1

Chapter 4 – Distance methods

Distance sampling (or called plotless sampling) is widely used in forestry and ecology

to study the spatial patterns of plants. Numerous mathematical models based on

distance sampling have been developed since the 50’s.

These models depend partly or wholly on distances from randomly selected points to

the nearest plant or from a randomly selected plant to its nearest neighbor.

The majority of the models are based on the assumptions that (1) the population of

interest is randomly distributed (Poisson distribution) within an infinitely large area and

(2) an observed distribution is a realization (or part) of the theoretical population.

Distance methods make use of precise information on the locations of events and have

the advantage of not depending on arbitrary choices of quadrat size or shape.

2

x

y

0 20 40 60 80 100

20

40

60

80

10

0

xri

ri

ri

1. Two types of distance measures: from tree to tree and from point to tree.

2. In general a buffer zone is needed to eliminate edge effect.

3

Nearest neighbor distance index

This index is the simplest one, based on the distance from a tree to its nearest

neighbor. It was first developed by Clark and Evans (1954). It is defined as

where R = the nearest neighbor index

= average distance from randomly selected plants to their nearest

neighbors

= expected mean distance between nearest neighbors. Under the Poisson

distribution with intensity , we have

* Clark, P.J. and Evans, F.C. 1954. Distance to nearest neighbor as a measure of spatial relationships in

populations. Ecology 35:445-453.

E

A

r

rR

Ar

Er

2

1Er

4

Testing the nearest neighbor distance index

The ratio R provides a method for detecting the degree to which the observed

distance departs from random expectation. In a regular distribution, R would be

significantly greater than 1, whereas in an aggregated distribution R would be

significantly less than 1. To test the null hypothesis (H0) that the observed

distance is from a randomly distributed population, we have

where and

1. Two-tail test: p-value = p(|z| zobs). Large |zobs| value has small p-value,

evidence against H0, suggesting aggregated or regular pattern.

2. One-tail test: p-value = p(z zobs) for testing regularity, or p-value = p(z zobs) for testing aggregated pattern.

E

A

r

rR

)1,0(~ Ns

rrz

r

EA

.26136.01

4

4

nnsr

2

1Er

5

Derivation of the nearest neighbor distance

We now go on to show how the nearest neighbor distance was derived. Assume a

population of organisms randomly distributed with intensity , the probability of x

individuals falling in any area of unit size is

Then, the number of individuals in a circle of radius r follows

a Poisson distribution with mean r2:

Similarly, the probability that the number of individuals in the annulus between the

concentric circles radii r and r1 is

.!

)(x

exp

x

.!

)()(

22

x

erxp

rx

r

r1•

•

•

•

.!

)]([)(

)(221

221

x

errxp

rrx

6

.!

)()(

212

1

x

erxp

rx

.!

)]([)(

)(21

22

21

22

x

errxp

rrx

The probability for the nearest neighbor distance r can be derived as follows.

p(r) p(circle r is empty, but individuals occur in the annulus)= p(circle r is empty) p(individuals occur in the annulus)

It is straightforward to compute the two probabilities:

The first probability is:

The second probability is:

Therefore,

.)0(2rexp

.1)0( )( 221 rrexp

.1)( )( 221

2

rrr eerp

r

r1•

•

•

•

7

The probability for the nearest neighbor distance r is obtained by assuming r1 r:

Thus, the pdf for the nearest neighbor distance r is a Weibull distribution:

Mean:

Variance:

)(111)( 221

)( 2221

2rreeerp rrrr

drerrrrrerre rrr 2222))(()( 11

221

22)( rrerp

2

12)(

0

2 2

drerrE r

4

4)(

rV

)2

1(

)(0

1 dxex x

Need to use the gamma function:

nrV

4

4)(

8

An example for the nearest neighbor distance

We test the spatial pattern for the western hemlock in the Victoria Watershed plot.

There are 982 hemlock stems in the 10387 m plot. The procedure is as follows.

Clark & Evans Nearest Neighbor Index

0 20 40 60 80 100

020

4060

80

1. Randomly choose 200 stems,

2. Measure the distance for each of these 200 stems to

its nearest neighbor,

3. Average these 200 distances (= 1.0458),

4. Calculate the expected mean distance (= 1.5104),

5. Compute the density = 0.1096,

6. The nearest neighbor index R = 0.6924,

7. Calculate the standard error sr = 0.05582,

8. Calculate the z-value = (1.0458-1.5104)/0.05582 =

-8.3232,

9. p-value = p(z zobs) = p(z -8.3232) = 0,

10. Conclusion: Reject null hypothesis of random

distribution; strong evidence for aggregated spatial

pattern.

11. R: distance.main(hl.xy,200,”event.event”)

9

The nth nearest neighbor distance

Thompson (1956) proved that the mean distance to the nth nearest neighbor is

5.08/15.02

11

!2

!21)( nen

n

nnrE n

nn

Thompson, H.R. 1956. Distribution of distance to nth neighbour in a population of randomly distributed individuals. Ecology 37:391-394.

