1. GLBIO ML workshop May 17, 2016 Ivan Kryukov and Jeff Wintersinger
Clustering
GLBIO ML workshop
May 17, 2016
Ivan Kryukov and Jeff Wintersinger

2. Introduction

3. Why cluster? Goal: given data points, group them by common properties
What properties do they share?
Example of unsupervised learning -- no ground truth against which we can compare
Sometimes we want small number of clusters broadly summarizing trends
Sometimes we want large number of homogeneous clusters, each with only a few members
Image source: Wikipedia

4. Our problem
We have single-cell RNA-seq data for 271 cells across 575 genes
Cells sampled at 0 h, 24 h, 48 h, 72 h
Do cells at same timepoints show same gene expression?
If each cluster consists of only cells from the same timepoint, then the answer is yes!
Image source: Trapnell (2014)

5. K-means

6. K-means clustering
Extremely simple clustering algorithm, but can be quite effective
One of two clustering algorithms we will discuss
You must define the number of clusters K you want
Image source: Wikipedia

7. K-means clustering: step 1
We’re going to create three clusters
So, we randomly place three centroids amongst our data
Image source: Wikipedia

8. K-means clustering: step 2
Assign every data point to its closest centroid
Image source: Wikipedia

9. K-means clustering: step 3
Move each centroid to the centre of all the data points belonging to its cluster
Now go back to step 2 and iterate
Image source: Wikipedia

10. K-means: step 4 When no data points change assignments, you’re done!
Note that, depending on where you place your centroids at the start, your results may differ
Image source: Wikipedia

11. Gaussian mixture models

12. Gaussian mixture model clustering
We will fit a mixture of Gaussians using expectation maximization
Each Gaussian has parameters describing mean and variance

13. GMM step 1
Initialize with a Gaussian for each cluster, using random means and variances

14. GMM step 2 Calculate expectation of cluster membership for each point
Not captured by figure: these are soft assignments

15. GMM step 3
Choose parameter values that maximize likelihood of observed assignment of points to clusters

16. GMM step 4
Once you converge, you’re done!

17. Let’s cluster simulated data using a GMM!
Once more, to the notebook!

18. Evaluating clustering success

19. Evaluating clustering success
How do we evaluate clustering?
For supervised learning, we can examine accuracy, precision-recall curve, etc.
Two types of evaluation: extrinsic and intrinsic
Extrinsic measure: compare your clusters relative to ground-truth classes
This is similar to supervised learning, in which you know the “correct” answer for some of your data
For our RNA-seq data, we know what timepoint each cell came from
But if gene expression isn’t consistent between cells in same timepoint, the data won’t cluster well -- this is a problem with the data, not the clustering algorithm
Intrinsic measure: examine structure of clusters without reference to external ground truth

20. Extrinsic metric: V-measure
V-measure: average of homogeneity and completeness, both of which are desireable
Homogeneity: for a given cluster, do all the points in it come from the same class?
Completeness: for a given class, are all its points placed in one cluster?
Achieving good V-measure scores:
Perfect homogeneity, perfect completeness: your clustering matches your classes perfectly
Perfect homogeneity, horrible completeness: every single point is placed in its own cluster
Perfect completeness, horrible homogeneity: all your points are placed in just one cluster

21. Calculating homogeneity
Homogeneity and completeness are defined in terms of entropies, which is a numeric measure of uncertainty
Both values occur on the [0, 1] interval
If I tell you what points went in a given cluster -- e.g., “for cluster 1, cells 19, 143, and 240 are in it” -- and you know with certainty the class of all points in that cluster -- “oh, that’s the T = 24 h timepoint”, then the cluster is homogeneous

22. Calculating completeness
If I tell you what points are in a given class -- “the T = 48 h timepoint has cells 131, 179, and 221” -- and you know with certainty what cluster they belong to -- “oh, those cells are all in the second cluster” -- then that class is complete with respect to the clustering

23. Now that we have homogeneity and completeness ...
V-measure is just the (harmonic) mean of homogeneity and completeness
Why the harmonic mean rather than the arithmetic mean?
If h = 1 and c = 0: then arithmetic mean is 0.5
This is the degenerate case where each point goes to its own cluster
But under same values, harmonic mean is 0, which better represents the quality of the clustering

24. Intrinsic measure: silhouette score

25. Example low silhouette score

26. Let’s see how well our simulated data is clustered!
Notebook time! Hooray!
Why does k-means do better than GMM? Our data were generated via Gaussians
Exercise: generate more complex simulated data, evaluate performance

27. The curse of dimensionality

28. What is the curse of dimensionality?
You have a straight line 100 metres long. Drop a penny on it. Easy to find!
You have a square 100 m * 100 m. Drop a penny inside it. Harder to find
Like two football fields put next to each other
You have a building 100 m * 100 m * 100 m. Drop a penny in it
Now you’re searching inside a 30-storey building the size of a football field
Your life sucks
The point: intuition of what works in two or three dimensions breaks down as we move to much higher-dimensional spaces
Gene data: 575 differentially expressed genes -- we’re working in 575 dimensions!
With so many dimensions, everything is “far” from everything else -- clustering based on distance breaks down