computational biology clustering parts taken from introduction to data mining by tan, steinbach,...
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Computational Biology
Clustering
Parts taken from
Introduction to Data Mining
by
Tan, Steinbach, Kumar
Lecture Slides Week 9
Clustering
• A clustering is a set of clusters
• Important distinction between hierarchical and partitional sets of clusters
• Partitional Clustering– A division data objects into non-overlapping subsets (clusters) such
that each data object is in exactly one subset
• Hierarchical clustering– A set of nested clusters organized as a hierarchical tree
Partitional Clustering
Original Points A Partitional Clustering
Hierarchical Clustering
p4p1
p3
p2
p4 p1
p3
p2
p4p1 p2 p3
p4p1 p2 p3
Traditional Hierarchical Clustering
Non-traditional Hierarchical Clustering Non-traditional Dendrogram
Traditional Dendrogram
Notion of a Cluster can be Ambiguous
How many clusters?
Four Clusters Two Clusters
Six Clusters
What is Cluster Analysis?
• Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups
Inter-cluster distances are maximized
Intra-cluster distances are
minimized
10 Clusters Example
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Starting with two initial centroids in one cluster of each pair of clusters
10 Clusters Example
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Iteration 1
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Iteration 2
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Iteration 3
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Iteration 4
Starting with two initial centroids in one cluster of each pair of clusters
10 Clusters Example
Starting with some pairs of clusters having three initial centroids, while other have only one.
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Iteration 1
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Iteration 2
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Iteration 3
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Iteration 4
10 Clusters Example
Starting with some pairs of clusters having three initial centroids, while other have only one.
0 5 10 15 20
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Iteration 2
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Iteration 3
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Iteration 4
Limitations of K-means: Differing Sizes
Original Points K-means (3 Clusters)
Limitations of K-means: Differing Density
Original Points K-means (3 Clusters)
Limitations of K-means: Non-globular Shapes
Original Points K-means (2 Clusters)
Hierarchical Clustering
• Produces a set of nested clusters organized as a hierarchical tree
• Can be visualized as a dendrogram– A tree like diagram that records the sequences of merges or
splits
1 3 2 5 4 60
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Starting Situation
• Start with clusters of individual points and a proximity matrix
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
. Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
Intermediate Situation
• After some merging steps, we have some clusters
C1
C4
C2 C5
C3
C2C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
Intermediate Situation
• We want to merge the two closest clusters (C2 and C5) and update the proximity matrix.
C1
C4
C2 C5
C3
C2C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
After Merging
• The question is “How do we update the proximity matrix?”
C1
C4
C2 U C5
C3? ? ? ?
?
?
?
C2 U C5C1
C1
C3
C4
C2 U C5
C3 C4
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
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Similarity?
• MIN• MAX• Group Average• Distance Between Centroids• Other methods driven by an objective function
– Ward’s Method uses squared error
Proximity Matrix
How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
• MIN• MAX• Group Average• Distance Between Centroids• Other methods driven by an objective function
– Ward’s Method uses squared error
How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
• MIN• MAX• Group Average• Distance Between Centroids• Other methods driven by an objective function
– Ward’s Method uses squared error
How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
• MIN• MAX• Group Average• Distance Between Centroids• Other methods driven by an objective function
– Ward’s Method uses squared error
How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
• MIN• MAX• Group Average• Distance Between Centroids• Other methods driven by an objective function
– Ward’s Method uses squared error
Cluster Similarity: MIN or Single Link
• Similarity of two clusters is based on the two most similar (closest) points in the different clusters
– Determined by one pair of points, i.e., by one link in the proximity graph.
I1 I2 I3 I4 I5I1 1.00 0.90 0.10 0.65 0.20I2 0.90 1.00 0.70 0.60 0.50I3 0.10 0.70 1.00 0.40 0.30I4 0.65 0.60 0.40 1.00 0.80I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5
Hierarchical Clustering: MIN
Nested Clusters Dendrogram
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• Two matrices – Proximity Matrix– “Incidence” Matrix
• One row and one column for each data point
• An entry is 1 if the associated pair of points belong to the same cluster
• An entry is 0 if the associated pair of points belongs to different clusters
• Compute the correlation between the two matrices– Since the matrices are symmetric, only the correlation between
n(n-1) / 2 entries needs to be calculated.
• High correlation indicates that points that belong to the same cluster are close to each other.
