Monotony and Surprise Algorithmic and Combinatorial Foundations of Pattern Discovery

1. Monotony and SurpriseAlgorithmic and Combinatorial Foundations of Pattern Discovery Alberto Apostolico University of Padovaand Georgia Inst. Of Tech.

2. http://www.cc.gatech.edu/~axa/papers • A) Specialized Material • A. Apostolico and G. Bejerano ``Optimal Amnesic Probabilistic Automata, or How to Learn and • Classify Proteins in linear Time and Space '', RECOMB 2000 and Journal of Computational Biology, • 7(3/4):381--393, 2000. • A.Apostolico, M.E. Bock, S. Lonardi and X. Xu. ``Efficient Detection of Unusual Words'', • Proceedings of RECOMB 2002 and Journal of Computational Biology, 7(1/2):71--94, 2000. • A. Apostolico, F. Gong and S. Lonardi. ``Verbumculus and the Detection of Unusual Words'', • Journal of Computer Science and Technology, 19:1 ( Special Issue on Bioinformatics), 22-41 (2004). • A. Apostolico, L. Parida. ``Incremental Paradigms of Motif Discovery'', • Journal of Computational Biology 11:1, 15--25 (2004). • A. Apostolico, M.E. Bock and S. Lonardi. ``Monotony of Surprise and Large Scale Quest for Unusual Words.'‘ Journal of Computational Biology, 10, 3-4, 283-311 (2003). • A. Apostolico, C. Pizzi.``Monotone Scoring of Patterns with Mismatches'‘ • Proceedings of the 4th Workshop on Algorithms in Bioinformatics, Bergen, Norway, Springer Verlag • LNCS 3240, 87-98, (2004) Alberto Apostolico - Erice05

3. http://www.cc.gatech.edu/~axa/papers • B) Introductory Material • A. Apostolico and M. Crochemore ``String Pattern Matching for a Deluge Survival Kit'' • Handbook of Massive Data Sets, J. Abello et al, Eds. Kluver Acad. Publishers, to appear. • A. Apostolico ``General Pattern Matching'', Handbook of Algorithms and Theory of Computation, • M.J. Atallah, ed., CRC Press Ch. 13, pp. 1--22 (1999). • A. Apostolico ``Of Maps Bigger than the Empire'', Keynote, SPIRE2001, IEEE Press (2001) • A. Apostolico ``Pattern Discovery and the Algorithmics of Surprise'' • Artificial Intelligence and Heuristic Methods for Bioinformatics, (P. Frasconi and R. Shamir, eds.) • IOS Press, 111--127 (2003). • Apostolico ``Pattern Discovery in the Crib of Procrustes'' • Imagination and Rigor, Essays on Eduardo R. Caianiello's Scientific Heritage Ten Years after his Death, • ( S. Termini, ed.), Springer-Verlag, to appear 2005. Alberto Apostolico - Erice05

4. Acknowledgements Gill Bejerano Dept. of Computer Science - The Hebrew University Mary Ellen Bock Dept. of Statistics - Purdue University Matteo Comin Univ. of Padova Jianhua Dong Dept. of Industrial Technology, Purdue University S. Lonardi Dept. of Comp. Science and Eng. - UC Riverside Fu Lu Celera FangCheng Gong Celera Laxmi Parida IBM Cinzia Pizzi Univ. of Padova Xuyan Xu CapitalOne Alberto Apostolico - Erice05

5. Form = Function A hemoglobin molecule consists of four polypeptide chains: two a globin chains (shown in green and blue) and two b globin chains (shown in yellow and orange). Each globin chain contains a heme (shown in red). Hemoglobin is the protein that carries oxygen from the lungs to the tissues and carries carbon dioxide from the tissues back to the lungs. In order to function most efficiently, hemoglobin needs to bind oxygen tightly in the oxygen-rich atmosphere of the lungs and be able to release oxygen rapidly in the relatively oxygen-poor environment of the tissues. Alberto Apostolico - Erice05

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7. Bioinformatics the Road Ahead • ‘’. . . more than any other single factor, the sheer volume of data poses the most serious challenge --many problems that are ordinarily quite manageable • become seemingly insurmountable when scaled up to these extents. For these reasons, it is evident that imaginative new applications of technologies • designed for dealing with problems of scale will be required. For example, it may be imagined that • data mining techniques will have to supplant manual • search, intelligent data base integration will be needed in place of hyperlink browsing, scientific visualization will replace conventional interface to the data, and knowledge-based systems will have to supervise high-throughput annotation of the [sequence] data’’ • [ D.B. Searls, Grand Challenges in Computational Biology Salzberg Searls Kasif eds Elsevier 1998] Alberto Apostolico - Erice05

8. At a joint EU - US panel meeting on large scientific data bases • held in Annapolis in 1999, I was invited with the physicists and earth observators • to represent the needs of computational biology • In honest to my duty, I said time and again that the kind of data available to • biology was a tiny fracvtion of what is produced in earth observation and • high energy physics. Just as the others were disposing of me saying that • swe did not need money, I said : don’t worry we will make up for it • with the data we will generate • biology is a natural science, it dissects and multiplies • formal sciences synthesize and cluster • there is no telling where these two will go together Alberto Apostolico - Erice05

