JSS Journal of Statistical Software August 2018, Volume 86, Code Snippet 1. doi: 10.18637/jss.v086.c01

Rectangular Statistical Cartograms in R: The recmap Package

Christian Panse Swiss Federal Institute of Technology Zurich

Abstract Cartogram drawing is a technique for showing geography-related statistical informa- tion, such as demographic and epidemiological data. The idea is to distort a map by resizing its regions according to a statistical parameter by keeping the map recognizable. This article describes an R package implementing an algorithm called RecMap which approximates every map region by a rectangle where the area corresponds to the given statistical value (maintain zero cartographic error). The package implements the compu- tationally intensive tasks in C++. This paper’s contribution is that it demonstrates on real and synthetic maps how package recmap can be used, how it is implemented and how it is used with other statistical packages.

Keywords: cartogram, spatial data analysis, geovisualization, demographics, R.

1. Introduction

The idea of generating a cartogram is to distort a map by resizing its regions according to a given statistical parameter, but in a way that keeps the map recognizable. These so-called cartograms or value-by-area maps may be used to visualize any geo-spatial related data, e.g., political, economic, or public health data. There exist several algorithms to compute so-called contiguous cartograms. An overview on historical, hand-drawn, and computer generated cartograms can be found in Tobler(2004) and Nusrat and Kobourov(2016). For using contiguous cartograms, the diffusion-based method of Gastner and Newman(2004) is available through the R packages Rcartogram and getcartr (Temple Lang 2016; Brunsdon and Charlton 2014). An alternative approach to contiguous cartograms is to entirely relax the map topology by approximating each map region by basic geometric objects like rectangles or circles (Dorling 1996). Such rectangular cartograms can be generated from geolocation and statistical data. 2 recmap: Rectangular Statistical Cartograms in R

Maine

New Hampshire Vermont North Dakota Michigan Washington Minnesota Wisconsin New New Massachusetts Montana South Michigan Dakota Jersey Idaho Oregon Wyoming Nebraska Iowa York Rhode Connecticut Indiana Island Nevada Utah Colorado Kansas Illinois Pennsylvania Ohio Delaware Oklahoma Arkansas Kentucky California West Arizona Virginia Maryland D.C. New Mexico Louisiana Missouri Texas Virginia Tennessee North Carolina

Mississippi Georgia South Alabama Carolina

Florida

Figure 1: A rectangular statistical cartogram of the US election in 2004 is drawn. The area corresponds to the number of electors. Color is indicating the outcome of the vote. Democrats are represented by the color blue and Republicans are represented through red coloring. Regions with low saturation, e.g., Ohio, Pennsylvania, and Florida, highlight states with a tight outcome of the vote (also known as swing states). The election cartogram was computed by using the original implementation of the construction heuristic RecMap MP2 introduced by Heilmann et al. (2004). Map source: US Census Bureau; election data source: http://www.electoral-vote.com/, November 2004.

Hence, they provide a useful alternative, even if there are no boundaries available or some statistical values are missing. First rectangular cartograms were drawn by hand following a system of construction (Raisz 1934). Recent research publications on rectangular cartogram drawing include Van Kreveld and Speckmann(2004, 2007); Buchin, Speckmann, and Ver- donschot(2012) and Buchin, Eppstein, Löffler, Nöllenburg, and Silveira(2016). However, according to a recent publication, both variants of RecMap (Heilmann, Keim, Panse, and Sips 2004) are the only rectangular cartogram algorithms that “maintain zero cartographic error” (Nusrat and Kobourov 2016, section 5.4). The R (R Core Team 2018) package recmap (Panse 2018) discussed in this article contains an implementation of the RecMap (Map Partition variant 2) algorithm (Heilmann et al. 2004) to draw maps according to given statistical parameter. A typical usage of a cartogram based vi- sualization is demonstrated in Figure1. Package recmap is available from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=recmap. The article is organized as follows: The next section defines the input and output of the RecMap algorithm and the objective functions. In Section3, the usage of the recmap pack- age using the R shell is demonstrated. Section4 discusses major implementation details and provides some benchmark and performance studies. Section5 describes how two metaheuris- tics can be used to find an optimized cartogram drawing. In Section6, some applications are presented. Section7 summarizes and discuss the approach. Journal of Statistical Software – Code Snippets 3

2. Problem definition and objective functions

The input consists of a map represented by overlapping rectangles R = (r1, . . . , rn). Each map region rj contains:

• a tuple of (x, y) values corresponding to the (longitude, latitude) position,

• a tuple of (dx, dy) of expansion along (longitude, latitude),

• and a statistical value z.

The (x, y) coordinates represent the center of the minimal bounding boxes (MBBs). The coordinates of the MBB are derived by adding or subtracting the (dx, dy) tuple accordingly. A tuple (dx, dy) also defines the ratio of the corresponding map region. The statistical values (z1, . . . , zn) determine the desired area of each map region. The ordering Π is an index vector taken from the permutation set Perm(n).

The output is a rectangular cartogram R where the map regions are:

• non-overlapping,

• connected,

• rectangles are placed parallel to the axes.

Furthermore, for each map region the following criteria have to be satisfied:

• the area is equal to the desired area derived from the as input given statistical value z,

• the ratio, dy/dx, is preserved.

The recmap construction heuristic is a function

n×2 n×3 n×2 n×2 f : R × R>0 × Perm(n) → R × R>0 , (1)

which transforms the set of input rectangles R and a permutation Π into a rectangular cartogram

R = f(R, Π), (2) so that important spatial constraints, in particular

• the topology of the dual graph G(R,E), defined by the overlapping input rectangles,

• the relative position of map region centers, 4 recmap: Rectangular Statistical Cartograms in R are tried to be preserved. If the output satisfies these criteria, the rectangular cartogram is denoted as a feasible solution. The following equations were introduced by Keim, North, and Panse(2004, Definition 1). The desired area A˜j of a map region rj is defined as Pn A r ˜ i=1 ( i) Aj = zj · Pn , (3) i=1 zi where the area of the rectangle r is defined by A(r) = 4 · dx · dy. (4)

The objective functions for area dA, shape dS (ratios of the MBBs), relative position dR, and map topology dT , are as defined and described by Heilmann et al. (2004, Equations 2–4):

dA = dA(R, R) (5) n X = |Aj − A˜j| (6) j=1

dS = dS(R, R) (7) n X = |(dyj/dxj) − (dyj/dxj)| (8) j=1

dT = dT (R, R) (9)

|Ea\Ea| + |Ea\Ea| = , (10) |Ea ∪ Ea|

dR = dR(R, R) (11) n−1 n 2 X X = · | (r , r ) − (r ˜ , r˜ )| (12) n · n − ] i j ] i j ( 1) i=1 j=i+1 | {z } | {z } ∈R ∈R

Note, if ∀j ∈ {1, . . . , n} : Aj = A˜j ⇒ dA = 0 and if ∀j ∈ {1, . . . , n} : dyj/dxj = dyj/dxj ⇒ dS = 0. n = |R| and E denotes the edges of the dual graph. dT determines the differences in the pseudo dual graphs. dR measures the angle differences between all pairs of rectangle 1 centers of the input map and output cartogram by using ](x, y) defined as follows :

](x, y) = arctan2(x, y) (13) To find an optimal rectangular statistical cartogram out of all feasible solutions, as it will be intended in this manuscript using recmap, the optimization problem can be expressed as follows:

minimize a · dR + b · dT with a, b ∈ R≥0 (14) Π∈Perm(n)

subject to dA = 0, dS = 0. (15) To find a sufficiently good solution for the optimization problem a metaheuristic, as described in Section5, will be applied. 1Implemented using the C++ method std::atan2 – “Computes the arc tangent of y/x using the signs of arguments to determine the correct quadrant.” http://en.cppreference.com/w/cpp/numeric/math/atan2, access: 2017-01-01. Journal of Statistical Software – Code Snippets 5

Term Description Equation R = (r1, . . . , rn) overlapping input rectangles (x, y) coordinates represent the center of a rectangle (dx, dy) expansion along x and y axes z statistical value of a rectangle G(R,E) dual graph of R Π permutation A˜j desired area of a map region j 3 A(r) area of a rectangle4 dA area error5 dS shape error8 dT topology error 10 dR relative position error 12 2 ](x, y) angle between two points x and y in R 13 f construction heuristic1 R output / cartogram2

Table 1: The table provides a glossary of the used algebraic terms.

3. The package usage

3.1. Input The US map on the state level is often used to compare cartogram algorithms. To generate a useful dual graph, which is a requirement of the algorithm, the input map regions have to overlap. For the state.x77 data used in this section this can be done by correcting lines of longitude derived from the square roots of the area values (see Figure2 left).

