Measures for In-Sample Evaluation

 


This passage describes the process of numerically evaluating regression models using two important measures: Mean Squared Error (MSE) and R-squared.

  1. Mean Squared Error (MSE): MSE is a measure of the average squared difference between the actual values (y) and the predicted values (ŷ). To calculate MSE, you square the difference between each actual and predicted value, sum them up, and then divide by the number of samples. In Python, you can use the mean_squared_error function from sklearn.metrics to calculate MSE. A lower MSE indicates a better fit of the model to the data.

  2. R-squared (Coefficient of Determination): R-squared is a measure of how close the data points are to the fitted regression line. It compares the performance of the regression model to a simple model (usually the mean of the data points). R-squared ranges between 0 and 1, where 1 indicates a perfect fit. It is calculated as 1 minus the ratio of the MSE of the regression line to the MSE of the mean of the data points. In Python, you can obtain the R-squared value using the score method in the linear regression object. A higher R-squared value indicates a better fit of the model to the data.

The passage also provides visual examples to illustrate the concept of MSE and R-squared. In the example where the regression line fits the data well, the MSE is small compared to the MSE of the mean of the data points, resulting in an R-squared value close to 1. Conversely, in the example where the regression line does not fit the data well, the MSE is comparable to the MSE of the mean, resulting in an R-squared value close to 0.

Overall, MSE and R-squared are important metrics for assessing the performance of regression models and determining how well they fit the data.

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