Optimization of Microphone Array Geometry with Evolutionary Algorithm

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1 JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 13 Otimization of Microhone Array Geometry with Evolutionary Algorithm Jingjing Yu Univerity of Kentucky, Lexington, USA Fahan Yu and Yinglei Li Henan Polytechnic Univerity, Jiaozuo, CHINA; Univerity of Kentucky, Lexington, USA Abtract The comlex relationhi between array atial gain attern and microhone ditribution limit the effective ue of traditional method to otimize the element lacement of microhone array. Thi aer reent a genetic algorithm (GA) for microhone array otimization roblem with the uroe of maximizing beamforming SNR for given oible ditribution of ound ource. Function baed on the tatitical geometry decritor are alied a the objective function of GA. Three otimization exeriment involving linear and lanar array in continuou and dicrete deign ace are erformed under variou acoutic cene. Reult demontrate that rooed algorithm ort out thee uerior geometrie with ignificant SNR imrovement over randomly generated array and regular array. High ucceful rate, raid convergence eed, and fat roceing time oberved in all the exeriment demontrate the feaibility of thi algorithm a a ractical tool for microhone array otimization. Index Term microhone array otimization, genetic algorithm, beamforming I. INTRODUCTION Ditributed microhone ytem are widely alied in ignal enhancing alication uch a teleconference, eech recognition, ound ource localization and acoutic urveillance ytem [1][]. Array erformance uually aeing by it ability to detect, locate, track and cature deired target ignal i greatly affect by microhone ditribution, acoutic condition, and roceing algorithm, while the ditribution of microhone i demontrated to be the critical one to limit the otential imrovement of erformance derived from other factor [3]. Therefore, thi aer focue on the otimization roblem of microhone array with fixed number of element and ecified acoutic cene (oible ditribution of target and noie ource) to earch for the otimal geometry roviding uerior noie ureing ability. Becaue the atial gain attern of microhone array i a comlex nonlinear function of microhone oition, traditional analytical otimization method uually aly linear aroximation and atial erturbation to imly thi roblem, whoe erformance may be limited by the rerequiite of erturbation level and unredictable aroximation error [4][5]. Random or exhautive earching method are alo ued in earch of otimal geometry by evaluating each candidate via Monte Carlo imulation []. However, it i time-conuming and not feaible for large deign ace of element lacement and comlex acoutic cene. Other numerical earching method baed on natural algorithm, uch a genetic algorithm, can be conidered a an alternation to overcome thee limitation [6][7]. Derived from natural election theory, genetic algorithm (GA) i a heuritic earching method exloiting the hitorical information of evolution rocedure to redict new generation with exected better erformance, which ha been demontrated a an effective tool for nonlinear otimization roblem [8][9][1][11]. Thi aer alie GA to the microhone array otimization roblem with the uroe to obtain uerior ability of noie reduction for eech alication. Intead of comuting the gain attern of every candidate via Monte Carlo imulation, a function baed on the relationhi between tatitical decritor of microhone geometry and array erformance metric i alied a the objective function of GA. Exeriment of linear and lanar array are erformed to validate the effectivene and feaibility of thi algorithm in earch of otimal microhone geometrie with re-known knowledge of acoutic cene (oible target and noie ace). The reult in term of SNR are comared to comarable uniform-aced array and randomly generated array via Monte Carlo imulation. II. PROBLEM FORMULATION Conider the acoutic cene with microhone array and ound ource ditributed in a three dimenional ace. Signal received by the th microhone can be exreed a: v ( t; r, r ) = u ( τ ; r ) h ( t τ ; r, r ) dτ. (1) 13 ACADEMY PUBLISHER doi:1.434/jc