R> R.version.string

[1] "R version 3.5.1 (2018-07-02)"

R> packageVersion("recmap")

[1] '0.5.37'

R> US.map <- data.frame(x = state.center$x,y = state.center$y, + dx = sqrt(state.area) / 2 / (0.7 * 60 * cos(state.center$y * + pi / 180)), dy = sqrt(state.area) / 2 / (0.7 * 60), + z = sqrt(state.area), name = state.name) R> head(US.map)

x y dx dy z name 1 -86.7509 32.5901 3.209890 2.704478 227.1761 Alabama 2 -127.2500 49.2500 14.005670 9.142338 767.9564 Alaska 3 -111.6250 34.2192 4.859044 4.017906 337.5041 Arizona 6 recmap: Rectangular Statistical Cartograms in R

4 -92.2992 34.7336 3.338204 2.743370 230.4431 Arkansas 5 -119.7730 36.5341 5.902177 4.742415 398.3629 California 6 -105.5130 38.6777 4.923602 3.843727 322.8730 Colorado

Please note, in general, recmap is not transforming the geodetic datum, e.g., WGS84 or Swissgrid. Furthermore, geospatial positions have to be mapped from the earth surface to the plane. This transformation has to be done prior the cartogram generation. An overview of map projection can be found in Snyder(1997). Map projections aim to optimize towards different objectives, e.g., conformal mapping (preservation of local angles) or area mapping (preservation of area). In cartogram publications, the map projection aspect is often ne- glected, and conformal errors are accepted. However, studying US cartograms, a cylindrical projection seems to be the projection of choice. The R user can find support for a wide va- riety of adequate map projections due to using the mapproj package by McIlroy, Brownrigg, Minka, and Bivand(2015).

3.2. Run The algorithm takes a data.frame object having the column names c("x", "y", "dx", "dy", "z", "name"), here the US.map object, as input. Additional control is given by the index order and the (dx, dy) values. The index order Π defines in which order the dual graph is explored and the (dx, dy) values define which map regions are adjacent.

R> library("recmap") R> US.cartogram <- recmap(US.map)

3.3. Output The recmap method returns an S3 object of class c("recmap", "data.frame") of the trans- formed map. Additional columns in the result contain information for topology and relative position error defined in Equations 10 and 12. The column dfn.num indicates in which order the dual graph has been explored.

R> head(US.cartogram)

x y dx dy z name dfs.num 1 -79.27226 10.40598 3.916894 3.300161 227.1761 Alabama 23 2 -129.22283 63.71115 8.181797 5.340748 767.9564 Alaska 49 3 -126.86923 30.51915 4.819169 3.984933 337.5041 Arizona 34 4 -87.19357 10.42302 3.994415 3.282650 230.4431 Arkansas 39 5 -137.00972 33.40013 5.311325 4.267664 398.3629 California 35 6 -115.35010 62.23315 4.851078 3.787109 322.8730 Colorado 48 topology.error relpos.error relposnh.error 1 6 0.3893223 0.7694393 2 6 0.2883036 0.5287275 3 3 0.2074725 0.2355095 4 8 0.5599091 1.0584316 5 4 0.1761236 0.1878248 6 10 0.6831601 1.1107418 Journal of Statistical Software – Code Snippets 7

North Dakota

South Dakota Alaska Colorado Iowa

Nebraska Nevada Missouri Illinois Indiana Ohio

AlaskaWashington North Dakota Washington Montana Minnesota Montana Minnesota Kentucky Maine Pennsylvania

South Dakota Wisconsin Vermont North Carolina Oregon New Hampshire Idaho Michigan New York Maine Wyoming Vermont Massachusetts Oregon Wisconsin Iowa Connecticut Nebraska Pennsylvania Wyoming Michigan New York Idaho New Hampshire Maryland IllinoisIndiana Ohio New Jersey Utah Rhode Island Maryland

Delaware Nevada Utah Colorado West Virginia Kansas Kansas Missouri Massachusetts Kentucky Virginia New Jersey California West Virginia California Oklahoma Tennessee Virginia Oklahoma North Carolina Arizona New Mexico Delaware Connecticut Arkansas New Mexico Hawaii Tennessee Arizona South Carolina Mississippi Alabama Georgia Hawaii Texas South Carolina Texas Louisiana Louisiana Georgia

Florida

Mississippi Florida

Arkansas Alabama

Figure 2: The “usage” example, generated from the “US State Facts and Figures” data in the datasets package, was used for drawing a rectangular map approximation. The input set of overlapping rectangles is shown left. A feasible solution thereof generated with recmap is on the right side. The state area, original size of the map region, is used as statistical value.

The output of the recmap function can be visualized using the S3 plot method for ‘recmap’ objects. The output of the R code snippet can be seen in Figure2. As default the plot method for ‘recmap’ objects places the name attribute in the center of each rectangle. The text is scaled by using cex = dx / strwidth(name) as an argument for the text function to avoid overplotting of the labels. However small statistical values result in small label areas. This problem can be circumvented by using an interactive visualization, e.g., using shiny (Chang, Cheng, Allaire, Xie, and McPherson 2016) the function hoverOpts enables the “mouseover” feature. Please note that while the input US.map is not classified as an object of class ‘recmap’, the plot method function for ‘recmap’ objects cannot be used through S3 method dispatch and function plot.recmap has to be explicitly called.

R> op <- par(mfrow = c(1, 2), mar = c(0, 0, 0, 0)) R> plot.recmap(US.map, col.text = "darkred") R> plot(US.cartogram, col.text = "darkred")

A summary method implements the calculation of some metadata including the objective functions.

R> summary(US.cartogram)

values number of map regions 50.000000 area error 0.000000 topology error 266.000000 relative position error 0.420000 screen filling [in %] 36.893585 xmin -146.967409 xmax -34.722706 ymin 7.105818 ymax 73.190904 8 recmap: Rectangular Statistical Cartograms in R

The S3 methods, as.recmap and as.SpatialPolygonsDataFrame, enable the exchange and the manipulation of geographic data contained in the classes of the sp package (Pebesma and Bivand 2005; Bivand, Pebesma, and Gómez-Rubio 2013). The following code displays how a ‘recmap’ object can be translated into a ‘SpatialPolygonsDataFrame’ object and back.

R> X <- checkerboard(8) R> all.equal(X, as.recmap(as.SpatialPolygonsDataFrame(X)))

[1] TRUE

4. Implementation

The RecMap algorithm is implemented in C++ using features provided by the C++-11 stan- dard. The input and output data transfer between R and the C++ ‘recmap’ class is handled by using the Rcpp (Eddelbuettel and François 2011) mechanism. The C++ ‘recmap’ class itself consists of a std::vector of map_region.A map_region contains all the (x, y, dx, dy, z) values, a std::vector of type int linking to its neighbor map regions, and some additional help variables to ease the error computation. In general, the construction algorithm follows the map partition 2 (MP2) procedure described in Heilmann et al. (2004). The local place- ment function can place any rectangle next to another rectangle as demonstrated in Figure3. The current implementation starts with the original bearing α of the two map region centers (see Figure3 where β = 0). If the placement does not lead to a feasible solution, the angle β π is added to α. β is iterating between [0, π] with a step size of 180 and a changing sign until a placement without overlap has been found. If no placement can be found, the algorithm considers all adjacent placed map regions. If also in a later step during the depth-first search (DFS) a map region cannot be placed, a non-feasible solution is accepted. This situation is often caused because the construction algorithm is hampered by the input configuration of the map regions. Solving this can be very compute-expensive and often the procedure leads to a solution which will be rejected by the metaheuristic due to the low fitness value. Furthermore, no genetic algorithm (metaheuristic) has been implemented. Here recmap will use the GA package (Scrucca 2013) available on CRAN as demonstrated in Section5. The most computationally expensive part is the computation of MBB intersections which has to be performed to achieve feasible solutions, multiple times, for each placement step. In the package version 0.2.1 these tests were performed by iterating over each map region. All later versions use a std::multiset data structure and a std::lower_bound algorithm of the C++ standard template library (STL) to reduce the search space. The time complexity for one recmap run is O(n2), where n is the number of regions. A DFS run is visiting each map region only once, and therefore it has time complexity O(n). For each map_region placement, a constant number of MBB intersection are called (max. 360). The MBB check is implemented using a std::multiset container, and the functions std::insert, std::upper_bound, and std::upper_bound. The time complexity for all of these functions is reported on http://www.cplusplus.com/reference/stl as O(log(n)). However, the worst case scenario for a range query is O(n), if and only if dx or dy cover the whole x or y range. The boxplots in Figure4 (left plot) show that the number of MBB intersection test calls could be reduced by using a std::multiset data structure. For this benchmark, synthetic Journal of Statistical Software – Code Snippets 9

6 2 β=0 β=−0.14 β=0.27 fpl : R × [− π, π] → R

B B

A A A B

β=−0.41 β=0.55 β=−0.69

A B B

A A A

y return value B

α argument −3.14 −2.36 β=0.82 β=−0.96 β=1.1 −1.57 −0.79 0 0.79 1.57 2.36 B 3.14 A A A

