Understanding K-Means Non-hierarchical Clustering
|
|
- Theresa Tucker
- 5 years ago
- Views:
Transcription
1 SUNY Albany - Techncal Report 0- Understandng K-Means Non-herarchcal Clusterng Ian Davdson State Unversty of New York, 1400 Washngton Ave., Albany, 105. DAVIDSON@CS.ALBANY.EDU Abstract The K-means algorthm s a popular approach to fndng clusters due to ts smplcty of mplementaton and fast executon. It appears extensvely n the machne learnng lterature and n most datamnng sutes of tools. However, ts mplementaton smplcty masks an algorthm whose behavor s complex. Understandng the algorthm s behavor s vtal to successful use and nterpretng the results. In ths paper we dscuss nne nsghts nto the behavor of the algorthm n the clusterng context and llustrate how they affect preparaton/representaton of the data, formulaton of the learnng problem and fnally, how we nterpret and use the resultant clusters. We dscuss mplct problems wth K-Means clusterng that cannot be overcome. 1. Introducton The feld of ntrnsc classfcaton attempts to take a collecton of homogeneously descrbed nstances and group them nto sub-populatons commonly known as classes or clusters. Intrnsc classfcaton s nherently densty estmaton as one tres to fnd the varous dense collectons of nstances n a m dmensonal space f m attrbutes represent each nstance. Intrnsc classfcaton s also parameter estmaton as we often dentfy the dstrbuton of each of the m attrbutes for each cluster. We wll also use the terms cluster and classes nterchangeably. Intrnsc classfcaton has two popular sub-felds: clusterng and mxture modelng though the names are often used nterchangeably. Clusterng attempts to maxmally separate the subpopulatons by exclusvely assgnng (known as hard assgnment) an nstance to only one class. Ths s effectvely fndng the best set partton of the nstances or nstallng hard hyperplanes that partton the m dmensonal space. Colloqually, clusterng attempts to fnd groups of nstances, so that the nstances wthn a group are smlar whlst beng dssmlar to those nstances n all other groups. The am of mxture modelng s to model the underlyng sub-populatons, not to fnd classes that maxmally separate the nstances. Dependng on the nstance's degree of belongng (or dstance) to a class, t s partally assgned (soft assgnment) to one or more classes. Mxture modelng technques are usually probablstc and often Bayesan. To llustrate the dfference consder Fgure. A clusterng technque (lke K-Means) wll most lkely fnd that there exsts only one cluster. A mxture modeler wll separate the two classes because there s an underlyng dfference n the classes even though the nstances are qute smlar. We can say that a mxture modeler attempts to fnd the mplct structure n the data whlst a clusterng tool attempts to explctly ntroduce a segmentaton scheme to group together smlar nstances. In ths paper, we wll descrbe the K-Means algorthm for clusterng. The algorthm s extremely popular and appears n at least fve popular commercal data mnng sutes [1]. We wll outlne and descrbe nne nsghts nto K-Means clusterng. Each nsght affects at least one of: a) How we prepare the data for clusterng b) How we formulate the problem and use the clusterng tool c) How we nterpret the clusters and use them. The nsghts are both notes of cauton (whch can be overcome) and nherent lmtatons of the approach (whch cannot be overcome). Bayesan mxture modelng tool that uses the MML prncple and MCMC samplng can n theory overcome the lmtatons of K-Means clusterng []. The nsghts we wll dscuss are: 1. The Effect Of Exclusve Assgnment. The Inconsstency of the learnng algorthm 3. The learnng algorthm s not nvarant to non-lnear 4. The learnng algorthm s not nvarant to scale 5. The learnng algorthm fnds the local mnma of ts loss functon, whch s the vector quantzaton error (dstorton). 6. The learnng algorthm provdes based class parameter estmates 7. The learnng algorthm requres the a-pror specfcaton of the number of classes
2 SUNY Albany - Techncal Report 0-8. Eucldean dstance measures can unequally weght attrbutes 9. Non-parametrc modelng of contnuous attrbutes. We begn the paper by outlnng the hstory of the algorthm, then descrbe the algorthm tself and dscuss ts computatonal behavor. We then llustrate and dscuss our nne nsghts and how each nsght effects applcaton of the clusterng algorthm. We conclude by llustratng the contrbuton of ths paper.. The Algorthm The K Means clusterng algorthm was postulated n a number of papers n the nneteen sxtes [3][4]. For a m attrbute problem, each nstance maps nto a m dmensonal space. The cluster centrod descrbes the cluster and s a pont n m dmensonal space around whch nstances belongng to the cluster occur. The dstance from an nstance to a cluster center s typcally the Eucldean dstance though varatons such as the Manhattan dstance (step-wse dstance) are common. As most mplementatons of K-Means clusterng use Eucldean dstance, ths paper wll focus on Eucldean space. The K-means algorthm consst of two prmary steps: 1) The assgnment step where the nstances are placed n the closest class. [1] The re-estmaton step where the class centrods are recalculated from the nstances assgned to the class. We repeat the two steps untl convergence occurs whch s when the re-estmaton step leads to mnmal change n the class centrods. Algorthmcally, the K-means algorthm n ts general form dffers only slghtly to the EM algorthm [5] though the loss functons and results obtaned usng them can dffer greatly. The frst step of the K Means algorthm nvolves exclusve assgnment of nstances to the closest class. The EM algorthm partally assgns nstances to clusters; the porton of the nstance assgned dependng on how probable (or lkely) the class generated the object. The Eucldean dstance between two nstances X and Y whch are represented by m contnuous attrbutes s: d ( X, Y) ( X 1 Y1 ) + ( X Y )... + ( X m Ym ) = ( 1 ) In the second step, the algorthm uses the attrbute values of the nstances assgned to a cluster to recalculate the cluster s centrod. We recompute the estmates from only those nstances currently assgned to the class. Suppose an nstance belongs to class A wth probablty a and class B wth probablty b. If a s larger than b under the K means algorthm the nstance would be assgned totally to class A even f the dfference between a and b s mnmal. The nstance would only contrbute to class A s centrod locaton. 