1. NEW SOLUTION FOR CONSTRUCTION OF RECTILINEAR AREA CARTOGRAM
Chile
19/Nov/2009
NEW SOLUTION FOR CONSTRUCTION OF RECTILINEAR AREA CARTOGRAM
Ryo Inoue, Kazuki Kitaura, and Eihan Shimizu
Department of Civil Enineering University of Tokyo

2. What are (area) cartograms?
Cartograms are transformed maps on which areas of regions are proportional to statistical data values
Annual population growth rate(%)
2.0～
1.0～ 2.0
0.5～ 1.0
0.0～ 0.5
-0.5～ 0.0
-1.0～-0.5
-2.0～-1.0
～-2.0
Choropleth map
Population cartogram of Japan in 2005
Deformation of the shape of regions assist map-readers to recognize the distribution of data represented on cartograms

3. Classification of cartograms
Contiguity
Continuous
Non-Continuous
Shape
Complex
Simple
“Most-common” continuous cartogram
(Inoue & Shimizu, 2006)
Non-contiguous (Olson, 1976)
Rectilinear (New York Times, 2008)
Circle (Dorling, 2006)
Rectangular (Kreveld et al., 2007)
Rectangular
(Heilmann et al., 2004)

4. Continuous cartogram
Cartogram for population of municipalities shows the election results
Easy to recognize which party won the election
However,
cartograms having complex region shapes are difficult to compare the sizes of regions
July 31st 2007
Tokyo Shinbun (Newspaper)

5. Rectilinear & rectangular cartogram
One type of continuous cartograms, on which all corners of polygons are right angles
Rectangular cartogram of European population (Kreveld & Speckmann, 2007)
Rectilinear cartogram of Electoral vote in the 2008 Presidential Election
(New York Times, 2008)
Simple shape makes comparison of area easy!

6. Objectives of the study
Expression of data by rectilinear and rectangular cartograms are easy to understand the spatial distribution of data
However, no computational solution has been proposed for rectilinear cartogram construction, and not many for rectangular
In this study, we propose a new solution for rectilinear cartogram construction

7. What is rectilinear cartogram construction problem?
Simplified rectilinear shape of regions
Rectilinear cartogram
Shape transformation
Data to visualize
Polygon A：100 Polygon B：200
Impossible to get unique answer Only from the information on area
Cartogram construction is ill-posed problem
Regularization of problem is necessary！

8. Approach for regularization
Same problem also exists in continuous cartogram construction!
Impossible to fix polygons’ shape ONLY from their size!
same size, but different shape
Rectilinear cartogram construction problem is quite similar to continuous cartogram construction problem…
Both deals with contiguous and complex shape polygons
Then, we try to apply our previous method for continuous cartogram construction (Inoue & Shimizu, 2006) to rectilinear cartogram construction

9. Main concept of our previous method for continuous cartogram construction
While reading a cartogram,
map readers note the differences in the size and shape of regions through a comparison with those on a geographical map
If the map distortion is too large,
data interpretation becomes difficult.
Area cartograms must retain the regional shape.
Add a regularization term that minimizes the shape changes
Regularization of construction problem by imposing a restriction on the bearing changes of the edges

10. Previous method for continuous cartogram construction
To construct continuous cartograms on which shape deformation is small in a simple algorithm, we proposed
Triangulation of polygons
Formulation in non-linear least square problem
Restriction on the bearing changes of the triangle edges
Linearization around the geographical coordinates

11. Previous method for continuous cartogram construction
To construct continuous cartograms on which shape deformation is small in a simple algorithm, we proposed
Triangulation of polygons
Formulation in non-linear least square problem
Restriction on the bearing changes of the triangle edges
Linearization around the geographical coordinates
Triangulation of polygons
（Delaunay triangulation）

12. Previous method for continuous cartogram construction
To construct continuous cartograms on which shape deformation is small in a simple algorithm, we proposed
Triangulation of polygons
Formulation in non-linear least square problem
Restriction on the bearing changes of the triangle edges
Linearization around the geographical coordinates
Transform shape of triangles according to data
Shape on geographical map
Shape on cartogram
Easy area calculation leads to easy formulation

