Toblers First Law of Geography

Everything is related to everything else, but near things are more related to each other.1

Physical Measurements

  • Location

  • Distance

  • Network

  • Neighborhoods & Regions

Emerging Properties

  • Spatial Heterogeneity

  • Spatial Dependence

  • Objects & Fields

A Projection

A projection is a mathematical mapping of points measured on this surface of this earth that can be represented on things like computer screens.

Ellipsoids

Models of the physical structure of the surface of the planet.

  • NAD83/GRS80 - Satellite measurements of distance from core to surface of earth.

  • WGS84 - Model built on global GPS system.

Example Data - Maps

For these examples, I’m going to be using the maps library.

library( ggplot2 )
library( maps )
states <- map_data( "state" )
head( states, n=500)
          long      lat group order   region subregion
1    -87.46201 30.38968     1     1  alabama      <NA>
2    -87.48493 30.37249     1     2  alabama      <NA>
3    -87.52503 30.37249     1     3  alabama      <NA>
4    -87.53076 30.33239     1     4  alabama      <NA>
5    -87.57087 30.32665     1     5  alabama      <NA>
6    -87.58806 30.32665     1     6  alabama      <NA>
7    -87.59379 30.30947     1     7  alabama      <NA>
8    -87.59379 30.28655     1     8  alabama      <NA>
9    -87.67400 30.27509     1     9  alabama      <NA>
10   -87.81152 30.25790     1    10  alabama      <NA>
11   -87.88026 30.24644     1    11  alabama      <NA>
12   -87.92037 30.24644     1    12  alabama      <NA>
13   -87.95475 30.24644     1    13  alabama      <NA>
14   -88.00632 30.24071     1    14  alabama      <NA>
15   -88.01778 30.25217     1    15  alabama      <NA>
16   -88.01205 30.26936     1    16  alabama      <NA>
17   -87.99486 30.27509     1    17  alabama      <NA>
18   -87.95475 30.27509     1    18  alabama      <NA>
19   -87.90318 30.28082     1    19  alabama      <NA>
20   -87.82870 30.28655     1    20  alabama      <NA>
21   -87.80006 30.28655     1    21  alabama      <NA>
22   -87.80006 30.32665     1    22  alabama      <NA>
23   -87.81724 30.34385     1    23  alabama      <NA>
24   -87.84016 30.38395     1    24  alabama      <NA>
25   -87.85162 30.40114     1    25  alabama      <NA>
26   -87.87453 30.41260     1    26  alabama      <NA>
27   -87.90318 30.42406     1    27  alabama      <NA>
28   -87.92610 30.44698     1    28  alabama      <NA>
29   -87.93183 30.49281     1    29  alabama      <NA>
30   -87.94329 30.52719     1    30  alabama      <NA>
31   -87.92037 30.56157     1    31  alabama      <NA>
32   -87.91464 30.58449     1    32  alabama      <NA>
33   -87.92610 30.61886     1    33  alabama      <NA>
34   -87.92037 30.67043     1    34  alabama      <NA>
35   -87.94902 30.69908     1    35  alabama      <NA>
36   -87.98913 30.79075     1    36  alabama      <NA>
37   -88.00632 30.79648     1    37  alabama      <NA>
38   -88.01778 30.80221     1    38  alabama      <NA>
39   -88.03497 30.79075     1    39  alabama      <NA>
40   -88.04642 30.75638     1    40  alabama      <NA>
41   -88.05215 30.72773     1    41  alabama      <NA>
42   -88.05215 30.71054     1    42  alabama      <NA>
43   -88.06361 30.68762     1    43  alabama      <NA>
44   -88.06934 30.68189     1    44  alabama      <NA>
45   -88.08080 30.63033     1    45  alabama      <NA>
46   -88.08080 30.61314     1    46  alabama      <NA>
47   -88.09799 30.60741     1    47  alabama      <NA>
48   -88.10944 30.59595     1    48  alabama      <NA>
49   -88.11518 30.58449     1    49  alabama      <NA>
50   -88.10944 30.55584     1    50  alabama      <NA>
51   -88.12091 30.48136     1    51  alabama      <NA>
52   -88.12091 30.44125     1    52  alabama      <NA>
53   -88.12664 30.38968     1    53  alabama      <NA>
54   -88.13237 30.36103     1    54  alabama      <NA>
55   -88.14383 30.34385     1    55  alabama      <NA>
56   -88.16102 30.33812     1    56  alabama      <NA>
57   -88.17248 30.33812     1    57  alabama      <NA>
58   -88.18966 30.35530     1    58  alabama      <NA>
59   -88.22404 30.37822     1    59  alabama      <NA>
60   -88.23550 30.39541     1    60  alabama      <NA>
61   -88.28706 30.40114     1    61  alabama      <NA>
62   -88.31571 30.40114     1    62  alabama      <NA>
63   -88.33290 30.40114     1    63  alabama      <NA>
64   -88.35010 30.42406     1    64  alabama      <NA>
65   -88.39021 30.41260     1    65  alabama      <NA>
66   -88.40739 30.73346     1    66  alabama      <NA>
67   -88.41885 30.99702     1    67  alabama      <NA>
68   -88.43031 31.12307     1    68  alabama      <NA>
69   -88.45322 31.43247     1    69  alabama      <NA>
70   -88.46468 31.69603     1    70  alabama      <NA>
