Download presentation

Presentation is loading. Please wait.

Published byBrian Mooney Modified over 3 years ago

1
Data Analysis Lecture 8 Tijl De Bie

2
Dimensionality reduction How to deal with high-dimensional data? How to visualize it? How to explore it? Dimensionality reduction is one way…

3
Projections in vector spaces wx Meaning: –||w||*||x||*cos(theta) –For unit norm w: projection of x on w –To express hyperplanes: wx=b –To express halfspaces: wx>b All these interpretations are relevant

4
Projections in vector spaces [Some drawings…]

5
Variance of a projection wx=xw is the projection of x on w Let X contain many points x as its rows Projection of all points in X is: –Xw = (x 1 w, x 2 w, …, x n w) Variance of projection on w: –sum i (x i w/||w||) 2 = (wXXw)/(ww) –Or, if ||w||=1, this is: sum i (x i w) 2 = wXXw

6
Principal Component Analysis Direction / unit vector w with largest variance? –max w wXXw subject to ww=1 Lagrangian: –L(w) = wXXw-lambda(ww-1) Gradient w.r.t. w equal to zero: –2*XXw=2*lambda*w –(XX)*w=lambda*w Eigenvalue problem!

7
Principal Component Analysis Find w as dominant eigenvector of XX! Then we can project the data on this w For no other projection the variance is larger This projection is the best 1-D representation of the data

8
Principal Component Analysis Best 1-D representation given by projection on dominant eigenvector Second best w: the second eigenvector and so on…

9
Technical but important… I havent mentioned: –The data should be centred –That is: the mean of each of the features should be 0 –If that is not the case: subtract from each feature its mean (centering)

10
Clustering Another way to make sense of high- dimensional data Find coherent groups in the data Points that are: –close to one another within a cluster, but –distant from points in other clusters

11
Distances between points Distance between points: ||x i -x j || Can we assign points to K different clusters –each of which is coherent –distant from each other? Define the clusters by means of cluster centres m k with k=1,2,…,K

12
K-means cost function Ideal clustering: –||x i -m k(i) || small for all x i if m k(i) is its cluster centre –sum i ||x i -m k(i) || 2 small Unfortunately: hard to minimise… Simultaneous optimisation of: –k(i) (which cluster centre for which point) –m k (where are the cluster centres) Iterative strategy!

13
K-means clustering Iteratively optimise centres and cluster assignments K-means algorithm: –Start with random choices of K centres m k –Set k(i)=argmin k ||x i -m k || 2 –Set m k =mean({x i : k(i)=k}) Do this for many different random starts, and pick the best result (with lowest cost)

Similar presentations

Presentation is loading. Please wait....

OK

Algorithms 2005 Ramesh Hariharan. Algebraic Methods.

Algorithms 2005 Ramesh Hariharan. Algebraic Methods.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Convert doc file to ppt online training Download free ppt on active and passive voice Ppt on entrepreneurship and small business management Ppt on idea cellular ltd Ppt on product specification document Ppt on flora and fauna of kerala Ppt on leverages resources Ppt on world environment day pictures Full ppt on electron beam machining products Ppt on animal kingdom classification class 9