2 JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 13 1 where u t; r ) i the reure wave tranmitted from the ( ource located at r, h(.) rereent the imule reone of roagation from r to r a ( t τ ) + a n ( t τ n ) n = 1 h( t; r, r ) = a. () where an i the reone related to the n th ath roagation, and n= rereent the direct ath from ource to microhone. τ i the correonding time delay. Then, the ower of delay and um beamformer focued on r i with a ound ource at r can be comuted a [1] S ( r, r i ) P P = = 1 q = 1 B i B Vˆ ( ω ; r, r iq ex( jω ( τ i ) Vˆ τ )) dω iq * q ( ω ; r., r q ). where B i i the beamformer coefficient related to deired target oition r i and microhone oition r. Normally in order to emhaize ignal received by microhone cloer to target oition, let B i = 1/ di, where di denote the ditance form r i to r. Auming coefficient of beamformer and roagation attenuation are indeendent with microhone oition and only conidering direct ath roagation, by taking mean value over all microhone air, (3) can be rewritten a below [1]: S ( r, r i ) = P ˆ U ( ω ; r d E ex( jπ ( ) q d λ B i B iq d + A i A * q d λ iq (3) )) dω. (4) where P i the number of microhone, and angular bracket denote mean value over all microhone air. For the ound ource different with deire target, r ri, which i conidered a an interference or noie, S( ri, r ) hould be a mall a oible to indicate uerior ability of noie ureion. From (4), it can be een that large an of exonential hae term from π to π can reult in ignificant incoherence and near zero ower gain for non-target oition, while limited delay value over the hae term make artial coherence more likely for ignal received from non-target ource. With the aumtion of contant ignal ower and beamformer coefficient, array ability to decorrelate ound ource at non-target oition i directly related to the differential ditance derived from microhone oition. Therefore, (4) demontrate the ignificant imact of microhone lacement on array beamforming erformance. In order to further formulate thi relationhi, aer [13] rooed everal tatitical decritor to characterize microhone geometry, which are roved to have trong correlation with key erformance metric, uch a Mainlobe Width (MLW) aociated with atial reolution or array ability to ditinguih ignal from cloe ource, and Mainlobe-to-eak-idelobe Ratio (MPSR) aociated with array ability to ure noie, a hown in Table I. Decritor Centroid offet Array dierion Entroy baed tatitic of inter-ath ditance TABLE I. GEOMETRY DESCRIPTORS Illumination The ditance from array centroid to focal oint of beamformer. It ha a direct imact on the mainlobe reolution and hae. The tandard deviation of microhone coordinate about array centroid. Inter-ath ditance i the differential ditance from microhone air to target and noie ource air. For ecified target and noie ource, the ditribution of inter-ath ditance over all microhone air i directly related to the atial gain for the ignal received from the noie ource when teering at target. Relationhi with erformance metric With fixed array dierion, increaed centroid offet bring large MLW and mall MPSR, rereenting degradation of array erformance. Small dierion reult in better MPSR. For MLW, when array dierion increae, the mainlobe area on horizontal lane of target decreae, while the mainlobe region along vertical direction grow. Inter-ath ditance with rich diverity and large read for non-target oition rereent uerior array ability to ure noie. The cloe form relationhi function between geometry decritor and erformance metric rovided by [13] can be alied a the objective function of GA. For a ecified acoutic cene, oible ource ditribution can be deicted by the robability denity function of ource location, related to the behavior attern of eaker and intereted target ace. Then, the objective function of GA can be defined a ' F ( G ) = F ( G, ri, r ) ( r ) dr i T arg etsace r NoieSace ( ri ) dri. (5) r where F G, r, r ) i the relationhi function rovided ( i by [13] for the focal oint r i and noie ource at r. Smaller value of F G, r, r ) rereent better ability to ( i 13 ACADEMY PUBLISHER

3 JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 13 decorrelate the ignal received from thi noie ource when teering at target. G i the et of geometry decritor to identify a cla of microhone array with imilar erformance. ( r i ) and ( r ) are the robability denity function rereenting exected location of focal oint for the target and noie ource occurrence, reectively. Therefore, the criterion to earch for otimal array geometry can be written a ' Got = argmin F ( G). (6) III. GENETIC ALGORITHM A a heuritic earching method, which ue emirical information to earch for better olution and reduce the comutational comlexity, GA ha been demontrated to be effective for olving nonlinear otimization roblem [6][7][8][9][1][11]. The main ideal of thi algorithm i to ue hitorical information of evolution rocedure to guide earching direction by redicting new generation with higher fitne value. Following the rule of urvival of the fittet, gene of the individual