71   -88.47614 31.89083     1    71  alabama      <NA>
72   -88.43031 32.22314     1    72  alabama      <NA>
73   -88.43031 32.30909     1    73  alabama      <NA>
74   -88.39021 32.57838     1    74  alabama      <NA>
75   -88.35010 32.92215     1    75  alabama      <NA>
76   -88.33863 32.98518     1    76  alabama      <NA>
77   -88.29853 33.28312     1    77  alabama      <NA>
78   -88.27560 33.52949     1    78  alabama      <NA>
79   -88.25269 33.73576     1    79  alabama      <NA>
80   -88.20686 34.06234     1    80  alabama      <NA>
81   -88.20686 34.07953     1    81  alabama      <NA>
82   -88.18394 34.31444     1    82  alabama      <NA>
83   -88.16675 34.46341     1    83  alabama      <NA>
84   -88.14955 34.58373     1    84  alabama      <NA>
85   -88.10944 34.89886     1    85  alabama      <NA>
86   -88.13237 34.90459     1    86  alabama      <NA>
87   -88.16675 34.95043     1    87  alabama      <NA>
88   -88.18966 34.99627     1    88  alabama      <NA>
89   -88.18394 35.01345     1    89  alabama      <NA>
90   -87.98913 35.01345     1    90  alabama      <NA>
91   -87.60525 35.00772     1    91  alabama      <NA>
92   -87.22710 35.00199     1    92  alabama      <NA>
93   -87.20417 35.00199     1    93  alabama      <NA>
94   -86.82603 34.99627     1    94  alabama      <NA>
95   -86.78019 34.99053     1    95  alabama      <NA>
96   -86.31036 34.99053     1    96  alabama      <NA>
97   -85.85200 34.98480     1    97  alabama      <NA>
98   -85.61135 34.97907     1    98  alabama      <NA>
99   -85.58844 34.84156     1    99  alabama      <NA>
100  -85.54260 34.61811     1   100  alabama      <NA>
101  -85.52541 34.58373     1   101  alabama      <NA>
102  -85.46812 34.26287     1   102  alabama      <NA>
103  -85.42801 34.07953     1   103  alabama      <NA>
104  -85.40509 33.95921     1   104  alabama      <NA>
105  -85.39363 33.90764     1   105  alabama      <NA>
106  -85.34779 33.66127     1   106  alabama      <NA>
107  -85.31342 33.49511     1   107  alabama      <NA>
108  -85.29623 33.43209     1   108  alabama      <NA>
109  -85.24466 33.11696     1   109  alabama      <NA>
110  -85.23893 33.10550     1   110  alabama      <NA>
111  -85.19309 32.87059     1   111  alabama      <NA>
112  -85.19309 32.85913     1   112  alabama      <NA>
113  -85.17018 32.84194     1   113  alabama      <NA>
114  -85.14725 32.79037     1   114  alabama      <NA>
115  -85.13007 32.76746     1   115  alabama      <NA>
116  -85.13007 32.75027     1   116  alabama      <NA>
117  -85.12434 32.73308     1   117  alabama      <NA>
118  -85.10142 32.67578     1   118  alabama      <NA>
119  -85.10142 32.63567     1   119  alabama      <NA>
120  -85.06704 32.60703     1   120  alabama      <NA>
121  -85.04986 32.55546     1   121  alabama      <NA>
122  -85.00975 32.52682     1   122  alabama      <NA>
123  -84.98683 32.49817     1   123  alabama      <NA>
124  -84.99256 32.46952     1   124  alabama      <NA>
125  -84.97537 32.42368     1   125  alabama      <NA>
126  -84.98109 32.37785     1   126  alabama      <NA>
127  -85.00402 32.33774     1   127  alabama      <NA>
128  -84.97537 32.30909     1   128  alabama      <NA>
129  -84.92381 32.28044     1   129  alabama      <NA>
130  -84.90089 32.25179     1   130  alabama      <NA>
131  -84.92953 32.23461     1   131  alabama      <NA>
132  -84.95245 32.22887     1   132  alabama      <NA>
133  -84.96964 32.20023     1   133  alabama      <NA>
134  -84.99256 32.18304     1   134  alabama      <NA>
135  -85.04986 32.16012     1   135  alabama      <NA>
136  -85.06131 32.12001     1   136  alabama      <NA>
137  -85.06131 32.10283     1   137  alabama      <NA>
138  -85.06131 32.08564     1   138  alabama      <NA>
139  -85.06131 32.06272     1   139  alabama      <NA>
140  -85.06131 32.02834     1   140  alabama      <NA>
141  -85.06131 31.99396     1   141  alabama      <NA>
142  -85.07851 31.95959     1   142  alabama      <NA>
143  -85.10715 31.93667     1   143  alabama      <NA>
144  -85.14153 31.86791     1   144  alabama      <NA>
145  -85.13580 31.78770     1   145  alabama      <NA>
146  -85.13580 31.77624     1   146  alabama      <NA>
147  -85.13007 31.73040     1   147  alabama      <NA>
148  -85.11861 31.68456     1   148  alabama      <NA>
149  -85.07851 31.61581     1   149  alabama      <NA>
150  -85.06131 31.55852     1   150  alabama      <NA>
151  -85.06131 31.52414     1   151  alabama      <NA>
152  -85.08424 31.47830     1   152  alabama      <NA>
153  -85.08424 31.44392     1   153  alabama      <NA>
154  -85.09570 31.40955     1   154  alabama      <NA>
155  -85.09570 31.34079     1   155  alabama      <NA>
156  -85.10142 31.30068     1   156  alabama      <NA>
