Graduate Research Award Program Application: 2O1O-2O11 Page 2 of 10

Application Checklist and Application Packaqe Gover Sheet

ApplicantfEnsms' Xue Mengran Date 0511212010

Last First Middle

txl A completed application checklist and application package cover sheet

[x] Personal information of applicant

txl Qualifications of applicant

txl Research project ProPosal

txl Reference letter #1

txl Reference letter #2

txl Research advisor form

txl Officialtranscripts from allcolleges and universities attended

Please list all institutions from which official transcripts have been requested. Transcripts may be

included in the submission packet if properly sealed from the registrar'

lf official transcripts will be sent separately, please indicate that they have been ordered and

include an unofficial copy of each of your transcripts in the application package.

txl Writing Sample

(The writing sample should be submitted with the application. An appropriate writingsample might be a previous publication, a paper written for a class assignment, a

research project report, or similar example of professional writing. lf papers were co-

authored, the role of the applicant must be clearly described. Writing samples may not

be longer than 25 pages.)

Graduate Research Award Program Application: 2O1O-2O11 Page 3 of 10

APPLICATION FORM

GRADUATE RESEARCH PROGRAM ON PUBLIC-SECTOR AVIATION ISSUES

Sponsored By: Administered By:FederalAviation Administration Airport Cooperative Research ProgramU.S. Department of Transportation Transportation Research Board, National Academies

PART I_ PERSONAL INFORMATION OF APPLICANT

(Please Type)

1. Fulllegalname:Xue MengranLast First Middle Former name (if any)

2. Date of birth: 09/08/1985 Ptace of birth: Tongling, China

3. Citizenship: China

4. Gender: [ ]Male [x] Female

5. Ethnicity (optional) :

I American lndian or Alaskan Native: origin in any of the original peoples of North AmericaI Black: origin in any of the black racial groupsI Hispanic: Mexican, Puerto Rican, Central or South America, or other Spanish culture or origin,

regardless of race[x] Asian or Pacific lslander: origin in any of the original peoples of the Far East, Southeast Asia,

or the pacific lslands. lncludes China, Japan, Korea, the Philippine lslands, Samoa, and thelndian Subcontinent

[ ] White: origin

6. Mailing address:

in any of the original peoples of Europe, North Africa, or the Middle East

1630 NE Valley Rd, Apt.4203, Pullman, WA, 99163, United States

7. Permanent address: 1630 NE Valley Rd, Apt.4203, Pullman, WA, 99163, United States

8.

9.

10.

11.

Telephone numbers - Mailing: 509-336-9680 Permanent: 509-336-9680

Email address: mxue@eecs.wsu.edu

College or University currently enrolled ¿1' Washington State University

Major"Field : Electricál Engineêring

Degree objective: [ ] Maste/s [x] DoctorateExpected month and year of graduation: 0612012

Names of two people from whom you are requesting reference letters.

a. Sandip Roy

x Ali Saberi

Name and title of faculty research advisor for this project:

Sandip Roy

12.

Graduate Research Award Program Appl¡cation: 2O1O-2O11 Page 4 of 10

PART II_ QUALIFICATIONS OF APPLICANT

College / University Location Major FieldDates

AttendedGPA Degree

Date degreeawarded/exoected

Washington State University Pullman,WA EE 08/2007-06/201 2 3.8 Ph.D 06t2012

Un¡versity ofScience and Technology of China Hefei,China EE 08i2003-07/2007 3.6 B.S 06t27t2007

(Please Type)

13. Education: ln reverse chronological order, list colleges or universities attended.

Please explain any interruption(s) of schooling, i.e., military training, illness, etc.)

14. Professional Experience. ln reverse chronological order, list professional experience, includingsummer and term-time work.

Name of Emolover Location Dates Nature of Work

15. Awards, honors, and publications: List fellowships, scholarships, and other academic and/orprofessional positions, held since entering college or university.

Please see the resume for publications.

Award, Honor, or publication Date(s) Description

Outstanding Student 2005-2007 from Dept.EE,USTC

Outstanding Leadership 06/2006 from School of lnformation Technology,USTC

General Chair's Recognition Award 12t2009 48th IEEE Conference on Decision and Control

Outstanding Ph.D Student 04t20't0 from Schoolof EECS,WSU

Mengran Xue <mxue@eecs . wsu. edu>(s09)336-9680Sloan 339, School ofEECS

EoucrrroN

t¡Ltp: / /www. eecs . i^¿su. edu/ -mxuePullman, WA,99163

Washington State University (WSU)Ph.D Candidate in Electrical Engineering

University of Science and Technology of China (USTC)Bachelor of Science in Electrical Engineering

Aw¡,nos

August 2007-present

August 2003-June 2007

Outstanding Student ScholarshipDepartment of Electrical Engineering, USTC

Outstanding LeadershipSchool of Information and Technology, USTC

General Chairos Recognition Award for Interactive Papers48th IEEE Conference on Decision and Control

Outstanding Ph.D StudentSchoot of Electrical Engineering and Computer Science, WSU

ExpnnrrNcn

2006-2007

2006

December 2009

2009-2010

Project AssistantIntelligent Administration Platform for Local County Government

o Built an administration system for the local government management in Zhejiang province.

o Created a series of toolboxes for government management, including rosidential database, communi-cations, emergency training.

Lab AssistantComplex System and Network Lab at USTC

July 2005-May 2007

¡ Studied dynamical evolutions and developed optimal searching algorithms on large scale networks.

Lab AssistantIntelligent Information Processing Lab at USTC

July 2006-Møy 2007

o Studied pattern recognition and approximate algorithms in Vehicle Routing problems.

o Completed the bachelor thesis "Shortest Path Algorithms: Optimality and Applications".

Teaching AssistantI)epartment of Electrical Engineering at WSU

August 2007-May 2009

o EE 221: Numerical Computing for Engineers (Fa112007, Spring 2008 and Spring 2009).

o EE 261: Electrical Circuits I (Fall 2008).

June 2 006-December 2 006

Research Assistant August 2007-present

Dynamical Network Group at WSU

o Focusing on the research direction of stochastic modeling and network desigr.

¡ Attended several academic conferences on control and decision and presented the related work.

o Published a Mathemetica demonstration on "Dynamical Network Design for Controlling Virus Spread"

for Wolfram.

Paper Reviewer

o 4th IFAC Symposium on System, Structure and Control (April 2010).

c 49th IEEE Conference on Decision and Control (May 2010).

PusLrcarroNs

l. Yan Wan, Sandip Roy, Xu Wang, Ali Saberi, Tao Yang, Mengran Xue, Babak Malek, "On the Structure

of Graph Edge Designs that Optimize the Algebraic C owrcctivityl' 47th IEEE Conference on Decision

and Control, Cancun, Mexico, Dec.9-l1,2008.

2. Sandip Roy, Yan Wan, Ali Saberi, and Mengran Xue, "Designing Linear Distributed Algorithms withMemory for Fast Convergence," 48th IEEE Conference on Decision and Control, Shanghai, China,

December 14-16,2009.

3. Mengran Xue, Sandip Roy, Ali Saberi, and Bernard Lesieutre, "On Generating Sets of Binary Random

Variables with Specified Firstand Second- Moments," to appear in 2010 American Control Conference,

Baltimore, MD, 2010. Extended version submitted to Automatica.

