| name | longitude | latitude | aqi |
|---|---|---|---|
| River Terrace | -76.95807 | 38.89557 | 42 |
| Rockville | -77.10688 | 39.11431 | 30 |
| Springfield | -77.18347 | 38.76835 | 34 |
MDI Data Workshop
Mapping Spatial Dependence:
An Introduction to Spatial Data Analysis
Le Bao
Massive Data Institute, Georgetown University
March 19, 2024
MDI Data Workshop
- Interdependence in data
- This time: spatial dependence
- April 15-16: Next Generation Generative Models for Vision Related Tasks with Dr. Rupayan Mallick
- Other MDI events: https://mdi.georgetown.edu/mdi-events/
- March 26: AI & Me: Equity and Representation
- April 9: AI & Me: Creativity and Content
Plan
- The basics of spatial data
- Exploring spatial dependence
- Spatial autocorrelation
- Modeling spatial dependence
- Spatial interpolation
- Spatial regression
- Workshop materials: https://baole.io/teaching/mdi-workshop-2024s/
R(vspython).
Basics: Types of Spatial Data
- Vector data
- Spatial objects are often depicted using vector data, which details the object’s “geometry” or “shape” and includes additional variables, or “attributes”.
- At its simplest level, vector data comprises of individual points stored as coordinate pairs that indicate a physical location in the world.
Vector Data
plot(dmv_air$longitude, dmv_air$latitude, pch=20,
col=heat.colors(3)[rank(dmv_air$aqi)],
xlim=c(-77.25, -76.75), main='Air Quality Index',
ylab = "Latitude", xlab = "Longitude", bg="white")
text(dmv_air$longitude, dmv_air$latitude, labels=dmv_air$name,
pos=4, cex=0.7, offset = 0.5)
legend("topright", legend=unique(aqi), fill=heat.colors(3)[order(unique(aqi))],
title="AQI", bg="white")
Basics: Types of Spatial Data
- Raster data
- Raster data is typically used to depict continuous spatial phenomena (e.g. altitude or temperature gradients).
- A raster dataset subdivides the spatial area into a grid of uniform rectangles, each holding one or more values for the variables of interest.
- In raster data, geometry is inherently encoded rather than explicitly stored as coordinates.
Raster Data
library(raster)
dc_extent <- extent(-77.12, -76.91, 38.79, 39.05)
ras <- raster(dc_extent, nrow=25, ncol=25)
values(ras) <- runif(ncell(ras))
rasclass : RasterLayer
dimensions : 25, 25, 625 (nrow, ncol, ncell)
resolution : 0.0084, 0.0104 (x, y)
extent : -77.12, -76.91, 38.79, 39.05 (xmin, xmax, ymin, ymax)
crs : NA
source : memory
names : layer
values : 0.0003433817, 0.9985121 (min, max)Raster Data
Vector Data
- Vector data consists of points, lines, and polygons, etc.
- For example, a point might represent a location of air quality sensor with attributes including the observation date, the observed air quality, and the pollutant’s information.
- A line, a series of connected coordinate pairs, could represent a river system.
- Polygons, which are closed polylines, are often used to represent areas such as geographical areas.
Line
library(sp)
line_coords <- data.frame(
longitude = c(-77.3, -77.2086, -77.2367, -77.2555, -77.2865, -76.8),
latitude = c(38.92, 38.9041, 38.8954, 38.8816, 38.8688, 38.85)
)
line <- SpatialLines(list(Lines(Line(line_coords), ID="segment")))
lineclass : SpatialLines
features : 1
extent : -77.3, -76.8, 38.85, 38.92 (xmin, xmax, ymin, ymax)
crs : NA Line
Polygon
polygon_coords <- rbind(
c(-77.11976, 38.93434),
c(-77.04102, 38.99555),
c(-77.00255, 38.96553),
c(-76.90939, 38.89285),
c(-76.97950, 38.83781),
c(-77.03901, 38.79164),
c(-77.03907, 38.84127),
c(-77.03910, 38.86811),
c(-77.09020, 38.90421),
c(-77.11976, 38.93434)
)
dc_polygons <- SpatialPolygons(list(Polygons(list(Polygon(polygon_coords)),
ID="DC")))Polygon
plot(dc_polygons, main="Washington D.C.", col="lightblue", border="blue",
xlim = c(-77.3865, -76.8094), ylim = c(38.76835, 39.11431))
points(dmv_air$longitude, dmv_air$latitude, pch=20, col=heat.colors(3)[rank(dmv_air$aqi)])
text(dmv_air$longitude, dmv_air$latitude, labels=dmv_air$name, pos=4, cex=0.7, offset = 0.5)
lines(line_coords, col="blue", lwd=2)
Shapefile and GeoJSON
- Shapefile
Reading layer `Wards_from_2022' from data source
`/Users/baole/Desktop/spatial-workshop/data/Wards_from_2022/Wards_from_2022.shp'
using driver `ESRI Shapefile'
Simple feature collection with 8 features and 325 fields
Geometry type: POLYGON
Dimension: XY
Bounding box: xmin: -77.1198 ymin: 38.79164 xmax: -76.90915 ymax: 38.99597
Geodetic CRS: WGS 84- GEOJSON
Reading layer `neighbourhoods' from data source
`/Users/baole/Desktop/spatial-workshop/data/neighbourhoods.geojson'
using driver `GeoJSON'
Simple feature collection with 39 features and 2 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: -77.11934 ymin: 38.81801 xmax: -76.90915 ymax: 38.99597
Geodetic CRS: WGS 84Projection
- Three-dimensional angular system to a two-dimensional planar system.
