Paper presentation: D. N. Reshef et al., “Detecting Novel Associations in Large Data Sets,” Science, vol. 334, no. 6062, pp. 1518-1524, 2011.

1. CS-GN-TEAM: internal presentation
detecting novel associations
in large data sets
Michele Filannino + You
Presented paper:
D. N. Reshef et al., “Detecting Novel Associations in Large Data Sets,” Science, vol. 334,
no. 6062, pp. 1518-1524, 2011.
Manchester, 05/03/2012

2. presentation my research taster project
where we are
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3. Introduction

4. presentation my research taster project
novel association
■ two variables, X and Y, are associated if there is a
relationship between them
● functional
▶
● non functional
▶
■ novel: unknown
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5. presentation my research taster project
example
f0 f1 f2 f3 f4 f5
s0 4.00 -0.76 5.00 12.00 8.22 1.83
s1 9.00 0.41 10.00 23.00 27.12 4.30
s2 3.00 0.14 4.00 0.00 0.56 -0.43
s3 10.00 -0.54 11.00 100.00 94.02 6.24
s4 5.00 -0.96 6.00 45.00 39.25 3.56
s5 2.00 0.91 3.00 123.00 125.73 2.97
s6 7.00 0.66 8.00 4.00 9.26 2.56
s7 8.00 0.99 9.00 -2.00 6.90 2.37
s8 1.00 0.84 2.00 36.00 37.68 1.58
s9 6.00 -0.28 7.00 0.00 -1.96 0.71
Data set 10x6 05/03/2012, Michele Filannino 5 / 36

6. presentation my research taster project
example
f0 f1 f2 f3 f4 f5
s0 4.00 -0.76 5.00 12.00 8.22 1.83
s1 9.00 0.41 10.00 23.00 27.12 4.30
s2 3.00 0.14 4.00 0.00 0.56 -0.43
s3 10.00 -0.54 11.00 100.00 94.02 6.24
s4 5.00 -0.96 6.00 45.00 39.25 3.56
s5 2.00 0.91 3.00 123.00 125.73 2.97
s6 7.00 0.66 8.00 4.00 9.26 2.56
s7 8.00 0.99 9.00 -2.00 6.90 2.37
s8 1.00 0.84 2.00 36.00 37.68 1.58
s9 6.00 -0.28 7.00 0.00 -1.96 0.71
Data set 10x6 05/03/2012, Michele Filannino 6 / 36

7. presentation my research taster project
scatter plot: f0 vs. f2
f2(x) = f0(x) + 1 05/03/2012, Michele Filannino 7 / 36

8. presentation my research taster project
example
f0 f1 f2 f3 f4 f5
s0 4.00 -0.76 5.00 12.00 8.22 1.83
s1 9.00 0.41 10.00 23.00 27.12 4.30
s2 3.00 0.14 4.00 0.00 0.56 -0.43
s3 10.00 -0.54 11.00 100.00 94.02 6.24
s4 5.00 -0.96 6.00 45.00 39.25 3.56
s5 2.00 0.91 3.00 123.00 125.73 2.97
s6 7.00 0.66 8.00 4.00 9.26 2.56
s7 8.00 0.99 9.00 -2.00 6.90 2.37
s8 1.00 0.84 2.00 36.00 37.68 1.58
s9 6.00 -0.28 7.00 0.00 -1.96 0.71
Data set 10x6 05/03/2012, Michele Filannino 8 / 36

9. presentation my research taster project
scatter plot: f0 vs. f1
no relation 05/03/2012, Michele Filannino 9 / 36

10. presentation my research taster project
correlation coefficients
Pearson Mutual Infor. MI norm.
f0-f5 0.63 2.45 0.74
f0-f1 -0.17 1.57 0.47
f0-f2 1.00 3.32 1.00
f2-f3 -0.08 3.12 0.94
f0-f3 -0.08 3.12 0.94
05/03/2012, Michele Filannino 10 / 36

11. presentation my research taster project
pros. & cons.
■ Pearson’s coeff. ■ Mutual Information
✔ closed interval result ✔ non linear relations
✖ only linear relations ✖ only categorical data
✖ feature independency ✖ biased towards higher
arity features
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12. the new measure

