More common spatial analysis tools

Rachel Franklin

Harvard University

Goals for this session

• Get comfortable with a few more foundational GIS spatial analysis tools
• Start to see how concepts and tools map onto research questions
• Building understanding of spatial analysis workflows

Without further ado, let’s dive in

Thiessen (or Voronoi) polygons

Deriving polygons from points

• What about the case where we might want to go from point features to polygons?
• For example, suppose we want to define “catchment areas” for each metro station in Rome.
  • The assumption being that each person will use the station closest to them

Thiessen (or Voronoi) polygons

Deriving polygons from points

Thiessen (or Voronoi) polygons

Allocating areas to their closest points

• In QGIS: Voronoi Polygons tool
  • Creates new polygon layer as output
  • Default extent of polygons is the bounded rectangle containing the points
    • Use buffer region option to expand boundaries of new polygon layer

Thiessen (or Voronoi) polygons

Allocating areas to their closest points

How to change boundaries of a layer to match another

Clip

• In QGIS located under Vector > Geoprocessing
  • Clips—or cuts—one layer, using boundaries of a second layer
  • So here, we can clip Voronoi polygons using the urban area polygon layer

How to change boundaries of a layer to match another

Clip

Other common tools

Near (distance to nearest hub in QGIS)

Calculates the straight line distance from the input feature to the nearest feature in a specified feature set

Grids

Creating our own polygons

• Useful for generating uniform polygons as inputs for spatial joins
  • Overcomes shortcomings of administrative geographies which can vary considerably in size
• Also useful for visualizing data in an appealing way
• In QGIS, under Research Tools > Create Grid
  • Choice of size and shape of output polygons (e.g., hexagons, squares, diamonds)

Grids

Creating our own polygons

Grids

Spatial join with grids: AirBnb

What other tools could we use to improve this visualization?

Density estimation

Take our standard approach so far

Visualising points

Take our standard approach so far

Point in polygon

The usual approach, where we reference counts of occurrences by some spatial unit and/or by area

Density estimation

Here, the darker the polygon, the higher the point count, which can be problematic

• Dependent on choice of spatial unit
• Assumes that points immediately outside the unit have no impact
• An alternative is to generate a surface that estimates the density of points across the study area
• We’ll talk about the most common method:
  • Kernel Density Estimation

Density estimation

• With density estimation, we start with a set of points
• These may be simple points (e.g., crime occurrences) or points with data associated with them (e.g., cities with a population attribute)
• In this case, we are looking for alternative ways to visualize and measure the density of points
• This can be useful for:
  • Hot spot identification
  • Alternatives to choropleth mapping

Kernel density

• Outputs a raster, where we decide the cell size
• Cell contents are the estimated density of points for that area
• Kernel density methods place a kernel of a particular shape and of a particular bandwidth over each point and distributes its value across the kernel
• This kernel is like the neighborhood, except that it surrounds the points and distributes their value underneath the curve
  • Cell densities are then the sum of the kernel intersections that occur

Kernel density

Airbnb density in Rome, March 2023

Kernel density

• Results will depend on cell size and bandwidth (or radius)
  • Larger cell sizes will be grainier
  • Larger radii will result in a smoother surface

Spatial statistics, a short overview

Spatial statistics

• Spatial statistics are like regular statistics
• We have numbers we calculate to describe spatial patterns or how values are distributed across space
• Common applications for spatial statistics:
  • to help us understand when a pattern really is unusual or where outliers really are
  • to estimate models that either measure the impact “space” has on our dependent variable or filter that effect out so we can focus on the variables we’re truly interested in

Spatial statistics

A big umbrella term

• Point pattern analysis
  • Is the distribution of points random? Uniform? Can we identify clusters?
• Measures of spatial autocorrelation or dependence
  • Global – Do we observe positive or negative autocorrelation across our study area
  • Local – Are values correlated with local neighbors?
    • House values
    • Crime
• Capturing spatial heterogenity in relationships and processes
  • e.g., Geographically Weighted Regression

Spatial autocorrelation

• Tobler’s First Law of Geography: “Everything is related to everything else, but near things are more related than distant things.”
• A variable’s values are related to each other in space – they’re correlated
• This means that observations are often not independent of each other
  • For example, house values: If I tell you how much a particular house is worth, does it affect your prediction of the neighboring house’s value?
• We distinguish between two types of autocorrelation: positive and negative

Spatial autocorrelation

GeoDa

• GeoDa((https://geodacenter.github.io) – An exploratory spatial data analysis package.
• It’s free and in some ways much more powerful than QGIS or ArcMap
• To use, open GeoDa
  • And then, under File, open geopackage or shapefile
• Native facility with spatial statistics
  • Good first choice for visualization, exploration, and model estimation
  • Plus cartograms, variable smoothing tools, and lots of other exploratory tools

Spatial autocorrelation

What is neighborhood?

• All measures of spatial association depend on scale
  • And how we define neighbors
• Neighborhoods can be defined based on distance, contiguity, or nearest neighbors
  • Distance: My neighbors are those who live within a mile of me, for example
  • Contiguity: Refers to polygons. My neighbors are those I share a border with:
    • Queen’s case: Shared borders and corners count for contiguity
    • Rook’s case: Only shared borders count for contiguity
  • k Nearest Neighbors: The k nearest units to mine are considered my neighbors
• 1st order versus 2nd order, etc: We could choose our immediate neighbors, or those that are neighbors of our neighbors.

Spatial autocorrelation

Quantifying neighborhood

• When we define our neighborhood, this is implemented using a “weights matrix”
  • Usually 1 and 0’s that indicate yes or no for whether a spatial unit is my neighbor
  • Units are not considered neighbors of themselves
  • These matrices are generally symmetric – If I’m your neighbor, then you’re my neighbor (but this isn’t always the case – e.g., k nearest neighbors)
• How you define your neighborhood can influence results
  • So, best to check how sensitive your results are to your choice of weights
• Ideally choice of weights is informed by data, theory, and previous research

Spatial autocorrelation

Global – Moran’s I

• The most commonly used measure of global spatial autocorrelation
• This is how we estimate the extent of spatial dependence across a set of units for a particular variable
• This is also how we might typically assess whether model residuals are spatially auto-correlated

Spatial autocorrelation

Moran’s I – Airbnb counts across Rome urban zones

• Urban Zones with counts of AirBnb listings
• Weights file
• Significance is generated through randomization
• Interpretation: The line indicates the correlation between numbers of AirBnbs and spatially lagged values
• Positive spatial autocorrelation in the distribution of AirBnbs across Urban Zones in Rome

Spatial autocorrelation

Local Indicators of Spatial Association (LISA)

Local Moran’s I

• Identifies the presence or absence of significant spatial clusters or outliers for each location. A randomization approach is used to generate a spatially random reference distribution to assess statistical significance
• Rather than looking at the entire spatial configuration of units and values at one time—like the global Moran’s I—local indicators of spatial autocorrelation look at each observation at a time, along with values of neighbors

Spatial autocorrelation

Getis-Ord Gi* Statistic

• The resulting Z score tells you where features with either high or low values cluster spatially. This tool works by looking at each feature within the context of neighboring features.
• A feature with a high value is interesting, but may not be a statistically significant hot spot. To be a statistically significant hot spot, a feature will have a high value and be surrounded by other features with high values as well.

Spatial autocorrelation

Counts of AirBnb Listings in Rome (GeoDa)

Next up: Tutorial 4!