4. Mengran Xue, Sandip Roy, Yan Wan, Ali Saberi, "Designing Asymptotics and Transients of Linear

Stochastic Automaton Networks," accepted by AIAA Guidance, Navigation and Control Conference,

August 2010, Toronto, Canada.

5. Mengran Xue, Anurag Rai, Enoch Yeung, Sandip Roy, Ali Saberi, Yan Wan, Sean Warnick, "Initial-Condition Estimation in Nefwork Synchronization Processes: Graphical Characterizatíons of Esti-

mator Structure and Performance", submittedto 49th IEEE Conference on Decision and Control.

Extended version submitted to Physical Review E.

6. Mengran Xue, Sandip Roy, Ali Saberi, "Graph-Theoretic Bounds on Steady-State-Probability Esti-

mation Performance for an Ergodic Markov Chain", submitted To 49th IEEE Conference on Decision

and Control. Extended version submitted to International Journal of Robust and Nonlinear Control.

Graduate Research Award Program Application: 2O1O-2O11 Page 5 of 10

16. Describe your career goals and how this research will contribute to achieving those goals (you

may add page(s) if necessary):

Please see the attached.

16. Describe your career goals and how this research will contribute to achieving those goals.

Personal Statement

I joined the Dynamical Network Group in the Department of Electrical Engineering at Washington

State Universify (WSU) formy Ph.D degree roughly three years ago (in August 2007). Since then, I

have been focused on a variety ofproblems on analysis, inference, and design of stochastic network

dynamics. The results of this research work can be used in many application areas, including air

traffic management, synchronization phenomenon in Physics, anci <iata generation. My acaciemic

experience at WSU has helped me to gain a deep understanding of the fields ofs¡ochastic modeling

and network design, and to develop a rich body of work toward my Ph.D thesis.

As one exciting application of my research, air traffic management particularly draws my focus

given that many practical problems in this area do require a lot of network modeling and design

work. For example, our ongoing work (in collaboration with the Mitre Corporation) on weather

impact modeling at the strategic time horizoz needs an accurate yet tractable network model to

describe/capture the dynamics of weather impacts and weather-impact uncertainties. I also strongly

believe that stochastic network modeling and design ideas are essential for many open research

problems not only in air traffic management, but also in several other application areas. In the

following couple years, I will continue research on both the core theory and applications, and so

complete my Ph.D progress research. In the long run, I would like to pursue a career in engineering

research, preferably in an industrial setting; I hope this research career will permit me to continue

advancing the field of stochastic networks, and its applications to aviation. Hopefully, my Ph.D.

research achievement would serve as a solid foundation for many problems in aviation and other

domains that I will encounter in this future career.

As a new aspect of my ongoing work on weather modeling, the project proposed here, namely

"Airport/Airspace Resource Planning under Weather (Jncertainty: A Stochastic Modeling and

Network Design Approach",will help me in pursuing my careff goals. First of all, it will broaden

and enhance my understanding of weather uncertainty impact on the National Airspace System

(NAS), and widen my research expertise regarding the airspace system. Secondly, I will have an

opportunity to learn more about the policy making and the financial side of this large scale system,

a direction that is of particular interest to me in my long-term career goals. Thirdly, the chance

to pursue independent research work is valuable in both my Ph.D progress and my future career.

Also, I may have some chance to interact with people from industry and gain some experience on

cooperation and leadership.

In brief, this project would provide me with a great opportunity and an important foundation

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could be developed through the project.

Graduate Research Award Program Application: 2O1O-2O11 Page 6 of 1O

APPLICATION FORM

GRADUATE RESEARCH AWARD PROGRAM ON PUBLIC-SECTOR AVIATION ISSUES

Sponsored By: Administered BY:

FederalAviation Administration Airport Cooperative Research ProgramU.S. Department of Transportation Transportation Research Board, NationalAcademies

PART III -RESEARCH PROJECT PROPOSAL

(Please Type)

Name: Xue Mengran DateLast First Middle

Title of Research Project:AirporUAirspace Resôurce Planning under Weather Uncertainty: A Stochastic Modeling and Network

Design Approach

ln 500 words or less, describe the proposed research project. lnclude project objectives,methodology, and expected outcomes. Also indicate how this research work could benefit theaviation community, and contribute to your career goals.

Please see the attached.

[Please attach additional sheets as needed.]

Research Project Proposal

As the National Airspace System (NAS) becomes increasingly stressed, weather uncertainties have

increasing impact on NAS performance. The role of weather in decision-making at the tactical (0-

2 hr) and strategic (2-24 how) horizons is being extensively studied, see e.g. our ongoing work

on weather-impact modeling for strategic flow management. Given the ubiquitous presence of

weather, we believe that weather- and weather-uncertainty must also be accounted for in long-range

planning of the NAS (at a time horizon of months to years). Yet, to the best of our knowledge,

no systematic study of weather in long-range planning has been attempted. The purpose of this

project is to study the role of weather and weather uncertainty in airport and airspace planning,

using stochastic modeling and resource-allocation design methods.

We expect this research to have three primary outcomes:

1) The relationship between weather/weather-uncertainty and airport/airspace performance (i.e.,

flight delay, safety) over a long time horizon will be characterized, for some prototypical lo-

cations and events (specifically, wind events in Chicago, winter weather in the Northeast).

2) Human and financial resources at the planning time horizon will be identified that can permit

improvement of airport/airspace performance by taking into account weather uncertainty.

3) A preliminary study on how these resources can be allocated/reallocated to optimize air-

port/airspace performance will be completed.

To achieve the first outcome, namely characterizafion of the relationship between weather/weather-

uncertainty and airport/airspace performance, two complementary approaches will be considered:

a) We will use the historical weather- and airporlairspace performance- data for the prototype

examples to obtain correlations between weather events and airporlairspace performance.

b) Motivated by the observation that often weather uncertainty rather than weather itself plays a

crucial role in performance, we will use a stochastic network model for weather uncertainty

(together with performance data) to correlate between uncertainty and performance.

To achieve the second outcome, we will complete a thorough literature review of airport and

airspace resource planning capabilities. We also plan to interface with industrial researchers and,

if possible, NAS and airport personnel to identiff available resources.

Finally, to meet the third outcome, we will use the results of Outcomes 1 and 2 to model the

benefit in performance that can be obtained through allocation of planning resources. Upon doing

so, \rye will pose the resource-allocation problem as a network design problem, and to address this

design problem using tools from graph- and control- theory.

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in stochastic nefwork modeling and network design/optimization, as well as ongoing collaborative

research with the Mitre Corporation on weather modeling for the NAS specifically. These efforts

on weather modeling for flow management will inform the statistical analyses proposed here, how-

ever we expect many new challenges to arise in addressing the planning problem: these include the

excess and variability of historical weather data over a planning time horizon, and the wide range

of technical and financial issues that arise in planning (including human-factor concerns, faimess

issues, and spread in performance even for good designs).

Graduate Research Award Program Application: 2O1O-2O11

APPLICATION FORM

GRADUATE RESEARCH PROGRAM ON PUBLIC-SECTOR AVIATION ISSUES

Sponsored By:Federal Aviation AdministrationU.S. Department of Transportation

Page 7 of 10

Administered By:Airport Cooperative Research ProgramTransportation Research Board, National Academies

PART IV - REFERENCE LETTER ON APPLICANT

(Please Type)

Thic canfinn fn ha nnmnlotod hr¡ Annlinanf ''"""ry

Name of Applicant: Xue Mengran ¡4¿0511212010Last

Applicant's major field of study

First Middle

Stochastic modeling and network design.

