Statistical analysis of geospatial data for environmental studies ... · Statistical analysis of...
Transcript of Statistical analysis of geospatial data for environmental studies ... · Statistical analysis of...
Statistical analysis of geospatial data
for environmental studies
CIHEAM, 11-22 June 2012
Fernando T. Maestre Área de Biodiversidad y Conservación, Universidad Rey Juan Carlos, Móstoles
TABLE OF CONTENTS
1) Introduction to spatial data analysis
1.1. Typical examples of spatial data and questions
1.2. Overview of spatial statistics for environmental studies
INTRODUCTION TO SPATIAL DATA ANALYSIS
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The world is a heterogeneous place
Semiarid vegetation
Maestre & Cortina. 2002. Plant Soil 241: 279-291
Maestre et al. 2003. Bol. R. Soc. Esp. Hist. Nat. (Sec. Biol.) 99: 159-172
The world is a heterogeneous place
Distribution of population
http://www.blog.designsquish.com/index.php?/site/world_population_map/
Sand content
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Soil compaction
Soil properties
Remote sensing data
Density of animals
http://www.fws.gov/kulmwetlands/duck_pair_maps.html
http://www.blm.gov/wo/st/en/prog/more/fish__wildlife_and/sage-grouse-conservation/bird_density.html
A spatial pattern is a perceptual structure, placement, or
arrangement of objects on Earth. It also includes the space in
between those objects. Patterns may be recognized because of their
arrangement. When dealing with spatial patterns, important questions
we may ask are the following:
* Is there an area that is more dense with objects than others?
* Is there an area that has fewer or no objects than others?
* Are there clusters of objects?
* Is there a randomness or uniformity to the location of the objects?
* Does there seem to be a relationship between individual objects (is
one object located where is its because of another)?
The term “spatial pattern analysis” encapsulate a wide array of
methodologies designed to quantitatively analyze spatial data
Spatial pattern
Broad types of spatial patterns
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Loarie et al. 2009. The Velocity of Climate Change, Nature
Broad types of spatial patterns
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Tiger bush landscape in western NSW (Australia)
Broad types of spatial patterns
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http://capita.wustl.edu/otag/Reports/aqafinvol_I/Html/Im
age151.gif
Ecology is the science that studies the relationships between organisms and
their environment. Given that both organisms and the abiotic factors show in
most cases non-random patterns, the study of such patterns is crucial to
understand these relationships.
Spatial ecology is a specialization of ecology and geography that is
concerned with the identification of spatial patterns and their relationships to
ecological events. These events can be explained through the detection of
patterns at a given spatial scale; local, regional, or global.
Through the application of spatial pattern analyses, factors leading to
ecological events can be determined and verified.
In environmental sciences, spatial statistics is important for mapping the risk
of natural hazards, in agriculture and pest control, in modeling the spread of
invasive species…
Why is important to study spatial patterns in ecology and
environmental sciences?
Spatial heterogeneity: Heterogeneity may be viewed as a continuum of
variability and complexity - from low to high - with homogeneity being the low
end (i.e., the minimum). When this variability has a spatial structure, spatial
heterogeneity can be used as a synonymous of spatial pattern.
Two basic strategies can be used to quantify heterogeneity: (1) directly, by
measuring complexity and variability and (2) indirectly, by measuring departure
from homogeneity. For example, heterogeneity in categorical
maps can be defined as complexity in number of patch types, proportion, patch
shape, and contrast between neighboring patches, and different methods can
be used to quantify these aspects of heterogeneity. Moreover, heterogeneity in
numerical maps can be measured as degree of departure from randomness
when homogeneity is defined as the randomness of the distribution of a system
property
Some important concepts
Li & Reynolds. 1995. Oikos 73: 280-284
Scale: this term refers to the spatial extent of ecological processes and the
spatial interpretation of the data.