13. presentation my research taster project
motivations
■ generality:
● capture a wide range of interesting associations, not
limited to specific function types
■ equitability:
● give similar scores to equally noisy relationships of
different types
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14. presentation my research taster project
definition of MIC
■ Given a finite set D of ordered pairs, we can
partition the X-values of D into x bins and the Y-
values of D into y bins
■ We obtain a pair of partitions called x-by-y grid
D = (F0, F1)
F0 = (1.00, 2.00, 3.00 | 4.00, 5.00, 6.00, | 7.00, 8.00, 9.00, 10.00)
F1 = (-0.96, -0.76 | -0.54, -0.28 | 0.14, | 0.41, | 0.66, 0.84, 0.91, 0.99)
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15. presentation my research taster project
x-by-y grid
2-by-4 grid 05/03/2012, Michele Filannino 15 / 36

16. presentation my research taster project
definition of MIC
■ given the grid we could calculate D|G, the frequency
distribution induced by the points in D on the cells
of G
● different grids G result in different distributions D|G
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17. presentation my research taster project
maximal MI over all grids
number of columns number of rows
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18. presentation my research taster project
characteristic matrix
Infinite matrix!
normalisation factor
(derived by MI definition)
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19. presentation my research taster project
Maximal Information Coeff.
max grid size
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20. presentation my research taster project
matrix computation
■ space of grids grows exponentially
● B(n) ≤ O(n1-ε) for 0 < ε < 1
■ approximation of MIC
● heuristic dynamic programming
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21. presentation my research taster project
MIC summary
✔ closed interval result
✔ non linear relations
✔ all types of data
✖ B(n) is crucial
✖ too high: non-zero scores even for random data
✖ too low: we are searching only for simple pattern
✖ still univariate
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22. presentation my research taster project
B(n) behaviour
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23. presentation my research taster project
B(n) behaviour
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24. how to use it

25. presentation my research taster project
python
import xstats.MINE as MINE
x = [40,50,None,70,80,90,100,110,120,130,140,150,
160,170,180,190,200,210,220,230,240,250,260]
y = [-0.07,-0.23,-0.1,0.03,-0.04,None,-0.28,-0.44,
-0.09,0.12,0.06,-0.04,0.31,0.59,0.34,-0.28,-0.09,
-0.44,0.31,0.03,0.57,0,0.01]
print "x y", MINE.analyze_pair(x, y)
https://github.com/ajmazurie/xstats.MINE 05/03/2012, Michele Filannino 25 / 36

26. presentation my research taster project
python: result
{'MCN': 2.5849625999999999,
'MAS': 0.040419996,
'pearson': 0.31553724,
'MIC': 0.38196000000000002,
'MEV': 0.27117000000000002,
'non_linearity': 0.28239626000000001}
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27. presentation my research taster project
correlation coefficients
Mutual
Pearson MI norm. MIC graph
Informat.
f0-f5 0.63 2.45 0.74 0.24
f0-f1 -0.17 1.57 0.47 0.24
f0-f2 1.00 3.32 1.00 1.00
f2-f3 -0.08 3.12 0.94 0.24
f0-f3 -0.08 3.12 0.94 0.24
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28. presentation my research taster project
MIC summary
✔ closed interval result
✔ non linear relations
✔ all types of data
✖ B(n) is crucial
✖ n is too low!
✖ still univariate
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29. presentation my research taster project
python
import xstats.MINE as MINE
import math
x = [n*0.01 for n in range(1,2000)]
y = [math.sin(n) for n in x]
result = MINE.analyze_pair(x, y)
print "MIC:", result[‘MIC’]
print "Pearson:", result[‘pearson’]
>>> MIC: 0.99999
>>> Pearson: -0.16366038
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30. conclusion

31. presentation my research taster project
relationship types
Source: paper 05/03/2012, Michele Filannino 31 / 36

32. presentation my research taster project
relationship types
Source: paper 05/03/2012, Michele Filannino 32 / 36

33. presentation my research taster project
real application
Source: paper 05/03/2012, Michele Filannino 33 / 36

34. presentation my research taster project
suggestions
■ use MIC only when you have lots of samples
● samples > 2000
■ use B(n) = n0.6
■ don’t use it for all the possible pairs of features
● it is slower than Pearson’s correlation coefficient or
Mutual Information
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35. Thank you.

36. presentation my research taster project
references
■ D. N. Reshef et al., “Detecting Novel Associations in
Large Data Sets,” Science, vol. 334, no. 6062, pp.
1518-1524, 2011.
■ D. N. Reshef et al., “Supporting Online Material for
Detecting Novel Associations in Large Data Sets”
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