Dogs and Cats

AMATH 582 – Computational Methods for Data Analysis

CatsDogs

Low-Dimensional Dogs and Cats: “Eigen-Dogs/Cats”

Linear Discrimination Analysis

R. A. Fisher 1936: Classification of Species

Find a projection that maximizes the statistical distance between two random data sets

Histogram of the Data

95% Accurate

Misclassified Dogs

It’s the ears!!!!

PDEsA backbone of AMATH (fluids)

• Getting the right basis- POD modes

• Minimal dynamics- dynamical systems of the cheap

How to solve a PDE: 3 easy steps

* Sad fact: This is all you know how to do!

PDE ODE Algebra*

Undo

1 2

3

Eigenfunction expansion (separation of variables)

- u(x,t) = Σ an(t)ϕn(x)

Two Soliton Solution of a nonlinear PDE

Kutz, Data Driven Modeling Sciand entific Computing (Oxford)

SVD on Data

Kutz, Data Driven Modeling Sciand entific Computing (Oxford)

Galerkin Expansion

Kutz, Data Driven Modeling Sciand entific Computing (Oxford)

Nonlinear Schroedinger Equation

Kutz, Data Driven Modeling Sciand entific Computing (Oxford)

Water Waves: Governing Equations

V. E. Zakharov. “Stability of periodic waves of finite amplitude on the surface of a deep fluid.” Zh. Prikl. Mekh. Tekh. Fiz. (1968).

Low Amplitude Waves

High Amplitude Waves

Mode Structure

PCA (POD) Convergence

P. Holmes, J. Lumley, and G. Berkooz. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge U. Press (1996)

The n mode POD basis of a set of data captures more of the L2 norm of that data than any other linear set of n modes:

www.mathworks/moler/eigs.pdf

See alsohttp://people.maths.ox.ac.uk/trefethen/text.html