Clustering methods: Part 4 Clustering Cost function Pasi Fränti 29.4.2014 Speech and Image Processing Unit School of Computing University of Eastern Finland Data types • • • • • Numeric Binary Categorical Text Time series Part I: Numeric data Distance measures Type Possible operations Example variable Example values Nominal == Major subject Computer science Mathematics Physics Ordinal ==, <, > Degree Bachelor Master Licentiate Doctor Interval ==, <, >, - Temperature 10 °C 20 °C 10 °F Ratio ==, <, >, -, / Weight 0 kg 10 kg 20 kg Definition of distance metric A distance function is metric if the following conditions are met for all data points x, y, z: • All distances are non-negative: d(x, y) ≥ 0 • Distance to point itself is zero: d(x, x) = 0 • All distances are symmetric: d(x, y) = d(y, x) • Triangular inequality: d(x, y) d(x, z) + d(z, y) Common distance metrics Xj = (xj1, xj2, …, xjp) dij = ? Xi = (xi1, xi2, …, xip) • Minkowski distance d (i , j ) q x i1 x j 1 q 1st dimension xi 2 x j 2 q 2nd dimension ... x ip x jp q pth dimension • Euclidean distance q=2 d (i , j ) x i1 x j 1 2 xi 2 x j 2 2 ... x ip x jp • Manhattan distance q=1 d ( i , j ) x i1 x j 1 x i 2 x j 2 ... x ip x jp 2 Distance metrics example 10 2D example x1 = (2,8) x2 = (6,3) X1 = (2,8) Euclidean distance 5 5 d (1, 2 ) X2 = (6,3) 26 2 83 2 Manhattan distance d (1, 2 ) 2 6 8 3 9 0 5 4 10 41 Chebyshev distance In case of q , the distance equals to the maximum difference of the attributes. Useful if the worst case must be avoided: d ( X , Y ) lim x i y i q i 1 n max x 1 q 1 q y1 , x 2 y 2 , , x n y n Example: d ( 2 ,8 ), ( 6 , 3 ) max 2 6 , 8 3 max 4 , 5 5 Hierarchical clustering Cost functions Single link: the smallest distance between vectors in clusters i and j: d (C i , C j ) d ( xi , x j ) min x i C i , x j C j , Complete-link: the largest distance between vectors in clusters i and j: d (C i , C j ) d ( xi , x j ) max x i C i , x j C j, Average link: the average distance between vectors in clusters i and j: d (C i , C j ) 1 Ci C d (x j x i C i x j C j i ,x j ) Single Link Complete Link Average Link Cost function example [Theodoridis, Koutroumbas, 2006] 1 Data Set x1 1.1 x2 1.2 x3 1.3 x4 1.4 x5 Single Link: x1 x2 x3 x4 x5 1.5 x6 x7 Complete Link: x6 x7 x1 x2 x3 x4 x5 x6 x7 Part II: Binary data Hamming Distance (Binary and categorical data) • • • • Number of different attribute values. Distance of (1011101) and (1001001) is 2. Distance (2143896) and (2233796) Distance between (toned) and (roses) is 3. 3-bit binary cube 100->011 has distance 3 (red path) 010->111 has distance 2 (blue path) Hard thresholding of centroid (0.40, 0.60, 0.75, 0.20, 0.45, 0.25) T h resh o ld 0 .4 0 0 0 .6 0 1 0 .7 5 0 0 .2 0 0 .4 5 0 0 0 .0 1 0 .2 5 0 .5 1 .0 Hard and soft centroids Bridge (binary version) 1.60 1.55 co n v erg en ce B inary 1.50 1.45 1.40 S oft 1.35 1.30 co n v erg en ce 1.25 0 1 2 3 4 5 6 Iteration s 7 8 9 10 Distance and distortion General distance function: K d L d x i , c j x ik c jk k 1 Distortion function: E X ,C 1 N N L d x i , c p i i 1 1 d Distortion for binary data Cost of a single attribute: D jk q d jk c jk r jk 1 c jk d The number of zeroes is qjk, the number of ones is rjk and cjk is the current centroid value for variable k of group j. Optimal centroid position Optimal centroid position depends on the metric. Given parameter: 1 / d 1 The optimal position is: c jk r jk q jk r jk Example of centroid location 1.00 0.90 c 0.80 0.70 q=10,r=90 q=20,r=80 q=30,r=70 0.60 0.50 q=40,r=60 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 d Centroid location d = ( = 0) d = 3 ( = 0.5) d = 2 ( = 1) d = 1 c = 0.5 c = 0.37 c = 0.25 c = 0 0.0 q = 75 1.0 r = 25 Part III: Categorical data Categorical clustering Three attributes t1 (Godfather II) t2 (Good Fellas) t3 (Vertigo) t4 (N by NW) director actor genre Coppol De Crime a Niro Scorses De Crime e Niro Hitchco Stewar Thriller ck t Hitchco Grant Thriller Categorical clustering Sample 2-d data: color and shape Model A Model B Model C Categorical clustering Methods: • • • • • • k-modes k-medoids k-distributions k-histograms k-populations k-representatives Histogram-based methods: Entropy-based cost functions Category utility: Entropy of data set: Entropies of the clusters relative to the data: Iterative algorithms K-modes clustering Distance function K-modes clustering Prototype of cluster K-medoids clustering Prototype of cluster Vector with minimal total distance to every other 2 A C E 3 B C F 2 Medoid: B D G B C F 2+3=5 2+2=4 2+3=5 K-medoids Example K-medoids Calculation K-histograms D 2/3 F 1/3 