Methods of spatial point pattern analysis applied in forest ecology

TIỂU BAN SINH THÁI HỌC VÀ MÔI TRƯỜNG

METHODS OF SPATIAL POINT PATTERN ANALYSIS APPLIED

IN FOREST ECOLOGY

Nguyen Hong Hai

Vietnam National University of Forestry

Spatial patterns of forest trees result from complex dynamic processes such as

establishment, dispersal, mortality, land use and climate (Franklin et al. 2010), especially in

tropical forests which are among the world‘s most species-rich terrestrial ecosystems. Spatial

correlation of trees may provide evidences of ecological interactions which are assumed to be

drivers of spatial pattern in plant communities. Spatial pattern analysis in ecology has received

increasing attention of ecologists and mathematicians over the last decades. Furthermore, it is

stimulated by the development of spatial point pattern methods and relevant computer

applications.

Several processes and mechanisms have been proposed to explain species coexistence and

community structure. For example, plant-plant interactions, such as competition or facilitation

(Bruno et al. 2003), limited dispersal (Nathan & Muller-Landau 2000), habitat preference

(Harms et al. 2001), density dependent mortality (Janzen 1970; Connell 1971) and neutral

theory ((Hubbell 2001; Chave 2004). Hence, understanding these underlying processes is a

central goal in ecology (Tilman 1994).

The neutral theory is proposed in order to find an explanation for the observed patterns of

species abundance across scales in space and time (Chave 2004). This theory assumes that all

individuals in a community are strictly equivalent regarding their prospects of reproduction and

death (Chave 2004). However, there is ample evidence showing that species are not equivalent

and differ in their ecological traits (Wiegand et al. 2007).

Janzen(1970) and Connell(1971) hypothesized that host specific pests reduce recruitment

near conspecific adults, thus freeing space for other plant species. Conditet al. (1994) suggested

that Janzen-Connell hypothesis is exhibited basically among those species with the highest

population densities, other studies show that density dependence is very common in tropical tree

species (Lan et al. 2009). Moreover, it is expected that aggregation should decrease with

increasing tree size (age) classes, due to competition (Sterner et al. 1986).

Callaway & Walker(1997) stated that competition has long been recognized as a main force

in structuring plant communities, while facilitation has not received much attention. They found

that the relative importance of these two processes can be evaluated by investigating the effects

of abiotic stress, consumer pressure, life stage, age and intensity of interaction strengths.

Facilitation has been observed to increase establishment of seedlings close to adults such as

nurse plants (Lan et al. 2012) or mycorrhizal fungi (Dickie et al. 2007). The species herd

protection hypothesis is an extension of the Janzen-Connell hypothesis (Peters 2003) and

suggests that hetero-specific neighbors can promote species coexistence by thwarting the

transmission of biotic plant pests (Lan et al. 2012). Thus, major mechanisms or processes

leading to aggregation or over-dispersion of plant distribution still remain controversial (Murrell

2009). However, tree species are long-lived, therefore long-term observations including period

censuses are needed to examine the effects of competition and facilitation in forest community

ecology.

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Colonization limitation, which is also called recruitment or dispersal limitation, is an

important factor in successional dynamics, community diversity and composition (Tilman

1994). Seeds can be dispersed in various spatial scales depending on the specific mechanisms or

agents, e.g. by wind, animals and/or gravity. Seed dispersal patterns vary among plant

individuals, species and populations and differ in distances from parents, micro-sites and times

(Nathan & Muller-Landau 2000).

Niche differentiation is a prominent hypothesis explaining the maintenance of tree species

diversity in tropical forests (Connell 1978). It suggests that different species are best suited to

different habitats in which they are completely dominant and more abundant than in less

suitable habitats (Harms et al. 2001). In addition, distribution patterns of tropical trees are

generally more clumped or aggregated than random (Condit et al. 2000). Furthermore,

environmental heterogeneity (difference in soil, elevation, slope, etc.) may obscure aggregated

distribution of plant species across spatial scales (Harms et al. 2001) . Therefore, it is difficult to

assess whether aggregated patterns are caused by local dispersal, local interaction or

environmental heterogeneity.

I. METHODS

Point processes are stochastic models of point patterns while a point pattern is a collection

of points which is typically interpreted as a sample from a point process (Diggle 2003). The

fundamental difference between the two terms is that a point process is a theoretical stochastic

model or random variable, whereas a pattern is a realization of the process (Perry et al. 2006). In

a simple case, each point pattern is defined by sets of Cartesian coordinates (x_i, y_i) and referred

to events. Moreover, an additional property of an event, a so-called mark, can be attached.

