Spatial Function Correlation: Measure It Like a Pro!

Understanding spatial relationships is critical in various fields, and ESRI’s ArcGIS tools provide powerful capabilities for spatial analysis. Tobler’s First Law of Geography highlights the fundamental principle that everything is related to everything else, but near things are more related than distant things. This principle informs the need to measure the spatial correlation of function effectively. When researching this correlation, it is essential to follow the principles of statistical analysis. Such work is often used in the context of urban planning to identify spatially correlated functions such as schools and parks

Image taken from the YouTube channel Intro_to_Spatial_Stats , from the video titled Spatial correlation functions – a primer .

Measuring Spatial Correlation of Functions: A Comprehensive Guide

Understanding how functions, specifically their values at different locations, relate to each other across space is crucial in various fields. This guide provides a structured approach to effectively measure the spatial correlation of functions. The core concept we will focus on is how to "measure the spatial correlation of function."

Understanding Spatial Correlation

Spatial correlation, in its simplest form, describes the degree to which things that are near each other are more related than things that are far apart. When applied to functions, it examines whether the values of a function at one location are statistically dependent on the values of the same or different functions at nearby locations. A high spatial correlation suggests a strong dependency.

Key Concepts

• Spatial Autocorrelation: Correlation of a single function with itself across space. If high values cluster together and low values cluster together, you have positive spatial autocorrelation. If high values are surrounded by low values (and vice versa), you have negative spatial autocorrelation.
• Spatial Cross-Correlation: Correlation between two or more different functions across space. This measures how changes in one function relate to changes in another function, considering their spatial locations.

Why Measure Spatial Correlation of Function?

• Pattern Identification: Reveals underlying spatial patterns and relationships.
• Variable Dependency: Helps understand how different variables (represented by functions) influence each other across a geographic area.
• Prediction: Enables better prediction of function values at unsampled locations based on nearby values.
• Model Building: Informs the development of spatially explicit models.

Methods for Measuring Spatial Correlation

There are various methods to measure the spatial correlation of function, each suited to different types of data and research questions.

1. Visual Inspection: Exploratory Data Analysis

Before diving into complex calculations, visually examining your data is vital.

• Scatterplots: Plotting function values at different locations against each other can provide an initial impression of correlation.
• Choropleth Maps: Displaying function values using different colors on a map reveals spatial patterns and clusters.
• Surface Plots: Representing functions as 3D surfaces can highlight spatial trends and variations.

2. Global Measures of Spatial Autocorrelation

These provide a single value representing the overall spatial autocorrelation in a dataset.

• Moran’s I: A widely used statistic that measures the overall spatial autocorrelation. It ranges from -1 to +1, where:

  • Positive values indicate positive spatial autocorrelation.
  • Negative values indicate negative spatial autocorrelation.
  • Values near zero suggest random spatial distribution.

  The formula for Moran’s I is:

  I = (N / S0) * Σi Σj w_ij (x_i - x̄)(x_j - x̄) / Σi (x_i - x̄)^2

  Where:

  • N is the number of spatial units
  • w_ij is the spatial weight between units i and j
  • x_i is the function value at location i
  • x̄ is the mean of the function values
  • S0 is the sum of all spatial weights

• Geary’s C: Another global measure that is inversely related to Moran’s I. Values range from 0 to 2, where:

  • Values near 0 indicate positive spatial autocorrelation.
  • Values near 2 indicate negative spatial autocorrelation.
  • Values near 1 suggest random spatial distribution.

  The formula for Geary’s C is:

  C = ((N - 1) / (2 * S0)) * Σi Σj w_ij (x_i - x_j)^2 / Σi (x_i - x̄)^2

  Where the variables are defined as in Moran’s I.

• Join Count Statistics: Suitable for binary data (e.g., presence/absence of a feature). It counts the number of times similar or dissimilar values are adjacent to each other.

3. Local Measures of Spatial Autocorrelation

These identify spatial clusters and outliers within a dataset.

• Local Moran’s I (LISA): Identifies locations with significant clustering of high values (High-High), low values (Low-Low), high values surrounded by low values (High-Low), and low values surrounded by high values (Low-High).

  The formula for Local Moran’s I for location i is:

  I_i = (x_i - x̄) / (Σj (x_j - x̄)^2 / (n - 1)) * Σj w_ij (x_j - x̄)

  Where:

  • x_i is the value of the function at location i.
  • x̄ is the average value of the function across all locations.
  • n is the number of locations.
  • w_ij is the spatial weight between locations i and j.

• *Getis-Ord Gi:** Identifies statistically significant hotspots (clusters of high values) and coldspots (clusters of low values).

4. Spatial Cross-Correlation Measures

These methods measure the spatial relationship between different functions.

• Cross-Correlograms: Visualize the correlation between two functions at different spatial lags (distances). This helps identify at which distances the functions are most strongly correlated.

  Method	Description	Suitable for
  Visual Inspection	Initial data exploration and pattern identification.	All data types.
  Moran’s I	Global measure of spatial autocorrelation.	Continuous data.
  Geary’s C	Global measure of spatial autocorrelation (inversely related to I).	Continuous data.
  Local Moran’s I	Local measure to identify clusters and outliers.	Continuous data.
  Getis-Ord Gi*	Local measure to identify hotspots and coldspots.	Continuous data.
  Cross-Correlogram	Measures spatial correlation between two functions at different lags.	Continuous data (often after standardization).

Workflow for Measuring Spatial Correlation of Function

1. Data Preparation:
   • Clean and preprocess your data.
   • Ensure the data is spatially referenced (e.g., coordinates).
   • Consider transforming the data (e.g., log transformation) if necessary.
2. Define Spatial Weights:
   • Choose an appropriate spatial weighting scheme (e.g., contiguity, distance-based).
   • Consider the scale of your analysis and the potential influence of distant locations.
3. Calculate Spatial Correlation Measures:
   • Select the appropriate methods based on your data type and research question.
   • Calculate global and/or local measures.
4. Interpret Results:
   • Assess the statistical significance of the results.
   • Consider the limitations of the chosen methods.
   • Visualize the results using maps and plots.
5. Sensitivity Analysis:
   • Test the robustness of your results by using different spatial weights or transformations.

Choosing the Right Method

Selecting the most appropriate method to "measure the spatial correlation of function" depends heavily on the characteristics of your data and the research question you are trying to answer.

Consider the following factors:

• Data Type: Continuous, binary, or categorical.
• Scale of Analysis: Global or local patterns.
• Spatial Weights: The influence of neighboring locations.
• Research Question: Are you interested in autocorrelation or cross-correlation? Are you looking for overall patterns or local clusters?

Hopefully, this helps you on your way to confidently measure the spatial correlation of function like a pro! Now go forth and analyze!