- A coordinate reference system (CRS) defines, with the help of coordinates, how the two-dimensional, projected map is related to real locations on the earth.
- A planar CRS comprises a projection, a datum, and a set of parameters that specify aspects like the map’s center.
- The most commonly used datum globally is WGS84 (World Geodesic System 1984), closely related to NAD83 (North American Datum of 1983)
- Most CRSs are assigned an “EPSG code” (European Petroleum Survey Group), which provides a unique identifier to a CRS.
- EPSG:4326, also known as the WGS84 projection (because it’s based on WGS84’s ellipsoid), is a coordinate system used in Google Earth and GSP systems. EPSG:4269 is based on NAD83.
Projection and CRS
Simple feature collection with 6 features and 325 fields
Geometry type: POLYGON
Dimension: XY
Bounding box: xmin: -77.08172 ymin: 38.79164 xmax: -76.90915 ymax: 38.9573
Geodetic CRS: WGS 84
WARD NAME REP_NAME
1 8 Ward 8 Trayon White, Sr.
2 6 Ward 6 Charles Allen
3 7 Ward 7 Vincent Gray
4 2 Ward 2 Brooke Pinto
5 1 Ward 1 Brianne Nadeau
6 5 Ward 5 Zachary Parker
WEB_URL
1 https://www.dccouncil.us/council/councilmember-trayon-white-sr
2 https://www.dccouncil.us/council/councilmember-allen
3 https://dccouncil.us/council/vincent-gray
4 https://www.dccouncil.us/council/ward-2-councilmember-brooke-pinto
5 https://dccouncil.us/council/brianne-nadeau
6 https://dccouncil.us/council/kenyan-mcduffie
REP_PHONE REP_EMAIL
1 (202) 724-8045 twhite@dccouncil.us
2 (202) 724-8072 callen@dccouncil.us
3 (202) 724-8068 vgray@dccouncil.us
4 (202) 724-8058 bpinto@dccouncil.us
5 (202) 724-8181 bnadeau@dccouncil.us
6 (202) 724-8028 zparker@dccouncil.gov
REP_OFFICE WARD_ID LABEL STUSAB SUMLEV
1 1350 Pennsylvania Ave, Suite 400, NW 20004 8 Ward 8 DC 610
2 1350 Pennsylvania Ave, Suite 110, NW 20004 6 Ward 6 DC 610
3 1350 Pennsylvania Ave, Suite 406, NW 20004 7 Ward 7 DC 610
4 1350 Pennsylvania Ave, Suite 106, NW 20004 2 Ward 2 DC 610
5 1350 Pennsylvania Ave, Suite 108, NW 20004 1 Ward 1 DC 610
6 1350 Pennsylvania Ave, Suite 102, NW 20004 5 Ward 5 DC 610
GEOID GEOCODE STATE POP100 HU100 P0010001 P0010002 P0010003 P0010004
1 610U600US11008 11008 11 85246 39164 85246 81405 8096 70959
2 610U600US11006 11006 11 84266 52768 84266 77187 50624 19562
3 610U600US11007 11007 11 85685 38968 85685 81893 7038 71628
4 610U600US11002 11002 11 89485 53217 89485 81571 57597 10196
5 610U600US11001 11001 11 85285 45694 85285 76084 42307 18741
6 610U600US11005 11005 11 89617 41794 89617 82833 22248 51307
P0010005 P0010006 P0010007 P0010008 P0010009 P0010010 P0010011 P0010012
1 305 751 34 1260 3841 3341 988 148
2 216 5028 54 1703 7079 6449 851 388
3 343 591 40 2253 3792 3299 935 140