Title of Proposed Research Project:

AiroorUAirsoace Resource Planninq under Weather Uncertaintv: A Stochastic Modeling and Network

Design Approach

This section to be completed bv reference respondent:

NOTE: This appticant has named you as one of two people who know his/her academic and professional

experience and ability. Your views will help us evaluate this applicant's qualifications for receiving anaward for conducting research on the above proiect.

Please complete this form, make another copy, and place each copy ín a separate envelope. Sealboth envelopes, sþn each across the seal, and return BOTH envelopes to the applicant forinctusion with the application to be submitted to the Transportation Researcfi Board.

Name of Rererence Respondent: Sqfi ffip Êi.OY

Title:

Phone: îoq" 7(ì"Ð11

rn what capacity do you know the appticant? P(liqAR.Y (ESnOneH åÐøSfHow long have you known the applicant? l YËAP.S

riEEÆN66 (r, rnr-ifru

S eIr/VG

Graduate Research Award Program Application: 2O1O-2O11 Page B of 10

PART lV: Page2

1. Please evaluate the applicant in the following areas as compared with other individuals of

comparable training, age and experience.

2. Please comment on the ability of the applicant to carry out the proposed research in a timelymanner'

PLEASÊ- 5ÈE- AITAq ft),3. Please add any other comments that you consider to be pertinent to the evaluation of the

applicant and that are not covered adequately by your other answers. Attach additional sheet(s) ifnecessarv.' PlHsr sm Afr¡c+lr¡.

CONFIDENTIALITY: The information contained in this reference letter shall not be available to the

applicant or othen¡rise publicly disclosed except as required by law.

sisnature or Rererence Respondent *':; V o^r.' 5lltrllO

'ìr rfclan¡lin¡ Aboveaverage

AverageBelow

average

lnsufficientopportunity to

observeJUtùto¡ rvil r\

ftowledge of major field

ìesearch skills)roblem solving skills

)reativity

-eadershipA/ritten communication

)ral communication

Sandip RoyAssociate Professor, School of EE&CSWashington State University

Graduate Research Award Program EvaluatorAirport Cooperative Research Programtansportation Research Board

Dear Madam/Sir:

I enthusiastically recommend Ms. Mengran Xue for the graduate research award on public-

sector aviation issues for 2010-2011. I believe that Mengran, a member of my research group, is an

ideal candidate fo¡ the research program: she is very well-suited to make significant advances invaluable airport and airspace planning research (as described in her research statement), and at the

same time involvement in the program will aid her scholarly and professional deveiopment. In thisletter, let me briefly describe her educational background and overview her research contributions,which will serve as a foundation for the research that she has proposed here.

I have had the pleasure of working with Mengran since Fall 2007, when she joined the graduateprogram at Washington State University under the joint supervision of Professor Ali Saberi and

myself, Sandip Roy. She is currently a Qualified Ph.D. student in good standing here, and I expect

her to complete an excellent thesis project in about 1.5 years. During the three years the she has

been at Washington State Universit¡ Mengran has pursued extensive educational training in the

broad area of Systems and Control Theory and its applications, and at the same time has made

significant research contributions in the particular fields of stochastic network modeling and

network design. With regard to coursework, Mengran has done very solid work in a body of15 graduate courses, maintaining a grade point average of above 3.8. With regard to her course

training, I am especially impressed by the breadth of coursework that she has pursued: her course

selection includes topics in mathematics, engineering, and computer science. I believe Mengran'sbroad course background wiil allow her to pursue the cross-disciplinary (mixed engineering and

policy-oriented)research topic that she has proposed.

Flom my standpoint, Mengran's exceptional success in completing scholarly research is the

highlight of her contribution to our graduate program, and strongly speaks to her ability to complete

the proposed research project. During her time at WSU, Mengran has completed a body of eight

significant scholarly works (which are all published or submitted for scholarly publication). These

efforts constitute major contributions to foundational research on stochastic network modeling

and network design, and/or applications of these methods in the engineering sciences. Recently,

she has been collaborating with researchers at the Mitre Corporation (under the auspices of ajoint research project between my group and the researchers at Mitre), to apply the results ofher analytical research to weather-impact modeling and flow-contingency management problems.

Through involvement in this project, Mengran has gained a deep understanding of the technicalchallenges involved in aviation-system design and air traffic management specifically, and she is

well on her way to completing publishable research in this domain. Part of her research effort on

the Mitre-led project has been to develop a weather-impact simulator. This research contributionwill directly inform the research that she has proposed on airport/airspace planning under weather

uncertainty. At the same time, completing the proposed research would complement her current

research on traffic management. In concluding discussion of her research, Iet me especially stress

that Mengran has shown a keen ability to abstract technical problems from real applications, and

then to add.ress these technical problems; this ability will serve her well as she aims to account for

weather uncertainty in the complex airspace planning problem.

Finally, Iet me stress that participating in the graduate research award program would benefrt

her scholarly and professional development. This opportunity would give her a chance to lead

a research project of her own choosing, which I believe is critical in a researcher's development.

Also, Mengran has shown a strong interest in aviation research as a career choice (most likely inan industrial setting), and the interfaces that she could develop through the program would be

invaluable to her as she pursues this career goal.

In short, I believe that Mengran is ideatly suited to study public-sector issues in aviation, and inparticula-r to complete the important problem on aviation resource planning that she has proposed.

Please do not hesitate to contact me, at 509-781-2299 or sroy@eecs.wsu.edu, if I can be of any

further assistance.

'With warm regards,

L,.rØ p"/Sandip Roy'

{ffwashington stare universirySchool of Electrical Engineering

and Computer SciencePullman, Washington 99L64-2752

Ali SaberiEmail : saberi@eecs.wsu.eduTEL: ßos\szs-szzzFAX : (sos) ¡¡s-¡sre

May 16, 2010

Graduate Research Awards ProgramATTN: Lawrence D. GoldsteinSenior Program OfficerAirrrort Coor¡erative Research Prosram-'_-_----o-

tansportation Research Board500 Fifth Street, NWWashington, DC 20001

Dear Mr Lawrence D. Goldstein

Ref: Letter Of Recommendation for Ms. Mengran Xuei

I write this letter on the request of Miss Mengran Xue in support of her applicationfor graduate research award. Mengran is a Ph.D. Candidate in the school of ElectricalEngineeing and Computer Science here at WSU, and is planning to graduate sometime inFall 2011. I am her Ph.D Co-advisor.

I would like to address Mengran's research accomplishments. Mengran's research capabilityand potential is very good. In a very short span, she inspired and collaborated in writingabout six technical papers some of which have already appeared, some are in press, whilethe others are under the review process. In her research, Mengran is involved in theanalysis and synthesis of control strategies for dynamical networks both in deteministicand stochastic frameworks.

In summary, Mengran is a very strong, creative and hard working student and I believethat she has a great potential in research. I strongly support her application.

With best regards,

Sincerely

"X.t5'-t-^^AIi SaberiProfessor

Graduate Research Award Program Application: ZO1O-2O11 Page 7 of 10

APPLICATION FORM

GRADUATE RESEARCH PROGRAM ON PUBLIC-SECTOR AVIATION ISSUES

Sponsored By: Administered By:FederalAviation Administration Airport Cooperative Research programU.S. Department of Transportat¡on Transportation Research Board, Nãtional Academies

PART IV - REFERENCE LETTER ON APPLICANT

(Please Type)

This section to be completed bv Aoplicant: !

