2. Principal Component Analysis Principles and Application

3. Examples: Satellite Data Digital Camera, Video Data Tomography Particle Imaging Velocimetry (PIV) Ultrasound Velocimetry (UVP)

4. Low resolution image Large Data Sets There are 400 x 600 = 240,000 pieces of information. Not all of this information is independent => information compression (data compression)

5. Experiment: Consider the flow past a cylinder, and suppose we position a cross-wire probe downstream of the cylinder. With a cross-wire probe we can measure two components of the velocity at successive time intervals and store the results in a computer. Example 1 Two component velocity measurement

6. As the previous slide suggests, the pair of velocities can be represented as a column vector: u is a vector at position x in physical space: The magnitude and angle of the vector changes with time. x y u x Mathematical Representation of Data

7. Mean velocity : Variance : Covariance : Correlation : Basic Statistics

8. Plot u vs v u v The data look correlated

9. Examine the Statistics Move to a data centered coordinate system u v v’v’ u’u’ Calculate the Covariance matrix Diagonal terms are the variances in the u ’ and v ’ directions

10. Examine the Statistics Move to a data centered coordinate system u v v’v’ u’u’ Calculate the Covariance matrix covariance or cross-correlation

11. Rotate coordinates to remove the correlations u v 11 v”v” 22 u”u” Covariance matrix in the (u ”,v ” ) coordinate system

12. We have just carried out a Principal Axis Transformation. This is the first step in a Principal Component Analysis (PCA).

13. Principal Component Analysis A procedure for transforming a set of correlated variables into a new set of uncorrelated variables. How do we do it??

14. Construction of the PCA coordinate system The PCA coordinate system is one that maximizes the mean squared projection of the data. In this sense it is an “optimal” orthogonal coordinate system. Its popularity is primarily due to its dimension reducing properties. The basic algorithm for constructing the PCA eigenvectors is: Find the best direction (line) in the space,  1. Find the best direction (line)  2 with the restriction that it must be orthogonal to  1. Find the best direction (line)  i with the restriction that  i is orthogonal to  j for all j < i.

15. How do we find this nice coordinate system?? Calculate the eigenvalues and eigenvectors of the Covariance Matrix

16. Experiment: Pipe Flow -- measurement of velocity profile. Example 2. Velocity Profile Measurement z u(z)

17. As before we represent the velocities in the form of a column vector, but this time the vector is not in physical space. The space in which our vector lives is one we shall call profile space or pattern space. Profile space has n dimensions. In this example, the position z k defines a direction in profile space. As time evolves, we measure a sequence of velocity profiles: Vectors in Profile Space

19. The Preliminary Calculations 1. UVP Data Matrix (n x m=128 x 1024) 2. Mean Profile Matrix (n x m) 3. Centered Data Matrix (n x m) 4. Covariance Matrix (n x n = 128 x 128)

20. The Diagonalization Eigenvalue Equation Eigenvalues Eigenvectors (eigenprofiles)

21. Example 3. Taylor-Couette Flow

22. UVP Example space time UVP data Before space After (diagonalisation) Covariance Matrix compression!!

23. The Eigenvalue Spectrum (Signal) Energy Spectrum Energy Fraction EkEk Mode Number 1281 1 0 cumulative sum of E k EkEk Mode Number 120 1 0 1

24. Filtering and Reconstruction Decompose X into signal and noise dominated components (subspaces): where X F is the Filtered data X Noise is the Residual Reconstruct filtered UVP velocity

25. U UFUF X Noise =U-U F

26. Eigenvalue Spectrum

27. Filtered Time Series (Channel 70) Raw data Filtered data Residual

28. Power Spectra (Integrated over all channels)

29. Superimpose the Spectra

30. Generalizations Generalise Response to a stimulus Comparison of multiple data sets obtained by varying a parameter to study a transition.