Scale has different meanings. Concepts such as cartographic ratio, grain,
extent, resolution, support, range, variance and footprint have all been used as
synonyms of scale in one context or another. Examples:
* In geography, scale is commonly used as cartographic ratio referring to the
relationship between the distance or area represented on a map to the
corresponding real-world distance or area.
* In landscape ecology, scale has the disjunctive definition of ‘‘grain and extent’’.
* The term ‘‘resolution,’’ commonly used in remote sensing, is defined as the
smallest object that can be reliably detected.
Some important concepts
Dungan et al. 2002. ECOGRAPHY 25: 626–640.
Some important concepts
García. 2008. In: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales
Grain
Extent
Interval
The response of an organism or a species to the environment is
particular to a specific scale, and may respond differently at a larger or
smaller scale.
Some important concepts
Regular
Clumped
Rosenberg & Anderson. 2011. Methods in Ecology & Evolution 2(3):229-232.
Inapropriate grain
Some important concepts
García. 2008. In: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales
Apropriate extent
Small extent
Apropriate grain
Some important concepts
Choosing a scale that is appropriate to the ecological process in question is very
important in accurately hypothesizing and determining the underlying cause.
Most often, ecological patterns are a result of multiple ecological processes, which
often operate at more than one spatial scale.
Spatial independence occurs when the value of a given sample do not depend
on its proximity to another.
Some important concepts
Xi-2 Xi-1 Xi Xi+1
ρ=0 ρ=0 ρ=0
A)
B) Xi-2 Xi-1 Xi+1
Zi-2 Zi-1 Zi+1
ρz=0
Xi
Zi
ρx=0 ρx=0 ρx=0
ρz=0 ρz=0
Maestre & Escudero. 2008. In: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales
Spatial autocorrelation occurs when the value of a given sample taken close
to each other are more likely to have similar magnitude than by chance alone.
When a pair of values located at a certain distance apart are more similar than
expected by chance, the spatial autocorrelation is said to be positive. When a
pair of values are less similar, the spatial autocorrelation is said to be negative.
It is common for values to be positively autocorrelated at shorter distances and
negative autocorrelated at longer distances. In ecology, there are two important
sources of spatial autocorrelation:
• True/inherent spatial autocorrelation arises from interactions among
individuals located in close proximity. This process is endogenous (internal) and
results in the individuals being spatially adjacent in a patchy fashion.
• Induced spatial autocorrelation (or ‘induced spatial dependence’) arises from
the response of a given species to the spatial structure of exogenous (external)
factors, which are themselves spatially autocorrelated.
Some important concepts
Xi-2 Xi-1
Xi
Xi+1
ρx
A)
B)
ρx ρx
Xi-2 Xi-1
Xi-2 Xi-1 Xi+1 rx rx rx
Zi-2 Zi-1 Zi+1
ρz ρz ρz
Xi
Zi
Xi-2 Xi-1 Xi+1 rx rx rx
Zi-2 Zi-1 Zi+1
ρz ρz ρz
Xi
Zi
C)
Zi-2 Zi-1 Zi
Zi-2 Zi-1 Zi Zi+1 Xi-2 Xi-1 Xi
Maestre & Escudero. 2008. In: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales
Most ecological/environmental data exhibit some degree of spatial
autocorrelation,depending on the scale of interest.
As the spatial arrangement of most ecological data is not random, traditional
random population samples tend to over-estimate the true value of a variable,
or infer significant correlation where there is none. This bias can be corrected
through the use of spatial pattern analyses.
Regardless of method, the sample size must be appropriate to the scale and
the spatial statistical method used in order to be valid.
Some important concepts
TYPICAL EXAMPLES OF SPATIAL DATA
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Point pattern data
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Point pattern data
These data are typically obtained when studying sessile organisms and
structures (e.g. plants, ant nests, sessile animals).