Name of Applicant Xue MengranLast First Middle

Applicant's major field of study Stochastic modeling and network design.

Title of Proposed flesearch Project:

¡4¿0511212010

Resource under Weather :A Stochastic and Network

Design Approach

This section to be completed bv reference respondent:

NOTE: This applicant has named you as one of two peopte who know his/her academic and professionalexperience and ability. Your views will help us evaluate this applicant's quatifications for re,ceiving anaward for conducting research on the above project.

PÍease complete this form, make another copy, and place each copy in a separate envelope. Sealboth envelopes, sign eacfi across the seal, and return BOTH envelopes to ine appticant forinclusion with the application to be submilted to the Transportation Research Aoara.

Name of

Title:

ln what capacity do you know the appticant? / c r c) I 's" ( q -T'-<c^c \How tong have you known rhe appticant? ffiO+ç I (, .s- T*, y . "rê

Graduate Research Award Program Appl¡cat¡on : 2O1O-2O11 Page I of 10

PART lV: Page2

1. Please evaluate the applicant in the following areas as compared with other individuals ofcomparable training, age and experience.

2. Please comment on the abilitymanner.

F/Æ-u.

Signature of Reference Respondent:

to carry out the proposed research in a timely

,&-trrÀ ,ft^._/4)of the applicant

(v<p

3. Please add any other comments that you consider to be pertinent to the evaluation of theapplicant and that are not covered adequately by your other answers. Attach additional sheet(s) ifnecessary.

F/'-r'-* (-arc-L AuTDA ^Ñt[*'CONFIDENTIALITY: The information contained in this reference letter shall not be available to theapplicant or othenryise publicly disclosed by law.

o"", 3/t/ /¿¿

ì, r+a+an¡lin¡ Aboveaverage

AverageBelow

avefage

lnsufficientopportunity to

observeftowledge of major field t,/ìesearch skills)roblem solving skills

)reativity v-eadership Y,IA/ritten communication l,/,)ral communication V

Graduate Research Award Program Application: 2O1O-2O11 Page 9 of 10

APPLICATION FORM

GRADUATE RESEARCH AWARD PROGRAM ON PUBLIC-SECTOR AVIATION ISSUES

Sponsored By:Federal Aviation AdministrationU.S. Department of Transportation

Administered By.Airport Cooperative Research ProgramTransportation Research Board, National Academies

PART V _ FACULTY RESEARCH ADVISOR

To be completed bv the aoolicant:

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necessary for you to request a faculty member from your university who is familiar with your researchproject to act as a research advisor to you during the course of the proiect. Please provide the followinginformation and ask the facutty member to complete the form. You should submit it with your application.The research advisor may also be a reference respondent.

ApplicantTHName: Xue Mengran 05t12t2010Last First Middle

Title of Proposed Research Project:

AirporUAirspace Resource Planning under Weather Uncertainty: A Stochastic Modeling and Network

Design Approach

To be comoleted bv the facultv research advisorNOTE: The research product of research award recipients can be considerably enhanced if a facultymember at the applicantffiuniversity acfs as an advisor to the applicant during the conduct of theresearch. Therefore, each appticant is required to designate such an advisor who will be available tohim/her throughout the course of the research project to provide advice as it progresses. When researchpapers by award winners are pubtished by the Transportation Research Board, the faculty member will be

identified as ffie research advisor.

Faculty Research Advjsor

Department:University:Mailing Address:Email: 5 f

?Yes/*o1.

2.

Have you examined the applicanttEproposed research plan

Do you consider the applicantlE research plan reasonable?lf no, please comment.

v""/ No

Graduate Research Award Program Application: 2O'lO-2O11 Page 10 of 1O

PART V: Page2

Do you believe that this applicant can complete the proposed research within the time frameindicated? Yes \-/ No

-.

lf no, please comment.

.,/4. Will the applicant receive academic credit for this work? Yes

-

tto / lf yes, pleaseinÀi¡a+a +ha na+¡ ¡¡a af +hi¡ a¡a¡lami¡ ¡ra¡ti+ ftrlafa' Þanairrin¡ anar{omin nrodif in nn rrrarr nnl lnfcll ¡u¡gqlg ll 19 I rqlul I vr rr r¡o qvqvv tr lvlvt I \vve. t rr tY

against the applicant.l

5. Please indicate briefly how you plan to monitor and advise on the work of the applicant on thisprojeôt.

PÆ*ss 5EÊ. {f,{dlÌJ)

6. I am willing to beaward. Yes ,/

theNo

research advisor to the applicant if the applicant receives this research

sionature: l,r^^¡r" ft, Date: s/iI,Jru

Faculty Research Advisor Form, Question 5:

As Mengran's primary research advisor, I already meet with her 2-3 times per week to discuss

her research progress. If this research project is supported, I expect to interact with her regularlyon this project at these frequent meetings. I believe that the project would provide her with a good

opportunity to lead a research effort (and she is very well prepared to do so), so I will serve in an

advisory role rather than guide the research. Precisel¡ I expect to provide her with backgroundmaterial on aviation issues, as well as on the technical methods that she chooses to use. I also expect

to serve as a sounding board for her ideas on the project, and to help her with the presentationand dissemination of the work. Finall¡ I will put her in touch with other researchers in the field,:f ^--^L ^^-+^^+^ ^-^ L^l^f,,I +^ l^^-rr Þuut¡ vulr!@vuÐ @aç uçrPlur uv ¡¡ç¡ y¡vó¡çÞù.

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uü iE æÀScholastic Record for Undergraduate Students of USTC

GPÀ Celcul,ati on :ce¡.tesimal Grade 100-95 s4-90 Bg-Bs 84-82 81-zE 'rl*rs74*12 zi*6g 6z-6s 64I,etter füaile A+ A A_ B+ B B_ C+ C C_ D+Foint Yarue 4.3 4 3,7 3.3 3 2.1 2.3 2 1.T l.s

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Section of Teachin

On Generating Sets of Binary Random Variables with Specified First-and Second- Moments

Mengran Xue, Sandip Roy, Ali Saberi, Bernard Lesieutre

Abstract-We study the problem of generating sets of binaryrandom variables with specified means and pairwise correla-tions (i.e., specified individual- and pairwise-joint- probabil-ities). We propose a low-complexity algorithm for generatingsuch correlated random variables, that involves first generatinga set of mutuaüy independent "source" binary random vari-ables and then constructing the desired random variables byrandcmly selectirag fro¡a alrd probabilictically copying or anti'copying the source variables. We show tùat the parameters ofthis data-generation algorithm can be easily designed to achieve

the desired statistics, under broad conditions.

I. INrnooucrloN

The generation of large sets of correlated binary random

variables is needed in numerous fields, including in image

processing, modeting of ferromagnetic materials, economics,

weather modeling, and computational ecology' Classically'

such sets of binary random variables have been generated

through stochastic iterations, with the aim of asymptoticallyachieving desired joint distributions ill, t2l. For ractability'ttre desired joint distribution is specified by a Marlcov random

field or MRF model (which imposes a graphical conditional-independence structure), with the generaiity of the achievable

distribution but also the computational difÊculty in formu-

lation/generation growing gracefully with the sizes of the

cliques in the Markov random field's graph. This powerful

data modeling and generation methodology has found very

wide application, see e.g. []-[3].In this article, we put forth an alternate methodology for

generating large sets of binary dat¿, that is based on achiev-

ing desirabte low-order statístics (in particular, probabilities

for individual random variables and pairwise correlations).

