In-class Exercise 2: Spatial

Published

November 25, 2023

Modified

November 30, 2023

Hidden: In-Class Lecture notes

In-Class Lecture Notes

“In real world, there is no concept of equality.” – Prof Kam

### Dependency Ratio Plot

-   First: check for statistical significance,
-   Then, plot/test for null hypothesis == normally distributed

-   In geospatial, we don't test for `normality`, we test for `spatial randomness` : whatever is observed in space is randomly distributed
    -   Similar to default assumption that distributions are Normal, default assumption
    -   Worth checking; not all

-   `Dependency Ratio`: Number of non-working people / Number of working people
    - High chance that dependency ratio is not randomly distributed;
    - Socio-economic factors (eg private homes, "good" schools, workers)
    - SES-locales can affect crime rates, electricity consumption, 
    => Concept of `Spatial Inequality`: This world is not equal


### Tobler's First Law of Geography: 

Everything is related to everything else, but near things are more related than distant things


### Spatial Weights

- Spatial weights help to parse spatial dependency
- Use of `adjacency` or `neighbours` as a frame to examine dependency
    - Neighbourhood search: neighbours or not neighbours
    
- How do we qualify 'neighbourliness'?
    - ADJACENCY - If vertex/edges touching, then neighbours
    - DISTANCE  - If within certain distance, then neighbours
        - binary, (1/0) neighbourhood stats
    - INVERSE DISTANCE  - 1/distance
        - gives higher weightage to nearer neighbours; Tobler's

- In the real world, not everything is adjacency -- islands are not touching
    - If using hex-based map, also 
    - Tobler's first law: not touching but also has direct effects
- Use of hexagon binning: MODIFIED AREA UNIT
    - Planning Subzone has huge irregularities e.g. large areas (deployment area), oddly shaped long areas etc
    - Hexagons are one of most compact tessellating shapes
    - Modern city design work uses grids -- rare, not very usual (eg Punggol) 


- QUEEN'S CONTIGUITY: "So long as they touch"; any point on edge is vertex

- LAGGED CONTINUITY
    - Lag 1: all adjacent neighbours
    - Lag 2: adjacent neighbours' neighbours PLUS Lag 1;
- All levels of lag include previous levels;
    - Hence, see correllogram: influence gradually wanes over distance
- EXAMPLE: Where to put Pizza Hut to minimise drive time, maximize food quality
    - Look at real estate to identify largest expected market share, penetration
    
- If region is considered water or unnecessary, region can be dropped as rows
    - Has to be excluded from consideration, otherwise will have geometric errors
    - Will also mess with Moran's I calculations
    
    
- Use of ROW-STANDARDISED WEIGHTS to avoid bias
    - Since weights matrix is a matrix, diagonals are zero, symmetric about diagonal
    - Also possible to calculate COLUMN-STANDARDISED METRICS with similar effect

- GLOBAL Measure are for mathematical understanding;
- LOCAL measures are more interesting

- GDPPC in any regions will be unevenly spatially distributed
    - Aside from capital, may have secondary city: second region of prosperity & development as growth centre
    
    
### Spatial Dependency

- When measuring rainfall, not every region has a rainfall station. How do we do this for all region?
    - Use interpolation for uncovered regions + concept of Spatial Dependency
    
### Spatial Autocorrelation
- Typical correlation compares 2 different features;
- Spatial Autocorrelation compares one feature, of self vs neighbour
    - Similar formula to correlation coefficient
    
- Spatial Autocorrelation tests (Moran's I, Geary's C) tries to reject the null hypothesis that "things are randomly distributed in space"
    - Positive Spatial Autocorrelation: Pattern of clustering, e.g. high is found near to high 
    - Negative Spatial Autocorrelation does not mean randomness! Instead, it follows some ordered pattern (e.g. checkerboard -- high surrounds low, low surrounds high); searching for outliers where different from surroundings
    - Spatial Autocorrelation suggests non-random patterns 
    
- in Spatial Data, go directly to monte carlo simulation


### LISA
- Helps to identify clusters (high-high, low-low) and outliers (high-low, low-high) ==> regions of positive/negative spatial autocorrelation
    - hence quadrant 5, "none"
- For all tests, we only highlight Spatial Autocorrelation AND high statistical significance