4 317 9853 64 3544 7914 7407 726 359
5 696 5273 88 8979 9201 8497 956 330
6 495 2731 87 5965 6784 6057 1189 241
P0010013 P0010014 P0010015 P0010016 P0010017 P0010018 P0010019 P0010020
1 290 22 608 513 156 26 483 4
2 1528 47 2847 228 157 11 274 13
3 219 17 654 591 150 13 486 5
4 1666 34 4056 120 95 20 193 10
5 1294 38 4558 130 197 3 638 10
6 838 22 2146 396 260 12 736 7
P0010021 P0010022 P0010023 P0010024 P0010025 P0010026 P0010027 P0010028
1 0 46 30 24 3 427 217 32
2 0 31 31 38 5 518 125 73
3 2 32 29 19 7 444 233 42
4 2 61 21 42 2 431 97 51
5 8 239 21 61 14 609 155 50
6 2 133 23 38 14 630 242 40
P0010029 P0010030 P0010031 P0010032 P0010033 P0010034 P0010035 P0010036
1 5 66 7 1 33 18 20 1
2 4 106 11 0 106 20 33 0
3 1 53 5 0 24 10 21 9
4 2 81 9 0 96 23 34 4
5 5 114 24 0 145 16 49 0
6 1 123 8 1 81 30 42 1
P0010037 P0010038 P0010039 P0010040 P0010041 P0010042 P0010043 P0010044
1 9 3 5 3 3 0 0 0
2 8 3 17 5 5 1 0 0
3 10 3 15 6 7 2 1 0
4 5 3 9 3 8 4 0 0
5 7 3 19 2 9 6 0 0
6 8 4 25 10 3 9 1 0
P0010045 P0010046 P0010047 P0010048 P0010049 P0010050 P0010051 P0010052
1 2 2 66 15 3 34 2 6
2 0 1 101 18 0 57 7 14
3 1 1 43 12 0 23 0 3
4 0 2 74 25 0 39 1 2
5 1 4 87 10 1 52 2 10
6 0 1 92 17 0 55 1 7
P0010053 P0010054 P0010055 P0010056 P0010057 P0010058 P0010059 P0010060
1 0 0 4 0 2 0 0 0
2 0 0 1 0 2 2 0 0
3 0 1 2 0 1 0 0 0
4 0 0 2 0 4 1 0 0
5 0 0 5 0 4 2 1 0
6 0 0 3 3 0 4 1 0
P0010061 P0010062 P0010063 P0010064 P0010065 P0010066 P0010067 P0010068
1 0 0 7 4 1 0 1 0
2 0 0 10 4 5 0 0 0
3 1 0 6 0 4 0 1 1
4 0 0 1 1 0 0 0 0
5 0 0 8 1 3 0 1 2
6 1 0 5 4 0 0 0 0
P0010069 P0010070 P0010071 P0020001 P0020002 P0020003 P0020004 P0020005
1 1 0 0 85246 3081 82165 79340 7670
2 1 1 1 84266 6122 78144 73945 49107
3 0 0 0 85685 4078 81607 78876 6624
4 0 1 1 89485 9744 79741 75843 55391
5 1 0 0 85285 17269 68016 64204 40017
6 1 0 0 89617 10419 79198 75285 21217
P0020006 P0020007 P0020008 P0020009 P0020010 P0020011 P0020012 P0020013
1 70357 213 731 29 340 2825 2475 935
2 19243 121 4977 50 447 4199 3885 800
3 70969 239 568 32 444 2731 2384 907
4 10000 120 9803 54 475 3898 3669 695
5 18330 163 5202 70 422 3812 3506 862
6 50554 218 2687 76 533 3913 3516 1122
P0020014 P0020015 P0020016 P0020017 P0020018 P0020019 P0020020 P0020021
1 132 272 19 104 482 152 23 297
2 338 1492 40 630 206 153 11 156
3 93 205 11 131 571 143 13 262
4 297 1633 27 662 99 87 20 85
5 238 1279 30 548 118 189 3 172
6 187 802 14 338 381 251 11 349
P0020022 P0020023 P0020024 P0020025 P0020026 P0020027 P0020028 P0020029
1 4 0 12 29 14 0 315 194
2 12 0 2 22 23 0 268 97
3 3 2 6 29 8 0 326 202
4 9 2 2 15 36 0 200 72
5 8 7 2 19 29 2 275 103
6 7 1 9 22 21 1 368 197
P0020030 P0020031 P0020032 P0020033 P0020034 P0020035 P0020036 P0020037