9. Which Information Anyway • - Greek ``eidos" is form, appearance, or, in Latin, species- information is modern, quantified version of what the Greek called ``eidos" - it is a measure of the amount of structure • the three dimensions of information: • syntactic (formal medium without meaning) • semantic (dualism of subject and object invented by modern philosophy) • pragmatic (attempt to describe the understanding of meaning as a natural process) Alberto Apostolico - Erice05

10. King Phillip Came Over For Green Soup(Kingdom, Phylum, Class, Order, Family, Genus, Species)biologists group organisms by morphology to represent similarities and propose relationships Linnaeus’ Taxonomy (partial) Alberto Apostolico - Erice05

11. The “Chinese” Taxonomy attributed by a Dr. Franz Kuhn to the Chinese Encyclopedia entitled Celestial Emporium of Benevolent Knowledge. Animals are divided into (a) those that belong to the Emperor, (b) embalmed ones, (c) those that are trained, (d) suckling pigs, (e) mermaids, (f) fabulous ones, (g) stray dogs, (h) those that are included in this classification, (i) those that tremble as if they were mad, (j) innumerable ones, (k) those drawn with a very fine camel's hair brush, (l) others, (m) those that have just broken a flower vase, (n) those that resemble flies from afar'' J.L. Borges, "The Analytical Language of John Wilkins," from Otras Inquisiciones (Other Inquisitions 1937-1952, London: Souvenir Press, 1973) Alberto Apostolico - Erice05

12. Summary Form and Information • To Classify and Generate • Of Free Lunches, Ugly Ducklings, and Little Green Men • Privileging Syntactic Information • Avoidable and Unavoidable Regularities • Periods, Palindromes, Squares, etc. • Theories Bigger than Life • Motifs, Profiles and Weigh Matrices • The Emperor’s New Map Alberto Apostolico - Erice05

13. Defining ‘’Class’’ From Watanabe’s pattern recognition as information compression in Frontiers in pattern ‘recognition Class can be defined by 1 intension ( = list of properties or predicates) or 2 extension ( = list of names of individual members) Class can be also defined by 3 paradigm ( = show a few members and, optionally, few non-members) This is what brain does well (and what pattern recognition does poorly) Finally, Class can be defined by 4 clustering ( = we are not even given paradigms but rather sets of objects and asked to isolate subsets with strong coherence) Alberto Apostolico - Erice05

14. Class by Intension From Watanabe’s pattern recognition as information compression in Frontiers in pattern ‘recognition Types of class intension: Vectorial approach (statistical pattern recognition) divides into two = In the conventional zone a class is characterized by a predicate of type: belongs to such and such volume of n-dimensional representation space I In the subspace method, a class is characterized by a predicate of type: belongs to such and such subspace in n-dimensional representation space Structural or grammatical approach = a class is characterized by a predicate of the type: consists of such and such elementary components which are arranged together in such and such ways Note: structural and vector description are not uncorrelated, on the contrary For example, multiple sequence alignment can be considered as a search for the discovery of dimensions along which paradigms of noisy vectors exhibit same value Alberto Apostolico - Erice05

15. Statistical Classification • A Class is formed by Objects with many Predicates in common • Theorem of the Ugly Duckling (S. Watanabe): as long as all of the predicates characterizing the objects to be classified are given the same importance or ``weight", then a swan will be found to be just as similar to a duck as to another swan. • Classification as experienced on an empirical basis is only possible to the extent that the various predicates characterizing objects are given non-uniform weights. Alberto Apostolico - Erice05

16. Statistical Classification Theorem of the Ugly Duckling (S. Watanabe): as long as all of the predicates characterizing the objects to be classified are given the same importance or ``weight", then a swan will be found to be just as similar to a duck as to another swan. Cannot measure similarity by # of shared features: a member with only the left eye is more similar to one with no eye than to one with only the right eye Must measure similarity by # of shared predicates But this number is irrespective of the number of objects and same for all pairs Alberto Apostolico - Erice05

17. Statistical Classification Cannot measure similarity by # of shared features: a member with only the left eye is more similar to one with no eye than to one with only the right eye Must measure similarity by # of shared predicates But this number is irrespective of the number of objects and same for all pairs n Total # of predicates =Sr=0()= (1 +1)n = 2 n d-2 Total # of predicates =Sr=2()= (1 +1) d-2 = 2 d-2 shared by ANY two patterns n r d-2 r-2 Theorem of the Ugly Duckling (S. Watanabe): as long as all of the predicates characterizing the objects to be classified are given the same importance or ``weight", then a swan will be found to be just as similar to a duck as to another swan. Alberto Apostolico - Erice05

18. Inferring Grammars grammatical inference problem: Input: a finite set of symbol strings from some language L and possibly a finite set of strings from the complement of L Output: a grammar for the language ``Precisely the same problem arises in trying to choose a model or theory to explain a collection of sample data. This is one of the most important information processing problems known and it is surprising that there has been so little work on its formalization.’’ ( Bierman- Feldman, 1972) Alberto Apostolico - Erice05

