Selection Procedures In Identifying EM Fields Following Log-normal Distributions 1, 2 Pinyuen Chen Interdisciplinary Statistics Program Syracuse University 1. Joint work with Elena Buzaianu and Tiee-Jian Wu 2. Presented at Cheng Kung University on August 11, 2011

Preface “Statistics is a subject of amazingly many uses and surprisingly few effective practitioners. The traditional road to statistical knowledge is blocked, for most, by a formidable wall of mathematics. Our approach here avoids that wall.” --- Efron and Tibshirani (1994) “Sometimes statisticians design new techniques by applying mathematical theories; other times they try to find the theoretical basis for an empirically correct method. The beauty of the field is that one seeks the unification of theoretical validity and empirical usefulness.” --- Hua Tang, 1997 Gertrude Cox Scholar

Abstract: We use selection methodology to characterize multiple targets in electromagnetic (EM) fields. Previous research concluded that the observed EM field power fluxes and cable powers follow either a chi-square distribution with two degrees of freedom or a log-normal distribution. That is, such EM field can be characterized by either an exponential distribution with mean μ or a log-normal distribution with parameters μ and σ. These cases exist also in a far field as a result of what the aircraft returns, while the distribution exists due to multiple reflecting surfaces instead of internal EM fields. In this paper, we propose subset selection procedures to identify EM fields which follow log-normal distributions with common σ. We illustrate the properties of our proposed procedures by numerical examples and we give simulation examples to demonstrate the procedures. The primary application of this research lies in the area of electromagnetic vulnerability (EMV) although the statistical theory developed in this article is as general as any other mathematical tools and the results can be applied to any electromagnetic problems in radar, sonar, and navigation systems.

Overview Introduction The Problem Single-Stage Procedures Two-Stage Procedure Sample Size Determination Simulation Results Concluding Comments

Introduction Indifference Zone Approach vs. Subset Selection Approach Have k experimental populations and a standard/control population with same distribution functions differing only in their unknown (location) parameter. Goal: Develop selection procedures to identify experimental populations that are “matching” or are “close to” the standard/control * Indifference Zone Approach (IZA) versus Subset Selection Approach (SSA): ** IZA selects a fixed size subset containing populations that are “matching” or are “close to” the standard/control. It is used more in the design of experiment. ** SSA selects a random size subset containing all populations that are “matching” or are “close to” the standard/control. It is used more in data analysis.

Introduction Motivation of the Work Statistical Characterization of Aircraft Electromagnetic Vulnerability (EMV) EMV: the characteristics of a system that cause it to suffer a definite degradation (incapability to perform the designated mission) as a result of having been subjected to certain level of electromagnetic environmental effects. (From Dictionary of Military and Associated Terms, US Department of Defense.) Applicable areas: Sensors, Electronics, Battle space environment RF components EMV measurements on aircrafts are obtained by: - using many different aspect angles of the field transmitting antenna - using mode-stirred techniques - …

Introduction The above work was the result of a Call for Proposal by Office of Navy Research: Title: Stochastic Characterization of Naval Aircrafts Electromagnetic Vulnerability Objective: Develop mathematical tools capable of characterizing the electromagnetic fields within naval aircraft and the associated currents on avionic systems and their interconnecting cables in the operational electromagnetic environment. A key component of the tool is its ability to quantify the results in a stochastic sense in order to facilitate weapon system performance risk assessments. Private Sector Use of Technology: The technology developed under the topic has direct impact to a wide variety of commercial EMC ( Electromagnetic Compatibility 電磁兼容性 ) and EMI Electromagnetic Interference 電磁干擾 ) problems.

Introduction Earlier Work: EM fields inside overmoded cavities projected over any axis follow either an exponential distribution or a log-normal distribution Holland, R. and St. John, R. “Statistical Electromagnetic” 1999, Taylor & Francis, Philadelphia, PA. (This book addresses the problem of treating interior responses of complex electronic enclosures or systems, and presents a probabilistic approach.) Recent Work: Exponential Distribution Case Chen, P., Osadciw, L., and Wu, T. J. (2010) Multiple Targets Characterization of Electromagnetic Vulnerability,Signal Processing, 90, Log-normal Distribution Case Buzaianu, E., Chen, P., and Wu, T. J. (2011) Subset Selection Procedures to Identify Electromagnetic Fields Following Lognormal Distributions, IET Radar Sonar Navigation, 5,

The problem

The Problem

Single-Stage Procedures Case 1: σ is known, µ 0 is known are iid following a noncentral Chi-square distribution with df = 1 and noncentrality parameter Procedure R 1 :

Single-Stage Procedures.

Case 2: σ is unknown, µ 0 is known (PQD was first introduced by Lehmann (1966) Some concepts of dependence. Annals of Mathematical Statistics, 37, Recent work in finance and risk management emphasizes the importance of PQD. )

Single-Stage procedures Procedure R 2 :

Single-Stage Procedures When µ 0 is unknown, we need to estimate µ 0. are not independent. are not PQD. Case 3: σ known, µ 0 unknown Procedure R 3 :

Single-Stage Procedures

Two-Stage Procedure Case 4: σ is unknown, µ 0 is unknown Procedure R 4 : Stage I: Stage II: For pre-assigned δ* and c 4, find n and h: Decision Rule:

Two-Stage Procedure

Sample Size Determination Procedure R 1

Sample Size Determination Procedure R 2 :

Sample Size Determination Procedure R 3

Sample Size Determination Procedure R 4

Sample Size Determination Procedure R 4 :

Sample Size Determination Procedure R 4 :

Simulation Results Example 1 (for R 1 and R 2 ):

Simulation Results Example 1 (for R 1 and R 2 ):

Simulation Results Example 2 (for R 3 ):

Simulation Results Example 3 (for R 4 )

Simulation Results Example 3 (for R 4 )

Concluding Comments The distribution of the observed EM fields resulting from clutter power fluxes and cable powers is either a chi-square distribution with two degrees of freedom or a log-normal distribution. We propose subset selection procedures to select among electromagnetic fields, whose characteristics can be modeled by lognormal distributions, the ones that match an existing standard or a reference. Simulation studies show that our procedures are reliable, i.e. the simulated probability of correct selection attains the specified level. Ranking and selection theory can be a useful tool in dealing with the stochastic characteristics and the comparison of electromagnetic fields.