• Not a good measure for some density or contiguity based clusters.
Measuring Cluster Validity Via Correlation
Measuring Cluster Validity Via Correlation
• Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets.
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Corr = -0.9235 Corr = -0.5810
• Order the similarity matrix with respect to cluster labels and inspect visually.
Using Similarity Matrix for Cluster Validation
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Points
Po
ints
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End Theory I
• 5 min mindmapping• 10 min break
Practice I
Clustering Dataset
• We will use the same datasets as last week
• Have fun clustering with Orange– Try K-means clustering, hierachial clustering, MDS– Analyze differences in results
K?
• What is the k in k-means again?
• How many clusters are in my dataset?
• Solution: iterate over a reasonable number of ks
• Do that and try to find out how many clusters there are in your data
End Practice I
• 15 min break
Theory II
Microarrays
• Gene Expression:– We see difference between cels because of differential
gene expression,– Gene is expressed by transcribing DNA intosingle-stranded
mRNA,– mRNA is later translated into a protein,– Microarrays measure the level of mRNA expression
Microarrays
• Gene Expression:– mRNA expression represents dynamic aspects of cell,– mRNA is isolated and labeled using a fluorescent material,– mRNA is hybridized to the target; level of hybridization
corresponds to light emission which is measured with a laser
Microarrays
Microarrays
Microarrays
Processing Microarray Data
• Differentiating gene expression:
– R = G not differentiated
– R > G up-regulated
– R < G down regulated
Processing Microarray Data
• Problems:– Extract data from microarrays,– Analyze the meaning of the multiple arrays.
Processing Microarray Data
Processing Microarray Data
• Problems:– Extract data from microarrays,– Analyze the meaning of the multiple arrays.
Processing Microarray Data
• Microarray data:
Processing Microarray Data
• Clustering:– Find classes in the data,– Identify new classes,– Identify gene correlations,– Methods:
• K-means clustering,
• Hierarchical clustering,
• Self Organizing Maps (SOM)
Processing Microarray Data
• Distance Measures:– Euclidean Distance:
– Manhattan Distance:
Processing Microarray Data
• K-means Clustering:– Break the data into K clusters,– Start with random partitioning,– Improve it by iterating.
Processing Microarray Data
• Agglomerative Hierarchical Clustering:
Processing Microarray Data
• Self-Organizing Feature Maps:– by Teuvo Kohonen, – a data visualization technique which helps to understand high
dimensional data by reducing the dimensions of data to a map.
Processing Microarray Data
• Self-Organizing Feature Maps:– humans simply cannot visualize high dimensional data as is,– SOM help us understand this high dimensional data.
Processing Microarray Data
• Self-Organizing Feature Maps:– Based on competitive learning,– SOM helps us by producing a map of usually 1 or 2 dimensions,– SOM plot the similarities of the data by grouping– similar data items together.
Processing Microarray Data
• Self-Organizing Feature Maps:
Processing Microarray Data
• Self-Organizing Feature Maps:
Input vector, synaptic weight vectorx = [x1, x2, …, xm]T
wj=[wj1, wj2, …, wjm]T, j = 1, 2,3, l
Best matching, winning neuroni(x) = arg min ||x-wj||, j =1,2,3,..,l
Weights wi are updated.
Figure 2. Output map containing the distributions of genes from the alpha30 database.
Chavez-Alvarez R, Chavoya A, Mendez-Vazquez A (2014) Discovery of Possible Gene Relationships through the Application of Self-Organizing Maps to DNA Microarray Databases. PLoS ONE 9(4): e93233. doi:10.1371/journal.pone.0093233http://www.plosone.org/article/info:doi/10.1371/journal.pone.0093233
Figure 5. Color-coded output maps representing the final weight of neurons from the samples of the alpha30 database.
Chavez-Alvarez R, Chavoya A, Mendez-Vazquez A (2014) Discovery of Possible Gene Relationships through the Application of Self-Organizing Maps to DNA Microarray Databases. PLoS ONE 9(4): e93233. doi:10.1371/journal.pone.0093233http://www.plosone.org/article/info:doi/10.1371/journal.pone.0093233
End Theory II
• 5 min mindmapping• 10 min break
Practice II
Microarray
• http://www.biolab.si/supp/bi-visprog/dicty/dictyExample.htm
• Use the data as described on the page