P.Mean: Exponential interpolation (created 2010-02-11).

Someone wanted an exponential interpolation formula. It's not quite a statistics question, but it caught my interest. Any interpolation starts with a range for x and for y. Let x be between a and b. Let y be between c and d. In a linear interpolation, an x-value halfway between a and b produces a y value halfway between c and d. The formula for any x would be

y = (x-a) / (b-a) * (d-c) + c

The exponential interpolation assumes a multiplicative relationship throughout the range. If x is halfway between a and b, then y has to be a factor of the square root of d/c between c and d. The formula  for any x would be

y = c * (d/c)^((x-a) / (b-a))

Note that both terms have a factor

(x-a) / (b-a)

which shows the proportion of the distance from a to b. In an exponential model, you use this proportion as an exponent. Here's a plot with circles showing the exponential interpolation and X's showing the linear interpolation.

This work is licensed under a Creative Commons Attribution 3.0 United States License. This page was written by Steve Simon and was last modified on 2010-04-01. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Statistical computing.