19. Regular, Anomalous, Entropy, Negentropy • Shannon: information is entropy • Brillouin: info is negentropy, entropy is chaos • Key to the paradox: actual versus potential information • How can we express gain in information? (difference between two distributions ?) • This measure is global and can be either positive or negative • A better measure (Alfred Renyi - always positive) Alberto Apostolico - Erice05

20. Random, Regular, Compressible • Measuring structure in finite objects presupposes the ability to measure randomness in such objects. • Defining randomness has been an elusive goal for statisticians since the turn of the last century. • Kolmogorov's definition of information (note resemblance to molecular evolution): information (alternatively, conditional information) is the length of the recorded sequence of zeroes and ones that constitute a shortest program by which a universal machine produces one string from scratch (alt., from another string). Alberto Apostolico - Erice05

21. Random, Regular, Compressible • Kolmogorov's definition of information (note resemblance to molecular evolution): information (alternatively, conditional information) is the length of the recorded sequence of zeroes and ones that constitute a shortest program by which a universal machine produces one string from scratch (alt., from another string). • The programs of length less than k are at most L, 0, 1, 00, 01, 10, 11, …, ..., 11…1 (or k `1’) • The number of strings with a program of length less than k is 1+2+…+4 + 2k-1 = 2k -1 < 2k Bad News: there is hardly such a notion as that of a finite random sequenceand yet most very long strings are complex – any given short sequence seems to exhibit some kind of regularity,however, in the limit, a great many sequences of sufficiently large length are seen to be incompressible and hence to appear as random It appears thus that we attribute and measure structure in finite objects only to the extent that we privilege (i.e., assign a high weight to) certain regularities and neglect others (is this the structural classification pendant to the theorem of the ugly duckling?) Alberto Apostolico - Erice05

22. Summary Form and Information To Classify and Generate Of Free Lunches, Ugly Ducklings, and Little Green Men Privileging Syntactic Information • Avoidable and Unavoidable Regularities • Periods, Palindromes, Squares, etc. • Theories Bigger than Life • Motifs, Profiles and Weigh Matrices Alberto Apostolico - Erice05

23. Privileging Syntactic Regularities in Strings Syntactic regularities in strings are pervasive notions in Computer Science and its applications. In Molecular Biology, regularities are variously implicated in diverse facets of biological function and structure Typical string regularities: -cadences -periods -squares or tandem repeats -repetitions -palindromes -episodes -motifs -other exact variants and approximate versions thereof • There are avoidable and unavoidable regularities ! Alberto Apostolico - Erice05

24. Unavoidable Regularities If N is partitioned into k classes, one of the classes contains arbitrarily long arithmetic progressions (Baudet-Artin-vanDer Waerden 1926-27 ) Alberto Apostolico - Erice05

25. Avoidable Regularities Periods, Borders periodicities are pervasive notions of string algorithmics, e.g., KMP string searching abaabaababaabaababaabaabaabaab A string can have many periods abacabacaba abac abacabac abacabacab The smallest one is THE period of the string Alberto Apostolico - Erice05

26. Periods cannot coexist too long • A string can have many periods abacabacaba abac abacabac abacabacab • Periodicity Lemma (Lyndon-Schutzemberger, 62) If w has two periods of length p and q and |w| is at least p+q, then w has period gcd(p,q) • Proof assume wlog p>q, take x[i] either 1) i-q is not smaller than 1 or 2) i+p is not larger than n case 1: x[i] = x[i-q] = x[i-q+p] case 2: x[i] = x[i+p] = x[i+p-q] • so p-q is a period ----> now repeat on q and p-q p q Alberto Apostolico - Erice05

27. Avoidable Regularities Periods and periodicities are pervasive notions of string algorithmics,, e.g., KMP string searching abaabaababaabaababaabaabaabaab A string can have many periods abacabacaba abac abacabac abacabacab The smallest one is THE period of the string Palindromes w = w R Once we know how to compute optimally ALL periods of a string we an also compute all initial palindromes • Proof: run the algorithm on w*w R abab ... * ... baba (In fact, all palindromes of a string can be computed in serial linear time: Manacher, 76) Alberto Apostolico - Erice05

28. Squares or Tandem Repeatsor why does genetic code need more than 2 characters • Square: a string in the form ww with w a primitive string • Primitive string: a string that cannot be rewritten in the form v k with k > 1 • Square free strings : a string that contains no square • Longest squarefree string on two symbols010 ? • Thue (1906): On an alphabet of at least 3 symbols we can write indefinitely long square free strings • square free morphism • rew(a) -> abcab • rew(b) -> acabcb • rew(c) -> acbcacb Istrail’s morphism (square free on ``a’’) rew(a) -> abc rew(b) -> ac rew(c) -> b there are about n2 ways of choosing indices i and j , thus n2 squares ? • i • j Alberto Apostolico - Erice05

29. Detecting Squares • How many squares? • there can be cnlogn squares in a string (Crochemore, 81) • Example: Fibonacci words Fo = a F1 = b Fi = Fi-1 Fi-2 a b ba bab babba babbabab babbababbabba ... Optimal nlogn algorithms since early 80's (Main-Lorentz, AA-Preparata, Rabin, Crochemore) Recent (Kosaraju, Gusfield) Parallel (AA, Crochemore-Rytter, AA-Breslauer) Alberto Apostolico - Erice05