157  -85.12434 31.28349     1   157  alabama      <NA>
158  -85.12434 31.25485     1   158  alabama      <NA>
159  -85.11288 31.19182     1   159  alabama      <NA>
160  -85.11288 31.17463     1   160  alabama      <NA>
161  -85.08997 31.14026     1   161  alabama      <NA>
162  -85.04413 31.11161     1   162  alabama      <NA>
163  -85.03840 31.09442     1   163  alabama      <NA>
164  -85.03267 31.07723     1   164  alabama      <NA>
165  -85.03267 31.03712     1   165  alabama      <NA>
166  -85.01548 31.00275     1   166  alabama      <NA>
167  -85.01548 30.99702     1   167  alabama      <NA>
168  -85.49677 31.00847     1   168  alabama      <NA>
169  -85.51968 31.00847     1   169  alabama      <NA>
170  -86.04680 31.00847     1   170  alabama      <NA>
171  -86.20150 31.00847     1   171  alabama      <NA>
172  -86.39057 31.00847     1   172  alabama      <NA>
173  -86.70570 31.00847     1   173  alabama      <NA>
174  -86.79737 31.00847     1   174  alabama      <NA>
175  -87.18126 31.00847     1   175  alabama      <NA>
176  -87.60525 31.00847     1   176  alabama      <NA>
177  -87.60525 30.95691     1   177  alabama      <NA>
178  -87.61097 30.93399     1   178  alabama      <NA>
179  -87.63390 30.89962     1   179  alabama      <NA>
180  -87.63963 30.87670     1   180  alabama      <NA>
181  -87.63390 30.85951     1   181  alabama      <NA>
182  -87.61097 30.84232     1   182  alabama      <NA>
183  -87.54795 30.77356     1   183  alabama      <NA>
184  -87.49065 30.72200     1   184  alabama      <NA>
185  -87.46201 30.71054     1   185  alabama      <NA>
186  -87.45055 30.69908     1   186  alabama      <NA>
187  -87.42190 30.67616     1   187  alabama      <NA>
188  -87.41044 30.64751     1   188  alabama      <NA>
189  -87.41044 30.61886     1   189  alabama      <NA>
190  -87.43336 30.58449     1   190  alabama      <NA>
191  -87.46201 30.54438     1   191  alabama      <NA>
192  -87.46774 30.52719     1   192  alabama      <NA>
193  -87.46201 30.51000     1   193  alabama      <NA>
194  -87.44482 30.49281     1   194  alabama      <NA>
195  -87.39899 30.45844     1   195  alabama      <NA>
196  -87.39326 30.44698     1   196  alabama      <NA>
197  -87.40472 30.42979     1   197  alabama      <NA>
198  -87.42190 30.42979     1   198  alabama      <NA>
199  -87.43909 30.42406     1   199  alabama      <NA>
200  -87.43909 30.41833     1   200  alabama      <NA>
201  -87.45055 30.40114     1   201  alabama      <NA>
202  -87.46201 30.38968     1   202  alabama      <NA>
203 -114.63739 35.01918     2   204  arizona      <NA>
204 -114.64313 35.10512     2   205  arizona      <NA>
205 -114.60302 35.12231     2   206  arizona      <NA>
206 -114.57436 35.17961     2   207  arizona      <NA>
207 -114.58583 35.23690     2   208  arizona      <NA>
208 -114.59728 35.28274     2   209  arizona      <NA>
209 -114.60874 35.35723     2   210  arizona      <NA>
210 -114.62594 35.40879     2   211  arizona      <NA>
211 -114.66605 35.45463     2   212  arizona      <NA>
212 -114.66031 35.50047     2   213  arizona      <NA>
213 -114.66605 35.54057     2   214  arizona      <NA>
214 -114.67178 35.60933     2   215  arizona      <NA>
215 -114.68324 35.66662     2   216  arizona      <NA>
216 -114.70042 35.71246     2   217  arizona      <NA>
217 -114.70042 35.79268     2   218  arizona      <NA>
218 -114.68896 35.88435     2   219  arizona      <NA>
219 -114.71188 35.91299     2   220  arizona      <NA>
220 -114.73480 35.95883     2   221  arizona      <NA>
221 -114.73480 36.01613     2   222  arizona      <NA>
222 -114.75771 36.06197     2   223  arizona      <NA>
223 -114.73480 36.10780     2   224  arizona      <NA>
224 -114.70042 36.11926     2   225  arizona      <NA>
225 -114.64313 36.14218     2   226  arizona      <NA>
226 -114.55718 36.15364     2   227  arizona      <NA>
227 -114.48843 36.13645     2   228  arizona      <NA>
228 -114.39675 36.13645     2   229  arizona      <NA>
229 -114.32800 36.11353     2   230  arizona      <NA>
230 -114.31081 36.04478     2   231  arizona      <NA>
231 -114.26498 36.02758     2   232  arizona      <NA>
232 -114.20194 36.02186     2   233  arizona      <NA>
233 -114.13892 36.03905     2   234  arizona      <NA>
234 -114.11600 36.09634     2   235  arizona      <NA>
235 -114.08163 36.13645     2   236  arizona      <NA>
236 -114.05298 36.21093     2   237  arizona      <NA>
237 -114.04725 36.84119     2   238  arizona      <NA>
238 -114.04152 36.99588     2   239  arizona      <NA>
239 -113.82380 36.99588     2   240  arizona      <NA>
240 -113.53732 37.00161     2   241  arizona      <NA>
241 -113.27949 37.00161     2   242  arizona      <NA>
242 -113.00447 37.00161     2   243  arizona      <NA>