13. Previous method for continuous cartogram construction
To construct continuous cartograms on which shape deformation is small in a simple algorithm, we proposed
Triangulation of polygons
Formulation in non-linear least square problem
Restriction on the bearing changes of the triangle edges
Linearization around the geographical coordinates
T ：Set of triangles
Dijk ：Data given to triangle ijk
Aijk ：Area of triangle ijk
xi, yi: Coordinates of vertex i
Objective function Fit area of triangles to data given to triangles

14. Previous method for continuous cartogram construction
To construct continuous cartograms on which shape deformation is small in a simple algorithm, we proposed
Triangulation of polygons
Formulation in non-linear least square problem
Restriction on the bearing changes of the triangle edges
Linearization around the geographical coordinates
Bearing of edge mn on geographical map
Bearing of edge mn on cartogram
Regularization term
tijk：Triangle with vertex i, j, and k
emn：Edge between vertex m and n
E：Set of edges
μ：weight parameter for regularization term n n m m 14

15. Previous method for continuous cartogram construction
To construct continuous cartograms on which shape deformation is small in a simple algorithm, we proposed
Triangulation of polygons
Formulation in non-linear least square problem
Restriction on the bearing changes of the triangle edges
Linearization around the geographical coordinates
Solve linear least squares iteratively
(x'i, y'i): Approximate values of the vertices’ coordinates 15

16. Characteristics of rectilinear cartograms
All corners are right angle, then
→ All edges are horizontal or vertical edges
→ x-or y-coordinates of next vertices are equal 16

17. Characteristics of rectilinear cartograms
All corners are right angle, then
→ All edges are horizontal or vertical edges
→ x-or y-coordinates of next vertices are equal
(Number of vertices) × 2 − 2
Number of parameters
(Number of edges) − 2 17

18. Application of previous method
Output Shape
Triangulation
Transformation
Input Shape
Add constraint condition to construct rectilinear cartograms
EPx：Set of vertical edges
EPy：Set of horizontal edges 18

19. Confirmation of uniqueness of answer
Triangulation
Without
With
Number of unknown variables 4
Number of observation equation 1 7
>
Ill-posed problem
<
unique answer

20. Formulation of rectilinear cartogram construction
Proposed method for rectilinear cartogram construction
Rewrite equation using coordinates of vertex i (xi, yi)

21. Formulation of rectilinear cartogram construction
Linearlize the formula near the input coordinates of vertices i (xi', yi‘)
Iteration of the linear least squares problem outputs cartograms

22. Application of proposed method
Apply the method to construct a rectilinear cartogram which represents the total population of states of the United States in 2005
Input rectilinear shape of 48 continental states
Dummy polygons
210 vertices
84 vertical edges
76 horizontal edges 22

23. Total population of states in 2005
Input shape
Calculation time: less than 2 sec. (1.2GHz Pentium)
RMSE: 167 people (0.003% of avg. state pop.) Almost perfectly represented

24. Application for rectangular cartogram construction
Apply the method to construct a rectangular cartogram which represents the total population of states of the United States in 2005
Input rectangular shape of 48 continental states
109 vertices
30 vertical edges
31 horizontal edges 24

25. Problems in rectangular cartogram construction
Input rectangular polygons
RMSE: around 2 million (40% of avg. state pop.)
Fails to represent data on rectangular cartogram since the degree of freedom is too small !!

26. Conclusions
We proposed a new construction method for rectilinear area cartograms by applying our previous method for continuous cartogram construction
The proposed solution can construct rectilinear cartograms that represent data accurately
However, it cannot be used to construct accurate rectangular cartograms 26

27. Future works
A development of algorithm for designing input rectilinear shapes automatically is necessary to improve usability of our proposed construction method

28. Input polygon for rectangular cartogram
Future works
For rectangular area cartogram construction, another approach which is similar to that of non-continuous area cartogram is necessary
Input polygon for rectangular cartogram 28

30. Presidential election in 2008
■Barack Obama
■John McCain
2008米大統領選挙 州選挙人数
直角カルトグラム
cf. New York Times