1 30 5 37 3 0 7 15 6
2 62 3 38 8 0 5 18 11
3 36 1 35 5 0 4 10 4
4 45 2 23 6 0 3 19 7
5 48 3 48 19 0 4 15 2
6 33 1 53 7 0 1 26 13
P0020038 P0020039 P0020040 P0020041 P0020042 P0020043 P0020044 P0020045
1 0 8 2 2 3 2 0 0
2 0 7 3 5 5 5 1 0
3 0 8 2 7 6 5 0 1
4 3 4 2 6 3 4 1 0
5 0 6 3 11 2 8 1 0
6 1 6 4 8 10 0 7 1
P0020046 P0020047 P0020048 P0020049 P0020050 P0020051 P0020052 P0020053
1 0 0 1 32 13 3 10 1
2 0 0 0 40 16 0 9 6
3 0 0 0 19 10 0 7 0
4 0 0 0 28 20 0 2 1
5 0 0 2 27 5 0 10 2
6 0 0 0 27 13 0 8 0
P0020054 P0020055 P0020056 P0020057 P0020058 P0020059 P0020060 P0020061
1 5 0 0 0 0 0 0 0
2 8 0 0 0 0 0 1 0
3 1 0 1 0 0 0 0 0
4 2 0 0 0 0 2 1 0
5 7 0 0 0 0 1 2 0
6 5 0 0 0 0 0 0 1
P0020062 P0020063 P0020064 P0020065 P0020066 P0020067 P0020068 P0020069
1 0 0 0 3 2 0 0 1
2 0 0 0 6 4 1 0 0
3 0 0 0 2 0 0 0 1
4 0 0 0 1 1 0 0 0
5 0 0 0 4 1 0 0 1
6 0 0 0 2 2 0 0 0
P0020070 P0020071 P0020072 P0020073 P0030001 P0030002 P0030003 P0030004
1 0 0 0 0 62507 59754 6972 50933
2 0 1 0 0 74262 68775 45847 16404
3 1 0 0 0 66683 63965 5876 55656
4 0 0 0 0 83628 76770 54915 9062
5 2 0 0 0 75254 67609 39386 15702
6 0 0 0 0 74096 69224 19777 41961
P0030005 P0030006 P0030007 P0030008 P0030009 P0030010 P0030011 P0030012
1 237 646 25 941 2753 2413 675 111
2 177 4765 50 1532 5487 4988 610 332
3 265 504 33 1631 2718 2375 667 90
4 272 9442 53 3026 6858 6476 630 322
5 543 4954 82 6942 7645 7102 757 291
6 414 2490 75 4507 4872 4360 798 193
P0030013 P0030014 P0030015 P0030016 P0030017 P0030018 P0030019 P0030020
1 177 19 464 378 104 19 375 4
2 966 42 2422 181 116 7 221 12
3 114 12 498 469 92 11 352 3
4 1372 31 3677 97 61 16 159 10
5 1001 31 3905 94 167 3 554 9
6 480 14 1605 327 181 10 582 6
P0030021 P0030022 P0030023 P0030024 P0030025 P0030026 P0030027 P0030028
1 0 41 25 19 2 293 161 14
2 0 24 24 30 1 404 101 48
3 1 21 24 14 7 306 159 33
4 1 48 20 30 2 317 73 32
5 6 203 19 49 13 469 129 35
6 2 100 22 28 12 449 181 23
P0030029 P0030030 P0030031 P0030032 P0030033 P0030034 P0030035 P0030036
1 5 45 4 1 24 10 9 0
2 4 91 4 0 96 11 18 0
3 1 36 2 0 15 6 14 8
4 1 58 7 0 79 16 20 4
5 4 87 15 0 134 8 29 0
6 0 95 3 1 59 23 25 1
P0030037 P0030038 P0030039 P0030040 P0030041 P0030042 P0030043 P0030044
1 8 3 3 2 2 0 0 0
2 5 3 12 4 5 1 0 0
3 7 2 12 2 7 2 0 0
4 5 3 7 2 5 3 0 0
5 4 2 8 1 4 6 0 0
6 5 2 20 7 0 2 1 0
P0030045 P0030046 P0030047 P0030048 P0030049 P0030050 P0030051 P0030052
1 0 2 43 10 3 24 1 2
2 0 1 88 15 0 52 5 12
3 0 0 32 7 0 20 0 3
4 0 2 64 24 0 33 1 0
5 0 3 68 4 1 44 2 8
6 0 1 60 11 0 40 0 3
P0030053 P0030054 P0030055 P0030056 P0030057 P0030058 P0030059 P0030060
1 0 0 1 0 2 0 0 0
2 0 0 1 0 2 1 0 0
3 0 0 1 0 1 0 0 0
4 0 0 2 0 3 1 0 0
5 0 0 3 0 4 2 0 0
6 0 0 2 3 0 1 0 0