243 -112.90707 36.99588     2   244  arizona      <NA>
244 -112.72945 37.00161     2   245  arizona      <NA>
245 -112.54037 36.99588     2   246  arizona      <NA>
246 -112.41432 36.99588     2   247  arizona      <NA>
247 -112.20805 36.99588     2   248  arizona      <NA>
248 -111.95023 37.00161     2   249  arizona      <NA>
249 -111.68667 37.00161     2   250  arizona      <NA>
250 -111.41164 37.00161     2   251  arizona      <NA>
251 -111.31997 37.00161     2   252  arizona      <NA>
252 -111.25121 36.99588     2   253  arizona      <NA>
253 -111.12516 36.99588     2   254  arizona      <NA>
254 -110.84441 37.00161     2   255  arizona      <NA>
255 -110.75847 36.99588     2   256  arizona      <NA>
256 -110.68399 36.99588     2   257  arizona      <NA>
257 -110.54648 36.99588     2   258  arizona      <NA>
258 -110.30583 37.00161     2   259  arizona      <NA>
259 -110.08810 37.00161     2   260  arizona      <NA>
260 -110.00790 37.00161     2   261  arizona      <NA>
261 -109.93341 36.99588     2   262  arizona      <NA>
262 -109.71568 36.99588     2   263  arizona      <NA>
263 -109.41203 36.99588     2   264  arizona      <NA>
264 -109.14846 36.99588     2   265  arizona      <NA>
265 -109.03960 36.99588     2   266  arizona      <NA>
266 -109.04533 35.99894     2   267  arizona      <NA>
267 -109.05106 34.95043     2   268  arizona      <NA>
268 -109.05106 34.57227     2   269  arizona      <NA>
269 -109.05679 33.77586     2   270  arizona      <NA>
270 -109.05106 33.20290     2   271  arizona      <NA>
271 -109.05679 32.77892     2   272  arizona      <NA>
272 -109.05679 32.42368     2   273  arizona      <NA>
273 -109.05679 31.35225     2   274  arizona      <NA>
274 -109.23441 31.35225     2   275  arizona      <NA>
275 -109.62402 31.34652     2   276  arizona      <NA>
276 -109.99644 31.35225     2   277  arizona      <NA>
277 -110.35167 31.35225     2   278  arizona      <NA>
278 -110.47772 31.35225     2   279  arizona      <NA>
279 -110.63242 31.35225     2   280  arizona      <NA>
280 -110.98766 31.34652     2   281  arizona      <NA>
281 -111.11943 31.35798     2   282  arizona      <NA>
282 -111.29705 31.41528     2   283  arizona      <NA>
283 -111.38873 31.44392     2   284  arizona      <NA>
284 -111.62936 31.51841     2   285  arizona      <NA>
285 -112.03616 31.65019     2   286  arizona      <NA>
286 -112.32838 31.74186     2   287  arizona      <NA>
287 -112.66642 31.83927     2   288  arizona      <NA>
288 -112.92998 31.92521     2   289  arizona      <NA>
289 -113.18781 32.00542     2   290  arizona      <NA>
290 -113.34824 32.05699     2   291  arizona      <NA>
291 -113.46856 32.09710     2   292  arizona      <NA>
292 -113.74358 32.18877     2   293  arizona      <NA>
293 -114.03580 32.27471     2   294  arizona      <NA>
294 -114.31654 32.36066     2   295  arizona      <NA>
295 -114.70042 32.48098     2   296  arizona      <NA>
296 -114.80928 32.51535     2   297  arizona      <NA>
297 -114.79781 32.57838     2   298  arizona      <NA>
298 -114.76917 32.67006     2   299  arizona      <NA>
299 -114.73480 32.73308     2   300  arizona      <NA>
300 -114.68896 32.74454     2   301  arizona      <NA>
301 -114.65459 32.73881     2   302  arizona      <NA>
302 -114.59156 32.73881     2   303  arizona      <NA>
303 -114.55145 32.76173     2   304  arizona      <NA>
304 -114.52280 32.82475     2   305  arizona      <NA>
305 -114.47697 32.87632     2   306  arizona      <NA>
306 -114.47697 32.93362     2   307  arizona      <NA>
307 -114.48270 32.99664     2   308  arizona      <NA>
308 -114.53999 33.04247     2   309  arizona      <NA>
309 -114.66031 33.05394     2   310  arizona      <NA>
310 -114.70042 33.08258     2   311  arizona      <NA>
311 -114.71188 33.13415     2   312  arizona      <NA>
312 -114.68896 33.26593     2   313  arizona      <NA>
313 -114.74052 33.31176     2   314  arizona      <NA>
314 -114.71761 33.35760     2   315  arizona      <NA>
315 -114.72906 33.40344     2   316  arizona      <NA>
316 -114.68896 33.43209     2   317  arizona      <NA>
317 -114.64885 33.45501     2   318  arizona      <NA>
318 -114.63739 33.48365     2   319  arizona      <NA>
319 -114.60874 33.50084     2   320  arizona      <NA>
320 -114.62020 33.52949     2   321  arizona      <NA>
321 -114.56864 33.54668     2   322  arizona      <NA>
322 -114.55718 33.57533     2   323  arizona      <NA>
323 -114.52853 33.63262     2   324  arizona      <NA>
324 -114.53999 33.69565     2   325  arizona      <NA>
325 -114.52280 33.73003     2   326  arizona      <NA>
326 -114.53999 33.78159     2   327  arizona      <NA>
327 -114.53999 33.92483     2   328  arizona      <NA>
328 -114.51707 33.96494     2   329  arizona      <NA>
