1. Principal Component Analysis

2. Philosophy of PCA
Introduced by Pearson (1901) and Hotelling (1933) to describe the variation in a set of multivariate data in terms of a set of uncorrelated variables
We typically have a data matrix of n observations on p correlated variables x1,x2,…xp
PCA looks for a transformation of the xi into p new variables yi that are uncorrelated

3. The data matrix case ht (x1) wt(x2) age(x3) sbp(x4) heart rate (x5) 1
175
1225 25
117 56 2
156
1050 31
122 63 n
202
1350 58
154 67

4. Reduce dimension
The simplet way is to keep one variable and discard all others: not reasonable!
Wheigt all variable equally: not reasonable (unless they have same variance)
Wheigted average based on some citerion.
Which criterion?

5. Let us write it first
Looking for a transformation of the data matrix X (nxp) such that
Y= T X=1 X1+ 2 X2+..+ p Xp
Where =(1 , 2 ,.., p)T is a column vector of wheights with
1²+ 2²+..+ p² =1

6. One good criterion
Maximize the variance of the projection of the observations on the Y variables
Find  so that
Var(T X)= T Var(X)  is maximal
The matrix C=Var(X) is the covariance matrix of the Xi variables

7. Let us see it on a figure
Good
Better

8. Covariance matrix C=

10. And so.. We find that
The direction of  is given by the eigenvector 1 correponding to the largest eigenvalue of matrix C
The second vector that is orthogonal (uncorrelated) to the first is the one that has the second highest variance which comes to be the eignevector corresponding to the second eigenvalue
And so on …

11. So PCA gives
New variables Yi that are linear combination of the original variables (xi):
Yi= ai1x1+ai2x2+…aipxp ; i=1..p
The new variables Yi are derived in decreasing order of importance;
they are called ‘principal components’

12. Calculating eignevalues and eigenvectors
The eigenvalues i are found by solving the equation
det(C-I)=0
Eigenvectors are columns of the matrix A such that
C=A D AT
Where D=

13. An example Let us take two variables with covariance c>0 C= C-I=
det(C-I)=(1- )²-c²
Solving this we find 1 =1+c
2 =1-c < 1

14. and eigenvectors Any eigenvector A satisfies the condition CA=A
Solving we find A=
CA= = = A1 A2

15. PCA is sensitive to scale
If you multiply one variable by a scalar you get different results
(can you show it?)
This is because it uses covariance matrix (and not correlation)
PCA should be applied on data that have approximately the same scale in each variable

16. Interpretation of PCA
The new variables (PCs) have a variance equal to their corresponding eigenvalue
Var(Yi)= i for all i=1…p
Small i  small variance  data change little in the direction of component Yi
The relative variance explained by each PC is given by li / li

17. How many components to keep?
Enough PCs to have a cumulative variance explained by the PCs that is >50-70%
Kaiser criterion: keep PCs with eigenvalues >1
Scree plot: represents the ability of PCs to explain de variation in data

19. Do it graphically

20. Interpretation of components
See the wheights of variables in each component
If Y1= 0.89 X X2-0.77X3+0.51X4
Then X1 and X3 have the highest wheights and so are the mots important variable in the first PC
See the correlation between variables Xi and PCs: circle of correlation

21. Circle of correlation

22. Normalized (standardized) PCA
If variables have very heterogenous variances we standardize them
The standardized variables Xi*
Xi*= (Xi-mean)/variance
The new variables all have the same variance (1), so each variable have the same wheight.

23. Application of PCA in Genomics
PCA is useful for finding new, more informative, uncorrelated features; it reduces dimensionality by rejecting low variance features
Analysis of expression data
Analysis of metabolomics data (Ward et al., 2003)

24. However
PCA is only powerful if the biological question is related to the highest variance in the dataset
If not other techniques are more useful : Independent Component Analysis
Introduced by Jutten in 1987

25. What is ICA?

26. That looks like that

27. The idea behind ICA

28. How it works?

29. Rationale of ICA
Find the components Si that are as independent as possible in the sens of maximizing some function F(s1,s2,.,sk) that measures indepedence
All ICs (except 1) should be non-Normal
The variance of all ICs is 1
There is no hierarchy between ICs

30. How to find ICs ? Many choices of objective function F
Mutual information
We use the kurtosis of the variables to approximate the distribution function
The number of ICs is chosen by the user

31. Difference with PCA It is not a dimensionality reduction technique
There is no single (exact) solution for components; uses different algorithms (in R: FastICA, PearsonICA, MLICA)
ICs are of course uncorrelated but also as independent as possible
Uninteresting for Normally distributed variables

32. Example: Lee and Batzoglou (2003)
Microarray expression data on 7070 genes in 59 Normal human tissue samples (19 types)
We are not interested in reducing dimension but rather in looking for genes that show tissue specific expression profile (what make tissue types differents)

33. PCA vs ICA
Hsiao et al (2002) applied PCA and by visual inspection observed three gene cluster of 425 genes: liver-specific, brain-specific and muscle-specific
ICA identified more tissue-specific genes than PCA