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Supervised and u nsupervised learning in high-throughput data

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## Supervised and u nsupervised learning in high-throughput data

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Supervised and unsupervised learning in high-throughput data

- General considerations
- Clustering
- Dimension reduction
- Feature selection and classification

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.

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 location parameter that minimize the expected error at each x:

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 … …

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 to evaluate the outcome;

Heuristic arguments are used to motivate the methods; Reasonable explanation of the outcome is expected from the subjective field of study.

Find co-regulated sets, infer hidden regulation signals, infer regulatory networks, …….

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:

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

Curse of Dimensionality

Just a reminder, the expected prediction error contains variance and bias components.

Under model:

Y=f(X)+ε

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 ……

The simplest workflow in biomarker study

Obtain high-throughput data

Unsupervised learning (dimension reduction/clustering) to show that sample from different treatment are indeed separated, and identify any interesting pattern.

Feature selection based on testing – find features that are differentially expressed between treatment. FDR is used here.

Experimental validation of the selected features, using more reliable biological techniques. (e.g. real-time PCR is used to validate microarray expression data.)

Classification model building.

From an independent group of samples, measure the feature levels using reliable technique.

Find the sensitivity/specificity of the model using the independent data.

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.

Hierarchical clustering

Start: every data point being one cluster;

Joint nearest clusters at each step.

Hierarchical clustering

Hierarchical clustering

Figure 14.12: Dendrogram from agglomerative hierarchical clustering with average linkage to the human tumor microarray data.

K-means

Assign cluster membership

Find cluster mean

Assign cluster membership based on distance to the means

N

Convergence? Y

Report clusters

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.

PCA sequentially seeks the subspace that explain the most variation in the data.

From the covariance matrix:

Find the eigen values & vectors:

By solving the characteristic equation:

Figure 14.20: The first linear principal component of a set of data. The line minimizes the total squared distance from each point to its orthogonal projection onto the line.

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.

Fisher Linear Discriminant Analysis

Find the lower-dimension space where the classes are most separated.

distance

Within-class scatter

Classification

- In the projection, two goals are to be fulfilled:
- Maximize between-class distance
- Minimize within-class scatter
- Maximize this function with all non-zero vectors w

Fisher Linear Discriminant Analysis

In the two-class case, we are projecting to a line to find the best separation:

Maximization yields:

mean1

mean2

Decision boundry:

Decision boundry

Maximum Likelihood discriminant rules

Assuming the forms of the class conditional densities are

Known.

If

The rule is

Same covariance matrix:

Diagonal cov

matrix

Same diagonal

cov matrix

across classes

An example classification tree.

Classification Trees

Every split (mostly binary)should increase node purity.

Drop of impurity as a criteria for variable selection at each split.

Tree should not be overly complex. May prune tree.

Choice or features.

Random forests

Grow an ensemble of classification trees, each based on a bootstrap sample from the original training data.

In each tree, the splitter at each node is determined partially at random.

Prediction is made by taking the votes from the ensemble of trees.

The random forest is a strong classifier; it can help estimate the importance of variables; it helps detect variable interactions.

Boosting

Sequentially apply a weak classification algorithm to reweighted data.

The final strong classifier is made from a weighted voting from the weak classifiers.

“A Tutorial on Boosting”,Yoav Freund and Rob Schapire

“A Tutorial on Boosting”,Yoav Freund and Rob Schapire

“A Tutorial on Boosting”,Yoav Freund and Rob Schapire

“A Tutorial on Boosting”,Yoav Freund and Rob Schapire

Support Vector Machine

Acknowledge two classes may be inseparable using linear boundary.

Maximize C,

Allow Slack Variables

Support Vector Machine

In computing SVM, its solution is special that it only involves the inner product of the input features, and hence for transformed features.

Thus the transform need not be explicitly specified. Only the kernel function is needed:

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