Ⅱ-4. Hadron Physics

RIKEN Accel. Prog. Rep. 47 (2014)

The Application of Gaussian Process Regression to Background
Spectrum Modeling at PHENIX
J. Seele,∗1 C. Gal,∗2 and A. Deshpande∗1,∗2in p+p collisions4) . This is a steeply falling spectrum
that isn’t modeled well by using power laws or exponentials. This particular problem requires the extension of the Gaussian process technique to multiscale problems, which required transforming the data3)
and applying the Gaussian process regression in thiswarped space. Our current application of Gaussian
processes to modeling this spectrum can be seen in
figure 2.

Often in nuclear and particle physics, we need to estimate the area under a peak which sits over an oddly
shaped background (see figure 1). Occasionally we areable to perform an analytical first principles calculation to calculate a shape for the background, but more
often we are faced with little or no information about
what the shape of the background should be. Facedwith this, we typically choose a polynomial and fit thisfunctional form to the background. The choice of thispolynomial and its fitting leaves an unquantified uncertainty.

Fig. 1. A sample background (sampled from a known thirdorder polynomial) and Gaussian peak spectrum.

Gaussian processes1,2) are a mathematical conceptthat allow for a method of data regression and predictive functional modeling using a minimal set of priorassumptions. A nice feature of this method is thatsimultaneously with the predictions, uncertainties areprovided.
Gaussian processes are a specific type of stochastic
process where the variance of each of the random variables comprising the process is Gaussian. An important feature of stochastic processes is that, mathematically, they sample over the space of functions similar
to how a random variable samples over a set spaceof possible outcomes. In defining the variances to beGaussian, we’ve narrowed down the space of possiblefunctions and simplified the math needed to specify theprocess. Specifically, with this requirement of Gaussian uncertainties, the expectation of the process canbe defined entirely in terms of a mean function and acovariance function similar to how the Gaussian distribution is defined purely by a mean and a variance.
The specific background spectrum that we are interested in applying this technique to is the backgroundsitting beneath the Jacobian peak, in W ± production∗1∗2Fig. 2. The W Jacobian peak and background spectra. The
black line and blue band represent the Gaussian process
best fit and uncertainty band.

The uncertainty band currently encompasses a large
space of functions, many of which we don’t expect
to be physical (e.g. an undulating, falling spectrum).
We are currently working to add shape constraints tothe Gaussian process modeling by sampling individual, though coarsely grained, functions from the constrained space and then accepting or rejecting thosefunctions based on their individual shapes.
References
1) MacKay, David J. C.: Information Theory, Inference,and Learning Algorithms (Cambridge University Press,2003)
2) Rasmussen, C. E. & Williams, K. I.: Gaussian Processes
for Machine Learning (The MIT Press, 2006)
3) Snelson, E., Rasmussen, C. E., Ghahramani, Z.:Warped Gaussian Processes, NIPS, 2003.
4) Adare, et al. : Phys.Rev.Lett. 106 (2011) 062001

RIKEN Nishina Center
Department of Physics, Stony Brook University- 95 -