K-distributions Cost function with ε addition Example of cluster allocation Change of entropy Problem of non-convergence Non-convergence Results with Census dataset Part IV: Text data Applications of text clustering • • • • Query relaxation Spell-checking Automatic categorization Document similarity Query relaxation Current solution Alternate solution Matching suffixes from database From semantic clustering Spell-checking Word kahvila (café) but with one correct and two incorrect spellings Automatic categorization Category by clustering String clustering • The similarity between every string pair is calculated as a basis for determining the clusters • A similarity measure is required to calculate the similarity between two strings. Approximate string matching Semantic similarity Document clustering Motivation: – Group related documents based on their content – No predefined training set (taxonomy) – Generate a taxonomy at runtime Clustering Process: – Data preprocessing: remove stop words, stem, feature extraction and lexical analysis – Define cost function – Perform clustering Exact string matching • Given a text string T of length n and a pattern string P of length m, the exact string matching problem is to find all occurrences of P in T. • Example: T=“AGCTTGA” • Applications: – Searching keywords in a file – Searching engines (like Google) – Database searching P=“GCT” Approximate string matching Determine if a text string T of length n and a pattern string P of length m partially matches. – Consider the string “approximate”. Which of these are partial matches? aproximate approximately appropriate proximate approx approximat apropos approxximate – A partial match can be thought of as one that has k differences from the string where k is some small integer (for instance 1 or 2) – A difference occurs if the string1.charAt(j) != string2.charAt(j) or if string1.charAt(j) does not appear in string2 (or vice versa) The former case is known as a revise difference, the latter is a delete or insert difference. What about two characters that appear out of position? For instance, approximate vs. apporximate? Approximate string matching Keanu Reeves Samuel Jackson Arnold Schwarzenegger H. Norman Schwarfkopf Schwarrzenger Bernard Schwartz … Query errors Limited knowledge about data Typos Limited input device (cell phone) input Data errors Typos Web data OCR Similarity functions: Edit distance Q-gram Cosine Edit distance Levenhstein distance • Given two strings T and P, the edit distance is the minimum number of substitutions, insertion and deletions, which will transform the string T into P. • Time complexity by dynamic programming: O(nm) Edit distance 1974 t m p 0 1 2 3 t 1 0 1 2 e 2 1 2 2 m 3 2 1 2 p 4 3 2 1 Dynamic programming: m[i][j] = min{ m[i-1][j]+1, m[i][j-1]+1, m[i-1][j-1]+d(i,j)} d(i,j) =0 if i=j, d(i,j)=1 else Q-grams b i n g o n 2-grams Fixed length (q) ed(T, P) <= k, then # of common grams >= # of T grams – k * q Q-grams T = “bingo”, P = “going” gram1 = {#b, bi, in, ng, go, o#} gram2 = {#g, go, oi, in, ng, g#} Total(gram1, gram2) = {#b,bi,in,ng,go,o#,#g, go,oi,in,ng,g#} |common terms difference|= sum{1,1,0,0,0,1,1,0,1,0,0,1} gram1.length = (T.length + (q - 1) * 2 + 1) – q gram2.length = (P.length + (q - 1) * 2 + 1) - q L = gram1.length + gram2.length=12 Similarity = (L- |common terms difference| )/ L =0.5 Cosine similarity • Two vectors A and B,θ is represented using a dot product and magnitude as • Implementation: Cosine similarity = (Common Terms) / (sqrt(Number of terms in String1) + sqrt(Number of terms in String2)) Cosine similarity T = “bingo right”, P = “going right” T1 = {bingo right}, P1 = {going right} L1 = unique(T1).length; L2 = unique(P1).length; Unique(T1&P1) = {bingo right going} L3 = Unique(T1&P1) .length; Common terms = (L1+L2)-L3; Similarity = common terms / (sqrt(L1)*sqrt(L2)) Dice coefficient • Similar with cosine similarity • Dices coefficient = (2*Common Terms) / (Number of terms in String1 + Number of terms in String2) Similarities for sample data Compared Strings Pizza Express Café Pizza Express Lounasravintola Pinja Ky – ravintoloita Lounasravintola Pinja Kioski Piirakkapaja Different Kioski Marttakahvio Kauppa Kulta Keidas Different Kauppa Kulta Nalle Ravintola Beer Stop Pub Baari, Beer Stop R-kylä Ravintola Foxie s Bar Foxie Karsikko Play baari Ravintola Bar Play – Ravintoloita Edit Q-gram Q-gram Q-gram Cosine Q=3 Q=4 distance distance Q=2 72% 79% 74% 70% 82% 54% 68% 67 % 65% 63 % 47% 45% 33% 32% 50% 68% 67% 63 % 60% 67% 39% 42% 36% 31% 50% 31% 25% 15% 12% 24% 21% 31% 17% 8% 