30. Tandem Repeats, Repeated Episodes (Myers ‘87, Kannan-Myers ‘92, Landau-Schmidt ‘93, Benson ’98, Ap.-Federico ’98, Myers-Sagot ’99, Ap-Atallah `99) Input: textstring Output: repeated episode (within constaints) (worst-case quadratic or nk with max k errors) Max 12 pos Max 30 pos Alberto Apostolico - Erice05

31. Pattern Discovery in WAKA alluded to: Kokin-shu #315 (Minamoto-no-Muneyuki) ya-ma-sa-to-ha fu-yu-so-sa-hi-shi-sa ma-sa-ri-ke-ru hi-to-me-mo-ku-sa-mo ka-re-nu-to-o-mo-he-ha allusive variation shugyoku-shu #3528 (Jien) ya-to-sa-hi-te hi-to-me-mo-ku-sa-mo ka-re-nu-re-ha so-te-ni-so-no-ko-ru a-ki-no-shi-ra-tsu-yu alluded to: Kokin-shu #315 A hamlet in mountain is the drearier in winter. I feel that there is no one to see and no green around allusive variation shugyoku-shu #3528 My home has been deserted Now in autumn, there is no one to see And no green around There is a pearl dew left in my sleeve Alberto Apostolico - Erice05

32. Discovering instances of poetic allusion from anthologies of classical Japanese poemsTheoretical Computer Science Volume 292 ,  Issue 2  Masayuki TakedaTomoko FukudaIchiro NanriMayumi YamasakiKoichi Tamari ABSTRACT Waka is a form of traditional Japanese poetry with a 1300-year history. In this paper, we attempt to semi-automatically discover instances of poetic allusion, or more generally, to find similar poems in anthologies of Waka poems. One reasonable approach would be to arrange all possible pairs of poems in two anthologies in decreasing order of similarity values, and to scrutinize high-ranked pairs by human effort. The means of defining similarity between Waka poems plays a key role in this approach. In this paper, we generalize existing (dis)similarity measures into a uniform framework, called string resemblance systems, and using this framework, we develop new similarity measures suitable for finding similar poems. Using the measures, we report successful results in finding instances of poetic allusion between two anthologies Kokin-Shu and Shin-Kokin-Shu. Most interestingly, we have found an instance of poetic allusion that has never before been pointed out in the long history of Waka research. Alberto Apostolico - Erice05

33. Cheating by Schoolteachers(the longest substring common to k of n strings) 112a4a342cb214d0001acd24a3a12dadbcb4a0000000 d4a2341cacbddad3142a2344a2ac23421c00adb4b3cb 1b2a34d4ac42d23b141acd24a3a12dadbcb4a2134141 dba23dad1abbac1db11acd24a3a12dadbcb4a21db200 dbbbd21d3aac11da42dadcc000adcd21c4b4421dd000 121a4a2dcc2cadc11a1acd24a3a12dadbcb4a11da011 1421acbbdba23dad121acd24a3a12dadbcb4aa000214 cacb1dadbc42dd11221acd24a3a12dadbcb4acacb1da dbbbd21d3aac11da421dadcc000adcd21c4b4421dd00 2baaab3dad2aadca221acd24a3a12dadbcb4a23421c0 1baaab3dcacb1dadbc42ac2cc31012dadbcb4ad40000 From: S.D.Levit and S.J Dubner, Freakanomics Morrow, 2005 Alberto Apostolico - Erice05

34. Summary Form and Information To Classify and Generate Of Free Lunches, Ugly Ducklings, and Little Green Men Privileging Syntactic Information Avoidable and Unavoidable Regularities Periods, Palindromes, Squares, etc. • Theories Bigger than Life • Motifs, Profiles and Weigh Matrices Alberto Apostolico - Erice05

35. General Form of Pattern Discovery • Find-exploit a priori unknown patterns or associations thereof • in a Data Base • With some prior domain-specific knowledge • Without any domain-specific prior knowledge • Tenet: a pattern or association (rule) that occurs more • frequently than one would expect is potentially informative • and thus interesting frequent = interesting Alberto Apostolico - Erice05

36. Data Compression by Textual Substitution 1 Detect Repeated Patterns 2 Set up Dictionary 3 Use Pointers to Dictionary to Encode Replicas • Redundancy (repetitiveness) is sought in order to remove it Alberto Apostolico - Erice05

37. Consumer Prediction (Data Mining)Intrusion Detection (Security)Protein Classification (Bio-Informatics) • Infer consistent behavior from protocol of past record • Use to predict future behavior or detect malicious practices 1) Collect a set of behavioral sequences (normal profile) into a repository or dictionary 2) Define measure(s) of sequence similarity 3) Compare any new sequence to the dictionary, using similarity to past behavior as a a basis for classification as normal or anomalous Anomaly is sought as a carrier of information Similarity or predictability equals fitness to the model • Learning from positive & negative samples Alberto Apostolico - Erice05