Our perspective is that many applications in the engineering

sciences naturally require such a group of binary random

numbers with speciûed low-order statistical properties (ustas engineering of continuous-valued stochastic processes is

largely based on first- and second-order statistics [4]), since

this statistical description naturally permits representations ofsfengths and extents of interactions. However, the classical

MRF-based methods are not well-suited for designing low-

order statistics in most cases, especially when the graph

of correlations is dense (because we need to obtain the

complicated joint distribution corresponding to large clique

sizes first, and then marginalize). Yeq enforcement of desi-red

This work was supported by grants NSF-0528882, NSF-0725589' NSF-

0901i37, and NASA NNA-06CN264. The fi¡st th¡ee authors a¡e withthe DeparEnent of Electrical Engineering, Washington Stâte University.The fou¡th author is with the Department of Electric¿l and Computer

Engineering, University of Wisconsin-Madison. Please send correspondence

to mxue@eecs.wsu.edu .

low-order statistics places limiæd constraints on a set of data,

and so we might expect that much simpler methods for data

generation can be obt¿ined.

Here, we develop a methodology for easily generating

binary data sets with desi¡ed fi¡st- and second-order statistics

regardless of how dense the correlntion graph is (and how

complicated dre joint distribution is). Unlik'e the stochastic

iterations used in the conventional methods, we take the

following path: we first generate a group of indepen'dent

and ídentically distributed (i.i.d.) binary random variables

with certain first moment as source random variøbles. We

then generate a set of target random variables to have the

desired statistical properties based on these source random

variables, as follows: to generate each target random variable,

we choose one source site stochastically based on a network

matríx, and then copy or anti-copy the value of this random

variable based on a state transition matix. We will show

that we can design a network matrix and state transition

matrices so that the target random variables will have the

desi¡ed statistical properties. We also exploit the connections

between this data generation method and other stochastic-

modeling and linea¡-algebra consûucts, to gain structural

insights into the method.The remainder of the article is organized as follows. In

Section II, we formulate the problem of generating binary

data with desi¡ed first- and second-order statistics in gener-

aliry. In Section III, we describe our methodology for data

generation, and bring forth algorithrns for several cases.

II. PRoBLEM FoRlr¿uLRuo¡¡

We consider the problem of generating n binary random

va¡iables with certain desired first- and second-order statis-

tics. In this article, we refer this problem as the second-

order binary data generation problem (BDG-2). Also, we

refer to these binary random variables to be generated as

the target random variables here, and use the notation y¡,

i: I,2,"' ,n, to rePresent them.

Læt us begin by discussing approaches for solving the

BDG-2 problem at a conceptual level, and hence motivate

a new approach. A classical approach to solving the BDG-2problem is to first frnd a joint distribution that achieves the

desired first- and second-order statistics, and then to gener-

ate data according to the joint distribution using standard

methods (e.g., [1]). Such a method can easily be shown

to be computationally complex even when n is moderately

large (specifically, the complexity is exponential in ¡t, see

the extended version [9]). However, when solving ttre BDG-2problem, we are only i¡terested in the first- and second-order

statistics, which admit a much more terse description than the

full joint distribution. Thus, one might guess that the tårgetrandom va¡iables could be lairly easily generated to meet thedesi¡ed first- and second-order statistics. The main purposeof this article is to show how to generate the target randomvariables at low computational cost, by avoiding solving fortheir joint distribution and generating data thereof.

Let us also give a formal statement of the BDG-2 problem.In the BDG-2 problem, the first- and second-order statisticsthat we desire to set are the individual and pairwise jointprobabilities of the target random variables, or equivalentlytheir first and second moments. Specifically, we aim togenerate the target random va¡iables so that øi is the firstmoment (or individual probability) of I¡, for i € {1,... ,n}and o¡¡ is the second moment (or pairwise joint probability)of i'¡ an<i ii for some pairs of ciisdnct i anci 7 e ii,...,n).That is, we seek to generate the target random variables so

that at: B¡ytl: P(Yi: 1) and oi,¡: EIY¡Y¡I: P(Y¡= I,Yj:1). We refer to the a¿ as the desired first moments and the

Q,.¡ as the desired second moments.In some cases, all correlations between pairs of target

random variables must be set, while in other cases only someof them a¡e of interest. We find it useful to describe theconelations between pairs of target random variables thatmust be set in a graphical way. Specificall¡ we consider a

graph where each vertex represents a target random variable.If a correlation between two random variables is to be set,

we connect them by an edge in the graph. Otherwise, if a

correlation need not be set, then there is no edge betweenthe conesponding two vertices. Here we use the notation Ifor the unweighted and undirected correlation graph.

III, OUR APPROACH

Our main goal is to solve the BDG-2 probiem in a

computationally effrcient way. To achieve this, we avoidcomputing a joint probability mass function (PMF) for thedesired random va¡iables, and instead suggest generatingthe target random variables I'¡ that have the desired low-order moments directly from a set of independent binarysource random variabies. Here, let us introduce this data-generation methodology in concept, and then explore howthe technique can be designed (parameterized) to achieve a

particular set of desired moments. Then we will present somedesign algorithms and corresponding structural results.

A. New Methodology and. Design Problem

In our new methodology, the key idea is to generate

the group of desi¡ed target random variables based on

another group of independent binary source random vari-ables. Specifically, we first generate a set of z independentbinary source random va¡iables, Xr,... ,Xr, with each X;independently equal to 1 with probability q¡, j : I,2,... ,n,and equal to 0 otherwise. Each target random va¡iable Y¡,

i: I,2,. . ' ,n, is generated as follows: first, a source randomvariable X;, is independently chosen with probability p¡¡;second, the value of 1¡ is determined by copying or anti-copying the value of X;, according to a 2 by 2 state

transition matrix A,;. (That is, if X¡ :0, then I¡ is setto 0 with probability A¿r.11 and to 1 with probability A¡¡,12.

Meanwhile, tf Xj :1, then Y¡ is set to 0 with probabilityA¡¡,21 and to 1 with probabiliry A¡j,zz. In other words, the

rows of A¡7 specif] probabilities for the value of Y¡, giventhe value of X¡.) We form an nby n maúix p : þ¡r], whichwe call the network matrix. Both the network matrix P andthe state transition matrices A¡¡ are row-stochnstic matríces.

Now let us consider using this generation rule to solve

the BDG-2 problem in Section II. We assume that a¡, i:7,2,. . ., z, is the desired fust moment of each target randomvariable I¡ and that 6¡¡, í:1,2,'.. ,n, j :1,2,... ,n, j *i,íSthe desired pairwise correlation between Y¡ ardY¡, for vertexI and vertex j that are connected in the correlation graph l.It turns out that if the source probabilities q¡, úe networkmatrix P, and the state transition matrices A¡¡ are properiydesigned, the target random variables will have the desiredfirst and second moments. Thus, in the following part of thisarticle, we focus on how to design proper q¡, P and A¡¡ so

that { has the given statistics. This design tums out to beparticularly simple when P and A¡¡ have certain special struc-tures, and these special structures can nevertheless achievea broad class of desi¡ed moments. Therefore, we primarilyfocus on these special cases: specifically, we limit P to a

lower-triangular matrix (Section ltr-8.1) or to a symmetricmatrix (Section III-B.2), and limit A¿¡ to the identity matrixor to be equal for all.generated target random variables.

