Principal Component Analysis Principles and Application

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Principal Component AnalysisPrinciples and
Application
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• Examples
• Satellite Data
• Digital Camera, Video Data
• Tomography
• Particle Imaging Velocimetry (PIV)
• Ultrasound Velocimetry (UVP)

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Large Data Sets
Low resolution image

• There are 400 x 600 240,000 pieces of
  information.
• Not all of this information is independent
• gt information compression (data compression)

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Example 1 Two component velocity measurement

• 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.

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Mathematical Representation of Data
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Basic Statistics

• Mean velocity
• Variance
• Covariance
• Correlation

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Plot u vs v
The data look correlated
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Examine the Statistics
Move to a data centered coordinate system
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Examine the Statistics
Move to a data centered coordinate system
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Rotate coordinates to remove the correlations
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We have just carried out a Principal Axis
Transformation. This is the first step in
a Principal Component Analysis (PCA).
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Principal Component Analysis A procedure for
transforming a set of correlated variables into a
new set of uncorrelated variables. How do we do
it??
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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
  lt i.

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How do we find this nice coordinate system??
Calculate the eigenvalues and eigenvectors of
the Covariance Matrix
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Example 2. Velocity Profile Measurement

• Experiment
• Pipe Flow -- measurement of velocity profile.

u(z)
z
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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 profile space 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

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The Preliminary Calculations
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The Diagonalization
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Example 3.Taylor-Couette Flow
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UVP Example
Covariance Matrix
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The Eigenvalue Spectrum(Signal) Energy Spectrum
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Filtering and Reconstruction

• Decompose X into signal and noise dominated
  components (subspaces)
• where XF is the Filtered data
• XNoise is the Residual
• Reconstruct filtered UVP velocity

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Eigenvalue Spectrum
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Filtered Time Series(Channel 70)
Raw data
Filtered data
Residual
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Power Spectra(Integrated over all channels)
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Superimpose the Spectra
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Generalizations

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