For Victoria HL:

1124.0

Observed HemlockCSR ex

pecta

tion

Observed Hemlock

CSR expectation

2/12

1

)!2(

)!2(1)( n

n

nnrE

nn

Hubbell, S.P. et al. 2008. How many tree species are there in the Amazon and how many of them will go extinct? PNAS 105:11498-11504

12

Index of point to plant distances

First proposed by Pielou (1959), is based on the distances from randomly chosen

points to their respective nearest events (trees). The index is defined as

where = Pielou’s index of non-randomness

= average density of events per unit area

= mean squared distance between randomly chosen points to their nearest

neighbors. For randomly distributed population, it is

For observed distances, it is calculated as

(ri is the distance from the ith point to its nn)

* Pielou, E.C. 1959. The use of point-to-plant distances in the study of the pattern of plant populations.

Journal of Ecology 47:607-613.

.1

)(

E

n

iirn 1

21

13

Test statistics for Pielou’s index

It can be shown that 2n ~ 22n. (Sketch of the derivation: Following the Weibull

distribution on p.7, it is easy to show that has an exponential distribution:

f() = e- = e- ( is the density per unit circle). Then the sum of ’s

follows a gamma distribution of which 2 is a special case.)

Thus,

Test for the hypothesis of random pattern:

1. p-value = p(22n > 2n) for testing aggregated pattern of distribution. Large

2n value has small p-value, evidence against H0, suggesting aggregated

patterns.

2. p-value = p(22n < 2n) for testing regularity. Small 2n value leads to small

p-value, evidence to suggest regular patterns.

11

)(n

nE

)1(2)(2)2()2( nEnnEnE

(Unbiased estimator)

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Hopkins and Skellam’s coefficient of aggregation

This test is based on the assumption that a population is randomly distributed if the

distribution of distances from a random point to its nearest neighbor is identical to

the distribution of distances from a random plant to its nearest neighbor. The index

is defined as the ratio of the sum of the squared distances from point-to-plant (1)

to the sum of the squared distances from plant-to-plant (2):

A = 1 for a randomly distributed population

A > 1 for an aggregated population

A < 1 for a regular population.

To test whether A departs significantly from its expectation of 1, the sampling

distribution for the following statistic is derived:

2

1

A

vu

u

A

Ax

21

1

1

Hopkins, B. (with an appendix by Skellam, J.G.) 1954. A new method for determining the type of distribustion of plant individuals. Ann. Bot., London, N.S. 18:213.

15

x ~ Beta distribution

It is not difficult to show that x follows a beta distribution.

That is

where

The mean and variance of the beta distribution are:

vu

u

A

Ax

21

1

1

.)1(),(

1)( 11 nn xx

nnBxf

.)2(

)()(),(

n

nnnnB

)12(4

1

)1()()(

5.0)(

2

nxV

xE

11 )1(),(

1)(

xx

Bxf

Standard beta distribution:

unn

eun

uf

1

)()(

Note (same for v):

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Test for x

x = 0.5 is for random distribution

x > 0.5 is for aggregated distribution

x < 0.5 is for regular distribution

For a large sample size n, x tends towards normality. We have

Therefore, a statistical decision can be made based on the size of p-value:

1. p-value = p(z > zobs) for testing aggregated pattern of distribution. Large zobs

value has small p-value, suggesting an aggregated pattern.

2. p-value = p(z < zobs) for testing regularity. Small zobs value leads to small p-

value, evidence for a regular pattern.

vu

u

A

Ax

21

1

1

).1,0(~12)5.0(2 Nnxz

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Spatial relationships between two species

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

Random pattern

Aggregated pattern

Unsegregated species Segregated species

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Index of species segregation

Segregation is the degree to which the individuals of two (or more) species tend

to separate from one another. We have learned that quadrat counts could be

used to test the association of two species, but the results are strongly influenced

by quadrat size. An alternative approach which overcomes this problem is

based on distance sampling.

Assume there are two species, we randomly select an individual plant and locate

its nearest neighbor and then record the species type. This process is repeated N

times. The data can be summarized in a contingency table similar to the one for

the quadrat counts.

Nearest neighbor

Species A Species B

Base SpeciesSpecies A a b m = a+b

Species B c d n = c+d

r = a+c s = b+d N

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Index of segregation (Kappa statistic)

Pielou (1961):

Nearest neighbor

Species A Species B

Base SpeciesSpecies A a (x11) b (x12) m = a+b (x1+)

Species B c (x21) d (x22) n = c+d (x2+)

r = a+c (x+1) s = b+d (x+2) N

)(

)(1

2 nsmrN

cbN

2

1

2

2

1

2

11

iii

i iii

xxN

xxN

4

2

224

21

32

32112

2

112

)1(

)4()1(

)1(

)2)(1(2

)1(

)1(1

N

2

11

i

ii

N

x

2

122

i

ii

N

xx

2

13

i

iiii

N

xx

N

x

2

1

2

1

24 )(

i j

jiij

N

xx

N

x

where

Cohen (1960):

Note: With a large sample size, ~ N(0,1)

* Pielou, E.C. 1961. Segregation and symmetry in two-species population as studied by nearest-neighbor relationships. Journal of Ecology 49:255-269

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What we have learned?

• The concept of nearest neighbor distances• Tree-to-tree (event-to-event) distances (Clark & Evans 1954)• Point-to-tree distances (Pielou 1959)• Hopkins and Skellam’s index of aggregation (Hopkins 1954)• Index of species aggregation (kappa statistics)