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CSCI568 Lecture 8: Similarity Metrics September 11, 2009 Monday, September 14, 2009

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  • CSCI568Lecture 8: Similarity Metrics

    September 11, 2009

    Monday, September 14, 2009

  • Dis/similarity Between Two Attributes

    Type Dissimilarity Similarity

    Nominal

    Ordinal

    Interval/Ratio

    Monday, September 14, 2009

  • Dis/similarities Between Data Objects

    Euclidean distance Pearson Correlation Coefficient Simple Matching Coefficient (SMC) Jaccard / Tanimoto Cosine Similarity Bregman Divergence

    Monday, September 14, 2009

  • Minkowski Distance Metric

    General distance calculation r=1 City Block r=2 Euclidean r=(infinity) Supremum (think lim(r->inf.))

    DM 70

    Monday, September 14, 2009

  • Euclidean Distance

    Simple! Linear distance between two points.

    d(x/y) = (xk - yk)2k=1

    n

    xk and yk are values of kth attribute of objects x and y

    DM 69 - 71

    Monday, September 14, 2009

  • Simple Matching Coefficient

    Linear distance is good for many things, but not necessarily binary data!

    Monday, September 14, 2009

  • Simple Matching Coefficient

    SMC = # of matching attributes# of attributes

    Simple!

    SMC = f11 + f00f01 + f10 + f11 + f00

    Monday, September 14, 2009

  • Jaccard Coefficient

    Jaccard = # of matching present attributes# of attributes w/ values present

    Simple!

    Jaccard = f11f01 + f10 + f11

    Monday, September 14, 2009

  • SMC vs. JaccardLike SMC, but for asymmetric binary attributes.

    (we only care about presence)

    Think: market basket data (sparse dataset, asymmetric/binary attributes)

    SMC --> most transactions are alike (everyone doesnt purchase most items

    Jaccard --> only compares attributes w/ existing values

    Monday, September 14, 2009

  • SMC / Jaccard Example

    x = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0)y = (0, 0, 0, 0, 0, 0, 1, 0, 0, 1)

    f01 = 2f10 = 1f00 = 7f11 = 0

    SMC = f11 + f00f01 + f10 + f11 + f00

    = 0+7

    2+1+0+7

    Jaccard = f11f01 + f10 + f11

    = 02+1

    Monday, September 14, 2009

  • Cosine Similarity

    Often used for document word-frequency.

    cos_sim(x,y) = x . y

    ||x|| ||y||

    Monday, September 14, 2009

  • Cosine Similarity Example

    x = (3, 2, 0, 5, 0, 0, 0, 2, 0, 0)y = (1, 2, 0, 0, 0, 0, 0, 1, 0, 2)

    cow pig dog cat log bug fox ape man carx 3 2 0 5 0 0 0 2 0 0y 1 2 0 5 0 0 0 1 0 2

    x . y = 3*1 + 2*0 + 0*0 + 5*0 + 0*0... 2*1...0*2 = 5

    ||x|| = sqrt(3*3+2*2...) = 6.48||y|| = sqrt(1*1+0*0...) = 2.24

    Monday, September 14, 2009

  • Extended Jaccardaka Tanimoto Coefficient

    (reduces to Jaccard for binary attributes)

    jaccard() --> compute similarities of binary attributes

    tanimoto() --> compute similarities of continuous attributes

    Monday, September 14, 2009

  • Extended Jaccard (Tanimoto Coefficient)

    EJ(x,y) =x . y

    ||x||2 + ||y||2 - x . y

    Monday, September 14, 2009

  • Pearson CorrelationThink: Like Euclidean, but corrects for grade inflation.

    eg: Movie ratings. Some users consistently give more stars than others. Euclidean is ok, Pearson is better.

    Monday, September 14, 2009

  • Pearson Correlation

    Always [-1, 1]

    For binary/continuous attributes.

    DM 77CI 11

    Monday, September 14, 2009

  • Example: Movie Recommendations

    CI chapter 2

    Monday, September 14, 2009


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