multivariate and high dimensional visualizations

Multivariate and High Dimensional Visualizations

Robert Herring

Articles Covered

Visualizing the Behavior of Higher Dimensional Dynamical Systems Rainer Wegenkittl, Helwig Loffelmann, and

Eduard Groller

Multivariate Visualization Using Metric Scaling Pak Chung Wong and R. Daniel Bergeron

Problem Addressed

Information gathered often contains multiple variables to be studied

Most visualization techniques focus on discrete statistical characteristics

These techniques are ill suited for visualizing continuous flow in high-dimensional space from dynamical systems

Statistical visualizations typically are not designed to show integral curves within a high-dimensional phase space

Visualizing the Behavior of Higher Dimensional Dynamical Systems

Visualizing Multidimensional Data

Multivariate data sets becoming common Data is either discrete of continuous Data can be spatially coherent or spatially

incoherent Data sets may consist of a collection of

sampled data Each sample is an n-dimensional data item Can be sampled from m-dimensional space Lm

n data set


Visualizing Multidimensional Data

Two important goals Identification of individual parameters Detection of regions and correlation of variables

Methods for visualizing high-dimensional data Attribute Mapping Geometric Coding Sonification Reduction of Dimension Parallel Coordinates


Attribute Mapping

Use geometric primitives, planes, etc. Most commonly used attribute used in

attribute mapping is color Most common color models RGB and HLS Advantages

Easy calculation/interpretation, many people familiar with color mapping

Disadvantages No unique order, requires legend Can only encode 3 variables 8% of population has some form of color

blindness


Geometric Coding

Use distinct geometric objects and map high-dimensional data to geometric features or attributes of these objects

Glyphs Icons Chernoff Faces Data Jacks m-Arm Glyph


Geometric Coding

Glyphs Utilized for interactive exploration of data sets Generic term for graphical entity whose

shape/appearance is modified by mapping data values to graphical attributes (length, shape, angle, color, transparency, et.)

Icons Use icons as basic primitives Attributes mapped to icon shape, color, and

texture to map multiple variables


Geometric Coding

Chernoff Faces Uses stylized faces where variables influence

appearance features like overall shape, mouth, eyes, nose, eyebrows, etc

Data Jacks Three-dimensional shapes with four different

limbs (length, color, etc modified) m-Arm Glyph

Two-dimensional structure with m arms attached (thickness, angle from main axis, etc modified)


Sonification

The use of sound to add a layer of dimensional that does not overload visual system

Sounds can vary over Pitch Volume Pulse


Reduction of Dimension

Focusing Selecting subsets, reduction of dimension

through projection Examples are panning, zooming, and slicing High dimensional data can be mapped to lower

dimensions with other dimensions being represented via attributes

Linking Showing multiple varying visualizations of data


Parallel Coordinates

Represent dimensions as parallel axes orthogonal to a horizontal line uniformly spaced on display

Each data set corresponds to a polyline that traverses/intersects these parallel axes


High Dimensional Dynamical Systems and Visualization

Many natural phenomenona can be described by differential equations

Each differential equation describes the change of one state variable n differential equations define behavior of n state

variables describing a n-dimensional dynamical system

n-dimensional vector from each sampled set of state variables

The discretized flow described by n differential equations forms a vector field of dimension n, where each vector itself is of dimension n

Lnn data set



Dynamical system typically describes a complex but smooth flow

Behavior of flow determined by topology To interpret behavior of system each point

within n-space cannot be investigated by itself but seen in respect to its neighborhood Derived from continuous flow field

Two basic approaches



Neighboring information can be calculated from vector field (interpreting Jacobian matrix) and the derived data displayed in n-space

Directional information at each point in n-space may be projected to an m-dimensional data object describing some local feature



Direct global flow visualization can be done by starting short integral curves (trajectories), which follow the flow, at the nodes of an n-dimensional regular grid

Features like separatrices can be detected visually by interpreting the flow directions of the trajectories.