38. Of Exactitude in Science • ...In that Empire, the craft of Cartography attained such Perfection that the Map • of a Single province covered the space of an entire City, and the Map of the Empire • itself an entire Province. In the course of Time, these Extensive maps were found • somehow wanting, and so the College of Cartographers evolved a Map of the Empire that was of the same Scale as the Empire and that coincided with it point • for point. • Less attentive to the Study of Cartography, succeeding Generations came to judge • a map of such Magnitude cumbersome, and, not without Irreverence, they abandoned it to the Rigours of Sun and Rain. • In the western Deserts, tattered Fragments of the Map are still to be found, Sheltering an occasional Beast or beggar; in the whole Nation, no other relic is left of the Discipline of Geography. • From Travels of Praiseworthy Men (1658) by J. A. Suarez Miranda • The piece was written by Jorge Luis Borges and Adolfo Bioy Casares. English translation • quoted from J. L. Borges, A Universal History of Infamy, Penguin Books, London, 1975. Alberto Apostolico - Erice05

39. Detection and Analysis of GeneRegulatory Regions(Jacques van Helden,http://copan.cifn.unam.mx/Computational_Biology/yeast-tools) `` Starting from the simple knowledge that a set of genes share some regulatory behavior, one can suppose that some elements are shared by their upstream region, and one would like to detect such elements. We implemented a simple and fast method to extract such elements, based on a detection ofover-represented oligonucleotides. J. Mol. Biol. (1998) 281, 827-842. ‘’ Alberto Apostolico - Erice05

40. http://www.ucmb.ulb.ac.be/bioinformatics/rsa-tools/ Index of /bioinformatics/rsa-tools/data/ Escherichia_coli_K12/oligo-frequencies A table of mono-mers only contains 4 lines seq observed_freq occ a 0.2879006655447 301075 c 0.2120993344553 221805 g 0.2120993344553 221805 t 0.2879006655447 301075 Alberto Apostolico - Erice05

41. ;seq observed_freq occ • aa 0.0996514874362 103508 • ac 0.0516799845961 53680 • ag 0.0522951766631 54319 • at 0.0840396649658 87292 • ca 0.0630865504958 65528 • cc 0.0474795417349 49317 • cg 0.0490959853663 50996 • ct 0.0522951766631 54319 • ga 0.0559112351978 58075 • gc 0.0573659381920 59586 • gg 0.0474795417349 49317 • gt 0.0516799845961 53680 • ta 0.069290459227971972 • tc 0.0559112351978 58075 • tg 0.063086550495865528 • tt 0.0996514874362 103508 http://www.ucmb.ulb.ac.be/bioinformatics/rsa-tools/ Index of /bioinformatics/rsa-tools/data/ Escherichia_coli_K12/oligo-frequencies A table of 2-mers contains 16 lines Alberto Apostolico - Erice05

42. ;seq observed_freqocc • gct 0.0161176513919 16629 • ctt 0.0163987337723 16919 • gaa 0.0180416118233 18614 • gac 0.0096450026461 9951 • gag 0.0108817651198 11227 • gat 0.0172361654160 17783 • gca 0.0166342614221 17162 • gcc 0.0133436590723 13767 • gcg 0.0147384092288 15206 • gct 0.0127214008370 13125 • gga 0.0123763479839 12769 • ggc 0.0133436590723 13767 • ggg 0.0103942325773 10724 • ggt 0.0114206678905 11783 • gta 0.0123288547541 12720 • gtc 0.0096450026461 9951 • gtg 0.0117036887701 12075 • gtt 0.0180639045638 18637 • taa 0.0259671657010 26791 • tac 0.0123288547541 12720 • tag 0.0088434332371 9124 • tat 0.0221735228152 22877 • tca 0.0190302464026 19634 • tcc 0.0123763479839 12769 • tcg 0.0102721071292 10598 • tct 0.0141936909606 14644 • tga 0.0190302464026 19634 • tgc 0.0166342614221 17162 • tgg 0.0113256814309 11685 • tgt 0.0162746698251 16791 • tta 0.0259671657010 26791 • ttc 0.0180416118233 18614 • ttg 0.0181356290333 18711 • ttt 0.0374140033303 38601 http://www.ucmb.ulb.ac.be/bioinformatics/rsa-tools/ RSA-tools - menu.htm • ;seq observed_freq occ • aaa 0.0374140033303 38601 • aac 0.0180639045638 18637 • aag 0.0163987337723 16919 • aat 0.0276555984825 28533 • aca 0.0162746698251 16791 • acc 0.0114206678905 11783 • acg 0.0118180602214 12193 • act 0.0121282200894 12513 • aga 0.0141936909606 14644 • agc 0.0127214008370 13125 • agg 0.0133359050756 13759 • agt 0.0121282200894 12513 • ata 0.0221735228152 22877 • atc 0.0172361654160 17783 • atg 0.0170946549762 17637 • att 0.0276555984825 28533 • caa 0.0181356290333 18711 • cac 0.0117036887701 12075 • cag 0.0161176513919 16629 • cat 0.0170946549762 17637 • cca 0.0113256814309 11685 • ccc 0.0103942325773 10724 • ccg 0.0122910540202 12681 • cct 0.0133359050756 13759 • cga 0.0102721071292 10598 • cgc 0.0147384092288 15206 • cgg 0.0122910540202 12681 • cgt 0.0118180602214 12193 • cta 0.0088434332371 9124 • ctc 0.0108817651198 11227 • ctg 0.0161176513919 16629 • ctt 0.0163987337723 16919 • gaa With increasing k, a table of k-mers grows rapidly out of proportions How many k-mers in total, for all k? Alberto Apostolico - Erice05

