Notation
(Classification discussed here according to Cressie (1993))
Consider a spatial process in \(d\)
dimensions: \(\{Z(s): s \in D \subset
\mathbb{R}^d\}\). \(Z\) is the
attribute we observe at spatial location \(s\) (a vector if \(d\) co-ordinates). Most often \(d=2\) - Cartesian co-ordinates.
1. Geospatial Data
- \(D\): continuous, fixed set
- continuous: \(Z\) can be
observed everywhere
- fixed: points in \(D\) are
not stochastic
- \(Z\) can be discrete of
continuous
Data measured only at certain locations - cannot be sampled
exhaustively. Can we construct a surface of \(Z\) over the entire domain?
1. Geospatial Data
a <- as.ppp(longleaf)
a1 <- as.data.frame(longleaf)
plot(a)
1. Geospatial Data
aa <- as.ppp(chorley)
plot(aa)
1. Geospatial Data
## [[1]]
## [[1]]$x
## [1] 359.46 359.41 359.56 360.78 361.35 362.31 363.30 363.91 364.71 366.45
## [11] 365.80 365.61 365.76 364.99 365.40 365.75 365.90 366.04 365.36 365.05
## [21] 365.20 364.48 364.21 364.47 364.32 363.93 363.97 363.67 363.32 363.28
## [31] 362.90 362.71 362.93 363.43 363.43 363.69 363.50 363.01 362.74 362.77
## [41] 363.69 364.22 364.03 363.45 362.54 361.70 361.35 360.86 360.40 359.76
## [51] 358.38 358.19 358.88 359.00 358.69 358.27 357.55 357.47 357.13 356.56
## [61] 356.22 355.73 355.23 354.77 354.05 353.13 352.83 352.60 351.99 350.04
## [71] 347.83 346.16 343.45 344.45 344.30 345.33 345.52 345.83 345.68 345.26
## [81] 345.23 345.69 345.99 345.65 346.03 346.07 345.66 345.96 345.77 346.27
## [91] 346.42 346.16 346.46 346.24 346.66 346.54 346.81 346.55 346.86 347.32
## [101] 347.17 347.66 347.93 347.51 348.38 348.35 349.30 349.84 350.94 351.36
## [111] 352.85 353.38 353.72 354.06 354.06 354.25 354.37 353.88 354.03 354.34
## [121] 354.72 355.67 356.13 357.08 358.15 358.42 358.23 358.42 358.61 358.39
## [131] 358.73
##
## [[1]]$y
## [1] 410.57 411.15 411.99 412.80 412.57 410.90 411.51 412.51 412.93 414.59
## [11] 415.25 416.21 416.91 419.08 420.12 420.16 420.50 422.01 422.01 422.29
## [21] 422.71 423.02 423.99 424.41 425.03 425.31 425.58 425.77 425.77 426.70
## [31] 426.97 427.48 427.90 428.28 428.63 429.09 429.21 428.98 429.14 429.56
## [41] 430.22 430.72 431.10 431.14 431.23 431.15 431.74 431.39 431.74 431.36
## [51] 431.79 431.56 431.01 430.55 430.24 430.01 430.02 429.55 429.63 428.71
## [61] 428.21 428.35 428.64 428.21 428.57 428.11 428.34 428.96 429.16 429.05
## [71] 428.48 427.91 427.00 426.92 426.38 425.40 424.90 423.28 423.12 423.12
## [81] 422.55 421.69 421.57 421.07 420.76 420.03 419.61 419.26 418.99 418.75
## [91] 418.21 418.02 417.82 417.55 416.86 416.66 416.35 415.04 414.96 414.22
## [101] 413.84 413.45 413.72 414.49 414.49 414.10 413.98 413.52 414.40 414.40
## [111] 413.77 413.77 414.31 414.38 414.69 414.57 413.84 413.42 412.37 412.10
## [121] 412.02 412.17 412.09 412.39 412.54 412.31 412.04 411.81 410.99 410.49
## [131] 410.41
2. Lattice Data (or Regional Data)
- \(D\): fixed and discrete
- fixed: points \(D\) are
not stochastic
- discrete: countable
For example (1) data collected at the ward level, (2) remotely sensed
data reported at the pixel level. This is spatially aggregated
data thus also called Regional Data. The data is usually
exhaustively observed.
