A Psychological Perspective on Similarity and Distance Measures. Daniel Müllensiefen Department of Psychology Goldsmiths University of London

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1 A Psychological Perspective on Similarity and Distance Measures Daniel Müllensiefen Department of Psychology Goldsmiths University of London

2 Structure 1 The Psychology of Similarity Perception 2 Similarity in Music Perception (Questions and Applications) 3 Evaluation on a Musical Dataset 4 Summary and Conclusion

3 Introduction: The Psychology Similarity Perception Geometrical Models Set-theortic Models Transformative Models

4 Geometrical Models Psychology of Perception (1960s-80s) Assumed mechanism of similarity perception: Human mind extracts parametric properties from object Computes distance/similarity between objects across properties Objects are located in mental space, but extraction and distance computation are often subconscious Consciously accessible: only similarity judgements Perceptual Similarity as a metric: Identity Symmetry Triangle inequality

5 Geometrical Models Tools: Psychological measurement theory for quantifying object properties (e.g. Stevens, 1951) Multi-Dimensional Scaling (e.g. Shepard, 1962; Kruskal & Wish, 1978) Data: Pair-wise similarity ratings Ranking Tri-angular ratings (ABX designs)

6 Geometrical Models Representation: Objects with coordinates in lowdimensional cognitive space From Shepard (1962) From Meyer & Eisenberg (1988)

7 Set-theoretic Models Tversky s critique of geometrical models (1977, Tversky & Hutchinson, 1986): Human similarity judgements often not symmetric Qualities of objects rather perceived as nominal features than continuous properties Conceptual data often better represented by cluster membership than geometrical space

8 Set-theoretic Models Tversky s ratio model of similarity (1977): "(s,t) = f (s n # t n ) f (s n # t n ) + $f (s n \ t n ) + %f (t n \ s n ),$,% & 0 Similarity depends on: Number of features objects s,t have in common / not in common Psychological salience of features f() Weights α and β to determine symmetry relation Note: Not a metric: no symmetry, no triangle inequality Makes use of statistical context information via salience function

9 Transformative Models Critique of geometrical and set-theoretic approaches (Markman & Gentner, 1993; Hahn et al., 2003): Real-world objects are more than sets of features or coordinates in space Relations between elements within objects are also important

10 Transformative Models Structural Mapping Similarity (Falkenhainer et al., 1994; Goldstone, 1994):

11 Transformative Models Representational Distortion (Chater & Hahn, 1997): Similarity between objects s,t is function of effort/complexity to transform s into t. Interpretation / Implementations: Levenshtein (edit) distance for symbol sequences Transportation distances Kolmogorov complexity > Normalised Information Distance (Li & Vitanyi, 1997) NID(s,t) = max{k(t s),k(s t)} max{k(s),k(t)}

12 Transformative Models Representational Distortion (RD) via Kolmogorov complexity: Often approximated by compression distance using standard compression algorithms (e.g. gzip, bzip2) Compression distance CD(s,t) is #bits of t compressed given s. Note: NCD(s,t) = Z(st) " min{z(s),z(t)} max{z(s),z(t)} NCD with appropriate compressor is a metric Works on digital files, perceptual and conceptual data NCD is context-free

13 Similarity in Music Research Research topics and applications: Tune classification in folk song research =>organisation of tune collections Music categorisation and search => Music Information Retrieval Identification of musical relations (e.g. theme and variations ) => music analysis and models of music perception Identification of cover songs and plagiarism detection => commercial relevance

14 Musical Plagiarism Huge public interest, important for pop industry - very little research Idea (Müllensiefen & Pendzich, 2009; Cason & Müllensiefen, submitted; Wolf & Müllensiefen, in prep): Measure similarity between melodies using different similarity models Compare similarity values to previous court decisions Compare both to listeners perception

15 Evaluation Dataset: 19 court cases from US and Commonwealth jurisdiction Binary dependent variable: Pro-plaintiff = plagiarism (8/19) Contra-plaintiff = no plagiarism (11/19)

16 Making Melodies Computable m-type of length 2: s1e_s1e m-type of length 4: s1q_s1l_s1q_s1l Symbol sequence encoding: s1e_s1e_s1q_u2q_d5l_s1q_s1l_s1q_s1l_s1q_s1q_s1l_s1q_s1l i.abs.std = $ i ("p i # "p) 2 N #1 = 2.83 Overlap in m-types between s, t (Tversky) Mutual compressability of s,t (Vitanyi) Euclidean distance of global features between s,t (Shepard)

17 Similarity Measures Euclidean Distance across global summary features Overlap of melodic motives (=nominal features) weighted by inverted document frequencies in large pop corpus; asymmetric plaintiff perspective Compression effort of distorting one symbol string into another

18 Experiment Implicit memory paradigm: Confusion matrix as proxy for cognitive similarity 32 participants Exposure phase: Listen 3x to 20 tunes, cover tasks Test phase: Listen to 30 tunes, indicate the ones form test phase (10 unrelated, 5 identical, 15 similar) Dependent variable: #confusions with similar item from exposure phase

19 Results Euclidean Distance (Shepard) Feature overlap (Tversky) Compression Distance (Vitanyi) Human listeners Human listeners (correlation) Court decisions (AUC)

20 Summary Different mathematical concepts of distance/similarity are at core of different psychological theories of similarity perception Appropriateness not clear apriori, may depend on: Perceptual / conceptual objects? Sequential vs. non-sequential objects? Purpose of distance calculations (i.e. similarity measurement, classification) Identification of good data representation Usefulness of statistical context information

21 Next steps and open questions Next steps: Comparison with data from ranking task (explicit) Cross-validation on other similarity datasets Comparison with similarity measures based on music theory Experiment with different compressors, e.g. PPMZ (prediction by partial match) Open questions: Do court judgements differ by country? Do subjects differ systematically from each other? How to approximate small GT datasets by larger dataset from lab / online surveys? How to compare the performance of different similarity measures?

22 A Psychological Perspective on Similarity and Distance Measures

23 Geometrical Models Distance <--> Similarity: Daniel Müllensiefen Department of Psychology The universal law of generalization (Shepard, 1987) Goldsmiths University of London s ij = e "c#d ij