Selection Procedures In Identifying EM Fields
Following Log-normal Distributions1, 2
Pinyuen Chen
Interdisciplinary Statistics Program
Syracuse University
Joint work with Elena Buzaianu and Tiee-Jian Wu
2.Presented at Cheng Kung University on August 11, 2011
1.
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, 344-351.
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, 458-465.
The problem
 i (i  1, 2,, k) follows a lognormal distribution with parameters i and 2 denoted (i, 2).
 1
(ln x  i ) 2
exp( 
)

2

x
2




f ( x)  
0



 i is equivalent to  0 if i  (
x0
x0
i  0 2
)   * , where  *  0 is pre-specified

When  i is equivalent to  0 ,  i is labeled to be good
The problem: Select a subset containing all the good populations (EM fileds)
The Problem
We assume that there is at least one good target clutter
Correct Selection (CS): select a subset containing all good EM fields
Requirement: P(CS | At least one good)  P* , where 1/(2 k  1)  P*  1 is pre-assigned
We only consider the case where  i ( i  0,1,..., k ) are all equal to 
1 ni
1 ni
Let Yi   j 1Yij   j 1 ln X ij .
ni
ni
Single-Stage Procedures
Case 1: σ is known, µ0 is known
Di  ni (Yi   0 ) 2 /  2 .
Di  Z i2 , where Z i  (Yi   0 ) /( / ni ) ~ N ( ni (  i   0 ) /  ,1)
Di are iid following a noncentral Chi-square distribution with df = 1 and noncentrality parameter ni i .
Procedure R1:
Select  i if Di  ni c1 (  (Yi   0 ) 2 /  2  c1 ), where c1 is to be determined.
.
Single-Stage Procedures
P(CS | R1 )  P( Di  ni c1 ; i  G )
  G1,ni i (ni c1 )
iG
  G1,n  * (ni c1 )
iG
i 1
k
  G1,n  * (ni c1 )
i 1
i 1
G p , ( x) is the c.d.f of non -central chi -square with p degrees of freedom and non -centrality
parameter 
If n1  n2  ...  nk  n , then the lower bound is PLNSK (c1 , n,  1* , k )  [G1,n * (nc1 )] k
1
c1  G1,n1 * ( k P * ) / n .
1
Single-Stage Procedures

Case 2: σ is unknown, µ0 is known
S 2  i 1 (ni  1)S i2 /( N  k ) , S i2   j 1 ( X ij  X i )( X ij  X i )' /( ni  1) N  i 1 ni .
k
Fi  ni (Yi   0 ) 2 / S 2
ni
k
Fi  (Vi / 1) /(W /( N  k ))
Vi is the non-central chi-square with 1 degree of freedom and the non-centrality parameter ni i
W is the central chi-square with ( N  k ) degrees of freedom and is the same for every Fi .
V1 , …, V k , and W are mutually independent
F1 , ..., Fk are positively quadrant dependent (PQD), they satisfy Šidák’s inequality:
P( F1   1 ,..., Fk   k )  i 1 P( Fi   i ) , ( 1 , ... ,  k )  R k .
k
(PQD was first introduced by Lehmann (1966) Some concepts of dependence. Annals of Mathematical
Statistics, 37, 1137-1153. Recent work in finance and risk management emphasizes the importance of
PQD. )
Single-Stage procedures
2
2
Procedure R2: Select  i if Fi  ni c 2 (  (Yi   0 ) / S  c 2 ), where c2 is to be determined.
P (CS | R2 )  P ( Fi  ni c2 ; i  G )
  F1, N k ,nii (ni c2 )
iG
  F1, N k ,n  * (ni c2 )
iG
i 2
  F1, N k ,n  * (ni c2 )
iK
i 2
F p ,v , is the c.d.f. of the non -central F with degrees of freedom p and v and the non -centrality
parameter 
If n1  n2  ...  nk  n , then the lower bound is PLNSUK (c 2 , n,  2* , k )  [ F1,k ( n1),n (nc2 )] k
*
2
c2  F1,k1( n 1), n * ( k P * ) / n
2
Single-Stage Procedures
When µ0 is unknown, we need to estimate µ0.
Di  ni n0 (Yi  Y0 ) 2 /[( ni  n0 ) 2 ] are not independent.
Fi  ni n0 (Yi  Y0 ) 2 /[( ni  n0 ) S 2 ] are not PQD.
Case 3: σ known, µ0 unknown
Procedure R3:
Single-Stage Procedures
Single-Stage Procedures
Two-Stage Procedure

Case 4: σ is unknown, µ0 is unknown
Redefine the characteristic of  i to be equivalent to  0 if
  *  i  0   *
where
 *  0 is pre-specified.
Procedure R4:
Stage I:
Stage II:
For pre-assigned δ* and c4, find n and h:
n
Take n  n 0 more observations from each population and calculate Yi   j 1Yij / n
Decision Rule:
If | Yi  Y0 | c 4 (or  c 4  Yi  Y0  c 4 ), then select EM field  i .
Two-Stage Procedure
Suppose there are q good EM fields in the good subset G
P(CS | R4 )  P(c 4  Yi  Y0  c 4 ; i  G )
 P((
Y0   0
/ n

  [ ( x 


  [ ( x 

)
n

n

n

(c 4   i   0 ) 
(c 4   4* ))  ( x 
(c 4   4* ))  ( x 
n

n

Yi   i
/ n
(
Y0   0
/ n
)
(c 4   4* )] q  ( x)dx
(c 4   4* )] k  ( x)dx
n

(c 4   i   0 ); i  G )
Two-Stage Procedure
Let
n  hS n0 / d  n  h 2 S n20 / d 2 , where d  c 4   3* , then

