An exponential regression is the process of finding the exponential function that fits best for a given set of data. We consider a function y = exp(a + bx), where parameters a and b are to be found in such a way that this function is the best approximation of the data. Enter the set of x and y coordinates of the input points in the appropriate fields of the Exponential Regression Calculator and calculate the regression function parameters. You can paste the data copied from a spreadsheet or csv-file or enter manually using comma, space or ‘enter’ as separators.
Exponential regression equation
The exponential regression is a form of nonlinear regression analysis, in which observational data are modeled by an exponential function. In particular, we consider the following exponential model:
$$y={ e }^{ (a+bx) }.$$
Taking the natural log of both sides of the equation, we have the following equivalent equation:
$$ln(y)=a+bx.$$
This equation has the form of a linear regression model, so we can apply a linear least squares method. The linear least squares regression line method is a mathematical procedure for finding the best-fitting straight line to a given set of points by minimizing the sum of the squares of the offsets of the points from the approximating line.
The condition for the sum of the squares of the offsets to be a minimum is that the derivatives of this sum with respect to the approximating line parameters are to be zero. These lead to the set of two linear equations with two variables. The solution to this system gives us the parameters \(a\) and \(b\):
$$b=\frac { n\sum _{ i=1 }^{ n }{ { x }_{ i } } ln( { y }_{ i })-\sum _{ i=1 }^{ n }{ { x }_{ i }\sum _{ i=1 }^{ n }{ ln( { y }_{ i } )} } }{ n\sum _{ i=1 }^{ n }{ { x }_{ i }^{ 2 }-{ \left( \sum _{ i=1 }^{ n }{ { x }_{ i } } \right) }^{ 2 } } }, $$
$$a=\frac { \sum _{ i=1 }^{ n }{ln( { y }_{ i } )} -b\sum _{ i=1 }^{ n }{ { x }_{ i } } }{ n }.$$
Our exponential regression calculator also calculates the correlation coefficient of the input data. The correlation coefficient measures the strength of exponential relationship between two variables and thus gives the quality of a least squares fitting to the original data.
The correlation coefficient \(r\) (sometimes also denoted \(R\)) is defined by the formula:
$$r=b\frac { \sqrt { n\sum _{ i=1 }^{ n }{ { x }_{ i }^{ 2 } } -(\sum _{ i=1 }^{ n }{ { x }_{ i })^{ 2 } } } }{ \sqrt { n\sum _{ i=1 }^{ n }{ (ln({ y }_{ i }))^{ 2 } } -(\sum _{ i=1 }^{ n }{ln( { y }_{ i }))^{ 2 } } } } .$$
The correlation coefficient has the following characteristics:
1. The range of \(r\) is between -1 and 1, inclusive.
2. If 0.7 ＜ |\(r\)| ≤ 1 the data points are in strong correlation.
3. If 0.4 ＜ |\(r\)| ≤ 0.7 the data points are in moderate correlation.
4. If 0.2 ＜ |\(r\)| ≤ 0.4 the data points are in weak correlation.
5. If 0 ≤ |\(r\)| ≤ 0.2 the data points are in no correlation.
Related calculators
Check out our other statistics calculators such as Linear Regression Calculator or Standard Deviation Calculator.