The Topography of Multivariate Normal Mixtures
The Topography of Multivariate Normal Mixtures
The Topography of Multivariate Normal Mixtures
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<strong>The</strong> <strong>Topography</strong> <strong>of</strong> <strong>Multivariate</strong><br />
<strong>Normal</strong> <strong>Mixtures</strong><br />
Post Doctoral Talk<br />
SAMSI<br />
Surajit Ray<br />
EMAIL: sray@bios.unc.edu<br />
Surajit Ray Nov 2, 2004
Mixture <strong>of</strong> Distributions<br />
■ Flexible way <strong>of</strong> modeling heterogeneous population.<br />
■ Data reduction through the number, location and shape <strong>of</strong> Mixture<br />
components.<br />
■ Can be directly used for model based clustering.<br />
Surajit Ray Nov 2, 2004
Mixture <strong>of</strong> Distributions<br />
-component mixture:<br />
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Surajit Ray Nov 2, 2004<br />
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Number <strong>of</strong> components Number <strong>of</strong> Modes<br />
0.0 0.05 0.10 0.15 0.20<br />
-6 -4 -2 0 2 4 6<br />
Mixture <strong>of</strong> normals 4 standard deviations apart<br />
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0.0 0.05 0.10 0.15 0.20 0.25<br />
-6 -4 -2 0 2 4 6<br />
Mixture <strong>of</strong> normals 2 standard deviations apart<br />
Bimodal Unimodal<br />
¤ number<br />
<strong>of</strong> modes.<br />
NOTE: Practical interest could lie in finding components that correspond<br />
to separate modes.<br />
Surajit Ray Nov 2, 2004
Detection <strong>of</strong> modality: Univariate Case<br />
■ Conditions for bimodality <strong>of</strong> two univariate normals<br />
<strong>The</strong>orem 1. (?).<br />
■ Conditions for arbitrary mixture in (??)<br />
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Surajit Ray Nov 2, 2004<br />
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Modal Structure...<strong>Topography</strong><br />
■ Modes are potentially symptomatic to the underlying population<br />
structure.<br />
Studying modal structures in high dimensions is extremely challenging.<br />
Our Goals<br />
■ Study the topography <strong>of</strong> normal mixtures in high dimension.<br />
■ Provide analytical and graphical solution for determining the number <strong>of</strong><br />
modes.<br />
■ Show how it can be used to find the number <strong>of</strong> clusters in high<br />
dimensional data � Modal Clusters.<br />
Surajit Ray Nov 2, 2004
Modality: <strong>Multivariate</strong> Distribution<br />
density<br />
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(1,1)<br />
(−1,−1)<br />
Bivariate normal with means (-1,-1) and (1,1)<br />
Surajit Ray Nov 2, 2004<br />
−2<br />
x<br />
0<br />
2<br />
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Modality: <strong>Multivariate</strong> Distribution<br />
density<br />
0.0 0.1 0.2 0.3 0.4 0.5<br />
−4 −2 0 2 4<br />
x−axis<br />
Marginal distribution in<br />
and£ -axis.<br />
Surajit Ray Nov 2, 2004
Modality: <strong>Multivariate</strong> Distribution<br />
z3<br />
0.00 0.05 0.10 0.15 0.20 0.25<br />
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Distribution along the line<br />
Surajit Ray Nov 2, 2004<br />
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Modality: Equal variance<br />
■ A mixture <strong>of</strong> multivariate normal is bimodal<br />
distribution along some line is bimodal.<br />
■ Get the axis <strong>of</strong> maximum separation.<br />
■ Use the bimodality condition in the univariate case.<br />
<strong>The</strong>orem 2. <strong>Multivariate</strong> Modality Condition: Equal variance Case<br />
<strong>The</strong> distribution <strong>of</strong><br />
is bimodal iff<br />
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Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004<br />
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<strong>The</strong> image <strong>of</strong> this map will be denoted by and called the ridgeline<br />
surface or manifold. If<br />
one-dimensional curve.<br />
will be called the ridgeline as it is a<br />
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hence modes, antimodes and saddlepoints, are points in .<br />
Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004<br />
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density<br />
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replacements<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
Surajit Ray Nov 2, 2004
Ridgeline Elevation Plots<br />
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Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004
Ridgeline Surface/Manifold<br />
Three components, three modes, equal variance: For<br />
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Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004<br />
α 2
<strong>The</strong> -plots<br />
Ridgeline elevation/contour plots<br />
■ Full information (location and height) <strong>of</strong> the modes and saddle-point.<br />
■ No dependence on � .<br />
-plots<br />
■ Only location <strong>of</strong> the modes and saddle-point not height<br />
■ But extra advantage <strong>of</strong> understanding the dependence on the mixing<br />
parameter �<br />
Surajit Ray Nov 2, 2004
<strong>The</strong> ‘ -equation’<br />
Two component case: If<br />
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Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004<br />
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Surajit Ray Nov 2, 2004
Example: -plot for unimodal density<br />
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Surajit Ray Nov 2, 2004
Example: -plot for example with 4 modes<br />
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Surajit Ray Nov 2, 2004
Analytic Solution<br />
From the properties <strong>of</strong> the<br />
by the zeroes <strong>of</strong><br />
Same as the zeroes in<br />
<strong>The</strong>orem 0.1. Let<br />
where<br />
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is determined<br />
normal densities. <strong>The</strong>n<br />
Surajit Ray Nov 2, 2004<br />
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Data Example:Iris Data<br />
§ � direct contour plotting <strong>of</strong> is<br />
Component 2<br />
−3 −2 −1 0 1 2<br />
not available.<br />
CLUSPLOT( iris )<br />
−3 −2 −1 0 1 2 3<br />
Component 1<br />
<strong>The</strong>se two components explain 95.81 % <strong>of</strong> the point variability.<br />
Projection <strong>of</strong> Iris Data on the first two principal-components plane<br />
Surajit Ray Nov 2, 2004
Iris Data: Ridgeline Contour plot<br />
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α 3<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
Surajit Ray Nov 2, 2004<br />
α 1<br />
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Iris Data: -plot<br />
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Surajit Ray Nov 2, 2004<br />
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Example: Egyptian Skull Data<br />
Egyptian Skull Data: This data consists <strong>of</strong> four measurements Maximal<br />
Breadth, Basibregmatic Height, Basialveolar Length , and Nasal Height <strong>of</strong><br />
male Egyptian skulls from five different time periods ( 4000 BC, 3300 BC,<br />
1850 BC, 200 BC, 150 AD). Thirty skulls were measured from each time<br />
period.<br />
Here we analyze the three earliest time periods<br />
Surajit Ray Nov 2, 2004
Egyptian Skull Data: Ridgeline Contour plot<br />
0.0 0.2 0.4 0.6 0.8<br />
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Surajit Ray Nov 2, 2004<br />
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Egyptian Skull Data: Iris Data: -plot<br />
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Surajit Ray Nov 2, 2004<br />
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Summary and Future Work<br />
■ For , the problem <strong>of</strong> determining the number <strong>of</strong> modes in the<br />
-dimensional space can be reduced to a 1-dimensional space<br />
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■ Can be used to determine the number <strong>of</strong> modal clusters<br />
■ Analytical results and dependence <strong>of</strong> modes on the mixing proportion<br />
are discussed in Ray and Lindsay, 2004.<br />
■ Several ways <strong>of</strong> generalization to more than two components.<br />
■ Relating the subspace reduction in discriminant analysis to the<br />
subspace reduction in modality determinantion.<br />
Surajit Ray Nov 2, 2004
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