Finding Outstanding Aspects and Contrast Subspaces Jian Pei School - - PowerPoint PPT Presentation
Finding Outstanding Aspects and Contrast Subspaces Jian Pei School of Computing Science Simon Fraser University jpei@cs.sfu.ca CHIRC Computational Health Intelligence Research Centre Population health powered by big data
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Symptoms:
high blood pressure, back pain, short of breadth, chest pain, cold sweat … In what aspect is he most similar to cases of coronary artery disease and, at the same time, dissimilar to adiposity?
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−distS (q,o)2 2h2 S
−distS (q,o)2 2h2 S+
−distS (q,o)2 2h2 S−
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S(q | O+) = 1 |O+| √ 2πσ0
minh0
distS (q,o)2 2(σS h0
σ0
min = min{σS0 | S0 ∈ S}, h0
h0
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For a query object q, a set of objects O, and subspaces S1, S2 such that S1 is an ancestor of S2 in the subspace set enumeration tree using the standard deviation descending order in O+, L∗
S1(q | O+) ≥ LS2(q | O+).
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S(q) = {o ∈ O | distS(q, o) ≤ ✏}
S(q | O) + Lrest
S
S(q | O) =
1 |O| √ 2πhS
S(q)
−distS (q,o)2 2h2 S
S
1 |O| √ 2πhS
S(q)
−distS (q,o)2 2h2 S
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|O|−|N ✏
S(q)|
|O| √ 2πhS · e − distS (q,O)2
2h2 S
S
S(q)|
|O| √ 2πhS · e −
✏2 2h2 S
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For a query object q, a set of objects O and ✏ ≥ 0, LL✏
S(q | O) ≤ LS(q | O) ≤ UL✏ S(q | O)
where LL✏
S(q | O) =
1 |O| √ 2⇡hS @ X
S(q)
e
−dist✏ S (q,o)2 2h2 S
+ (|O| − |N ✏
S(q)|)e − distS (q,O)2
2h2 S
1 A and UL✏
S(q | O) =
1 |O| √ 2⇡hS @ X
S(q)
e
−dist✏ S (q,o)2 2h2 S
+ (|O| − |N ✏
S(q)|)e −
✏2 2h2 S
1 A For a query object q, a set of objects O+, a set of objects O−, and ✏ ≥ 0, LCS(q) ≤ UL✏
S(q|O+)
LL✏
S(q|O−).
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Assist Personal foul
Joe
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Assist Points/game
Joe 1 2 3 4 5 10 15 20 25 30
Personal foul Points/game
Joe
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Given a set of objects O in a multidimensional space D, a query object q 2 O and a maximum dimensionality threshold 0 < ` |D|, a subspace S ✓ D (0 < |S| `) is called a minimal outlying subspace of q if
such that rankS0(q) < rankS(q); and
that rankS00(q) = rankS(q). The problem of outlying aspect mining is to find the minimal outlying subspaces of q.
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− P
Di∈S (q.Di−o.Di)2 2h2 Di
Given a set of objects O in space S = {D1, . . . , Dd}, define a linear transfor- mation g(o) = (a1o.D1+b1, . . . , ado.Dd+bd) for any o ∈ O, where a1, . . . , ad and b1, . . . , bd are real numbers. Let O0 = {g(o)|o ∈ O} be the transformed data set. For any objects o1, o2 ∈ O such that ˜ fS(o1) > ˜ fS(o2) in O, ˜ fS(g(o1)) > ˜ fS(g(o2)) if the product kernel is used and the bandwidths are set using H¨ ardle’s rule of thumb
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