3. Computatonal Behavor of the Algorthm The K Means algorthm ams to fnd the mnmum dstorton wthn each cluster for all clusters. The dstorton s also known as the vector quantzaton error. Let the k classes partton the nstances nto the subsets C 1 k, the cluster centrods be represented by w 1 k and the n elements to cluster be S 1 n. The mnmum dstorton or vector quantzaton error that the K means algorthm attempts to mnmze s: E = ( ) The mathematcal trval soluton that mnmzes ths expresson s to have a cluster for each nstance. Recent work has nvestgated the algorthm from an nformaton theoretc perspectve [6]. The major nsght from ths work s that any algorthm that attempts to mnmze the dstorton must manage a trade-off the authors to refer to as the nformaton-modelng trade-off. The trade-off s between dstrbutng the nstances amongst the k clusters and how well each cluster models the nstances assgned to t. For a two-class problem, the expectaton of the partton loss wth respect to the samplng densty Q (the true dstrbuton of the objects) s: Ε x Q[ F where K N j= 1 = 1 S C j S w Class( S ) χ ( x)] = ω KL( Q P ) + ω KL( Q P ) I( Q ) ( 3 ) Q 0, Q 1 are the samplng densty of the frst and second sub-populatons respectvely P 0, P 1 are the densty estmaton of the frst and second clusters respectvely ω 0 ω 1 are the weghts of the frst and second clusters respectvely. F s the set partton mparted by the clusterng soluton The frst two terms of ths expresson are the Kullback- Lebler dstance between the samplng densty and the hypotheszed sub-populaton densty for each cluster. Ths measures how well the clusters model the two subpopulatons ndvdually. The thrd expresson s how much the partton reduces the entropy of Q and measures
3 SUNY Albany - Techncal Report 0- the effectveness of dstrbutng the nstances amongst the clusters. From equaton ( 3 ) we see that the ndvdual classes are modeled separately as s the dstrbuton of the nstances amongst the classes. There are no terms consderng the nteracton between the classes. 4. Nne Insghts nto K-Means Clusterng 4.1 The Use of Exclusve Assgnment The assgnment mechansm of a K-means clusterng requres that an nstance belong to only one of the clusters. Consder the followng stuatons: 1. If an nstance could belong to two classes, t must only be assgned to one.. Instances that do not belong to any partcular cluster are assgned to one. These nusance or outler nstances can affect the poston of class centrods n relatvely small szed clusters. Ths has two mplcatons. Frstly, f there are two hghly overlappng classes (.e. Fgure ) then K-Means wll not be able to resolve that there are two clusters. Secondly, the class centrods wll be based away from the true class centrods (ths wll be descrbed n a later nsght). If the sze of a class s small or the number of nusance nstances large then the cluster centrods maybe dstorted from ther true value. The nablty to model overlappng classes s nherent to the K-Means technque and cannot be overcome. We can see ths clearly n equaton ( 3 ) as the classes are modeled separately and an nstance can only belong to one class. Therefore, nstances n the regons where the classes overlap are problematc. The effect of nusance nstances on the cluster centrods can be partally overcome f the cluster szes are relatvely large 4. The learnng algorthm s Inconsstent Consder a model space Θ k, whch contans models of only k classes, n whch θ TRUE s the true model that generated the nstances. Intally, there maybe only a small number of nstances so θ TRUE s not the most probable model. If a learnng algorthm s consstent then we fnd that: lm P( θ TRUE ) = 1 n where ns the number of observatons ( 4 ) That s, as the amount of data ncreases the probablty that the true model s the most probable model approaches certanty. An nconsstent learnng algorthm does not have ths property and results n overlookng the true model n favor of ncreasngly complex (more classes) models. Consder the loss functon of K-means ( ), the (trval) optmal soluton s to have a cluster for each nstance. It s precsely ths bas whch leads the learnng algorthm to consstently favor ncreasngly (as more data s avalable) complcated models. 4.3 The learnng algorthm s not nvarant to nonlnear Learnng algorthms that are nvarant to non-lnear have the desrable property that regardless the data representaton, the most probable model wll be the same. Consder the representaton of the same data usng Eucldean co-ordnates and polar coordnates. We would hope that n both representatons the best model would be the same as the data s the same. However, ths s not necessarly the case wth all learnng algorthms. Eucldean dstance s not nvarant to non-lnear and as the K-means algorthm uses ths dstance to assgn nstances we fnd that dfferent representatons of the same data can gve dfferent results. Therefore, the result of the K-means clusterng algorthm wll depend on the representaton of the data as well as the ntrnsc structure amongst the nstances. 4.4 The learnng algorthm s not nvarant to scale From the defnton of the Eucldean dstance ( 1 ) we can see that the attrbutes wth a larger scale provde a larger contrbuton to the dstance. Ths means attrbutes wth a larger range of values contrbute more to the dstance functon than those wth smaller scales. A common soluton to ths s to transform each attrbute to Z scores (x j =(µ j -x j )/σ j ) or 0-1 scores (x j =(x j - mn j )/max j ). Ths transforms all attrbutes to be wthn the approxmate range [-1,1] and [0,1] respectvely. However, ths transformaton effectvely provdes more weght to those attrbutes wth a smaller standard devaton. 4.5 The learnng algorthm provdes the local optma of the vector quantzaton error (dstorton) The vector quantzaton error, equaton ( ), s locally maxmzed by the K-means algorthm. The vector quantzaton error s the error surface (or objectve functon) whch K-Means tres to mnmze. The algorthm performs a gradent descent of ths error functon and can therefore become stuck n local mnma. For most nterestng practcal problems the error surface wll contan many local mnma [7]. 4.6 The learnng algorthm provdes based class parameter estmates The use of exclusve assgnment means the algorthm does not model overlappng classes well. Consder a
