On a fast growing online platform arise numerous metrics. With increasing amount of metrics methods of exploratory data analysis are becoming more and more important. We will show how recognition of similar metrics and clustering can make monitoring feasible and provide a better understanding of their mutual dependencies.

1. Extracting relevant Metrics with Spectral Clustering
Evelyn Trautmann
PyData Berlin
6-8 July, 2018
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2. Introduction
Introduction
platform monitoring means keeping track of various metrics
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3. Introduction
Introduction
platform monitoring means keeping track of various metrics
some effects are visible only in sub-dimensions like single countries, or
device types
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4. Introduction
Introduction
platform monitoring means keeping track of various metrics
some effects are visible only in sub-dimensions like single countries, or
device types
dimension reduction without losing important details
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5. Algorithm
Clustering
metrics considered as points in a vector space
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6. Algorithm
Clustering
metrics considered as points in a vector space
similarity defines (inverse) distance measure
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7. Algorithm
Clustering
metrics considered as points in a vector space
similarity defines (inverse) distance measure
find clusters of closely related points
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8. Algorithm
Toy Example - Gaming App
metric 0 game a: count of games played by paying users
metric 1 game a: count of games played by non paying users
metric 2 game b: count of games played by paying users
metric 3 game b: count of games played by non paying users
...
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9. Algorithm
Similarity Graph
Figure: Similarity Graph
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10. Algorithm
Matrix Representation
Figure: Matrix Representation
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11. Algorithm
Matrix Representation
Figure: Matrix Representation
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12. Algorithm
Toy Example
Toy Example.ipynb
https://github.com/metterlein/spectral clustering.
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13. Algorithm
Spectral Clustering
1: procedure SpectralClustering(X, k)
2: compute similarity matrix with pairwise similarities
3: Transform to Graph Laplacian L
4: v1, ..., vk = eig(L, k) first k eigenvectors
5: U = V T k n-dimensional → n k-dimensional vectors
6: clusterAssignment = kmeans(U, k)
7: return clusterAssignment
8: end procedure[vL07]
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14. Parameters
Similarity Function
Similarity Function φ : Rn × Rn → [0, 1]
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15. Parameters
Similarity Function
Similarity Function φ : Rn × Rn → [0, 1]
radial basis function [PVG+11]
φ(xi , xj ) = exp(−γ xi − xj
2
)
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16. Parameters
Similarity Function
Similarity Function φ : Rn × Rn → [0, 1]
radial basis function [PVG+11]
φ(xi , xj ) = exp(−γ xi − xj
2
)
correlation based similarity
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17. Parameters
Similarity Function
Similarity Function φ : Rn × Rn → [0, 1]
radial basis function [PVG+11]
φ(xi , xj ) = exp(−γ xi − xj
2
)
correlation based similarity
handle negative correlations
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18. Parameters
Number of Clusters
with clear block structure eigenvalue gap is obvious
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19. Parameters
Number of Clusters
with clear block structure eigenvalue gap is obvious
if connectivity amongst blocks is too large, determination of cluster
count is getting complex
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20. Parameters
Number of Clusters
with clear block structure eigenvalue gap is obvious
if connectivity amongst blocks is too large, determination of cluster
count is getting complex
possible approach:
- PCA explained variance
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21. Application
Data Description
metrics representing user activities, payments, etc
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22. Application
Data Description
metrics representing user activities, payments, etc
each metric is divided in several dimensions like country, gender, etc
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23. Application
Data Description
metrics representing user activities, payments, etc
each metric is divided in several dimensions like country, gender, etc
combination of metrics and dimensions generates around 300
timeseries in our example.
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24. Application
Data Preparation
aggregation per day
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25. Application
Data Preparation
aggregation per day
normalization X = 1
σ (X − µ), with µ mean value and σ standard
deviation
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26. Application
Data Preparation
aggregation per day
normalization X = 1
σ (X − µ), with µ mean value and σ standard
deviation
rolling mean of 7 days to smooth weekly periodicity
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27. Application
Measure Clustering Quality
compute variance for each timepoint over all cluster members
optimal clustering minimizes variance for each cluster
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28. Demo
Real Data Example
Real Data Example.ipynb
https://github.com/metterlein/spectral clustering
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29. Conclusion
Conclusion
clustering metrics reduces dimensionality of observation space
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30. Conclusion
Conclusion
clustering metrics reduces dimensionality of observation space
small number of interpretable cluster representatives help keeping
track of main platform dynamics
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31. Conclusion
Conclusion
clustering metrics reduces dimensionality of observation space
small number of interpretable cluster representatives help keeping
track of main platform dynamics
by cluster assignments can be discovered unexpected relations
between several metrics
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32. Conclusion
Outlook
investigate cluster assignment change over time
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33. Conclusion
Outlook
investigate cluster assignment change over time
hierarchical approach: clustering sub-blocks
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34. Conclusion
Outlook
investigate cluster assignment change over time
hierarchical approach: clustering sub-blocks
add time shifted series to recognize Granger causalities
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35. Conclusion
References I
F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion,
O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg,
J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and
E. Duchesnay, Scikit-learn: Machine learning in Python, Journal of
Machine Learning Research 12 (2011), 2825–2830.
Evelyn Trautmann, Spectral clustering demo,
https://github.com/metterlein/spectral_clustering, 2018.
Ulrike von Luxburg, A tutorial on spectral clustering, CoRR
abs/0711.0189 (2007).
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36. Conclusion
Thank You!
Questions?
evelyn.trautmann@lovoo.com
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37. Conclusion
Graph Laplacian
Graph Laplacian with similarities as off-diagonal entries
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38. Conclusion
Graph Laplacian
Graph Laplacian with similarities as off-diagonal entries
negative row sums on diagonal
L =
lij ≥ 0, for i = j
lii = − k=i lik, for i = j.
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39. Conclusion
Graph Laplacian
Graph Laplacian with similarities as off-diagonal entries
negative row sums on diagonal
L =
lij ≥ 0, for i = j
lii = − k=i lik, for i = j.
matrix exponential is a stochastic matrix
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40. Conclusion
Graph Laplacian
Graph Laplacian with similarities as off-diagonal entries
negative row sums on diagonal
L =
lij ≥ 0, for i = j
lii = − k=i lik, for i = j.
matrix exponential is a stochastic matrix
results from Markov chain theory applicable
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41. Conclusion
Clustering Methods
[PVG+11]
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42. Conclusion
PCA
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43. Conclusion
PCA
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44. Conclusion
Extract Platform Dynamics by means of Cluster
Centers
compute average timeseries per cluster
illustrate average cluster timeseries with variance corridor
visualize metrics types and subdimensions entering respective clusters
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