Multi-Dimensional Visualization
November 3, 2011
InfoVis 583
Multi-Dimensional Visualization
1. Venn Diagrams
– Many Boolean dimensions.
2. Projection / Multivariate Scaling
– Spatial layout for data with >3 dimensions.
3. Data Cubes
– Data in cubes (2D + time + colour).
Venn Diagrams
Venn Diagrams
Venn Diagrams
Venn Diagrams
• Logical relationships
between a finite
collection of sets.
Hotel
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Pool
Golf
X
X
X
X
X
X
X
X
X
X
X
X
Restaurant
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Venn Diagrams
• Logical relationships
between a finite
collection of sets.
• Many boolean variables.
C
Pool
A
L
F
E
Golf
I
H G
K
B
D
J
VWXY
M
N
U
Q T
S O
R
P
Restaurant
What about more than three variables?
Hotel
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Pool
Golf
X
X
X
X
X
X
X
X
X
X
X
X
Restaurant
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Venn Diagrams
Venn Diagrams
4
5
6
Venn Diagrams
Venn Diagrams - Starbucks
Source: http://postwhatever.com/2010/09/coffee-venn-diagram-starbucks-survival-guide/
Cluster Maps
Cluster Maps
Projections
Have:
• data with many, many dimensions
Want:
• to “see” the data as positions
• i.e., we need some sort of spatial layout
Projection: 3D to 2D
B. Taylor, 1811, in New Principles of Linear Perspectives.
Projection
Projections
Have:
• data with many, many dimensions
Want:
• to “see” the data as positions
• i.e., we need some sort of spatial layout
What might we want in a layout?
• Objects close to one another more similar than
objects further way.
Property
Name
Location
Bedrooms
Baths
Lot Size
Asking
Price
Karang
Estate
Museum
Ready
Valley
Estate
Trout Pond Meadow Lane Oceanfront Pell Mansion
San
Santa Fe, Manhattan,
Southampton, Palm Beach, Francisco,
NM
NY
Hudson, NY
NY
FL
CA
Napa, CA
9
11
6
7
7
10
7
6
7
6
7
8
8
9
London, Monterrey, Monterrey,
England
CA
CA
8
5
5
7
6
6
12.5 acres
12.5 acres
0.75 acres
8,500,000 3,000,000
90.5 acres
3.5 acres
6.5 acres
10,425,000
15,500,000
13,290,000
4.25 acres
62 acres
Bon Air
Royal Star
Runners
Canyon
2.25 acres
1.5 acres
14,500,000 6,995,000 5,460,000 6,000,000 6,250,000
Projections
Have:
• data with many, many dimensions
Want:
• to “see” the data
• i.e., we need some sort of spatial layout
What might we want in a layout?
• Objects close to one another should be more
similar than objects further away.
Multi-Dimensional Scaling
• One Dimension: dissimilarity (d)
𝑘
𝑎, 𝑏 ∈ 𝑅𝑘 ,
𝑑 𝑎, 𝑏 =
𝛿 𝑎𝑖 , 𝑏𝑖
𝑖=1
2
Multi-Dimensional Scaling
• One Dimension: dissimilarity (d)
𝑘
𝑎, 𝑏 ∈ 𝑅𝑘 ,
𝑑 𝑎, 𝑏 =
𝛿 𝑎𝑖 , 𝑏𝑖
𝑖=1
Numeric Values:
𝛿 𝑎, 𝑏 = 𝑎 − 𝑏
2
Multi-Dimensional Scaling
• One Dimension: dissimilarity (d)
𝑘
𝑎, 𝑏 ∈ 𝑅𝑘 ,
𝑑 𝑎, 𝑏 =
𝛿 𝑎𝑖 , 𝑏𝑖
𝑖=1
Numeric Values: 𝛿 𝑎, 𝑏 = 𝑎 − 𝑏
Non-numeric Values: ????
2
Multi-Dimensional Scaling
• One Dimension: dissimilarity (d)
𝑘
Property
Name
Location
Bedrooms
Baths
Lot Size
Asking
Price
𝑎, 𝑏 ∈ 𝑅𝑘 ,
Karang
Estate
Museum
Ready
𝑑 𝑎, 𝑏 =
𝛿 𝑎𝑖 , 𝑏𝑖
Valley
Trout Pond Meadow Lane Oceanfront Pell Mansion Estate
San
Santa Fe, Manhattan,
Southampton, Palm Beach, Francisco,
NM
NY
Hudson, NY
NY
FL
CA
Napa, CA
9
11
6
7
7
10
7
6
7
6
7
8
8
9
𝑖=1
Bon Air
0.75 acres
8,500,000 3,000,000
90.5 acres
3.5 acres
6.5 acres
10,425,000
15,500,000
13,290,000
4.25 acres
62 acres
Royal Star
Runners
Canyon
London, Monterrey, Monterrey,
England
CA
CA
8
5
5
7
6
6
Numeric Values: 𝛿 𝑎, 𝑏 = 𝑎 − 𝑏
Non-numeric Values: ????