4 SUNY Albany - Techncal Report 0- unvarate example of two classes whch are created from the Gaussan dstrbutons N(µ=0,σ=1) and N(µ=,σ=1). We wll fnd that nstances generated from each of the nstances wll overlap n the same area of nstance space. Dagrammatcally we can llustrate the stuaton n Fgure 1. Due to the requrement of exclusve assgnment, we wll fnd nstances generated from the rght hand tal of class 1 wll be assgned to class and the nstances from the left hand tal of class wll be assgned to class 1. When the class centrods are then calculated they wll devate from the generaton mechansms. The mean of class 1 wll be under-estmated and the mean of class over-estmated. The algorthm s estmate of the two sub-populatons means s based. Class 1 Class Fgure 1 Two overlappng classes centered on N(µ=0,σ=1) and N(µ=,σ=1). 4.7 The learnng algorthm requres the a-pror specfcaton of the number of classes The K means algorthm requres the apror specfcaton of the number of classes. The model space t explores s all possble models wth k classes and effectvely removes k (an mportant unknown n ntrnsc classfcaton) from the problem. One can select a desrable range of k and use the algorthm for each value wthn the range, but due to the algorthm fndng only a local mnmum, ths process would need to be completed many tmes for each value to get the best model for each k. However, we cannot easly compare models obtaned for dfferent values of k. The dstorton for models wth a large k wll have a greater potental to be lower to than those models wth a small k. We are somewhat stuck. The algorthm requres a specfcaton of k but we cannot compare the loss functon across dfferent values of k. Of course, the models could be compared qualtatvely n terms of ftness for busness purposes. 4.8 Eucldean dstance measures can unequally weght underlyng factors. Consder the stuaton where our nstances are represented by say ten attrbutes. Suppose that these attrbutes are manfestatons of two underlyng factors A and B and that eght of the attrbutes represent A and two B. That s the attrbutes representng a factor are hghly correlated. As the number of attrbutes representng the factors s not equal, then the algorthms dependence on the Eucldean dstance ( 1 ) wll weght factor A more hghly than B n the analyss. There are two methods of overcomng ths problem. Frstly, one can perform factor analyss to reduce the ten attrbutes to two attrbutes (one for each factor) to use n the analyss. That s the nput to the clusterng process s two attrbutes not ten. Alternatvely, one may drectly compute the correlaton between each and every attrbute and use a modfed form of Eucldean dstance called the Mahalanobs dstance [8]. The Mahalanobs dstance effectvely performs a transformaton on a Eucldean space so that correlated attrbutes are closer together and therefore do not contrbute overwhelmngly to the dstance measure. In Eucldean dstance the axes are at rght angles to each other. However, the Mahalanobs dstance effectvely transforms the axes so that the angle between two axes s nversely proportonal to the correlaton between the two attrbutes. However, the Mahalanobs dstance s computatonally ntensve to compute as the number of attrbutes ncrease. 4.9 Non-parametrc modelng of contnuous attrbutes. As earler stated ntrnsc classfcaton s densty estmaton wth the am of fndng patterns whch solate dense sub-populatons of nstances. How we represent the sub-populatons lmts whether we wll be able to fnd them. The K-means algorthm specfes a class by descrbng ts centrod. However, t s well establshed that contnuous attrbutes can be successfully modeled as beng drawn from some parametrc populaton such as the Gaussan, Posson or bnomal. Ths provdes a more rcher class descrpton language and hence the ablty to specfy more complex patterns. As ths method of modelng attrbutes s gnored n K-Means clusterng then some patterns whch exst n the populaton are overlooked. Consder a unvarate two class stuaton shown n Fgure.
5 SUNY Albany - Techncal Report 0- Fgure. Two overlappng classes wth the same mean. The classes only dffer by ther standard devaton as ther means are the same. However, the K-Means algorthm would not dfferentate between the two classes because t does not model the specfc aspect whch dfferentates them. 5. Concluson Class 1 Class A classc problem dlemma posed by phlosophers can be paraphrased as, do the theores/models we dscover descrbe the mplct structure that exsts n nature or s the structure we force onto t. If we use K-Means clusterngs to nduce the theory/model then the answer to the queston s the later suggeston. K-Means clusterng s not orented towards fndng the mplct structure n the nstances, rather, t fnds the structure wthn the bounds we explctly state for t. Ths s an mportant message to understand, that the K-Means algorthm produces results whch are an nteracton between the ntrnsc structure n the nstances and how we represent the data and defne the clusterng problem. To use the K-Means clusterng algorthm successfully, care must be taken to totally understand the behavor of the algorthm. Our nne nsghts help to provde a more complete understandng of the algorthm for practoners. We rase several problems wth the algorthm. Some can be overcome at least partally and we suggest ways to acheve ths, the remanng problems cannot be overcome and ther affects need to be consdered when representng the data, formulatng the problem and nterpretng the results of the clusterng process. References [1] [1] The Two Crows Report: Avalable at [] Davdson, I., Mnmum Message Length Clusterng Usng Gbbs Samplng, The 16th Uncertanty n Artfcal Intellgence Conference, Stanford Unversty, 000. [3] Max, J., Quantzng for Mnmum Dstorton, IEEE Transactons on Informaton Theory, 6, pages 7-1, 1960 [4] MacQueen, J., Some Methods for classfcaton and analyss of multattrbute nstances, Proceedngs of the Ffty Berkeley Symposum on Mathematcs, Statstcs and Probablty, volume 1, pages 81-96, [5] Dempster, A.P et al, Maxmum Lkelhood from ncomplete data va the EM algorthm, Journal of the Royal Statstcal Socety B, Vol 39 pages 1-39, [6] Kearns, M., Mansour, Y., Ng, A., An Informaton-Theoretc Analyss of Hard and Soft Assgnment Methods for Clusterng, Proceedngs of the Internatonal Conference on Uncertanty n Artfcal Intellgence, [7] Glks, W.R., Rchardson, S. and Spegelhalter D., Markov Chan Monte Carlo In Practce, Chapman and Hall, [8] Mahalanobs. P.C., On Tests and Measures of Groups Dvergence, Journal of the Asatc Socety of Benagal, 6:541, 1930.
