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Dimension reduction : PCA and Clustering

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Presentation on theme: "Dimension reduction : PCA and Clustering"— Presentation transcript:

1 Dimension reduction : PCA and Clustering
Slides by Agnieszka Juncker and Chris Workman

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

3 Motivation: Multidimensional data
Pat1 Pat2 Pat3 Pat4 Pat5 Pat6 Pat7 Pat8 Pat9 209619_at 32541_at 206398_s_at 219281_at 207857_at 211338_at 213539_at 221497_x_at 213958_at 210835_s_at 209199_s_at 217979_at 201015_s_at 203332_s_at 204670_x_at 208788_at 210784_x_at 204319_s_at 205049_s_at 202114_at 213792_s_at 203932_at 203963_at 203978_at 203753_at 204891_s_at 209365_s_at 209604_s_at 211005_at 219686_at 38521_at 217853_at 217028_at 201137_s_at 202284_s_at 201999_s_at 221737_at 205456_at 201540_at 219371_s_at 205297_s_at 208650_s_at 210031_at 203675_at 205255_x_at 202598_at 201022_s_at 218205_s_at 207820_at 202207_at

4 Dimension reduction methods
Principal component analysis (PCA) Singular value decomposition (SVD) Multidimensional scaling Correspondence analysis Cluster analysis Can be thought of as a dimensionality reduction method as clusters summarize data

5 Principal Component Analysis (PCA)
Used for visualization of high-dimensional data Projects high-dimensional data into a small number of dimensions Typically 2-3 principle component dimensions Often captures much of the total data variation in a only few dimensions Exact solutions require a fully determined system (matrix with full rank) i.e. A “square” matrix with independent entries

6 PCA

7 Singular Value Decomposition

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

9 PCA: Variance by dimension

10 PCA dimensions by experiment

11 PCA projections (as XY-plot)

12 PCA: Leukemia data, precursor B and T
Plot of 34 patients, dimension of 8973 genes reduced to 2

13 PCA of genes (Leukemia data)
Plot of 8973 genes, dimension of 34 patients reduced to 2

14 Why do we cluster? Organize observed data into meaningful structures
Summarize large data sets Used when we have no a priori hypotheses Optimization: Minimize within cluster distances Maximize between cluster distances

15 Many types of clustering methods
K-class Hierarchical, e.g. UPGMA Agglomerative (bottom-up) Divisive (top-down) Graph theoretic Information used: Supervised vs unsupervised Final description of the items: Partitioning vs non-partitioning fuzzy, multi-class

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

17 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

18 Hierarchical clustering

19 Hierarchical clustering

20 Hierarchical clustering
Data with clustering order and distances Dendrogram representation

21 Leukemia data - clustering of patients

22 Leukemia data - clustering of patients on top 100 significant genes

23 Leukemia data - clustering of genes

24 K-means clustering Partition data into K clusters
Parameter: Number of clusters (K) must be chosen Randomized initialization: Different clusters each time Non-deterministic

25 K-means - Algorithm

26 K-mean clustering, K=3

27 K-mean clustering, K=3

28 K-mean clustering, K=3

29 K-means clustering of Leukemia data

30 K-means clustering of Cell Cycle data

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

32 SOM - example

33 SOM - example

34 SOM - example

35 SOM - example

36 SOM - example

37 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

38 Distance measures Euclidian distance Vector angle distance
Pearsons distance

39 Comparison of distance measures

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

41 Next: Exercises in PCA and clustering
Coffee break Next: Exercises in PCA and clustering

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