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Metrics.ppt

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Metrics.ppt
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Metrics,Mikko Kolehmainen Department of Environmental Science University of Eastern Finland,Norm (http://mathworld.wolfram.com/Norm.html),The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object. Most commonly, the unqualified term “norm“ refers the flavor of vector norm technically known as the L2-norm.,16.5.2019,Mikko Kolehmainen,Example of calculating the norm,16.5.2019,Mikko Kolehmainen,x =20191714109101187989131316172122 norm(x)ans =61.6036,What is a metric?,A metric is a dissimilarity measure that satisfies three conditions (Hand et. al, 2001): d(i,j) = 0 for all i and j, and d(i,j) = 0 if and only if i = j d(i,j) = d(j,i) for all i and j d(i,j) = d(i,k) + d(k,j) for all i, j and k (triangle inequality),16.5.2019,Mikko Kolehmainen,Definition of metric example (similarity in this case),16.5.2019,Mikko Kolehmainen,1: A-T-C-G-C 2: A-C-G-G-C,”Rules” for creating a metric: Directly opposite = 1 point Slantwise opposite = 0.5 points Otherwise = 0 points,1) A-T-C-G-C A-A, G-G, C-C = 3 x 1.0 = 3.0A-C-G-G-C,2) A-T-C-G-C C-C, G-G = 2 x 0.5 = 1.0A-C-G-G-C,Total: d(1,2) = 3.0 + 1.0 = 4.0,Euclidean distance metrics,If two vectors (xi and xj) are given the corresponding Euclidean distance of them can be defined as follows (two ways):,16.5.2019,Mikko Kolehmainen,,,,,Euclidean distance example,16.5.2019,Mikko Kolehmainen,a =1425 bb =1346 dEUC = sqrt((a-b)'*(a-b))dEUC =2.4495,Euclidean distance metrics,Assumes some commensurability between the variables This must usually be assured by pre-processing using some suitable transformation Special case of Minkowski Lλ metrics with λ = 2 L1 is so called Manhattan or city-block distance What is L∞ ?,16.5.2019,Mikko Kolehmainen,Inner product,Also called dot product Describes similarity whereas Euclidean is a distance measure,16.5.2019,Mikko Kolehmainen,,Example of calculating the inner product,16.5.2019,Mikko Kolehmainen,a =1425 bb =1346 IProd = a'*bIProd
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