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Clustering training

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What is clustering? …

What is clustering?
Distance: Similarity and dissimilarity
Data types in cluster analysis
Clustering methods
Evaluation of clustering
Summary

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  • 1. CLUSTERING TUTORIAL GABOR VERESS 2013.10.16
  • 2. CONTENTS What is clustering? Distance: Similarity and dissimilarity Data types in cluster analysis Clustering methods Evaluation of clustering Summary 2
  • 3. WHAT IS CLUSTERING? Grouping of objects 3
  • 4. CLUSTERING I. (BY TYPE) Fruit Veggie 4
  • 5. CLUSTERING II. (BY COLOR) Yellow Green 5
  • 6. CLUSTERING III. (BY SHAPE) Bushy Longish Ball Chili shape 6
  • 7. ANOTHER CLUSTERING EXAMPLE 7
  • 8. IMAGE PROCESSING EXAMPLE Figure from “Image and video segmentation: the normalised cut framework”by Shi and Malik, copyright IEEE, 1998 8
  • 9. YET ANOTHER EXAMPLE Original Clustering 1 Clustering 2 9
  • 10. CLUSTERING BY COLOR EXAMPLE Item Cian Magenta Yellow Black Chili 72 0 51 57 Cucamber 11 0 45 19 Broccoli 15 0 23 31 Apple 25 0 74 20 Paprika 0 52 100 11 Lemon 0 20 93 5 Orange 0 18 65 3 Banana 0 1 100 1 10
  • 11. CLUSTERING BY COLOR EXAMPLE Item Cian Magenta Yellow Black Cluster Chili 72 0 51 57 Cluster 1 Cucamber 11 0 45 19 Cluster 1 Broccoli 15 0 23 31 Cluster 1 Apple 25 0 74 20 Cluster 1 Paprika 0 52 100 11 Cluster 2 Lemon 0 20 93 5 Cluster 2 Orange 0 18 65 3 Cluster 2 Banana 0 1 100 1 Cluster 2 11
  • 12. WHAT IS CLUSTERING? Grouping of objects into classes such a way that • Objects in same cluster are similar • Objects in different clusters are dissimilar Segmentation vs. Clustering • Clustering is finding borders between groups, • Segmenting is using borders to form groups Clustering is the method of creating segments 12
  • 13. SUPERVISED VS. UNSUPERVISED CLASSIFICATION VS. CLUSTERING Classification – Supervised Classes are predetermined we know in advance the stamping For example if we already diagnosed some disease Or we know who has churned Clustering – Unsupervised Classes are not known in advance we don’t know in advance the stamping Market behaviour segmentation Or Gene analysis 13
  • 14. APPLICATIONS OF CLUSTERING Marketing: segmentation of the customer based on behavior Banking: ATM Fraud detection (outlier detection) ATM classification: segmentation based on time series Gene analysis: Identifying gene responsible for a disease Chemistry: Periodic table of the elements Image processing: identifying objects on an image (face detection) Insurance: identifying groups of car insurance policy holders with a high average claim cost Houses: identifying groups of houses according to their house type, value, and geographical location 14
  • 15. TYPICAL DATABASE id age sex ID12101 48 FEMALE ID12102 40 MALE ID12103 51 FEMALE ID12104 23 FEMALE ID12105 57 FEMALE ID12106 57 FEMALE ID12107 22 MALE ID12108 58 MALE ID12109 37 FEMALE ID12110 54 MALE ID12111 66 FEMALE region income married children car INNER_CITY 17,546 NO 1 NO TOWN 30,085 YES 3 YES INNER_CITY 16,575 YES 0 YES TOWN 20,375 YES 3 NO RURAL 50,576 YES 0 NO TOWN 37,870 YES 2 NO RURAL 8,877 NO 0 NO TOWN 24,947 YES 0 YES SUBURBAN 25,304 YES 2 YES TOWN 24,212 YES 2 YES TOWN 59,804 YES 0 NO save_act NO NO YES NO YES YES NO YES NO YES YES current_act NO YES YES YES NO YES YES YES NO YES YES mortgage NO YES NO NO NO NO NO NO NO NO NO pep YES NO NO NO NO YES YES NO NO NO NO How we define similarity or dissimilarity? Especially for categorical variables? 15
