Spatial Statistics in Ecology: Point Pattern Analysis

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Spatial Statistics in EcologyPoint Pattern
Analysis

• Lecture Two

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Re-Cap and Introduction to Point Pattern Analysis

• First-order effects look at trends over space
• Second order effects look at PAIRS OF POINTS i.e.
  the spatial dependence or covariance structure of
  pairs of variables over space
• Spatial dependence gives rise to different types
  of processes

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Types of Processes

• HOMOGENEOUS or stationary processes
• Mean is constant over R (space)
• Variance is constant over R
• Covariance is dependant on DISTANCE AND DIRECTION
• Sothere is NO global trend!
• .there is NO first order effect!

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Types of Processes

• HETEROGENEOUS processes or non-stationarity
• Constant mean
• Constant variance
• Covariance is ONLY dependant on DISTANCE
• SO.your process is ISOTROPIC!

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Point Patterns

• Recall that point patterns deal with events that
  occur in discrete locations
• Point patterns consist of events with variation
  in the mean value of the event
• Can you think of what types of point patterns
  ecologists deal with? They will all be either
• RANDOM, CLUMPED or HOMOGENEOUS

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Is the distribution clustered or regular?

• s1, s2, s3, s4 are events which have
  coordinates (x,y) in area R.
• events are objects with various intensity in
  this case height

Study region R
s1
s3
1.2m
1.8m
s2
s4
1.6m
1.6m
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Isotropic versus Stationary

• Intensity ? (s) for a first-order property.
  For a stationary process this is constant over R.
• For second-order intensity ? (si, sj). If its
  isotropic spatial dependence is a function of
  length h (distance). If its stationary it
  depends on only vector distance (both direction
  and distance) not on absolute location

s1
s3
isotropic
1.8m
1.2m
h
s2
s4
h
1.6m
1.6m
stationary
Study region R
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Visualizing point patterns

• Visualization is the simplest method to use
  for point pattern analysis. These are called dot
  maps. What would happen to the observed patterns
  of the scale (grain or extent) changed?
• A random pattern can look ordered if the scale
  in too small.

What type of patterns are these?
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Exploring spatial point patterns

• Statistics and plots can be derived for point
  patterns. These can be used to describe the
  pattern or how mean values of points change
  across space. The simplest is the QUADRAT METHOD.
  The of events per unit area are counted and
  divided by area of each square to get a measure
  of the intensity of each quadrat

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Quadrat Methods

• This can tell is something about how the
  processes changes over R. We have now transformed
  our data into area data. There are obvious
  problems with this type of approach however. We
  throw away a lot of spatial detail and the edge
  effects may give us a different pattern to the
  one we observe.

Doesnt take into account relative position of
points and is dependant on this size of the grid.
Close points also count as much as far points!
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Kernel Estimation

• Kernel estimation weights points that are
  further away less than those that are close.
  Point A will count less than Point B.

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Bandwidth is important
With a very large bandwidth no pattern is seen.
With a very small bandwidth there is too fine a
resolution to see patterns. Which bandwidth picks
up the trend?
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Second Order-effects

• Recall that second order effects deal with PAIRS
  OF POINTS
• How do variables covary at each point in space
• Nearest-neighbor techniques are the most commonly
  used

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Nearest-neighbor techniques
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1
3
6
5
4
7
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Distribution functions

• NN can be used to create an event-event plot
• If the slope rises fast the points are dense
  (ie. the pairs of NN are clustered together).
  This is subjective though and makes no
  corrections for edge effects OR for points other
  than the NN

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Second-Order Point Pattern Analysis The K
function

• The K analysis provides a measure of the reduced
  second moment measure or K function of the
  observed process. This provides a more effective
  summary at a wider range of scales. However, care
  must be taken that within the scale of interest
  the data is homogeneous or isotropic.

• ?K(h) E
• measure of mean intensity
• n
• R
• n events
• R (area of observation)
• K k function
• h distance
• Sothis tells you the expected
• of events within distance h
• of a randomly selected event

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How does it work?

• To get the k function you visit all 120 events
  and find how many are within a distance of 2 km
  from each event. This is done for each ?

2 events within distance h from event i
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Edge Effects

• Edge effects can seriously degrade
  distance-based statistics, and there are at least
  two ways to deal with these. One way is to invoke
  a buffer area around the study area, and to
  analyze only a smaller area nested within the
  buffer. By common convention, the analysis is
  restricted to distances of half the smallest
  dimension of the study area. This, of course, is
  expensive in terms of the data not used in the
  analysis. A second approach is to apply an edge
  correction to the indicator function for those
  points that fall near the edges of the study
  area Ripley and others have suggested a variety
  of geometric corrections.

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Confidence Limits

• Ripley derived approximations of the test of
  significance for normal data. But data are often
  not normal, and assumptions about normality are
  particularly suspect under edge effects. So in
  practice, the K function is generated from the
  test data, and then these data are randomized to
  generate the test of significance as confidence
  limits. For example, if one permuted the data 99
  times and saved the smallest and largest values
  of L(d) for each d, these extremes would indicate
  the confidence limits at alpha0.01 that is, an
  observed value outside these limits would be a
  1-in-a-hundred chance. Likewise, 19
  randomizations would yield the 95 confidence
  limits Note that these estimates are actually
  rather imprecise simulations suggest that it
  might require 1,000-5,000 randomizations to yield
  precise estimates of the 95 confidence limits,
  and gt10,000 randomizations to yield precise 99
  limits.

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Modelling Spatial Point Patterns First-order --
CSR

• Modelling of spatial point patterns is done using
  the COMPLETE SPATIAL RANDOMNESS (CSR) model.
• Events follow a homogenous Poisson process over
  the study region (which as we know is normally
  violated)
• CSR provides a baseline of complete randomness
  from which we can quantify deviations as regular
  or clustered

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How does it work?

• Regularity in the first map and clustering in
  the second can be quantified as departures from
  randomness. Either event-event or point-event
  distances are used. This can only tell us that
  there is a departure from CSR. The K function can
  also be extended with its focus on dependence
  over a range of scales however if second-order
  effects are present (in our case there are
  obvious spatial effects) other methods should be
  used

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Locations of redwood seedlings in a forest
Population Level

• Many spatial techniques have their origins in
  plant ecology where describing and analyzing the
  spatial distribution of plants, frequently within
  small areas of only a few square meters, can
  yield interesting ecological information. This
  example uses a small set of data comprising the
  locations of 62 redwood seedlings distributed in
  an area of 23m2. From our standpoint we might
  expect evidence of clustering around existing
  parent trees.

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Locations of the seedlings
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Nearest-Neighbor Analysis
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Cumulative Distribution Function
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The k-function
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Test of CSR (complete spatial randomness)
The test statistic indicates a strong departure
from randomness towards clustering
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Kernel Bandwidth 4 km
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Kernel Bandwidth 2km
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Kernel Bandwidth 0.5km
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Lecture Two Summary

• Point patterns can be analyzed to determine the
  TREND of a variable over space or the spatial
  dependence of the pattern over space.
• To look at first-order effects use quadrat
  methods or kernel estimation
• To look at second-order effects use NN techniques
  or K-functions