B B

x return value

Figure 3: The rainbow colored rectangles graph the feasible positions of a local placement function fpl to place a rectangle B around a given rectangle A of any angle α between [−π, π] (left figure). There are special cases for quadrant I, II, III, and IV indicated by different colors π in the graphic. On the right figure, B should be placed around A starting with an angle of 4 (this is the initial relative position in the input map). Since the rectangle cannot be placed without overlap (indicated by red), an angle β is added.

checkerboards with map regions in the interval of 22,..., 202 were generated using the R function checkerboard. For each checkerboard size recmap was called 100 times using different index orderings of the checkerboard input. The index order of the input records has a direct impact how the DFS is traversing the map. This characteristic will later be used for the metaheuristic. The benchmark was performed on an Apple MacBookPro Laptop having a 2.9 GHz Intel Core i7 CPU from 2017 using only one core. The experiments were performed on MacOSX 10.13.4, Linux Debian 9 running on a 4.9.06-amd64 kernel and a Windows 10 platform. The Linux and Windows OS instances were running as virtual machines using VirtualBox. The middle plot in Figure4 displays the resulting mean aggregated measured data of the three (hardware/OS) systems using the two different implementations of the MBB intersection test. Besides the fact that the Linux system cannot benefit from the more efficient implementation of the MBB intersection test, the plot in the middle shows that even for an input size of 202 = 400 map regions a rectangular cartogram can be computed in less than a second. The right plot in Figure4 was derived from the performance study (plot in the middle). It shows the number of rectangular cartograms which can be generated within one second depending on the number of map regions. The gray vertical line indicates the number of the US map regions. The ability to generate a high number of cartogram candidates in a short time period is an important requirement for any metaheuristic as is demonstrated in the next section. 10 recmap: Rectangular Statistical Cartograms in R

A ● 1e+00 A WL C++ STL multiset / MacOSX 10.13.4 / 2.9 GHz Intel Core i7 WL L A L 1e+07 WA WL A L L W L A WAL W L A ● WAL ● ● W L WA W ● ● ● A ● ● L ● ● WA ● ● ● ● ● AL 5000 ● ● ● WL A ● ● W ● ● ● L A WAL W ● ● ● ● 1e−01 ● ● ● ● ● ● A WAL W ● ● ● ● ● L ● ● AL W ● ● W A ● ● L ● ● L W ●

1e+05 A ● L WA ● ● ● ● L W ● A ● WA 500 ● L W ● A ● 1e−02 L L WA W ● ● ● A ● W ● L L ● ● A ● W ● ● ● A ● ● ● ● L W ● ● W ● ● A ●

50 ● ● A ● ● L ● ● W ●

1e+03 ● ● ● ● ● 1e−03 ● ● ● ● ● ● WA A ● ●

number of MBB intersection calls number ● ● ● ● ● ● ● ● ● ● L A ● 10 ● ● L A ● ● ● ● C++ STL list ● ●

● median aggregated process time [in seconds] ● ● ●

A C++ STL multiset 5 −A− MacOSX 10.13.4 / 2.9 GHz Intel Core i7

● of rectangular cartogramsnumber per second generated

● 1e−04 C++ STL list A −L− Debian deb9.3 / 2.9 GHz Intel Core i7 (VM) ● A

1e+01 C++ STL multiset −W− Windows 10 / 2.9 GHz Intel Core i7 (VM)

5x5 US 10x10 15x15 20x20 5x5 US 10x10 15x15 20x20 5x5 US 10x10 15x15 20x20

number of map regions number of map regions number of map regions

Figure 4: The boxplot (left) graphs the number of MBB intersection test calls for a given checkerboard size. The graph in the middle displays the mean aggregated measured com- putation time. The scatterplot (right) displays how many rectangular cartograms can be generated within one second based on the board size. All y-axes are on logarithmic scale.

5. Choose a metaheuristic

The design of the RecMap algorithm is as follows: First, compute a set of feasible solutions. In a second evaluation step, choose the best solution. In the current implementation, variations can be introduced by changing the index order Π of the input data. This order has a direct impact on the DFS traversal and leads to different cartogram layouts. The objective functions, Equations5 to 12, can be used to evaluate the result and to define a fitness function which has to be maximized. The following R code defines the fitness function which will be used as default.

R> recmap:::.recmap.fitness

function (idxOrder, Map, ...) { Cartogram <- recmap(Map[idxOrder, ]) if (sum(Cartogram$topology.error == 100) > 0) { return(0) } 1/sum(Cartogram$relpos.error) }

Other variants of fitness functions will lead to different results as shown in Heilmann et al. (2004, Figure 4). Since it is not possible to compute and evaluate all permutations, which is n!, random exper- iments are conducted. In the following section, it is demonstrated how two metaheuristics, GRASP and GA, can be used to find an optimal layout for the rectangular statistical car- togram. For a visual evaluation of the metaheuristics, proposed in this section, an 8 × 8 checkerboard will be used as input map. The map is generated using the checkerboard method. Journal of Statistical Software – Code Snippets 11

R> Checkerboard <- checkerboard(8) R> summary(Checkerboard)

values number of map regions 64.00 area error 0.27 topology error NA relative position error NA screen filling [in %] 100.00 xmin 0.50 xmax 8.50 ymin 0.50 ymax 8.50 and can be seen in Figure5 (left). If the assumption is made, that for each vertex, the cyclic order of edges in the contiguous cartogram remains the same as in the input map, checkerboards provide examples of sets of map regions which do not have ideal cartogram solutions (Keim et al. 2004, Definition 2, Lemma 1, Figure 3).

5.1. Greedy randomized adaptive search procedures One group of optimizers that is “trivial to efficiently implement” (Feo and Resende 1995) and can directly benefit from a parallel environment is called greedy randomized adaptive search procedures (GRASP). The R method recmapGRASP defines a generic GRASP implementation as described in Feo and Resende(1995, Figure 1). The recmapGRASP function generates a set of rectangular cartograms which have different layouts caused by the random sampling. Each cartogram is evaluated. The best candidate is saved. The following command will generate a cartogram solution based on a GRASP metaheuristic.

R> set.seed(1) R> res.GRASP <- recmapGRASP(Checkerboard) R> plot(res.GRASP$Cartogram, + col = c("white", "white", "white", "black")[res.GRASP$Cartogram$z])

A drawing of the cartogram can be found in Figure5 (middle). For some types of input maps, GRASP can generate amazing results in a short time. As it can be seen in Figure5, for the checkerboard, GRASP is outperformed by the genetic algorithm, introduced in the next section. The plot in Figure6 (right) shows that the solution process runs too fast into saturation.

5.2. Lessons learned from biological evolution In this paragraph, a constraint-based genetic algorithm (GA) as discussed in Heilmann et al. (2004) is used as metaheuristic. Here the construction heuristic benefits from the existence of the GA package by Scrucca(2013). The GA configuration used for recmap was directly derived from the traveling salesperson problem (TSP) example (Scrucca 2013, Section 4.8) using the permutation type of the ga method. As genotype the index order Π of the input 12 recmap: Rectangular Statistical Cartograms in R map is used. The following command generates an almost perfect rectangular cartogram for the checkerboard map having 64 map regions on the author’s laptop (MacBook Pro from 2017, 2.9 GHz Intel Core i7) within 60 seconds. R> recmap.GA <- ga(type = "permutation", fitness = recmap:::.recmap.fitness, + Map = Checkerboard, min = 1, max = nrow(Checkerboard), popSize = 50, + maxiter = 300, maxFitness = 1.7, maxiter = 300, pmutation = 0.25)

The metaheuristic stops when a maximum number of iteration has been performed or the fit- ness value is higher than 1.7. Having reached a fitness value of 0.342 using the recmap.fitness function the result in Figure5 (right) looks like an almost “optimal” solution. The recmapGA function is a higher level wrapper function to glue the recmap construction heuristic with the metaheuristic ga. R> res.GA <- recmapGA(Checkerboard, popSize = 50, run = 300, maxiter = 300, + seed = 3) R> summary(res.GA$Cartogram)

values number of map regions 64.0000000 area error 0.0000000 topology error 294.0000000 relative position error 0.0500000 screen filling [in %] 65.7237172 xmin 0.4088612 xmax 9.9656942 ymin -0.7320998 ymax 9.4571887

R> plot(res.GA$Cartogram, + col = c("white", "white", "white", "black")[res.GA$Cartogram$z])