In addition, we find it convenient and explicit to use

a network illustration to describe the generation rule. Letus introduce some further notation regarding this network.First, we use n sites/nodes tn rhe network (labelled l, . . . ,n)to represent the r? target randorn va¡iables, and another zsites/nodes (labelled 1,...,n) to represent ,? source ran-dom variables. Moreove¡ we view each site as having twodifferent states (0 and l), that represent the value of thecorresponding (source or target) random variable. Then, sinceeach target site i chooses a source site j with probabiliry p¡;to generate its state, we draw a di¡ected edge from sourcesite j to target site i with weight p¡; Gig. 1).

Target sites: ill|.i = I.2.....n

Source sites: i-ï,).i = 1.2...-.n

Fig. 1. The network within the two groups of sites.

The generation rule can be viewed as a network dynamicsinvolving two levels of interaction. At the network level,each target site can be influenced by a source site withprobability p¡j (since each l¡ independently chooses a X;with probabiliry ø¡).Meanwhile, at the local level, eachI¡'s value is selected from the chosen X;, based on the state

transition matrix A¡;. Such a network dynamics is similar

to that of the influence model, although without a temporal

evolution t5l, t6l. Like the influence model, this rule istractable in that site statistics can be obtained easily.

Let us now pusue design of the generation rule (equiv-

alently, the network dynamics) to obtain target random

variables with desi¡ed statistics.

B. Design Methods for the Homogeneous Case

Let us first consider design for the case where each desired

û¡st moment is identical, which we call the homogeneous

cfise.To start, we study design of the first moment of each target

random va¡iable. The following theorem presents a conditionon the data-generation rule, which guarantees that each Y¡

achieves the desired first moment d.t tlgulerla l. jÆt uù vuvvùu u49r¡ ùuuvvr4ruvlr¡ v@¡4vrç1rJ

to have first moment ø, i.e. let us choose qj : d. We also

assume that, for each i€ {1,...,n}, every A¡¡ is identical:

At : A¡z - " ' : A¡n:: A¡.If the left eigenvector of each A¡

corresponding to its unity eigenvatue is lt-o ø] r,

ttt"neach Y¡ will have first moment ø, for any valid P.

In Theorem 1, we see that arbitrary P and many A¡¡ (otA¡) can achieve the desired fi¡st moment. We will exploitthis remaining freedom upon designing the first moment toachieve specified second moments. To simplify this design

(and because often design specifications can be met even

with this simplification), we will actually limit ourselves to

the case that all A¡¡ are the identity matrix (i.e., each Y¡ can

only copy the state of its neighboring source sites in the

network). By doing this, we limit ourselves to the design

of the network maûix P, which is central in specifying the

interaction structure among the target random variables.

To continue, let us design the network matrix P to achieve

the desi¡ed second moments. Before doing so, let us firstintroduce some further notation here. We assume that o¡t:o;i is the desi¡ed pairwise correlation between two target

sites i and j that are connected in the correlation graph.

Then we deûne a symmetric matrix Cy to contain the desüed

correlations (second moments) between the target random

variables. The second moment of each Yt, EIY?1, is fixed at a(because the desired fust moment of Y¡ is a andYl: I¡), and

so the diagonal entries of Cy are all a. Also the off-diagonalentries are Cy,¡¡ : Cy,ji : o¡¡ 1f Y¡ and Y j are connected in the

correlation graph; meanwhile, if Y¡ and Y¡ are not connected,

the corresponding entry of Cy is variable and we view it as

a free parameter. We also define a symmetric matrix Cx as

containing the source random variables' correlations. Since

the X,'s are i.i.d., Cy has diagonal entries at o and off-diagonal entries at a,2, or

Cx : az1.tr + (a- az)\, (1)

where 1- is a column vector of l's and I is the identity matrix.For convenience, we also define two state vectors: Y:

(Yt,Yz,... ,Yr)r and X : (Xr,Xz,"' ,Xr)r . Noting that each

f¡ is determined by choosing one X; with probabiliry p¡¡

and copying the state of X;, we see that I can be writtenas I : QX, where p is an n by n permutation matrix.

We also see that the irÈ row of Q is the standard basis

vector ej (i.e., the basis vector with unity entry at ,h" i'hcomponent and 0 elsewhere) with probability pi;. We then

have Cy : EIYYr] : EIQXXT Qrl : n¡n¡zxxr QrllSl:E[ZEWxr]Qr lQ) : ElQCxOr lQl. combining it together

with (1), we obtain

cv : øfg@zttr +(a-d.lDQrlQJ: a2n¡gttr grl+ (ø- az)Eleerl: aztLr +1a-uL¡n¡ggrl. (2)

Notice that by designing P, we can design Cg : EIQQT) and

hence hope to achieve the desired correlation matrix Cr. To

achieve this design, it helps to re-write (2) as

ce:g# (3)

Let us now further simplify C¿. Since each row of p is

a standard basis vector, the diagonal entries of QOr arrd

hence Cg are all 1. The (i, j)rå off-diagonal entry of QQr isI with probability X:i pikpjk arLd 0 otherwise' Hence, the

(i, j)'e off-diagonal entry of Cp is Ð'fr:lPtrrP¡r, identical to

the (i, j)'å off-diagonat entry oi PPz. Thus, Ca: PPr + n,

where  - diag{t - tÍ:' r?;}. No* we can re-write (3) as

ppr + L_ cy - azl"Lr . (4)d,- d:

Also we know that the diagonal entries in the desired Cy are

all a. Therefore, in (4), tne aiagonat entries of qã# *"all 1. Hence, if we can find a stochastic matrix P such that the

off-diagonal terms in the matrix equation (4) conespondingto edges in the correlation graph I are satisfied, then the

desired second moments (or Cy) can be achieved.

Meanwhile, in (4), if such a stochastic P exists, all the off-diagonal entries in Cy must be within the range laz,ul. Inthe rest of article, we will denote a correlation within [42, a]as a positive correlation and the correlation witfiin [0, a2)as a negative correlation. In the rest of Section Itr-B, we

will first assume that all the desi¡ed second moments have

positive co¡relations, and considering designing P to achieve

these correlations (in two ways); we will then develop a

generalized algorithm for the negative-correlation case.

We note that some correlations cannot be achieved by any

set of binary random variables, and our designs also do notachieve all allowable correlations, but they provide simplegenerations for a fairly broad class. More specifrcally, inthe sequel we present explicit designs for achieving desired

moments using two classes of (stochastic) network matricesP: lower-triangular P and symmetric P. We also present some

structual conditions on C¡ under which we can obtain such

P. We note that several generalizations of our designs are

possible, that allow for achievement of a more general set

of desi¡ed moments (including negative moments); possiblegeneralizations include allowing more general P or A¡;, and

considering architecfures with a larger number of source

sites. As mentioned above, we shall briefly consider using

such designs to achieve desired negative correlations, butleave a det¿iled study to future work.

Here is the basic design (for positive correlations), usinglower-triangular a¡d symmetric P.

I) Lower-tiangulør P. The design can be achieved sim-ply in the case where P is chosen to be lower-triangular, i.e.p¡j:0 for all i < j (o¡ in other words, there is no connectionbetween X.¡ and Y¡ for i < j, see Fig.2). Note that the matrixP now automatically has a lower-triangula¡ structure. For thiscase, the design problem reduces to solving a set of linearalgebraic equations to achieve the desired second moments.