  • 16. WHAT TO DERIVE FORM THE DATABASE? id age sex ID12101 48 FEMALE ID12102 40 MALE ID12103 51 FEMALE ID12104 23 FEMALE ID12105 57 FEMALE ID12106 57 FEMALE ID12107 22 MALE ID12108 58 MALE ID12109 37 FEMALE ID12110 54 MALE ID12111 66 FEMALE region income married children car INNER_CITY 17,546 NO 1 NO TOWN 30,085 YES 3 YES INNER_CITY 16,575 YES 0 YES TOWN 20,375 YES 3 NO RURAL 50,576 YES 0 NO TOWN 37,870 YES 2 NO RURAL 8,877 NO 0 NO TOWN 24,947 YES 0 YES SUBURBAN 25,304 YES 2 YES TOWN 24,212 YES 2 YES TOWN 59,804 YES 0 NO Upper: Original database of the objects (customers) Right: Similarity or dissimilarity measure of the objects (similarity of customers) save_act NO NO YES NO YES YES NO YES NO YES YES current_act NO YES YES YES NO YES YES YES NO YES YES mortgage NO YES NO NO NO NO NO NO NO NO NO pep YES NO NO NO NO YES YES NO NO NO NO id ID12101 ID12102 ID12103 ID12104 ID12105 ID12101 0 12 23 19 13 ID12102 12 0 25 13 17 ID12103 23 25 0 9 21 ID12104 19 13 9 0 12 ID12105 13 17 21 12 0 16
  • 17. REQUIREMENTS OF CLUSTERING • Ability to deal with different types of attributes • Discovery of clusters with arbitrary shape • Able to deal with noise and outliers • Insensitive to order of input records • High dimensionality • Scalability • Minimal requirements for domain knowledge to determine input parameters • Incorporation of user-specified constraints • Interpretability and usability 17
  • 18. DISTANCE: SIMILARITY AND DISSIMILARITY
  • 19. SIMILARITY AND DISSIMILARITY There is no single definition of similarity or dissimilarity between data objects The definition of similarity or dissimilarity between objects depends on • the type of the data considered • what kind of similarity we are looking for 19
  • 20. DISTANCE MEASURE Similarity/dissimilarity between objects is often expressed in terms of a distance measure d(x,y) Ideally, every distance measure should be a metric, i.e., it should satisfy the following conditions: 1. d(x,y) ≥ 0 2. d(x,y) = 0 if x = y 3. d(x,y) = d(y,x) 4. d(x,z) ≤ d(x,y) + d(y,z) 20
  • 21. TYPE OF VARIABLES id age sex ID12101 48 FEMALE ID12102 40 MALE ID12103 51 FEMALE ID12104 23 FEMALE ID12105 57 FEMALE ID12106 57 FEMALE ID12110 54 MALE region income married children car INNER_CITY 17,546 NO 1 NO TOWN 30,085 YES 3 YES INNER_CITY 16,575 YES 0 YES TOWN 20,375 YES 3 NO RURAL 50,576 YES 0 NO TOWN 37,870 YES 2 NO TOWN 24,212 YES 2 YES save_act NO NO YES NO YES YES YES current_act NO YES YES YES NO YES YES mortgage NO YES NO NO NO NO NO pep YES NO NO NO NO YES NO Interval-scaled variables: Age Binary variables: Car, Mortgage Nominal, Ordinal, and Ratio variables Variables of mixed types Complex data types: Documents, GPS coordinates 21
  • 22. INTERVAL-SCALED VARIABLES Continuous measurements of a roughly linear scale for example, age, weight and height The measurement unit can affect the cluster analysis To avoid dependence on the measurement unit, we should standardize the data 22
  • 23. STANDARDIZATION To standardize the measurements: • calculate the mean absolute deviation s f  1 (| x1 f  m f |  | x2 f  m f | ... | xnf  m f |) n where m f  1 (x1 f  x2 f  ...  xnf ), n and • calculate the standardized measurement (z-score) xif  m f zif  sf 23