The recmapGA function returns a list of the input Map, the solution of the GA, and a ‘recmap’ object containing the cartogram. The resulting cartograms using a GA can be seen in Figure5 (right). The red line in Figure6 (left) indicates in which order the rectangles were placed using the DFS numbering. The red • symbol marks the first placed rectangle and the  the last one. Figure7 illustrates the variability in solutions, dependent on the initial seed value. The experiment was repeated twice to demonstrate the effect that the same seed values lead to the same permutation order Π and finally to the same cartogram construction. As shown above, the results of the recmap implementation are reproducible on the same platform. However, due to the numerical ill-condition of the problem special care has to be taken for the computation of the fitness value. Small differences in the fitness values on different platforms cause error propagation through iterations of the metaheuristic and derive a different solution sequence Π and cartogram drawing. The rectangular map approximations in Figure8 demonstrate the continuous improvement of the feasible solutions with an increasing number of generations using the GA as metaheuristic for the data used in Section3. The code below defines a weighted fitness function. Journal of Statistical Software – Code Snippets 13

a8 b8 d8 e8 g8 h8 b8 d8 e8 g8 h8 a8 b8 c8 d8 g8 h8 c8 f8 a8 c8 f8 e8 f8

b7 d7 a7 b7 d7 e7 g7 h7 a7 c7 e7 f7 g7 h7 a7 b7 c7 e7 f7 g7 h7 c7 f7 d7

c6 a6 b6 d6 e6 g6 h6 a6 b6 c6 d6 e6 g6 h6 a6 b6 d6 e6 h6 c6 f6 f6 f6 g6

b5 d5 a5 b5 c5 d5 e5 g5 h5 a5 c5 e5 f5 g5 a5 b5 c5 d5 e5 f5 g5 f5 h5 h5

a4 b4 d4 e4 g4 h4 a4 b4 c4 d4 e4 h4 a4 b4 d4 e4 g4 h4 c4 f4 f4 g4 c4 f4

a3 b3 c3 d3 e3 g3 h3 b3 c3 d3 e3 f3 g3 h3 a3 b3 c3 d3 e3 g3 f3 f3 h3 a3 g2 a2 a2 b2 d2 e2 g2 h2 b2 c2 d2 e2 h2 b2 c2 d2 g2 h2 c2 f2 a2 f2 e2 f2 h1 b1 f1 a1 b1 c1 d1 e1 f1 g1 h1 a1 b1 c1 d1 e1 f1 g1 a1 c1 d1 e1 g1 h1

Figure 5: A comparison of the input map, GRASP, and GA is displayed. As input map, a 8 × 8 checkerboard (left) has been generated. The area of a black box needs to be four times as large as the area of a white box. The cartogram in the middle proposes a solution generated by a GRASP metaheuristic. The right cartogram graphs a solution computed by a genetic algorithm within one minute.

a8 c8 g8 ●●●●●●●●●●● b8 d8 h8 0.35 e8 f8 ●●●●●●●●●● 47 46 45 44 5 42 40 ●●● 43 ● d7 ●●● a7 b7 c7 53 e7 f7 g7 h7 ● 48 49 55 4 50 6 7 ●●●●●●●● 0.30 ●●●● c6 ● a6 b6 58 d6 e6 h6 ●●●●●●●●● 63 62 3 54 f651 g6 8 ●●●●●●●●● start 41 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

● 0.25 a5 b5 c5 d5 e5 f5 g5 ●●● 60 1 2 56 52 37 9 h5 39 a4 b4 d4 e4 f4 g4 h4 c4 best fitness value

64 61 57 35 10 38 36 0.20 59 ●●●●●●●●●● ●●● a3 b3 c3 d3 e3 g3 ●● 24 26 27 31 11 f3 34 h3 33 16 ● a2 Evolutionary based Genetic Algorithm (GA) 0.15 ● ● Greedy Randomized Adaptive Search Procedures (GRASP) 25 b2 c2 d2 g2 h2 22 29 32 e2 f212 17 15 30 b1 f1 0 20 40 60 80 a1 28 c1 d1 e1 18 g1 h1 23 21 20 19 13 14 elapsed time [in seconds]

Figure 6: On the checkerboard cartogram drawn on the left, the red lines trace the order of each rectangle placement step of the construction algorithm. The right scatterplot graphs the best fitness value computed by recmap.fitness versus the elapsed time of the two meta- heuristics. Each • represents an iteration. The GRASP reaches the plateau after a few iterations.

R> fitness.weighted <- function (idxOrder, Map, ...) { + Cartogram <- recmap(Map[idxOrder, ]) + if (sum(Cartogram$topology.error == 100) > 0) { + return(0) + } + 14 recmap: Rectangular Statistical Cartograms in R

a4 b4 c4 d4 a4 b4 c4 d4 b4 c4 d4 a4 b4 c4 d4 a4 d3 a3 b3 c3 b3 b3 a3 c3 d3 a3 b3 c3 d3 a3 c3 d2 d3 a2 c2 a2 c2 a2 c2 a2 c2 a1 b1 c1 d1 b2 d2 b2 d2 b2 d2

b1 d1 b1 d1 b2 a1 c1 a1 c1 d1 a1 b1 c1

Πforward = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16) Π1 = (2, 14, 10, 4, 12, 8, 3, 5, 6, 16, 7, 9, 1, 13, 11, 15) Π2 = (13, 3, 8, 7, 1, 6, 9, 2, 15, 4, 10, 14, 12, 16, 11, 5) Π3 = (5, 15, 7, 12, 13, 9, 14, 16, 2, 8, 1, 10, 11, 3, 4, 6)

c3 a4 b4 c4 d4 b4 c4 d4 a4 b4 c4 d4 a4

a4 b4 c4 d4 b3 b3 a3 c3 d3 a3 b3 c3 d3 a3 c3 d3 b3 d3 a3 a2 c2 a2 c2 a2 c2 b2 c2 d2 b2 d2 b2 d2 b2 d2 a2 b1 d1 b1 d1 a1 b1 c1 d1 a1 c1 a1 c1 d1 a1 b1 c1

Πreverse = (16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1) Π1 = (2, 14, 10, 4, 12, 8, 3, 5, 6, 16, 7, 9, 1, 13, 11, 15) Π2 = (13, 3, 8, 7, 1, 6, 9, 2, 15, 4, 10, 14, 12, 16, 11, 5) Π3 = (5, 15, 7, 12, 13, 9, 14, 16, 2, 8, 1, 10, 11, 3, 4, 6)

Figure 7: This graphical example illustrates the variability in solutions, dependent on the initial seed value in Πseed. The left column displays forward- and reverse index orders. All other columns show the results from duplicate seeds {1, 2, 3} to demonstrate that the same seed will lead to the same index order and the identical layout of the cartogram.

+ S <- summary(Cartogram) + dT <- max(Cartogram$topology.error) + dR <- S[4, ] + dE <- (100 - S[5, ]) / 100 + + 1 / (c(0.2, 0.6, 0.2) %*% c(dT, dR, dE)) + } R> set.seed(2) R> US.map.best <- recmapGA(Map = US.map, fitness = fitness.weighted, + maxiter = 100, maxFitness = 100, popSize = 50, keepBest = TRUE, + pmutation = 0.35, parallel = TRUE)

Note that the metaheuristic of the space filling quad tree (Finkel and Bentley 1974) based RecMap MP1 variant could also be realized by using the GA package. Here, instead of a permutation, a binary representation of decision variables has to be chosen. This can be done by setting the type attribute of the ga function to "binary". The genotype, given as a binary vector, is defining the split type of the quad tree data structure. A 1 is applying a vertical split, while a 0 triggers a horizontal split.

6. Application

This section applies package recmap to some real world maps having numbers of map regions of different magnitudes and different kind of topology. Beside Figures1 and9 the examples fo- cus more on the demonstration of the drawing characteristics of the recmap method itself and Journal of Statistical Software – Code Snippets 15

●●●●●●●●●●●●●●●●●●●●●●● 50 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●● 40 0.48

●●●●● ● 30

● ● 20 0.44

● order Index Fitness value Best ●● ●● ●● ● ● ● ●●● ● Mean 10 ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●● ● ● ●●● ● ● ● ●● ●● ● ●● ● ●● ● ●● ●● ● ● ● ● ● ● ●● ●● ● Median● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 0.40

0 20 40 60 80 100 20 40 60 80 100

Generation Generation

South Dakota Montana North Dakota Wisconsin Maine Vermont Idaho Iowa Montana Pennsylvania Nebraska South Dakota Illinois Indiana Ohio Massachusetts Alaska Colorado Michigan Vermont Maine Alaska Nevada Washington Kentucky Idaho North Dakota Pennsylvania Washington Minnesota New York Minnesota Massachusetts New Hampshire Wyoming Wisconsin Wyoming Mississippi Utah Maryland Oregon Connecticut Rhode Island Nevada Oregon Nebraska Michigan Iowa Connecticut Rhode Island New Jersey

New Jersey Colorado Illinois Ohio West Virginia Missouri Utah West Virginia Delaware

California Delaware Missouri New Mexico Arizona New Mexico California Kentucky North Carolina Virginia Kansas Alabama Hawaii North Carolina Arizona New Hampshire Hawaii New York Kansas Arkansas South Carolina Oklahoma South Carolina Alabama Georgia Oklahoma Arkansas Florida Maryland

Louisiana

Tennessee Virginia Texas Tennessee Indiana Florida Generation 1 Texas Generation 6 Louisiana Mississippi Georgia dT = 10 dT = 9 dR = 0.25 dR = 0.39 dE = 43 dE = 39