Therefore, a set of events typically takes the form {[x_i, y_i, m_i]}, giving the locations (x_i, y_i) and

marks m_iin the region of observation (Stoyan & Penttinen 2000). For example, mapped data of

trees contain the positions of stems and the marks (such as species, diameter, height, etc.).

100

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20

0

20

40

60

80

100

X (m)

Fig. 1: A map of trees with locations and diameters proportional to size of circles

A fundamental property of a point process is the point intensity λ, which may be interpreted

as the mean number of points per unit area. A point process N is called homogeneous

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(stationary) if N and its translated point processes have the same distribution for all translations

(Diggle 2003). The simplest case is complete spatial randomness (CSR) and termed the

homogeneous Poisson process with intensity λ. This point process has two important properties

(Stoyan & Penttinen 2000): (1) the number of events in any area unit follows the Poisson

distribution with mean λ. (2) Given n events in the observation region, their positions follow an

independent sample from the uniform distribution in this region.

A spatial point pattern can be characterized by its first-order and second-order properties.

The first-order property describes the mean number of events per unit area while the second-

order property is related to the variance of the number of events per unit of observed area (Perry

et al. 2006). A point pattern may deviate from stationarity in cases (Diggle 2003): (i) The

intensity function λ(x, y) or point density is not constant but varies spatially; (ii) The local point

configurations may be location-dependent. This generalizes to the inhomogeneous Poisson

process in which the constant intensity in CSR is replaced by an intensity function λ(x, y) whose

value varies with the location (x, y) (Diggle 2003). The different intensities are shown in fig. 2.

Homogeneity

Heterogeneity

100

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60

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0

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X (m)

Fig. 2: Examples of spatial distributions of points with constant and varying intensities.

Ripley’s K-function and pair-correlation function

The Ripley‘s K-function is the expected number of points in a circle of radius r around an

arbitrary point, divided by the intensity λ of the pattern (Ripley 1976). Thus, Ripley‘s K is

cumulative up to distance r meaning that point intensity is calculated within entire circle with

radius r.

( )

∫ ( )

λK(r) is the mean number of points within a distance r from an arbitrary point, particularly

K(r) = πr²for a homogeneous Poisson process. Let L(r) = (K(r)/π)^0.5- r, r ≥ 0; thus L(r) = 0 for a

homogeneous Poisson process; i.e., a straight line with slope 0 (Mateu 2000).

For computation of the pair-correlation function g(r), the circle is replaced by a ring. g(r) is

the expected density of points at distance r from an arbitrary point, divided by the intensity λ of

the pattern (Stoyan & Stoyan 1994). The difference in computation of K- and g-functions is

presented in fig. 3. Therefore, we can determine whether a pattern is random, clumped or

regular at a specific distance r. Under CSR, g(r) = 1, under aggregation g(r)> 1 and under

regularity g(r) < 1. An example is shown in fig. 4.

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Fig. 3: The difference in computation between K-function (A) and g-function (B)

Both Ripley‘s K-function and the g-function can be used as bivariate functions when

considering the spatial relation of 2 point patterns. Hence, the bivariate K-function K₁₂(r) is

defined as the expected number of points of pattern 2 within a given distance r of an arbitrary

point of pattern 1, divided by the intensity λ₂of points of pattern 2. Similarly, the pair-

correlation function g₁₂(r) gives the expected density of points of pattern 2 at distance r from an

arbitrary point of pattern 1, divided by the intensity λ₂of points of pattern 2 (Wiegand &

Moloney 2004). g₁₂(r) indicates whether pattern 2 is characterized by (1) independence (g₁₂(r) =

1), (2) repulsion (g₁₂(r) < 1) and (3) attraction (g₁₂(r) > 1) from pattern 1 at distance r. The pair-

correlation function g₁₂(r) is related to K₁₂(r) by:

( )

∫ ( )

Null models and hypothesis testing

To answer specific biological questions related to dynamics of plant distribution, one may

test the observed data based on an appropriate null hypothesis to find departure from the null

model. To choose an appropriate null model, a proposed approach is based on the mathematical

form of K(r) or g(r) functions: (i) inspection of the estimated K(r) or g(r) to find appropriate

models and parameters for the point process, (ii) construction of confidence envelopes via

Monte Carlo simulations of the stochastic process (Wiegand & Moloney 2004).

