Determine if a set of data is from a linear or logarithmic function?

Question

I have a set of data points and am curious if the data represents a linear function or a logarithmic function.

The data set is 2 dimensional.

Let's say an ideal set of data points followed the function f(x) = x. If I plotted the data point I would be able to tell it is linear.

Similarly if the data points followed the function f(x) = log(x), I would be able to visually tell it is logarithmic.

On the other hand, having the program determine if a set of data is linear or logarithmic is nontrivial. How would I approach this?

Answer 1

One option would be to do a linear regression on the data set to get a best-fit line. If the data is linear, you'll get a very good fit and the mean squared error should be low. Otherwise, you'll get an okay fit and a reasonable error.

Alternatively, you could consider transforming the data set by converting each point (x₀, x₁, ..., x_n, y) to (x₀, x₁, ..., x_n, e^y). If the data was linear, now it will be exponential, and if the data was logarithmic, now it will be linear. Running a linear regression and getting the mean-squared error now will have a low error for the logarithmic data and a staggeringly huge error for the linear data, since the exponential function blows up extremely quickly.

To actually implement the regression, one option would be to use a least-squares regression. This would have the added benefit of giving you a correlation coefficient in addition to the model, which could also be used to distinguish between the two data sets.

Because you've asked for how to do this in Java, a quick Google search turned up this Java code to do a linear regression. However, you might have a better fit in a language like Matlab that is specifically optimized to do these sorts of queries. For example, in Matlab, you can do this regression in one line of code by writing

linearFunction = inputs / outputs

Hope this helps!