Week 2: How Do Maps Work?

PPOL 6805 / DSAN 6750: GIS for Spatial Data Science
Fall 2026

Author

Affiliation

Jeff Jacobs

jj1088@georgetown.edu

Open slides in new tab →

Schedule

Today’s Planned Schedule:

	Start	End	Topic
Lecture	6:30pm	7:00pm	Logistics 1: Positron →
	6:50pm	7:15pm	Logistics 2: Reducing Fear →
	7:20pm	7:50pm	Building Our First Map! →
Break!	7:50pm	8:00pm
	8:00pm	8:30pm	Raster Data →
	8:30pm	9:00pm	Finding GIS Data →

source("../dsan-globals/_globals.r")
set.seed(6805)

\[ \DeclareMathOperator*{\argmax}{argmax} \DeclareMathOperator*{\argmin}{argmin} \newcommand{\bigexp}[1]{\exp\mkern-4mu\left[ #1 \right]} \newcommand{\bigexpect}[1]{\mathbb{E}\mkern-4mu \left[ #1 \right]} \newcommand{\definedas}{\overset{\small\text{def}}{=}} \newcommand{\definedalign}{\overset{\phantom{\text{defn}}}{=}} \newcommand{\eqeventual}{\overset{\text{eventually}}{=}} \newcommand{\Err}{\text{Err}} \newcommand{\expect}[1]{\mathbb{E}[#1]} \newcommand{\expectsq}[1]{\mathbb{E}^2[#1]} \newcommand{\fw}[1]{\texttt{#1}} \newcommand{\given}{\mid} \newcommand{\green}[1]{\color{green}{#1}} \newcommand{\heads}{\outcome{heads}} \newcommand{\iid}{\overset{\text{\small{iid}}}{\sim}} \newcommand{\lik}{\mathcal{L}} \newcommand{\loglik}{\ell} \DeclareMathOperator*{\maximize}{maximize} \DeclareMathOperator*{\minimize}{minimize} \newcommand{\mle}{\textsf{ML}} \newcommand{\nimplies}{\;\not\!\!\!\!\implies} \newcommand{\orange}[1]{\color{orange}{#1}} \newcommand{\outcome}[1]{\textsf{#1}} \newcommand{\param}[1]{{\color{purple} #1}} \newcommand{\pgsamplespace}{\{\green{1},\green{2},\green{3},\purp{4},\purp{5},\purp{6}\}} \newcommand{\pedge}[2]{\require{enclose}\enclose{circle}{~{#1}~} \rightarrow \; \enclose{circle}{\kern.01em {#2}~\kern.01em}} \newcommand{\pnode}[1]{\require{enclose}\enclose{circle}{\kern.1em {#1} \kern.1em}} \newcommand{\ponode}[1]{\require{enclose}\enclose{box}[background=lightgray]{{#1}}} \newcommand{\pnodesp}[1]{\require{enclose}\enclose{circle}{~{#1}~}} \newcommand{\purp}[1]{\color{purple}{#1}} \newcommand{\sign}{\text{Sign}} \newcommand{\spacecap}{\; \cap \;} \newcommand{\spacewedge}{\; \wedge \;} \newcommand{\tails}{\outcome{tails}} \newcommand{\Var}[1]{\text{Var}[#1]} \newcommand{\bigVar}[1]{\text{Var}\mkern-4mu \left[ #1 \right]} \]

Logistics 1: Positron

• For doing HW1, now released!

GIS as an “Umbrella Term”

• Libraries and tools we’ll use: specific systems/methods for geospatial analysis
• GIS is an “umbrella term”, which just vaguely refers to this entire universe of libraries/tools/techniques/approaches

Umbrella Term	Concepts	Specific Skills
Coding	Variables Control Flow Algorithms	Python R JavaScript
GIS	Projections Vector vs. Raster Spatial Data Formats (shapefiles, .geojson)	ArcGIS GeoPandas (Python) sf (R)

ArcGIS…

• For info on Georgetown’s provision of ArcGIS (Online, Pro, and Desktop), see the Library Guide

Ukraine Level-1 Administrative Regions Map (see CDTO talk)

Then… Why Can’t We Just Use ArcGIS?

Analogy from non-geospatial data science:

Text Drawn Map	→	Speadsheet Digital Map	→	Equations Maps w/ArcGIS	→	Code This Class
Start writing info.txt I gave Ana $3, then Ana paid me back $2. [...] Realize there’s regularity/structure 🤔		Start entering info in rows Fr To Amt Bal Me Ana $3 -$3 Ana Me $2 -$1 Realize you’re manually computing things that could be automated 🤔		Start using equations Fr To Amt Bal Me Ana $3 =0-C1 Ana Me $2 =D1-C2 Realize you need fancier equations, and/or need to coordinate with inputs (APIs), outputs (plotting libraries) 🤔		Write code plot_balance.py import pandas as pd df = pd.read_csv(...) calc_weekly_balance() df.plot() Profit 💲💰🤑💰💲

Course Policy Things

• How To Not Be Scared of Prerequisites
• AI Things
• Learning How To Learn

Pedagogical Principles

• There’s literally no such thing as “intelligence”
• Anyone is capable of learning anything (neural plasticity)
• Growth mindset: “I can’t do this” \(\leadsto\) “I can’t do this yet!”
• The point of a class is learning: understanding something about the world, either (a) For its own sake (end in itself) or (b) Because it’s relevant to something you care about (means to an end)

Our teaching should be governed, not by a desire to make students learn things, but by the endeavor to keep burning within them that light which is called curiosity. (Montessori 1916)

AI and Whatnot

• If you feel like AI things will help you learn something in the course, then use them!
• If you feel like you’re using them as a “crutch”, try to hold yourself accountable for not using it!

Take the time/energy you're using to worry about...	Use it instead to worry about...
AI Policies Collaboration Policies Plagiarism	Learning GIS

On Not Worrying About Prereqs

• I genuinely believe that I can make the course accessible to you, meeting you wherever you’re at, no matter what!
• Everyone learns at their own pace (who says 14 weeks is “correct” amount of time to learn GIS?), and I structure my courses as best as I possibly can to adapt to your pace
• \(\Rightarrow\) Assessments (HW, Midterm) valuable in two ways:
• [Valuable for you] As an accountability mechanism to make sure you’re learn the material (how do we know when we’ve learned something? When we can answer questions about it / use it to accomplish things!)
• [Valuable for me] For assessing and updating pace

R and/or Python and/or JS

• My Geometry vs. Algebra Rant… Euclid’s Elements, Book VI, Proposition 28.
• The problem: Divide a given straight line so that the rectangle contained by its segments may be equal to a given area, not exceeding the square of half the line.

Geometers solved w/geometry (300 BC)…

…Algebraists solved w/algebra (2000 BC)…

\[ \begin{align*} &ax^2 + bx + c = 0 \\ \Rightarrow \; & x_+ = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \end{align*} \]

…From 1637 onwards, whichever is easier! 🤯🤯🤯 (Isomorphism)

Figure 1: Circle with radius 1? Or \((x,y)\) satisfying \(x^2 + y^2 = 1\)?

Learning How To Learn

Figure 2: From The Carter (Documentary)

He’s Literally Extremely Correct!

From Elsevier Osmosis: Spaced Repetition

Let’s Make Some Dang Maps!