Minnesota

Wisconsin Montana

South Dakota Washington Iowa Alaska North Dakota North Dakota Vermont Massachusetts Maine Washington Alaska Idaho Wyoming New York Idaho Indiana Wyoming New Hampshire Minnesota Michigan Kansas Nebraska Iowa New Hampshire Nevada Maryland Utah Colorado Nebraska Oregon Connecticut Rhode Island Nevada Wisconsin Maine Michigan Vermont Missouri Maryland New Jersey Utah Colorado West Virginia Montana New York Indiana Massachusetts Kansas Oregon South Dakota Illinois Virginia Ohio Pennsylvania California Connecticut Rhode Island Alabama Missouri Tennessee New Jersey Arizona Kentucky West Virginia Hawaii Louisiana California Virginia Oklahoma Delaware Texas Arizona New Mexico Tennessee Florida Arkansas Hawaii Pennsylvania

South Carolina North Carolina New Mexico Louisiana Ohio Mississippi Georgia Delaware Mississippi Oklahoma Illinois Texas Arkansas

Alabama Kentucky North Carolina Florida Generation 12 Generation 25 South Carolina Georgia dT = 9 dT = 9 dR = 0.3 dR = 0.2 dE = 44 dE = 40

Washington Michigan

Washington North Dakota New Hampshire North Dakota Montana Minnesota Alaska Maine Idaho Wyoming Indiana Vermont

New York Maryland Minnesota Alaska Nebraska Kansas Nevada Colorado Iowa Massachusetts Iowa New Hampshire Ohio Utah Colorado Wisconsin Nevada Nebraska Wisconsin Rhode Island Maine Connecticut Michigan Pennsylvania Maryland Vermont

West Virginia New York South Dakota Montana Oregon Wyoming New Jersey Indiana Massachusetts Idaho Oregon South Dakota Illinois Illinois Pennsylvania Ohio Delaware Connecticut Rhode Island Missouri Kentucky Virginia Missouri Tennessee New Jersey Kentucky West Virginia Utah California Kansas Oklahoma Virginia California Oklahoma Delaware Tennessee South Carolina Arizona New Mexico Arkansas Georgia North Carolina Arkansas Hawaii Hawaii South Carolina North Carolina New Mexico Louisiana Mississippi Georgia Alabama Arizona Texas Louisiana Florida Texas

Alabama Florida Mississippi Generation 50 Generation 100 dT = 9 dT = 9 dR = 0.2 dR = 0.19 dE = 40 dE = 50

Figure 8: The plot on the top left displays the fitness value during the increasing genera- tion of the genetic algorithm. The image plot on the top right visualizes the genotype (Π) change from one generation to the other using a gray colormap for encoding the index order. The six phenotypes visualize the improvement of the solution with an increasing number of generations. 16 recmap: Rectangular Statistical Cartograms in R less on the information visualization scopes. Examples of combining cartogram drawings with pixel visualization techniques can be found in Panse, Sips, Keim, and North(2006). Figure 13 demonstrates how ‘recmap’ objects can be transformed into ‘SpatialPolygonsDataFrame’ objects.

US state facts and figures based cartograms are displayed in Figure9. The data are available from the data frame state.x77. On the cartograms, two statistical data are displayed using the area and the color of a map region. The colormap was generated by using the heat_hcl function of the colorspace package by Zeileis, Hornik, and Murrell(2009) (red is low; white is high). The code below is a wrapper function for the recmap and the ga functions. A tuple of state.x77 column names is given as input.

R> recmap_state_x77 <- function(input, Map = US.map, DF = state.x77, + cm = heat_hcl(10)) { + # Join map and data.frame + Map <- cbind(Map, DF, match(Map$name, row.names(DF))) + attr(Map, "Map.name") <- "U.S." + attr(Map, "Map.area") <- input$area + + # Filter + Map <- Map[!Map$name %in% c("Hawaii", "Alaska"), ] + + # Set attribute for desired area + Map$z <- Map[, input$area] + + res <- recmapGA(Map = Map, popSize = 300, maxiter = 30, run = 10) + + # Set attribute for the coloring + S <- Map[res$GA@solution[1, ], input$color] + col.idx <- round((length(cm) - 1) * (S - min(S))/(max(S) - min(S))) + 1 + + # Have fun + plot(res$Cartogram, col = cm[col.idx], col.text = "black") + legend("bottomleft", c(paste("area:", input$area), + paste("color:", input$color)), cex = 1.5) + + res + } As input map, the US.map defined in Section3 is used. The lines below generate the car- tograms. R> op <- par(mfrow = c(4, 1), mar = rep(0.25, 4), bg = "white") R> set.seed(1) R> recmapGA.x77 <- lapply(list(list(color = "Area", area = "Population"), + list(color = "HS Grad", area = "Murder"), list(color = "HS Grad", + area = "Income"), list(color = "Life Exp", area = "Illiteracy")), + recmap_state_x77) R> par(op) Journal of Statistical Software – Code Snippets 17

The fitness values versus the generations are graphed using the plot method of the ‘GA’ class.

R> op <- par(mar = c(5, 5, 3, 3), mfrow = c(4, 1)) R> res <- lapply(cartogram.x77, function(x) { + plot(x$GA) + }) R> par(op)

US population cartograms on county level showing cartograms of California, Col- orado, Florida, New Jersey, and New York are displayed in Figure 10. The map material was extracted from the maps package by Becker, Wilks, Brownrigg, Minka, and Deckmyn(2016) and the population data were retrieved from the noncensus package by Ramey(2014). The map regions were joined over the FIPS (Federal Information Processing Standard) county codes using the counties data frame. The cartograms were generated by using the genetic algorithm as metaheuristic. An interactive shiny (Chang et al. 2016) web application is available by running the code snippet below. It provides more combinations of parameter settings, attributes, and maps drawn in Figures5,8,9, and 10.

R> library("shiny") R> recmap_shiny <- system.file("shiny-examples", package = "recmap") R> shiny::runApp(recmap_shiny, display.mode = "normal")

The last three application examples use the recmapGA function and the following main pa- rameter setting: one iteration and a population size of 64. Table2 lists other significant operational parameters. The seed values were derived by a greedy heuristic to have a proper starting construction sequence and thereby avoid intensive computing of the replication code of the manuscript. In praxis, it turned out that a population size similar to the number of map regions and the maximum number of iteration set to no more than a couple of hundred are useful initial values.

A population cartogram of Switzerland on community (Gemeinde) level is drawn in Figure 11. The rectangles of the original map were extracted from an ESRI shape file of the map data Landschaftsmodelle: GG25 from the Federal Office of Topography (swisstopo) us- ing shapefiles by Stabler(2013). The following attributes were extracted for each map region: box, Gemeindecode, and Gemeindename. There are 2300 rectangles to place. The statistical values (population 2013, published in 2015) were downloaded from Swiss Statistics (Region- alporträts: Kennzahlen aller Gemeinden (je-d-21.03.01) Bundesamt für Statistik BFS; http: //www.bfs.admin.ch/bfs/portal/de/index/regionen/02/daten.html) and joined by the Gemeindecode attribute with the swisstopo map.

A Swiss railway passenger frequency cartogram is graphed on the bottom of Fig- ure 12. The visualization above shows the overlapping rectangles of all 724 geo-locations which define the pseudo dual of the map. The data were retrieved from https://data.sbb. ch/explore/ and contain already the longitude and latitude coordinates of the railway main 18 recmap: Rectangular Statistical Cartograms in R 13 and 12 , 11 , having 10 , 3.30GHz @ 9 , CPU 8 , i5-2500 5 , Core Intel Figures on in processed drawn were GNU/Linux. cartograms cartograms rectangular 9 rectangular listed Debian statistical All running the regions. cores of map four of summary number a the provides by ordered spreadsheet The 2: Table

Hpplto 3005 4102 .7TU .0115 8_4Linux x86_64 151.50 0.40 Linux x86_64 Linux TRUE 9.40 x86_64 0.17 Linux Linux 3.70 0.25 6.80 x86_64 x86_64 1 Linux 60.30 Linux 17.10 27.90 TRUE x86_64 x86_64 157.90 Linux 64 0.46 Linux 84.70 179.20 105.60 TRUE FALSE x86_64 Linux 0.25 FALSE x86_64 0.59 60.60 Linux 0.01 0.10 166.20 20.50 x86_64 Linux 1 0.50 41.30 2300 x86_64 0.25 FALSE 0.25 FALSE 142.40 x86_64 0.35 220.00 31.00 1 0.06 28 210.70 0.21 Linux 143.70 36.30 64 FALSE 100 FALSE 0.25 0.25 FALSE 222.70 x86_64 340 0.12 50 64 199.50 0.67 0.12 44 10 FALSE 0.10 18.70 0.44 0.25 FALSE 724 0.25 0.17 0.12 0.57 0.25 320 68 15 640 0.07 224.90 population 29 50 0.25 370 66 0.68 0.27 FALSE 0.25 Linux 300 23 frequency 300 64 passagier 64 electorates 0.12 310 25 x86_64 0.40 of number 0.29 300 0.25 0.67 4.70 48 290 48 0.37 14 62 0.62 48 562.40 300 58 FALSE population 0.41 0.26 CH 48 population 0.25 SBB 25 area UK population illiteracy 105 florida US 1:4 income 0.38 colorado population US 8 21 x murder 8 checkerboard york population new US california US US US population US US US jersey new US Map.name