329 -114.45405 33.99932     2   330  arizona      <NA>
330 -114.47124 34.01650     2   331  arizona      <NA>
331 -114.44832 34.03942     2   332  arizona      <NA>
332 -114.41967 34.07953     2   333  arizona      <NA>
333 -114.39675 34.11391     2   334  arizona      <NA>
334 -114.29935 34.15401     2   335  arizona      <NA>
335 -114.23060 34.19985     2   336  arizona      <NA>
336 -114.19049 34.24569     2   337  arizona      <NA>
337 -114.13319 34.26287     2   338  arizona      <NA>
338 -114.13892 34.29725     2   339  arizona      <NA>
339 -114.16756 34.33736     2   340  arizona      <NA>
340 -114.28789 34.42330     2   341  arizona      <NA>
341 -114.37383 34.46914     2   342  arizona      <NA>
342 -114.40249 34.58946     2   343  arizona      <NA>
343 -114.41967 34.61238     2   344  arizona      <NA>
344 -114.44832 34.70978     2   345  arizona      <NA>
345 -114.57436 34.80719     2   346  arizona      <NA>
346 -114.55145 34.83583     2   347  arizona      <NA>
347 -114.54572 34.84729     2   348  arizona      <NA>
348 -114.56864 34.86448     2   349  arizona      <NA>
349 -114.60874 34.88740     2   350  arizona      <NA>
350 -114.63167 34.96188     2   351  arizona      <NA>
351 -114.63739 35.01918     2   352  arizona      <NA>
352  -94.05103 33.03675     3   354 arkansas      <NA>
353  -94.05103 33.30031     3   355 arkansas      <NA>
354  -94.05676 33.57533     3   356 arkansas      <NA>
355  -94.07967 33.58678     3   357 arkansas      <NA>
356  -94.09113 33.59824     3   358 arkansas      <NA>
357  -94.11977 33.59252     3   359 arkansas      <NA>
358  -94.14843 33.58678     3   360 arkansas      <NA>
359  -94.15416 33.58678     3   361 arkansas      <NA>
360  -94.18854 33.60970     3   362 arkansas      <NA>
361  -94.20572 33.61544     3   363 arkansas      <NA>
362  -94.22291 33.60970     3   364 arkansas      <NA>
363  -94.22864 33.59824     3   365 arkansas      <NA>
364  -94.24010 33.57533     3   366 arkansas      <NA>
365  -94.26302 33.58678     3   367 arkansas      <NA>
366  -94.28020 33.58678     3   368 arkansas      <NA>
367  -94.29739 33.57533     3   369 arkansas      <NA>
368  -94.30885 33.56960     3   370 arkansas      <NA>
369  -94.36615 33.57533     3   371 arkansas      <NA>
370  -94.40053 33.56960     3   372 arkansas      <NA>
371  -94.40625 33.57533     3   373 arkansas      <NA>
372  -94.42345 33.59824     3   374 arkansas      <NA>
373  -94.45209 33.59824     3   375 arkansas      <NA>
374  -94.46355 33.61544     3   376 arkansas      <NA>
375  -94.47501 33.63262     3   377 arkansas      <NA>
376  -94.48074 33.65554     3   378 arkansas      <NA>
377  -94.49792 33.66700     3   379 arkansas      <NA>
378  -94.49220 33.67846     3   380 arkansas      <NA>
379  -94.49220 33.96494     3   381 arkansas      <NA>
380  -94.48647 34.21131     3   382 arkansas      <NA>
381  -94.48074 34.53217     3   383 arkansas      <NA>
382  -94.46928 34.74989     3   384 arkansas      <NA>
383  -94.45782 34.98480     3   385 arkansas      <NA>
384  -94.44064 35.39733     3   386 arkansas      <NA>
385  -94.44636 35.44317     3   387 arkansas      <NA>
386  -94.48074 35.65517     3   388 arkansas      <NA>
387  -94.50366 35.76403     3   389 arkansas      <NA>
388  -94.56096 36.11353     3   390 arkansas      <NA>
389  -94.57241 36.17656     3   391 arkansas      <NA>
390  -94.62398 36.50887     3   392 arkansas      <NA>
391  -94.08540 36.50314     3   393 arkansas      <NA>
392  -94.07967 36.49741     3   394 arkansas      <NA>
393  -93.89059 36.50314     3   395 arkansas      <NA>
394  -93.87341 36.49741     3   396 arkansas      <NA>
395  -93.59839 36.50314     3   397 arkansas      <NA>
396  -93.59265 36.49741     3   398 arkansas      <NA>
397  -93.33482 36.50314     3   399 arkansas      <NA>
398  -93.32336 36.49741     3   400 arkansas      <NA>
399  -93.30618 36.50314     3   401 arkansas      <NA>
400  -93.30045 36.49741     3   402 arkansas      <NA>
401  -92.84781 36.50314     3   403 arkansas      <NA>
402  -92.84208 36.50314     3   404 arkansas      <NA>
403  -92.77332 36.50314     3   405 arkansas      <NA>
404  -92.76759 36.50314     3   406 arkansas      <NA>
405  -92.54987 36.50314     3   407 arkansas      <NA>
406  -92.16026 36.49741     3   408 arkansas      <NA>
407  -92.13161 36.49741     3   409 arkansas      <NA>
408  -91.67325 36.50314     3   410 arkansas      <NA>
409  -91.46698 36.49741     3   411 arkansas      <NA>
410  -91.46125 36.49741     3   412 arkansas      <NA>
411  -91.42114 36.49741     3   413 arkansas      <NA>
412  -91.41541 36.50314     3   414 arkansas      <NA>
413  -91.14613 36.50314     3   415 arkansas      <NA>
414  -90.79089 36.50314     3   416 arkansas      <NA>