Our First Map: Polygons!

library(tidyverse)
library(rmarkdown)
library(sf)
# Load DC tracts data
dc_sf_fpath <- "data/DC_Census_2020/Census_Tracts_in_2020.shp"
dc_sf <- st_read(dc_sf_fpath);

Reading layer `Census_Tracts_in_2020' from data source 
  `/Users/jpj/gtown-local/ppol6805/w02/data/DC_Census_2020/Census_Tracts_in_2020.shp' 
  using driver `ESRI Shapefile'
Simple feature collection with 206 features and 315 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: -8584933 ymin: 4691871 xmax: -8561515 ymax: 4721078
Projected CRS: WGS 84 / Pseudo-Mercator

cols_to_keep <- c(
  "OBJECTID", "TRACT", "GEOID", "ALAND", "AWATER",
  "STUSAB", "SUMLEV", "GEOCODE", "STATE", "NAME",
  "POP100", "HU100", "geometry"
)
dc_sf <- dc_sf |> select(cols_to_keep)

Warning: Using an external vector in selections was deprecated in tidyselect 1.1.0.
ℹ Please use `all_of()` or `any_of()` instead.
  # Was:
  data %>% select(cols_to_keep)

  # Now:
  data %>% select(all_of(cols_to_keep))

See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.

sf Objects

dc_sf is an R object of type sf (short for “simple features”), which extends data.frame by adding a special column named geometry (containing POLYGONs)

class(dc_sf)

[1] "sf"         "data.frame"

dim(dc_sf)

[1] 206  13

rmarkdown::paged_table(dc_sf, options=list(rows.print=6))

