1 / 29

Principal Component Analysis Principles and Application

Principal Component Analysis Principles and Application. Examples: Satellite Data Digital Camera, Video Data Tomography Particle Imaging Velocimetry (PIV) Ultrasound Velocimetry (UVP). Large Data Sets. Low resolution image. There are 400 x 600 = 240,000 pieces of information.

lawrence-sanders
Download Presentation

Principal Component Analysis Principles and Application

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Principal Component AnalysisPrinciples and Application

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

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

  4. 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

  5. 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. u y x x Mathematical Representation of Data

  6. Basic Statistics • Mean velocity : • Variance : • Covariance : • Correlation :

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

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

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

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

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

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

  13. 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.

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

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

  16. Vectors in Profile Space • 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 profilespace or pattern space. • Profile space has n dimensions. In this example, the position zk defines a direction in profile space. • As time evolves, we measure a sequence of velocity profiles:

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

  18. Eigenvectors (eigenprofiles) Eigenvalue Equation Eigenvalues The Diagonalization

  19. Example 3.Taylor-Couette Flow

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

  21. 1 Energy Fraction Ek 0 1 Mode Number 128 1 1 Ek cumulative sum of Ek 0 1 20 Mode Number The Eigenvalue Spectrum(Signal) Energy Spectrum

  22. Filtering and Reconstruction • Decompose X into signal and noise dominated components (subspaces): where XF is the Filtered data XNoise is the Residual • Reconstructfiltered UVP velocity

  23. U UF XNoise=U-UF

  24. Eigenvalue Spectrum

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

  26. Power Spectra(Integrated over all channels)

  27. Superimpose the Spectra

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

More Related