Transcript Lecture 8 Unsupervised Learning

Unsupervised learning
Supervised and Unsupervised learning
General considerations
Clustering
Dimension reduction
The lecture is partly based on:
Hastie, Tibshirani & Friedman. The Elements of Statistical Learning. 2009. Chapter 2.
Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wiley & Sons,
2000. Chapter 2.
Dudoit, S., Fridlyand, J., & Speed, T. (2000). Comparison of discrimination methods for the
classification of tumors using gene expression data. JASA 2000.
General considerations
This is the common structure of microarray gene expression data
from a simple cross-sectional case-control design.
Data from other high-throughput technology are often similar.
Control 1 Control 2
Gene 1
Gene 2
Gene 3
Gene 4
Gene 5
Gene 6
Gene 7
Gene 8
Gene 9
Gene 10
Gene 11
……
Gene 50000
9.25
6.99
4.55
7.04
2.84
6.08
4
4.01
6.37
2.91
3.71
……
3.65
9.77
5.85
5.3
7.16
3.21
6.26
4.41
4.15
7.2
3.04
3.79
……
3.73
……
……
……
……
……
……
……
……
……
……
……
……
……
……
Control 25 Disease 1 Disease 2
9.4
5
4.73
6.47
3.2
7.19
4.22
3.45
8.14
3.03
3.39
……
3.8
8.58
5.14
3.66
6.79
3.06
6.12
4.42
3.77
5.13
2.83
5.15
……
3.87
5.62
5.43
4.27
6.87
3.26
5.93
4.09
3.55
7.06
3.86
6.23
……
3.76
……
Disease 40
……
……
……
……
……
……
……
……
……
……
……
……
……
6.88
5.01
4.11
6.45
3.15
6.44
4.26
3.82
7.27
2.89
4.44
……
3.62
General considerations
Supervised learning
In supervised learning, the problem is well-defined:
Given a set of observations {xi, yi},
estimate the density Pr(Y|X)
Usually the goal is to find the parameter that
minimize the expected classification error, or some loss
derived from it.
Objective criteria exists to measure the success of a
supervised learning mechanism:
Error rate from testing (or cross-validation) data
Disease classification, predict survival, predict cost … …
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General considerations
Unsupervised learning
There is no output variable, all we observe is a set {xi}.
The goal is to infer Pr(X) and/or some of its properties.
When the dimension is low, nonparametric density estimation is
possible; When the dimension is high, may need to find simple
properties without density estimation, or apply strong assumptions
to estimate the density.
There is no objective criteria from the data itself; to justify a result:
> Heuristic arguments,
> External information,
> Reasonable explanation of the outcome
Find co-regulated sets, infer hidden regulation signals, infer
regulatory networks, …….
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General considerations
Correlation structure
There is always correlations between features (genes, proteins,
metabolites …) in biological data. This is caused by the intrinsic
biological interactions and regulations.
The problem is:
(1) We don’t know what the correlation structure is
(in some cases we have some idea, e.g. DNA)
(2) We cannot reliably estimate it because the dimension is too
high and there is not enough data
General considerations
Curse of Dimensionality
Bellman R.E.,
1961.
In p-dimensions, to get a hypercube with volume r, the edge length
needed is r1/p.
In 10 dimensions, to capture 1% of the data to get a local average,
we need 63% of the range of each input variable.
General considerations
Curse of Dimensionality
In other words,
To get a “dense” sample, if we need N=100 samples in 1
dimension, then we need N=10010 samples in 10 dimensions.
In high-dimension, the data is always sparse and do not support
density estimation.
More data points are closer to the boundary, rather than to any
other data point  prediction is much harder near the edge of the
training sample.
General considerations
General considerations
Curse of Dimensionality
We have talked about the curse of dimensionality in the
sense of density estimation.
In a classification problem, we do not necessarily need
density estimation.
Generative model --- care about class density function.
Discriminative model --- care about boundary.
Example: Classifying belt fish and carp. Looking at the
length/width ratio is enough. Why should we care other
variables such as shape of fins, or number of teeth?
General considerations
N<<p problem
We talk about “curse of dimensionality” when N is not >>>p.
In bioinformatics, usually N<100, and p>1000.
How to deal with this N<<p issue?
Dramatically reduce p before model-building.
Filter genes based on: variation, normal/disease test statistic,
projection……
Use methods that are resistant to large numbers of nuisance
variables: Support vector machines, random forests, boosting
……
Borrow other information: functional annotation, meta-analysis
……
A typical workflow
(1) Obtain high-throughput data
(2) Unsupervised learning (dimension reduction/clustering)
to show that sample from different treatment are indeed
separated, and identify any interesting pattern.
(3) Feature selection based on testing – find features that
are differentially expressed between treatment. FDR is
used here.
(4) Experimental validation of the selected features, using
more reliable biological techniques. (e.g. real-time PCR
is used to validate microarray expression data.)
(5) Classification model building.
(6) From an independent group of samples, measure the
feature levels using reliable technique.
(7) Find the sensitivity/specificity of the model using the
independent data.
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Clustering
Finding features/samples that are similar.
Can tolerate n<p.
Irrelevant features contribute random noise
that shouldn’t change strong clusters.
Some false clusters may be due to noise.
But their size should be limited.
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Hierarchical clustering
Agglomerative: build tree by joining nodes;
Divisive: build tree by dividing groups of objects.
Hierarchical clustering
Single linkage，d SL (G , H )  min d ii ;
iG
i H
Complete linkage，d CL (G , H )  max d ii ;
iG
i H
1
Average linkage，d GA (G , H ) 
NG N H
 d
iG i H
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ii 
.