These data can be obtained using a gps, a total station or knowing the distances
of sampling points to a reference point (P). If we know the distance between two
fixed points (O, X), and between these and P, the coordinates (x, y) can be
obtained as:
= cos-1[(b2 + c2 – a2)/2bc], donde x = b cos y y = b sin
P
XO
c
bay
x
Point pattern data
These data can also contain additional information in addition to the coordinaes
(e.g. plant size, species identity, age…).
Perry et al. 2002. ECOGRAPHY 25: 578–600.
Examples of forms of point-referenced data. The censussed mapped individuals of
(a) with additional additional continuous attribute with magnitude indicated by size
of symbol (b)
Point pattern data
Perry et al. Plant Ecol (2006) 187:59–82
Quadrat data (area-referenced data)
These data are typically used to study variables that vary continously in space
(e.g. soil properties), as well as other discrete variables (e.g. seedlings recruited
in a forest, insects per plant).
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Albeit the use of regularly spaced quadrats is common, other spatial
configurations (nested, irregular) can also be found.
Perry et al. 2002. ECOGRAPHY 25: 578–600.
Quadrat data (area-referenced data)
Dungan et al. 2002. ECOGRAPHY 25: 626–640.
Quadrat data (area-referenced data)
Quadrat data (area-referenced data)
When using quadrat data, there are important things that must be carefully
considered:
* The size of the sampling quadrat defines the minimum spatial resolution at
which data can be obtained. This size is very important, as the results will be
largely dependent on quadrat size.
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* The spatial location of quadrats
depends on the objectives of the
study, the size of the area being
studied, the resource available, the
characteristics of the study area…
* The distance between consecutive
quadrats should be smaller than the
size of the spatial structures that we
aim to detect. The use of nested
approaches can be a good solution if
these distances are unknown.
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Quadrat data (area-referenced data)
Overton & Levin. 2003. Ecological Research18, 405–421.
Quadrat data (area-referenced data)
* The number of sampling quadrats will depend on the objectives,
resources and characteristics of the study area and question to be asked.
* Some authors recommend a minimum of 30 units to detect the
presence of spatial autocorrelation (Legendre & Fortin 1989), albeit other
authors recommend to use at least 100 sampling units if we aim to use
geostatistics (Webster & Oliver 1992).
* The use of quadrat data requires a series of decissions that must be
taken a priori and that strongly condition the results of the analyses that
will be made. Thus, it is very important to have clear objectives, know the
biology of the organisms/characteristics of the phenomenon that we are
studying and estimate the costs of the sampling before it can be
conducted.
Legendre, P. y Fortin, M.-J. 1989. Vegetatio 80: 107-138.
Webster, R. y Oliver, M. A. 1992. Journal of Soil Science 43: 177-192
TYPICAL QUESTIONS WHEN DEALING WITH SPATIAL
DATA
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Characterizing the spatial patterns of organisms
Rossi. Pedobiologia 47, 490–496, 2003
Characterizing the spatial patterns of organisms
Rossi. Pedobiologia 47, 490–496, 2003
Characterizing the spatial patterns of organisms
Maestre et al. Plant Ecology (2005) 179:133–147
Characterizing the spatial patterns of abiotic factors
Maestre et al. Ecosystems (2003) 6: 630–643
Exploring spatio-temporal variations in species population data
Conrad et al. Journal of Insect Conservation (2006) 10: 53–64
Exploring spatio-temporal variations in species population data
Conrad et al. Journal of Insect Conservation (2006) 10: 53–64
Inferring ecological processes by exploring spatial patterns (plant-
plant interactions)
Tirado & Pugnaire. 2003. Oecologia 136: 296-301.
Inferring ecological processes by exploring spatial patterns (plant-plant
interactions)
Tirado & Pugnaire. 2003. Oecologia 136: 296-301.
Inferring ecological processes by exploring spatial patterns (predator-
prey interactions)
Winder et al. Ecology Letters, (2001) 4: 568±576
Inferring ecological processes by exploring spatial patterns (predator-
prey interactions)
Winder et al. Ecology Letters, (2001) 4: 568±576
Inferring ecological processes by exploring spatial patterns (predator-
prey interactions)
Winder et al. Ecology Letters, (2001) 4: 568±576
Inferring ecological processes by exploring spatial patterns
* Evaluating the factors driving the
abundance of species
Overton & Levin. 2003. Ecological Research18, 405–421.