Source sites: {-I,l.i = I-2..,..n

Fig. 2. The network D with lower-triangujar P.

The following algorithm shows how the remaining freeoff-diagonal entries can be determined through a set of linearalgebraic equations, and hence the diagonal entries can bedetermined as well, to achieve desirable second moments.We will also give a pseudocode that describes the process ofsolving the linear equations.

To develop the algorithm, let us characterize the correla-tion q; in terms of the designable entries of P. Since all thetarget sites (target random variables) independently updatethei¡ values, given a certain desired pairwise correlation o¡¡,we need ot¡ : EIYÍ ¡) : P (Y¡ : l,Y t : 1¡ : d,Pc t a2 çt - fr¡,where P. is the probability of Y¡ and Y¡ choosing a cornmonsource random va¡iable for copying. Since the cornmonneighboring source sites ofboth Y¡ andY¡ can only be X¿ forsome k < j, then we have

o'i : d (P-'*'r)* *' (r - i,o*o'r) (s)

Also since random variable Y1 caî only be influencedby the random variable X1, we immediately have p11 : l.Let us now sequentially consider generation of Y¡ for i:2,3,...,n, and focus on obtaining the corresponding p¡;'s,j < l, from a set of equations in (5). A key insight is thatthese equations are linea¡, since p;¿, j < i, k Sj, have alreadybeen determined. We present the process of solving for theseprobabilities in the following aigorithm.

Algorithm 1:Initialize pn: I; Ái : I for all j.Form the correlation graph f;For i:2 : n 70 outer loop

For;:1: (i- 1) 7o inner looPIf Y¡ is not a neighbor of I'¡ in f

Pi¡ : o;else

t /o,,-n2 ; 1 \pii: h (ä -Xí:i p,*p¡p);end

end

P¡¡: | -t¡lp¡¡end

When a particular pi¡ is being found in the above pseu-docode, note that all the p¡¿'s, for j : 1,... ,i- I and È < j,have already been obtained from the previous steps in theouter loop. All the pr¿'s where kl j are also solved fromthe previous steps in the inner loop. Hence, if Y¿ and Y¡ areneighbors in l, pij is the only unknown and can be uniquelyobtained f¡om (5). Evenhrally this process determines all theoff-diagonal entries, and in tum the diagonal entries, in P.

An interesting note is that the above algorithm is closelyrelated to the Cholesþ-decomposition algorithm L1l, Lïl,which factorizes a synrmetric matrix into a product of a

lower-triangular matrix and its transpose. What is particu-larly interesting, in considering this connection, is that ouralgorithm achieves such a factorization in the case that someof the entries in the factorized matrix are indeterminate (as

per the correlation graph). Our methodology shows that onlya subset of the entries in the lower-triangular matrix needbe solved for in this case, while the others can be set tonil. In this sense, our algorithm generalizes the Cholesþdecomposition to the indeterminate-entry case (and simplifiesthe computation in this case).

We note that Algorithm I cannot achieve all possibledesired correlation matrices Cy. This is because, for ourproblem, every entry in P needs to be nonnegative. Algorithm1 does not guarantee this condition: we may obtai¡l somenegative pi¡ or pt, after solving the set of linear algebraicequations (see the extended version of this paper [9] for an

example). Despite this limitation of Algorithm 1, we stillconsider it as an efrective way of data generation, not oniybecause it works for a fairly broad class of desired correlationmatrices, br¡.t also because the computational complexity ismuch lower than the conventional methods.

2) Symmetric P; In this section, we will consider designto achieve desired second moments, for another limited classof network matrices P: ones that are symmetric. In contrastto Section III-8.1, we will not consider a recursive algorithmto solve for P in this section. Instead, here, we will use thesymmetry properly to compute P directly. For convenience,let us first focus on the case where the correlation graphis complete, i.e. all second-moments must be set to desiredvalues, and then briefly discuss the more general version.

To present the computation, let us consider the equation (4)derived in Section III-8. We use the notation M to represent

^utti^ Eff in (4). Further noting that P is symmetric,

(4) now becomes

PL:M-L, (6)

where we recall that P is a stochastic matrix, A :dtus {t -!,=, n?¡}, u"a M is specified by the desi¡ed firstand second moments. Our goal is to design a P (and hencealso Â) so that 1) P is stochastic and 2) Equation 6 holds.

We claim that the following process can be used to find anetwork matrix P that achieves the design:

1) Construct the matrix M. Then, construct a matrix M whoseoff-diagonal entries equal those of M, and whose diagonalentries are chosen so that each row of û sums to 1. Continueif the matrix ú ir no*"gative def.nite and elementwisenonnegative.2) Find a matrix P that is a square root of the matrixû; a good choice is often the positive deû¡ite square root

P: t/ M.3) If atl entries of F a¡e nonnegative, then P : F is a

stochastic matrix that solves (6).

Let us argue that the result obtained through the aboveprocess is a valid solution to (6). To prove this, first notethat diagonal entries of the left and right sides of (6) a¡e

identical, regardless of P; to see this, simply note that the

diagona! entries of M are 1, ùe i'fr diagonal entry of A isi - )tr p?¡, efild, the itå diagonal enffry of P2 is ütp?¡.Next, we nóte that the off-diagonal entries of û are identicalto those of M and of M - L. Thus, from the construction ofP, it is automatic that the off-diagonal entries of the matrixequation (6) hold. Finally, let us argue that the obtained Pis a stochastic matrix. To do so, notice that M is stochasticand so has a dominant eigenvalue at I with correspondingright eigenvector 1. Taking the square root, we automaticallyfind that the result has an eigenvalue at 1 and correspondingeigenvector 1, i.e. each row of the square root surns to 1.

Together with the fact that the resulting entries have been

found to be non-negative, we obtain that the solution isa stochastic mat¡ix. Thus, the obt¿ined result is a validsolution.

Although we can always seek for a valid solution throughthe above process, it is helpful to have some simple necessary

or sufficient conditions for whether or not a valid solutioncan be obtained, to 1) simplify computation and 2) give

some insight into how generally the above process works.Fi¡st, we note the following necessary condition: in orderto use the process, we require Lj:r,¡¡tMii < I for all i(since otherwise û is not a stochastic mâtrix). This necessary

condition highlights that a syrnmetric solution ca¡ be foundusing the above process only if the desired correlations are

sufflciently small. Meanwhile, the following theorem (see

t101, tlu) presents a strong suffrcient condition for findinga symmetric solution.

Theorem 2.' Given a desired correlation matrix Cy, let us

frnd. û as in the process. If û-r exists and is an M-matrix,

then P : Jñ is a valid solution.

Theorem 2 phrases sufficiency for a valid solution in terms

of whether or not the inverse of a certain symmetric matrixis an M - matrix. Much is known about which symmetricmatrices have inverses that are M-matrices [10], so the above

theorem identiûes a class of correlation matrices that permitdesign of symmetric M. Also, for the case that Cy has some

free entries (i.e., the co¡relation graph is not complete), ifwe set these free entries to some paficular values so thatmatix û satisfies the conditions in Theorem 2, then we can

also obtai¡ a symmetric design.

