FixedNoiseGaussianLikelihood with z-scored data

Howdy folks,

I am doing some sensor/data fusion using an exact single task regression GP. The targets come from various sensors, each of which has its own 1-sigma uncertainty. I am grouping the targets according to the sensors Y = [y1, y1, ... y1, y2, ...y2, .... yn, yn, ...., yn] for sensors 1 through n.

Then I am grouping my 1-sigma noises as Y_std = [d1, d1, ..., d1, d2, d2, ..., d2, ..., dn, dn, ... dn], where d1 is the 1-sigma noise of sensor 1, d2 the 1-sigma noise of sensor 2, etc.

Then I am z-scoring my data:

Y'= (Y-mean(Y))/std(Y)
X'= (X-mean(X))/std(X)

Then before passing Y_std**2 to the noise of the FixedNoiseGaussianLikelihood constructor, I obviously need to correctly adjust for the target z-scoring done on the targets.

The right way of doing this is the main premise of my thread and is what I am searching for.

I started out with z-scoring the sensor standard deviations using the mean and std of the target data

Y_std' = (Y_std-mean(Y))/std(Y)

However, this doesn’t feel right to me, and I get counter intuitive results when I experiment with different settings for the sensor noise values.

I changed this by only dividing the noise by the std of the targets :

Y_std'=Y_std/std(Y)

which appears to work a lot better and both noise and target is undergoing the same scaling.

z-scoring the sensor noise using it’s own mean and std didn’t feel like the right thing to do, as I was afraid I would loose the correct relationship to the target data (essentially I’d be scaling the targets and the sensor noise by different values which doesn’t feel right to me)

I guess I am asking what the right way is for adjusting the sensor noises before passing them off as variances in the FixedNoiseGaussianLikelihood?

Thanks

Galto

[Balandat]

If I use a standard GaussianLikelihood won’t I loose influence on how I weigh my sources?

Yes, that’s true. I guess Geoff is wondering to what extent that’s necessary (if the sensors aren’t all that different it may not be).

is s_n^2, i.e. the variance of the noise, the same noise we are talking about for the FixedNoiseGaussianLikelihood?

Yes, it is.

[Galto2000]

Thanks @Balandat

Gotcha, so yes, in my case, there are very good reasons to “weigh” one data source over another 😃 (or so, at least I think so).