Converting Power Law Distribution to a Linear graph

Whenever we are working on ML projects, we have to deal with the high dimensionality of the dataset. There is a very particular term to refer here i.e. Curse of dimensionality. Although there are many ways to handle this problem one of the solutions can be to change the coordinate systems. This solution is somewhat interesting and new from a data scientist perspective.
When working on model training we sort of try to remove the linearity of the hypothesis to increase the accuracy of the model. But here in this article, we are trying to understand how to visualise the data easily and can understand it with the help of simple yet innovative maths.

Given below is the hypothesis:

  f(x)=b x^{-a}

Power Law is a very important concept in statistics and gives information about two variables. And these two variables are relatively proportional to each other which means that the change in quantity of one variable will reflect the change in the quantity of other variables. One quantity varies as a power of another.
Now lets’s solve this equation with the help of natural logarithm (ln).

Power Law Graph


  y=b x^{-a}
  \ln (y) = \ln \left(b x^{-a}\right)

Using the logarithm property,

  \ln y=\ln (b)-a \ln (x)

Now, lets assume ln(y) to be \widehat{y} and ln(x) to be \widehat{x}. Substituting the values in the the equation above and then we have the equation.
\hat{y}=-a \hat{x}+\ln (b)

Linear Graph from Power Law Graph


So now, it is very evident that the hypothesis is now been converted to a linear equation. And this will make our task easy to analyse how the parameters are affecting each other.

Attention geek! Strengthen your foundations with the Python Programming Foundation Course and learn the basics.

To begin with, your interview preparations Enhance your Data Structures concepts with the Python DS Course.

My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.