OBJECTID	TRACT	GEOID	ALAND	AWATER	STUSAB	SUMLEV	GEOCODE	STATE	NAME	POP100	HU100	geometry
1	002002	11001002002	849376	0	DC	140	11001002002	11	Census Tract 20.02	4072	1532	POLYGON ((-8575655 4714476,…
2	002101	11001002101	600992	0	DC	140	11001002101	11	Census Tract 21.01	5687	2335	POLYGON ((-8574745 4715676,…
3	002102	11001002102	725975	0	DC	140	11001002102	11	Census Tract 21.02	5099	2221	POLYGON ((-8573824 4715684,…
4	002201	11001002201	415173	0	DC	140	11001002201	11	Census Tract 22.01	3485	1229	POLYGON ((-8574654 4714781,…
5	002202	11001002202	698895	566	DC	140	11001002202	11	Census Tract 22.02	3339	1454	POLYGON ((-8573792 4714811,…
6	000101	11001000101	199776	5261	DC	140	11001000101	11	Census Tract 1.01	1406	999	POLYGON ((-8577962 4708867,…
7	000102	11001000102	1706484	516665	DC	140	11001000102	11	Census Tract 1.02	3417	2053	POLYGON ((-8579312 4707442,…
8	000201	11001000201	505004	0	DC	140	11001000201	11	Census Tract 2.01	4108	11	POLYGON ((-8580425 4709172,…
9	000202	11001000202	776435	439661	DC	140	11001000202	11	Census Tract 2.02	4672	2169	POLYGON ((-8580496 4708084,…
10	000300	11001000300	1042157	2305	DC	140	11001000300	11	Census Tract 3	6161	2845	POLYGON ((-8580799 4710391,…
11	000400	11001000400	1541239	69	DC	140	11001000400	11	Census Tract 4	1643	998	POLYGON ((-8579749 4711309,…
12	000501	11001000501	940249	29503	DC	140	11001000501	11	Census Tract 5.01	3829	2489	POLYGON ((-8578168 4709976,…
13	000502	11001000502	581507	0	DC	140	11001000502	11	Census Tract 5.02	3418	1881	POLYGON ((-8578993 4711789,…
14	000600	11001000600	1441865	71	DC	140	11001000600	11	Census Tract 6	4676	2435	POLYGON ((-8580118 4713399,…
15	000702	11001000702	308967	0	DC	140	11001000702	11	Census Tract 7.02	3452	2472	POLYGON ((-8580787 4710465,…
16	000703	11001000703	232137	0	DC	140	11001000703	11	Census Tract 7.03	3022	2136	POLYGON ((-8580690 4711115,…
17	000704	11001000704	332346	0	DC	140	11001000704	11	Census Tract 7.04	2753	1724	POLYGON ((-8580477 4712145,…
18	000802	11001000802	1688582	669985	DC	140	11001000802	11	Census Tract 8.02	3327	1541	POLYGON ((-8582829 4708962,…
19	000803	11001000803	308153	0	DC	140	11001000803	11	Census Tract 8.03	3174	1692	POLYGON ((-8581367 4712478,…
20	000804	11001000804	2566768	167978	DC	140	11001000804	11	Census Tract 8.04	3832	1347	POLYGON ((-8583308 4709826,…
21	000902	11001000902	1867106	307257	DC	140	11001000902	11	Census Tract 9.02	2317	894	POLYGON ((-8584932 4712273,…
22	000903	11001000903	928101	0	DC	140	11001000903	11	Census Tract 9.03	4202	484	POLYGON ((-8582848 4712389,…
23	000904	11001000904	2024710	75151	DC	140	11001000904	11	Census Tract 9.04	3148	1216	POLYGON ((-8584133 4713114,…
24	001002	11001001002	889722	0	DC	140	11001001002	11	Census Tract 10.02	3643	2544	POLYGON ((-8581134 4712828,…
25	001003	11001001003	982460	0	DC	140	11001001003	11	Census Tract 10.03	3111	1129	POLYGON ((-8582803 4714393,…
26	001004	11001001004	1471925	0	DC	140	11001001004	11	Census Tract 10.04	4480	1810	POLYGON ((-8581944 4713535,…
27	001100	11001001100	1676650	0	DC	140	11001001100	11	Census Tract 11	5300	2414	POLYGON ((-8581145 4716056,…
28	001200	11001001200	1219204	0	DC	140	11001001200	11	Census Tract 12	5371	2672	POLYGON ((-8580376 4714016,…
29	001301	11001001301	2855501	26550	DC	140	11001001301	11	Census Tract 13.01	4160	2187	POLYGON ((-8579329 4715134,…
30	001303	11001001303	889581	5274	DC	140	11001001303	11	Census Tract 13.03	3763	2532	POLYGON ((-8578912 4714212,…
31	001304	11001001304	707155	9432	DC	140	11001001304	11	Census Tract 13.04	4018	3087	POLYGON ((-8578455 4713220,…
32	001401	11001001401	788095	0	DC	140	11001001401	11	Census Tract 14.01	3258	1477	POLYGON ((-8580220 4716990,…
33	001402	11001001402	895209	0	DC	140	11001001402	11	Census Tract 14.02	3514	1791	POLYGON ((-8579762 4716090,…
34	001500	11001001500	4684714	39895	DC	140	11001001500	11	Census Tract 15	6156	2282	POLYGON ((-8579577 4717632,…
35	001600	11001001600	2667590	7820	DC	140	11001001600	11	Census Tract 16	4471	1720	POLYGON ((-8577348 4719831,…
36	001702	11001001702	939295	0	DC	140	11001001702	11	Census Tract 17.02	3356	1737	POLYGON ((-8574419 4717383,…
37	001803	11001001803	1136289	31909	DC	140	11001001803	11	Census Tract 18.03	4102	1640	POLYGON ((-8576903 4716627,…
38	001804	11001001804	601755	0	DC	140	11001001804	11	Census Tract 18.04	5500	2225	POLYGON ((-8575321 4716259,…
39	001901	11001001901	769304	0	DC	140	11001001901	11	Census Tract 19.01	4206	1644	POLYGON ((-8574721 4716094,…
40	001902	11001001902	877346	0	DC	140	11001001902	11	Census Tract 19.02	2117	883	POLYGON ((-8573816 4715855,…
41	002001	11001002001	632724	1132	DC	140	11001002001	11	Census Tract 20.01	2926	1130	POLYGON ((-8576430 4716153,…
42	002301	11001002301	407679	0	DC	140	11001002301	11	Census Tract 23.01	3266	1254	POLYGON ((-8573822 4713535,…
43	002302	11001002302	2010924	5298	DC	140	11001002302	11	Census Tract 23.02	1762	686	POLYGON ((-8573742 4712625,…
44	002400	11001002400	556408	0	DC	140	11001002400	11	Census Tract 24	4095	1724	POLYGON ((-8574562 4713958,…
45	002501	11001002501	543460	0	DC	140	11001002501	11	Census Tract 25.01	2688	1099	POLYGON ((-8575656 4713502,…
46	002503	11001002503	292105	0	DC	140	11001002503	11	Census Tract 25.03	3031	1523	POLYGON ((-8574908 4712629,…
47	002504	11001002504	311721	0	DC	140	11001002504	11	Census Tract 25.04	3349	1375	POLYGON ((-8575658 4712757,…
48	002600	11001002600	2106150	60195	DC	140	11001002600	11	Census Tract 26	2592	1013	POLYGON ((-8577402 4712795,…
49	002702	11001002702	479022	7175	DC	140	11001002702	11	Census Tract 27.02	5825	2806	POLYGON ((-8577254 4712081,…
50	002703	11001002703	163863	0	DC	140	11001002703	11	Census Tract 27.03	2614	1289	POLYGON ((-8576520 4712081,…
51	002704	11001002704	295339	2427	DC	140	11001002704	11	Census Tract 27.04	2993	1689	POLYGON ((-8577217 4712120,…
52	002801	11001002801	171910	0	DC	140	11001002801	11	Census Tract 28.01	4398	1999	POLYGON ((-8575659 4712241,…
53	002802	11001002802	229696	0	DC	140	11001002802	11	Census Tract 28.02	4685	2430	POLYGON ((-8575687 4711238,…
54	002900	11001002900	300632	0	DC	140	11001002900	11	Census Tract 29	4443	1944	POLYGON ((-8575245 4711935,…
55	003000	11001003000	210636	0	DC	140	11001003000	11	Census Tract 30	3561	1748	POLYGON ((-8575237 4711674,…
56	003100	11001003100	296819	0	DC	140	11001003100	11	Census Tract 31	3873	1818	POLYGON ((-8574747 4711911,…
57	003200	11001003200	447929	0	DC	140	11001003200	11	Census Tract 32	5099	2174	POLYGON ((-8574305 4712551,…
58	003301	11001003301	442120	0	DC	140	11001003301	11	Census Tract 33.01	3824	1847	POLYGON ((-8573264 4710315,…
59	003302	11001003302	204237	0	DC	140	11001003302	11	Census Tract 33.02	2434	1216	POLYGON ((-8573190 4709512,…
60	003400	11001003400	911458	159593	DC	140	11001003400	11	Census Tract 34	5120	1572	POLYGON ((-8574157 4711254,…
61	003500	11001003500	382021	0	DC	140	11001003500	11	Census Tract 35	4801	2587	POLYGON ((-8574612 4711186,…
62	003600	11001003600	305616	0	DC	140	11001003600	11	Census Tract 36	4775	2609	POLYGON ((-8575209 4711160,…
63	003701	11001003701	172840	0	DC	140	11001003701	11	Census Tract 37.01	2916	1628	POLYGON ((-8575664 4710117,…
64	003702	11001003702	119532	0	DC	140	11001003702	11	Census Tract 37.02	2906	1612	POLYGON ((-8575663 4710709,…
65	003801	11001003801	148928	0	DC	140	11001003801	11	Census Tract 38.01	1800	1211	POLYGON ((-8576244 4710171,…
66	003802	11001003802	207186	0	DC	140	11001003802	11	Census Tract 38.02	3911	2378	POLYGON ((-8576353 4710601,…
67	003901	11001003901	120715	0	DC	140	11001003901	11	Census Tract 39.01	2598	1769	POLYGON ((-8576449 4710723,…
68	003902	11001003902	266513	11205	DC	140	11001003902	11	Census Tract 39.02	2022	1379	POLYGON ((-8577145 4711348,…
69	004001	11001004001	271037	2414	DC	140	11001004001	11	Census Tract 40.01	3941	2494	POLYGON ((-8577187 4710397,…
