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• Objectives: Normal Random Variables Support Regions Whitening Transformations Resources:

• Objectives: Normal Random Variables Support Regions Whitening Transformations Resources: DHS – Chap. 2 (Part 2) K.F. – Intro to PR X. Z. – PR Course S.B. – Statistical Norm. Java Applet. ECE 8443 – Pattern Recognition. LECTURE 07: WHITENING TRANSFORMATIONS.

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• Objectives: Normal Random Variables Support Regions Whitening Transformations Resources:

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  1. • Objectives: Normal Random Variables Support Regions Whitening Transformations Resources: DHS – Chap. 2 (Part 2) K.F. – Intro to PR X. Z. – PR Course S.B. – Statistical Norm. Java Applet ECE 8443 – Pattern Recognition LECTURE 07: WHITENING TRANSFORMATIONS • URL: .../publications/courses/ece_8443/lectures/current/lecture_07.ppt

  2. 07: WHITENING TRANSFORMATIONS UNIVARIATE NORMAL DISTRIBUTION • A normal or Gaussian density is a powerful model for modeling continuous-valued feature vectors corrupted by noise due to its analytical tractability. • Univariate normal distribution: where the mean and covariance are defined by: and the entropy is given by:

  3. The peak is at: • 66% of the area is within one ; 95% is within two ; 99% is within three . 07: WHITENING TRANSFORMATIONS MEAN AND VARIANCE • A normal distribution is completely specified by its mean and variance: • A normal distribution achieves the maximum entropy of all distributions having a given mean and variance. • Central Limit Theorem: The sum of a large number of small, independent random variables will lead to a Gaussian distribution.

  4. where  represents the mean (vector) and  represents the covariance (matrix). • Note the exponent term is really a dot product or weighted Euclidean distance. 07: WHITENING TRANSFORMATIONS MULTIVARIATE NORMAL DISTRIBUTIONS • A multivariate distribution is defined as: • The covariance is always symmetric and positive semidefinite. • How does the shape vary as a function of the covariance?

  5. 07: WHITENING TRANSFORMATIONS SUPPORT REGIONS • A support region is the obtained by the intersection of a Gaussian distribution with a plane. • For a horizontal plane, this generates an ellipse whose points are of equal probability density. • The shape of the support region is defined by the covariance matrix.

  6. 07: WHITENING TRANSFORMATIONS DERIVATION

  7. 07: WHITENING TRANSFORMATIONS IDENTITY COVARIANCE

  8. 07: WHITENING TRANSFORMATIONS UNEQUAL VARIANCES

  9. 07: WHITENING TRANSFORMATIONS OFF-DIAGONAL ELEMENTS

  10. 07: WHITENING TRANSFORMATIONS UNCONSTRAINED ELEMENTS

  11. 07: WHITENING TRANSFORMATIONS COORDINATE TRANSFORMATIONS • Why is it convenient to convert an arbitrary distribution into a spherical one? (Hint: Euclidean distance) • Consider the transformation: Aw=  -1/2 where  is the matrix whose columns are the orthonormal eigenvectors of  and  is a diagonal matrix of eigenvalues (=   t). Note that  is unitary. • What is the covariance of y=Awx? E[yyt] = (Awx)(Awx)t =( -1/2x)( -1/2x)t =  -1/2xxt-1/2 t =  -1/2 -1/2 t =  -1/2   t -1/2 t =  t -1/2  -1/2 (t) = I

  12. 07: WHITENING TRANSFORMATIONS MAHALANOBIS DISTANCE • The weighted Euclidean distance: is known as the Mahalanobis distance, and represents a statistically normalized distance calculation that results from our whitening transformation. • Consider an example using our Java Applet.

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