Spatially continuous data

  • Previously, we explored point data and area data, which are associated with discrete events, such as store locations or individual residents.
  • In contrast, spatial phenomena can be continuous rather than discrete.
  • Examples of continuous phenomena include elevation and temperature.
  • Continuous data are typically stored in raster formats but can also be represented as vector points.

Spatially continuous data

  • Here, each point represents a measurement of the underlying continuous process, rather than a discrete event.
  • The methods we’ll discuss this week are specific to continuous data and are not suitable for point pattern analysis, and vice versa.

Motivation for spatial interpolation

  • Since it is impossible to observe an entire continuous process, there are infinitely many points that remain unobserved.
  • To understand the process at a specific, unknown location, we aim to estimate its value based on available data.
  • Spatial interpolation techniques address this challenge by providing methods to predict values at these unmeasured points.

Spatial interpolation

Voronoi Polygons/Tessellation:

  • Predict the value of an unknown point by assigning it the value of the nearest known point.

Inverse distance weighting (IDW): \(\hat{z_p}=\frac{\sum_i\frac{z_i}{d_{pi}^\gamma}}{\sum_i\frac{1}{d_{pi}^\gamma}}\)

  • Here, \(d_{pi}\) represents the (Euclidean) distance between the unknown point \(p\) and a known point \(i\).

\(k\)-Point Means/\(k\)-Nearest Neighbours:

  • Predict the value of an unknown point using the average value of the \(k\)-nearest known points.

Point estimate and uncertainty

The theoretical spatial continuous process: \(z_i = f(u_i, v_i) + \epsilon_i\)

Where \(\hat{f}(u_i, v_i)\) represents the point estimate for location \(i\).

Uncertainty is the interval that is likely to capture the true value, \(z_i\).

Accuracy and precision

Activities for today

  • We will work on the following chapter from the textbook:
    • Chapter 32: Activity 15: Spatially Continuous Data I
    • Chapter 34: Activity 16: Spatially Continuous Data II
  • The hard deadline is Friday, March 21.

Reference