with higher fitne value will have more chance to be inherited by offring, while erturbation are introduced randomly to the oulation to enhance diverity of evolution. In our cae, all coordinate of microhone array are conidered a an individual. The fitne value i aeed by the objective function defined in (5). Then arent are elected baed on fitne value undergoing croover and mutation to give birth to new generation. The evolution rocedure continue until reaching accetable fitne value or the limitation of generation number. General flow chart i hown in Fig. 1 and detail of GA are alied a below: Coding: individual of GA i defined a { x 1, x,..., xp, y1, y,..., yp, z1, z,..., z P}, where { x, y, z } i the coordinate of th microhone, P i the number of microhone. Selection: evaluate and rank the fitne of individual in the oulation by the objective function. According to the ideal of urvival of the fittet, uually to % individual are elected a elite to give birth to the offring by croover and mutation. Croover: a hown in Fig., croover i imlemented by randomly chooing one coordinate of offring from the correonding arent' coordinate in each dimenion. In our cae, 6% of elite are devoted into croover roce to generate new individual. Mutation: 4% of elite are ued for mutation by adding random atial erturbation to each dimenion of arent, a hown in Fig.. Thee erturbation are generated from a zero mean normal ditribution. The tandard deviation of thi normal ditribution can control the level of erturbation, and further affect the convergence of GA. Generally, large erturbation can increae the diverity of evolution, which i good for avoiding GA traed in local otimum. However, it will reduce the convergence eed by ditracting the evolution from current otimal direction. In thi aer, the tandard deviation of erturbation i derived from [1] ( n 1) σ ( n) = σ ρ. (7) where n i the iteration number, σ i the initial tandard deviation related to the wavelength of imortant ignal band. ρ i a contant to control the hrinkage rate of erturbation along generation. The idea here i to add large erturbation at the beginning of iteration to tart global earch of otimum. Along with the evolution rocedure, when the bet earching direction become more ecified, the erturbation level i reduced to eed u the convergence to the otimal olution. Relacement: in order to inure the urvival of highfitne individual, offring are ranked together with the old oulation baed on fitne. Illegal individual, which are outide deign ace of microhone oition or reeat with the other individual, are relaced by random generated one. Then the new generation i orted out with the ame ize of initial oulation. Figure 1. Flow chart of genetic algorithm Figure. Croover and mutation After the termination of iteration, to verify the actual erformance of the otimal individual, Monte Carlo imulation are erformed over the lat generation of GA to ick three array with to SNR reult a the outut of otimization rocedure. Note that maintaining the balance between inheritance and exloration i critical for the ucce of GA otimization. It mean that the tradeoff between earching diverity to enure global otimum (comuting comlexity) and convergence rate need to be 13 ACADEMY PUBLISHER

4 JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 13 3 taken into account when electing GA arameter, uch a the ize of initial oulation, ratio of arent election, ratio of croover and mutation, and the level of erturbation. IV. EXPERIMENTS In thi ection, otimization exeriment for 1D linear array and D lanar array with variou acoutic cene are erformed to evaluate the erformance of GA. The SNR of the otimal geometrie derived from the lat generation of GA are comuted and comared to comarable regular array via Monte Carlo imulation. A. Setting All the exeriment are erformed in a 1 1 m room. Colored noie generated by the band imortance function from the SII model [3] i conidered a the ource ignal for both target and noie, which emhaize the mot imortant frequency band for eech intelligibility. A hown in Table II, thee three exeriment involve different microhone deign ace and ource ditribution. To ae the erformance in an indeendent manner for all the exeriment, Monte Carlo imulation uing delay and um beamformer are erformed to evaluate the SNR of the otimal geometrie from the lat generation of GA, comared with randomly generated geometrie from the firt generation of GA and comarable regular array. Table III give the GA arameter ued for each otimization roblem. A robut otimization algorithm need to maintain the balance between inheritance and