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What’s residual sum of squares?



Residual sum of squares (RSS) is a commonly used concept in statistics and regression analysis that represents the sum of squared differences between observed values and predicted values from a regression model. This implies the clear meaning of residuals: residuals refer to the differences or deviations between observed values and predicted values from the regression model. They indicate the degree of mismatch between the actual values and the model.

In regression analysis, we use a mathematical model to describe the relationship between independent variables and dependent variables. Through the regression model, we can predict the values of the dependent variable based on the values of the independent variables. However, due to various reasons such as random errors and inaccuracies in the model assumptions, there will be differences between the predicted values and the actual observed values, i.e.: Residual = Observed value - Predicted value

Residuals can be positive or negative, with positive residuals indicating that the observed values are higher than the predicted values, and negative residuals indicating that the observed values are lower than the predicted values. Residuals can be used to measure the goodness of fit of the model to the data and the accuracy of predictions.

The calculation and analysis of residuals are crucial for the evaluation and diagnosis of regression models. By analyzing the distribution, statistical properties, and patterns of residuals, we can test the assumptions of the regression model, evaluate the goodness of fit of the model, and identify outliers, influential points, or nonlinearity in the model. For an appropriate regression model, residuals should exhibit characteristics such as random distribution, mean close to zero, and constant variance.

In summary, residuals serve as quantitative indicators for measuring the differences between observed values and predicted values from a regression model, and they have significant implications for understanding the accuracy of the model, goodness of fit, and identifying abnormal situations.

Residual sum of squares, as the name suggests, is the calculation of the square of each residual for every observed value, and then summing them up. The specific calculation steps are as follows:
Calculate the residual for each observed value (subtract the predicted value from the observed value).

Square each residual value.
Sum up all the squared residual values to obtain the residual sum of squares.
The residual sum of squares can measure the goodness of fit of the regression model to the data. If the residual sum of squares is small, it indicates that the differences between the predicted values and the actual observed values are minimal, suggesting a good fit of the model to the data. On the other hand, if the residual sum of squares is large, it indicates that the differences between the predicted values and the actual observed values are significant, indicating a poor fit of the model to the data.
In regression analysis, we typically seek the best-fitting line or curve that minimizes the residual sum of squares to find the optimal model. This can be achieved through methods such as the least squares method. The goal of minimizing the residual sum of squares is to make the model's predictions as close as possible to the actual observed values, thereby improving the accuracy and reliability of the model.

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