43. caaa 0.0064882479779 6650 • caac 0.0038051379119 3900 • caag 0.0024791937010 2541 • caat 0.0053652444557 5499 • caca 0.0034704809109 3557 • cacc 0.0029543481018 3028 • cacg 0.0021738069917 2228 • cact 0.0031114320002 3189 • caga 0.0043485896598 4457 • cagc 0.0036441513079 3735 • cagg 0.0048276467661 4948 • cagt 0.0033348618930 3418 • cata 0.0038753866118 3972 • catc 0.0046481223108 4764 • catg 0.0027670182354 2836 • catt 0.0058403988565 5986 • ccaa 0.0022635692194 2320 • ccac 0.0023425990068 2401 • ccag 0.0035407296108 3629 • ccat 0.0031836320529 3263 • ccca 0.0019981852419 2048 • cccc 0.0026196911009 2685 • cccg 0.0028801966964 2952 • ccct 0.0028811723727 2953 • ccga 0.0022460070444 2302 • ccgc 0.0035026782317 3590 • ccgg 0.0040002731894 4100 • ccgt 0.0025670045759 2631 • ccta 0.0015962065702 1636 • cctc 0.0026206667772 2686 • cctg 0.0048276467661 4948 • cctt 0.0042851706946 4392 • ccta 0.0015962065702 1636 • cctc 0.0026206667772 2686 • cctg 0.0048276467661 4948 • cctt 0.0042851706946 4392 • cgaa 0.0031855834057 3265 • cgac 0.0022645448957 2321 • cgag 0.0016557228299 1697 • cgat 0.0031348482335 3213 • cgca 0.0042002868489 4305 • cgcc 0.0039397812534 4038 • cgcg 0.0029309318685 3004 • cgct 0.0036841540398 3776 • cgga 0.0030977725308 3175 • cggc 0.0036060999288 3696 • cggg 0.0028801966964 2952 • cggt 0.0027182344160 2786 • cgta 0.0027201857688 2788 • cgtc 0.0025152937274 2578 • cgtg 0.0021738069917 2228 • cgtt 0.0044237167416 4534 • ctaa 0.0030499643878 3126 • ctac 0.0022577151610 2314 • ctag 0.0004624706077 474 • ctat 0.0030704535920 3147 • ctca 0.0031904617876 3270 • ctcc 0.0029133696935 2986 • ctcg 0.0016557228299 1697 • ctct 0.0031094806475 3187 • ctga 0.0050071712214 5132 • ctgc 0.0036968378328 3789 • ctgg 0.0035407296108 3629 • ctgt 0.0038831920229 3980 • ctta 0.0044656708263 4577 • cttc 0.0032207077557 3301 • cttg 0.0024791937010 2541 • cttt 0.0062443288810 6400 • gcgc 0.0039378299006 4036 • gcgg 0.0035026782317 3590 • gcgt 0.0039446596353 4043 • gcta 0.0028333642298 2904 • gctc 0.0022313718986 2287 • gctg 0.0036441513079 3735 • gctt 0.0039875893963 4087 • ggaa 0.0039446596353 4043 • ggac 0.0014869308148 1524 • ggag 0.0029133696935 2986 • ggat 0.0040002731894 4100 • ggca 0.0042198003766 4325 • ggcc 0.0023182070971 2376 • ggcg 0.0039397812534 4038 • ggct 0.0028704399325 2942 • ggga 0.0029016615769 2974 • gggc 0.0024147990594 2475 • gggg 0.0026196911009 2685 • gggt 0.0024606558497 2522 • ggta 0.0028021425853 2872 • ggtc 0.0017367039700 1780 • ggtg 0.0029543481018 3028 • ggtt 0.0039183163728 4016 • gtaa 0.0050003414867 5125 • gtac 0.0017210931478 1764 • gtag 0.0022577151610 2314 • gtat 0.0033397402749 3423 • gtca 0.0034831647039 3570 • gtcc 0.0014869308148 1524 • gtcg 0.0022645448957 2321 • gtct 0.0023933341789 2453 • gtga 0.0040032002186 4103 • gtgc 0.0027260398271 2794 • gtgg 0.0023425990068 2401 • gtgt 0.0026733533022 2740 • gtta 0.0051515713268 5280 • gttc 0.0028587318158 2930 • gttg 0.0038051379119 3900 • gttt 0.0061848126213 6339 • gaaa 0.0069331564107 7106 • gaac 0.0028587318158 2930 • gaag 0.0032207077557 3301 • gaat 0.0049993658103 5124 • gaca 0.0031611914960 3240 • gacc 0.0017367039700 1780 • gacg 0.0025152937274 2578 • gact 0.0022216151347 2277 • gaga 0.0032968105139 3379 • gagc 0.0022313718986 2287 • gagg 0.0026206667772 2686 • gagt 0.0027279911799 2796 • gata 0.0051232767116 5251 • gatc 0.0022206394583 2276 • gatg 0.0046481223108 4764 • gatt 0.0052569443767 5388 • gcaa 0.0055896500249 5729 • gcac 0.0027260398271 2794 • gcag 0.0036968378328 3789 • gcat 0.0045973871386 4712 • gcca 0.0035651215205 3654 • gccc 0.0024147990594 2475 • gccg 0.0036060999288 3696 • gcct 0.0037583054452 3852 • gcga 0.0033680348902 3452 • gcgc 0.0039378299006 4036 • gcgg 0.0035026782317 3590 • gcgt 0.0039446596353 4043 • gcta 0.0028333642298 2904 • gctc 0.0022313718986 2287 • gctg 0.0036441513079 3735 • gctt 0.0039875893963 4087 • agaa 0.0047310548037 4849 • agac 0.0023933341789 2453 • agag 0.0031094806475 3187 • agat 0.0039202677256 4018 • agca 0.0040002731894 4100 • agcc 0.0028704399325 2942 • agcg 0.0036841540398 3776 • agct 0.0021835637556 