Ellipse and Gaussian Distribution Prof. Seungchul Lee Industrial AI Lab.
Coordinates 2
Coordinates with Basis basis ΰ· π¦ 1 ΰ· π¦ 2 basis ΰ· π§ 1 ΰ· π§ 2 3
Coordinate Transformation 4
Equation of an Ellipse 5
Equation of an Ellipse β’ Unit circle 6
Equation of an Ellipse β’ Independent ellipse 7
Equation of an Ellipse β’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse 8
Equation of an Ellipse β’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse β’ Coordinate changes 9
Equation of an Ellipse β’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse β’ Coordinate changes 10
Equation of an Ellipse β’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse β’ Coordinate changes 11
Equation of an Ellipse β’ Dependent ellipse (Rotated ellipse) To find the equation of dependent ellipse 12
Equation of an Ellipse 13
Equation of an Ellipse 14
Question (Reverse Problem) β1 (or Ξ£ π§ ), β’ Given Ξ£ π§ β How to find π (major axis) and π (minor axis) or β How to find the Ξ£ π¦ or β How to find the proper matrix π 15
Question (Reverse Problem) β1 (or Ξ£ π§ ), β’ Given Ξ£ π§ β How to find π (major axis) and π (minor axis) or β How to find the Ξ£ π¦ or β How to find the proper matrix π 16
Question (Reverse Problem) β1 (or Ξ£ π§ ), β’ Given Ξ£ π§ β How to find π (major axis) and π (minor axis) or β How to find the Ξ£ π¦ or β How to find the proper matrix π β’ Eigenvectors of Ξ£ 17
Question (Reverse Problem) 18
Question (Reverse Problem) 19
Summary β’ Independent ellipse in ΰ· π¦ 1 , ΰ· π¦ 2 β’ Dependent ellipse in ΰ· π§ 1 , ΰ· π§ 2 β’ Decouple β Diagonalize β Eigen-analysis 20
Gaussian Distribution 21
Standard Univariate Normal Distribution β’ It is a continuous pdf, but β Parameterized by only two terms, π = 0 and π = 1 β This is a big advantage of using Gaussian 22
Standard Univariate Normal Distribution 23
Standard Univariate Normal Distribution β’ How to generate data from Gaussian distribution 24
Univariate Normal Distribution β’ Gaussian or normal distribution, 1D (mean π , variance π 2 ) β’ It is a continuous pdf, but parameterized by only two terms, π and π Affine transformation 25
Univariate Normal Distribution 26
Multivariate Gaussian Models β’ Similar to a univariate case, but in a matrix form β’ Multivariate Gaussian models and ellipse β Ellipse shows constant Ξ 2 valueβ¦ β The contours of equal probability is ellipse β’ Ellipsoidal probability contours β’ Bell shaped 27
Two Independent Variables β’ In a matrix form β Diagonal covariance 28
Two Independent Variables β’ Geometry of Gaussian β’ Summary in a matrix form 29
Two Independent Variables 30
Two Dependent Variables in π π , π π β’ Compute π π π§ from π π π¦ β’ Relationship between π§ and π¦ 31
Two Dependent Variables in π π , π π β’ Ξ£ π¦ : covariance matrix of π¦ β’ Ξ£ π§ : covariance matrix of π§ β’ If π£ is an eigenvector matrix of Ξ£ π§ , then Ξ£ π¦ is a diagonal matrix 32
Two Dependent Variables in π π , π π β’ Remark 33
Two Dependent Variables in π π , π π 34
Decouple using Covariance Matrix β’ Given data, how to find Ξ£ π§ and major (or minor) axis (assume π π§ = 0 ) β’ Statistics 35
Decouple using Covariance Matrix 36
Nice Properties of Gaussian Distribution 37
Properties of Gaussian Distribution β’ Symmetric about the mean β’ Parameterized β’ Uncorrelated β independent β’ Gaussian distributions are closed to β Linear transformation β Affine transformation β Reduced dimension of multivariate Gaussian β’ Marginalization (projection) β’ Conditioning (slice) β Highly related to inference 38
Affine Transformation of Gaussian β’ Suppose π¦~πͺ(π π¦ , Ξ£ π¦ ) β’ Consider affine transformation of π¦ β’ Then it is amazing that π§ is Gaussian with 39
Component of Gaussian Random Vector β’ Suppose π¦~πͺ(0, Ξ£) , π β β π be a unit vector β’ π§ is the component of π¦ in the direction π β’ π§ is Gaussian with πΉ π§ = 0, cov π§ = π π Ξ£π β’ So E π§ 2 = π π Ξ£π β’ The unit vector that minimizes π π Ξ£π is the eigenvector of Ξ£ with the smallest eigenvalue β’ Notice that we have seen this in PCA 40
Marginal Probability of Gaussian β’ Suppose π¦~πͺ(π, Ξ£) β’ Letβs look at the component π¦ 1 β’ In fact, the random vector π¦ 1 is also Gaussian. β (this is not obvious) 41
Marginalization (Projection) 42
Conditional Probability of Gaussian β’ The conditional pdf of π¦ given π§ is Gaussian β’ The conditional mean is β’ The conditional covariance is β’ Notice that conditional confidence intervals are narrower. i.e., measuring π§ gives information about π¦ 43
Conditioning (Slice) 44
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