Map.area

Map.number.regions

Map.error.area

GA.population.size

GA.number.generation

GA.pmutation

GA.fitness GA.parallel

GA.number.recmaps_a_second

Sys.compute.time Sys.machine

Sys.sysname Journal of Statistical Software – Code Snippets 19

Vermont Maine ● ● ● ● ● ● ● ● ● ● New Hampshire 0.12 New York North Dakota Massachusetts ● ● ● ● Minnesota Michigan Montana

South Dakota

Wisconsin

Rhode Island

Connecticut Washington

Wyoming Nebraska Iowa Pennsylvania 0.10 Idaho Nevada Utah Indiana Ohio Oregon Colorado New Jersey

Illinois West Virginia Tennessee Maryland Kentucky Delaware Kansas Missouri Fitness value California 0.08

Virginia North Carolina Oklahoma Alabama South Carolina Arizona New Mexico ● Best ● Mean Mississippi ● Georgia ● ● Texas ● ● ● ● ● ● ● ● ● ● Median

Arkansas 0.06 ● Florida Louisiana 2 4 6 8 10 12 14 area: Population color: Area Generation

Vermont Maine

● ● ● ● ● ● ● ● ● ●

New Hampshire 0.12 ● ● ● ● ● ● ●

Michigan New York Massachusetts

North Dakota Minnesota

Wisconsin Washington Wyoming Connecticut Rhode Island Nebraska ● ● ● ● ● ● Ohio Pennsylvania Oregon Idaho New Jersey Montana

South Dakota 0.10 Iowa West Virginia Delaware Nevada Utah Colorado Kentucky Indiana Kansas Illinois

North Carolina Maryland Fitness value California Arizona New Mexico Arkansas Missouri Tennessee 0.08 ● Best South Carolina Oklahoma Virginia ● Mean Mississippi ● Georgia ● ● ● ● ● ● ● ● ● ● ● Median● ● ● ● ● ● ● ● ● ● ●

Texas 0.06 Louisiana 5 10 15 20 Alabama Florida area: Murder color: HS Grad Generation

South Dakota Maine ● ● ● ● ● ● ● ● ● ● Iowa Washington Michigan New Hampshire 0.12 Montana ● ● ● ● ● ● ● ● ●

North Dakota Minnesota New York ● ● ● ● ● ● ● Oregon Massachusetts Idaho Wyoming Wisconsin Indiana ● ●

Vermont Nebraska Missouri 0.10 Nevada Utah Connecticut Rhode Island Colorado Illinois Ohio Pennsylvania ●

Kansas California West Virginia New Jersey Kentucky Arizona Maryland Fitness value

North Carolina 0.08 Oklahoma Delaware Arkansas Tennessee Virginia ● Best ● Mississippi Mean

South Carolina ● ● New Mexico ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●Median● ● ● Alabama Georgia Texas 0.06 Louisiana ●

Florida 0 5 10 15 20 25 30 area: Income color: HS Grad Generation

Maine

Montana Minnesota Vermont ● ● ● ● ● ● ● ● ● ●

North Dakota Wyoming Wisconsin Massachusetts Nebraska New Hampshire ● ● ● ● Michigan New York 0.11 ● South Dakota Iowa Washington

Kansas Missouri Pennsylvania Indiana Ohio Connecticut Rhode Island Idaho Oregon Colorado Illinois 0.10 Nevada

Oklahoma New Jersey Utah Arkansas West Virginia Tennessee Maryland California 0.09

Delaware Arizona New Mexico Virginia North Carolina

Mississippi Fitness value

Kentucky 0.08 Texas ● Best South Carolina 0.07 ● Mean ● Alabama ● ● Georgia ● ● ● ●Median ● ● ● ● ● ● ●

Louisiana 0.06 ●

Florida 2 4 6 8 10 12 14 area: Illiteracy color: Life Exp Generation

Figure 9: Rectangular statistical cartograms using the “US State Facts and Figures” dataset are drawn. The plots on the right column display the fitness values versus the generation of the genetic algorithm during the optimization process. 20 recmap: Rectangular Statistical Cartograms in R

Siskiyou County

Shasta County Del Norte County Modoc County Tehama County Siskiyou County Glenn County Butte County Sutter County Placer County Nevada County Yuba County

El Dorado County Yolo County ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Humboldt County Shasta County Sonoma County Napa County Trinity County ●

Lassen County Solano County 0.070 Tehama County Marin County Sacramento County Plumas County Contra Costa County ● ● ● ●

Butte County San Francisco County Glenn County Sierra County

Mendocino County San Joaquin County Yuba County Nevada County Alameda County Colusa County Lake County Sutter County Placer County

El Dorado County San Mateo County Madera County Stanislaus County Yolo County Alpine County

Napa County Sonoma County Amador County Santa Clara County Santa Cruz County Sacramento County 0.060 Solano County Fresno County Merced County Calaveras County Marin County Mono County Tuolumne County San Luis Obispo County

Contra Costa County San Joaquin County Kings County

Alameda County Stanislaus County Monterey County Tulare County Mariposa County

San Mateo County

Madera County Santa Clara County Merced County Santa Barbara County Kern County Santa Cruz County

Fresno County Fitness value San Benito County

Inyo County 0.050 Ventura County Monterey County Tulare County San Bernardino County Kings County ● Los Angeles County Best ● ● San Luis Obispo County ● ● ● ● Kern County ● Mean ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Riverside County ● San Bernardino County ● Median Santa Barbara County

0.040 ● Imperial County Ventura County

Los Angeles County

Orange County

Orange County Riverside County 5 10 15 20 25

San Diego County San Diego County Imperial County Generation

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.065

Sedgwick County ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Logan County Jackson County Larimer County Larimer County Moffat County Phillips County Logan County Routt County Weld County Weld County Broomfield County

Morgan County Morgan County Boulder County Grand County Boulder County ● ● ● ● ● ● ● ● Washington County Yuma County Rio Blanco County Adams County Gilpin County Adams County Denver County

Denver County

Clear Creek County Garfield County 0.055 Eagle County Summit County Arapahoe County Routt County

Jefferson County Garfield County Moffat County Grand County Rio Blanco County

Eagle County Summit County Douglas County Kit Carson County Arapahoe County Lake County Mesa County Elbert County Pitkin County Lake County Pitkin County Delta County Jefferson County Park County Lincoln County Elbert County Montrose County Gunnison County Delta County Mesa County Teller County El Paso County Cheyenne County San Miguel County Chaffee County Douglas County Gunnison County Fitness value

Fremont County Montrose County Kiowa County 0.045 Crowley County

Park County

Teller County ● Ouray County Pueblo County Chaffee County Saguache County Custer County El Paso County Best San Miguel County Otero County Bent County Prowers County ● Fremont County Mean San Juan County Hinsdale County Dolores County Montezuma County La Plata County ● Archuleta County Prowers County ● Mineral County ● ● ● ● Huerfano County ● ● ● ● ● ● ● ● ● ● ● ● ● ● Rio Grande County ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Median Alamosa County ● ● ● ● Las Animas County Pueblo County ● Montezuma County La Plata County Costilla County Baca County Otero County Archuleta County Alamosa County Conejos County Las Animas County 0.035

Conejos County 0 10 20 30 40

Generation

Holmes County Jackson County Okaloosa County Nassau County Santa Rosa County Escambia County Walton County Washington County

Gadsden County Okaloosa County Nassau County Santa Rosa County Gadsden County Escambia County Jackson County Leon County Madison County Hamilton County Calhoun County Jefferson County Baker County Duval County Leon County Duval County ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Bay County Liberty County Okaloosa County Bay County Columbia County Wakulla County Suwannee County Wakulla County

Lafayette County Union County Alachua County Taylor County Gulf County Clay County Bradford County Clay County St. Johns County Franklin County St. Johns County 0.10 Gilchrist County ● Levy County ● Alachua County Marion County ● ● ●

Flagler County Dixie County Putnam County Putnam County

Flagler County Citrus County

Levy County ● ● Sumter County Marion County Hernando County Volusia County ● Volusia County Pasco County 0.09 Lake County Citrus County Lake County Seminole County Seminole County

Sumter County Hernando County Orange County

Orange County Pasco County Pinellas County Brevard County

Hillsborough County 0.08

Polk CountyOsceola County Pinellas County Brevard County Hillsborough County

Manatee County Polk County Indian River County

Osceola County Fitness value St. Lucie County

Hardee County Indian River County 0.07

Manatee County Okeechobee County St. Lucie County Highlands County Sarasota County Highlands County Martin County