P(CS | R4 )   [ ( x  h

  

  [ ( x  h
0 
S n0

)  ( x  h
S n0

)] k  ( x)dx
y )   ( x  h y )] k  ( x) f N 0 ( y )dxdy
where f N 0 ( y ) is the p.d.f of  N2 0
The minimum sample size: n  max{ h 2 S n20 / d 2  1, n 0 } , where h  h(k , n 0 , P * ) is such that

PLNCUK 

*
k
[

(
x

h
y
)


(
x

h
y
)]

(
x
)
f
0 ( y ) dxdy = P

N
0 
Sample Size Determination
Procedure R1
Given  1* , k and c1 , we determine the smallest n such that [G1,n * (nc1 )] k  P *
1
Table 1 Minimum sample size n required for Procedure R1
k
2
3
4
5
P*
 1*
0.5
1
0.5
1
0.5
1
0.5
1
(1+0.2)  1
*
585
293
726
363
830
415
911
456
(1+0.4)  1
159
80
197
99
225
113
248
124
(1+0.2)  1
*
839
420
988
494
1096
548
1181
591
(1+0.4)  1
228
114
269
135
448
249
321
161
c1
0.9
*
0.95
*
Sample Size Determination
Procedure R2:
Given  2* , k and c2 , we determine the smallest n such that the [ F1,k ( n 1),n * (nc 2 )] k  P *
2
Table 2 Minimum sample size n required for Procedure R2
k
2
3
4
5
P*
 2*
0.5
1
0.5
1
0.5
1
0.5
1
(1+0.2)  2
*
676
383
801
438
893
479
967
512
(1+0.4)  2
*
189
109
222
123
246
134
266
142
(1+0.2)  2
*
968
548
1089
595
1180
632
1253
663
(1+0.4)  2
269
156
301
167
325
176
344
184
c2
0.9
0.95
*
Sample Size Determination
Procedure R3
Given  ,  3* , k and n, we determine c3 from PLNCK (c3 ,  3* , n,  , k )  P *
Table 3 Minimum sample size n required for Procedure R3 when   1
k
2
3
4
5
P*
 3*
0.5
1
0.5
1
0.5
1
0.5
1
(1+0.2)  3
*
368
92
426
107
467
117
299
125
(1+0.4)  3
*
184
46
213
54
234
59
250
63
(1+0.2)  3
*
490
125
552
138
597
150
631
158
(1+0.4)  3
245
63
276
69
299
75
316
79
c3
0.9
0.95
*
Sample Size Determination
Procedure R4
n 0 = initial common sample size
Sample Size Determination
Procedure R4:
n  max{ h 2 S n20 / d 2  1, n 0 } is a random variable
En = estimated final common sample size
Sample Size Determination
Procedure R4:
Table 6 Better initial sample size n 0 based on Table 5
K
 =0.1
d
Best n 0
3
4
5
0.05
0.06
0.05
0.06
0.05
0.06
40
20
40
30
40
30
Simulation Results

Example 1 (for R1 and R2):
 *  1 , k  3 , c1 = c2 = (1+0.4)  * =1.4, P *  0.95 .
Minimum n  135 for Procedure R1 (Table 1)
Minimum n  167 for Procedure R2 (Table 2)
Let   1,  0  1
1  1 and 2  3  1.1
Case 1: 1  2 ,  2   3  2.1 ;
1  2  1 and 3  1.1 CS: select a subset containing  1 and  2
CS: select a subset containing all EM fields
1  2  3  1 .
Case 2: 1   2  2 and  3  2.1
Case 3: 1   2   3  2 .
CS: select a subset containing  1
Simulation Results
Example 1 (for R1 and R2):
Simulation Results

Example 2 (for R3):
We suppose that   1 is known and  0  1 is unknown
c3  (1  0.4)  3*  1.4
Simulated Results for Example 2
Median PCS
Case 1 95% C.I. for PCS
Median PCS
0.9906
[0.9888, 0.9923]
0.9808
Case 2 95% C.I. for PCS
[0.9789, 0.9839]
Case 3 Median PCS
0.9734
95% C.I. for PCS
[0.9698, 0.9753]
Simulation Results

Example 3 (for R4)
We suppose that the unknown   0.1 .
We set up
P *  0.95 , k  3 , d  0.05 and  3*  0.01 , 0.001, 0.0001 and 0.00001, hence
c6  0.06 , 0.051, 0.0501 and 0.05001.
with respect to  3*  0.01 , we set up  0  1.5 and the following cases:
Case 1 (only one good): 1  1.49 and  2   3  1.48
Case 2 (two good): 1   2  1.49 and  3  1.48
Case 3 (all good): 1   2   3  1.49 .
(CS) is to select a s ubset containing EM field  1 for Case 1,  1 and  2 for Case 2, all EM
fields for Case 3
Simulation Results

Example 3 (for R4)
Median PCS in 40 simulated PCSs in Simulation Example 3
 3*  0.01
 3*  0.001
 3*  0.0001  3*  0.00001
Case 1
0.9920
0.9840
0.9820
0.9815
Case 3
0.9840
0.9710
0.9680
0.9660
Case 3
0.9760
0.9590
0.9555
0.9545
 3*
Case
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.