exloration, which could be greatly affected by the relative GA arameter. Therefore, in our exeriment, the initial oulation ize, ratio of arent election and the maximum iteration number are adjuted according to the knowledge of acoutic environment and deign ace. B. Reult Analyi Table IV how SNR reult of 3 bet array in the lat and firt generation of GA, rereenting the otimal geometrie and random generated geometrie, reectively. The mean SNR of each generation are alo rovided with ± one tandard deviation. By comaring the mean SNR of the firt and lat generation, ignificant imrovement i oberved to demontrate the effectivene of GA iteration and the objective function rule. It i noted that by roviding a moderate mean value of SNR the firt generation of GA include uerior array outerforming regular array and inferior array howing lower SNR than regular array. Through the GA otimization rocedure including the objective function of geometry decritor, thoe uerior array are orted out to comoe the lat generation with much higher SNR reult. For roblem 1 and roblem, where the target ace i comletely overlaed with the noie ace, the imrovement of mean SNR for the lat generation of GA i over three tandard deviation greater than the other array. For roblem 3, where the continuou ace for oible target and noie ditribution do not overla, the imrovement of SNR i even more ignificant, which almot double the SNR of randomly generated irregular array and regular array. 1D linear array D lanar array D lanar array TABLE II. OPTIMIZATION PROBLEMS Acoutic cene 5 dicrete ource are randomly ditributed in the room. Each ource ha equal oortunity to be elected a the target, while the other act a interference with ¾ chance to make noie. 5 dicrete ource are randomly ditributed in the room. Two of them are uniformly conidered a the intereted target, while the other act a interference. Target ace i a m region located in the center of room. Each atial oint in that region ha an equal oortunity to be conidered a the target oition. Noie ace i a. m region, where 1 dicrete noie ource are read uniformly inide the noie ace. Initial oulation. Ratio of elite Croover robability Mutation robability Iteration threhold TABLE III GA PARAMETERS Deign ace of microhone Equally aced 16 hole along x axi located in the middle of wall. (dicrete deign ace) Ceiling of the room. (continuou deign ace) Ceiling of the room. (continuou deign ace) Problem 1 Problem Problem % % % 6% 6% 6% 4% 4% 4% Iter = 1 Iter = Iter = Mic number The gain attern of the otimal array and comarable regular array for all the roblem are rovided in Fig. 3 and Fig. 4. It i aeared that through the otimization of GA, thee high idelobe which will everely degrade array erformance are uccefully teered away from the noie ource region, rereenting uerior ability for noie ureion. Although for roblem 1 the main lobe area of GA otimal array i larger than the regular array, it circumvent the noie ource oition. With lower ower gain for all the noie ource when teering at target, the otimal array till how better SNR erformance than the regular array. Furthermore, by viually inecting the otimal microhone ditribution in ACADEMY PUBLISHER

5 4 JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 13 each roblem, it conclude that for dicrete ource the microhone are clutered near the ource oition, while for the continuou ace of target and noie ditribution the microhone denitie are very high near the target ace but arely ditributed over the noie ace. Problem 1 Problem Problem 3 TABLE IV. SNR RESULTS OF ARRAY GEOMETRIES The lat generation of GA To 3 SNR Mean SNR 9.1 (±.47) 5.3 (±1.63) 3.5 (±.36) The firt generation of GA To 3 SNR Mean SNR 6.59 (±.69) 4.67 (±3.85) (±.88) Regular array SNR A mentioned before, the ize of initial oulation and maximum iteration number of the otimization trategy are the imortant factor to influence the earching diverity (reliability) and convergence of iteration (efficiency). Poor balance between the reliability and efficiency of GA will make the evolution rocedure eaily traed in local otimum or ditracted away from the otimal earching direction. Fig. 5 how the evolution of minimum and average value of the objective function along generation. Three different initial oulation are alied to comare the convergence of iteration. It can be een that for roblem and roblem 3, the objective function with different initial oulation converge to different value, which cannot be eliminated by the iteration. And for roblem 1 with le oible olution, large initial oulation doen t reult in better convergence value. Therefore, it conclude