1. Characterizing the Density of Chemical Spaces and
its Use in Outlier Analysis and Clustering
Rajarshi Guha
School of Informatics
Indiana University
17th August, 2007
Cambridge, MA

2. Outline
R-NN curves for diversity analysis
Counting clusters with R-NN curves
Density and domain applicability

3. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Nearest Neighbor Methods
Traditional kNN methods are q
simple, fast, intuitive q q
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Applications in q
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regression & classification q
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diversity analysis q
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Can be misleading if the nearest
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neighbor is far away q
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R-NN methods may be more q
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suitable

4. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Diversity Analysis
Why is it Important?
Compound acquisition
Lead hopping
Knowledge of the distribution of compounds in a descriptor
space may improve predictive models

5. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Approaches to Diversity Analysis
Cell based q q
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Divide space into bins q
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Compounds are mapped to bins
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Disadvantages q
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Not useful for high dimensional
data
Choosing the bin size can be tricky
Schnur, D.; J. Chem. Inf. Comput. Sci. 1999, 39, 36–45
Agrafiotis, D.; Rassokhin, D.; J. Chem. Inf. Comput. Sci. 2002, 42, 117–12

6. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Approaches to Diversity Analysis
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Distance based q q
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Considers distance between q
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compounds in a space q q
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Generally requires pairwise distance q
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calculation q q q
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Can be sped up by kD trees, MVP q
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trees etc. q
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Agrafiotis, D. K. and Lobanov, V. S.; J. Chem. Inf. Comput. Sci. 1999, 39, 51–58

7. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Generating an R-NN Curve
Observations
Consider a query point with a hypersphere, of radius R,
centered on it
For small R, the hypersphere will contain very few or no
neighbors
For larger R, the number of neighbors will increase
When R ≥ Dmax , the neighbor set is the whole dataset
The question is . . .
Does the variation of nearest neighbor count with radius allow us
to characterize the location of a query point in a dataset?

8. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Generating an R-NN Curve
Observations
Consider a query point with a hypersphere, of radius R,
centered on it
For small R, the hypersphere will contain very few or no
neighbors
For larger R, the number of neighbors will increase
When R ≥ Dmax , the neighbor set is the whole dataset
The question is . . .
Does the variation of nearest neighbor count with radius allow us
to characterize the location of a query point in a dataset?

9. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Generating an R-NN Curve
Algorithm q
q 50%
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Dmax ← max pairwise distance q q
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30%
for molecule in dataset do q q
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R ← 0.01 × Dmax
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q 10%
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q 5%
while R ≤ Dmax do
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q qq q q q
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Find NN’s within radius R
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Increment R
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end while q
end for q
Plot NN count vs. R
Guha, R. et al.; J. Chem. Inf. Model., 2006, 46, 1713–1772

10. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Generating an R-NN Curve
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R R
Sparse Dense

11. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Characterizing an R-NN Curve
Converting the Plot to Numbers
Since R-NN curves are sigmoidal, fit them to the logistic
equation
1 + me −R/τ
NN = a ·
1 + ne −R/τ
m, n should characterize the curve
Problems
Two parameters
Non-linear fitting is dependent on the starting point
For some starting points, the fir does not converge and
requires repetition

12. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Characterizing an R-NN Curve
Converting the Plot to a Number
Determine the value of R where the
lower tail transitions to the linear
portion of the curve S=0
S’’
Solution
Determine the slope at various
points on the curve S’
S=0
Find R for the first occurence of
the maximal slope (Rmax(S) )
Can be achieved using a finite
difference approach

13. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Characterizing an R-NN Curve
Converting the Plot to a Number
Determine the value of R where the
lower tail transitions to the linear
portion of the curve S=0
S’’
Solution
Determine the slope at various
points on the curve S’
S=0
Find R for the first occurence of
the maximal slope (Rmax(S) )
Can be achieved using a finite
difference approach

14. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Characterizing Multiple R-NN Curves
Problem
Visual inspection of curves is useful
3904
for a few molecules
For larger datasets we need to
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1861
summarize R-NN curves
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Rmax S
Solution
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Plot Rmax(S) values for each
molecule in the dataset
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Points at the top of the plot are
0 1000 2000 3000 4000
located in the sparsest regions Serial Number
Points at the bottom are located in
the densest regions
Kazius, J. et al.; J. Med. Chem. 2005, 48, 312–320

15. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
R-NN Curves and Outliers
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q H2N
HN N•
Number of Neighbors
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HO
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H HO
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NH NH
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H2N S
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OH NH
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16. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
How Can We Use It For Large Datasets?
Breaking the O(n2 ) barrier
Traditional NN detection has a time complexity of O(n2 )
Modern NN algorithms such as kD-trees
have lower time complexity
restricted to the exact NN problem
Solution is to use approximate NN algorithms such as Locality
Sensitive Hashing (LSH)
Bentley, J.; Commun. ACM 1980, 23, 214–229
Datar, M. et al.; SCG ’04: Proc. 20th Symp. Comp. Geom.; ACM Press, 2004
Dutta, D.; Guha, R.; Jurs, P.; Chen, T.; J. Chem. Inf. Model. 2006, 46, 321–333

17. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
How Can We Use It For Large Datasets?
−1.0
−1.5
log [Mean Query Time (sec)]
Why LSH?
−2.0
Theoretically sublinear
−2.5
Shown to be 3 orders of
−3.0
LSH (142 descriptors)
LSH (20 descriptors)
magnitude faster than kNN (142 descriptors)
kNN (20 descriptors)
−3.5
traditional kNN
0.02 0.03 0.04 0.05 0.06 0.07 0.08
Very accurate (> 94%) Radius
Comparison of NN detection speed
on a 42,000 compound dataset
using a 200 point query set
Dutta, D.; Guha, R.; Jurs, P.; Chen, T.; J. Chem. Inf. Model. 2006, 46, 321–333

18. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Alternatives?
Why not use PCA?
R-NN curves are fundamentally a
form of dimension reduction
4
Principal Components Analysis is
2
also a form of dimension reduction
Principal Component 2
0
Disadvantages
−2
Eigendecomposition via SVD is
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O(n3 )
−6
Difficult to visualize more than 2 or 0 5 10
Principal Component 1
3 PC’s at the same time
We are no longer in the original
descriptor space

19. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
R-NN Curves and Clusters
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Smoothed R-NN Curves
R-NN curves are indicative of the number of clusters

20. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
R-NN Curves and Clusters
Counting the steps Approaches
Essentially a curve matching Hausdorff / Fr´chet distance
e
problem - requires canonical curves
All points will not be RMSE from distance matrix
indicative of the number of Slope analysis
clusters
Not applicable for radially
distributed clusters

21. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
R-NN Curves and Their Slopes
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Slope of the RNN Curve
Slope of the RNN Curve
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Slope of the RNN Curve
Slope of the RNN Curve
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Smoothed first derivative of the R-NN Curves
Guha, R. et al., J. Chem. Inf. Model., 2007, 47, 1308–1318

22. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
R-NN Curves and Their Slopes
251 84 251 84
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Slope of the RNN Curve
Slope of the RNN Curve
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88 137 88 137
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Slope of the RNN Curve
Slope of the RNN Curve
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Smoothed first derivative of the R-NN Curves
Identifying peaks identifies the number of clusters
Automated picking can identify spurious peaks
Guha, R. et al., J. Chem. Inf. Model., 2007, 47, 1308–1318

23. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Slope Analysis of R-NN Curves
RNN Curve
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Procedure
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for i in molecules do
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D'
Evaluate R-NN curve
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F ← smoothed R-NN curve
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Evaluate F
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Smooth F
Nroot,i ← no. of roots of F
0
0 20 40 60 80 100
end for D''
Ncluster = [max(Nroot ) + 1]/2
0.5
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24. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
Simulated Data
Simulated 2D data using a Thomas point process
Predicted k, followed by kmeans clustering using k
Investigated similar values of k
k ASW k ASW k ASW
4 0.61 3 0.44 4 0.64
3 0.74 2 0.65 3 0.48
5 0.70 4 0.47 5 0.56
ASW - average silhouette width, higher is better; k - number of clusters

25. R-NN Curves for Diversity Analysis
Linking R-NN Curves to Cluster Counts
Density & Domain Applicability
A Mixed Dataset
Dataset composition
277 DHFR inhibitors based on
substituteed pyrimidinediamine and
diaminopteridine scaffolds
277 molecules from the DIPPR project
mainly simple hydrocarbons
boiling point was modeled
Evaluated 147 Molconn-Z descriptors,
reduced to 24
We expect at least 3 clusters
Sutherland, J. et al.; J. Chem. Inf. Comput. Sci., 2003, 43, 1906–1915
Goll, E. and Jurs, P.; J. Chem. Inf. Comput. Sci., 1999, 39, 974–983