2. Lattice Data (or Regional Data)
The term used is sites instead of points to refer
to the spatial location of lattice data. Usually a polygon (ward
boundary) with some representative location as the centroid (for
example).
Notation: \(Z(A_i)\)
- \(A_i\)’s in close proximity may
have similar values (positive spatial autocorrelation)
- Identify region clustering of ‘high’ values
- Where are the spatial risks? (Correlate with another
covariate.)
Measures on Lattices
Need a measure of spatial connectivity.How is distace between
‘representative points’ determined? For each pair \(s_i\) and \(s_j\) associate a weight \(w_{ij}\) for sites considered spatially
connected \[w_{ij} = \left\{\begin{eqnarray}
1 & \textrm{ if connected} \\
0 & \textrm{ otherwise.}
\end{eqnarray}\right.\]
For pixels: 4-connectivity, diagonal neighbours, 8-connectivity
Measures on Lattices
For polygons: if points share a common border or if representative
points are less than a certain critical distance apart
knitr::include_graphics('scale1to10.jpg')
Measures on Lattices
## List of spatial objects
##
## fine:
## window: binary image mask
## 1570 x 778 pixel array (ny, nx)
## enclosing rectangle: [0, 9.88] x [0, 19.94] metres
##
## medium:
## window: binary image mask
## 512 x 256 pixel array (ny, nx)
## enclosing rectangle: [0, 10] x [0, 20] metres
##
## coarse:
## window: binary image mask
## 200 x 100 pixel array (ny, nx)
## enclosing rectangle: [0, 10] x [0, 20] metres
Measures on Lattices
Measures on Lattices
Measures on Lattices
## binary image mask
## 1570 x 778 pixel array (ny, nx)
## pixel size: 0.0127 by 0.0127 metres
## enclosing rectangle: [0, 9.88] x [0, 19.94] metres
## Window area = 97.0189 square metres
## Unit of length: 1 metre
## Fraction of frame area: 0.492
Measures on Lattices
Can <- st_read("scottish_lip_cancer.shp")
## Reading layer `scottish_lip_cancer' from data source
## `G:\My Drive\My Documents\Work\Conferences and Seminars\Courses\3MC 2024\Inger's Notes\scottish_lip_cancer.shp'
## using driver `ESRI Shapefile'
## Simple feature collection with 56 features and 11 fields
## Geometry type: MULTIPOLYGON
## Dimension: XY
## Bounding box: xmin: -8.621389 ymin: 54.62722 xmax: -0.7530556 ymax: 60.84444
## Geodetic CRS: GCS_Assumed_Geographic_1
## Warning: plotting the first 10 out of 11 attributes; use max.plot = 11 to plot
## all
Measures on Lattices
## NAME ID COUNT SMR
## Length:56 Min. : 1.00 Min. : 0.000 Min. : 0.0
## Class :character 1st Qu.:14.75 1st Qu.: 4.750 1st Qu.: 49.6
## Mode :character Median :28.50 Median : 8.000 Median :111.5
## Mean :28.50 Mean : 9.571 Mean :152.6
## 3rd Qu.:42.25 3rd Qu.:11.000 3rd Qu.:223.0
## Max. :56.00 Max. :39.000 Max. :652.2
## LONG LAT PY EXP_
## Min. :54.94 Min. :1.430 Min. : 27075 Min. : 1.100
## 1st Qu.:55.78 1st Qu.:3.288 1st Qu.: 100559 1st Qu.: 4.050
## Median :56.04 Median :4.090 Median : 182333 Median : 6.300
## Mean :56.40 Mean :4.012 Mean : 267498 Mean : 9.575
## 3rd Qu.:57.02 3rd Qu.:4.730 3rd Qu.: 313845 3rd Qu.:10.125
## Max. :60.24 Max. :6.800 Max. :2316353 Max. :88.700
## AFF X_COOR Y_COOR geometry
## Min. : 0.000 Min. :112892 Min. : 561163 MULTIPOLYGON :56
## 1st Qu.: 1.000 1st Qu.:256624 1st Qu.: 649520 epsg:NA : 0
## Median : 7.000 Median :287577 Median : 681524 +proj=long...: 0
## Mean : 8.661 Mean :288524 Mean : 723127
## 3rd Qu.:11.500 3rd Qu.:333949 3rd Qu.: 794380
## Max. :24.000 Max. :442245 Max. :1168904
Measures on Lattices
