The relationship between the size of the residuals and the fitting effect

There is no relationship between the size of the residuals and the fitting effect. However, the smaller the sum of squares of the residuals of a set of data, the better the fit. In order to determine the effect of each explanatory variable and random error, the difference between the data point and its corresponding position on the regression line is statistically called the residual, and the sum of the squares of each residual is called the sum of squares of the residuals, which represents the effect of the random error.

The sum of squares of the residuals is a quantity to measure the degree of fit of the model in the best model, and it is a data processing method that uses continuous curves to approximate or compare the discrete point groups on the plane to represent the functional relationship between coordinates. A method of approximating discrete data with analytic expressions.

In scientific experiments or social activities, a set of data pairs (x,y) of variables x and y (e=1,2,...) are obtained through experiments or observationswhere the x's are different from each other.

A class of analytic expressions that are compatible with the background material rules of the data, y=f(x,c) is used to reflect the dependence between the variable x and y, that is, to approximate or fit the known data "optimally" in a certain sense. f(x,c) is often referred to as the fitting model, where c=(c,c,...c) are some pending parameters.

Apparently the smaller the better... Because the residuals are equal to the sum of the squares of the errors at all points.

ResidualsThe smaller the sum of squares, the better the fitting effect. In order to clearly explain the variables andRandom errorWhat is the effect of each, statistically the difference between the data point and its corresponding position on the regression line is called the residual, and the sum of the squares of each residual is called the sum of squares of the residuals, which represents the effect of random error. Vertical withering

The property of the sum of squares of the residuals1. Regression equations with only constant terms and no other explanatory variables.

The RSS and TSS are equal and their coefficient of determination.

Increasing the explanatory variables inevitably leads to a decrease in RSS. Therefore, if you want to reduce RSS, you can do so by adding as many explanatory variables as possible to the regression equation.

3. When the number k of all explanatory variables including constant terms is equal to the number of samples n, the rss is 0 and the coefficient of determination is 1.

The residual fitting effect is judged by analyzing the size and distribution of the residual circle, whether it has independence and randomness. The residual refers to the difference between the ** value and the true value, if the residual is small, it means that the ** value is close to the true value, and the fitting effect of the model is better. Otherwise, the fitting effect of the model is poor.

In addition, the distribution of the residuals can be analyzed, and if the distribution is approximately normal, it means that the model fits well, because the normal distribution is one of the most common forms of distribution in nature, and in statistical analysis, many assumptions are based on the assumption of the normal distribution of the collapse.

In addition, the independence and randomness of the residuals are also important indicators to judge the fitting effect. The independence of residuals means that there is no correlation between the residuals, while randomness means that the distribution of residuals does not have a clear trend or periodicity, and the residuals on the scatter plot show a random distribution.

In short, the judgment of the residual fitting effect needs to consider multiple indicators, including the size of the residuals, distribution characteristics, independence and randomness. A good fitting model should have small residuals, residuals that are approximately normally distributed, no correlation between residuals, and good randomness. <>

Summary. Residual plots are an effective way to judge the effect of fitting. The residual diagram is a graph that draws the difference between the actual value and the fitted value, if the fitting effect is good, the points on the residual plot will be distributed near a straight line, and if the fitting effect is not good, the points on the residual plot will be distributed near multiple straight lines, or show a certain regular distribution.

To improve the fitting effect, you should first analyze the residual plot to see if the points on the residual plot show some kind of regular distribution, and if so, you can consider adding more parameters to fit a more complex model. If not, you can consider changing the form of the fitting function, such as changing the order of the fitting function, or changing the parameters of the fitting function to better fit the data. In addition, you can also consider changing the fitting method, such as from least squares to least squares improvements, or from least squares to least square roots.

Residual plots are an effective way to judge the effect of fitting. The residual diagram is a graph that draws the difference between the actual value and the fitted value, if the fitting effect is good, the points on the residual graph will be distributed near a straight line, and if the fitting effect is not good, the points on the residual plot will be distributed near multiple straight lines, or show a certain regular distribution. In order to improve the fitting effect, we should first analyze the residual plot to see if the points on the residual plot show some kind of regular distribution, and if so, you can consider adding more parameters to fit a more complex model. If not, you can consider changing the form of the fitting function, such as changing the order of the fitting function, or changing the parameters of the megavariable fitting function to better fit the data.

In addition, you can also consider changing the fitting method, such as from least squares to least squares improvements, or from least squares to least square roots.

I'm sorry I don't understand, but can you elaborate on that?

Residual plots are an effective way to judge the effect of fitting. A residual plot is a graph that plots the difference between the actual observed value and the fitted value, which can reflect the quality of the fit. If the point distribution in the residual plot is closer to a straight line, the better the pseudo-bright muffling effect. Conversely, if the distribution of points in the residual plot is less than a straight line, the worse the fitting effect.

The residual plot can help us determine the fitting effect, but it also has some limitations. First of all, the residual plot can only reflect the fitting effect, not the accuracy of the fit. Secondly, the residual plot can only reflect the result of the fitting, but not the process of fitting.

Finally, the residual plot can only reflect the results of the fit, but not the reliable rubber bond of the fit. In short, the residual diagram is an effective method to judge the fitting effect, which can reflect the quality of the fit, but it also has certain limitations.