Overview of NonParametric Probability Density Estimation Methods Sherry

Overview of Non-Parametric Probability Density Estimation Methods Sherry Towers State University of New York at Stony Brook

All kernal PDF estimation methods (PDE’s) are developed from a simple idea… If a data point lies in a region where clustering of signal MC is tight, and bkgnd MC is loose, the point is likely to be signal S. Towers

® To estimate a PDF, PDE’s use the idea that any continuous function can be modelled by sum of some “kernal” function ® Gaussian kernals are a good choice for particle physics ® So, a PDF can be estimated by sum of multi -dimensional Gaussians centred about MC generated points S. Towers

® Best form of Gaussian kernal is a matter of debate: ® Static-kernal PDE method uses a kernal with covariance matrix obtained from entire sample ® The Gaussian Expansion Method (GEM), uses an adaptive kernal; the covariance matrix used for the Gaussian at each MC point comes from “local” covariance matrix. S. Towers

S. Towers

GEM vs Static-Kernal PDE ® GEM gives unbiased estimate of PDF, but slower to use because local covariance must be calculated for each MC point ® Static-kernal PDE methods have smaller variance, and are faster to use, but yield biased estimates of the PDF S. Towers

Comparison of GEM and static-kernal PDE: S. Towers

PDE vs Neural Networks ® Both PDE’s and Neural Networks can take into account non-linear correlations in parameter space ® Both methods are, in principle, equally powerful ® For most part they perform similarly in an “average” analysis S. Towers

PDE vs Neural Networks ® But, PDE’s have far fewer parameters, and algorithm is more intuitive in nature (easier to understand) S. Towers

Plus, PDE estimate of PDF can be visually examined: S. Towers

PDE’s vs Neural Nets… ® There are some problems that are particularly well suited to PDE’s: S. Towers

PDE’s vs Neural Nets… S. Towers

PDE’s vs Neural Nets… S. Towers

PDE’s vs Neural Nets… S. Towers

Summary ® PDE methods are as powerful as neural networks, and offer an interesting alternative ® Very few parameters, easy to use, easy to understand, and yield unbinned estimate of PDF that user can examine in the multidimensional parameter space! S. Towers