Spatial Sampling

How do we collect geospatial information?

TopHat Question 1

“Everything is related to everything else, but near things are more related than distant things.” This statement is known as:

  • Tobler’s First Law of Geography
  • Tobler’s Last Law of Geography
  • Geography’s First Principal
  • The Only Rule of GIS

The First Law of Geography

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

  • Object/areas near each other are more likely to be similar Objects/areas near that are distant from each other are more likely to be different
  • This aspect of nature keeps coming in GIS!

Sampling

The process of selecting points from within an area or population, called a sample frame.

  • We collect information for a subset of objects/locations sample frame
    • But we ignore most objects/locations
    • Think back to Bonini’s Paradox

Sampling

The process of selecting points from within an area or population, called a sample frame.

  • How we define the sample frame and choose samples can determine the quality of our data
    • We want to maximize representativeness of the sample
    • But also minimize effort and expense associated with sampling

Scientific Sampling

Requires each element in the sample frame have a known and pre-specified chance of selection.

  • Biased Sampling: some elements have a greater or lower chance of being selected
  • Unbiased: every element has an equal chance of being selected

Random Sampling

In theory, a random sample is best. Its the “gold standard”.

  • Unbiased: each location has equal chance of selection
    • Easy to do, randomly select 𝑥,𝑦 coordinates
  • A key assumption of many statistical tests

Random Sampling

Can be difficult to implement in practice.

  • Chance that all samples miss important features
  • May be barriers to access
    • Remote or rugged terrain
    • Private property
    • Systematic under-responses

Random Sampling

We have some options to account for the drawbacks

  • “Law of large numbers”: as we collect more information, our sample will become increasingly representative of actual population values
    • Larger sample sizes or “Bootstrapping”
  • Not always practical
    • Requires more time and resources

TopHat Question 2

To collect a random sample, every object or location must:

  • Have an equal chance of selection
  • Be approximately the same size
  • Not be close to other samples (i.e., equally dispersed)
  • Not have a predefined chance of selection

Alternate Approaches to Sampling

Biased sampling

  • Create a sample design that trades a sampling scheme for randomness
  • Induce bias to the sample to:
    • Save time or resources
    • Account for relevant information about the sample frame

Systematic Sampling

A random starting point is chosen and a fixed sampling interval is used.

  • Randomly select first of 3 students
    • Select every 3rd after

Systematic Sampling

A random starting point is chosen and a fixed sampling interval is used.

  • Choose a random starting point
    • Then draw equally spaced grid

Systematic Sampling

A random starting point is chosen and a fixed sampling interval is used.

  • Premise behind satellite data collection
    • Often good for continuous fields
      • eg., land cover

Systematic Sampling

A random starting point is chosen and a fixed sampling interval is used.

  • Not ideal for discrete objects that exhibit periodicity
    • City blocks, Roads, etc.

Stratified Sampling

Address the issues with systematic sampling by sampling at random locations, while applying a “systematic bias”

  • Create a systematic sampling grid, then take random samples within cells

  • Can avoid over/under sampling regularly repeating features

Stratified Sampling

Divide a population by certain attributes, then take random samples from sub-populations

  • Account for important factors
    • Gender, race, age, political party, etc.

Cluster Sampling

Intense sampling of features in clusters around a number of selected locations

  • Locations can be selected for specific features, e.g.,:
    • Shopping centers
    • Known history of invasive species
  • Or, locations can be selected at random across the grid

Cluster Sampling

Intense sampling of features in clusters around a number of selected locations

  • More efficient use of time and resources
    • May not be representative

Transect Sampling

Commonly used along line features like roads & rivers.

  • Focused effort on features of interest
  • Requires understanding of spatial structure for maximum effectiveness

TopHat Question 3

Which of these sampling methods are unbiased?

  • Cluster
  • Stratified
  • Transect
  • Random

How Many Samples?

The number of samples required is a function of how similar units of that population are.

  • Spatial structure can vary wildly across a landscape
    • Knowledge of your study area will help to establish how to best sample
  • Maximize returns minimize effort

Spatial Autocorrelation

When the values of objects are related to the values of nearby objects.

  • If you know the value of one object, you can make a reasonable guess at the value of nearby objects

Spatial Autocorrelation

Spatial Autocorrelation

Correlation does not imply causation!

  • There could be relationship between features
    • Or a relationship to a third object that determines the values of both
    • Or a completely random coincidence

Statistical Assumptions

Spatial autocorrelation is a problem when it comes to spatial statistics.

  • Most tests assume that there is no relationship between objects by default!
    • By violating this assumption, we “break” many common statistics!
    • Spatial statistics explores ways of analyzing statistical relationships across space

TopHat Question 4

Which number completes the sequence: 2, 4, 6, __, 10?

  • 3
  • 8
  • 11
  • 100

Statistical Interpolation

The process of “filling in the blanks” that you just performed is called interpolation

  • If you know the value of one object, you can make a reasonable guess at the value of nearby objects

  • Over a 2D or 3D surface we call this spatial interpolation

  • Intelligent guesswork in which we attempt to make reasonable estimates of the values of a continuous field at places where we do not have measurements

Spatial Interpolation

Spatial interpolation only makes sense for a continuous field with numeric values.

  • Rainfall, temperature, pressure, elevation
    • Estimate between measured locations
  • Can be problematic with qualitative data

Spatial Interpolation

Continuous fields tend to exhibit strong positive spatial autocorrelation

  • Reasonable to assume missing values are similar to those around them
    • Methods incorporate distance to known samples.
  • Sound familiar? This is Tobler’s First Law!
    • Closer samples given more weight than distant ones
    • A threshold is usually set, to determine the maximum distance to take samples from

Inverse Distance Weighting

Calculates cell values based on nearby observations.

  • Weight cells by distance from observation points
  • Mathematical expression of Tobler’s Law

Inverse Distance Weighting

Best applied to discrete samples of continuous quantitative variables.

  • Elevation
  • Temperature
  • Precipitation

Kernel Density

Calculates the “density” of discrete objects and converts to a raster surface

  • Probability of occurrence across space
    • Weight by “value” of points
  • Often applied to counts of qualitative data
    • Disease outbreaks & crime data

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