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4. Molecular Similarity Similarity and Searching • • • • • • • • Historical Progression Similarity Measures Fingerprint Construction “Pathological” Cases MinMax- Counts Pruning Search Space Aggregate Queries LSH 2 Historical Progression • Maximum Common Subgraph-Isomorphism (MCS) – maximum common substructure between to molecules. – “NP-complete” • Structural Keys – dictionary of predetermined, domain-specific sub-structures keyed to particular positions in a bit-vector constructed for each molecule – similarity computed between bit-vectors (fast O(D) scan) • 2D Compressed Fingerprints – ALL substructures stored in a bit-vector using a hashing scheme plus lossy compression (modulo operator) – Similarity computed between bit-vectors or count vectors • Faster Searches – database pruning – locality sensitive hashing (LSH): towards O(log n) similarity searching 3 Superstructure and Substructure Searches A B • A is a superstructure of B (ignoring H) • B is a substructure of A • Tversky similarity 4 The Similarity Problem How similar? 5 Spectral Similarity 1. Count substructures 2. Compare the count/bit vectors 6 2D Graph Substructures • For chemical compounds – atom/node labels: A = {C,N,O,H, … } – bond/edge labels: B = {s, d, t, ar, … } • • • • Trace ALL Paths O(N*dl) Cycles and trees Combinatorial Space (CsNsCdO) 7 Mapping Structures to Bits • Compact data representation • Hash each path to bit vector Feature space → Bit space • Resolve clashes with OR operator (i.e 1+1=1) 8 Similarity Measures • There are many ways of measuring similarity (or distance) between bit/count vectors: – – – – – – – Euclidean Cosine Exponentials Tanimoto/Jaccard Tversky MinMax And many more (L1,L2,Lp,Hamming, Manhattan,….) 9 10 11 Similarity Measures: Tanimoto • Tally features: – Unique (a,b) – Both on (c) – Both off (d) A B a c b • Similarity Formula – Tanimoto=c/(a+b+c) 12 The Fingerprint Approximation • Fingerprint bit similarity approximates chemical feature similarity. 13 Similarity Measures: Tversky • Tally features: – Unique (a,b) – Both on (c) – Both off (d) A B a c b • Similarity Formula – Tanimoto=c/(a+b+c) – Tversky(α,β)=c/(αa+βb+c) 14 Pathological Cases On the Properties of Bit String-Based Measures of Chemical Similarity. Flower DR, J. Chem. Inf. Comput. Sci. 1998, 38, 379-386 15 Pathological Cases Issue of labeling scheme. 16 Counts • MinMax similarity is a generalization of Tanimoto which uses the counts. • MinMax can work better than Tanimoto. 17 Pruning Search Space Using Bounds • Linear speedup (search CxD) for fixed threshold, often by one order of magnitude or more. • Sub-linear speedup (search CxD0.6) for top K. 18 19 Speedup from Pruning Speedup depends on: – – – – Threshold Query Fingerprint length Database size 20 21 22 Bias in Query Distribution 23 24 25 Aggregate Queries (“Profiles”) 26 Two Basic Strategies • Similar to bioinformatics 1. Aggregate individual pairwise measures 2. Build a fingerprint profile – Linear approaches – Non-linear approaches (consensus, modal, etc) • • Hybrid (profile + aggregation/”scaling”)) Profile-profile 27 Aggregations 28 Consensus Fingerprints • Create consensus fingerprint • Search database using the consensus & = 29 Local Sensitive Hashing • Bin fingerprints based on projections onto randomly directed vectors • log D random vectors → O(log D) • Search for neighbors by returning bin corresponding to the query’s projection • Has been used for clustering. May be useful for building diverse data sets. Not yet developed for searching 30 Outline • • • • • • • • Historical Progression Similarity Measures Fingerprint Construction Pathologic Cases MinMax- Counts Pruning Search Space Aggregate Queries LSH 31