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

0 10 20 30 40 50 60 70 80 90 100

Este (m)

0

5

10

15

20

25

No

rte

(m

)

404142434445464748

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

0 10 20 30 40 50 60 70 80 90 100

Este (m)

0

5

10

15

20

25

Nor

te (

m)

384246505458626668

0 10 20 30 40 50 60 70 80 90 100

Este (m)

0

5

10

15

20

25

Nor

te (

m)

1002504005507008501000

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

0 1 2 3 4 5 6

Este (m)

0

1

2

3

4

5

6

No

rte

(m

)

Gradient

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Loarie et al. 2009. The Velocity of Climate Change, Nature


0 1 2 3 4 5 6

Este (m)

0

1

2

3

4

5

6

No

rte

(m

)

Random

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5


0 1 2 3 4 5 6

Este (m)

0

1

2

3

4

5

6

No

rte

(m

)

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Regular

Tiger bush landscape in western NSW (Australia)


0 1 2 3 4 5 6

Este (m)

0

1

2

3

4

5

6

No

rte

(m

)

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Clustered

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.


Dungan et al. 2002. ECOGRAPHY 25: 626–640.


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.


Regular

Clumped

Rosenberg & Anderson. 2011. Methods in Ecology & Evolution 2(3):229-232.

Inapropriate grain


García. 2008. In: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales

Apropriate extent

Small extent

Apropriate grain


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.


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.


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.


TYPICAL EXAMPLES OF SPATIAL DATA

0 10 20 30 40 50 60 70 80 90 100

Este (m)

0

5

10

15

20

25

No

rte

(m

)

404142434445464748

Point pattern data

0 5 10 15 20 25 30 35 40 45 50

Este (m)

5

10

15

20

25

30

35

40

45

50

Nort

e (

m)

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).

0 10 20 30 40 50 60 70 80 90 100

Este (m)

0

5

10

15

20

25

No

rte

(m

)

404142434445464748

Albeit the use of regularly spaced quadrats is common, other spatial

configurations (nested, irregular) can also be found.

Dungan et al. 2002. ECOGRAPHY 25: 626–640.



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.

0 1 2 3 4 5 6 7 8 9 10

Este (m)

0

1

2

3

4

5

6

7

8

9

10

Nort

e (

m)


* 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.

0 1 2 3 4 5 6 7 8 9 10

Este (m)

0

1

2

3

4

5

6

7

8

9

10

Nort

e (

m)


Overton & Levin. 2003. Ecological Research18, 405–421.


* 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

0 10 20 30 40 50 60 70 80 90 100

Este (m)

0

5

10

15

20

25

No

rte

(m

)

404142434445464748

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

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

* Evaluating the factors driving the

abundance of species

Overton & Levin. 2003. Ecological Research18, 405–421.


Wilson et al. 2004. Nature 432: 393-396

* Predicting biodiversity changes


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

0 10 20 30 40 50 60 70 80 90 100

Este (m)

0

5

10

15

20

25

No

rte

(m

)

404142434445464748

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.



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)



Mapping

(interpolation)

“Krigging” (this course)

Trend surface analysis (Legendre y

Legendre 1998)

Voronoi polygons (Fortin & Dale

2005)

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

http://www.passagesoftware.net/webhelp/Bibliography.htm




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’’


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


Testing for spatial autocorrelation (univariate data): Moran´s I

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.


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

http://home.cogeco.ca/~sadieexplained/reprints/pwha_ecol_lett_99_rb.pdf


Dynamic tutorial

http://home.cogeco.ca/~kfconrad/SADIE2008/index.html

SADIE software and resources:

http://home.cogeco.ca/~sadieexplained/index.html

http://home.cogeco.ca/~kfconrad/SADIE2008/spatial_pattern_1.html


Stipa tenacissima

Ia = 2.15, P < 0.001

0 5 10 15 20 25 30 35 40 45 50

Este (m)

0

5

10

15

20

25

30

35

40

45

50

No

rte

(m

)

Biological crusts

Ia = 1.75, P < 0.001

Maestre & Cortina. 2002. Plant Soil 241: 279-291


Cocu et al. 2005. Bulletin of Entomological Research, 32, 47-56.1.


0 5 10 15 20 25 30 35 40 45 50

Este (cm)

0

5

10

15

20

25

30

35

40

45

50

No

rte

(cm

)

0 5 10 15 20 25 30 35 40 45 50

Este (cm)

0

5

10

15

20

25

30

35

40

45

50

No

rte

(cm

)

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

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.

Rosenberg, M.S., and C.D. Anderson (2011) PASSaGE: Pattern Analysis, Spatial Statistics and

Geographic Exegesis. Version 2. Methods in Ecology & Evolution 2(3):229-232

Dale, M. R. T. y Mah, M. 1998. The use of wavelets for spatial pattern analysis in ecology. Journal of

Vegetation Science 9: 805-814.

Plotnick, R. E., Gardner, R. H. y O’Neill, R. V. 1993. Lacunarity indices as measures of landscape texture.

Landscape Ecology 8: 201–211.

Perry, J. N., Winder, L., Holland J. M. y Alston R. D. 1999. Red-blue plots for detecting clusters in count

data. Ecology Letters 2: 106-113.

Winder, L., Alexander, C., Holland, J. M., Woolley C. y Perry, J. N. 2001. Modelling the dynamic spatio-

temporal response of predators to transient prey patches in the field. Ecology Letters 4: 568-576.

Renshaw, E. 1997. Spectral techniques in spatial analysis. Forest Ecology and Management 94: 165-174.

Haining, R. P. 1990. Spatial Data Análisis in the Social and Environmental Sciences. Cambridge

University Press, Cambridge.

Reading Material

Cressie, N. A. C. 1993. Statistics for spatial data. John Wiley & Sons, Nueva York, Estados Unidos.

Perry, J. N., Liebhold, A. M., Rosenberg, M. S., Dungan, J. L., Miriti, M., Jakomulska, A. y Citron-Pousty,

S. 2002. Illustrations and guidelines for selecting statistical methods for quantifying spatial pattern in

ecological data. Ecography 25: 578-600.

Dixon, P. M. 2002. Nearest-neighbor contingency table analysis of spatial segregation for several species.

Ecoscience 9: 142-151.

Dale, M. R. T., Dixon, P., Fortin, M.-J., Legendre, P., Myers, D. E. y Rosenberg, M. S. 2002. Conceptual

and mathematical relationships among methods for spatial analysis. Ecography 25: 558-577.

Legendre, P. 1993. Spatial autocorrelation: trouble or new paradigm? Ecology 74: 1659-1673.

Legendre, P. y Fortin, M.-J. 1989. Spatial pattern and ecological analysis. Vegetatio 80: 107-138.

Legendre, P., Dale, M. R. T., Fortin, M.-J., Casgrain, P. y Gurevitch, J. 2004. Effects of spatial structures

on the results of field experiments. Ecology 85: 3202-3214.

Legendre, P., Dale, M. R. T., Fortin, M.-J., Gurevitch, J. y Myers, D. E. 2002. The consequences of spatial

structure for the design and analysis of ecological field surveys. Ecography 25: 601-615.

Goovaerts, P. 1997. Geostatistics for natural resources evaluation. Oxford University Press, Nueva York.

Lichstein, J. W., Simons, T. R., Shriner, S. A. y Franzreb, K. E. 2002. Spatial autocorrelation and

autoregressive models in ecology. Ecological Monographs 72: 445-463.

statistical analysis of geospatial data for environmental studies ... · statistical analysis of...

Documents