3) Generalized Algorithm: So far, we have limited the

state transition matrix A¡¡ to the identity matrix. Althoughthis is sufficient to achieve many desi¡ed Cy, some cases

require a more general algorithm. One way to enlarge the

range of Cy is to permit differing A¡¡. Another way is toadd more source sites to the network. The strategy of addingmore source sites increases the complexity of computing and

we do not focus on it in this article. In this subsection, weexplore the first way to enlarge the correlation range: usingdiffering and non-identity A¡¡.

Let us consider the following model for the ^4¡¡. For each

target site i, we Set A1: A¡z: "' : A¡¡¿:: A¡, i.e., each

target site i has the same transition matrix,4¡, for any source

site j. To achieve the desired first moment a for each targetrandom variable, according to Theo-rem 1, we.still require

that each A¡ has a Ieft eigenvector it - ø al' associated

with its unity eigenvalue.

The following leruna clarifies that many valid state transi-tion matrices have the desired left eigenvector correspondingto the unity eigenvalue.

I¿mma1.' Assume that [1-a a]r is the left eigen-vector of a stochastic matrix A corresponding to the unityeigenvalue. Then A is not uniquely determined from the lefteigenvector.

Therefore, we can use different staþ fansition matrices A¡

of each site to achieve fr's with the desired first moments.

Next, let us show that, if we set Ar properly, some negativecorrelations can also be achieved.

Theorem3.' Consider applying the generalized dat¿-generation rule (with non-identity A¡) to obtain a set of targetrandom variables with identical first-lnoment ø. Throughproper selection of P and A¡, we can achieve negativecorrelations among the target random variables, i.e. we can

achieve correlations ElYiYjl that are less than cr2.

In summary, Lemma I guarantees ttrat, when differentstate transition matrices are used, the desired first momentcan still be achieved; meanwhile, Theorem 3 shows thatnegative correlations among the target random variables also

can be achieved by designing proper state transition matricesand network mafiix. V/ith these observations in mind, let us

present a generalization of Algorithm 1 which allows us tosimply find a lower-triangular P, so as to generate binarydata with desirable first- and second- moments.

Algorithm 2:Initialize pt : L.

Form the correlation graph l;For i:2: n 70 outer loop

For j: I : (i- 1) 7o inner iooP

If Y; is not a neighbor of 1¡ in fP¡¡ : O;

elseif o¡; > a2Set A, : J;

-..- I (gi¡-o' -l-l- - \.Pij: pjj tã:ã7- - LL:tPikP jk)ielseif q; < ø2

Set Ptl :0'

A¡,zz : a6Èî ¡,õ (tÆP - oe i,,,),A¡,21 :l-A¡,zz;A¡,rz: &0-A¡,zz);A¡Jt: I - A¡,tz;

endendp¡¡: r -Ðj;-lrpt¡;

end

Again, Algorithm 2 does not provide a general solutionfor all Cy. However, this design generalizes the class ofcorrelation matrices that could be achieved using Algorithm1, including to some cases with negative correlations.

4) Example: To illustate our designs of the data-

generation rule, we have considered designing binary data fora parlicuiar dense correiarion graph (one wirh iarge cliques).Desired correlation structures like this one often do not lendthemselves to classical data-generation methods, because ofthe computational cost of specifying/using joint PMFs fo¡large cliques, and (if the specification is given via moments)the diffrculty in computing joint PMFs. In the interest ofspace, we have omitted the example from this article, and

ask the reader to see [9] for the details.

C. Design Methods for the Inhomogeneous Case

In many applications, we may desire that each targetrandom variable has a different fust moment, í.e., the in-homogeneous case is of interest. Thus, in this subsection,

we consider design of each Y¡ to achieve (an in general

different) first moment q, as well as a set of second-

moment requirements. Our design technique tums out todepend on the value of the largest desi¡ed first moment,so we find it convenient to denote this largest moment as

dmax: max{a¡,dzr"' ,dn) -

Again, by adding more source nodes to the network ordesigning proper state transition matrices, we may achieve

the desi¡ed fi¡st and second moments for the inhomogeneous

case. However, this design process can become quite com-plex. Instead, our main focus here is to maintain the lowcompiexity by only slightly changing the generation rule

for the homogeneous case, and so apply a similar design

algorithm for the inhomogeneous design. Specifically, we

only modify the generation rule by adding one more source

random variable Xs, which we call the balancing randomvariable. We simply set X0 equal to 0. Meanwhile, for the

other source nodes, we independently generate the source

random variable X¡ so that its first moment is the largest

desired fust moment d,r*r. ln the modified update rule(shown in Fig. 3), each target random variableY¡, i:1,... ,h,can additionaily be determined through influence by the

balancing node Xs with probabili$ p¡0, upon which it takes

on the vaiue of Xo (i.e. 0).

Next, we will describe how this new generation rule witha balancing node can be used to achieve target randomvariables with different first moments. As st¿ted in the

following theorem, we see that if proper p¿s is selected, I'¡ can

achieve the desired first moment ø¡. Befo¡e presenting this

Source sites: {-.r*,},i = 1,2.....ndú dú au* 4u

Fig. 3. The network for the inhomogeneous câse.

result, we note that in the new generation rule, the source-

selection probabilities satisfy p¡s + )i:l pi¡ : l.Theorem 4: If we choose pn: L - # i" the new gener-

ation rule (and choose Qi: dmax, AU:1, and p¡¡ arbitrarilybut so that pn-l\:1pi¡:1), each target site i will have

the desired frrst moment dr.Next, let us consider designing the algorithm to achieve

the desi¡ed pairwise correlations. This design turns out to be

staightforward, for the following reason: the expression forthe desired pairwise correlation E[liIJ remains exactly as

in the original generation rule (for the homogeneous case),

since X6 is nil and so does not play a role in the second-

moment expression. Therefore, (5) still holds here. Hence we

can still apply Atgorithm I to construct a lower-trianguiarP, with only a slight difference in computing the diagonalentries since P is no longer stochastic. We ieave the detailsto future work.

Rrpenewces

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t2l W. K. Hastings, 'Monte C¿¡io Sampling Methods Using MarkovChains and Their Applications", Biometrika, vol. 57, no. 1. (Apr.,1970), pp.97-109-

[3] S. Geman, C. Grafñgne, "Markov Random Field Image Models and

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[5] C. Asavathiratham, "The In.fluence Model: A Trâctâble Representationfor the Dynamics of Networked Markov Chains", Ph.D. thesis, Mas-

sachusetts I¡stitute of Technolog¡ Cambridge, MA, USA, October2000.

[6] C. Asavathirathar¡u S. Roy, B. Lesieutre, G. Verghese, "The influencemodel", IEEE Control Systems, December 2001.

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eds. M. G. Cox a¡d S. Hammarling, 1990.

[8] K. Ta¡abe and M. Sagae, "An exact Cholesky decomposiúon and

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B 54 (1992), no. 1, pp. 211-219.

[9] M. Xue, S. Roy, A. Saberi, B. Lesieutre, "On Generating Sets ofBina¡y Random Va¡iabies with Specifred First- and Second- Moments"(extended version), wwweecs. wsu.edr¡./-mxudpublications .

[10] M. Fiedler, "Some cha¡acteriz¿tiors of symmetric inverse M-matrices", Linear Algebra and its Applications, (1998)179 -I87, pp.

275-276.

tl1l N. J. Higham and L. Lin, "On pth Roots of Stochastic Matrices", 2009.Available: http://epriûts.ma.ma¡.ac.ulc/view/subjects/MsC- 1 5.html.

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