  • 24. DISTANCE MEASURE I. One group of popular distance measures for intervalscaled variables are Minkowski distances d (i, j)  q (| x  x |q  | x  x |q ... | x  x |q ) i1 j1 i2 j 2 ip jp where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is a positive integer 24
  • 25. DISTANCE MEASURES II. If q = 1, the distance measure is Manhattan (or city block) distance d (i, j) | x  x |  | x  x | ... | x  x | i1 j1 i2 j2 ip jp If q = 2, the distance measure is Euclidean distance d (i, j)  (| x  x |2  | x  x |2 ... | x  x |2 ) i1 j1 i2 j 2 ip jp 25
  • 26. EXAMPLE: DISTANCE MEASURES point p1 p2 p3 p4 x 0 2 3 5 y 2 0 1 1 Manhattan Distance 0, 2 y 2 1 3, 1 0 5, 1 2, 0 0 1 2 3 4 5 6 p2 p3 p4 p1 p2 p3 p4 3 p1 0 4 4 6 4 0 2 4 4 2 0 2 6 4 2 0 Euclidean Distance p1 p2 p3 p4 p1 p2 p3 p4 0 2.828 3.162 5.099 2.828 0 1.414 3.162 3.162 1.414 0 2 5.099 3.162 2 0 x Distance Matrix 26
  • 27. WHY STANDARDIZATION? Age and Income No standardization Income >> Age No separation on age With standardization Separation based on both age and income 27
  • 28. RATIO-SCALED VARIABLES A positive measurement on a nonlinear scale, approximately at exponential scale AeBt or Ae-Bt Methods: 1. treat them like interval-scaled variables is not a good choice! 2. apply logarithmic transformation yif = log(xif) 3. treat them as continuous ordinal data and treat their rank as interval-scaled 4. create a better variable Object On-net Off-net Ratio Log-Ratio On-net/Total 1 95 6 0.06 -1.20 94% 2 56 15 0.27 -0.57 79% Dist 1-2 0.04 0.39 0.02 3 12 23 1.92 0.28 34% 4 12 29 2.42 0.38 29% Dist 3-4 0.25 0.01 0.00 28
  • 29. ORDINAL VARIABLES An ordinal variable can be discrete or continuous Order of values is important, e.g., rank Can be treated like interval-scaled rif {1, ..., M f } • replacing x by their rank if • map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by rif 1 zif  M f 1 • compute the dissimilarity using methods for intervalscaled variables 29
  • 30. BINARY VARIABLES I. Binary variables has 2 outcomes 0/1, Y/N, … Distances Symmetric binary variable: FEMALE MALE FEMALE 0 1 MALE 1 0 No preference on which outcome should be coded 0 or 1 like gender Asymmetric binary variable: Outcomes are not equally important, or based on one outcome the objects are similar but based on the other outcome we can’t tell Like Has Mortgage or HIV positive Mortgage No Mortgage Mortgage 0 1 No Mortgage 1 undef 30
  • 31. BINARY VARIABLES II. If we have more binary variables in the database we can calculate the distance based on the contingency table A contingency table for binary data Object i 1 0 SUM 1 a c a+c Object j 0 b d b+d SUM a+b c+d t 31
  • 32. BINARY VARIABLES III. Object i 1 0 SUM 1 a c a+c Object j 0 b d b+d SUM a+b c+d t Simple matching coefficient (invariant similarity, if the binary variable is symmetric): bc ad sim(i, j)  d (i, j)  a bc  d a bc  d Jaccard coefficient (non-invariant similarity, if the binary variable is asymmetric): a sim(i, j)  d (i, j)  b  c a bc a bc 32
  • 33. NOMINAL VARIABLES Generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue Distance matrix More variables Method 1: simple matching Distance Red Yellow Blue Red Yellow 0 1 1 Blue 1 0 1 1 1 0 • m: # of matches, p: total # of variables sim(i, j)  m p d (i, j)  p  m p Method 2: use a large number of binary variables • create new binary variable for each of the k nominal states 33