12.5 acres
2
12.5 acres
2.25 acres
1.5 acres
14,500,000 6,995,000 5,460,000 6,000,000 6,250,000
Multi-Dimensional Scaling
• One Dimension: dissimilarity (d)
𝑘
Property
Name
Location
Bedrooms
Baths
Lot Size
Asking
Price
𝑎, 𝑏 ∈ 𝑅𝑘 ,
Karang
Estate
Museum
Ready
𝑑 𝑎, 𝑏 =
𝛿 𝑎𝑖 , 𝑏𝑖
Valley
Trout Pond Meadow Lane Oceanfront Pell Mansion Estate
San
Santa Fe, Manhattan,
Southampton, Palm Beach, Francisco,
NM
NY
Hudson, NY
NY
FL
CA
Napa, CA
9
11
6
7
7
10
7
6
7
6
7
8
8
9
𝑖=1
Bon Air
0.75 acres
8,500,000 3,000,000
90.5 acres
3.5 acres
6.5 acres
10,425,000
15,500,000
13,290,000
4.25 acres
62 acres
Royal Star
Runners
Canyon
London, Monterrey, Monterrey,
England
CA
CA
8
5
5
7
6
6
Numeric Values: 𝛿 𝑎, 𝑏 = 𝑎 − 𝑏
Non-numeric Values: ????
12.5 acres
2
12.5 acres
2.25 acres
1.5 acres
14,500,000 6,995,000 5,460,000 6,000,000 6,250,000
Note: We can also weight some dimensions more than others!
Big Matrix of Dissimilarity
D=
How to lay this out in 2 or 3D?
Multi-Dimensional Scaling
• Optimization Problem:
– Total dissimilarity:
𝑚
𝑖=1
𝑚
𝑗=1
𝑝𝑖 − 𝑝𝑗 𝑑𝑖,𝑗
Multi-Dimensional Scaling
• Optimization Problem:
– Total dissimilarity:
𝑚
𝑖=1
𝑚
𝑗=1
𝑝𝑖 − 𝑝𝑗 𝑑𝑖,𝑗
Euclidean distance
Multi-Dimensional Scaling
• Optimization Problem
– Total dissimilarity
𝑚
𝑖=1
𝑚
𝑗=1
𝑝𝑖 − 𝑝𝑗 𝑑𝑖,𝑗
Dissimilarity
Steerable, Progressive,
Multidimensional Scaling
Matt Williams & Tamara Munzner
UBC
InfoVis 2004
Steerable, Progressive MDS
Problem: cannot interactively explore highdimensional data sets
– Huge time cost for data sets with large #s of
dimensions and points
Solution: focus computational power on areas
of interest
Layout a random subset of the data set
Divide bin in two
Apply high-dimensional distance
A new random subset of points are added into the layout
Focus is placed on user defined bin
A new subset of random points selected from the unplaced points in the selected region
are added
The process is repeated as the user refines his selection
Data Cubes
Data: SEED and Forest Management
• Land use planning
• Assessing alternative
harvesting techniques
• Impact assessments
– e.g. habitat suitability
Initial State
Landscape Events
Harvesting
Succession
Fires
SELES
Output State
Landscape Events
Models of processes responsible
for landscape change
• human or natural
• continuous or periodic
Data:
• Spatial dimensions
• Temporal dimensions
• spatio-temporal relationships
• temporal complexity
x, y
z
• Attribute dimensions
• multiple static layers
• multiple dynamic layers
colour
Temporal Landscape Event Data
burnt
young douglas fir
mature douglas fir
mixed fir & hemlock
old growth hemlock
Slices
Using height to show subsequent states through time
Spatial shapes show differences in
event type
fire
harvesting
Opening the Spatio-Temporal
Block as a Book
Picking a moment
in the simulation
Opening the Spatio-Temporal
Block as a Book
Softcover: opening by compressing the front
Causes stretch in the current layer
Opening the Spatio-Temporal
Block as a Book
Hardcover: opening by bending the back
View two consecutive layers
Opening the Spatio-Temporal
Block as a Book
Hardcover: opening by
bending the back
less distortion in
current layers
Constraining Information
Multi-Dimensional Visualization
1. Venn Diagrams
– Many Boolean dimensions.
2. Projection / Multivariate Scaling
– Spatial layout for data with >3 dimensions.
3. Data Cubes
– Data in cubes (2D + time + colour).