria is based on the likelihood function and it is closely related to the AIC. The penalty term in BIC is larger than in AIC.
$$BIC=\ln(n)k-2\ln(\hat{L})$$
where $\hat{L}$ is the maximized value of the likelihood function of the regression model.

Cross-Validation

Cross-validation is a technique where the data is split into training and testing sets. The model is trained on the training data and then evaluated on the unseen testing data. This helps assess how well the model generalizes to unseen data and avoids overfitting.

Note that no one of these criteria is necessarily superior to the others.

Read more about Correlation and Regression Analysis

Learning R Language Programming

Coefficient of Determination Formula: Quick Guide 2019

by

In this post, we will discuss not only the coefficient of determination formula but also the use and computation of the coefficient of determination. Coefficient of Determination as a Link between Regression and Correlation Analysis.

Coefficient of Determination $R^2$ in Statistics

The R squared ($r^2$; the square of the correlation coefficient) shows the percentage of the total variation of the dependent variable ($Y$) that can be explained by the independent (explanatory) variable ($X$). For this reason, $r^2$ (r-squared) is sometimes called the coefficient of determination.

The coefficient of Determination (R-squared is commonly used in various fields like Social Science, Finance, and Economics to evaluate the performance of the regression models. It helps the researchers to understand how well their models capture the relationship between the variables being studied.

Since

\[r=\frac{\sum x_i y_y}{\sqrt{\sum x_i^2} \sqrt{\sum y_i^2}},\]

Coefficient of Determination Formula

\begin{align*}
r^2&=\frac{(\sum x_iy_i)^2}{(\sum x_i^2)(\sum y_i^2)}=\frac{\sum \hat{y}^2}{\sum y^2}\\
&=\frac{\text{Explained Variation}}{\text{Total Variation}}
\end{align*}

where $r$ shows the degree of covariability of $X$ and $Y$. Note that the formula used here is in deviation form, that is, $x=X-\mu$ and $y=Y-\mu$.

The link of $r^2$ between regression and correlation analysis can be considered from these points.

Coefficient of Determination Formula
  • If all the observations lie on the regression line then there will be no scattered points. In other words, the total variation of variable $Y$ is explained completely by the estimated regression line, which shows that there would be no scatterness in the data points(or no unexplained variation). That is
    \[\frac{\sum e^2}{\sum y^2}=\frac{\text{Unexplained Variation}}{\text{Total Variation}}=0\]
    Hence, $r^2=r=1$.
  • If the regression line explains only part of the variation in variable $Y$ then there will be some explained variation, that is,
    \[\frac{\sum e^2}{\sum y^2}=\frac{\text{Unexplained Variation}}{\text{Total Variation}}>0\]
    then, $r^2$ will be smaller than 1.
  • If the regression line does not explain any part of the variation of variable $Y$, that is,
    \[\frac{\sum e^2}{\sum y^2}=\frac{\text{Unexplained Variation}}{\text{Total Variation}}=1\Rightarrow=\sum y^2 = \sum e^2\]
    then, $r^2=0$.

Because $r^2=1-\frac{\text{unexlained variation}}{\text{total variation}}$

Key Points about Coefficient of Determination

  • Overfitting: A model can achieve a high $R^2$ value by simply memorizing the training data, but the model might not perform well on unseen data.
  • Number of Predictors: Adding more independent variables to a model will tend to increase the $R^2$ value, but it does not necessarily mean the additional variables are statistically significant.
  • Alternative Metrics: To assess the nuance of the model fit, use other metrics like adjusted R-squared or residual analysis.

Keeping in mind the limitations of R-squared, the data analysts can use the coefficient of determination as a valuable tool to assess how well their models capture real-world relationships between variables.

Note that there are two main ways to calculate R-squared value:

  1. Squared Correlation Coefficient: R-squared is the square of the correlation coefficient ($r$) between the predicted values ($\hat{y}$) from the model and the actual values of the dependent variable ($y$).
  2. Analysis of Variance (ANOVA): R-squared can also be calculated using the ratio of the explained variance to the total variance (variance in the dependent variable).
Coefficient of determination formula

FAQs about Coefficient of Determination

  1. For a simple linear regression model, what is the link between the coefficient of correlation and the coefficient of determination?
  2. How Coefficient of Determination is interpreted?
  3. How Coefficient of determination can be obtained from the ANOVA table?
  4. How overfitting can be identified from the value of $R^2$?
  5. What are alternatives to $R^2$?
  6. What is the link between total variation, explained variation, and unexplained variation?
  7. What is the impact of adding extra/ more explanatory variables in the linear regression model?
  8. What is the link between explained and unexplained variation?
  9. Give real-life examples of coefficient of determination in which it is high enough.

Learn more about the Coefficient of Determination Formula and Definition in Statistics

https://itfeature.com

Regression Model in R Programming Language

Coefficient of Determination: Model Selection (2012)

by

$R^2$ pronounced R-squared (Coefficient of determination) is a useful statistic to check the regression fit value. $R^2$ measures the proportion of total variation about the mean $\bar{Y}$ explained by the regression. R is the correlation between $Y$ and $\hat{Y}$ and is usually the multiple correlation coefficient. The coefficient of determination ($R^2$) can take values as high as 1 or  (100%) when all the values are different i.e. $0\le R^2\le 1$.

Coefficient of Determination

When repeat runs exist in the data the value of $R^2$ cannot attain 1, no matter how well the model fits, because no model can explain the variation in the data due to the pure error. A perfect fit to data for which $\hat{Y}_i=Y_i$, $R^2=1$. If $\hat{Y}_i=\bar{Y}$, that is if $\beta_1=\beta_2=\cdots=\beta_{p-1}=0$ or if a model $Y=\beta_0 +\varepsilon$ alone has been fitted, then $R^2=0$. Therefore we can say that $R^2$ is a measure of the usefulness of the terms other than $\beta_0$ in the model.

Note that we must be sure that an improvement/ increase in $R^2$ value due to adding a new term (variable) to the model under study should have some real significance and is not because the number of parameters in the model is getting else to saturation point. If there is no pure error $R^2$ can be made unity.

\begin{align*}
R^2 &= \frac{\text {SS due to regression given}\, b_0}{\text{Total SS corrected for mean} \, \bar{Y}} \\
&= \frac{SS \, (b_1 | b_0)}{S_{YY}} \\
&= \frac{\sum(\hat{Y_i}-\bar{Y})^2} {\sum(Y_i-\bar{Y})^2}r \\
&= \frac{S^2_{XY}}{(S_{XY})(S_{YY})}
\end{align*}

where summation are over $i=1,2,\cdots, n$.

Coefficient of Determination
Coefficient of Determination

Interpreting R-Square $R^2$ does not indicate whether:

  • the independent variables (explanatory variables) are a cause of the changes in the dependent variable;
  • omitted-variable bias exists;
  • the correct regression was used;
  • the most appropriate set of explanatory variables has been selected;
  • there is collinearity (or multicollinearity) present in the data;
  • the model might be improved using transformed versions of the existing explanatory variables.

Learn more about

https://itfeature.com