2238 • agga 0.0036792756578 3771 • aggc 0.0037583054452 3852 • aggg 0.0028811723727 2953 • aggt 0.0030333778892 3109 • agta 0.0030324022128 3108 • agtc 0.0022216151347 2277 • agtg 0.0031114320002 3189 • agtt 0.0037729405911 3867 • ataa 0.0092581932425 9489 • atac 0.0033397402749 3423 • atag 0.0030704535920 3147 • atat 0.0065097128584 6672 • atca 0.0061487125950 6302 • atcc 0.0040002731894 4100 • atcg 0.0031348482335 3213 • atct 0.0039202677256 4018 • atga 0.0052588957295 5390 • atgc 0.0045973871386 4712 • atgg 0.0031836320529 3263 • atgt 0.0041339408545 4237 • atta 0.0072736674700 7455 • attc 0.0049993658103 5124 • attg 0.0053652444557 5499 • attt 0.0099860478277 10235 • ;seq observed_freq occ • aaaa 0.0149249217020 15297 • aaac 0.0061848126213 6339 • aaag 0.0062443288810 6400 • aaat 0.0099860478277 10235 • aaca 0.0059106475564 6058 • aacc 0.0039183163728 4016 • aacg 0.0044237167416 4534 • aact 0.0037729405911 3867 • aaga 0.0044959167943 4608 • aagc 0.0039875893963 4087 • aagg 0.0042851706946 4392 • aagt 0.0036529323954 3744 • aata 0.0082503195340 8456 • aatc 0.0052569443767 5388 • aatg 0.0058403988565 5986 • aatt 0.0083478871728 8556 • acaa 0.0055476959402 5686 • acac 0.0026733533022 2740 • acag 0.0038831920229 3980 • acat 0.0041339408545 4237 • acca 0.0031475320266 3226 • accc 0.0024606558497 2522 • accg 0.0027182344160 2786 • acct 0.0030333778892 3109 • acga 0.0026128613661 2678 • acgc 0.0039446596353 4043 • acgg 0.0025670045759 2631 • acgt 0.0026694505966 2736 • acta 0.0023845530914 2444 • actc 0.0027279911799 2796 • actg 0.0033348618930 3418 • actt 0.0036529323954 3744 • tatg 0.0038753866118 3972 • tatt 0.0082503195340 8456 • tcaa 0.0047778872704 4897 • tcac 0.0040032002186 4103 • tcag 0.0050071712214 5132 • tcat 0.0052588957295 5390 • tcca 0.0026372532758 2703 • tccc 0.0029016615769 2974 • tccg 0.0030977725308 3175 • tcct 0.0036792756578 3771 • tcga 0.0020567258252 2108 • tcgc 0.0033680348902 3452 • tcgg 0.0022460070444 2302 • tcgt 0.0026128613661 2678 • tcta 0.0020537987960 2105 • tctc 0.0032968105139 3379 • tctg 0.0043485896598 4457 • tctt 0.0044959167943 4608 • tgaa 0.0062082288547 6363 • tgac 0.0034831647039 3570 • tgag 0.0031904617876 3270 • tgat 0.0061487125950 6302 • tgca 0.0042441922863 4350 • tgcc 0.0042198003766 4325 • tgcg 0.0042002868489 4305 • tgct 0.0040002731894 4100 • tgga 0.0026372532758 2703 • tggc 0.0035651215205 3654 • taaa 0.0090659849941 9292 • taac 0.0051515713268 5280 • taag 0.0044656708263 4577 • taat 0.0072736674700 7455 • taca 0.0037748919438 3869 • tacc 0.0028021425853 2872 • tacg 0.0027201857688 2788 • tact 0.0030324022128 3108 • taga 0.0020537987960 2105 • tagc 0.0028333642298 2904 • tagg 0.0015962065702 1636 • tagt 0.0023845530914 2444 • tata 0.0049661928132 5090 • tatc 0.0051232767116 5251 • tatg 0.0038753866118 3972 • tatt 0.0082503195340 8456 • tcaa 0.0047778872704 4897 • tcac 0.0040032002186 4103 • tcag 0.0050071712214 5132 • tcat 0.0052588957295 5390 • tcca 0.0026372532758 2703 • tccc 0.0029016615769 2974 • tccg 0.0030977725308 3175 • tcct 0.0036792756578 3771 • tcga 0.0020567258252 2108 • tcgc 0.0033680348902 3452 • tcgg 0.0022460070444 2302 • tcgt 0.0026128613661 2678 • tcta 0.0020537987960 2105 • tctc 0.0032968105139 3379 • tctg 0.0043485896598 4457 • tctt 0.0044959167943 4608 • http://www.ucmb.ulb.ac.be/bioinformatics/rsa-tools/ • RSA-tools - menu.htm • tggg 0.0019981852419 2048 • tggt 0.0031475320266 3226 • tgta 0.0037748919438 3869 • tgtc 0.0031611914960 3240 • tgtg 0.0034704809109 3557 • tgtt 0.0059106475564 6058 • ttaa 0.0086815684974 8898 • ttac 0.0050003414867 5125 • ttag 0.0030499643878 3126 • ttat 0.0092581932425 9489 • ttca 0.0062082288547 6363 • ttcc 0.0039446596353 4043 • ttcg 0.0031855834057 3265 • ttct 0.0047310548037 4849 • ttga 0.0047778872704 4897 • ttgc 0.0055896500249 5729 • ttgg 0.0022635692194 2320 • ttgt 0.0055476959402 5686 • ttta 0.0090659849941 9292 • tttc 0.0069331564107 7106 • tttg 0.0064882479779 6650 • tttt 0.0149249217020 15297 A table of k-mers grows rapidly out of proportions or out of sight How many k-mers in total, for all k? Alberto Apostolico - Erice05

