• Cholera outbreak on Broad Street in Soho London in 1854
• On the Mode of Communication of Cholera in 1849 and 1855
• Predates understanding and acceptance of bacteriology and germ theory

http://ecodevoevo.blogspot.com/2014/10/was-john-snow-more-of-empiricist-than.html

https://commons.wikimedia.org/w/index.php?curid=473104

https://www1.udel.edu/johnmack/frec682/cholera/snow_map.png

https://scienceblogs.com/significantfigures/index.php/2013/03/11/200-years-of-dr-john-snow-a-significant-figure-in-the-world-of-water

http://geoawesomeness.com/how-often-does-a-map-change-the-world-london-cholera-map-of-dr-john-snow-from-1854/

Voronoi Diagrams

AKA Thiessen Polygons

Numerous polygons are created:

• Each polygon contains a point, \(p_K\)
• Its corresponding polygon consists of every point whose distance to \(p_K\) is less than or equal to the distance to any other \(p_K\)

Feel free to try it out on your own!

http://qgissextante.blogspot.com/2012/10/analyzing-john-snows-cholera-dataset.html

Spatial Allocation: walking distance

Dotted line: the map on the right is a later version.
https://johnsnow.matrix.msu.edu/images/online_companion/chapter_images/fig12-7.jpg

John Snow’s Cholera Study

• Snow had a hypothesis that cholera was a waterborne disease before he even drew his maps
• Spatial analysis did not prove his theory. That required microbiology and pathology (in 1883, Rober Kock isolated Vibrio cholerae).
• But … spatial analysis did provide supporting evidence that had an important impact on policy.

Steven Johnson

Four Types of Spatial Analysis

1. Spatial data manipulation
2. Spatial data analysis
3. Spatial statistical analysis
4. Spatial Modeling

1: Spatial data manipulation

• Typically done via GIS
• E.g., merging, overlay, dissolve.

2: Spatial data analysis

• First steps in analysis: exploratory and descriptive

3: Spatial statistical analysis

• Hypothesis testing

4: Spatial Modeling

• Forecasts and predictions
• E.g., climate change modeling

What will we do?

That is a lot, considering how long you’ve studied just #1 (and maybe a bit of #2)

Spatial data analysis is concerned with that branch of data analysis where the geographical referencing of objects contains important information. In many areas of data collection, especially in some areas of experimental science, the indexes that distinguish different cases can be exchanged without any loss of information. All the information relevant to understanding the variation in the data set is contained in the observations and no relevant information is contained in the indexing. In the case of spatial data the indexing (by location and time) may contain crucial information. A definition of spatial analysis (of which spatial data analysis is one element) is that it represents a collection of techniques and models that explicitly use the spatial referencing of each data case.

Goodchild, Michael, and Haining. 2004. “GIS and spatial data analysis: converging perspectives”. Papers in Regional Science 83:363-385

Spatial data analysis is concerned with that branch of data analysis where the geographical referencing of objects contains important information. In many areas of data collection, especially in some areas of experimental science, the indexes that distinguish different cases can be exchanged without any loss of information. All the information relevant to understanding the variation in the data set is contained in the observations and no relevant information is contained in the indexing. In the case of spatial data the indexing (by location and time) may contain crucial information. A definition of spatial analysis (of which spatial data analysis is one element) is that it represents a collection of techniques and models that explicitly use the spatial referencing of each data case.

Goodchild, Michael, and Haining. 2004. “GIS and spatial data analysis: converging perspectives”. Papers in Regional Science 83:363-385

Spatial Data - A refresher

• What are spatial data?
• How do we represent the world in GIS?

www.in.gov/gis/gis101.htm

Vector (object) vs. Raster (field)

• Vector (object): Describes the world as a space filled with discrete, identifiable units that have some type of spatial reference
• Raster (field): Describes the world as a collection of continuous spatial distributions of different phenomena

Vector (object) vs. Raster (field)

Raster and Vector Data

Fields

• Useful when the phenomenon is measured at all locations … such as?
• Common in physical geography
• Often used to visualize patterns
• Usually represented on a grid