Subspace clustering. Clustering. Fundamental to all clustering techniques is the choice of distance measure between data points;
Subspace clusterng Clusterng Fundamental to all clusterng technques s the choce of dstance measure between data ponts; D q ( ) ( ) 2 x x = x x, j k = 1 k jk Squared Eucldean dstance Assumpton: All features
More informationOutline. Type of Machine Learning. Examples of Application. Unsupervised Learning
Outlne Artfcal Intellgence and ts applcatons Lecture 8 Unsupervsed Learnng Professor Danel Yeung danyeung@eee.org Dr. Patrck Chan patrckchan@eee.org South Chna Unversty of Technology, Chna Introducton
More informationCS 534: Computer Vision Model Fitting
CS 534: Computer Vson Model Fttng Sprng 004 Ahmed Elgammal Dept of Computer Scence CS 534 Model Fttng - 1 Outlnes Model fttng s mportant Least-squares fttng Maxmum lkelhood estmaton MAP estmaton Robust
More informationUnsupervised Learning
Pattern Recognton Lecture 8 Outlne Introducton Unsupervsed Learnng Parametrc VS Non-Parametrc Approach Mxture of Denstes Maxmum-Lkelhood Estmates Clusterng Prof. Danel Yeung School of Computer Scence and
More informationParallelism for Nested Loops with Non-uniform and Flow Dependences
Parallelsm for Nested Loops wth Non-unform and Flow Dependences Sam-Jn Jeong Dept. of Informaton & Communcaton Engneerng, Cheonan Unversty, 5, Anseo-dong, Cheonan, Chungnam, 330-80, Korea. seong@cheonan.ac.kr
More informationMachine Learning. Topic 6: Clustering
Machne Learnng Topc 6: lusterng lusterng Groupng data nto (hopefully useful) sets. Thngs on the left Thngs on the rght Applcatons of lusterng Hypothess Generaton lusters mght suggest natural groups. Hypothess
More informationSupport Vector Machines
/9/207 MIST.6060 Busness Intellgence and Data Mnng What are Support Vector Machnes? Support Vector Machnes Support Vector Machnes (SVMs) are supervsed learnng technques that analyze data and recognze patterns.
More informationUnsupervised Learning and Clustering
Unsupervsed Learnng and Clusterng Why consder unlabeled samples?. Collectng and labelng large set of samples s costly Gettng recorded speech s free, labelng s tme consumng 2. Classfer could be desgned
More informationMachine Learning: Algorithms and Applications
14/05/1 Machne Learnng: Algorthms and Applcatons Florano Zn Free Unversty of Bozen-Bolzano Faculty of Computer Scence Academc Year 011-01 Lecture 10: 14 May 01 Unsupervsed Learnng cont Sldes courtesy of
More informationCluster Analysis of Electrical Behavior
Journal of Computer and Communcatons, 205, 3, 88-93 Publshed Onlne May 205 n ScRes. http://www.scrp.org/ournal/cc http://dx.do.org/0.4236/cc.205.350 Cluster Analyss of Electrcal Behavor Ln Lu Ln Lu, School
More informationHelsinki University Of Technology, Systems Analysis Laboratory Mat Independent research projects in applied mathematics (3 cr)
Helsnk Unversty Of Technology, Systems Analyss Laboratory Mat-2.08 Independent research projects n appled mathematcs (3 cr) "! #$&% Antt Laukkanen 506 R ajlaukka@cc.hut.f 2 Introducton...3 2 Multattrbute
More informationSLAM Summer School 2006 Practical 2: SLAM using Monocular Vision
SLAM Summer School 2006 Practcal 2: SLAM usng Monocular Vson Javer Cvera, Unversty of Zaragoza Andrew J. Davson, Imperal College London J.M.M Montel, Unversty of Zaragoza. josemar@unzar.es, jcvera@unzar.es,
More informationX- Chart Using ANOM Approach
ISSN 1684-8403 Journal of Statstcs Volume 17, 010, pp. 3-3 Abstract X- Chart Usng ANOM Approach Gullapall Chakravarth 1 and Chaluvad Venkateswara Rao Control lmts for ndvdual measurements (X) chart are
More informationAn Entropy-Based Approach to Integrated Information Needs Assessment
Dstrbuton Statement A: Approved for publc release; dstrbuton s unlmted. An Entropy-Based Approach to ntegrated nformaton Needs Assessment June 8, 2004 Wllam J. Farrell Lockheed Martn Advanced Technology
More informationDetermining the Optimal Bandwidth Based on Multi-criterion Fusion
Proceedngs of 01 4th Internatonal Conference on Machne Learnng and Computng IPCSIT vol. 5 (01) (01) IACSIT Press, Sngapore Determnng the Optmal Bandwdth Based on Mult-crteron Fuson Ha-L Lang 1+, Xan-Mn
More informationBiostatistics 615/815
The E-M Algorthm Bostatstcs 615/815 Lecture 17 Last Lecture: The Smplex Method General method for optmzaton Makes few assumptons about functon Crawls towards mnmum Some recommendatons Multple startng ponts
More informationLearning the Kernel Parameters in Kernel Minimum Distance Classifier
Learnng the Kernel Parameters n Kernel Mnmum Dstance Classfer Daoqang Zhang 1,, Songcan Chen and Zh-Hua Zhou 1* 1 Natonal Laboratory for Novel Software Technology Nanjng Unversty, Nanjng 193, Chna Department
More informationA Robust Method for Estimating the Fundamental Matrix
Proc. VIIth Dgtal Image Computng: Technques and Applcatons, Sun C., Talbot H., Ourseln S. and Adraansen T. (Eds.), 0- Dec. 003, Sydney A Robust Method for Estmatng the Fundamental Matrx C.L. Feng and Y.S.