70	004002	11001004002	194755	0	DC	140	11001004002	11	Census Tract 40.02	3291	2251	POLYGON ((-8576755 4709692,…
71	004100	11001004100	788701	7303	DC	140	11001004100	11	Census Tract 41	2836	1757	POLYGON ((-8578147 4709619,…
72	004201	11001004201	204529	0	DC	140	11001004201	11	Census Tract 42.01	3548	2367	POLYGON ((-8576239 4709387,…
73	004202	11001004202	207646	0	DC	140	11001004202	11	Census Tract 42.02	2850	2060	POLYGON ((-8576721 4709433,…
74	004300	11001004300	236694	0	DC	140	11001004300	11	Census Tract 43	4436	3030	POLYGON ((-8575665 4709399,…
75	004401	11001004401	246895	0	DC	140	11001004401	11	Census Tract 44.01	3430	2302	POLYGON ((-8575158 4710216,…
76	004402	11001004402	274746	0	DC	140	11001004402	11	Census Tract 44.02	2755	1629	POLYGON ((-8575158 4709586,…
77	004600	11001004600	437422	0	DC	140	11001004600	11	Census Tract 46	3543	1844	POLYGON ((-8573617 4709454,…
78	004702	11001004702	326418	0	DC	140	11001004702	11	Census Tract 47.02	3934	2762	POLYGON ((-8574041 4707719,…
79	004703	11001004703	124124	0	DC	140	11001004703	11	Census Tract 47.03	3331	2356	POLYGON ((-8574041 4707779,…
80	004704	11001004704	198509	0	DC	140	11001004704	11	Census Tract 47.04	1945	881	POLYGON ((-8573250 4708134,…
81	004801	11001004801	317222	0	DC	140	11001004801	11	Census Tract 48.01	2820	1636	POLYGON ((-8574041 4709003,…
82	004802	11001004802	288793	0	DC	140	11001004802	11	Census Tract 48.02	3634	2070	POLYGON ((-8574041 4707813,…
83	004901	11001004901	271866	0	DC	140	11001004901	11	Census Tract 49.01	3380	2031	POLYGON ((-8574611 4708884,…
84	004902	11001004902	263879	0	DC	140	11001004902	11	Census Tract 49.02	3449	2070	POLYGON ((-8574612 4707955,…
85	005001	11001005001	200969	0	DC	140	11001005001	11	Census Tract 50.01	2012	1273	POLYGON ((-8575158 4708916,…
86	005003	11001005003	94136	0	DC	140	11001005003	11	Census Tract 50.03	2143	1381	POLYGON ((-8575158 4708394,…
87	005004	11001005004	143156	0	DC	140	11001005004	11	Census Tract 50.04	3861	2524	POLYGON ((-8574977 4708742,…
88	005202	11001005202	246292	0	DC	140	11001005202	11	Census Tract 52.02	3408	2309	POLYGON ((-8575667 4708550,…
89	005203	11001005203	93419	0	DC	140	11001005203	11	Census Tract 52.03	2979	2450	POLYGON ((-8575449 4708395,…
90	005302	11001005302	117635	0	DC	140	11001005302	11	Census Tract 53.02	2518	1832	POLYGON ((-8576240 4709009,…
91	005303	11001005303	168769	0	DC	140	11001005303	11	Census Tract 53.03	3215	2484	POLYGON ((-8576385 4708809,…
92	005501	11001005501	164749	0	DC	140	11001005501	11	Census Tract 55.01	2303	1776	POLYGON ((-8577325 4707809,…
93	005502	11001005502	251302	0	DC	140	11001005502	11	Census Tract 55.02	2968	2016	POLYGON ((-8577033 4708467,…
94	005503	11001005503	213338	4682	DC	140	11001005503	11	Census Tract 55.03	2492	1799	POLYGON ((-8577830 4708009,…
95	005601	11001005601	185309	1275	DC	140	11001005601	11	Census Tract 56.01	2900	2437	POLYGON ((-8578015 4707737,…
96	005602	11001005602	273491	6261	DC	140	11001005602	11	Census Tract 56.02	4100	2355	POLYGON ((-8578124 4707549,…
97	005801	11001005801	232951	0	DC	140	11001005801	11	Census Tract 58.01	1672	1312	POLYGON ((-8574041 4706439,…
98	005802	11001005802	682488	0	DC	140	11001005802	11	Census Tract 58.02	2145	1616	POLYGON ((-8575346 4706979,…
99	005900	11001005900	641993	0	DC	140	11001005900	11	Census Tract 59	2617	1705	POLYGON ((-8573819 4706443,…
100	006400	11001006400	691410	243648	DC	140	11001006400	11	Census Tract 64	2481	1956	POLYGON ((-8573310 4701938,…
101	006500	11001006500	509297	0	DC	140	11001006500	11	Census Tract 65	2609	1630	POLYGON ((-8572616 4704770,…
102	006600	11001006600	285156	0	DC	140	11001006600	11	Census Tract 66	2164	1297	POLYGON ((-8571992 4705886,…
103	006700	11001006700	428358	0	DC	140	11001006700	11	Census Tract 67	3927	2062	POLYGON ((-8571041 4705369,…
104	006801	11001006801	244750	0	DC	140	11001006801	11	Census Tract 68.01	2238	1173	POLYGON ((-8569781 4705521,…
105	006802	11001006802	280108	0	DC	140	11001006802	11	Census Tract 68.02	2580	1206	POLYGON ((-8569782 4704455,…
106	006804	11001006804	1542277	477741	DC	140	11001006804	11	Census Tract 68.04	2429	17	POLYGON ((-8569544 4704345,…
107	006900	11001006900	398513	0	DC	140	11001006900	11	Census Tract 69	2799	1793	POLYGON ((-8570907 4705154,…
108	007000	11001007000	439473	0	DC	140	11001007000	11	Census Tract 70	2697	1346	POLYGON ((-8571829 4704590,…
109	007100	11001007100	602223	149901	DC	140	11001007100	11	Census Tract 71	3218	1939	POLYGON ((-8570659 4704101,…
110	007201	11001007201	781457	312419	DC	140	11001007201	11	Census Tract 72.01	2985	2672	POLYGON ((-8572614 4703874,…
111	007202	11001007202	183977	0	DC	140	11001007202	11	Census Tract 72.02	4303	3563	POLYGON ((-8572620 4704474,…
112	007203	11001007203	384178	0	DC	140	11001007203	11	Census Tract 72.03	3748	2246	POLYGON ((-8572240 4704619,…
113	007301	11001007301	4684643	5139082	DC	140	11001007301	11	Census Tract 73.01	3841	1272	POLYGON ((-8576945 4698961,…
114	007304	11001007304	1232841	8983	DC	140	11001007304	11	Census Tract 73.04	4410	1765	POLYGON ((-8570958 4699307,…
115	007401	11001007401	1209113	200980	DC	140	11001007401	11	Census Tract 74.01	1259	644	POLYGON ((-8572202 4702896,…
116	007403	11001007403	330904	0	DC	140	11001007403	11	Census Tract 74.03	2613	1056	POLYGON ((-8569516 4699726,…
117	007404	11001007404	826388	0	DC	140	11001007404	11	Census Tract 74.04	3637	1453	POLYGON ((-8570741 4700704,…
118	007406	11001007406	360152	0	DC	140	11001007406	11	Census Tract 74.06	3528	1286	POLYGON ((-8570894 4701168,…
119	007407	11001007407	608700	0	DC	140	11001007407	11	Census Tract 74.07	3491	1570	POLYGON ((-8570960 4701752,…
120	007408	11001007408	372349	0	DC	140	11001007408	11	Census Tract 74.08	2867	1205	POLYGON ((-8568918 4700728,…
121	007409	11001007409	404873	5767	DC	140	11001007409	11	Census Tract 74.09	3835	1613	POLYGON ((-8569441 4699556,…
122	007502	11001007502	648317	0	DC	140	11001007502	11	Census Tract 75.02	4687	1997	POLYGON ((-8568807 4701881,…
123	007503	11001007503	515179	0	DC	140	11001007503	11	Census Tract 75.03	2528	1152	POLYGON ((-8570776 4702348,…
124	007504	11001007504	764081	0	DC	140	11001007504	11	Census Tract 75.04	2730	1321	POLYGON ((-8570263 4701676,…
125	007601	11001007601	1115359	141529	DC	140	11001007601	11	Census Tract 76.01	4693	2350	POLYGON ((-8570458 4703300,…
126	007603	11001007603	1226616	0	DC	140	11001007603	11	Census Tract 76.03	4185	2374	POLYGON ((-8568061 4701482,…
127	007604	11001007604	1369580	0	DC	140	11001007604	11	Census Tract 76.04	3803	2056	POLYGON ((-8568410 4703627,…
128	007605	11001007605	453794	0	DC	140	11001007605	11	Census Tract 76.05	3663	1834	POLYGON ((-8569859 4702590,…
129	007703	11001007703	1017741	1059	DC	140	11001007703	11	Census Tract 77.03	5227	2418	POLYGON ((-8566855 4705043,…
130	007707	11001007707	842851	0	DC	140	11001007707	11	Census Tract 77.07	3927	1757	POLYGON ((-8565463 4704369,…
131	007708	11001007708	745809	129100	DC	140	11001007708	11	Census Tract 77.08	2574	1275	POLYGON ((-8568423 4704589,…
132	007709	11001007709	828872	82685	DC	140	11001007709	11	Census Tract 77.09	2072	1096	POLYGON ((-8569155 4704052,…
133	007803	11001007803	986677	3755	DC	140	11001007803	11	Census Tract 78.03	4590	2301	POLYGON ((-8565969 4706900,…
134	007804	11001007804	838428	5705	DC	140	11001007804	11	Census Tract 78.04	3499	1421	POLYGON ((-8564832 4707278,…
135	007806	11001007806	671434	0	DC	140	11001007806	11	Census Tract 78.06	3457	1122	POLYGON ((-8565240 4707651,…
136	007807	11001007807	522749	0	DC	140	11001007807	11	Census Tract 78.07	2114	926	POLYGON ((-8563393 4706879,…
137	007808	11001007808	936852	7722	DC	140	11001007808	11	Census Tract 78.08	3792	1703	POLYGON ((-8563582 4705915,…
