# Gaussian Process Regression With Multiple Inputs?

Is it possible to use a Gaussian Process to relate multiple independent input variables (X1, X2, X3) to an output variable (Y)?

More specifically, I would like to produce a regression graph like the example shown below where confidence interval reduces around clusters of data (i.e. variance is high at x = 1 where there is no data, but x = 0.3 the regression is tight due to the clustering of input variables) and Instead of having one input variable on the x-axis, where would be multiple inputs. For example, is it possible to develop a regression relationship that relates the price of houses (HPrice = [125000, 63000, 500000]) to Floor Area (FArea = [856,497,1300]) and the Number of Bedrooms (BedR = [2,2,4])?

Ideally I would like to do this in R, and wonder if there are any recommendations/example available?

Thanks!

• Could you please explain what you have in mind by "using a Gaussian process" and how that would differ from multiple regression? – whuber Feb 27 '14 at 18:58
• Thanks @whuber - I've updated my post so that it is more specific. – hoof_hearted Feb 27 '14 at 19:42

## 1 Answer

Yes, it is entirely possible. Recall the Gaussian Process model is defined by a kernel function, $K(\bf x,\bf x')$, yours is a case where you need some function that exploits the vector $\bf x$ in an appropriate way.

The graphs in books tend to be univariate input (not always) just because it's straightforward to see what's going on. This is an example of one draw from a GP where the inputs are coordinates $\bf{x}$$=(x_1,x_2)$ on a $30 \times 30$ grid with a kernel I hoped would induce a spatial relationship. I was reading Bishop 2006 at the time and found it helpful if you want more on the theory - which covers the regression also.

• Could you provide the actual link to Bishop 2006? I'm unable to find this resource. Could you give some pointers on what adaptations are necessary for multi-input GP? – Yu Chen Aug 26 '17 at 23:21
• The full citation for the book is: Bishop, Christopher M. Pattern recognition and machine learning. Springer, 2006. There is probably a pdf floating around somewhere. You might also want to look at the site: gaussianprocess.org/gpml which has more info online. – conjectures Aug 29 '17 at 10:59
• Would you happen to be able to share a test example of code in Python that produces the plot above? – Mathews24 Nov 22 '18 at 6:02