---
abstract: |
    We consider the problem of estimating a spatially varying density function, motivated by problems that arise in large-scale radiological survey and anomaly detection. In this context, the density functions to be estimated are the background gamma-ray energy spectra at sites spread across a large geographical area, such as nuclear production and waste-storage sites, military bases, medical facilities, university campuses, or the downtown of a city. Several challenges combine to make this a difficult problem. First, the spectral density at any given spatial location may have both smooth and non-smooth features. Second, the spatial correlation in these density functions is neither stationary nor locally isotropic. Finally, at some spatial locations, there is very little data. We present a method called multiscale spatial density smoothing that successfully addresses these challenges. The method is based on recursive dyadic partition of the sample space, and therefore shares much in common with other multiscale methods, such as wavelets and Pólya-tree priors. We describe an efficient algorithm for finding a maximum a posteriori (MAP) estimate that leverages recent advances in convex optimization for non-smooth functions.

    We apply multiscale spatial density smoothing to real data collected on the background gamma-ray spectra at locations across a large university campus. The method exhibits state-of-the-art performance for spatial smoothing in density estimation, and it leads to substantial improvements in power when used in conjunction with existing methods for detecting the kinds of radiological anomalies that may have important consequences for public health and safety.

    Key words: radiological survey, density estimation, spatial statistics, Bayesian nonparametrics, total-variation denoising, fused lasso
author:
- 'Wesley Tansey[^1]'
- 'Alex Athey[^2]'
- 'Alex Reinhart[^3]'
- 'James G. Scott[^4]'
bibliography:
- 'spatial\_density\_multiscale.bib'
date: 'This version: '
title: 'Multiscale spatial density smoothing: an application to large-scale radiological survey and anomaly detection'
---

Detecting radiation anomalies {#sec:introduction}
=============================

Overview of approach {#sec:preliminaries}
====================

Spatial smoothing via graph-based denoising {#sec:spatial_smoothing}
===========================================

Simulations {#sec:simulations}
===========

Radiological survey and anomaly detection at UT-Austin {#sec:anomaly_example}
======================================================

Conclusions {#sec:conclusions}
===========

#### Acknowledgements.

The authors thank Patrick Vetter of the UT Applied Research Laboratories for his assistance with the pilot studies described here; the University of Texas Police Department for their ongoing collaboration with data collection; and Ryan Tibshirani of CMU for sharing his expertise on algorithms for the graph-fused lasso.

Pre-processing {#app:remarks}
==============

Protocol for estimating empirical spectra {#app:protocol}
=========================================

Further details of Bayesian method {#app:bayes}
==================================

[^1]: Department of Computer Science, University of Texas at Austin.

[^2]: Applied Research Laboratories, University of Texas at Austin.

[^3]: Department of Statistics, Carnegie Mellon University, <areinhar@stat.cmu.edu> (corresponding author).

[^4]: Department of Information, Risk, and Operations Management; Department of Statistics and Data Sciences, University of Texas at Austin.