415  -90.57317 36.50887     3   417 arkansas      <NA>
416  -90.22366 36.50887     3   418 arkansas      <NA>
417  -90.17210 36.50314     3   419 arkansas      <NA>
418  -90.13772 36.47449     3   420 arkansas      <NA>
419  -90.12626 36.40574     3   421 arkansas      <NA>
420  -90.10334 36.40001     3   422 arkansas      <NA>
421  -90.06897 36.38855     3   423 arkansas      <NA>
422  -90.06897 36.34844     3   424 arkansas      <NA>
423  -90.09188 36.28542     3   425 arkansas      <NA>
424  -90.12054 36.27396     3   426 arkansas      <NA>
425  -90.13772 36.21666     3   427 arkansas      <NA>
426  -90.18929 36.19947     3   428 arkansas      <NA>
427  -90.18929 36.17656     3   429 arkansas      <NA>
428  -90.26377 36.13072     3   430 arkansas      <NA>
429  -90.37836 36.00467     3   431 arkansas      <NA>
430  -90.38409 36.00467     3   432 arkansas      <NA>
431  -90.29815 35.99894     3   433 arkansas      <NA>
432  -89.97157 35.99894     3   434 arkansas      <NA>
433  -89.96583 36.00467     3   435 arkansas      <NA>
434  -89.79395 36.00467     3   436 arkansas      <NA>
435  -89.75384 36.00467     3   437 arkansas      <NA>
436  -89.71946 36.00467     3   438 arkansas      <NA>
437  -89.71373 35.98748     3   439 arkansas      <NA>
438  -89.68508 35.97029     3   440 arkansas      <NA>
439  -89.65644 35.94737     3   441 arkansas      <NA>
440  -89.65070 35.92445     3   442 arkansas      <NA>
441  -89.66217 35.91299     3   443 arkansas      <NA>
442  -89.70801 35.90726     3   444 arkansas      <NA>
443  -89.73665 35.90154     3   445 arkansas      <NA>
444  -89.73665 35.87862     3   446 arkansas      <NA>
445  -89.73092 35.85570     3   447 arkansas      <NA>
446  -89.73092 35.83278     3   448 arkansas      <NA>
447  -89.74238 35.82132     3   449 arkansas      <NA>
448  -89.78249 35.80413     3   450 arkansas      <NA>
449  -89.79967 35.79268     3   451 arkansas      <NA>
450  -89.79967 35.77549     3   452 arkansas      <NA>
451  -89.82832 35.76403     3   453 arkansas      <NA>
452  -89.90854 35.74684     3   454 arkansas      <NA>
453  -89.93719 35.73538     3   455 arkansas      <NA>
454  -89.94292 35.70673     3   456 arkansas      <NA>
455  -89.93719 35.68954     3   457 arkansas      <NA>
456  -89.91427 35.66662     3   458 arkansas      <NA>
457  -89.87416 35.66662     3   459 arkansas      <NA>
458  -89.86270 35.65517     3   460 arkansas      <NA>
459  -89.86843 35.64370     3   461 arkansas      <NA>
460  -89.89708 35.59787     3   462 arkansas      <NA>
461  -89.90281 35.57495     3   463 arkansas      <NA>
462  -89.90854 35.55203     3   464 arkansas      <NA>
463  -89.90854 35.53484     3   465 arkansas      <NA>
464  -89.91427 35.52338     3   466 arkansas      <NA>
465  -89.96010 35.51765     3   467 arkansas      <NA>
466  -89.98302 35.51765     3   468 arkansas      <NA>
467  -89.99448 35.49474     3   469 arkansas      <NA>
468  -90.01167 35.43744     3   470 arkansas      <NA>
469  -90.02886 35.42025     3   471 arkansas      <NA>
470  -90.04604 35.40879     3   472 arkansas      <NA>
471  -90.06897 35.41452     3   473 arkansas      <NA>
472  -90.08042 35.42598     3   474 arkansas      <NA>
473  -90.08615 35.47182     3   475 arkansas      <NA>
474  -90.09761 35.47755     3   476 arkansas      <NA>
475  -90.12054 35.47182     3   477 arkansas      <NA>
476  -90.12626 35.44890     3   478 arkansas      <NA>
477  -90.13772 35.43744     3   479 arkansas      <NA>
478  -90.16064 35.42598     3   480 arkansas      <NA>
479  -90.18356 35.41452     3   481 arkansas      <NA>
480  -90.18929 35.39733     3   482 arkansas      <NA>
481  -90.18929 35.38588     3   483 arkansas      <NA>
482  -90.16637 35.36868     3   484 arkansas      <NA>
483  -90.14918 35.36868     3   485 arkansas      <NA>
484  -90.10908 35.39733     3   486 arkansas      <NA>
485  -90.10334 35.36868     3   487 arkansas      <NA>
486  -90.09188 35.36296     3   488 arkansas      <NA>
487  -90.09761 35.34577     3   489 arkansas      <NA>
488  -90.09761 35.30566     3   490 arkansas      <NA>
489  -90.08615 35.25409     3   491 arkansas      <NA>
490  -90.06897 35.21972     3   492 arkansas      <NA>
491  -90.05750 35.19680     3   493 arkansas      <NA>
492  -90.06897 35.17961     3   494 arkansas      <NA>
493  -90.07470 35.16242     3   495 arkansas      <NA>
494  -90.06324 35.13950     3   496 arkansas      <NA>
495  -90.08615 35.11086     3   497 arkansas      <NA>
496  -90.10334 35.08794     3   498 arkansas      <NA>
497  -90.12054 35.08220     3   499 arkansas      <NA>
498  -90.14918 35.08220     3   500 arkansas      <NA>
499  -90.17783 35.08794     3   501 arkansas      <NA>
500  -90.19501 35.08220     3   502 arkansas      <NA>