32% Thesaurus-based WordNet WordNet An extensive lexical network for the English language • Contains over 138,838 words. • Several graphs, one for each part-of-speech. • Synsets (synonym sets), each defining a semantic sense. • Relationship information (antonym, hyponym, meronym …) • Downloadable for free (UNIX, Windows) • Expanding to other languages (Global WordNet Association) • Funded >$3 million, mainly government (translation interest) • Founder George Miller, National Medal of Science, 1991. moist watery parched wet dry damp anhydrous arid synonym antonym Example of WordNet object artifact instrumentality conveyance, transport wheeled vehicle car, auto ware table ware vehicle automotive, motor article bike, bicycle truck cutlery, eating utensil fork Examples of probabilities Entity (40%) Inanimate-object (17 %) Natural-object (1.6%) Geological-formation (0.17%) Natural-elevation (0.011%) Hill (0.0019%) Shore ( 0.008%) Coast (0.002%) Hierarchical clustering by WordNet Performance of WordNet Word Pair Human Judgment Edge Counting Based Measures Path WUP Information Content Based Measures LIN Jiang&Conrath Car Automobile 3.92 1 1 1 1 Gem Journey Boy Coast Asylum Jewel Voyage Lad Shore Madhouse 3.84 3.84 3.76 3.70 3.61 1 0.97 0.97 0.97 0.97 1 0.92 0.93 0.91 0.94 1 0.84 0.86 0.98 0.97 1 0.88 0.88 0.99 0.97 Magician Wizard 3.50 1 1 1 1 Midday Furnace Food Bird Bird 3.42 3.11 3.08 3.05 2.97 1 0.81 0.81 0.97 0.92 1 0.46 0.22 0.94 0.84 1 0.23 0.13 0.6 0.6 1 0.39 0.63 0.73 0.73 Noon Stove Fruit Cock Crane Part V: Time series Clustering of time-series Dynamic Time Warping Align two time-series by minimizing distance of the aligned observations Solve by dynamic programming! Example of DTW Prototype of a cluster Sequence c that minimizes E(Sj,c) is called a Steiner sequence. Good approximation to Steiner problem, is to use medoid of the cluster (discrete median). Medoid is such a time-series in the cluster that minimizes E(Sj,c). Calculating the prototype Can be solved by dynamic programming. Complexity is exponential to the number of time-series in a cluster. Averaging heuristic Calculate the medoid sequence Calculate warping paths from the medoid to all other time series in the cluster New prototype is the average sequence over warping paths Local search heuristics Example of the three methods E(S) = 159 E(S) = 138 E(S) = 118 • LS provides better fit in terms of the Steiner cost function. • It cannot modify sequence length during the iterations. In datasets with varying lengths it might provide better fit, but non-sensitive prototypes Experiments Part VI: Other clustering problems Clustering of GPS trajectories Density clusters Swim hall Walking street Market place Science park Shop Homes of users Image segmentation Objects of different colors Literature 1. S. Theodoridis and K. Koutroumbas, Pattern Recognition, Academic Press, 2nd edition, 2006. 2. P. Fränti and T. Kaukoranta, "Binary vector quantizer design using soft centroids", Signal Processing: Image Communication, 14 (9), 677-681, 1999. 3. I. Kärkkäinen and P. Fränti, "Variable metric for binary vector quantization", IEEE Int. Conf. on Image Processing (ICIP’04), Singapore, vol. 3, 3499-3502, October 2004. Literature Modified k-modes + k-histograms: M. Ng, M.J. Li, J. Z. Huang and Z. He, On the Impact of Dissimilarity Measure in k-Modes Clustering Algorithm, IEEE Trans. on Pattern Analysis and Machine Intelligence, 29 (3), 503-507, March, 2007. ACE: K. Chen and L. Liu, The “Best k'' for entropy-based categorical dataclustering, Int. Conf. on Scientific and Statistical Database Management (SSDBM'2005), pp. 253-262, Berkeley, USA, 2005. ROCK: S. Guha, R. Rastogi and K. Shim, “Rock: A robust clustering algorithm for categorical attributes”, Information Systems, Vol. 25, No. 5, pp. 345-366, 200x. K-medoids: L. Kaufman and P. J. Rousseeuw, Finding groups in data: an introduction to cluster analysis, John Wiley Sons, New York, 1990. K-modes: Z. Huang, Extensions to k-means algorithm for clustering large data sets with categorical values, Data mining knowledge discovery, Vol. 2, No. 3, pp. 283-304, 1998. K-distributions: Z. Cai, D. Wang and L. Jiang, K-Distributions: A New Algorithm for Clustering Categorical Data, Int. Conf. on Intelligent Computing (ICIC 2007), pp. 436-443, Qingdao, China, 2007. K-histograms: Zengyou He, Xiaofei Xu, Shengchun Deng and Bin Dong, K-Histograms: An Efficient Clustering Algorithm for Categorical Dataset, CoRR, abs/cs/0509033, http://arxiv.org/abs/cs/0509033, 2005.

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# Clustering Methods