44. n 1 i j How many distinct substrings in a string of n symbols A: no more than (n x n)/2 ( n ways to choose beginning or i, then n-i ways to choose end or j ) Alberto Apostolico - Erice05

45. How many surprising substrings in a string of n symbols • Agree on a model for the source: e.g., the source emits symbols independently with identical distribution • Agree on some measure of surprise, e.g., departure from • expected number of occurrences exceeds a certain threshold • For a given observed string of n symbols, how many substrings may turn out to be surprising? A: possibly, all (n x n)/2 of them ! Alberto Apostolico - Erice05

46. Order-2 Markov Chain 0.75 Probabilistic Suffix Automaton 0.25 0.25 10 00 0.25 0.25 10 00 0.75 0.25 0.75 0.5 0.25 0.75 11 01 1 1 0.5 0.25 0.5 Prob Suffix Tree 00 (0.5, 0.5) (0.75, 0.25) 0 (0.5, 0.5) 10 (0.25, 0.75) 1 (0.5, 0.5) Source Modeling by Probabilistic Finite State Automata Alberto Apostolico - Erice05

47. Approximate Patterns Finding surprising substrings with mismatches • Input: a sequence or set of sequences, integers m and k • Out: all substrings of length m that occur unusually often, up to k mismatches, as a replica of the same pattern • NOTE: the pattern might never occur exactly in the input How many patterns should one try ? Alberto Apostolico - Erice05

48. From the Special Issue for the 50th Shannon Anniversary of IEEE Trans. IT ``Perhaps as a consequence of the fact that approximate matches abound whereas exact matches are unique, it is inherently much faster to look for an exact match that it is to search from a plethora of approximate matches looking for the best, or even nearly the best, among them. The right way to trade off search effort in a poorly understood environment against the degree to which the product of the search possesses desired criteria has long been a human enigma.'' T. Berger and J.D. Gibson, ``Lossy Source Coding,'‘ IEEE Trans. on Inform. Theory, vol. 44, No. 6, pp. 2693--2723, 1998. Alberto Apostolico - Erice05

49. T A G A G G T A G A T AG T ``don’t care’’ characters solid character T A G A G G T A G A T AG T Syntactic Motif: a recurring pattern with some solid characters and some characters that are a subset of the alphabet, or a ‘’don’t care’’ or ‘’gap’’ PROBLEM Input: textstring Output: repeated motifs T A G A G G T A G A T AG T T A G A G G T A G A T A T Motifs may be rigid or extensible (sometimes also called flexible) Alberto Apostolico - Erice05

50. From Syntax to Stat: Extracting a Profile Matrix & Consensus (From Hertz-Stormo 99) A A T T G A A G G T C C A G G A T G A G G C G T 4 1 0 1 0 1 Alignment Matrix 0 0 0 1 1 1 0 3 3 0 2 1 0 0 1 2 1 1 A G G T G ? (Consensus - by majority rule ) ni,j = times letter i is observed at jth position in alignment N = number of sequences = 4 NOTE: While each sequence is a ``realization’’ of the consensus the consensus itself might not be any of the sequences Alberto Apostolico - Erice05