More informationA mathematical programming approach to the analysis, design and scheduling of offshore oilfields
17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 A mathematcal programmng approach to the analyss, desgn and
More informationHybridization of Expectation-Maximization and K-Means Algorithms for Better Clustering Performance
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 16, No 2 Sofa 2016 Prnt ISSN: 1311-9702; Onlne ISSN: 1314-4081 DOI: 10.1515/cat-2016-0017 Hybrdzaton of Expectaton-Maxmzaton
More informationAnalysis of Continuous Beams in General
Analyss of Contnuous Beams n General Contnuous beams consdered here are prsmatc, rgdly connected to each beam segment and supported at varous ponts along the beam. onts are selected at ponts of support,
More informationLecture 5: Multilayer Perceptrons
Lecture 5: Multlayer Perceptrons Roger Grosse 1 Introducton So far, we ve only talked about lnear models: lnear regresson and lnear bnary classfers. We noted that there are functons that can t be represented
More information6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour
6.854 Advanced Algorthms Petar Maymounkov Problem Set 11 (November 23, 2005) Wth: Benjamn Rossman, Oren Wemann, and Pouya Kheradpour Problem 1. We reduce vertex cover to MAX-SAT wth weghts, such that the
More informationProper Choice of Data Used for the Estimation of Datum Transformation Parameters
Proper Choce of Data Used for the Estmaton of Datum Transformaton Parameters Hakan S. KUTOGLU, Turkey Key words: Coordnate systems; transformaton; estmaton, relablty. SUMMARY Advances n technologes and
More informationOutline. Self-Organizing Maps (SOM) US Hebbian Learning, Cntd. The learning rule is Hebbian like:
Self-Organzng Maps (SOM) Turgay İBRİKÇİ, PhD. Outlne Introducton Structures of SOM SOM Archtecture Neghborhoods SOM Algorthm Examples Summary 1 2 Unsupervsed Hebban Learnng US Hebban Learnng, Cntd 3 A
More informationIntelligent Information Acquisition for Improved Clustering
Intellgent Informaton Acquston for Improved Clusterng Duy Vu Unversty of Texas at Austn duyvu@cs.utexas.edu Mkhal Blenko Mcrosoft Research mblenko@mcrosoft.com Prem Melvlle IBM T.J. Watson Research Center
More informationThe Research of Support Vector Machine in Agricultural Data Classification
The Research of Support Vector Machne n Agrcultural Data Classfcaton Le Sh, Qguo Duan, Xnmng Ma, Me Weng College of Informaton and Management Scence, HeNan Agrcultural Unversty, Zhengzhou 45000 Chna Zhengzhou
More informationEXTENDED BIC CRITERION FOR MODEL SELECTION
IDIAP RESEARCH REPORT EXTEDED BIC CRITERIO FOR ODEL SELECTIO Itshak Lapdot Andrew orrs IDIAP-RR-0-4 Dalle olle Insttute for Perceptual Artfcal Intellgence P.O.Box 59 artgny Valas Swtzerland phone +4 7
More informationGSLM Operations Research II Fall 13/14
GSLM 58 Operatons Research II Fall /4 6. Separable Programmng Consder a general NLP mn f(x) s.t. g j (x) b j j =. m. Defnton 6.. The NLP s a separable program f ts objectve functon and all constrants are
More informationNUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS
ARPN Journal of Engneerng and Appled Scences 006-017 Asan Research Publshng Network (ARPN). All rghts reserved. NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS Igor Grgoryev, Svetlana
More informationWishing you all a Total Quality New Year!
Total Qualty Management and Sx Sgma Post Graduate Program 214-15 Sesson 4 Vnay Kumar Kalakband Assstant Professor Operatons & Systems Area 1 Wshng you all a Total Qualty New Year! Hope you acheve Sx sgma
More informationS1 Note. Basis functions.
S1 Note. Bass functons. Contents Types of bass functons...1 The Fourer bass...2 B-splne bass...3 Power and type I error rates wth dfferent numbers of bass functons...4 Table S1. Smulaton results of type
More informationFEATURE EXTRACTION. Dr. K.Vijayarekha. Associate Dean School of Electrical and Electronics Engineering SASTRA University, Thanjavur
FEATURE EXTRACTION Dr. K.Vjayarekha Assocate Dean School of Electrcal and Electroncs Engneerng SASTRA Unversty, Thanjavur613 41 Jont Intatve of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents
More informationAn Optimal Algorithm for Prufer Codes *
J. Software Engneerng & Applcatons, 2009, 2: 111-115 do:10.4236/jsea.2009.22016 Publshed Onlne July 2009 (www.scrp.org/journal/jsea) An Optmal Algorthm for Prufer Codes * Xaodong Wang 1, 2, Le Wang 3,
More informationHierarchical clustering for gene expression data analysis
Herarchcal clusterng for gene expresson data analyss Gorgo Valentn e-mal: valentn@ds.unm.t Clusterng of Mcroarray Data. Clusterng of gene expresson profles (rows) => dscovery of co-regulated and functonally
More informationK-means and Hierarchical Clustering
Note to other teachers and users of these sldes. Andrew would be delghted f you found ths source materal useful n gvng your own lectures. Feel free to use these sldes verbatm, or to modfy them to ft your
More informationContent Based Image Retrieval Using 2-D Discrete Wavelet with Texture Feature with Different Classifiers
IOSR Journal of Electroncs and Communcaton Engneerng (IOSR-JECE) e-issn: 78-834,p- ISSN: 78-8735.Volume 9, Issue, Ver. IV (Mar - Apr. 04), PP 0-07 Content Based Image Retreval Usng -D Dscrete Wavelet wth
More informationCompiler Design. Spring Register Allocation. Sample Exercises and Solutions. Prof. Pedro C. Diniz
Compler Desgn Sprng 2014 Regster Allocaton Sample Exercses and Solutons Prof. Pedro C. Dnz USC / Informaton Scences Insttute 4676 Admralty Way, Sute 1001 Marna del Rey, Calforna 90292 pedro@s.edu Regster
More informationR s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges
More informationSupport Vector Machines
Support Vector Machnes Decson surface s a hyperplane (lne n 2D) n feature space (smlar to the Perceptron) Arguably, the most mportant recent dscovery n machne learnng In a nutshell: map the data to a predetermned
More informationClassifier Selection Based on Data Complexity Measures *
Classfer Selecton Based on Data Complexty Measures * Edth Hernández-Reyes, J.A. Carrasco-Ochoa, and J.Fco. Martínez-Trndad Natonal Insttute for Astrophyscs, Optcs and Electroncs, Lus Enrque Erro No.1 Sta.