Can you name this location? Hint: it is very close to downtown Portland!

https://www.crwr.utexas.edu/gis/gishydro05/Time/daymet.htm

An example in R

require(raster)
require(tmap)
rgbi <- stack('C:/data/rgbi.tif') %>% 
  `names()<-`(c("R","G","B","I"))

dhm <- raster('C:/data/dhm.tif')

dhm
rgbi

## class      : RasterLayer 
## dimensions : 948, 1005, 952740  (nrow, ncol, ncell)
## resolution : 3, 3  (x, y)
## extent     : 744597.5, 747612.5, 1379506, 1382350  (xmin, xmax, ymin, ymax)
## crs        : +proj=lcc +lat_0=41.75 +lon_0=-120.5 +lat_1=43 +lat_2=45.5 +x_0=400000 +y_0=0 +ellps=GRS80 +units=ft +no_defs 
## source     : dhm.tif 
## names      : dhm 
## values     : 0, 203.74  (min, max)

## class      : RasterStack 
## dimensions : 948, 1005, 952740, 4  (nrow, ncol, ncell, nlayers)
## resolution : 3, 3  (x, y)
## extent     : 744597.5, 747612.5, 1379506, 1382350  (xmin, xmax, ymin, ymax)
## crs        : +proj=lcc +lat_0=41.75 +lon_0=-120.5 +lat_1=43 +lat_2=45.5 +x_0=400000 +y_0=0 +ellps=GRS80 +units=ft +no_defs 
## names      : R, G, B, I

RGBI

tmap_options(legend.outside=T,legend.outside.size = 0.2)
tm_shape(rgbi) +
  tm_rgb(r = 4,g = 2,b = 3) +
  tm_layout(main.title = "False Color Infrared")

DHM

tm_shape(dhm) +
  tm_raster(style = 'cont',title="Height",palette = "-Greys") +
  tm_layout(main.title = "Digital Height Model",outer.margins = c(0.01,0.03,0.03,-0.1))

Convert to a dataframe

rgbi_df <- as.data.frame(rgbi)
head(rgbi_df,100)

##      R  G  B   I
## 1   56 65 51 140
## 2   34 42 31 113
## 3   36 44 34 127
## 4   15 19 15  79
## 5    3  6  9  41
## 6    3  5  9  33
## 7    3  5  7  31
## 8    8 14 16  56
## 9   22 32 31 100
## 10   6 11 11  52
## 11   3  6  9  36
## 12   3  7  9  44
## 13   3  6 10  44
## 14   4  9 12  46
## 15   3  5 10  43
## 16  34 45 36 143
## 17  34 44 29 138
## 18  50 65 45 177
## 19  41 53 38 166
## 20  12 15 10  94
## 21  59 75 52 191
## 22  38 49 32 162
## 23  50 64 45 177
## 24  36 48 32 146
## 25  26 36 26 120
## 26  46 62 51 170
## 27  26 35 25 120
## 28  20 27 18  94
## 29  65 78 60 162
## 30  53 62 48 139
## 31  38 46 33 119
## 32  56 65 48 135
## 33  66 79 62 148
## 34  70 77 59 138
## 35  20 25 17  68
## 36  10 15 11  52
## 37   8 13 11  46
## 38   6 10 11  39
## 39  13 18 15  61
## 40  29 36 26 103
## 41  29 37 24 113
## 42  14 22 16  90
## 43  12 19 17  77
## 44  11 18 17  79
## 45  36 47 36 139
## 46   7 11  9  61
## 47  40 53 38 133
## 48   6 10 13  13
## 49   5  9 14   7
## 50   5 10 15   5
## 51   7 12 18   8
## 52   6 12 16   6
## 53   7 10 17   7
## 54  11 16 22  10
## 55  25 27 32  19
## 56  50 48 48  39
## 57  15 16 14  13
## 58  83 79 77  66
## 59  84 80 77  60
## 60  81 78 76  59
## 61  75 71 71  52
## 62  70 67 67  55
## 63  65 57 52  65
## 64  60 51 48  58
## 65  64 52 48  66
## 66  63 54 50  64
## 67  60 53 51  59
## 68  38 35 38  37
## 69  15 16 21  24
## 70   8 12 14  31
## 71  43 54 32 135
## 72  70 82 50 185
## 73  64 72 45 175
## 74  39 53 34 171
## 75  44 60 39 169
## 76  42 55 40 174
## 77  32 46 28 159
## 78  46 65 35 173
## 79  24 36 18 116
## 80  33 44 26 115
## 81  30 40 29 109
## 82  61 73 57 149
## 83  56 68 53 145
## 84  46 57 44 128
## 85  37 47 38 113
## 86  61 74 61 149
## 87  57 72 56 143
## 88  64 77 61 153
## 89  55 68 51 147
## 90  56 67 55 135
## 91  50 62 51 138
## 92  44 56 47 128
## 93  45 55 44 120
## 94  37 48 38 118
## 95  48 60 48 132
## 96  62 75 59 149
## 97  52 64 50 133
## 98  64 79 65 154
## 99  63 79 65 153
## 100 59 75 61 151

