Find Function with r^2 Near Zero & Approaching 0

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In summary, the conversation discusses finding a function with a behavior of r^2 near zero and quickly approaching zero as r tends to infinity. The suggestion is to multiply two functions together, one behaving like r^2 near zero and the other behaving like 1 near zero and decaying faster than r^2 increases. Various options for the second function are suggested, such as r^2/(1+r^3) or r^2e^(-r). The goal is to make the product of the two functions fit specific constraints.
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intervoxel
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I'm looking for a function which has a behavior r^2 near zero and approaching fast to zero when r tends to infinity. Thanks for any hints.
 
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Multiply two functions together where one behaves like ##r^2## near zero (how about ##r^2##) and the other behaves like 1 near zero and decays to zero more quickly than ##r^2## increases. After you get the basic second function you can manipulated it to make the product of the two functions fit other constraints you have.
 
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You want to pick the "second function" to make life easy for whatever you are doing. There are plenty of options, like
$$\frac{r^2}{1 + r^3}, \quad r^2 e^{-r}, \quad \text{etc.}$$
 
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FAQ: Find Function with r^2 Near Zero & Approaching 0

1. What does it mean when r^2 is close to zero?

When r^2 (the coefficient of determination) is close to zero, it means that there is little to no linear relationship between the two variables being compared. This means that there is no predictive power in the relationship and the data points are widely scattered.

2. How do you interpret r^2 when it is approaching zero?

As r^2 approaches zero, it indicates that there is no correlation between the variables being compared. This means that the changes in one variable do not cause or predict changes in the other variable.

3. Can r^2 ever be negative or greater than 1?

No, r^2 can never be negative or greater than 1. A negative r^2 value indicates that there is a negative correlation between the variables, while a value greater than 1 would suggest that there is a non-linear relationship between the variables.

4. Why is it important to consider r^2 when analyzing data?

r^2 is important to consider because it provides a measure of how well the data points fit a regression model. It helps to determine the strength and direction of the relationship between the variables being compared.

5. Is r^2 the only measure of a good fit for a regression model?

No, r^2 is not the only measure of a good fit for a regression model. Other measures, such as the root mean square error (RMSE) and the mean absolute error (MAE), should also be considered when evaluating the overall fit of a regression model.

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