  • 34. VARIABLES OF MIXED TYPES Database usually contains different types of variables • symmetric binary, asymmetric binary, nominal, ordinal, interval Approaches 1. Group each type of variable together, performing a separate cluster analysis for each type. 2. Bring different variables onto a common scale of the interval [0.0, 1.0], performing a single cluster analysis 34
  • 35. WEIGHTED FORMULA ( (  p  1 ij f ) dij f ) f d (i, j)  (  p  1 ij f ) f Weight δij (f) = 0 • if xif or xjf is missing • or xif = xjf =0 and variable f is asymmetric binary, Otherwise Weight δij (f) = 1 Another option is to choose the weights based on business aspects 35
  • 36. VECTOR OBJECTS: COSINE SIMILARITY Vector objects: keywords in documents, gene features in micro-arrays, … Applications: information retrieval, biologic taxonomy, ... Cosine measure: If d1 and d2 are two vectors, then cos(d1, d2) = (d1  d2) /||d1|| ||d2|| , where  indicates vector dot product, ||d||: the length of vector d Example: d1 = 3 2 0 5 0 0 0 2 0 0 d2 = 1 0 0 0 0 0 0 1 0 2 d1d2 = 3*1+2*0+0*0+5*0+0*0+0*0+0*0+2*1+0*0+0*2 = 5 ||d1||= (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5=(42)0.5 = 6.481 ||d2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2)0.5=(6) 0.5 = 2.245 cos( d1, d2 ) = .3150 36
  • 37. COMPLEX DATA TYPES All not relational objects => complex types of data • examples: spatial data, location data, multimedia data, genetic data, time-series data, text data and data collected from Web We can define our own similarity or dissimilarity measures than the previous • can, for example, mean using of string and/or sequence matching, or methods of information retrieval 37
  • 38. CLUSTERING METHODS
  • 39. MAJOR CLUSTERING APPROACHES Partitioning algorithms: Construct various partitions and then evaluate them by some criterion Hierarchy algorithms: Create a hierarchical decomposition of the set of data (or objects) using some criterion Density-based: based on connectivity and density functions Grid-based: based on a multiple-level granularity structure Model-based: A model is hypothesized for each of the clusters and the idea is to find the best fit of that model to each other 39
  • 40. PARTITIONING BASIC CONCEPT Partitioning method: Construct a partition of a database D of n objects into k clusters • each cluster contains at least one object • each object belongs to exactly one cluster Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion (min distance from cluster centers) • Global optimal: exhaustively enumerate all partitions Stirling(n,k) (S(10,3) = 9.330, S(20,3) = 580.606.446,…) • Heuristic methods: k-means and k-medoids algorithms • k-means: Each cluster is represented by the center of the cluster • k-medoids or PAM (Partition around medoids): Each cluster is represented by one of the objects in the cluster 40
  • 41. PARTITIONING K-MEANS ALGORITHM Input: k clusters, n objects of database D. Output: A set of k clusters which minimizes the squared-error function Algorithm: 1. Choose k objects as the initial cluster centers 2. Assign each object to the cluster which has the closest mean point (centroid) under squared Euclidean distance metric 3. When all objects have been assigned, recalculate the positions of k mean point (centroid) 4. Repeat Steps 2. and 3. until the centroids do not change any more 41
  • 42. PARTITIONING K-MEANS ALGORITHM Source: Clustering: A survey 2008, R. Capaldo F. Collovà 42