There are two commonly used null models for simulating a univariate point pattern:

complete spatial randomness (CSR) and heterogeneous Poisson process (HP). The CSR null

model is implemented as a homogeneous Poisson process where the intensity λ is constant over

the study region (Wiegand & Moloney 2004). Inversely, the HP null model is applied when a

point pattern is not homogeneous, therefore varying values of point intensity are quantified by a

function λ(x,y).

For analyzing a bivariate point pattern, two null models are mainly used: independence and

random labeling. The independence null model assumes two patterns are generated by two

different processes, and therefore is used to test the independence of two point patterns, for

example two point patterns of two different tree species. The random labeling null model

hypothesizes that two patterns are created by the same stochastic processes, for instance two

point patterns of a tree species (e.g. dead and alive). In practice, Goreaud & Pelissier(2003)

gave detailed suggestions when to use which null model and how to avoid misinterpretations.

Alternatively, antecedent conditions may be useful to choose as an appropriate null model in

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some practical cases. In this null model, pattern 1 (e.g., adult trees) is kept unchanged but for the

locations of pattern 2 (e.g., saplings) is randomized (CSR is assumed), because adults do not

change their positions over time but saplings may be found in the entire observed region.

Clumping/ Attraction

Random/ Independence

Regularity/ Repulsion

Fig. 4: Typical forms of pair-correlation functions g₁₁(r) and g₁₂(r)

A principal advantage of Monte Carlo testing is that the investigator is not constrained to

know distribution theory and can use informative statistics (Diggle 2003). Once distribution

theory is known, Monte Carlo testing can be used to check its applicability. Due to

mathematically unknown or intractable distribution theory of stochastic point processes,

significance tests for spatial measures are often carried out by Monte Carlo simulation (Diggle

2003; Perry et al. 2006). Based on a null hypothesis, the data sets were simulated by calculating

the statistic values (Marriott 1979) and rejection limits via confidence envelopes were

estimated. Wiegand & Moloney(2004) provided a detailed guideline for choosing an appropriate

null model for observed point data.

Models for marked point processes

Marks are used as properties of the objects (e.g., trees) and may be qualitative (e.g., species,

damage level) or quantitative (e.g., diameter of tree, tree height). Therefore, marked point

processes are models for random point patterns where marks are attached to the points (Illian et

al. 2008). In mathematical notation, a marked point process M is a sequence of random marked

points, M={[x_n;m(x_n)]}, where m(x_n) is the mark of the point x_n.

Similar to pure point processes, marked point processes can be used to consider

relationships of two types of marks (Mateu 2000). We consider qualitatively marked point

processes as sub-processes of point processes for aggregation or repulsion to find correlation

structure in the marks, conditional on spatial pattern of the trees carrying the marks. The

analysis of quantitative marks addresses questions concerning the numerical difference among

the marks that is dependent on the distances of the corresponding points, for example, why

neighboring points tend to have smaller (larger) marks than the mean mark (Getzin et al. 2008;

Getzin et al. 2011).

2

The mark correlation function k_mm(r) is defined as k_mm(r) = c_mm(r)/ for r> 0, where c_mm(r)

is the conditional mean of the product of the marks of a pair of points with distance r; is the

mean mark (Illian et al. 2008). This normalization allows comparing the strength of mark

correlation between different processes. If the empirical mark correlation functions are not

constantly equal 1, there is reason to assume that marks are not independent. Applied to forest

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ecology, k_mm(r) < 1 is assumed to indicate inhibition, individual trees compete against each other

and thus have smaller than average marks if they are distance r apart. Inversely, k_mm(r) > 1

indicates that points at distance r have average marks larger than the mean mark.

Fig. 5: Typical forms of the mark correlation function

II. DISCUSSION

Even though applications of point process methods have been developed and widely

implemented in various scientific fields, these tools are bounded in practical uses. There are

three main reasons: (1) requirement of mapped data, (2) pairwise-based second-order

characteristics, and (3) snapshot analysis of pure spatial patterns (Comas & Mateu 2007).

Basically, in point process models and spatial statistical tools, pair-wise analysis is a major part

of second-order characteristics and analysis tools for multi-specific interaction do not analyze

more than two variables. However, spatial correlation provides a sensitive indicator of

ecological interactions structuring spatial patterns of plant species in communities (Wiegand et

al. 2007). Moreover, snapshot observations combined with time series analyses have specific

advantages (e.g., less time consuming and cheaper) and are appropriate approaches for dynamic

assessments of long-lived tree species (Wiegand et al. 2000; Halpern et al. 2010).