Extruded Parallel Coordinates

Instead of using same coordinate system for each sample, move parallel coordinate system along third spatial axis

Polylines viewed as cross sections of a moving plane with a complex surface that defines the trajectory



Geometry of surface can be generated and modified quickly

Clustering and correlation visually detectable

Convergence and divergence pbserved by varying the starting coordinates of the trajectory slightly


Linking with Wings

Two dimensions of high-dimensional system selected and displayed as a two-dimensional trajectory within a base plane

Third dimension of display can now be use to display third variable over base trajectory

If resulting three-dimensional trajectory is connected with base trajectory, thought of as a wing on the base trajectory Wing can be tilted at each point within a plane

normal to base trajectory


Linking with Wings


Linking with Wings

Any number of wings can be added to display high-dimensional trajectories (occlusion problem)

Wings can be textured with a grid texture allowing exact measurement of wing dimensions

Self intersection can be a problem Wing size chosen to be small with respect to size

of base trajectories Angles of wings kept must not be too big


Three-dimensional Parallel Coordinates

Based on parallel coordinate method One-dimensional spaces put together within

two-dimensional space (planes) and linked with polylines

Positioning of planes is more flexible Can be moved, rotated within three-dimensional

space


Three-dimensional Parallel Coordinates


Results


Problem Addressed

Large multivariate can be difficult to navigate

Need a low dimensional representation to easy navigation

Metric scaling used as basis for creating low-dimensional overview

Multivariate Visualization Using Metric Scaling

Metric Scaling

Start with set of n records with v variables and dissimilarities rs measured between all pairs of records in n dimensional space

Configure graph of n vertices in d dimensional space Each vertex represents one record Distances drs measured all pairs of vertices in display

space match rs in variate space as closely as possible

Goal is to determine the dissimilarities between all pairs in v space and map them to coordinates in d dimensional display space


Metric Scaling


Data Dissimilarity Measurement

Compute dissimilarity between all pairs of input records

Euclidean distance in v space most common metric

Dissimilarity rs between records r and s


Data Dissimilarity Measurement

Dataset with n records generates n x n real symmetric dissimilarity matrix


Recovery of Coordinates

Represent data as points in new p dimensional space where p <= n

Create inner product matrix from rs in variate space, find its non-negative eigenvalues and the corresponding eigenvectors Yield the Euclidean coordinates of n vertices in p

dimensional space


Principal Coordinates

Let Euclidean coordinates of n vertices in n dimensional Euclidean space be a matrix X = [x1, x2, …, xn] such that

xr = [xr1, …, xrn]T where r = 1, 2, …, n

Euclidean distance between vertices r and s


(1)

(2)


Standardize data to have zero mean and unit variance, center of mass of the vertices is the origin




since

Equation (2) becomes

(3)


Similarly

(4)




Furthermore from (4)

(5)



Defining inner product matrix B such that



Substituting (3), (4), and (5) into (1) gives the inner productMatrix B in terms of drs


Use principle components to recover Euclidean coordinates of the n dimensional space denoted by the matrix X from

B = XXT

Since B is symmetric and positive semi-definite it has p positive eigenvalues

Let be the eigenvalue matrix where the diagonals are the sorted eigenvalues


(6)


Let the corresponding normalized eigenvector of be V By definition of eigenvectors matrix B can be described

as

B = VVT

Since there are only p positive eigenvalues B can be expressed as

B = V11V1T

= V111/21

1/2V1T

Where 1 is the eigenvalue matrix with the diagonal with 1 – p eigenvalues and V1 is the corresponding eigenvalue of 1


(7)


From (6) and (7)X = V11

1/2



If eigenvalues are sorted in desending order the first principal component associated with the first eigenvalue is more important than the second

The distance between vertices r and s is

Where xr and xs are the distance vectors associated with points r and s respectively



A smaller eigenvalue contributes much less weight to the distance drs Smaller eigenvalues can be truncated with less

error Suppose d is selected as most significant

eigenvalue to display data overview, the degree of accuracy of the approximation can be measured by


Strengths and Weaknesses

Data clustering



Display Density



Outlier Detections



Multiresolution Visualization Prograssive refinement to visualize datasets with

many variates



Multiresolution Visualization



Individual variate values are lost


Integration of Techniques

Linking



Merging Merge Euclidean coordinates of data overview

and the data into one visualization display Brings new perspective to the conventional

icon/glyph visualizations


Questions?

multivariate and high dimensional visualizations

Documents

layer of dimensional

highdimensional space

ndimensional data itemcan

mapping data values

geometric features

graphical attributes

overall shape

icon shape