Sarasota County DeSoto County ● Charlotte County Martin County DeSoto County Best Glades County Charlotte County ● Palm Beach County Lee County Mean Palm Beach County ● Broward County 0.06 ● ● ● Hendry County ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Lee County Hendry County ● ● Median Collier County ● ●

Collier County Broward County Monroe County Miami−Dade County 0 5 10 15 20 25

Miami−Dade County

Monroe County Generation

Sussex County

Sussex County Passaic County ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Passaic County Bergen County Bergen County 0.26 Warren County Morris County Essex County Essex County Hudson County ● ●

Union County

Warren County Morris County Hunterdon County Somerset County ● Hudson County ● ● Middlesex County Union County 0.22

Somerset County

Mercer County Monmouth County

Hunterdon County

Middlesex County Fitness value

Burlington County Ocean County 0.18 Camden County ● Mercer County ● Monmouth County Best● Gloucester County ● ● ● ● ● ● ● ● ● ● ● ● ● Mean ● ● ● ● Salem County ● ● ● ● ● ● Atlantic County Median Burlington County ● Ocean County Cumberland County Camden County 0.14

Gloucester County Atlantic County 5 10 15 20 25

Cape May County Salem County Cumberland County

Cape May County Generation

Clinton County St. Lawrence County ●●●●●●●●●●●●●●●●●●●●

Jefferson County

Essex County Warren County Orleans County Niagara County Monroe County

Clinton County Oswego County

Fulton County Oneida County Franklin County Saratoga County St. Lawrence County Montgomery County Genesee County Wayne County ●●●●●●●●●●●●●●●●●● Rensselaer County Onondaga County

Madison County Albany County Erie County Ontario County Livingston County Schenectady County Wyoming County 0.09 ●●●●●●●●●●●●●●●● Otsego County ●● Essex County Yates County Columbia County Tompkins County ●●●●●●● Jefferson County Tioga County Greene County Steuben County Chautauqua County ● Chemung County

Cattaraugus County Broome County Lewis County Ulster County ● Dutchess County

Hamilton County Delaware County Sullivan County ● Warren County Herkimer County Oswego County Orange County Putnam County Washington County 0.08

Orleans County Oneida County Niagara County Wayne County Monroe County Fulton County Rockland County Saratoga County Westchester County Cayuga County Onondaga County Genesee County Madison County Montgomery County Suffolk County Ontario County Schenectady County Erie County Seneca County Livingston County Rensselaer County Wyoming County

Yates County Cortland County Otsego County Schoharie County Albany County

Chenango County Tompkins County

Schuyler County 0.07 Fitness value Chautauqua County Cattaraugus County Steuben County Greene County Allegany County Columbia County Bronx County Tioga County Broome County Delaware County Chemung County Kings County Ulster County ● Dutchess County Sullivan County Best Richmond County New York County ● 0.06 Putnam County Mean Orange County ● ● ● ● ● ● ● ● ● ●● ●● ●●●● ● ● ● ●● ● ● ● ●● ● ● ●● ●● ● ● ● ● Rockland County ● ●● ●●● ● ● ● ● ● ● Westchester County ● ●● ●● ● Median● ●● ● ●●● ●● ● Nassau County ● Suffolk County Queens County Nassau County

Queens County 0 10 20 30 40 50 60

Generation

Figure 10: US input maps (left) of California, Colorado, Florida, New Jersey, and New York on county level were used as input to compute 2010 census population cartograms (middle; top to bottom). On the right column, the fitness values versus the generations are displayed. Journal of Statistical Software – Code Snippets 21

Schaffhausen

Baden

Aarau

Basel Bülach Kreuzlingen

Winterthur

Allschwil

Frauenfeld

Dübendorf

Kloten

Rapperswil−Jona

Reinach (BL) Uster Wettingen

Dietikon

Riehen Kriens Adliswil Zürich Zug Wetzikon (ZH) Luzern

Emmen

Horgen St. Gallen

Neuchâtel Biel/Bienne

Bern Wädenswil La Chaux−de−Fonds Baar Bulle

Köniz Yverdon−les−Bains

Meyrin Thun Chur

Vevey Montreux Renens (VD)

Lausanne Nyon Sion Fribourg

Genève Lugano

Vernier

Carouge (GE)

Lancy

Figure 11: A rectangular population cartogram of Switzerland is shown. Map data source: Swiss Federal Office of Topography using Landscape Models/Boundaries GG25 (http:// www.toposhop.admin.ch/en/shop/products/landscape/gg25_1, downloaded 2016-05-01); statistical data: Bundesamt für Statistik (BFS), Website Statistik Schweiz (http://www. bfs.admin.ch/bfs/portal/de/index.html), downloaded file je-d-21.03.01.xls (http: //www.bfs.admin.ch/bfs/portal/de/index/regionen/02/daten.html) on 2016-05-26. station and stops. The fitness function below weights the relative position error of regions with a higher traveler frequency more than map regions with a lower travel frequency.

R> fitness.SBB <- function(idxOrder, Map, ...) { + Cartogram <- recmap(Map[idxOrder, ]) + if (sum(Cartogram$topology.error == 100) > 1){return (0)} + 1 / sum(Cartogram$z / (sqrt(sum(Cartogram$z^2))) * + Cartogram$relpos.error) + }

The UK Brexit EU-referendum is shown as a final example in Figure 13. The UK boundary file was downloaded from https://census.edina.ac.uk/ and joined by the col- umn name geo_code and Area_Code with the outcome of the referendum downloaded through 22 recmap: RectangularBasel SBB Statistical Cartograms in R Brugg AGBaden Winterthur Zürich Flughafen Wil Zürich Oerlikon St. Gallen AarauLenzburg ZürichZürichZürich AltstettenHardbrückeStettbach HB Olten Zürich StadelhofenUster Wetzikon

Rapperswil Zug Biel/Bienne Basel SBB Luzern Neuchâtel Brugg AGBaden Winterthur Bern Zürich Flughafen Wil Zürich Oerlikon St. Gallen AarauLenzburg ZürichZürichZürich AltstettenHardbrückeStettbach HB Olten Zürich StadelhofenUster Chur Fribourg Wetzikon Thun Rapperswil Zug Biel/Bienne Luzern Lausanne Neuchâtel Vevey Bern

Chur Fribourg Thun Genève

Lausanne Vevey

Genève

Schaffhausen

Bülach Brugg AG Effretikon Winterthur Zürich Oerlikon Wil Baden Zürich Flughafen Wetzikon Zürich Altstetten Olten St. Gallen Basel SBB Lenzburg Aarau Thalwil Bern SchaffhausenDietikon Uster

Bülach Zug Brugg AG Zürich HB Effretikon Luzern Zürich Hardbrücke Chur Rapperswil Winterthur Thun Biel/Bienne Zürich Oerlikon Wil Baden Zürich Flughafen Wetzikon Zürich Altstetten Zürich Stadelhofen Neuchâtel Wädenswil Stettbach St. Gallen Liestal Olten Basel SBB Lenzburg Aarau Thalwil BernFribourg Dietikon Uster

Nyon Zug Zürich HB Luzern Zürich Hardbrücke Chur Lausanne Rapperswil Thun Biel/Bienne Visp Zürich Stadelhofen Neuchâtel Wädenswil Stettbach Liestal Renens VD Genève FribourgVevey

Nyon

Lausanne Figure 12: A Swiss railwayVisp passenger frequency cartogram is shown on the lower map. The

Renens VD Genève graphic on the topVevey displays the overlapping rectangles of the input map. Source: http: //sbb.ch/, 2016-05-12.

http://www.electoralcommission.org.uk/ on July 3rd. This example also demonstrates the usage of the sp package by Bivand et al. (2013). Through using the S3 method as.SpatialPolygonsDataFrame the ‘recmap’ instance, UK$Map, has been transformed into a ‘SpatialPolygonsDataFrame’ object.