138	007809	11001007809	535254	0	DC	140	11001007809	11	Census Tract 78.09	3155	1369	POLYGON ((-8565198 4707650,…
139	007901	11001007901	402041	0	DC	140	11001007901	11	Census Tract 79.01	4148	1895	POLYGON ((-8569781 4707029,…
140	007903	11001007903	247945	0	DC	140	11001007903	11	Census Tract 79.03	1865	917	POLYGON ((-8569069 4706521,…
141	008001	11001008001	322795	0	DC	140	11001008001	11	Census Tract 80.01	3103	1448	POLYGON ((-8570516 4706973,…
142	008002	11001008002	578961	0	DC	140	11001008002	11	Census Tract 80.02	3366	1708	POLYGON ((-8570659 4705900,…
143	008100	11001008100	290284	0	DC	140	11001008100	11	Census Tract 81	3153	1645	POLYGON ((-8571041 4706062,…
144	008200	11001008200	338424	0	DC	140	11001008200	11	Census Tract 82	2695	1664	POLYGON ((-8572261 4706190,…
145	008301	11001008301	303973	0	DC	140	11001008301	11	Census Tract 83.01	3220	1761	POLYGON ((-8571998 4707388,…
146	008302	11001008302	260891	0	DC	140	11001008302	11	Census Tract 83.02	3026	1560	POLYGON ((-8571429 4706527,…
147	008402	11001008402	274180	0	DC	140	11001008402	11	Census Tract 84.02	2447	1804	POLYGON ((-8571041 4706986,…
148	008410	11001008410	219939	0	DC	140	11001008410	11	Census Tract 84.10	1843	898	POLYGON ((-8571040 4707548,…
149	008701	11001008701	382686	0	DC	140	11001008701	11	Census Tract 87.01	2814	1498	POLYGON ((-8572606 4708949,…
150	008702	11001008702	486411	0	DC	140	11001008702	11	Census Tract 87.02	3574	2073	POLYGON ((-8572189 4708697,…
151	008802	11001008802	468578	0	DC	140	11001008802	11	Census Tract 88.02	4360	2240	POLYGON ((-8570837 4707931,…
152	008803	11001008803	1168294	0	DC	140	11001008803	11	Census Tract 88.03	4036	2026	POLYGON ((-8571871 4708552,…
153	008804	11001008804	778052	0	DC	140	11001008804	11	Census Tract 88.04	2417	1551	POLYGON ((-8570358 4708471,…
154	008903	11001008903	305651	0	DC	140	11001008903	11	Census Tract 89.03	3062	1661	POLYGON ((-8569704 4707476,…
155	008904	11001008904	449343	0	DC	140	11001008904	11	Census Tract 89.04	3137	1806	POLYGON ((-8569738 4707408,…
156	009000	11001009000	1427114	31550	DC	140	11001009000	11	Census Tract 90	4676	2548	POLYGON ((-8567723 4710972,…
157	009102	11001009102	1816287	0	DC	140	11001009102	11	Census Tract 91.02	4608	2173	POLYGON ((-8571767 4708853,…
158	009201	11001009201	629781	0	DC	140	11001009201	11	Census Tract 92.01	3103	1549	POLYGON ((-8572298 4711224,…
159	009203	11001009203	581087	0	DC	140	11001009203	11	Census Tract 92.03	2775	1352	POLYGON ((-8572603 4709949,…
160	009204	11001009204	308465	0	DC	140	11001009204	11	Census Tract 92.04	2889	1414	POLYGON ((-8571976 4710795,…
161	009301	11001009301	1109091	0	DC	140	11001009301	11	Census Tract 93.01	3853	1385	POLYGON ((-8571026 4712187,…
162	009302	11001009302	391905	0	DC	140	11001009302	11	Census Tract 93.02	1750	878	POLYGON ((-8571159 4710418,…
163	009400	11001009400	1582882	0	DC	140	11001009400	11	Census Tract 94	4424	1800	POLYGON ((-8569419 4713523,…
164	009503	11001009503	1423609	0	DC	140	11001009503	11	Census Tract 95.03	3106	1282	POLYGON ((-8570435 4714552,…
165	009504	11001009504	1033168	0	DC	140	11001009504	11	Census Tract 95.04	3208	1442	POLYGON ((-8571503 4713979,…
166	009505	11001009505	1033208	0	DC	140	11001009505	11	Census Tract 95.05	3796	1674	POLYGON ((-8572794 4716282,…
167	009507	11001009507	295290	0	DC	140	11001009507	11	Census Tract 95.07	1525	623	POLYGON ((-8571851 4715497,…
168	009508	11001009508	962212	0	DC	140	11001009508	11	Census Tract 95.08	4243	2068	POLYGON ((-8572608 4714990,…
169	009509	11001009509	648848	0	DC	140	11001009509	11	Census Tract 95.09	3194	1377	POLYGON ((-8571825 4714664,…
170	009510	11001009510	841510	0	DC	140	11001009510	11	Census Tract 95.10	4525	1939	POLYGON ((-8572810 4713747,…
171	009511	11001009511	719468	0	DC	140	11001009511	11	Census Tract 95.11	2066	10	POLYGON ((-8572532 4713276,…
172	009601	11001009601	1905262	189666	DC	140	11001009601	11	Census Tract 96.01	2086	819	POLYGON ((-8566913 4708256,…
173	009602	11001009602	1306709	67073	DC	140	11001009602	11	Census Tract 96.02	4481	2046	POLYGON ((-8567395 4707287,…
174	009603	11001009603	607400	0	DC	140	11001009603	11	Census Tract 96.03	3971	1892	POLYGON ((-8566744 4705958,…
175	009604	11001009604	503381	65762	DC	140	11001009604	11	Census Tract 96.04	1996	1005	POLYGON ((-8567512 4705978,…
176	009700	11001009700	403051	3427	DC	140	11001009700	11	Census Tract 97	3188	1355	POLYGON ((-8571028 4697243,…
177	009801	11001009801	431384	7884	DC	140	11001009801	11	Census Tract 98.01	1848	898	POLYGON ((-8572031 4697492,…
178	009802	11001009802	200745	0	DC	140	11001009802	11	Census Tract 98.02	2064	880	POLYGON ((-8571665 4697488,…
179	009803	11001009803	531060	3756	DC	140	11001009803	11	Census Tract 98.03	2968	1480	POLYGON ((-8572537 4697872,…
180	009804	11001009804	518363	5609	DC	140	11001009804	11	Census Tract 98.04	2517	1133	POLYGON ((-8572070 4698838,…
181	009807	11001009807	1037012	0	DC	140	11001009807	11	Census Tract 98.07	3523	1682	POLYGON ((-8573535 4696479,…
182	009810	11001009810	398419	14058	DC	140	11001009810	11	Census Tract 98.10	2461	1243	POLYGON ((-8572456 4697551,…
183	009811	11001009811	467511	0	DC	140	11001009811	11	Census Tract 98.11	4438	1939	POLYGON ((-8571871 4696978,…
184	009901	11001009901	2102150	0	DC	140	11001009901	11	Census Tract 99.01	2364	1066	POLYGON ((-8567684 4703986,…
185	009902	11001009902	1294556	0	DC	140	11001009902	11	Census Tract 99.02	2774	1289	POLYGON ((-8566359 4702556,…
186	009903	11001009903	383679	0	DC	140	11001009903	11	Census Tract 99.03	2004	743	POLYGON ((-8563815 4705903,…
187	009904	11001009904	426198	0	DC	140	11001009904	11	Census Tract 99.04	2886	1276	POLYGON ((-8564451 4705351,…
188	009905	11001009905	431558	0	DC	140	11001009905	11	Census Tract 99.05	2790	1344	POLYGON ((-8563733 4704608,…
189	009906	11001009906	252684	0	DC	140	11001009906	11	Census Tract 99.06	1773	896	POLYGON ((-8564983 4705437,…
190	009907	11001009907	466954	0	DC	140	11001009907	11	Census Tract 99.07	2259	1146	POLYGON ((-8565564 4704623,…
191	010100	11001010100	579230	0	DC	140	11001010100	11	Census Tract 101	2699	2022	POLYGON ((-8575670 4707461,…
192	010201	11001010201	174884	0	DC	140	11001010201	11	Census Tract 102.01	3341	2471	POLYGON ((-8574057 4704372,…
193	010202	11001010202	1069479	136323	DC	140	11001010202	11	Census Tract 102.02	2631	2042	POLYGON ((-8575241 4704871,…
194	010300	11001010300	1084591	0	DC	140	11001010300	11	Census Tract 103	3604	1483	POLYGON ((-8575650 4717824,…
195	010400	11001010400	2602393	7006	DC	140	11001010400	11	Census Tract 104	4616	1894	POLYGON ((-8572624 4698663,…
196	010500	11001010500	749891	0	DC	140	11001010500	11	Census Tract 105	3712	2440	POLYGON ((-8573556 4704827,…
197	010601	11001010601	224811	0	DC	140	11001010601	11	Census Tract 106.01	2130	1486	POLYGON ((-8572603 4708182,…
198	010602	11001010602	542509	0	DC	140	11001010602	11	Census Tract 106.02	6594	3730	POLYGON ((-8572095 4707718,…
199	010603	11001010603	464270	0	DC	140	11001010603	11	Census Tract 106.03	3068	2004	POLYGON ((-8572611 4707052,…
200	010700	11001010700	891588	0	DC	140	11001010700	11	Census Tract 107	2296	1556	POLYGON ((-8577137 4707783,…
201	010800	11001010800	661580	0	DC	140	11001010800	11	Census Tract 108	6879	1501	POLYGON ((-8577183 4707206,…
202	010900	11001010900	2381015	2933566	DC	140	11001010900	11	Census Tract 109	3333	1378	POLYGON ((-8575964 4693136,…
203	011001	11001011001	152450	0	DC	140	11001011001	11	Census Tract 110.01	2423	1636	POLYGON ((-8573556 4703992,…
204	011002	11001011002	498122	367023	DC	140	11001011002	11	Census Tract 110.02	1859	1471	POLYGON ((-8574037 4703998,…
205	011100	11001011100	5718693	366232	DC	140	11001011100	11	Census Tract 111	5903	1952	POLYGON ((-8569278 4711323,…
206	980000	11001980000	6514228	4996439	DC	140	11001980000	11	Census Tract 9800	176	12	POLYGON ((-8578803 4706193,…