In spatial point patterns analysis, an observed spatial pattern from the K-statistics is

compared to a hypothetical model and evaluated via confidence envelopes, which are

constructed by the maximum and minimum results computed across the simulated patterns.

However, results from this approach are problematic because of violation of Monte Carlo

methods and incorrect type I error performance rate (Loosmore & Ford 2006). A proposed

solution is goodness-of-fit test as implemented in the software Programita

(http://programita.org/). However, other authors stated that Monte Carlo method is appropriate

and can be used to assess whether the spatial pattern is significantly different from random

(Dale et al. 2002).

In computing point-pattern statistics, edge correction is required since events near the edge

of the study region have fewer neighbors than centered events, leading to incorrectly calculated-

intensities.Therefore, circle or ring samples will produce a biased estimation of the point pattern

if used without edge correction (Wiegand & Moloney 2004). Three approaches are proposed for

dealing with edge effect: Ripley‘s weighted correction, a toroidal correction and a guard area

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correction (Haase 1995; Yamada & Rogerson 2003). The major finding of Yamada &

Rogerson(2003) is that the K-function method adjusted by either the Ripley or toroidal edge is

more powerful than the guard area method. Among two alternatives to estimate the bivariate K-

function (numeric and analytical methods), numeric approaches are integrated and use an

underlying grid of cells. Therefore, analyses using Programita software do not require edge

correction (Wiegand & Moloney 2004).

Generally, plant community dynamics are driven by spatially dependent birth, death and

growth processes and closely embedded in a heterogeneous landscape (Law et al. 2009). From

multispecies spatial patterns analyzed, ecologists may use spatio-temporal information to tackle

basic questions. Moreover, plants are obviously not points, marks characterized for individual

plants (e.g., species, biomass, height, so on) and environmental data need to be considered as

they are highly relevant (Illian & Burslem 2007). Therefore, theoretical and empirical issues are

closely connected with their estimations and infer to the real dynamic processes generating

patterns of plant communities. Spatial point pattern analysis is stimulated by large and technical

literature in mathematics and plant ecology, moreover by strongly developed applications in

computer science (Law et al. 2009).

III. CONCLUSION

Analysis of spatial point pattern has a long history in plant ecology and there are a number

of tests available to characterise and explore such data. However, these tests do not all perform

equally and all have their weaknesses and strengths. As a result, it is suggested that a suite of

statistics is used to characterise spatial point patterns, otherwise there is a risk that the

description of the pattern will be partially determined by the test chosen. We have to note that

point-pattern analysis is a descriptive analysis. Even if a particular null model describes our

pattern well, it is not appropriate to conclude that the mechanism behind the null model is the

mechanism responsible for our pattern. Other mechanisms may lead to exactly the same pattern.

However, point-pattern analysis helps to characterize our pattern and to put forward hypotheses

on the underlying mechanisms that should be tested in subsequent steps in the field.

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CÁC PHƢƠNG PHÁP PHÂN TÍCH MÔ HÌNH ĐIỂM KHÔNG GIAN

ỨNG DỤNG TRONG SINH THÁI RỪNG

Nguyễn Hồng Hải

Đại học Lâm nghiệp Việt Nam

TÓM TẮT

Rất nhiều các phương pháp phân tích về mô hình điểm đã được phát triển trong nhiều lĩnh

vực khoa học. Thống kê bậc nhất mô tả sự biến động của mật độ các điểm trên phạm vi lớn của

vùng nghiên cứu, trong khi tính chất của bậc hai là tổng hợp thống kê của khoảng cách giữa các

điểm và cung cấp khả năng nhận dạng các loại mô hình ở các phạm vi khác nhau. Thống kê bậc

hai dựa vào hàm Ripley‘s K được sử dụng rộng rãi trong sinh thái để mô tả mô hình không gian

và để phát triển giả thuyết về quá trình đang diễn ra. Mục tiêu của bài báo này là thống kê lại

các phương pháp phân tích mô hình điểm không gian có thể ứng dụng trong sinh thái rừng.

Chúng tôi đã tổng hợp các vấn đề liên quan trong phần thảo luận và kết luận. Chúng tôi cũng

giới thiệu phần mềm Programita để ứng dụng tất cả các phương pháp được trình bày trong bài

báo.

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