Groups & States - Contiguity

colSums( table( states$group, states$region ) > 0 ) 
             alabama              arizona             arkansas 
                   1                    1                    1 
          california             colorado          connecticut 
                   1                    1                    1 
            delaware district of columbia              florida 
                   1                    1                    1 
             georgia                idaho             illinois 
                   1                    1                    1 
             indiana                 iowa               kansas 
                   1                    1                    1 
            kentucky            louisiana                maine 
                   1                    1                    1 
            maryland        massachusetts             michigan 
                   1                    3                    2 
           minnesota          mississippi             missouri 
                   1                    1                    1 
             montana             nebraska               nevada 
                   1                    1                    1 
       new hampshire           new jersey           new mexico 
                   1                    1                    1 
            new york       north carolina         north dakota 
                   4                    3                    1 
                ohio             oklahoma               oregon 
                   1                    1                    1 
        pennsylvania         rhode island       south carolina 
                   1                    1                    1 
        south dakota            tennessee                texas 
                   1                    1                    1 
                utah              vermont             virginia 
                   1                    1                    3 
          washington        west virginia            wisconsin 
                   5                    1                    1 
             wyoming 
                   1

Example Data - Maps

A basic map, notice the use of group.

 

ggplot( states, aes(x=long, y=lat, group=group) ) + 
  geom_polygon( fill="lightgray", color="black", lwd=0.25) + 
  theme_void() -> p

 

This groups_by() for the purpose of plotting each individual polygon.

Azimuth Projections

Projected onto a 2-dimensional plane that is tangential to a single point on the surface of the earth (commonly a pole, though not a necessity).

 

p + coord_map( "azequalarea")

Cylindrical Projections

 

p + coord_map( "cylindrical" )

Parallels are straight lines and horizontal up and down the plot from latitude = 0

Conic Projections

Symmetric around the Prime Meridian and parallels are segments of concentric circles.

 

p + coord_map( "conic", lat0 = 30)

Coordinate Systems

The Datum are the coordinate system used on the ellipsoid.

 

For real data, you must have a datum specified.

Longitude & Latitude

The East/West & North/South position on the surface of the earth.

  • Prime Meridian (0° Longitude) passes thorugh the Royal Observatory in Greenwich England, with positive values of longitude to the east and negative to the west.

  • Equator (0° Latitude) and is defined as the point on the planet where both northern and southern hemisphers have equal amounts of day and night at the equinox (Sept. 21 & March 21).

  • Richmond, Virginia: 37.53°Lat, -77.47°Lon

Universal Trans Mercator

A division of the earth into 60 zones (~6°longitude each, labeled 01 - 60) and 20 bands each of which is ~8° latitude (labeled C-X excluding I & O with A & B dividing up Antartica). See image here.

  • Coordinates include Zone & band designation as well as coordinates in Easting and Northing (planar coordinates within the zone) measured in meters.

  • Richmond, Virginia: 18S 282051 4156899

Dyer’s First Law of Geospatial Analysis

Beetle Data

Back to our old friend, the Sonoran Desert bark beetle.

Araptus attenuatus

Euphorbia lomelii

Sampling Sites

library( readr )
url <- "https://raw.githubusercontent.com/dyerlab/ENVS-Lectures/master/data/Araptus_Disperal_Bias.csv"
read_csv( url ) |>
  select( Site, Longitude, Latitude, 
          Males, Females, Suitability ) |>
  arrange( Latitude ) -> data 
summary( data )
     Site             Longitude         Latitude         Males      
 Length:31          Min.   :-114.3   Min.   :23.29   Min.   : 9.00  
 Class :character   1st Qu.:-113.1   1st Qu.:24.95   1st Qu.:16.00  
 Mode  :character   Median :-112.0   Median :26.64   Median :21.00  
                    Mean   :-112.0   Mean   :26.44   Mean   :25.68  
                    3rd Qu.:-110.8   3rd Qu.:27.78   3rd Qu.:31.50  
                    Max.   :-109.3   Max.   :29.33   Max.   :64.00  
    Females       Suitability    
 Min.   : 5.00   Min.   :0.0563  
 1st Qu.:15.50   1st Qu.:0.2732  
 Median :21.00   Median :0.3975  
 Mean   :23.52   Mean   :0.4276  
 3rd Qu.:29.00   3rd Qu.:0.5442  
 Max.   :63.00   Max.   :0.9019

A Simple Map - Leaflet

The leaflet library allows you to make quick, interactive maps. Here I also add a message that we will use to present to the user in the web interface when they click on a location.

data |>
  mutate( Label = paste( "Site:", Site, 
                         "<hr>\nFemales:", Females, 
                         "<br>Males: ", Males,
                         "<br>Suitability:", Suitability) ) -> df

The Data

 

This is HTML markup where <hr> makes a horizontal line and <br> makes a line break (with no extra space like you see between paragraphs).