More informationParameter estimation for incomplete bivariate longitudinal data in clinical trials
Parameter estmaton for ncomplete bvarate longtudnal data n clncal trals Naum M. Khutoryansky Novo Nordsk Pharmaceutcals, Inc., Prnceton, NJ ABSTRACT Bvarate models are useful when analyzng longtudnal data
More informationClustering. A. Bellaachia Page: 1
Clusterng. Obectves.. Clusterng.... Defntons... General Applcatons.3. What s a good clusterng?. 3.4. Requrements 3 3. Data Structures 4 4. Smlarty Measures. 4 4.. Standardze data.. 5 4.. Bnary varables..
More informationSHAPE RECOGNITION METHOD BASED ON THE k-nearest NEIGHBOR RULE
SHAPE RECOGNITION METHOD BASED ON THE k-nearest NEIGHBOR RULE Dorna Purcaru Faculty of Automaton, Computers and Electroncs Unersty of Craoa 13 Al. I. Cuza Street, Craoa RO-1100 ROMANIA E-mal: dpurcaru@electroncs.uc.ro
More informationFor instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)
Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A
More informationEmpirical Distributions of Parameter Estimates. in Binary Logistic Regression Using Bootstrap
Int. Journal of Math. Analyss, Vol. 8, 4, no. 5, 7-7 HIKARI Ltd, www.m-hkar.com http://dx.do.org/.988/jma.4.494 Emprcal Dstrbutons of Parameter Estmates n Bnary Logstc Regresson Usng Bootstrap Anwar Ftranto*
More informationA Binarization Algorithm specialized on Document Images and Photos
A Bnarzaton Algorthm specalzed on Document mages and Photos Ergna Kavalleratou Dept. of nformaton and Communcaton Systems Engneerng Unversty of the Aegean kavalleratou@aegean.gr Abstract n ths paper, a
More informationComplexity Analysis of Problem-Dimension Using PSO
Proceedngs of the 7th WSEAS Internatonal Conference on Evolutonary Computng, Cavtat, Croata, June -4, 6 (pp45-5) Complexty Analyss of Problem-Dmenson Usng PSO BUTHAINAH S. AL-KAZEMI AND SAMI J. HABIB,
More informationThree supervised learning methods on pen digits character recognition dataset
Three supervsed learnng methods on pen dgts character recognton dataset Chrs Flezach Department of Computer Scence and Engneerng Unversty of Calforna, San Dego San Dego, CA 92093 cflezac@cs.ucsd.edu Satoru
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 15
CS434a/541a: Pattern Recognton Prof. Olga Veksler Lecture 15 Today New Topc: Unsupervsed Learnng Supervsed vs. unsupervsed learnng Unsupervsed learnng Net Tme: parametrc unsupervsed learnng Today: nonparametrc
More informationChapter 6 Programmng the fnte element method Inow turn to the man subject of ths book: The mplementaton of the fnte element algorthm n computer programs. In order to make my dscusson as straghtforward
More informationA Fast Content-Based Multimedia Retrieval Technique Using Compressed Data
A Fast Content-Based Multmeda Retreval Technque Usng Compressed Data Borko Furht and Pornvt Saksobhavvat NSF Multmeda Laboratory Florda Atlantc Unversty, Boca Raton, Florda 3343 ABSTRACT In ths paper,
More informationMeta-heuristics for Multidimensional Knapsack Problems
2012 4th Internatonal Conference on Computer Research and Development IPCSIT vol.39 (2012) (2012) IACSIT Press, Sngapore Meta-heurstcs for Multdmensonal Knapsack Problems Zhbao Man + Computer Scence Department,
More informationConcurrent Apriori Data Mining Algorithms
Concurrent Apror Data Mnng Algorthms Vassl Halatchev Department of Electrcal Engneerng and Computer Scence York Unversty, Toronto October 8, 2015 Outlne Why t s mportant Introducton to Assocaton Rule Mnng
More informationFusion Performance Model for Distributed Tracking and Classification
Fuson Performance Model for Dstrbuted rackng and Classfcaton K.C. Chang and Yng Song Dept. of SEOR, School of I&E George Mason Unversty FAIRFAX, VA kchang@gmu.edu Martn Lggns Verdan Systems Dvson, Inc.
More informationSVM-based Learning for Multiple Model Estimation
SVM-based Learnng for Multple Model Estmaton Vladmr Cherkassky and Yunqan Ma Department of Electrcal and Computer Engneerng Unversty of Mnnesota Mnneapols, MN 55455 {cherkass,myq}@ece.umn.edu Abstract:
More informationEECS 730 Introduction to Bioinformatics Sequence Alignment. Luke Huan Electrical Engineering and Computer Science
EECS 730 Introducton to Bonformatcs Sequence Algnment Luke Huan Electrcal Engneerng and Computer Scence http://people.eecs.ku.edu/~huan/ HMM Π s a set of states Transton Probabltes a kl Pr( l 1 k Probablty
More informationUser Authentication Based On Behavioral Mouse Dynamics Biometrics
User Authentcaton Based On Behavoral Mouse Dynamcs Bometrcs Chee-Hyung Yoon Danel Donghyun Km Department of Computer Scence Department of Computer Scence Stanford Unversty Stanford Unversty Stanford, CA
More informationLecture #15 Lecture Notes
Lecture #15 Lecture Notes The ocean water column s very much a 3-D spatal entt and we need to represent that structure n an economcal way to deal wth t n calculatons. We wll dscuss one way to do so, emprcal
More informationClassifying Acoustic Transient Signals Using Artificial Intelligence
Classfyng Acoustc Transent Sgnals Usng Artfcal Intellgence Steve Sutton, Unversty of North Carolna At Wlmngton (suttons@charter.net) Greg Huff, Unversty of North Carolna At Wlmngton (jgh7476@uncwl.edu)
More informationFuzzy Filtering Algorithms for Image Processing: Performance Evaluation of Various Approaches
Proceedngs of the Internatonal Conference on Cognton and Recognton Fuzzy Flterng Algorthms for Image Processng: Performance Evaluaton of Varous Approaches Rajoo Pandey and Umesh Ghanekar Department of
More informationInvestigating the Performance of Naïve- Bayes Classifiers and K- Nearest Neighbor Classifiers
Journal of Convergence Informaton Technology Volume 5, Number 2, Aprl 2010 Investgatng the Performance of Naïve- Bayes Classfers and K- Nearest Neghbor Classfers Mohammed J. Islam *, Q. M. Jonathan Wu,
More informationIMAGE MATCHING WITH SIFT FEATURES A PROBABILISTIC APPROACH
IMAGE MATCHING WITH SIFT FEATURES A PROBABILISTIC APPROACH Jyot Joglekar a, *, Shrsh S. Gedam b a CSRE, IIT Bombay, Doctoral Student, Mumba, Inda jyotj@tb.ac.n b Centre of Studes n Resources Engneerng,