Inferring ecological processes by exploring spatial patterns
Wilson et al. 2004. Nature 432: 393-396
* Predicting biodiversity changes
Inferring ecological processes by exploring spatial patterns
Wilson et al. 2004. Nature 432: 393-396
* Predicting biodiversity changes
Inferring ecological processes by exploring spatial patterns
Gallardo et al. Plant and Soil 222: 71–82, 2000
* Evaluating effects of plants on soil properties
Providing accurate maps
http://blogs.scientificamerican.com/observations/2012/02/01/new-map-shows-that-most-lyme-infected-
ticks-are-in-northeast-northern-midwest/
Providing accurate maps
OVERVIEW OF SPATIAL STATISTICS FOR
ENVIRONMENTAL STUDIES
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Methods for analyzing spatial pattern have been developed independently in a
wide variety of disciplines, including geology, ecology, geography, physics, and
engineering.
The motivation behind spatial analysis can vary widely. For example, traditionally
geographers were often interested in hypothesis testing, while geologists were
often interested in estimation and prediction.
These led to differing philosophies of spatial analysis, even if their methods often
show remarkable convergence.
Biologists who study spatial patterns tend to follow the philosophy of geographers
or geologists or plant ecologists (who developed their own set of methods, largely
based on the point locations of plants), but rarely are familiar with more than one
approach.
Objective Type of data
Quadrat data Point data
Exploration
(characterization of
spatial structure)
Joint-count (Rosenberg & Anderson 2011)
Correlograms (Moran´s I , Geary´s c, Legendre
& Legendre 1998)
Local Moran´s I, Getis´s G, Ord´s O
(Rosenberg & Anderson 2011)
Semivariograms (this course)
Mantel correlograms, Mantel test & partial
Mantel test (Legendre & Legendre 1998)
SADIE (Perry et al.1999, Winder et al. 2001)
Block variance quadrat (Dale 1999)
Fractal dimension (Alados et al. 2003)
Lacunarity (Plotnick et al. 1996)
Spectral analysis (Renshaw 1997)
Wavelets (Dale & Mah 1998)
Nearest neighbors and related
methods (Dale 1999)
Ripley´s K and related methods
(this course)
Joint-count (Rosenberg & Anderson
2011)
Dixon´s method (Dixon 2002)
Fractal dimension (Alados et al.
2003)
The different methods available can be classified according to the objectives
of the study. The same method can be applied to multiple objectives.
Objective Type of data
Quadrat data Point data
Statistical
inference
(hypothesis
testing, parameter
estimation)
Mantel test & partial Mantel test
(Legendre & Legendre 1998)
Semivariograms (this course)
SADIE (Perry et al. 1999, Winder et al.
2001)
Markov Hidden Models (Baldi et al. 1994)
Autorregressive models (Haining 1990)
Conditional annealing (Cressie 1993)
Ripley´s K and related
methods (this course)
Objective Type of data
Quadrat data Point data
Mapping
(interpolation)
“Krigging” (this course)
Trend surface analysis (Legendre y
Legendre 1998)
Voronoi polygons (Fortin & Dale
2005)
Perry et al. 2002. ECOGRAPHY 25: 578–600.
Many methods are quite similar, both logically and mathematically; others can be
quite distinct (Dale et al. 2002).