  • 43. PARTITIONING K-MEANS + Easy to implement + The K-means method is is relatively efficient: O(tkn), where n is objects number, k is clusters number, and t is iterations number. Normally, k, t << n. - Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms - Not applicable in categorical data Need to specify k, the number of clusters, in advance Unable to handle noisy data and outliers Not suitable to discover clusters with non-convex shapes To overcome some of these problems is introduced the K-medoids or PAM 43
  • 44. PARTITIONING K-MEDOID ALGORITHM The method K-medoid or PAM ( Partitioning Around Medoids ) is the same as k-means but instead of mean it uses medoid mq (q = 1,2,…,k) as object more representative of cluster medoid is the most centrally located object in a cluster Source: Clustering: A survey 2008, R. Capaldo F. Collovà 44
  • 45. PARTITIONING K-MEDOID OR PAM + PAM is more robust than K-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean - PAM works efficiently for small data sets but does not scale well for large data sets. Infact: O( k(n-k)2 ) for each iteration where n is data numbers, k is the clusters numbers To overcome these problems is introduced: CLARA (Clustering LARge Applications) - > Sampling based method CLARANS - > A Clustering Algorithm based on Randomized Search. 45
  • 46. PARTITIONING CLARA CLARA (Clustering LARge Applications) (Kaufmann and Rousseeuw in 1990) draws multiple sample of the dataset and applies PAM on the sample in order to find the medoids. + Deals with larger data sets than PAM + Experiments show that 5 samples of size 40+2k give satisfactory results - Efficiency depends on the sample size, should also determine that parameter - A good clustering based on samples will not necessarily represent a good clustering of the whole data set if the sample is biased, but to avoid this we use multiple sampling 46
  • 47. PARTITIONING CLARANS CLARANS (CLustering Algorithm based on RANdomized Search) (Ng and Han’94) A clustering method that draws sample of neighbors dynamically There are 2 parameters: maxneighbour the maximum number of neighbours examined, numlocal the number of local minimum obtained The algorithm is searching for new neighbours and replaces the current setup with a lower cost setup until the number of examined neighbours reaches the maxneighbour or the number of new local minimum obtained is reaches numlocal + + + - It is more efficient and scalable than both PAM and CLARA returns higher quality clusters has the benefit of not confining the search to a restricted area Depending on parameters can be very time consuming (close to PAM) 47
  • 48. HIERARCHICAL BASIC CONCEPT Hierarchical clustering Construct a hierarchy of clusters not just a single partition of objects • Use distance matrix as clustering criteria • Does not require the number of clusters as an input, but needs a termination condition, e. g., number of clusters or a distance threshold for merging 48
  • 49. HIERARCHICAL CLUSTERING TREE, DENDOGRAM Agglomerative Divisive The hierarchy of clustering is given as a clustering tree or dendrogram • leaves of the tree represent the individual objects • internal nodes of the tree represent the clusters Two main types of hierarchical clustering • agglomerative (bottom-up) • place each object in its own cluster (a singleton) • merge in each step the two most similar clusters until there is only one cluster left or the termination condition is satisfied • divisive (top-down) • start with one big cluster containing all the objects • divide the most distinctive cluster into smaller clusters and proceed until there are n clusters or the termination condition is satisfied 49
