Daniel Müllensiefen

Department of Psychology

GoldsmithsUniversity of London

A PsychologicalPerspective on Similarity

and Distance Measures

StructureStructure1 The Psychology of Similarity Perception2 Similarity in Music Perception (Questions

and Applications)3 Evaluation on a Musical Dataset4 Summary and Conclusion

IntroductionIntroduction: : The PsychologyThe PsychologySimilarity PerceptionSimilarity Perception

Geometrical ModelsSet-theortic ModelsTransformative Models

Geometrical Geometrical ModelsModels 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

Geometrical Geometrical ModelsModels Tools:

Psychological measurement theory forquantifying 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)

Geometrical Geometrical ModelsModels Representation: Objects with coordinates in low-

dimensional cognitive space

From Shepard (1962) From Meyer & Eisenberg (1988)

Tversky’s critique of geometrical models(1977, Tversky & Hutchinson, 1986):

Human similarity judgements often notsymmetric

Qualities of objects rather perceived asnominal features than continuousproperties

‘Conceptual’ data often betterrepresented by cluster membership thangeometrical space

Set-theoretic Set-theoretic ModelsModels

Tversky’s ratio model of similarity (1977):

Set-theoretic Set-theoretic ModelsModels

!

"(s,t) =f (sn # tn )

f (sn # tn ) +$f (sn \ tn ) + %f (tn \ sn ),$,% & 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

Transformative Transformative ModelsModels

Critique of geometrical and set-theoreticapproaches (Markman & Gentner, 1993; Hahn et al., 2003):

Real-world objects are more than sets offeatures or coordinates in space

Relations between elements within objects arealso important

Transformative Transformative ModelsModels

Structural Mapping Similarity (Falkenhainer etal., 1994; Goldstone, 1994):

Transformative Transformative ModelsModels

Representational Distortion (Chater & Hahn, 1997):

Similarity between objects s,t is function ofeffort/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)}

Transformative Transformative ModelsModels

Representational Distortion (RD) viaKolmogorov 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.

!

NCD(s,t) =Z(st) "min{Z(s),Z(t)}

max{Z(s),Z(t)}

Note: NCD with appropriate compressor is a metric Works on digital files, perceptual and conceptual data NCD is context-free

Similarity Similarity in Musicin Music ResearchResearchResearch topics and applications:

Tune classification in folk song research =>organisationof tune collections

Music categorisation and search => Music InformationRetrieval

Identification of musical relations (e.g. ‘theme andvariations’) => music analysis and models of musicperception

Identification of cover songs and plagiarism detection=> commercial relevance

Musical Musical PlagiarismPlagiarism Huge public interest,

important for popindustry - very littleresearch

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

EvaluationEvaluation Dataset: 19 court cases from US and

Commonwealth jurisdiction Binary dependent variable:

Pro-plaintiff = plagiarism (8/19) Contra-plaintiff = no plagiarism (11/19)

Making Melodies ComputableMaking Melodies Computable

!

i.abs.std ="pi # "p( )

2

i$

N #1= 2.83

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”

Overlap in m-typesbetween s, t (Tversky)

Mutualcompressability of s,t

(Vitanyi)

Euclidean distance ofglobal features between

s,t (Shepard)

Similarity MeasuresSimilarity Measures Euclidean Distance across global summary

features Overlap of melodic motives (=nominal

features) weighted by inverted documentfrequencies in large pop corpus; asymmetric“plaintiff perspective”

Compression effort of distorting one symbolstring into another

ExperimentExperimentImplicit 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, 15similar)

Dependent variable: #confusions with similar itemfrom exposure phase

ResultsResults

0.730.70.720.63Courtdecisions(AUC)

10.810.910.42Humanlisteners(correlation)

Humanlisteners

CompressionDistance(Vitanyi)

Featureoverlap(Tversky)

EuclideanDistance(Shepard)

SummarySummary Different mathematical concepts of

distance/similarity are at core of differentpsychological 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

Next steps Next steps and and open questionsopen questionsNext 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?

A PsychologicalPerspective on Similarity

and Distance Measures

Distance <--> Similarity:The universal law of generalization (Shepard, 1987)

Geometrical Geometrical ModelsModels

!

sij = e"c#d ij

Daniel Müllensiefen

Department of Psychology

GoldsmithsUniversity of London