Book Image

Scientific Computing with Python 3

By : Claus Führer, Jan Erik Solem, Olivier Verdier
Book Image

Scientific Computing with Python 3

By: Claus Führer, Jan Erik Solem, Olivier Verdier

Overview of this book

Python can be used for more than just general-purpose programming. It is a free, open source language and environment that has tremendous potential for use within the domain of scientific computing. This book presents Python in tight connection with mathematical applications and demonstrates how to use various concepts in Python for computing purposes, including examples with the latest version of Python 3. Python is an effective tool to use when coupling scientific computing and mathematics and this book will teach you how to use it for linear algebra, arrays, plotting, iterating, functions, polynomials, and much more.
Table of Contents (23 chapters)
Scientific Computing with Python 3
Credits
About the Authors
About the Reviewer
www.PacktPub.com
Acknowledgement
Preface
References

Spectral clustering


An interesting application of eigenvectors is for clustering data. Using the eigenvectors of a matrix derived from a distance matrix, unlabelled data can be separated into groups. Spectral clustering methods get their name from the use of the spectrum of this matrix. A distance matrix for n elements (for example, the pairwise distance between data points) is an n × n symmetric matrix. Given such an n × n distance matrix M with distance values mij , we can create the Laplacian matrix of the data points as follows:

Here, I is the identity matrix and D is the diagonal matrix containing the row sums of M,

 

The data clusters are obtained from the eigenvectors of L. In the simplest case of data points with only two classes, the first eigenvector (that is, the one corresponding to the largest eigenvalue) is often enough to separate the data.

Here is an example for simple two-class clustering. The following code creates some 2D data points and clusters them based on the first...