  • 50. HIERARCHICAL CLUSTER DISTANCE MEASURES Single link (nearest neighbor). The distance between two clusters is determined by the distance of the two closest objects (nearest neighbors) in the different clusters. Complete link (furthest neighbor). The distances between clusters are determined by the greatest distance between any two objects in the different clusters (i.e., by the "furthest neighbors"). Pair-group average link. The distance between two clusters is calculated as the average distance between all pairs of objects in the two different clusters. Pair-group centroid. The distance between two clusters is determined as the distance between centroids. Centroid link 50
  • 51. HIERARCHICAL EXAMPLE WITH DENDOGRAM 51
  • 52. HIERARCHICAL + Conceptually simple + Theoretical properties are well understood + When clusters are merged/split, the decision is permanent => the number of different alternatives that need to be examined is reduced - Merging/splitting of clusters is permanent => erroneous decisions are impossible to correct later - Divisive methods can be computational hard - Methods are not (necessarily) scalable for large data sets 52
  • 53. EVALUATION
  • 54. EVALUATION BASICS Business • Segment sizes • Meaningful segments Technical • Compactness • Separation 54
  • 55. COMPACTNESS AND SEPARATION Compactness intra-cluster variance Separation inter-cluster distance Sometimes the two measures leads to different results Dens_bw 0,5 0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 Scatt_orig Compactness Separation 2 3 4 5 6 55
  • 56. INDEX FUNCTIONS DB Number of clusters • Finding the minimum/maximum of a function we can determine the optimal number of clusters 1,6 1,4 1,2 1 0,8 0,6 0,4 0,2 0 2 3 4 5 7 8 SD KM Comparing clustering methods • Using the index functions we can compare the results of different clustering methods of the same database 6 9 10 SD TS 100 80 60 40 20 0 2 3 4 5 6 7 8 9 10 56
  • 57. SAMPLE DATABASE We generated a sample with 4 clusters • 2dimensions • Real values between (–10;15) With outliers 57
  • 58. TWO-STEP AND K-MEANS CLUSTERING K-means Two-step 3 4 3 4 5 6 5 6 7 8 7 8 58
  • 59. DB (DAVIES-BOULDIN) INDEX DB index summarizes the similarity of a given cluster and the most dissimilar cluster and then take the average of them DB TS DB KM 1 DB TS 0,8 1 0,6 0,4 0,8 0,2 0 2 3 4 5 6 7 8 9 10 DB KM 0,6 0,4 1 0,8 0,2 0,6 0,4 0 0,2 2 0 2 3 4 5 6 7 8 9 3 4 5 6 7 8 9 10 10 59
  • 60. S_DBW INDEX S_Dbw KM S_Dbw TS 4 7 0,6 2 components 0,5 • Dens_bw: cluster separation • Scatt: the average variance of the clusters divided by the variance of all objects 0,4 0,3 0,2 0,1 0 2 Dens_bw TS 3 Dens_bw KM Scatt TS 0,6 0,2 0,1 0,1 0 Scatt KM 0,3 0,2 9 0,4 0,3 8 0,5 0,4 6 0,6 0,5 5 0 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 60 10
  • 61. SD INDEX SD KM SD TS 80 70 60 50 40 30 20 10 0 100 80 2 components : 60 • Scatt: compactness of the clusters • Dis: Function of the centroids of the clusters We should know the maximum number of clusters 20 40 0 2 3 4 5 6 7 8 9 2 10 3 SD KM 4 5 6 7 8 9 SD TS 100 80 60 40 20 0 2 3 4 5 6 7 8 9 10 61 10