that inufficient ize of initial oulation cannot be comenated by the increae of maximum iteration number. It will retrict the evolution in local otimum from early iteration. On the other hand, exceive initial oulation only reult in limited imrovement of otima, but greatly increae the comutational comlexity during the earching rocedure (a) (b) (c) Figure 3. To view gain attern of otimal geometrie of GA when targeting at the center ource of the field of view. Circle rereent microhone oition. Cro or quare area rereent ource ace. (a) Problem 1. (b) Problem. (c) Problem 3 with the target ace in the center of the field of view. Therefore, chooing a moderate ize of initial oulation i critical for the ucce of the otimization trategy. C. Reliability and Convergence Seed In thi ection, ucceful rate and convergence eed are rovided to validate the reliability and efficiency of GA. 3 run for each roblem are alied to comute thee arameter. Reult are hown in Table V. Succeful rate i a good meaure to evaluate the reliability or robune of otimization trategy. High ucceful rate i neceary for the real-cae alication ACADEMY PUBLISHER

6 JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY (a) (b) (c) Figure 4. To view gain attern of regular geometrie when targeting at the center ource of the field of view. Circle rereent microhone oition. Cro or quare area rereent ource ace. (a) Problem 1. (b) Problem. (c) Problem 3 with the target ace in the center of the field of view. In thi aer, 3 exeriment for each roblem are erformed, while the ource location are randomly hifted inide the room. The criterion for the ucce of otimization i that SNR reult of GA otimal array outerform regular array. It can be een that GA how atifying ucceful rate ( 8%) in all the otimization roblem. Convergence eed to global otimum reflect the efficiency of evaluation. It i uually meaured by the average iteration number needed to achieve convergence Objective Function Objective Function Objective Function Iteration (a) Initial oulation 5 Initial oulation 1 Initial oulation Iteration (b) Initial oulation 5 Initial oulation 1 Initial oulation Iteration (c) Initial oulation Initial oulation 4 Initial oulation 6 Figure 5. Evolution of minimum (in red) and average (in blue) value of objective function along generation. (a) Problem 1. (b) Problem. (c) Problem 3 13 ACADEMY PUBLISHER

7 6 JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 13 TABLE V. SUCCESSFUL RATES AND CONVERGENCE SPEEDS OF GA Problem 1 Problem Problem 3 Succeful rate 83% 1% 8% Convergence eed Time conuming (comared with random earch method ) %.5%.5% A hown in Table V, raid convergence eed are oberved in all the exeriment, demontrating the feaibility of thi algorithm a a ractical tool for microhone array otimization. In addition, the time conuming feature of GA i comared to reviou random earch method via Monte Carlo imulation. A hown in Table V, it can be een that by alying the objective function of array geometry decritor and evolutionary algorithm during otimization rocedure, intead of directly earching via Monte Carlo imulation for every candidate, 99.5% running time i aved, while GA howing much higher robability to catch thee uerior array. Thi advantage of fat running time would be articularly alicable to dynamic ituation where the acoutic cene or environment cannot be known well in advance, uch a audio urveillance ytem. V. CONCLUSION Thi aer reented a genetic algorithm to otimize element lacement of microhone array with the uroe of maximizing beamforming SNR for given knowledge of acoutic cene or environment. Three otimization exeriment related to correonding actual real-world roblem wa erformed. Simulation reult of Monte Carlo exeriment demontrate that thi algorithm effectively ort out uerior geometrie with ignificant SNR imrovement over randomly generated irregular array and regular array. Furthermore, high ucceful rate and raid convergence eed are oberved in all the exeriment, a well a fat roceing time which i 99.5% le than the traditional random earching method, demontrating the feaibility of thi algorithm a an effective tool for microhone array deign. Thi otimization algorithm can be eecially ueful for immerive environment alication, uch a audio urveillance ytem where the field of interet i fixed and the ituation of ound ource cannot be known well in advance. Future work will extend thi algorithm to the three dimenional deign ace and comlex acoutic environment with reverberation. The imact of GA arameter will alo be tudied to imrove the effectivene and robutne of thi algorithm for different otimization