More informationA New Token Allocation Algorithm for TCP Traffic in Diffserv Network
A New Token Allocaton Algorthm for TCP Traffc n Dffserv Network A New Token Allocaton Algorthm for TCP Traffc n Dffserv Network S. Sudha and N. Ammasagounden Natonal Insttute of Technology, Truchrappall,
More informationDetection of an Object by using Principal Component Analysis
Detecton of an Object by usng Prncpal Component Analyss 1. G. Nagaven, 2. Dr. T. Sreenvasulu Reddy 1. M.Tech, Department of EEE, SVUCE, Trupath, Inda. 2. Assoc. Professor, Department of ECE, SVUCE, Trupath,
More informationMULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION
MULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION Paulo Quntlano 1 & Antono Santa-Rosa 1 Federal Polce Department, Brasla, Brazl. E-mals: quntlano.pqs@dpf.gov.br and
More information12/2/2009. Announcements. Parametric / Non-parametric. Case-Based Reasoning. Nearest-Neighbor on Images. Nearest-Neighbor Classification
Introducton to Artfcal Intellgence V22.0472-001 Fall 2009 Lecture 24: Nearest-Neghbors & Support Vector Machnes Rob Fergus Dept of Computer Scence, Courant Insttute, NYU Sldes from Danel Yeung, John DeNero
More informationPositive Semi-definite Programming Localization in Wireless Sensor Networks
Postve Sem-defnte Programmng Localzaton n Wreless Sensor etworks Shengdong Xe 1,, Jn Wang, Aqun Hu 1, Yunl Gu, Jang Xu, 1 School of Informaton Scence and Engneerng, Southeast Unversty, 10096, anjng Computer
More informationBAYESIAN MULTI-SOURCE DOMAIN ADAPTATION
BAYESIAN MULTI-SOURCE DOMAIN ADAPTATION SHI-LIANG SUN, HONG-LEI SHI Department of Computer Scence and Technology, East Chna Normal Unversty 500 Dongchuan Road, Shangha 200241, P. R. Chna E-MAIL: slsun@cs.ecnu.edu.cn,
More informationImage Alignment CSC 767
Image Algnment CSC 767 Image algnment Image from http://graphcs.cs.cmu.edu/courses/15-463/2010_fall/ Image algnment: Applcatons Panorama sttchng Image algnment: Applcatons Recognton of object nstances
More informationReview of approximation techniques
CHAPTER 2 Revew of appromaton technques 2. Introducton Optmzaton problems n engneerng desgn are characterzed by the followng assocated features: the objectve functon and constrants are mplct functons evaluated
More informationSteps for Computing the Dissimilarity, Entropy, Herfindahl-Hirschman and. Accessibility (Gravity with Competition) Indices
Steps for Computng the Dssmlarty, Entropy, Herfndahl-Hrschman and Accessblty (Gravty wth Competton) Indces I. Dssmlarty Index Measurement: The followng formula can be used to measure the evenness between
More informationAn Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation
17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 An Iteratve Soluton Approach to Process Plant Layout usng Mxed
More informationRange images. Range image registration. Examples of sampling patterns. Range images and range surfaces
Range mages For many structured lght scanners, the range data forms a hghly regular pattern known as a range mage. he samplng pattern s determned by the specfc scanner. Range mage regstraton 1 Examples
More informationApplying EM Algorithm for Segmentation of Textured Images
Proceedngs of the World Congress on Engneerng 2007 Vol I Applyng EM Algorthm for Segmentaton of Textured Images Dr. K Revathy, Dept. of Computer Scence, Unversty of Kerala, Inda Roshn V. S., ER&DCI Insttute
More informationFeature Reduction and Selection
Feature Reducton and Selecton Dr. Shuang LIANG School of Software Engneerng TongJ Unversty Fall, 2012 Today s Topcs Introducton Problems of Dmensonalty Feature Reducton Statstc methods Prncpal Components
More information3D vector computer graphics
3D vector computer graphcs Paolo Varagnolo: freelance engneer Padova Aprl 2016 Prvate Practce ----------------------------------- 1. Introducton Vector 3D model representaton n computer graphcs requres
More informationTerm Weighting Classification System Using the Chi-square Statistic for the Classification Subtask at NTCIR-6 Patent Retrieval Task
Proceedngs of NTCIR-6 Workshop Meetng, May 15-18, 2007, Tokyo, Japan Term Weghtng Classfcaton System Usng the Ch-square Statstc for the Classfcaton Subtask at NTCIR-6 Patent Retreval Task Kotaro Hashmoto
More informationTsinghua University at TAC 2009: Summarizing Multi-documents by Information Distance
Tsnghua Unversty at TAC 2009: Summarzng Mult-documents by Informaton Dstance Chong Long, Mnle Huang, Xaoyan Zhu State Key Laboratory of Intellgent Technology and Systems, Tsnghua Natonal Laboratory for
More informationSmoothing Spline ANOVA for variable screening
Smoothng Splne ANOVA for varable screenng a useful tool for metamodels tranng and mult-objectve optmzaton L. Rcco, E. Rgon, A. Turco Outlne RSM Introducton Possble couplng Test case MOO MOO wth Game Theory
More informationA Fast Visual Tracking Algorithm Based on Circle Pixels Matching
A Fast Vsual Trackng Algorthm Based on Crcle Pxels Matchng Zhqang Hou hou_zhq@sohu.com Chongzhao Han czhan@mal.xjtu.edu.cn Ln Zheng Abstract: A fast vsual trackng algorthm based on crcle pxels matchng
More informationMachine Learning 9. week
Machne Learnng 9. week Mappng Concept Radal Bass Functons (RBF) RBF Networks 1 Mappng It s probably the best scenaro for the classfcaton of two dataset s to separate them lnearly. As you see n the below
More informationEdge Detection in Noisy Images Using the Support Vector Machines
Edge Detecton n Nosy Images Usng the Support Vector Machnes Hlaro Gómez-Moreno, Saturnno Maldonado-Bascón, Francsco López-Ferreras Sgnal Theory and Communcatons Department. Unversty of Alcalá Crta. Madrd-Barcelona
More informationActive Contours/Snakes
Actve Contours/Snakes Erkut Erdem Acknowledgement: The sldes are adapted from the sldes prepared by K. Grauman of Unversty of Texas at Austn Fttng: Edges vs. boundares Edges useful sgnal to ndcate occludng
More informationFitting & Matching. Lecture 4 Prof. Bregler. Slides from: S. Lazebnik, S. Seitz, M. Pollefeys, A. Effros.