Dale et al. 2002. ECOGRAPHY 25: 558-577
Using non-spatial data: variance/mean ratios and related measures
The simplest and oldest measures of ‘‘spatial pattern’’ are based on the counts
of individuals in some kind of sampling units (e.g. quadrats). In many
instances, the aim is to distinguish among three categories of spatial point
patterns: random; underdispersed or clumped; and overdispersed or ‘‘regular’’
Dale et al. 2002. ECOGRAPHY 25: 558-577
It is sometimes suggested that as a statistical test of randomness, (n−1)D can
be compared to the 2 distribution on n−1 degrees of freedom because if the
points are random, the counts come from a Poisson distribution for which the
variance equals the mean. In the presence of spatial correlation, the sample
variance is not an unbiased estimator of the variance, but the sample mean is
an unbiased estimator of the mean
Using non-spatial data: Morisita´s index
It is a measure of the deviation from randomness based on the Simpon´s
diversity index, rather than on the Poisson distribution. It is a measure of
diversity among sampling units. Is calculated as:
where q is the number of squares, n is the number of points in the i-th square
and N is the total number of points. When the index is not different from 1, the
spatial pattern is random. If the index is significantly greater than 1 the pattern
is added, while if less than 1, the pattern is regular. The proof of deviation from
randomness, in the case of Iδ, is made by reference to a table of the F
distribution with n1 = q-1 and n2 = .
)1(
)1(1
NN
nnq
I
q
iii
Methods based on non-spatial data have strong limitations
In many applications the principal interest may not be merely to determine
which of the three categories (random, under-and overdispersed) a point
pattern falls for a particular scale of study. Frequently, if the points are
overdispersed we may want to know the average spacing between the points.
If the points are underdispersed, forming clumps of higher density separated
by gaps of lower density, we may want to know the average sizes of the
patches and gaps and whether there is a single scale of clumping or several.
For those kinds of questions, the spatial locations of the sampling units must
be included as information in the analysis
Testing for spatial autocorrelation
Dale et al. 2002. ECOGRAPHY 25: 558-577
Testing for spatial autocorrelation (univariate data): Moran´s I
Legendre & Legendre 1998. Numerical Ecology. Elsevier
Legendre & Legendre 1998. Numerical Ecology. Elsevier
Legendre & Legendre 1998. Numerical Ecology. Elsevier
Legendre & Legendre 1998. Numerical Ecology. Elsevier
Analyzing contiguous data: quadrat variance methods
Contiguous data analyses use sets of continuously made measurements as
input. In one dimension this may be thought of as a transect, in two
dimensions a surface, and in three dimensions a rectangular solid. It differs
from point analyses and scattered data analyses by the underlying
assumption that all of the data is regularly spaced and complete
These methods are all based on a similar principle. They calculate the
variance of differences among blocks of different sizes or scales and use
the pattern of the variance estimates to determine the scale of pattern. The
methods differ primarily in the number and distribution of blocks being
compared (the shape of the logical spatial template). There are many
variations of these methods, including: Blocked Quadrat Variance (BQV),
Local Quadrat Variances (TTLQV), Paired Quadrat Variances (PQV), New
Local Variances (NLV)….
See Usher (1975), Ludwig and Goodall (1978), Lepš (1990), and Dale
(1999) for comparisons and contrasts among quadrat variance and related
methods.
Dale et al. 2002. ECOGRAPHY 25: 558-577
Analyzing contiguous data: quadrat variance methods
Contiguous data analyses use sets of continuously made measurements as
input. In one dimension this may be thought of as a transect, in two
dimensions a surface, and in three dimensions a rectangular solid. It differs
from point analyses and scattered data analyses by the underlying
assumption that all of the data is regularly spaced and complete
These methods are all based on a similar principle. They calculate the
variance of differences among blocks of different sizes or scales and use
the pattern of the variance estimates to determine the scale of pattern. The
methods differ primarily in the number and distribution of blocks being
compared (the shape of the logical spatial template). There are many
variations of these methods, including: Blocked Quadrat Variance (BQV),
Local Quadrat Variances (TTLQV), Paired Quadrat Variances (PQV), New
Local Variances (NLV)….
See Usher (1975), Ludwig and Goodall (1978), Lepš (1990), and Dale
(1999) for comparisons and contrasts among quadrat variance and related
methods.