What Are These POLYGON Objects?

head(dc_sf)

OBJECTID	TRACT	GEOID	ALAND	AWATER	STUSAB	SUMLEV	GEOCODE	STATE	NAME	POP100	HU100	geometry
1	002002	11001002002	849376	0	DC	140	11001002002	11	Census Tract 20.02	4072	1532	POLYGON ((-8575655 4714476,…
2	002101	11001002101	600992	0	DC	140	11001002101	11	Census Tract 21.01	5687	2335	POLYGON ((-8574745 4715676,…
3	002102	11001002102	725975	0	DC	140	11001002102	11	Census Tract 21.02	5099	2221	POLYGON ((-8573824 4715684,…
4	002201	11001002201	415173	0	DC	140	11001002201	11	Census Tract 22.01	3485	1229	POLYGON ((-8574654 4714781,…
5	002202	11001002202	698895	566	DC	140	11001002202	11	Census Tract 22.02	3339	1454	POLYGON ((-8573792 4714811,…
6	000101	11001000101	199776	5261	DC	140	11001000101	11	Census Tract 1.01	1406	999	POLYGON ((-8577962 4708867,…

Zoom… Enhance…

Let’s try printing just the first row?

rmarkdown::paged_table(dc_sf[1,])

OBJECTID	TRACT	GEOID	ALAND	AWATER	STUSAB	SUMLEV	GEOCODE	STATE	NAME	POP100	HU100	geometry
1	002002	11001002002	849376	0	DC	140	11001002002	11	Census Tract 20.02	4072	1532	POLYGON ((-8575655 4714476,…

Printing the special geometry column?

rmarkdown::paged_table(dc_sf[1, "geometry"])

geometry
POLYGON ((-8575655 4714476,…

Nope, we’ll have to open the “black box” using the st_coordinates() function from the sf library →

sf::st_coordinates(dc_sf[1,"geometry"])