Create NDVI in the dataframe

\(NDVI = \frac{Red - NIR}{Red + NIR}\)

rgbi_df$NDVI <- (rgbi_df$R - rgbi_df$I) / (rgbi_df$R + rgbi_df$I)
head(rgbi_df, 100)

##      R  G  B   I         NDVI
## 1   56 65 51 140 -0.428571429
## 2   34 42 31 113 -0.537414966
## 3   36 44 34 127 -0.558282209
## 4   15 19 15  79 -0.680851064
## 5    3  6  9  41 -0.863636364
## 6    3  5  9  33 -0.833333333
## 7    3  5  7  31 -0.823529412
## 8    8 14 16  56 -0.750000000
## 9   22 32 31 100 -0.639344262
## 10   6 11 11  52 -0.793103448
## 11   3  6  9  36 -0.846153846
## 12   3  7  9  44 -0.872340426
## 13   3  6 10  44 -0.872340426
## 14   4  9 12  46 -0.840000000
## 15   3  5 10  43 -0.869565217
## 16  34 45 36 143 -0.615819209
## 17  34 44 29 138 -0.604651163
## 18  50 65 45 177 -0.559471366
## 19  41 53 38 166 -0.603864734
## 20  12 15 10  94 -0.773584906
## 21  59 75 52 191 -0.528000000
## 22  38 49 32 162 -0.620000000
## 23  50 64 45 177 -0.559471366
## 24  36 48 32 146 -0.604395604
## 25  26 36 26 120 -0.643835616
## 26  46 62 51 170 -0.574074074
## 27  26 35 25 120 -0.643835616
## 28  20 27 18  94 -0.649122807
## 29  65 78 60 162 -0.427312775
## 30  53 62 48 139 -0.447916667
## 31  38 46 33 119 -0.515923567
## 32  56 65 48 135 -0.413612565
## 33  66 79 62 148 -0.383177570
## 34  70 77 59 138 -0.326923077
## 35  20 25 17  68 -0.545454545
## 36  10 15 11  52 -0.677419355
## 37   8 13 11  46 -0.703703704
## 38   6 10 11  39 -0.733333333
## 39  13 18 15  61 -0.648648649
## 40  29 36 26 103 -0.560606061
## 41  29 37 24 113 -0.591549296
## 42  14 22 16  90 -0.730769231
## 43  12 19 17  77 -0.730337079
## 44  11 18 17  79 -0.755555556
## 45  36 47 36 139 -0.588571429
## 46   7 11  9  61 -0.794117647
## 47  40 53 38 133 -0.537572254
## 48   6 10 13  13 -0.368421053
## 49   5  9 14   7 -0.166666667
## 50   5 10 15   5  0.000000000
## 51   7 12 18   8 -0.066666667
## 52   6 12 16   6  0.000000000
## 53   7 10 17   7  0.000000000
## 54  11 16 22  10  0.047619048
## 55  25 27 32  19  0.136363636
## 56  50 48 48  39  0.123595506
## 57  15 16 14  13  0.071428571
## 58  83 79 77  66  0.114093960
## 59  84 80 77  60  0.166666667
## 60  81 78 76  59  0.157142857
## 61  75 71 71  52  0.181102362
## 62  70 67 67  55  0.120000000
## 63  65 57 52  65  0.000000000
## 64  60 51 48  58  0.016949153
## 65  64 52 48  66 -0.015384615
## 66  63 54 50  64 -0.007874016
## 67  60 53 51  59  0.008403361
## 68  38 35 38  37  0.013333333
## 69  15 16 21  24 -0.230769231
## 70   8 12 14  31 -0.589743590
## 71  43 54 32 135 -0.516853933
## 72  70 82 50 185 -0.450980392
## 73  64 72 45 175 -0.464435146
## 74  39 53 34 171 -0.628571429
## 75  44 60 39 169 -0.586854460
## 76  42 55 40 174 -0.611111111
## 77  32 46 28 159 -0.664921466
## 78  46 65 35 173 -0.579908676
## 79  24 36 18 116 -0.657142857
## 80  33 44 26 115 -0.554054054
## 81  30 40 29 109 -0.568345324
## 82  61 73 57 149 -0.419047619
## 83  56 68 53 145 -0.442786070
## 84  46 57 44 128 -0.471264368
## 85  37 47 38 113 -0.506666667
## 86  61 74 61 149 -0.419047619
## 87  57 72 56 143 -0.430000000
## 88  64 77 61 153 -0.410138249
## 89  55 68 51 147 -0.455445545
## 90  56 67 55 135 -0.413612565
## 91  50 62 51 138 -0.468085106
## 92  44 56 47 128 -0.488372093
## 93  45 55 44 120 -0.454545455
## 94  37 48 38 118 -0.522580645
## 95  48 60 48 132 -0.466666667
## 96  62 75 59 149 -0.412322275
## 97  52 64 50 133 -0.437837838
## 98  64 79 65 154 -0.412844037
## 99  63 79 65 153 -0.416666667
## 100 59 75 61 151 -0.438095238

