Dimension reduction : PCA and Clustering Agnieszka S. Juncker Slides: Christopher Workman and Agnieszka S. Juncker Center for Biological Sequence Analysis DTU

Sample Preparation Hybridization Sample Preparation Hybridization Array design Probe design Array design Probe design Question Experimental Design Question Experimental Design Buy Chip/Array Statistical Analysis Fit to Model (time series) Statistical Analysis Fit to Model (time series) Expression Index Calculation Expression Index Calculation Advanced Data Analysis ClusteringPCA Classification Promoter Analysis Meta analysisSurvival analysisRegulatory Network Advanced Data Analysis ClusteringPCA Classification Promoter Analysis Meta analysisSurvival analysisRegulatory Network Normalization Image analysis The DNA Array Analysis Pipeline Comparable Gene Expression Data Comparable Gene Expression Data

Motivation: Multidimensional data Pat1Pat2Pat3Pat4Pat5Pat6Pat7Pat8Pat _at _at _s_at _at _at _at _at _x_at _at _s_at _s_at _at _s_at _s_at _x_at _at _x_at _s_at _s_at _at _s_at _at _at _at _at _s_at _s_at _s_at _at _at _at _at _at _s_at _s_at _s_at _at _at _at _s_at _s_at _s_at _at _at _x_at _at _s_at _s_at _at _at

PCA

Principal Component Analysis (PCA) Numerical method Dimensionality reduction technique Primarily for visualization of arrays/samples Performs a rotation of the data that maximizes the variance in the new axes Projects high dimensional data into a low dimensional sub-space (visualized in 2-3 dims) Often captures much of the total data variation in a few dimensions (< 5) Exact solutions require a fully determined system (matrix with full rank) –i.e. A “square” matrix with independent rows

Principal components 1 st Principal component (PC1) –Direction along which there is greatest variation 2 nd Principal component (PC2) –Direction with maximum variation left in data, orthogonal to PC1

Principal components General about principal components –summary variables –linear combinations of the original variables –uncorrelated with each other –capture as much of the original variance as possible

Principal components - Variance

Singular Value Decomposition

Requirements: –No missing values –“Centered” observations, i.e. normalize data such that each gene has mean = 0

PCA of ALPS patients vs. healthy controls

PCA of leukemia patients

PCA of treated cell lines 4 conditions, 3 batches

PCA projections (as XY-plot)

Eigenvectors (eigenarrays, rows)

PCA of cell cycle data - based on only 500 cell cycle regulated genes

PCA of cell cycle data

PCA of cell cycle data broken scanner (sample 12-16)

Clustering

Why do we cluster? Organize observed data into meaningful structures Summarize large data sets Used when we have no a priori hypotheses

Many types of clustering methods Method: –K-class –Hierarchical, e.g. UPGMA Agglomerative (bottom-up) Divisive (top-down) –Graph theoretic

Hierarchical clustering Representation of all pair-wise distances Parameters: none (distance measure) Results: –One large cluster –Hierarchical tree (dendrogram) Deterministic

Hierarchical clustering – UPGMA Algorithm Assign each item to its own cluster Join the nearest clusters Re-estimate the distance between clusters Repeat for 1 to n

Hierarchical clustering

Hierarchical Clustering Data with clustering order and distances Dendrogram representation 2D data is a special (simple) case!

Hierarchical clustering example: leukemia patients (based on all genes)

Hierarchical clustering example: leukemia data, significant genes

K-mean clustering Partition data into K clusters Parameter: Number of clusters (K) must be chosen Randomilized initialization: –different clusters each time

K-mean - Algorithm Assign each item a class in 1 to K (randomly) For each class 1 to K –Calculate the centroid (one of the K- means) –Calculate distance from centroid to each item Assign each item to the nearest centroid Repeat until no items are re-assigned (convergence)

K-mean clustering, K=3

Self Organizing Maps (SOM) Partitioning method (similar to the K-means method) Clusters are organized in a two-dimensional grid Size of grid must be specified –(eg. 2x2 or 3x3) SOM algorithm finds the optimal organization of data in the grid

SOM - example

K-means clustering Cell cycle data

K-means clustering example: Cluster profiles, treated cell lines

Comparison of clustering methods Hierarchical clustering –Distances between all variables –Time consuming with a large number of gene –Advantage to cluster on selected genes K-means clustering –Faster algorithm –Does only show relations between all variables SOM –Machine learning algorithm

Distance measures Euclidian distance Vector angle distance Pearsons distance

Comparison of distance measures

Summary Dimension reduction important to visualize data Methods: –Principal Component Analysis –Clustering Hierarchical K-means Self organizing maps (distance measure important)

Coffee break Next: Exercises in PCA and clustering