              X       Y L1 L2
  [1,] -8575655 4714476  1  1
  [2,] -8575655 4714588  1  1
  [3,] -8575655 4714628  1  1
  [4,] -8575655 4714781  1  1
  [5,] -8575655 4714876  1  1
  [6,] -8575636 4714900  1  1
  [7,] -8575622 4714920  1  1
  [8,] -8575613 4714934  1  1
  [9,] -8575595 4714962  1  1
 [10,] -8575564 4715009  1  1
 [11,] -8575519 4715077  1  1
 [12,] -8575502 4715104  1  1
 [13,] -8575452 4715179  1  1
 [14,] -8575408 4715248  1  1
 [15,] -8575362 4715318  1  1
 [16,] -8575359 4715322  1  1
 [17,] -8575320 4715400  1  1
 [18,] -8575294 4715424  1  1
 [19,] -8575272 4715454  1  1
 [20,] -8575268 4715459  1  1
 [21,] -8575242 4715499  1  1
 [22,] -8575237 4715507  1  1
 [23,] -8575226 4715523  1  1
 [24,] -8575221 4715531  1  1
 [25,] -8575205 4715556  1  1
 [26,] -8575198 4715567  1  1
 [27,] -8575190 4715579  1  1
 [28,] -8575167 4715614  1  1
 [29,] -8575142 4715652  1  1
 [30,] -8575132 4715668  1  1
 [31,] -8575116 4715692  1  1
 [32,] -8575105 4715709  1  1
 [33,] -8575094 4715724  1  1
 [34,] -8575069 4715764  1  1
 [35,] -8575031 4715822  1  1
 [36,] -8575021 4715836  1  1
 [37,] -8575008 4715855  1  1
 [38,] -8574983 4715894  1  1
 [39,] -8574959 4715928  1  1
 [40,] -8574954 4715933  1  1
 [41,] -8574952 4715935  1  1
 [42,] -8574943 4715943  1  1
 [43,] -8574939 4715947  1  1
 [44,] -8574907 4715975  1  1
 [45,] -8574904 4715976  1  1
 [46,] -8574901 4715977  1  1
 [47,] -8574898 4715978  1  1
 [48,] -8574893 4715980  1  1
 [49,] -8574889 4715984  1  1
 [50,] -8574886 4715986  1  1
 [51,] -8574883 4715990  1  1
 [52,] -8574857 4716021  1  1
 [53,] -8574802 4716086  1  1
 [54,] -8574783 4716108  1  1
 [55,] -8574779 4716114  1  1
 [56,] -8574778 4716117  1  1
 [57,] -8574775 4716123  1  1
 [58,] -8574774 4716127  1  1
 [59,] -8574775 4716129  1  1
 [60,] -8574776 4716136  1  1
 [61,] -8574735 4716130  1  1
 [62,] -8574724 4716125  1  1
 [63,] -8574719 4716120  1  1
 [64,] -8574720 4716116  1  1
 [65,] -8574721 4716094  1  1
 [66,] -8574721 4716090  1  1
 [67,] -8574724 4716050  1  1
 [68,] -8574730 4715948  1  1
 [69,] -8574736 4715850  1  1
 [70,] -8574742 4715760  1  1
 [71,] -8574743 4715730  1  1
 [72,] -8574744 4715709  1  1
 [73,] -8574744 4715699  1  1
 [74,] -8574745 4715676  1  1
 [75,] -8574745 4715665  1  1
 [76,] -8574744 4715654  1  1
 [77,] -8574744 4715631  1  1
 [78,] -8574742 4715608  1  1
 [79,] -8574742 4715596  1  1
 [80,] -8574741 4715585  1  1
 [81,] -8574737 4715555  1  1
 [82,] -8574736 4715539  1  1
 [83,] -8574733 4715518  1  1
 [84,] -8574731 4715495  1  1
 [85,] -8574726 4715446  1  1
 [86,] -8574721 4715403  1  1
 [87,] -8574720 4715389  1  1
 [88,] -8574718 4715373  1  1
 [89,] -8574713 4715331  1  1
 [90,] -8574706 4715261  1  1
 [91,] -8574705 4715250  1  1
 [92,] -8574703 4715235  1  1
 [93,] -8574699 4715194  1  1
 [94,] -8574695 4715157  1  1
 [95,] -8574688 4715097  1  1
 [96,] -8574687 4715085  1  1
 [97,] -8574670 4714929  1  1
 [98,] -8574669 4714917  1  1
 [99,] -8574669 4714914  1  1
[100,] -8574654 4714781  1  1
[101,] -8574651 4714745  1  1
[102,] -8574638 4714628  1  1
[103,] -8574637 4714615  1  1
[104,] -8574636 4714603  1  1
[105,] -8574634 4714590  1  1
[106,] -8574633 4714578  1  1
[107,] -8574631 4714565  1  1
[108,] -8574630 4714553  1  1
[109,] -8574625 4714503  1  1
[110,] -8574622 4714475  1  1
[111,] -8574616 4714418  1  1
[112,] -8574612 4714382  1  1
[113,] -8574611 4714369  1  1
[114,] -8574608 4714347  1  1
[115,] -8574607 4714336  1  1
[116,] -8574602 4714293  1  1
[117,] -8574601 4714283  1  1
[118,] -8574600 4714275  1  1
[119,] -8574599 4714264  1  1
[120,] -8574586 4714137  1  1
[121,] -8574583 4714111  1  1
[122,] -8574569 4713998  1  1
[123,] -8574562 4713958  1  1
[124,] -8574609 4713959  1  1
[125,] -8574659 4713958  1  1
[126,] -8574695 4713958  1  1
[127,] -8574741 4713958  1  1
[128,] -8574821 4713959  1  1
[129,] -8574901 4713958  1  1
[130,] -8574921 4713996  1  1
[131,] -8574923 4713999  1  1
[132,] -8574946 4714027  1  1
[133,] -8574966 4714053  1  1
[134,] -8574989 4714082  1  1
[135,] -8575017 4714117  1  1
[136,] -8575019 4714120  1  1
[137,] -8575029 4714133  1  1
[138,] -8575049 4714158  1  1
[139,] -8575057 4714171  1  1
[140,] -8575060 4714175  1  1
[141,] -8575062 4714178  1  1
[142,] -8575065 4714181  1  1
[143,] -8575103 4714228  1  1
[144,] -8575122 4714253  1  1
[145,] -8575129 4714261  1  1
[146,] -8575244 4714265  1  1
[147,] -8575259 4714265  1  1
[148,] -8575284 4714264  1  1
[149,] -8575358 4714264  1  1
[150,] -8575432 4714264  1  1
[151,] -8575443 4714264  1  1
[152,] -8575540 4714264  1  1
[153,] -8575639 4714264  1  1
[154,] -8575656 4714264  1  1
[155,] -8575655 4714476  1  1

Working With sf Objects

head(dc_sf$NAME) # Select column by name  

[1] "Census Tract 20.02" "Census Tract 21.01" "Census Tract 21.02"
[4] "Census Tract 22.01" "Census Tract 22.02" "Census Tract 1.01" 

head(dc_sf[,4]) # Select column by number

ALAND	geometry
849376	POLYGON ((-8575655 4714476,…
600992	POLYGON ((-8574745 4715676,…
725975	POLYGON ((-8573824 4715684,…
415173	POLYGON ((-8574654 4714781,…
698895	POLYGON ((-8573792 4714811,…
199776	POLYGON ((-8577962 4708867,…

And… Actually Displaying the Map!

# We can extract the geometry with the st_geometry function
dc_geo <- st_geometry(dc_sf)

# Plot the geometry with base R's plot() function
plot(dc_geo)

And with ggplot!

dc_sf |>
  ggplot() +
  geom_sf() +
  theme_classic()

Looking Ahead: Types of Geospatial Data

Key Notation / Definition

• 2-Dimensional “Spatial Process” (Schabenberger and Gotway 2004, 6)

\[ \text{Data} = \left\{Z(\mathbf{s}) \mid \mathbf{s} \in D \subset \mathbb{R}^2\right\} \]

	Geostatistical Data	Lattice/Region Data	Point Pattern
Criteria	Fixed \(D\), Continuous	Fixed \(D\), Discrete	Random subset \(D^* \subseteq D\)
Interest	Infer non-observed parts of \(D\)	Autocorrelation, clustering	Point-generating process
Example	\(N\) trees \(\mathbf{s}_1, \mathbf{s}_2, \ldots, \mathbf{s}_N\), observed within a sample window \(D \subset \mathbb{R}^2\), (\(D\) some finite plot of land) \(Z(\mathbf{s}_i)\): Attribute(s) at site \(\mathbf{s}_i\) Example: Height. \(Z(\mathbf{s}_1) = 500\text{m}\), \(Z(\mathbf{s}_2) = 850\text{m}\), \(\ldots\)	\(Z(\mathbf{s})\) observed over \(N \times N\) grid of plots \(\Rightarrow\) Contiguity, Neighbors (next section of slides!) \(\Rightarrow\) Autocorrelation: Are points around \(\mathbf{s}_i\) likely to have values similar to \(Z(\mathbf{s}_i)\)?	Unknown number of lightning strikes \(\mathbf{s}_1, \mathbf{s}_2, \ldots\) Contrast with geostatistical: all of \(D\) is observed, but what determines the subset \(D^*\) where events occur? “Unmarked”: Just locations “Marked”: Locations+info (e.g., intensity of strike)

Spatial Randomness

library(tidyverse)
library(spatstat)
set.seed(6805)
N <- 60
r_core <- 0.05
obs_window <- square(1)
# Regularity via Inhibition
#reg_sims <- rMaternI(N, r=r_core, win=obs_window)
cond_reg_sims <- rSSI(r=r_core, N)
# CSR data
#csr_sims <- rpoispp(N, win=obs_window)
cond_sr_sims <- rpoint(N, win=obs_window)
### Clustered data
#clust_sims <- rMatClust(kappa=6, r=2.5*r_core, mu=10, win=obs_window)
#clust_sims <- rMatClust(mu=5, kappa=1, scale=0.1, win=obs_window, n.cond=N, w.cond=obs_window)
#clust_sims <- rclusterBKBC(clusters="MatClust", kappa=10, mu=10, scale=0.05, verbose=FALSE)
# Each cluster consist of 10 points in a disc of radius 0.2
nclust <- function(x0, y0, radius, n) {
    #print(n)
    return(runifdisc(10, radius, centre=c(x0, y0)))
}
cond_clust_sims <- rNeymanScott(kappa=5, expand=0.0, rclust=nclust, radius=2*r_core, n=10)
# And PLOT
plot_w <- 400
plot_h <- 400
plot_scale <- 2.25
cond_reg_plot <- cond_reg_sims |> sf::st_as_sf() |>
  ggplot() +
  geom_sf() +
  dsan_theme() +
  transparent_bg()
ggsave("images/cond_reg.png", cond_reg_plot, width=plot_w, height=plot_h, units="px", scale=plot_scale)
cond_sr_plot <- cond_sr_sims |> sf::st_as_sf() |>
  ggplot() +
  geom_sf() +
  dsan_theme() +
  transparent_bg()
ggsave("images/cond_sr.png", cond_sr_plot, width=plot_w, height=plot_h, units="px", scale=plot_scale)
cond_clust_plot <- cond_clust_sims |> sf::st_as_sf() |>
  ggplot() +
  geom_sf() +
  dsan_theme() +
  transparent_bg()
ggsave("images/cond_clust.png", cond_clust_plot, width=plot_w, height=plot_h, units="px", scale=plot_scale)