Looks like we created NDVI correctly

rgbi_df$NDVI %>% 
  hist(main="NDVI Distribution",xlab='NDVI Value')
abline(v=0,col='red',lwd=2)

# Also, convert dhm to data.frame
dhm_df <- as.data.frame(dhm)

Let’s make a model!

Linear/OLS models are very easy in R:

mod <- lm(mpg ~ hp + wt, data = mtcars)
summary(mod)

## 
## Call:
## lm(formula = mpg ~ hp + wt, data = mtcars)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -3.941 -1.600 -0.182  1.050  5.854 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 37.22727    1.59879  23.285  < 2e-16 ***
## hp          -0.03177    0.00903  -3.519  0.00145 ** 
## wt          -3.87783    0.63273  -6.129 1.12e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.593 on 29 degrees of freedom
## Multiple R-squared:  0.8268, Adjusted R-squared:  0.8148 
## F-statistic: 69.21 on 2 and 29 DF,  p-value: 9.109e-12

“MPG as a function of Horsepower and Weight”

Let’s make a model!

Our raster data is also now a data.frame!

mod <- lm(dhm_df$dhm ~ rgbi_df$NDVI)
summary(mod)

## 
## Call:
## lm(formula = dhm_df$dhm ~ rgbi_df$NDVI)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -108.801  -27.089    0.794   27.809  158.684 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   23.24802    0.07695   302.1   <2e-16 ***
## rgbi_df$NDVI -85.55340    0.12767  -670.1   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 39.03 on 952386 degrees of freedom
##   (352 observations deleted due to missingness)
## Multiple R-squared:  0.3204, Adjusted R-squared:  0.3204 
## F-statistic: 4.49e+05 on 1 and 952386 DF,  p-value: < 2.2e-16

Wouldn’t it be great if we could see where our model error is?