- Why might region not be statistically significant?
    - Not enough neighbours; test is uncertain
    - Enough neighbours, but data just not stat-significant
    
    
### Gi*
- For Gi*, should always be distance-based;
    - Moran's I, Geary's C can be contiguity or distance-based; but Getis-Ord should always be distance-based
    - Evolved from work on dengue transmissions; hence distance is more valuable
- Gi does not count itself, Gi* counts itself


### Mann-Kendall 
- Useful for time-series data;
    - Hunan GDPPC is single-instance, but most effects happen over space and time; thus, it's useful to consider time 
- A "sign" statistic:
    - compares value at time j vs at time k;
    - outputs either +1 if decreasing, -1 if increasing, or 0 if no change
    - does not consider output, only test for monotonicity and "interprets the relationship"
    - cannot have breaks in timeseries; eg if COVID, add datapoint of 0 to keep it going
- Note: this is nonspatial, only statistical


### Emerging Hotspot EHSA
- Space & time analysis improvement over Mann-Kendall
- Instead using Gi* instead of value
    - Compares Gi* over time -- increasing or decreasing
    
- Need to transform data into SPACETIME CUBE
    - x, y dimensions of attribute a
    - z-dimension of time 
    - `sfdep` allows construction of spacetime cube

Spatial Weights

Overview

See also In-Class Exercise 2: GLSA and In-Class Exercise 2: EHSA

Getting Started - Import packages

This function calls pacman to load sf, tidyverse, tmap, knitr packages;

tmap : For thematic mapping; powerful mapping package
sf : for geospatial data handling, but also geoprocessing: buffer, point-in-polygon count, etc
- batch processing over GIS packages; can handle tibble format
sfdep : creates space-time cube, EHSA; replaces spdep
tidyverse : for non-spatial data handling; commonly used R package
knitr : generates html table

pacman::p_load(tmap, sf, sfdep, tidyverse, knitr)

Loading the data

Hunan: geospatial dataset in ESRI shapefile format
- use of st_read() to import assf data.frame
  - $geometry column is actually a list inside the df cell; that’s the power of the tibble dataframe
  - “features” of simple features refers to geometric features eg point line curve etc
- note projection is WGS84; see `88
hunan2012: attribute format in csv format
- use of read_csv() astbl_df data.frame
!IMPORTANT! to retain geometry, you must left join to the sf dataframe (eg you can also hunan2012 right join hunan)
- without sf dataframe, normal tibble dataframe will drop the geometry column

show code

hunan <- st_read(dsn = "data/geospatial", 
                 layer = "Hunan")

Reading layer `Hunan' from data source 
  `C:\1darren\ISSS624\In-class_Ex\In-class_Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 88 features and 7 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 108.7831 ymin: 24.6342 xmax: 114.2544 ymax: 30.12812
Geodetic CRS:  WGS 84

show code

hunan2012 <- read_csv("data/aspatial/Hunan_2012.csv")
hunan_GDPPC <- left_join(hunan,hunan2012)%>%
  select(1:4, 7, 15)

Plot a chloropleth of GDPPC

show code

#qtm(hunan, "GDPPC") +
#  tm_layout(main.title = "GDPPC", main.title.position = "right")


tm_shape(hunan_GDPPC) +
  tm_fill(col = "GDPPC", 
          style = "pretty",
          palette = "Blues",
          title = "GDPPC") +
  tm_borders(alpha = 0.5) +
  tm_layout(main.title = "GDPPC",
            inner.margins = c(0.1, 0.1, 0.1, 0.1),
            outer.margins = c(0.1, 0.1, 0.1, 0.1)
            ) + 
  tm_grid(alpha = )

Deriving QUEEN contiguity weights

mutate is function that creates new column from previous column datas
- st_contiguity creates nb neighbour matrix (QUEEN contiguity, by default)
- st_weights creates row-standardised weights (style="W") from nb object
- One-step function using sfdep; a wrapper for spdep but writes output into sf dataframe

show code

wm_q <- hunan_GDPPC %>%
  mutate(nb = st_contiguity(geometry), 
         wt = st_weights(nb, style = "W"),
         .before = 1)