Quadrat Analysis

1
Quadrat Analysis

• Method of analyzing point patterns
• A uniform grid is placed over the points.
• The number of points in each cell (quadrat) is
  counted and an observed distribution found, i.e.,
  how many quadrats have 0 points, 1 point, 2
  points, etc.?
• The actual distribution then compared to a
  random distribution, e.g., Poisson

2
Quadrat Analysis

• Another way of looking at it the variance
  compares the number of points in each grid cell
  with the average number over all the grid cells
• The variance-to-mean ratio (VTMR) used to assess
  nature of point pattern
• if points dispersed, observed VTMR low (variance
  small cells have about the same of pts)
• if points clustered, VTMR high (a few cells have
  many pts., most have none or few)
• if points random, VTMR 1 (Poisson distrib.,
  which described freq. of random pt pattern, has
  varmean)

3
Quadrat Analysis

• Problems
• really measures dispersion, not pattern
• results in a single measure for the entire
  distribution and variations within the distrib.
  are not considered organization of the cells
  also important (spatial autocorrelation)

4
Nearest Neighbor Analysis

• Another method to analyze point patterns
• The distance between each point and its nearest
  neighbor is found
• The mean distance over all points is calculated
• This calculated distance is compared to expected
  value if distribution were random

5
Nearest Neighbor Analysis

• Expected Mean
• where D density of pts in study area
  Area of study area
• R degree of randomness observed/expected

6
Nearest Neighbor Analysis

• R value calculated
• clustered R 0 (points all on top of each
  other)
• uniform R 2.1491 (points in lattice pattern)
• random R 1 (actual value equals expected
  random value)
• 2nd, 3rd, etc. nearest neighbors can be analyzed
  to look at periodicity, hierarchy

7
Nearest Neighbor Analysis

• Advantages
• no quadrat size to worry about
• takes distances between points into account
• Problems
• boundary needed to calculate density for expected
  value
• boundary can affect whether pts are clustered or
  random
• possible edge effects if distance to nearest
  neighbor is greater than distance to boundary
• an R 1 may not necessarily indicate a random
  pattern

8
Thiessen Polygons

• Polygons generated from a point layer such that
  any location within a given polygon is closer to
  the enclosed point than to a point within any
  other polygon
• i.e., all points within each polygon are closer
  to the point inside the polygon than to any other
  point
• Thus, they divide the space between the points
  as evenly as possible

9
Thiessen Polygons

• Used in market area analysis and several
  environmental applications (e.g., watersheds,
  contours, rain gauge area assignment, etc.)
• Answers questions such as "Which (bank/
  school/grocery store/mall/etc.) is the nearest to
  each house?
• The ability to create these polygons tends to be
  mainly in higher-end GIS packages

10
Thiessen polygons
11
Interpolation

• Method of estimating unknown values at unsampled
  locations using known values of neighboring
  sampled sites
• in most cases the value must be measured at the
  interval or ratio level
• Rationale behind spatial interpolation is the
  observation that points close together in space
  are more likely to have similar values than
  points far apart (Tobler's First Law of
  Geography)

12
Uses of Interpolation

• Provide contours for displaying data graphically,
  for example
• elevation
• air pollution data from monitoring stations
• contamination of harbor using sampled sites
• Calculate some property of the surface at a given
  point
• Assist the spatial decision making process in
  such areas as mineral prospecting, hydrocarbon
  exploration, etc.

13
Interpolation Methods

• Manual Interpolation ("eyeballing")
• traditionally not a highly regarded method, but
  an important procedure in practice
• some practitioners distrust the more mathematical
  algorithms, feel they can use their expert
  knowledge and experience to generate a more
  realistic interpolation
• attempts now being made to use this expert
  knowledge and build it into an expert system
  for interpolation
• different methods may be used on diff. parts of
  the map
• tend to honor data points
• abrupt changes such as faults are more easily
  modeled
• the surfaces subjective and vary from expert to
  expert

14
Interpolation Methods

• Proximal
• all values assumed to be equal to the nearest
  known pt.
• is a local interpolator, not computer intensive
• creates Thiessen polys w/abrupt changes at
  boundaries
• ecological applications, e.g., territories,
  influence zones
• best for nominal data (but originally used by
  Thiessen for computing areal estimates from
  rainfall data)
• robust (always produces a result), but has no
  "intelligence" about the system being analyzed
• available in few mapping packages

15
Interpolation Methods

• Averaging functions
• avg of closest n (e.g., 2, 3, 4...) pts (ignores
  distance)
• fit line between closest 2
• fit surface between closest 3
• Moving average/distance-weighted average
• Estimates are averages of the values at n known
  pts
• z Sum wizi/Sum wi, where w is some function of
  distance, such as w1/d
• Problem with averaging/distance-weighted avg
• the range of interpolated values is limited by
  the range of the data...no interpolated value
  will be higher or lower than any value in the
  data set.