P0030061 P0030062 P0030063 P0030064 P0030065 P0030066 P0030067 P0030068
1 0 0 4 3 1 0 0 0
2 0 0 7 3 4 0 0 0
3 0 0 5 0 4 0 0 1
4 0 0 1 1 0 0 0 0
5 0 0 6 1 3 0 0 2
6 0 0 3 3 0 0 0 0
P0030069 P0030070 P0030071 P0040001 P0040002 P0040003 P0040004 P0040005
1 0 0 0 62507 2195 60312 58274 6649
2 0 0 0 74262 5257 69005 65905 44499
3 0 0 0 66683 2938 63745 61796 5529
4 0 0 0 83628 8618 75010 71714 52844
5 0 0 0 75254 13901 61353 58271 37464
6 0 0 0 74096 7776 66320 63538 18976
P0040006 P0040007 P0040008 P0040009 P0040010 P0040011 P0040012 P0040013
1 50546 178 628 23 250 2038 1779 651
2 16156 102 4722 48 378 3100 2854 581
3 55244 197 484 25 317 1949 1718 648
4 8908 111 9397 46 408 3296 3121 605
5 15389 129 4890 65 334 3082 2860 690
6 41437 180 2453 69 423 2782 2492 763
P0040014 P0040015 P0040016 P0040017 P0040018 P0040019 P0040020 P0040021
1 99 168 17 83 358 100 16 241
2 303 947 36 514 169 114 7 131
3 65 106 10 98 454 86 11 201
4 271 1344 26 585 83 59 16 80
5 218 993 24 473 86 163 3 150
6 151 462 7 261 317 175 9 292
P0040022 P0040023 P0040024 P0040025 P0040026 P0040027 P0040028 P0040029
1 4 0 7 25 10 0 233 152
2 11 0 2 18 21 0 207 82
3 1 1 5 24 8 0 218 136
4 9 1 1 15 26 0 150 56
5 7 5 2 18 26 2 203 88
6 6 1 9 21 17 1 273 157
P0040030 P0040031 P0040032 P0040033 P0040034 P0040035 P0040036 P0040037
1 14 5 26 3 0 6 7 4
2 40 3 34 4 0 5 10 9
3 27 1 21 2 0 2 6 2
4 26 1 17 5 0 3 15 7
5 34 2 37 12 0 3 8 2
6 17 0 45 3 0 1 20 9
P0040038 P0040039 P0040040 P0040041 P0040042 P0040043 P0040044 P0040045
1 0 7 2 2 2 2 0 0
2 0 4 3 3 4 5 1 0
3 0 7 1 6 2 5 0 0
4 3 4 2 5 2 4 0 0
5 0 3 2 4 1 4 1 0
6 1 3 2 7 7 0 0 1
P0040046 P0040047 P0040048 P0040049 P0040050 P0040051 P0040052 P0040053
1 0 0 1 25 9 3 10 1
2 0 0 0 35 13 0 8 5
3 0 0 0 12 6 0 5 0
4 0 0 0 24 20 0 1 1
5 0 0 2 16 1 0 5 2
6 0 0 0 16 7 0 6 0
P0040054 P0040055 P0040056 P0040057 P0040058 P0040059 P0040060 P0040061
1 2 0 0 0 0 0 0 0
2 8 0 0 0 0 0 1 0
3 1 0 0 0 0 0 0 0
4 0 0 0 0 0 1 1 0
5 5 0 0 0 0 1 2 0
6 3 0 0 0 0 0 0 0
P0040062 P0040063 P0040064 P0040065 P0040066 P0040067 P0040068 P0040069
1 0 0 0 1 1 0 0 0
2 0 0 0 4 3 1 0 0
3 0 0 0 1 0 0 0 0
4 0 0 0 1 1 0 0 0
5 0 0 0 3 1 0 0 0
6 0 0 0 1 1 0 0 0
P0040070 P0040071 P0040072 P0040073 H0010001 H0010002 H0010003 P0050001
1 0 0 0 0 39164 34608 4556 3362
2 0 0 0 0 52768 45285 7483 1991
3 1 0 0 0 38968 34958 4010 4528
4 0 0 0 0 53217 46543 6674 13218
5 2 0 0 0 45694 41035 4659 5066
6 0 0 0 0 41794 37211 4583 6759
P0050002 P0050003 P0050004 P0050005 P0050006 P0050007 P0050008 P0050009
1 1054 23 32 750 249 2308 0 563
2 194 31 29 134 0 1797 655 255
3 2557 2197 34 289 37 1971 0 0
4 205 27 0 178 0 13013 11470 0
5 305 0 173 132 0 4761 4009 0
6 1007 0 40 967 0 5752 3053 0
P0050010 OBJECTID GLOBALID CREATED_US