Plotting Residuals

# Save residuals/error to new object
residuals <- mod$residuals

# Create a raster the same size as DEM
resid <- dhm

# For all of the complete rows (no NA) in the imagery, add residuals
resid[complete.cases(rgbi_df)] <- residuals

tm_shape(resid) +
  tm_raster(style = "cont", palette = "RdBu",title = "Residuals") +
  tm_layout(main.title = "Residual Map",outer.margins = c(0.01,0.03,0.03,-0.03))

So… what are we looking at?

Residuals Refresher

Predicting With a Model

dhm_model <- dhm
dhm_model[complete.cases(rgbi_df)] <- predict(mod)

tm_shape(dhm_model) +
  tm_raster(style = "cont", palette = "-Greys",title = "Height") +
  tm_layout(main.title = "Predicted Heights",outer.margins = c(0.01,0.03,0.03,-0.03))

So… what are we looking at?

Steven’s Level of Measurement

Steven’sStevens’ Level of Measurement

S.S. Stevens’ paper is ancient… 1946! https://marces.org/EDMS623/Stevens%20SS%20(1946)%20On%20the%20Theory%20of%20Scales%20of%20Measurement.pdf

Commonly used in database design

Wait a moment…

… it’s ‘tidy’!

Those of you with database experience will recognize this.

Most of the legwork in spatial analysis is battling the entity-attribute model:

• data formatting
• data transformation
• data munging
• geoenrichment (more on this week 8!)

Form of Data Influences Results

Young, and Gotway. 2007. “Linking spatial data from different sources: The effects of change of support.” Stochastic Environmental Research & Risk Assessment 21:589-600

Study assesses the interpolation results of risk based on ZIP polygon centroids and based on point samples.

Results are completely different, with addition of data

Young, and Gotway. 2007. “Linking spatial data from different sources: The effects of change of support.” Stochastic Environmental Research & Risk Assessment 21:589-600

Some Issues

Recommended Required Reading

( … down the rabbit hole … )

Bian, L. (2007). Object-Oriented Representation of Environmental Phenomena: Is Everything Best Represented as an Object? Annals of the Association of American Geographers, 97(2), 267-281. https://doi.org/10.1111/j.1467-8306.2007.00535.x

• GIS usually claims to offer spatial analysis … but this doesn’t mean statistical spatial analysis
• Spatial data manipulation is readily available (buffer, overlay, spatial join)
• GIS has increased the need for spatial analysis, and lends some support

Spatial Analysis

Spatial Analysis is concerned with investigating the patterns that arise as a result of processes that may be operating in space. Techniques and methods to enable the representation, description, measurement, comparison and generation of spatial patterns are central to the study of geographic information analysis.

O’Sullivan and Unwin, page 4

Spatial Analysis

Spatial Analysis is concerned with investigating the patterns that arise as a result of processes that may be operating in space. Techniques and methods to enable the representation, description, measurement, comparison and generation of spatial patterns are central to the study of geographic information analysis.

O’Sullivan and Unwin, page 4

Why isn’t spatial analysis in GIS?

GIS is built around the entity-attribute model. Spatial analysis uses these data, and sees data as patterns that are the outcomes of processes. This is similar… but different than spatial data manipulation.

In short, I’m trying to make you think like a data analyst who can utilize spatial information and not a GIS analyst who knows some data analysis techniques (or worse, some buttons in ArcGIS)

Summary

1. We usually distinguish spatial analysis from GIS operations and spatial modeling
2. Spatial data in the entity-attribute model combines the entity with its attributes in various combinations
3. … which has a many issues
4. … and is always reductive (read Bian 2007!)
5. Spatial analysis is still poorly supported by GIS, but it is improving.

DataCamp

Do some exercises! Do a bunch!

You could easily knock out several before Monday, and free yourself up later in the term (or just learn more).

Lab

There’s no lab, because we haven’t even really learned anything yet!

Don’t worry, we’re just laying a sturdy foundation and will dive in next week.

Bian (2007)

Bian, L. (2007). Object-Oriented Representation of Environmental Phenomena: Is Everything Best Represented as an Object? Annals of the Association of American Geographers, 97(2), 267-281.