Working with count data: SADIE
SADIE is a class of methods designed to detect spatial pattern in the form of
clusters, either of patches or gaps. The calculations also involve comparisons of
local density with those elsewhere, but made across the whole study arena
simultaneously. Each sample unit is ascribed an index of clustering, and the
overall degree of clustering into patches and gaps is assessed by a
randomization test.
A specific extension to spatial association is made by comparing the clustering
indices of two sets of data across the sample units. A local index of association,
k, may be derived at each sample unit, k, and these may be combined to give an
overall value, X.
The power of these methods comes from the ability to describe and map local
variation of spatial pattern and association.
Perry, J.N. et al. 1999. Ecology Letters, 2, 106-113.
Winder, L. et al. 2001. Ecology Letters, 4, 568-576
Working with count data: SADIE
Dynamic tutorial
http://home.cogeco.ca/~kfconrad/SADIE2008/index.html
SADIE software and resources:
http://home.cogeco.ca/~sadieexplained/index.html
Working with count data: SADIE
Stipa tenacissima
Ia = 2.15, P < 0.001
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Ia = 1.75, P < 0.001
Maestre & Cortina. 2002. Plant Soil 241: 279-291
Working with count data: SADIE
Cocu et al. 2005. Bulletin of Entomological Research, 32, 47-56.1.
Working with count data: SADIE
Cocu et al. 2005. Bulletin of Entomological Research, 32, 47-56.1.
Working with count data: SADIE
Cocu et al. 2005. Bulletin of Entomological Research, 32, 47-56.1.
Working with count data: SADIE
Cocu et al. 2005. Bulletin of Entomological Research, 32, 47-56.1.
Working with count data: SADIE
Cocu et al. 2005. Bulletin of Entomological Research, 32, 47-56.1.
Working with count data: SADIE
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Working with count data: SADIE
Dale et al. 2002. ECOGRAPHY 25: 558-577
Working with multivariate data: Mantel test
The Mantel test (Mantel 1967; Mantel and Valand 1970) is an extremely
versatile statistical test that has many uses, including spatial analysis.
The Mantel test examines the relationship between two square matrices (often
distance matrices) X and Y. The values within each matrix (Xij or Yij) represent
a relationship between points i and j. The relationship represented by a matrix
could be geographic distance, a data distance, an angle, a binary matrix, or
almost any other conceivable data.
Often one matrix is a binary matrix representing a hypothesis of relationships
among the points or some other relationship (e.g., Xij may equal 1 if points i
and j are from the same country and 0 if they are not). By definition, the
diagonals of both matrices are always filled with zeros.
Rosenberg. 2011. PASSAGE Manual
Legendre & Legendre 1998. Numerical Ecology. Elsevier
Legendre & Legendre 1998. Numerical Ecology. Elsevier
Reading Material
Spatial Analysis, M.-J. Fortin and M. Dale (2005, Cambridge University Press)
Spatial Pattern Analysis in Plant Ecology, M.R.T. Dale (1999, Cambridge University Press)
Spatial Data Analysis by Example (2 volumes), G.J.G. Upton and B. Fingleton (1985, John Wiley & Sons)
Spatial Processes, A.D. Cliff and J.K. Ord (1981, Pion)
An Introduction to Applied Geostatistics, E.H. Issaks and M.R. Srivastava (1989, Oxford University Press)
Numerical Ecology, P. Legendre and L. Legendre (1998, Elsevier)
Baldi, P., Chauvin, Y., Hunkapiller, T. y McClure, M. A. 1994. Hidden Markov models of biological primary
sequence information. Proceedings of the National Academy of Science USA 91: 1059-1063.
Alados, C. L., Pueyo, Y., Giner, M. L., Navarro, T., Escos, J., Barroso, F., Cabezudo, B. y Emlen, J. M.
2003. Quantitative characterization of the regressive ecological succession by fractal analysis of plant
spatial patterns. Ecological Modelling 163: 1-17.
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