Autocorrelation	\(I = -1\)	←	\(I = 0\)	→	\(I = 1\)
Description	Negative Autocorr		No Autocorr		Positive Autocorr
Event at \(\mathbf{s} = (x,y)\) Implies	Less likely to find another point nearby		No information about nearby points		More likely to find another point nearby
Resulting Pattern	Regularity		Reg/Clustered Mix		Clustering
Process(es) Which Could Produce Pattern	1st Order: Random within even-spaced grid 2nd Order: Competition		1st Order: i.i.d. points 2nd Order: i.i.d. distances		1st Order: Tasty food at clust centers 2nd Order: Cooperation
Fixed \(N\)	60		60		60

Complete Spatial Randomness (CSR)

library(tidyverse)
library(spatstat)
set.seed(6807)
lambda <- 60
r_core <- 0.05
obs_window <- square(1)
# Regularity via Inhibition
# Regularity via Inhibition
reg_sims <- rMaternI(lambda, r=r_core, win=obs_window)
# CSR data
csr_sims <- rpoispp(N, win=obs_window)
### Clustered data
clust_mu <- 10
clust_sims <- rMatClust(kappa=lambda / clust_mu, scale=2*r_core, mu=10, win=obs_window)
# And PLOT
plot_w <- 400
plot_h <- 400
plot_scale <- 2.25
reg_plot <- reg_sims |> sf::st_as_sf() |>
  ggplot() +
  geom_sf() +
  labs(title=paste0("N = ",reg_sims$n)) +
  dsan_theme() +
  transparent_bg()
ggsave("images/reg.png", reg_plot, width=plot_w, height=plot_h, units="px", scale=plot_scale)
csr_plot <- csr_sims |> sf::st_as_sf() |>
  ggplot() +
  geom_sf() +
  labs(title=paste0("N = ",csr_sims$n)) +
  dsan_theme() +
  transparent_bg()
ggsave("images/csr.png", csr_plot, width=plot_w, height=plot_h, units="px", scale=plot_scale)
clust_plot <- clust_sims |> sf::st_as_sf() |>
  ggplot() +
  geom_sf() +
  labs(title=paste0("N = ",clust_sims$n)) +
  dsan_theme() +
  transparent_bg()
ggsave("images/clust.png", clust_plot, width=plot_w, height=plot_h, units="px", scale=plot_scale)

Autocorrelation	\(I = -1\)	←	\(I = 0\)	→	\(I = 1\)
Description	Negative Autocorr		No Autocorr		Positive Autocorr
Event at \(\mathbf{s} = (x,y)\) Implies	Less likely to find another point nearby		No information about nearby points		More likely to find another point nearby
Resulting Pattern	Regularity		Reg/Clustered Mix		Clustering
Process(es) Which Could Produce Pattern	1st Order: Random within even-spaced grid 2nd Order: Competition		1st Order: i.i.d. points 2nd Order: i.i.d. distances		1st Order: Tasty food at clust centers 2nd Order: Cooperation
Fixed Intensity \(\lambda\)	60		60		60
Random \(N\)

On Not Worrying About Prereqs

• I genuinely believe that I can make the course accessible to you, meeting you wherever you’re at, no matter what!
• Everyone learns at their own pace (who says 14 weeks is “correct” amount of time to learn GIS?), and I structure my courses as best as I possibly can to adapt to your pace
• \(\Rightarrow\) Assessments (HW, Midterm) valuable in two ways:
• [Valuable for you] As an accountability mechanism to make sure you’re learn the material (how do we know when we’ve learned something? When we can answer questions about it / use it to accomplish things!)
• [Valuable for me] For assessing and updating pace

R and/or Python and/or JS

• My Geometry vs. Algebra Rant… Euclid’s Elements, Book VI, Proposition 28.
• The problem: Divide a given straight line so that the rectangle contained by its segments may be equal to a given area, not exceeding the square of half the line.

Geometers solved w/geometry (300 BC)…

…Algebraists solved w/algebra (2000 BC)…

\[ \begin{align*} &ax^2 + bx + c = 0 \\ \Rightarrow \; & x_+ = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \end{align*} \]

…From 1637 onwards, whichever is easier! 🤯🤯🤯 (Isomorphism)

Figure 3: Circle with radius 1? Or \((x,y)\) satisfying \(x^2 + y^2 = 1\)?

Learning How To Learn

Figure 4: From The Carter (Documentary)

He’s Literally Extremely Correct!

From Elsevier Osmosis: Spaced Repetition

Vector \(\rightarrow\) Raster Data

Raster Data

• Each DC Census Tract has its own (odd) shape, which can be described by discrete coordinates forming a POLYGON
• For geospatial analysis, however, we often need to compute over evenly-spaced grids rather than this odd collection of shapes
  • Most common example: photos taken from an airplane/satellite! [Remote sensing]
• POLYGONs may make sense for demographers, but how about someone studying air pollution in DC? (Smog, for example, does not confine itself to census tracts!)

Step 1: Union of All Tracts

dc_union_sf <- sf::st_union(dc_sf)
dc_union_sf |>
  ggplot() +
  geom_sf() +
  theme_classic()

Step 2: Rasterize (terra)

library(terra)

terra 1.7.78

Attaching package: 'terra'

The following object is masked from 'package:tidyr':

    extract

dc_SpatVector <- terra::vect(dc_union_sf)
rast_template <- rast(ext(dc_SpatVector), resolution = 1000, crs = crs(dc_SpatVector))
dc_SpatRaster <- terra::rasterize(dc_SpatVector, rast_template)
dim(dc_SpatRaster)

[1] 29 23  1

plot(dc_SpatRaster)

Rasters From Scratch

Welcome to Gridtown!

set.seed(6805)
library(terra)
gridtown <- terra::rast(
  nrows = 4, ncols = 4,
  xmin = 0, xmax = 4, ymin = 0, ymax = 4,
  vals = sample(1:16)
)
plot(gridtown)
text(
  gridtown,
  labels=1:16,
  halo=TRUE, hc="black", col="white", hw=0.2
)

Figure 5: Gridtown Indices

• Raster indices vs. values: The above plot displays indices for each cell: since a raster is a regular grid, can achieve memory-efficient representation with a single index (rather than, e.g., \((x, y)\) coords). But what we really care about are…

Raster Layer Values

plot(gridtown)
text(gridtown, halo=TRUE, hc="black", col="white", hw=0.2)

Figure 6: Gridtown Values

References

Montessori, Maria. 1916. Spontaneous Activity in Education: A Basic Guide to the Montessori Methods of Learning in the Classroom. Lulu Press.

Schabenberger, Oliver, and Carol A. Gotway. 2004. Statistical Methods for Spatial Data Analysis. CRC Press.