1 1745 1 {E31550AE-6FAE-4B74-909F-52B283BFAF68} <NA>
2 887 2 {765C4F49-9292-4BDB-AA24-39F4EE43359F} <NA>
3 1971 3 {73F07042-7D7F-452B-9BB3-0F87B0EC5418} <NA>
4 1543 4 {7F8C2A51-427C-45FC-91EB-9693656AED9C} <NA>
5 752 5 {C3C6E2E7-E68D-49B2-970C-D60675EA7B4B} <NA>
6 2699 6 {6C10DD95-DE70-4F26-94BA-79F378FA74E0} <NA>
CREATED_DA LAST_EDITE LAST_EDI_1 SHAPEAREA SHAPELEN
1 <NA> <NA> <NA> 0 0
2 <NA> JLAY 2023-12-07 0 0
3 <NA> <NA> <NA> 0 0
4 <NA> <NA> <NA> 0 0
5 <NA> JLAY 2023-12-07 0 0
6 <NA> JLAY 2023-12-07 0 0
geometry
1 POLYGON ((-76.99392 38.8777...
2 POLYGON ((-77.00908 38.8716...
3 POLYGON ((-76.94186 38.9185...
4 POLYGON ((-77.0327 38.88262...
5 POLYGON ((-77.03523 38.9374...
6 POLYGON ((-76.99144 38.9573...Coordinate Reference System:
User input: WGS 84
wkt:
GEOGCRS["WGS 84",
DATUM["World Geodetic System 1984",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
CS[ellipsoidal,2],
AXIS["geodetic latitude (Lat)",north,
ORDER[1],
ANGLEUNIT["degree",0.0174532925199433]],
AXIS["geodetic longitude (Lon)",east,
ORDER[2],
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4326]]Exploring Spatial Dependence
Tobler’s first law of geography
“Everything is related to everything else, but near things are more related than distant things.”
- Tobler (1970)Examples of spatial dependence: crime patterns, disease spread, housing market, political orientations, transportation, environmental equity, etc.
We oftentimes even use spatial dependence without realizing it:
Spatial Weights Matrix
A vast number of spatial analysis methods are based on a spatial matrix \mathbf{W}_{ij} of size n \times n with i an observation and j the “neighbours” of that observation. \mathbf{W}_{ij} represents the degree of spatial relationship between i and j.
Classically one can define :
- Neighbouring matrix (\mathbf{W}_{ij} = 1 if j is a neighbour of i, 0 otherwise)
- Distance matrix (\mathbf{W}_{ij} = the distance between i and j, modified by a function like \frac{1}{distance} or \frac{1}{distance^2})
- Interaction matrix (\mathbf{W}_{ij} = the degree of interaction between i and j, the measure of the interaction depends on the subject of the analysis)
Spatial Weights Matrix: Neighborhood
Neighborhood
Neighborhood
id <- 25 # select an area
columbus$neighbors <- "other"
columbus$neighbors[id] <- "area"
columbus$neighbors[nb[[id]]] <- "neighbors"
cols <- c("gray30", "gray", "white")
plot(st_geometry(columbus), col=cols[as.factor(columbus$neighbors)], main="Spatial Neighborhoods")
legend("topright", legend=c("Selected Area", "Neighbors", "Other"), fill=cols, bty="n")
Neighborhood
- Neighborhood Matrix
k Nearest Neighbors
- E.g.: neighbors based on 3 nearest neighbors