## [1] 5 3 9 39 2 9 16 6 17 19 15 16 2 6 4 1 8 1 8 1 9 3 10 1 28
## [26] 8 6 11 19 3 2 11 10 9 7 7 5 7 0 7 0 8 3 7 20 31 8 7 6 11
## [51] 15 13 9 26 11 11
Measures on Lattices
## [1] 279.3 277.8 162.7 450.3 186.9 197.8 153.0 30.6 216.8 122.8 120.1 111.3
## [13] 46.3 83.3 75.9 27.6 50.7 29.1 93.8 14.2 89.1 36.6 53.3 17.4
## [25] 31.6 85.6 41.0 107.8 37.5 32.1 35.8 89.3 111.6 355.7 157.7 99.6
## [37] 105.3 124.6 0.0 167.5 0.0 241.7 104.2 115.9 301.7 136.7 333.3 304.3
## [49] 303.0 361.8 352.1 295.5 652.2 320.6 125.4 86.8
Measures on Lattices
## [1] 37521 29374 162867 231337 27075 111665 263205 547016 185472
## [10] 432132 378946 346041 141294 231227 156924 110707 426519 179194
## [19] 233125 246744 296238 238170 617413 146112 2316353 319072 449231
## [28] 382702 1287561 312103 246849 319316 231185 51710 102697 249667
## [37] 139148 163818 38704 94145 103412 86444 65448 163703 165554
## [46] 583327 53199 62603 59183 83190 129271 87815 28324 245513
## [55] 190816 391513
Measures on Lattices
count <- data.frame(Can$X_COOR, Can$Y_COOR, Can$COUNT)
w <- owin(xrange = c(min(Can$X_COOR), max(Can$X_COOR)), yrange = c(min(Can$Y_COOR), max(Can$Y_COOR)))
plot(w)
countpp <- as.ppp(count,w)
plot(countpp)
Measures on Lattices
## Warning: plotting the first 10 out of 11 attributes; use max.plot = 11 to plot
## all
3. Point Patterns (Unmarked patterns)
A point pattern is a collection of points \(I(s), s \in D^*\). The random
domain \(D^*\) is obtained as the
locations in fixed \(D\) for which
\(I(s) =1\) (locations where \(I(s)=0)\) removed from \(D\)). Each realisation of the point process
produces a \(D^*\). The indicator
function could be something like the following, but the focus is on
\(D^*\) more than on \(I(\cdot)\): \[I(s) = \left\{\begin{eqnarray}
1 & \textrm{ if } Z(s) \ge c \\
0 & \textrm{ otherwise.}
\end{eqnarray}
\right.\]
Point patterns are effectively unmarked spatial data.
3. Point Patterns (Unmarked patterns)
We ask: Are the points random or is there a spatial
pattern?
data(murchison)
murchison
## List of spatial objects
##
## gold:
## Planar point pattern: 255 points
## window: rectangle = [352782.9, 682589.6] x [6699742, 7101484] metres
##
## faults:
## planar line segment pattern: 3252 line segments
## window: rectangle = [352782.9, 682589.6] x [6699742, 7101484] metres
##
## greenstone:
## window: polygonal boundary
## enclosing rectangle: [352782.9, 681699.6] x [6706467, 7100804] metres
plot(murchison$greenstone)
# 3. Point Patterns (Unmarked patterns)
mpp <- as.ppp(murchison$gold)
mpp
## Planar point pattern: 255 points
## window: rectangle = [352782.9, 682589.6] x [6699742, 7101484] metres
3. Point Patterns (Unmarked patterns)
lpp <- as.psp(murchison$faults)
lpp
## planar line segment pattern: 3252 line segments
## window: rectangle = [352782.9, 682589.6] x [6699742, 7101484] metres
3. Point Patterns (Unmarked patterns)
ppp <- as.polygonal(murchison$greenstone)
ppp
## window: polygonal boundary
## enclosing rectangle: [352782.9, 681699.6] x [6706467, 7100804] metres