Hierarchical clustering
Example data:
Hierarchical clustering
Single linkage: find the distance between any two nodes
by nearest neighbor distance.
Hierarchical clustering
Single linkage:
Hierarchical clustering
Complete linkage: find the distance between any
two nodes by farthest neighbor distance.
Average linkage: find the distance between
any two nodes by average distance.
Hierarchical clustering
Comments:
Hierarchical clustering generates a tree; to
find clusters, the tree needs to be cut at a
certain height;
Complete linkage method favors compact,
ball-shaped clusters; single linkage method
favors chain-shaped clusters; average
linkage is somewhere in between.
Hierarchical clustering
Average linkage on
microarray data.
Row: genes
Column: samples
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Hierarchical clustering
Figure 14.12: Dendrogram from agglomerative hierarchical clustering with average linkage to
the human tumor microarray data.
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Center-based clustering
Have objective functions which define how good a
solution is;
 The goal is to minimize the objective function;
Efficient for large/high dimensional datasets;
The clusters are assumed to be convex shaped;
 The cluster center is representative of the cluster;
Some model-based clustering, e.g. Gaussian
mixtures, are center-based clustering.
Center-based clustering
K-means clustering. Let
disjoint clusters.
be k
Error is defined as the sum of the distance
from the cluster center
Center-based clustering
The k-means algorithm:
Center-based clustering
Understanding k-means as an optimization
procedure:
The objective function is:
Minimize the P(W,Q) subject to:
Center-based clustering
The solution is iteratively solving sub-problems:
When
only if:
is fixed,
When
is fixed,
and only if
is minimized if and
is minimized if
Center-based clustering
In terms of optimization, the k-means procedure is
greedy.
 Every iteration decreases the value of the
objective function; The algorithm converges to a
local minimum after a finite number of iterations.
Results depend on initiation values.
The computational complexity is proportional to the
size of the dataset  efficient on large data.
The clusters identified are mostly ball-shaped.
Works only on numerical data.
Center-based clustering
K-means
How to decide the number of clusters?
If there are truly k* groups, for k<k*, some groups are
merged, and within-cluster dissimilarity should be big; it
should drop substantially when increasing k; when k>k*,
some true groups are partitioned. Increasing k should not
bring much improvement on within-cluster dissimilarity.
Figure 14.8: Total within cluster sum of squares for K-means clustering applied to the
human tumor microarray data.
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Center-based clustering
Automated selection of k?
The x-means algorithm based on AIC/BIC.
A family of models at different k:
Is the likelihood of the data given the
jth model. pj is the number of parameters.
We have to assume a model to get the
likelihood. The convenient one is Gaussian.
Center-based clustering
Under the assumption of identical spherical
Gaussian assumption, (n is sample size; k is
number of centroids)
μ(i) is the centroid associated with xi.
The likelihood is:
The number of parameters is (d is dimension):
(class probabilities + parameters for mean & variance)
Dimension Reduction
The purpose of dimension reduction:
Data simplification
Data visualization
Reduce noise (if we can assume only the
dominating dimensions are signals)
Variable selection for prediction (in supervised
learning)
PCA
Explain the variance-covariance structure
among a set of random variables by a few
linear combinations of the variables;
Does not require normality!
PCA
First PC :
a' X that maximizes Var(a' X),
subject to a'a = 1
ith PC :
a 'i X that maximizes Var(a 'i X),
subject to a 'ia i = 1 and Cov(a 'i X, a 'k X) = 0, " k < i
PCA
PCA
The eigen values are the variance components:
Proportion of total variance explained by the kth PC:
PCA
PCA
The geometrical interpretation of PCA:
PCA
PCA using the correlation matrix, instead of the
covariance matrix?
This is equivalent to first standardizing all X
vectors.
PCA
Using the correlation matrix avoids the domination
from one X variable due to scaling (unit changes),
for example using inch instead of foot. Example:
é1 4 ù
é 1 0.4ù
S=ê
ú, r = ê
ú
4
100
0.4
1
ë
û
ë
û
PCA from S :
l1 = 100.16, e1' = [0.040 0.999]
l2 = 0.84, e'2 = [0.999 -0.040]
PCA from r :
l1 = 1+ r = 1.4, e1' = [0.707 0.707]
l2 = 1- r = 0.6, e'2 = [0.707 -0.707]
PCA
Selecting the number of
components?
Based on eigen values (% variation
explained). Assumption: the small
amount of variation explained by
low-rank PCs is noise.
PCA
Figure 14.21: The best rank-two linear
approximation to the half-sphere data. The
right panel shows the projected points with
coordinates given by U2D2, the first two
principal components of the data.
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Factor Analysis
If we take the first several PCs that explain most of
the variation in the data, we have one form of
factor model.
L: loading matrix
F: unobserved random vector (latent variables).
ε: unobserved random vector (noise)
Factor Analysis
Rotations in the m-dimensional subspace defined by
the factors make the solution non-unique:
PCA is one unique solution, as the vectors are
sequentially selected. Maximum likelihood estimator
is another solution:
Factor Analysis
As we said, rotations within the m-dimensional subspace
doesn’t change the overall amount of variation explained. Do
rotation to make the results more interpretable:
Factor Analysis
Orthogonal simple factor
rotation:
Rotate the orthogonal
factors around the origin
until the system is
maximally aligned with the
separate clusters of
variables.
Oblique Simple Structure
Rotation:
Allow the factors to
become correlated. Each
factor is rotated individually
to fit a cluster.
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