R> DF <- data.frame(Pct_Leave = UK$Map$Pct_Leave, row.names = UK$Map$name) R> spplot(as.SpatialPolygonsDataFrame(UK$Map, DF), + col.regions = diverge_hcl(16, alpha = 0.5), + main = "Input England/Wales/Scottland") Journal of Statistical Software – Code Snippets 23

Input England/Wales/Scottland

Pct_Leave Pct_Turnout Pct_Rejected 80 70

70

60 60

50

50 40

30 40

20

30 10

0

20

East Lothian Orkney Islands Perth & Kinross

Falkirk Highland West Lothian Fife Edinburgh, City of

Angus North Lanarkshire Moray Dundee City West Dunbartonshire East Dunbartonshire

Midlothian Eilean Siar

Aberdeenshire

Argyll & Bute Aberdeen City

Renfrewshire Glasgow City

South Lanarkshire

Stirling Clackmannanshire Newcastle upon Tyne North Tyneside

East Renfrewshire Inverclyde Scottish Borders

Dumfries & Galloway North Ayrshire

South Tyneside Eden Sunderland

East Ayrshire Carlisle

Craven Oldham Calderdale

South Ayrshire County Durham Allerdale Hartlepool Hyndburn

Preston Bolton Rochdale

Redcar and Cleveland Richmondshire Copeland Stockton−on−Tees Broxtowe Middlesbrough Blackburn with Darwen Kirklees Darlington North East Derbyshire

Chorley South Lakeland Sheffield

West Lancashire Scarborough Salford Barrow−in−Furness Ryedale Hambleton Bury Harrogate Burnley Lancaster East Riding of Yorkshire

Pendle Wakefield Knowsley York Selby

Blackpool Warrington Trafford Ribble Valley Bradford Stockport Wyre North East Lincolnshire Doncaster Kingston upon Hull, City of

Halton Derbyshire Dales Amber Valley Flintshire Stoke−on−Trent Fylde

South Ribble Rossendale Leeds North Lincolnshire Wrexham Wigan Derby Powys Sefton St. Helens Stafford

South Derbyshire

Cannock Chase Tamworth Tameside Barnsley Charnwood

Cheshire East Liverpool Manchester Leicester High Peak Mansfield Blaby Rotherham Staffordshire Moorlands Bolsover

Nottingham Corby Harborough Birmingham Newark and Sherwood West Lindsey

Chesterfield Bassetlaw Shropshire Cheshire West and Chester South Kesteven Lincoln Gedling

Ashfield Erewash East Lindsey Newcastle−under−Lyme Boston North Kesteven Wyre Forest Melton Worcester Mid Suffolk Bromsgrove St Edmundsbury Rushcliffe Dudley Sandwell East Staffordshire South Holland Rutland

Lichfield King's Lynn and West Norfolk Telford and Wrekin North West Leicestershire Kettering South Norfolk Peterborough Fenland Wellingborough South Staffordshire Daventry Rugby Walsall Bedford South Cambridgeshire Cambridge Wolverhampton Wirral North Warwickshire East Northamptonshire Oadby and Wigston

Nuneaton and Bedworth Huntingdonshire

Hinckley and Bosworth

Northern Stratford−on−Avon Wychavon Solihull South Northamptonshire Malvern Hills Denbighshire Milton Keynes Conwy

Ireland Isle of Anglesey Coventry

Central Bedfordshire Redditch Monmouthshire Warwick Stroud North Hertfordshire Gwynedd

Caerphilly Aylesbury Vale Rhondda Cynon Taf

Northampton Carmarthenshire Ceredigion Cotswold Vale of White Horse Cherwell

Cheltenham Hertsmere Three Rivers North Norfolk Pembrokeshire Chiltern Tewkesbury Hillingdon Neath Port Talbot Enfield Blaenau Gwent Swansea West Oxfordshire Broadland Merthyr Tydfil Oxford Forest of Dean Torfaen Breckland South Bucks Barnet Luton East Cambridgeshire Great Yarmouth Norwich Bridgend Haringey

South Oxfordshire Slough Forest Heath Newport Dacorum Uttlesford South Gloucestershire

Waltham Forest Waveney North Somerset Swindon Bristol, City of Braintree Ealing Redbridge Camden Bracknell Forest Havering Cardiff West Berkshire Wycombe Watford The Vale of Glamorgan Hart Basildon Bath and North East Somerset Babergh Ipswich Suffolk Coastal South Somerset

Taunton Deane Hackney Southampton Barking and Dagenham Test Valley Islington Reading Windsor and Maidenhead Harrow Broxbourne Spelthorne Epping Forest Sedgemoor Mendip Wokingham West Somerset Colchester Tendring

North Devon Mid Devon West Dorset East Dorset Woking North Dorset Brent Harlow Newham Chelmsford Eastleigh Torridge Surrey Heath East Devon Mole Valley Maldon Rushmoor Christchurch Tower Hamlets Teignbridge Brentwood Southend−on−Sea

Bournemouth Kensington and Chelsea West Devon Poole Rochford Fareham Exeter Chichester Purbeck Bexley Castle Point Greenwich

Havant Hammersmith and Fulham Dartford Torbay Portsmouth Richmond upon Thames Medway Plymouth South Hams Southwark Thanet Lambeth Swale

Gravesham Canterbury Isle of Wight Maidstone Hounslow Lewisham Wandsworth Tunbridge Wells Dover

Runnymede

Merton Rother Elmbridge Lewes Herefordshire, County of Shepway Ashford Gloucester Brighton and Hove Kingston upon Thames Adur Hastings Worthing Eastbourne

Epsom and Ewell Croydon

Guildford

Wealden Sevenoaks Tonbridge and Malling Basingstoke and Deane Wiltshire Waverley

Reigate and Banstead Thurrock

Winchester East Hampshire

Bromley New Forest Sutton Gosport Horsham

Crawley Tandridge Arun

Mid Sussex

Figure 13: The outcome of the UK Brexit EU-referendum is displayed. Northern Ireland was manually added. The overlapping MBBs of the input map are displayed in the top left. On all other plots, the region areas represent the number of the electorates. The colors are indicating the outcome of the referendum (blue: remain/red: leave; the lower the color intensity the closer is the outcome to 50%:50%). Other attributes represented as percentages (Pct) are displayed using the spplot of the sp package (top right). Copyright: Contains National Statistics data © Crown copyright and database right 2016. Contains NRS data © Crown copyright and database right 2016. Source: NISRA: Website: http://www.nisra.gov.uk/. Contains OS data © Crown copyright [and database right] (2016). 24 recmap: Rectangular Statistical Cartograms in R

The following code snippet applies the sp package’s summary and spplot methods after adding the NI record.

R> DF <- rbind(data.frame(Pct_Leave = UK$Map$Pct_Leave, + Pct_Turnout = UK$Map$Pct_Turnout, Pct_Rejected = UK$Map$Pct_Rejected, + row.names = UK$Map$name), + data.frame(Pct_Leave = 44.22, Pct_Turnout = 62.69, Pct_Rejected = 0.05, + row.names = "Northern\nIreland")) R> UK.sp <- as.SpatialPolygonsDataFrame(add_NI(UK.recmap), DF) R> summary(UK.sp)

Object of class SpatialPolygonsDataFrame Coordinates: min max x -554449.5 1073563 y -596291.6 1072578 Is projected: NA proj4string : [NA] Data attributes: Pct_Leave Pct_Turnout Pct_Rejected Min. :21.38 Min. :56.25 Min. :0.03000 1st Qu.:47.38 1st Qu.:70.19 1st Qu.:0.06000 Median :54.34 Median :74.30 Median :0.07000 Mean :53.29 Mean :73.69 Mean :0.07283 3rd Qu.:60.49 3rd Qu.:77.89 3rd Qu.:0.08000 Max. :75.56 Max. :83.57 Max. :0.24000

R> spplot(UK.sp, col.regions = diverge_hcl(19)[1:16], layout = c(3, 1))

7. Summary

This article introduces the CRAN recmap package which implements the RecMap MP2 algo- rithm. This method generates rectangular statistical cartograms. Two outstanding features of the implemented algorithm are: the areas of the map regions represent the exact statis- tical value without any area error, and the ratios of the map regions are preserved. These constraints are important for the correct interpretation of the geography-related statistical data. It is evident that using these restrictions the map topology cannot be preserved. The implementation allows the generation of rectangular statistical cartograms having less than one hundred map regions within a few seconds with the support of a metaheuristic. The implementation enables an interactive exploratory data analysis. All necessary steps can be done on the R command line or by using web applications on a modern computer. It has been demonstrated, how the drawing of the cartogram can be optimized according to a fitness function by using a metaheuristic and benefiting from today’s multi-core hardware and R’s parallel environment. Most promising is using a fitness function which is derived from the rel- ative position error objective function. It should also be highlighted that the method can read a spreadsheet containing the geographic location. It does not require any complex polygon Journal of Statistical Software – Code Snippets 25 mesh as input. The potential of the method is shown by using real world maps covering a map size of three orders magnitude and synthetic data (8 × 8 checkerboard). Table2 provides an overview of some rectangular cartogram specification drawn in this manuscript. The recmap package is a powerful tool in the hand of data analysts, cartographers, or statisticians using R who want to draw their own statistical rectangular cartograms.

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Affiliation: Christian Panse Functional Genomics Center Zurich UZH|ETHZ Winterthurerstr. 190 CH-8057, Zürich, Switzerland Telephone: +41/44/63-53912 E-mail: [email protected] URL: http://www.fgcz.ch/the-center/people/panse.html

Journal of Statistical Software http://www.jstatsoft.org/ published by the Foundation for Open Access Statistics http://www.foastat.org/ August 2018, Volume 86, Code Snippet 1 Submitted: 2016-06-01 doi:10.18637/jss.v086.c01 Accepted: 2017-06-12