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Ellipse and Gaussian Distribution Prof. Seungchul Lee Industrial AI Lab. Coordinates 2 Coordinates with Basis basis 1 2 basis 1 2 3 Coordinate Transformation 4 Equation of an Ellipse 5 Equation of an

  1. Ellipse and Gaussian Distribution Prof. Seungchul Lee Industrial AI Lab.

  2. Coordinates 2

  3. Coordinates with Basis basis ො 𝑦 1 ො 𝑦 2 basis ො 𝑧 1 ො 𝑧 2 3

  4. Coordinate Transformation 4

  5. Equation of an Ellipse 5

  6. Equation of an Ellipse β€’ Unit circle 6

  7. Equation of an Ellipse β€’ Independent ellipse 7

  8. Equation of an Ellipse β€’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse 8

  9. Equation of an Ellipse β€’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse β€’ Coordinate changes 9

  10. Equation of an Ellipse β€’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse β€’ Coordinate changes 10

  11. Equation of an Ellipse β€’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse β€’ Coordinate changes 11

  12. Equation of an Ellipse β€’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse 12

  13. Equation of an Ellipse 13

  14. Equation of an Ellipse 14

  15. Question (Reverse Problem) βˆ’1 (or Ξ£ 𝑧 ), β€’ Given Ξ£ 𝑧 – How to find 𝑏 (major axis) and 𝑐 (minor axis) or – How to find the Ξ£ 𝑦 or – How to find the proper matrix 𝑉 15

  16. Question (Reverse Problem) βˆ’1 (or Ξ£ 𝑧 ), β€’ Given Ξ£ 𝑧 – How to find 𝑏 (major axis) and 𝑐 (minor axis) or – How to find the Ξ£ 𝑦 or – How to find the proper matrix 𝑉 16

  17. Question (Reverse Problem) βˆ’1 (or Ξ£ 𝑧 ), β€’ Given Ξ£ 𝑧 – How to find 𝑏 (major axis) and 𝑐 (minor axis) or – How to find the Ξ£ 𝑦 or – How to find the proper matrix 𝑉 β€’ Eigenvectors of Ξ£ 17

  18. Question (Reverse Problem) 18

  19. Question (Reverse Problem) 19

  20. Summary β€’ Independent ellipse in ො 𝑦 1 , ො 𝑦 2 β€’ Dependent ellipse in ො 𝑧 1 , ො 𝑧 2 β€’ Decouple – Diagonalize – Eigen-analysis 20

  21. Gaussian Distribution 21

  22. Standard Univariate Normal Distribution β€’ It is a continuous pdf, but – Parameterized by only two terms, 𝜈 = 0 and 𝜏 = 1 – This is a big advantage of using Gaussian 22

  23. Standard Univariate Normal Distribution 23

  24. Standard Univariate Normal Distribution β€’ How to generate data from Gaussian distribution 24

  25. Univariate Normal Distribution β€’ Gaussian or normal distribution, 1D (mean 𝜈 , variance 𝜏 2 ) β€’ It is a continuous pdf, but parameterized by only two terms, 𝜈 and 𝜏 Affine transformation 25

  26. Univariate Normal Distribution 26

  27. Multivariate Gaussian Models β€’ Similar to a univariate case, but in a matrix form β€’ Multivariate Gaussian models and ellipse – Ellipse shows constant Ξ” 2 value… – The contours of equal probability is ellipse β€’ Ellipsoidal probability contours β€’ Bell shaped 27

  28. Two Independent Variables β€’ In a matrix form – Diagonal covariance 28

  29. Two Independent Variables β€’ Geometry of Gaussian β€’ Summary in a matrix form 29

  30. Two Independent Variables 30

  31. Two Dependent Variables in 𝒛 𝟐 , 𝒛 πŸ‘ β€’ Compute 𝑄 𝑍 𝑧 from 𝑄 π‘Œ 𝑦 β€’ Relationship between 𝑧 and 𝑦 31

  32. Two Dependent Variables in 𝒛 𝟐 , 𝒛 πŸ‘ β€’ Ξ£ 𝑦 : covariance matrix of 𝑦 β€’ Ξ£ 𝑧 : covariance matrix of 𝑧 β€’ If 𝑣 is an eigenvector matrix of Ξ£ 𝑧 , then Ξ£ 𝑦 is a diagonal matrix 32

  33. Two Dependent Variables in 𝒛 𝟐 , 𝒛 πŸ‘ β€’ Remark 33

  34. Two Dependent Variables in 𝒛 𝟐 , 𝒛 πŸ‘ 34

  35. Decouple using Covariance Matrix β€’ Given data, how to find Ξ£ 𝑧 and major (or minor) axis (assume 𝜈 𝑧 = 0 ) β€’ Statistics 35

  36. Decouple using Covariance Matrix 36

  37. Nice Properties of Gaussian Distribution 37

  38. Properties of Gaussian Distribution β€’ Symmetric about the mean β€’ Parameterized β€’ Uncorrelated β‡’ independent β€’ Gaussian distributions are closed to – Linear transformation – Affine transformation – Reduced dimension of multivariate Gaussian β€’ Marginalization (projection) β€’ Conditioning (slice) – Highly related to inference 38

  39. Affine Transformation of Gaussian β€’ Suppose 𝑦~π’ͺ(𝜈 𝑦 , Ξ£ 𝑦 ) β€’ Consider affine transformation of 𝑦 β€’ Then it is amazing that 𝑧 is Gaussian with 39

  40. Component of Gaussian Random Vector β€’ Suppose 𝑦~π’ͺ(0, Ξ£) , 𝑑 ∈ ℝ π‘œ be a unit vector β€’ 𝑧 is the component of 𝑦 in the direction 𝑑 β€’ 𝑧 is Gaussian with 𝐹 𝑧 = 0, cov 𝑧 = 𝑑 π‘ˆ Σ𝑑 β€’ So E 𝑧 2 = 𝑑 π‘ˆ Σ𝑑 β€’ The unit vector that minimizes 𝑑 π‘ˆ Σ𝑑 is the eigenvector of Ξ£ with the smallest eigenvalue β€’ Notice that we have seen this in PCA 40

  41. Marginal Probability of Gaussian β€’ Suppose 𝑦~π’ͺ(𝜈, Ξ£) β€’ Let’s look at the component 𝑦 1 β€’ In fact, the random vector 𝑦 1 is also Gaussian. – (this is not obvious) 41

  42. Marginalization (Projection) 42

  43. Conditional Probability of Gaussian β€’ The conditional pdf of 𝑦 given 𝑧 is Gaussian β€’ The conditional mean is β€’ The conditional covariance is β€’ Notice that conditional confidence intervals are narrower. i.e., measuring 𝑧 gives information about 𝑦 43

  44. Conditioning (Slice) 44

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