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Title: ON COMPUTING UPPER LIMITS TO SOURCE INTENSITIES

Abstract

A common problem in astrophysics is determining how bright a source could be and still not be detected in an observation. Despite the simplicity with which the problem can be stated, the solution involves complicated statistical issues that require careful analysis. In contrast to the more familiar confidence bound, this concept has never been formally analyzed, leading to a great variety of often ad hoc solutions. Here we formulate and describe the problem in a self-consistent manner. Detection significance is usually defined by the acceptable proportion of false positives (background fluctuations that are claimed as detections, or Type I error), and we invoke the complementary concept of false negatives (real sources that go undetected, or Type II error), based on the statistical power of a test, to compute an upper limit to the detectable source intensity. To determine the minimum intensity that a source must have for it to be detected, we first define a detection threshold and then compute the probabilities of detecting sources of various intensities at the given threshold. The intensity that corresponds to the specified Type II error probability defines that minimum intensity and is identified as the upper limit. Thus, an upper limit is amore » characteristic of the detection procedure rather than the strength of any particular source. It should not be confused with confidence intervals or other estimates of source intensity. This is particularly important given the large number of catalogs that are being generated from increasingly sensitive surveys. We discuss, with examples, the differences between these upper limits and confidence bounds. Both measures are useful quantities that should be reported in order to extract the most science from catalogs, though they answer different statistical questions: an upper bound describes an inference range on the source intensity, while an upper limit calibrates the detection process. We provide a recipe for computing upper limits that applies to all detection algorithms.« less

Authors:
;  [1]; ;  [2];  [3];  [4];  [5]
  1. Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138 (United States)
  2. Department of Statistics, University of California, Irvine, CA 92697-1250 (United States)
  3. Eureka Scientific, 2452 Delmer Street, Suite 100, Oakland, CA 94602-3017 (United States)
  4. Department of Statistics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213 (United States)
  5. Physics Department, University of Crete, P.O. Box 2208, GR-710 03, Heraklion, Crete (Greece)
Publication Date:
OSTI Identifier:
21454971
Resource Type:
Journal Article
Journal Name:
Astrophysical Journal
Additional Journal Information:
Journal Volume: 719; Journal Issue: 1; Other Information: DOI: 10.1088/0004-637X/719/1/900; Journal ID: ISSN 0004-637X
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ALGORITHMS; ASTROPHYSICS; DATA ANALYSIS; MATHEMATICAL SOLUTIONS; PROBABILITY; MATHEMATICAL LOGIC; PHYSICS

Citation Formats

Kashyap, Vinay L, Siemiginowska, Aneta, Van Dyk, David A, Jin, Xu, Connors, Alanna, Freeman, Peter E, and Zezas, Andreas. ON COMPUTING UPPER LIMITS TO SOURCE INTENSITIES. United States: N. p., 2010. Web. doi:10.1088/0004-637X/719/1/900.
Kashyap, Vinay L, Siemiginowska, Aneta, Van Dyk, David A, Jin, Xu, Connors, Alanna, Freeman, Peter E, & Zezas, Andreas. ON COMPUTING UPPER LIMITS TO SOURCE INTENSITIES. United States. https://doi.org/10.1088/0004-637X/719/1/900
Kashyap, Vinay L, Siemiginowska, Aneta, Van Dyk, David A, Jin, Xu, Connors, Alanna, Freeman, Peter E, and Zezas, Andreas. 2010. "ON COMPUTING UPPER LIMITS TO SOURCE INTENSITIES". United States. https://doi.org/10.1088/0004-637X/719/1/900.
@article{osti_21454971,
title = {ON COMPUTING UPPER LIMITS TO SOURCE INTENSITIES},
author = {Kashyap, Vinay L and Siemiginowska, Aneta and Van Dyk, David A and Jin, Xu and Connors, Alanna and Freeman, Peter E and Zezas, Andreas},
abstractNote = {A common problem in astrophysics is determining how bright a source could be and still not be detected in an observation. Despite the simplicity with which the problem can be stated, the solution involves complicated statistical issues that require careful analysis. In contrast to the more familiar confidence bound, this concept has never been formally analyzed, leading to a great variety of often ad hoc solutions. Here we formulate and describe the problem in a self-consistent manner. Detection significance is usually defined by the acceptable proportion of false positives (background fluctuations that are claimed as detections, or Type I error), and we invoke the complementary concept of false negatives (real sources that go undetected, or Type II error), based on the statistical power of a test, to compute an upper limit to the detectable source intensity. To determine the minimum intensity that a source must have for it to be detected, we first define a detection threshold and then compute the probabilities of detecting sources of various intensities at the given threshold. The intensity that corresponds to the specified Type II error probability defines that minimum intensity and is identified as the upper limit. Thus, an upper limit is a characteristic of the detection procedure rather than the strength of any particular source. It should not be confused with confidence intervals or other estimates of source intensity. This is particularly important given the large number of catalogs that are being generated from increasingly sensitive surveys. We discuss, with examples, the differences between these upper limits and confidence bounds. Both measures are useful quantities that should be reported in order to extract the most science from catalogs, though they answer different statistical questions: an upper bound describes an inference range on the source intensity, while an upper limit calibrates the detection process. We provide a recipe for computing upper limits that applies to all detection algorithms.},
doi = {10.1088/0004-637X/719/1/900},
url = {https://www.osti.gov/biblio/21454971}, journal = {Astrophysical Journal},
issn = {0004-637X},
number = 1,
volume = 719,
place = {United States},
year = {Tue Aug 10 00:00:00 EDT 2010},
month = {Tue Aug 10 00:00:00 EDT 2010}
}