 

head(df$Label)
[1] "Site: Aqu <hr>\nFemales: 9 <br>Males:  12 <br>Suitability: 0.7217" 
[2] "Site: 73 <hr>\nFemales: 5 <br>Males:  11 <br>Suitability: 0.1455"  
[3] "Site: 157 <hr>\nFemales: 30 <br>Males:  26 <br>Suitability: 0.881" 
[4] "Site: 153 <hr>\nFemales: 41 <br>Males:  35 <br>Suitability: 0.7325"
[5] "Site: 163 <hr>\nFemales: 21 <br>Males:  21 <br>Suitability: 0.4329"
[6] "Site: 48 <hr>\nFemales: 27 <br>Males:  18 <br>Suitability: 0.6195"

Leaflet Map

library( leaflet )
df |>
  leaflet() |>
  addMarkers( ~Longitude, ~Latitude, popup = ~Label ) |>
  addProviderTiles( "OpenTopoMap" )

The Simple Features (sf) library.

Simple Features are an open standard developed by the Open Geospatial Consortium (OGC). They define the basic types we use in geospatial analyses.

 

Basic Geometry Types

  • POINT

  • LINESTRING

  • POLYGON

Corresponding ‘multi’ versions

  • MULTIPOINT

  • MULTILINESTRING

  • MULTIPOLYGON

  • GEOMETRYCOLLECTION

Each of these basic types can be represented within a data.frame as a normal data object.

Setting sf objects

To convert from a normal data.frame with columns of numerical values to an sf object, you must indicate:
- Names of the x and y coordinates, and
- A reference to the Coordinate Reference System

data |> 
  st_as_sf( coords=c("Longitude","Latitude"), crs=4326 ) -> data

The Data

head( data )
Simple feature collection with 6 features and 4 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: -110.951 ymin: 23.2855 xmax: -109.8507 ymax: 24.21441
Geodetic CRS:  WGS 84
# A tibble: 6 × 5
  Site  Males Females Suitability             geometry
  <chr> <dbl>   <dbl>       <dbl>          <POINT [°]>
1 Aqu      12       9       0.722  (-110.1043 23.2855)
2 73       11       5       0.146 (-109.8507 24.00789)
3 157      26      30       0.881   (-110.096 24.0195)
4 153      35      41       0.732 (-110.4624 24.13389)
5 163      21      21       0.433   (-110.951 24.2115)
6 48       18      27       0.620 (-110.2725 24.21441)

Reprojecting Points

We can reproject data (that already has a coordinate reference system) into any other projection.

data |>
  st_transform( 6372 ) |>
  st_bbox()
   xmin    ymin    xmax    ymax 
1307745 1274010 1773676 1968473

Wait, What?

What is that 6372 EPSG thing you just mentioned?

EPGS.io

 

Eash of the data columns are associated with the geometry column we made. If we plot the whole data.frame, it makes replicate plots of each data column displayed in the coordinate space of the geometry.

  • Color will represent gradient in values for each plot.

  • Normal plot() parameters can be used to customize.

plot( data , pch=16, cex=2)

Linked data and geometry

plot( data$Suitability, pch=16, cex=2, bty="n" )

Linked data and geometry

plot( data["Suitability"], pch=16, cex=2)

Hello ggplot My Old Friend

ggplot( data ) + geom_sf( aes(size=Suitability) )

GGPlot and Labels

ggplot( data ) +  geom_sf_label( aes(label=Site)) + coord_map() + theme_void()

Maps

As with other ggplot activities, we can mix the data being plot by changing data and aes mappings.

 

map_data("world") |>
  filter( region == "Mexico") -> map

ggplot( ) + 
  geom_polygon( aes( x=long, 
                     y=lat, 
                     group=group ), 
                data=map, 
                fill="grey" ) + 
  geom_sf( data=data, 
           aes(color=Suitability), 
           size=2) +
  xlab("Longitude") + 
  ylab("Latitude") + 
  theme_bw( base_size = 12 ) +
  coord_sf( xlim = c(-115, -105),
            ylim = c(20, 30) )

Natural Earth

Loading sf object for background map from naturalearth libraries and then plotting. Notice: xlim and ylim are configured withing coord_sf().

library( rnaturalearth )
library( rnaturalearthdata )

world <- ne_countries( scale = "medium", returnclass = "sf")

ggplot() + 
  geom_sf( data=world ) + 
  geom_sf( data=data, 
           aes(color=Suitability)) +
  coord_sf( xlim = c(-115, -105),
            ylim = c(20, 30) ) + 
  xlab("Longitude") + 
  ylab("Latitude") + 
  theme( axis.text.x=element_text( angle=45,  
                                   hjust=1) )

Natural Earth

Answering Questions Easily

Once the data are in sf features, we can leverage tidyverse and spatial functions to easily get information from the data. Think of this as a tidyverse implementation of the Spatial Analysis Toolbox without all the menus and buttons.

Here are some examples.

Geospatial Operations

What are the bounding coordinates for all sampling locations (aka bounding box)?

data |> 
  st_bbox()
      xmin       ymin       xmax       ymax 
-114.29353   23.28550 -109.32700   29.32541

Geospatial Operations

Create a polygon with minimal area that encompasses all the sampling locations (aka convex hull).

data |> 
  st_union() |> 
  st_convex_hull() -> hull

Geospatial Operations

Where is the center of all the sampling locations?

hull |>
  st_centroid()
Geometry set for 1 feature 
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: -111.3417 ymin: 26.37741 xmax: -111.3417 ymax: 26.37741
Geodetic CRS:  WGS 84

Geospatial Operations

How much area is encompassed by all the points for these sampling locations (in km^2)?

library( units )
hull |>
  st_area() |>
  set_units( km^2 )
122130.5 [km^2]

Questions

If you have any questions, please feel free to either post them as an “Issue” on your copy of this GitHub Repository, post to the Canvas discussion board for the class, or drop me an email.

Peter Sellers looking bored