  • 62. RS, RMSSTD INDEXEK RD (R-squared) = variance between clusters / total variance RMSSTD (Root-mean-square standard deviation) = within cluster variance RS KM RS TS 1 RS_diff TS RS_diff KM RS TS 0,08 1 1,2 0,07 1,2 0,95 RS KM 1 0,15 0,1 0,06 0,8 0,05 0,6 0,04 0,03 0,4 0,02 0,2 0,01 0 0 2 3 4 5 6 7 8 RMSSTD_diff TS 9 0,8 0 2 1,5 1 0,5 0 4 5 6 7 8 9 10 4 5 6 7 8 9 0,6 2 3 10 4 5 6 7 RMSSTD KM RMSSTD KM 12 8 0,05 0 -0,05 2 3 RMSSTD_diff KM 0,3 0,25 0,2 0,15 0,1 3 0,7 8 9 10 -0,1 RMSSTD TS 2,5 0,8 0,75 0,65 -0,05 0,2 0,45 0,4 0,35 3 0 0,4 10 3,5 2 0,05 0,6 0,9 0,85 2 12 5 4 10 3 2 6 1 0 4 -1 -2 0 -3 2 3 4 5 6 7 8 9 10 RMSSTD TS 10 8 6 4 2 0 2 3 4 5 6 7 8 9 10 62
  • 63. SEGMENTATION IN BANK Needs based segmentation for new tariff plans When the number of cluster is 4 or 5 then we have a too big segment (cca. 60 000 customer) Above 6 segments we can not to identify more significant segment Balance decrease is the cutting variable Szeparáltság Átmérő Separation 0,12 0,8 0,7 0,6 0,5 0,35 0,1 0,9 0,3 0,08 0,06 0,4 0,3 0,04 0,2 0,1 0,02 0 0 2 3 4 5 6 7 8 9 10 Diameter 0,05 0,045 0,04 0,035 0,03 0,025 0,02 0,015 0,01 0,005 0 0,25 0,2 0,15 0,1 0,05 0 2 3 4 5 6 7 8 9 10 63
  • 64. BANK SEGMENTATION – INDEXES RS_diff TS DB TS RS TS SD TS 0,9 1,4 0,12 0,8 1,2 0,1 0,7 1 0,6 0,8 0,5 0,6 0,08 0,4 0,06 0,3 0,4 0,04 0,2 0,2 0,02 0,1 0 0 2 3 4 5 6 Dens_bw TS 7 8 9 10 0 2 Scatt_orig TS 3 4 5 6 7 8 RMSSTD_diff TS 9 4 3,5 3 2,5 2 1,5 1 0,5 0 2 10 0,04 0,03 0,02 0,01 0,1 0 0 6 7 8 9 10 8 9 10 RMSSTD TS 0,4 0,2 5 7 0,45 0,3 4 6 0,6 0,05 0,4 3 5 RMSSTD KM 0,5 2 4 RMSSTD TS 0,6 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 3 -0,01 2 3 4 5 6 7 8 9 10 0,55 0,5 0,35 0,3 0,25 0,2 2 3 Based on the indexes there are 4-6 really different segments 4 5 6 7 8 9 64 10
  • 65. LITERATURE I. J. Han and M. Kamber. Data Mining: Concepts and Techniques. Morgan Kaufmann Publishers, August 2000. J. Han and M. Kamber. Data Mining: Concepts and Techniques. Morgan Kaufmann Publishers, August 2000 (k-means, k-medoids or PAM ) L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, 1990 (CLARA, AGNES, DIANA). R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94 (CLARANS). J. Han and M. Kamber. Data Mining: Concepts andTechniques. Morgan Kaufmann Publishers, August 2000 (deterministic annealing, genetic algorithms). T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method for very large databases. SIGMOD'96 (BIRCH). S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large databases. SIGMOD'98 (CURE). 65
  • 66. LITERATURE II. Karypis G., Eui-Hong Han, Kumar V. Chameleon: hierarchical clustering using dynamic modeling (CHAMELEON). M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial databases. KDD'96 (DBSCAN). M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to identify the clustering structure, SIGMOD’99 (OPTICS). A. Hinneburg D., A. Keim: An Efficient Approach to Clustering in Large Multimedia Database with Noise. Proceedings of the 4-th ICKDDM, New York ’98 (DENCLUE). Abramowitz, M. and Stegun, I. A. (Eds.). "Stirling Numbers of the Second Kind." §24.1.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 824-825, 1972. Introduction to Data Mining Pang-Ning Tan, Michigan State University Michael Steinbach,Vipin Kumar, University of Minnesota Publisher: Addison-Wesley Copyright: 2006. 66
  • 67. THANK YOU! GABOR VERESS LYNX ANALYTICS