1. Principal Components

2. Karl Pearson

4. Principal Components (PC) Objective: Given a data matrix of dimensions nxp (p variables and n elements) try to represent these data by using r variables (r<p) with minimum lost of information

5. We want to find a new set of p variables, Z, which are linear combinations of the original X variable such that : r of them contains all the information The remaining p-r variables are noise

6. First interpretation of principal components Optimal Data Representation

7. xixi a zizi riri Proyection of a point in direction a: minimize the squared distance Implies maximizing the variance (assuming zero mean variables) x i T x i = r i T r i + z T i z i

11. Optimal Prediction Find a new variable z i =a’X i which is optimal to predict The value of X i in each element. In general, find r variables, z i =A r X i, which are optimal to forecast All X i with the least squared error criterion It is easy to see that the solution is that z i =a’X i must have maximum variance Second interpretation of PC:

12. The line which minimizes the orthogonal distance provides the axes of the ellipsoid Third interpretation of PC Find the optimal direction to represent the data. Axe of the ellipsoid which contains the data This is idea of Pearson orthogonal regression

16. Second component

21. Properties of PC

27. Standardized PC

29. Example Inves

32. Example Medifis

35. Example mundodes

36. Example Mundodes

37. Example for image analysis

39. The analysis have been done with 16 images. PC allows that Instead of sending 16 matrices of N2 pixels we send a vector 16x3 with the values of the components and a matrix 3xN2 with the values of the new variables. We save If instead of 16 images we have 100 images we save 95%