Evolution Strategies for Optimizing Rectangular Cartograms
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1 Evolution Strategies for Optimizing Rectangular Cartograms Kevin Buchin 1, Bettina Speckmann 1, and Sander Verdonschot 2 1 TU Eindhoven, 2 Carleton University September 20, 2012 Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
2 Cartograms Map with administrative boundaries Visualizes a quantitative value per region (e.g. population, GDP) Regions are deformed to make their area proportional to this variable c Copyright 2006 SASI Group (University of Sheffield) and Mark Newman (University of Michigan) Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
3 Cartograms c 2008 M. E. J. Newman, Department of Physics and Center for the Study of Complex Systems, University of Michigan Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
4 Types of Cartograms Contiguous area cartogram Non-contiguous area cartogram Dorling / Demers cartogram Rectangular cartogram Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
5 Contiguous area cartogram Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
6 Non-contiguous area cartogram Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
7 Dorling / Demers cartogram c 2002 Ian Bortins and Steve Demers Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
8 Rectangular Cartogram Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
9 Rectangular Cartogram Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
10 Workflow Given a geometric map and a corresponding variable Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
11 Workflow Convert it to the correct format Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
12 Workflow Assign a direction to each adjacency Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
13 Workflow Assign a direction to each adjacency Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
14 Workflow Build the cartogram [Speckmann, van Kreveld, and Florisson, 2006] Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
15 Workflow Build the cartogram [Speckmann, van Kreveld, and Florisson, 2006] Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
16 Workflow Our focus is on the direction of the adjacencies Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
17 Regular Edge Labelings Colouring of interior edges of the dual graph Red: vertical adjacency, Blue: horizontal adjacency Must satisfy local constraints Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
18 Regular Edge Labelings Flip: invert colour of all edges inside alternating 4-cycle Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
19 Regular Edge Labelings Flip: invert colour of all edges inside alternating 4-cycle Flip graph: Take all labelings, add an edge if two are connected by a flip Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
20 Regular Edge Labelings Flip: invert colour of all edges inside alternating 4-cycle Flip graph: Take all labelings, add an edge if two are connected by a flip This graph is a distributive lattice [Fusy, 2009] Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
21 Challenges Maps of EU and US have over 100 million labelings The best labeling can vary between data sets A local change in the labeling can cause global changes in the cartogram Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
22 Evolution Strategy 1 Initialize a random population of labelings Start at the minimum labeling Follow a random path towards the maximal labeling Stop at a random height Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
23 Evolution Strategy 1 Initialize a random population of labelings Start at the minimum labeling Follow a random path towards the maximal labeling Stop at a random height 2 Copy a few of the best labelings to the next generation Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
24 Evolution Strategy 1 Initialize a random population of labelings Start at the minimum labeling Follow a random path towards the maximal labeling Stop at a random height 2 Copy a few of the best labelings to the next generation 3 Use selection and mutation to fill the rest of the population Select a random labeling, favouring higher quality With low probability, take a long random walk in the lattice With high probability, take one random step in the lattice Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
25 Evolution Strategy 1 Initialize a random population of labelings Start at the minimum labeling Follow a random path towards the maximal labeling Stop at a random height 2 Copy a few of the best labelings to the next generation 3 Use selection and mutation to fill the rest of the population Select a random labeling, favouring higher quality With low probability, take a long random walk in the lattice With high probability, take one random step in the lattice 4 Repeat steps 2 and 3 a fixed number of times, keeping track of the best labeling found Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
