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Dogs and Cats. Dogs. Cats. Low-Dimensional Dogs and Cats: “Eigen-Dogs/Cats”. Linear Discrimination Analysis. 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!!!!. - PowerPoint PPT Presentation
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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