roblem. [1] J. Benety, J. Chen, and Y. Huang, Microhone Array Signal Proceing, Sringer, 8. [] D. Rabinkin, R. Renomeron, and J. French, Otimum enor lacement for array ound cature, in Proc. Int. Soc. for Ot. Eng., [3] J. Townend, K. Donohue, Beamfield analyi for tatitically decribed lanar microhone array, Proceeding of the IEEE Southeatcon, 9. [4] C. Yu, Sidelobe reduction of aymmetric linear array by acing erturbation, IEEE Electronic Letter, [5] C. Yu, Sidelobe reduction of a ymmetric broadide array by acing erturbation, Microwave and Otical Technology Letter, [6] S. Rao, Engineering Otimization Theory and Practice, John Wiley & Son, [7] G. Reklaiti, A. Ravindran, and K. Ragdell, Engineering Otimization Method and Alication, John Wiley & Son, [8] M. Simon, C. Hanen, Ue of genetic algorithm to otimize vibration actuator lacement for active control of harmonic interior noie in a cylinder, Noie Control Eng. J, vol. 44, [9] A. Montazeri, J. Pohtan, and M. Kahaei, Otimal lacement of loudeaker and microhone in an encloure uing genetic algorithm, Control Alication, 3. [1] I. Haya, B. Peteren, and B. Colitt, Otimum -D LOS MIMO erformance uing omni-directional antenna attained through genetic algorithm, Com. Network Svc Re. Conf., 8. [11] R. Akbari, K. Ziarati, A multilevel evolutionary algorithm for otimizing numerical function, International J. of Indutrial Eng. Comutation, 1. [1] K. Donohue, S. SaghaianNejadEfahani, and J. Yu, Contant fale alarm rate ound ource detection with ditributed microhone, EURASIP J. on Advance in Signal Proceing, 11. [13] J. Yu, K. Donohue, Performance for randomly decribed array, IEEE Workho on Alication of Signal Proceing to Audio and Acoutic, 11. [14] H. Schjaer-Jacoben, K. Maden, Synthei of nonuniformly aced array uing a general nonlinear minimax otimization method, IEEE Tranaction on Antenna and Proagation, [15] S. Shanan, C. Pomalaza-Raez, The ue of nonuniform element acing in array roceing algorithm, Journal of the Acoutical Society of American, [16] K. Donohue, K. McReynold, and A. Ramamurthy, Sound ource detection threhold etimation uing negative coherent ower, Proceeding of the IEEE Southeatcon, 8. [17] S. Gazor, Y. Grenier, Criteria for oitioning of enor for a microhone array, IEEE. Tranaction on Seech and audio roceing, [18] D. Cheng, Otimization technique for antenna array, Proceeding of IEEE, [19] D. Dudgeon, Fundamental of digital array roceing, Proceeding of IEEE, [] C. Balani, Antenna Theory and Deign, New York: Wiley, [1] M. Brandtein, D. Ward, Microhone array ignal roceing technique and alication, Sringer, 1. REFERENCES 13 ACADEMY PUBLISHER

8 JOURNAL OF COMPUTERS, VOL. 8, NO. 1, JANUARY 13 7 Jingjing Yu wa born in Henan, China. She received the B.S. and M.S. degree in communication engineering from Beijing Jiaotong Univerity, Beijing, China, in 5 and 7, reectively. She i currently a Ph.D. candidate in Electrical Engineering Deartment at the Univerity of Kentucky, Lexington, KY. And he i alo a reearch aitant in the Center for Viualization & Virtual Environment at the Univerity of Kentucky. Her current reearch interet include communication and information ytem, audio ignal roceing, and engineering otimization. M. Yu i a tudent member of the IEEE. Fahan Yu wa born in Hubei, China. He received the B.S. degree in automation from Henan Polytechnic Univerity, Jiaozuo, China, in July He i currently a Profeor and Dean of the School of Electrical Engineering & Automation at Henan Polytechnic Univerity, Jiaozuo, China. He i alo aointed a the Director of National Electrical & Electronic Exeriment Center at Henan Polytechnic Univerity. He i the author and coauthor of numerou aer and of everal book in the field of Control Theory and Control Engineering. Hi current reearch interet include indutrial roce control, PLC control, comuter imulation, AC DC eed control ytem, etc. Prof. Yu i a member of the Mine Automation Committee of China, and a member of the teaching guide committee on Automation organized by the Minitry of Education of China. Yinglei Li wa born in Zhejiang, China. She received the B.S. degree in textile trading from Donghua Univerity, Shanghai, China, in 3, the M.S. degree in economic and management from Tongji Univerity, Shanghai, China, in 8, and the M.S. degree in tatitic from the Univerity of Kentucky, Lexington, USA, in 1. She i currently a Ph.D. candidate in Statitic Deartment at the Univerity of Kentucky, Lexington, USA. Her current reearch interet include biotatitic, multivariate tatitic and tatitical modeling. M. Li i a member of American Statitical Aociation. 13 ACADEMY PUBLISHER