26 Quality Criteria Correct adjacencies Relative positions of the regions d d d d Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
27 Quality Criteria Correct adjacencies Relative positions of the regions Cartographic error of the regions A c A s A s A c : Area in cartogram A s : Area specified 50 (20) Error: (30) Error: 0.33 Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
28 Quality Criteria Correct adjacencies Relative positions of the regions Cartographic error of the regions A c A s A s A c : Area in cartogram A s : Area specified Error: 1.5 Error: 0.33 Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
29 Quality Criteria Correct adjacencies Relative positions of the regions (30 %) Cartographic error of the regions (70 %) A c A s A s A c : Area in cartogram A s : Area specified Error: 1.5 Error: 0.33 Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
30 Experimental Results Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
31 United States (63 regions) Made New Mexico adjacent to Utah to resolve 4-state point All US Census data sets with a positive value for each state Generating one cartogram took 8 minutes Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
32 United States - Population WA OR MT MN WI ID ND IN IL MI OH NY NJ ME VT NH MA CT RI WY SD IA NV UT NE CO KS MO KY WV PA VA DE MD CA OK AR TN NC AZ NM TX LA MS AL GA SC FL Average error: 0.5 % Maximum error: 2.2 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
33 United States - High School Graduates (%) WA OR ID MT ND MN ME NH VT CA NV UT WY SD NE IA WI MI OH PA MA AZ NM CO KS MO IL KY IN WV MD DE NY NJ CT RI OK AR TN VA NC TX LA MS AL GA SC FL Average error: 0.0 % Maximum error: 0.0 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
34 United States - Population per square mile ME WI MN MI VT NH ND WA MT ID IA SD OR WY NE NV IL UT COKS CA MO AZ IN OH WV MD PA DE NJ NY MA NM OK KY VA CT RI AR TN NC TX LA MS AL GA SC FL Average error: 2.0 % Maximum error: 11.3 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
35 United States - Population per square mile Negative correlation with land area Large variation ME VT NH WI MI MN ND WA MT ID IA SD OR WY NE NV IL UT COKS CA MO AZ IN OH WV MD PA DE NJ NY MA NM OK KY VA CT RI AR TN NC TX LA MS AL GA SC FL Average error: 2.0 % Maximum error: 11.3 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
36 United States - Non-Hispanic White Population (%) Negative correlation with land area Large variation ME WA MT ND MN VT NH MA ID WY SD IA WI MI PA OR NE OH NJ NY CT RI UT CO KS MO IL IN WV MD DE CA NV NM OK AR TN KY VA NC AZ TX LA MS AL GA SC Average error: 0.0 % Maximum error: 0.0 % FL Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
37 United States - Businesses w/o Paid Employees Negative correlation with land area Large variation WA OR ND ID MT MN WI MI ME VT NH SD WY NE NV UT CO IA KS MO IL IN OH PA NY MA CA OK KY WV AZ NM AR TN VA MD DE NJ CT RI NC TX SC LA MS GA AL FL Average error: 0.7 % Maximum error: 3.1 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
38 United States - Variation vs Maximum Error Negative correlation with land area Large variation maximum cartographic error (min) coefficient of variation Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
39 United States Presidential Elections ME WA OR ID ND MT MN WI MI DC NH VT CA SD WY NE NV UT CO KS OK AZ NM IA MO AR TN IL IN KY VA NC OH WV MD PA DE NJ NY MA CT RI TX LA MS AL GA SC AK HI FL Average error: 0.0 % Maximum error: 0.0 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
40 Europe (65 regions) Merged Luxembourg with Belgium, and Moldova with Ukraine All CIA World Fact Book ranked data sets with data for each country Generating one cartogram took 6 minutes Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
41 Europe - Population IS SE FI NO DK EE IE GB NL BE + LU DE SK PL LV LT BY UA + MD RU FR CZ HU MK RO BG PT ES CH IT AT RS KO SI ME GR BA HR AL MT TR CY Average error: 0.0 % Maximum error: 0.0 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
42 Europe - Account Balance IS NO IE GB DK SE FI EE LV EE LV LT LT BY RU NL DE PL CZ AT CH SK UA + MD HU RO SI SI HR RS BG BG MK MK BA MEKO AL BA ME ES IT GR TR PT BE BE + LU + LU FR MT MT CY Average error: 4.0 % Maximum error: 19.7 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
43 Europe - Exports IS IS NO GB NO SE FI IE BE + LU NL DK EE RU SE FI IE LV RU NL GB LT DK PL BY DE UA + MD DE EE LV FR LT BE + LU CZ SK RO BY HU ES PL CZ RS BG CH AT SK UA + MD HR MK HU RS RO CH AT KO SI AL ME SI HR ME KO BA BG AL MK FR BA TR PT ES PT GR IT TR MT GR IT MT CY CY Max error: 0.0 % BBSD: Max error: 1.6 % BBSD: Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
44 World (275 regions) All countries with population over 1 million Generating one cartogram took 3.5 hours Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
45 World - Population Average error: 1.17 % Maximum error: 18.5 % Sander Verdonschot (Carleton University) Optimizing Rectangular Cartograms September 20, / 33
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