Distributional Tests for LIGO Detections
- 1. Sophia G. Schwalbe Michele Zanolin, Marek Szczepanczyk Undergraduate Research Symposium – Embry-Riddle Aeronautical University, Prescott 30 October 2015 Distributional Tests for LIGO Detections
- 2. Overview • Purpose • Definition of Distributional Tests • Examples of Distributional Tests • Example Populations for LIGO
- 3. Purpose • Two types of triggers • Foreground: obtained from the detector data • Background: obtained from detectors after time shifts are applied • Ranked with respect to quantities related to energy (Signal-to-Noise Ratio) • Current detection relies on only the loudest event in the foreground (by calculating its False Alarm Probability using the background triggers) • Want to include possible weaker events. • i.e. low energy events
- 4. Distributional Tests • Distributional tests compare a distribution of events with respect to a reference distribution and produce a probability that the distribution of events is compatible with the reference • There are two kinds of distributional tests: • Parametric: the reference distribution has a known analytical shape • Non-parametric: the reference distribution is determined with background runs • In LIGO, we would like to use non-parametric tests
- 5. Non-Parametric Distributions • Non-parametric distributions do not assume shape or parameters. • Parametric assume a shape (ex. to be normal) or parameter. • Examples: • Kolmogorov-Smirnov • Mann-Whitney • Chi-squared • Binomial
- 6. Kolmogorov-Smirnov • The maximum distance between background and foreground distributions is found • If D is small, likely to be same distribution • If D is large, likely to be two different distributions • Probability distribution of the possible values of D is found to find probability of having a value larger than observed • If consistent regardless of distribution, then distributions are same (foreground is background, no gravitational waves) • If probability of a value of D larger than the observed is small, not from same distribution
- 7. Kolmogorov-Smirnov • Consider two unknown distributions SN1 (x) and SN2 (x) that are the fractions of the original distributions equal to or below a given value of x • Sizes of the distributions are N1 and N2, respectively • x: measure of the amplitude of the event • Null Hypothesis H0 : SN1 = SN2 for -∞<x<∞ • Computes the maximum distance over x between distributions • The probability of D being greater than the observed value is given as P • QKS: compliment of the cumulative distribution function (cdf) for D • Ne: average size of the distributions • P: derived from an approximation of the probability of D for the upper tail of the cdf to replace the tables for confidence and constants • λ: constant chosen from confidence level • If less than a given percentage, the null hypothesis is rejected
- 8. Kolmogorov-Smirnov • To chose cdf of D from two unknown distributions, consider rank • Combining the distributions and ordering the events by a given parameter “ranks” the events from 1 to N1+N2 • Can normalize rank by dividing individual rank by N1+N2 to give a value from 0 to 1 • Assuming distributions are equivalent, the probability distribution of the normalized rank will be level • Level probability distributions are known • Since area under level probability distribution is same as area under cdf of D, the cdf of D can be written in terms of the probability distribution of the normalized rank • 𝑑𝑦 𝑑𝑥 is the Jacobian of y with respect to x • py(y)=probability of D • px(x)=probability of rank
- 9. Mann-Whitney • First combines two distributions (foreground and background) and ranks the events from lowest to highest with respect to parameter • Assumes that if it is the same distribution, all the events will be dispersed uniformly instead of concentrated at smaller/larger values • Then computes sum of ranks for each distribution • If sum deviates too much from the mean value of the ranks, then the distributions are different
- 10. Symmetric Chi-Squared • Unlike Kolmogorov-Smirnov and Mann-Whitney, orders data into bins • Compares two data populations • Computes the χ2 statistic based on the number of events in the given bin • The probability for having a χ2 statistic greater than the observed value is found • If less than a given percentage, not from the same distribution
- 11. Asymmetric Chi-Squared • Compares a data set (foreground) with a known distribution (background) • Computes the χ2 statistic based on the number of events in the foreground and expected number of events in the background • Then computes probability of having a χ2 statistic greater than the observed • Gives a higher χ2 statistic than the symmetric, so gives a higher chance of distinguishing distributions
- 12. Populations • Same amplitude repeaters • Assumes all the triggers have the same parameter • Background-like • Distribution is exponential under logarithmic scale, where exponential constant is correlated to that of the noise (background) triggers • Galactic center • Distribution mimics that of distribution of same transients but random polarizations located in galactic center would produce • Power law decay • Assumes distributions of parameter proportional to negative integer powers
- 13. Summary • Using different distributional tests, we can find which test is most sensitive to different populations of signals • Want to use non-parametric tests instead of parametric in order to reduce assumptions on the distributions • Some possible non-parametric tests include Kolmogorov- Smirnov, Mann-Whitney, and chi-squared • Next, need to analyze the difference in efficiency between parametric and non-parametric tests for LIGO
- 14. References