Fttng & Matchng Lecture 4 Prof. Bregler Sldes from: S. Lazebnk, S. Setz, M. Pollefeys, A. Effros. How do we buld panorama? We need to match (algn) mages Matchng wth Features Detect feature ponts n both
More informationNAG Fortran Library Chapter Introduction. G10 Smoothing in Statistics
Introducton G10 NAG Fortran Lbrary Chapter Introducton G10 Smoothng n Statstcs Contents 1 Scope of the Chapter... 2 2 Background to the Problems... 2 2.1 Smoothng Methods... 2 2.2 Smoothng Splnes and Regresson
More informationMulti-view 3D Position Estimation of Sports Players
Mult-vew 3D Poston Estmaton of Sports Players Robbe Vos and Wlle Brnk Appled Mathematcs Department of Mathematcal Scences Unversty of Stellenbosch, South Afrca Emal: vosrobbe@gmal.com Abstract The problem
More informationA Statistical Model Selection Strategy Applied to Neural Networks
A Statstcal Model Selecton Strategy Appled to Neural Networks Joaquín Pzarro Elsa Guerrero Pedro L. Galndo joaqun.pzarro@uca.es elsa.guerrero@uca.es pedro.galndo@uca.es Dpto Lenguajes y Sstemas Informátcos
More informationA PATTERN RECOGNITION APPROACH TO IMAGE SEGMENTATION
1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volume 4, Number 2/2003, pp.000-000 A PATTERN RECOGNITION APPROACH TO IMAGE SEGMENTATION Tudor BARBU Insttute
More informationCell Count Method on a Network with SANET
CSIS Dscusson Paper No.59 Cell Count Method on a Network wth SANET Atsuyuk Okabe* and Shno Shode** Center for Spatal Informaton Scence, Unversty of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
More informationLearning-Based Top-N Selection Query Evaluation over Relational Databases
Learnng-Based Top-N Selecton Query Evaluaton over Relatonal Databases Lang Zhu *, Wey Meng ** * School of Mathematcs and Computer Scence, Hebe Unversty, Baodng, Hebe 071002, Chna, zhu@mal.hbu.edu.cn **
More informationA MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS
Proceedngs of the Wnter Smulaton Conference M E Kuhl, N M Steger, F B Armstrong, and J A Jones, eds A MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS Mark W Brantley Chun-Hung
More informationEcient Computation of the Most Probable Motion from Fuzzy. Moshe Ben-Ezra Shmuel Peleg Michael Werman. The Hebrew University of Jerusalem
Ecent Computaton of the Most Probable Moton from Fuzzy Correspondences Moshe Ben-Ezra Shmuel Peleg Mchael Werman Insttute of Computer Scence The Hebrew Unversty of Jerusalem 91904 Jerusalem, Israel Emal:
More informationSkew Angle Estimation and Correction of Hand Written, Textual and Large areas of Non-Textual Document Images: A Novel Approach
Angle Estmaton and Correcton of Hand Wrtten, Textual and Large areas of Non-Textual Document Images: A Novel Approach D.R.Ramesh Babu Pyush M Kumat Mahesh D Dhannawat PES Insttute of Technology Research
More informationAPPLICATION OF IMPROVED K-MEANS ALGORITHM IN THE DELIVERY LOCATION
An Open Access, Onlne Internatonal Journal Avalable at http://www.cbtech.org/pms.htm 2016 Vol. 6 (2) Aprl-June, pp. 11-17/Sh Research Artcle APPLICATION OF IMPROVED K-MEANS ALGORITHM IN THE DELIVERY LOCATION
More informationLECTURE : MANIFOLD LEARNING
LECTURE : MANIFOLD LEARNING Rta Osadchy Some sldes are due to L.Saul, V. C. Raykar, N. Verma Topcs PCA MDS IsoMap LLE EgenMaps Done! Dmensonalty Reducton Data representaton Inputs are real-valued vectors
More informationAn Accurate Evaluation of Integrals in Convex and Non convex Polygonal Domain by Twelve Node Quadrilateral Finite Element Method
Internatonal Journal of Computatonal and Appled Mathematcs. ISSN 89-4966 Volume, Number (07), pp. 33-4 Research Inda Publcatons http://www.rpublcaton.com An Accurate Evaluaton of Integrals n Convex and
More informationClassification / Regression Support Vector Machines
Classfcaton / Regresson Support Vector Machnes Jeff Howbert Introducton to Machne Learnng Wnter 04 Topcs SVM classfers for lnearly separable classes SVM classfers for non-lnearly separable classes SVM
More informationPerformance Evaluation of Information Retrieval Systems
Why System Evaluaton? Performance Evaluaton of Informaton Retreval Systems Many sldes n ths secton are adapted from Prof. Joydeep Ghosh (UT ECE) who n turn adapted them from Prof. Dk Lee (Unv. of Scence
More informationHierarchical agglomerative. Cluster Analysis. Christine Siedle Clustering 1
Herarchcal agglomeratve Cluster Analyss Chrstne Sedle 19-3-2004 Clusterng 1 Classfcaton Basc (unconscous & conscous) human strategy to reduce complexty Always based Cluster analyss to fnd or confrm types
More information