# Multi-dimensional data vis

Multi-Dimensional Data Visualization Information Types

Multi-dimensional: databases, 1D: timelines, 2D: maps, 3D: volumes, Hierarchies/Trees: directories, Networks/Graphs: web, communications, Document collections: digital libraries,

The Simple Stuff Univariate Bivariate Trivariate Univariate

Dot plot Bar chart (item vs. attribute) Tukey box plot Histogram Bivariate Scatterplot

Trivariate 3D scatterplot, spin plot 2D plot + size (or color) Multi-Dimensional Data Each attribute defines a dimension Small # of dimensions easy What about many dimensional data? n-D

What does 10-D space look like? Projection map n-D space onto 2-D screen Glyphs: Chernoff Faces 10 Parameters:

Head Eccentricity Eye Eccentricity Pupil Size Eyebrow Slope Nose Size Mouth Vertical Offset Eye Spacing Eye Size Mouth Width

Mouth Openness http://hesketh.com/schampeo/projects/Faces/chernoff.html Glyphs: Stars d1 d7 d2 d3

d6 d5 d4 Multiple Views with Brushing-and-linking Scatterplot Matrix

All pairs of attributes Brushing and linking http://noppa5.pc.helsinki.fi/koe/3d3.html on steroids

Different Arrangements of Axes Axes are good Lays out all points in a single space position is 1st in Clevelands rules Uniform treatment of dimensions Space > 3D ? Must trash orthogonality

Parallel Coordinates Inselberg, Multidimensional detective (parallel coordinates) Parallel Coordinates Bag cartesian (0,1,-1,2)= x

0 y 0 z 0

w 0 Star Plot 1 8 2

7 3 4 6 5

Parallel Coordinates with axes arranged radially Star Coordinates Kandogan, Star Coordinates Star Coordinates Cartesian Star Coordinates

P=(v1,v2,v3,v4,v5,v6,v7,v8) P=(v1, v2) d1 d1 d8 d2

v3 p v4 v2 v1 v2

v5 d7 d2 p v1

v8 v7 d6 Mapping: Items dots attribute vectors (x,y) v6

d5 d4 d3 Analysis Table Lens

Rao, Table Lens FOCUS / InfoZoom Spenke, FOCUS VisDB Keim, VisDB

Pixel Bar Charts Keim Comparison of Techniques Comparison of Techniques ParCood: <1000 items, <20 attrs Relate between adjacent attr pairs

StarCoord: <1,000,000 items, <20 attrs Interaction intensive TableLens: similar to par-coords more items with aggregation Relate 1:m attrs (sorting), short learn time Visdb: 100,000 items with 10 attrs Items*attrs = screenspace, long learn time, must query

Spotfire: <1,000,000 items, <10 attrs (DQ many) Filtering, short learn time Multi-Dimensional Functions Multi-Dimensional Functions y = f(x1, x2, x3, , xn) Continuous: E.g. y = x13 + 2x22 - 9x3

Discrete: xi are uniformly sampled in a bounded region E.g. xi = [0,1,2,,100] E.g. measured density in a 3D material under range of pressures and room temperatures. Relations vs. Functions Relations: R(A, B, C, D, E, F)

All dependent variables (1 ind.var.?) Sparse points in multi-d dep.var. space Functions: R(A, B, C, D, E, F, Y) : Y=f(A, B, C, D, E, F)

Many independent variables Defined at every point in multi-d ind.var. space (onto) Huge scale: 6D with 10 samples/D = 1,000,000 data points Multi-D Relation Visualizations Dont work well for multi-D functions Example: Parallel coords

5D func sampled on 1-9 for all ind.vars. Typically want to encode ind.vars. as spatial attrs 1-D: Easy b = f(a) ax

by b a 2-D: Easy c = f(a, b)

c Height field: ax by cz

b a 2-D: Easy c = f(a, b)

b Heat map: ax by c color a c

3-D: Hard d = f(a, b, c) Color volume: ax

by cz d color c b a

Whats inside? 4D: Really Hard y = f(x1, x2, x3, x4, , xn) What does a 5D space look like? Approaches:

Hierarchical axes (Mihalisin) Nested coordinate frames (Worlds within Worlds) Slicing (HyperSlice) Radial Focus+Context (PolarEyez, Sanjini) Hierarchical Axes 1D view of 3D function: f(x1, x2, x3)

x3 x2 x1 (Mihalisin et al.) as in TableLens 5D 9 samp/D

Hierarchical Axes 2D view of 4D function (using heat maps) y = f(x1, x2, x3, x4) Discrete: xi = [0,1,2,3,4] x3 x1 x2 y = f(x1,x2,0,0) as color

x4 Hierarchical Axes Scale? 6d = 3 levels in the 2d approach 10 samples/d = 1,000,000 data points = 1 screen For more dimensions: zoom in on blocks

reorder dimensions 5D 9 sample/D Nested Coordinate Frames Feiner, Worlds within Worlds Slicing

Van Wijk, HyperSlice Radial Focus+Context Jayaraman, PolarEyez x2 x3 x4

infovis.cs.vt.edu x1 x5 -x5 -x1 -x4

-x2 Comparison Hierarchical axes (Mihalisin): Nested coordinate frames (Worlds in Worlds) Slicing (HyperSlice):

Radial Focus+Context (PolarEyez) Comparison Hierarchical axes (Mihalisin): < 6d by 10 samples, ALL slices, view 2d at a time Nested coordinate frames (Worlds in Worlds)

< 5-8d, continuous, no overview, 3d hardware Slicing (HyperSlice): < 10d by 100 samples, 2d slices Radial Focus+Context (PolarEyez) < 10d by 1000 samples, overview, all D uniform, rays Dynamic Queries

HomeFinder Spotfire Limitations Scale: Scatterplot screen space: 10,000 1,000,000 Data structures & algorithms: < 50,000 Poor screen drawing on Filter-out

A Solution: Query Previews! AND queries only Arbitrary boolean queries? A solution: Filter Flow DQ Algorithm Idea: incremental algorithm only deal with data items that changed state

When slider moves: Calculate slider delta Search in data structure for data items in the delta region If slider moved inward (filter out): Erase data items from visualization Else slider moved outward (filter in): Draw data items on visualization Problem!

Overlapped items, erases items underneath too DQ Data Structures (1) Sorted array of the data for each slider Year: Delta

Need counter for each data item = # sliders that filter it Attribute Explorer visualizes these counters too! O(delta) DQ Data Structures (2) Multi-dimensional data structure E.g.: K-d tree, quad-tree, Recursively split space, store in tree structure Enables fast range search, O()

DQ Data Structures (2) Multi-dimensional data structure E.g.: K-d tree, quad-tree, Recursively split space, store in tree structure Enables fast range search, O(logn) Delta Erasure Problem

Each pixel has counter = number of items Can visualize this for density! Z-buffer? Redraw local area only Filter-Flow Betty Catherine Edna

Freda Grace Hilda Judy Marcus Tom Influence/Attribute Explorer Tweedie, Spence, Externalizing Abstract Mathematical Models (Influence/Attribute

Explorer) Query Previews Doan, Query Previews

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