Distributions for Fraser's spatial indicators

J. theor. Biol. (1981) 89,513-522

Distributions

for Fraser’s Spatial Indicators HOWARD

B. STAUFFER

Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6 W 1 W5, Canada (Received 18 February 1980, and in revised form 28 October 1980) A. R. Fraser has developed some spatial indicators from a triangle based polygonal construction utilized in forestry to measure spatial distribution. Significant correlation exists between Fraser’s indicators and Pielou’s index of non-randomness. The distribution for Pielou’s index is well known. Regressing Pielou’s index on Fraser’s indicators, distributions and confidence intervals are derived for Fraser’s spatial indicators. 1. Introduction

A. R. Fraser has developed a triangle based polygonal construction useful in a forestry context (Fraser&van den Driessche, 1972; Fraser, 1977). Given a point spatial pattern on the plane (e.g. of tree locations), Fraser constructs a triangulation of non-overlapping triangles with edges connecting least diagonal neighbors in the pattern. He next constructs a dual network of non-overlapping polygons surrounding each point. These polygons prove useful for defining areas of influence around trees and competition indices. They can also be utilized as probability polygons in forestry sampling. Fraser observes that these polygons and triangles also provide information about the spatial pattern. Spatial pattern is becoming increasingly important in forestry. Sampling techniques now evaluate the distribution of trees as well as their density. Indices of non-randomness which measure the extent of regularity, randomness, and aggregation (clumping) prove useful. Fraser defines three such indicators using the coefficients of variation of the edges of triangles (Fl), the areas of triangles (Fz), and the areas of polygons (F3) in his construction. These indicators increase in value as the spatial pattern varies from regular to random to aggregated. This paper derives distributions and confidence intervals for these spatial indicators Fl, Fz, and F3. Other indices of non-randomness, based upon quadrat and distance sampling, measure point spatial pattern (Clark & Evans, 1954; David & Moore, 1954; Hopkins & Skellam, 1954; Lloyd, 1967; Morisita, 1959; Pielou, 1969; Stauffer, 1977a). Pielou’s index of non-randomness (a) is one 513

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Press Inc. (London)

Ltd.

514

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B. STAUFFER

of the most commonly used distance sampling techniques (Pielou, 1959; Stauffer, 1979). Distributions and confidence intervals for Pielou’s index have been derived for random spacing. There is significant correlation between Fraser’s spatial indicators and Pielou’s index. Pielou’s index can be regressed on Fraser’s indicators to yield linear functional relationships. Distributions and confidence intervals will be derived for Fraser’s spatial indicators using these regression functions and known information about Pielou’s index. 2. Fraser’s

Spatial Indicators

Fraser has developed three spatial indicators to measure distribution in a point spatial pattern. His spatial indicators F,, F2, and F3 are the coefficients of variation (standard deviation/mean ratios) of the sides of triangles, the areas of triangles, and the areas of polygons, respectively, which arise from a triangulation of the point spatial pattern. This triangulation is constructed as follows. A pair of distinct points in a point spatial pattern are least diagonal neighbors (LCD) if and only if the interior of their connecting line segment cannot be intersected by a shorter line segment connecting another pair of distinct points in the spatial pattern. Given a point spatial pattern on the plane, the least diagonal neighbors generate a network of non-overlapping (a)

.

.

.

.

.

.

. .

*

.

.

(b)

FIG. 1. Fraser’s triangle based polygonal triangulation using least diagonal neighbors.

construction. (a)The spatial (c) The polygonal construction.

pattern.

(b)The

FRASER’S

triangles [see Fig. l(a)-(b)]?. given by

SPATIAL

INDICATORS

515

Th e coefficients of variation F1 and FZ are then Fl = sllml,

where ml and s1 are the mean and standard deviation, edges of the triangles, and F2 =

s2/mr

(1) respectively, of the (2)

where m2 and s2 are the mean and standard deviation, respectively, of the areas of the triangles. This triangulation of the plane generates a dual network of non-overlapping polygons constructed as follows. In each triangle, the line segments connecting the weighted midpoints* of each edge with the opposite vertices intersect in a point in the interior and partition the triangle into three quadrilaterals. The union of the quadrilaterals around each point yields a polygon. These form a network of non-overlapping polygons centered about the points in the original spatial pattern [see Fig. l(c)]§. The coefficient of variation F3 is given by

where m3 and s3 are the mean and standard deviation, respectively, of the areas of the polygons. Fraser has noted increasing variation in the triangles and polygons as the spatial pattern varies from regular to random to aggregated. This increases the variance of the edges and areas of the triangles and areas of the polygons giving larger coefficients of variation. Thus, Fraser’s spatial indicators furnish measures of the distribution in the point spatial pattern. t Note that the triangulation is not uniquely defined. Two pairs of points may be candidates for least diagonal neighbors with intersecting line segments of equal length. In such cases, the selection of the least diagonal neighbors can be imposed from an ordering of the pairs of points in the spatial pattern. t The weighting is based upon weight values assigned to each of the points in the point spatial pattern. In forestry application where the points represent tree positions, the weights are usually estimates of tree height, diameter, volume, crown length, competitive status, etc. The weighted midpoint then between points a and b with weights W, and tvb, respectively, is given by m=a+[~,/(w,,+~~)](b-a).

For the purposes of this paper, equal weights (w, = u+, = 1) are assumed. 5 There is topological terminology in mathematics describing this construction. The triangulation, defines a two-dimensional simplicial complex. The polygonal construction then is just the dual cell complex of the original simplicial complex.

516

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STAUFFER

3. Pielou’s Index of Non-randomness Pielou’s index of non-randomness (cr)t is a distance sampling technique used to measure point spatial pattern. For points on a plane Ly =7T.p.W,

(41

where p = the point density (number of points per unit area) and (3 = the average squared distance between a randomly chosen position and the nearest neighboring point. The average (5 is obtained from a designated number (n) of samples. Note that (Y may be considered to be the product of the density (p) and the average area of a “hole” (r. W) in the plot. Consequently, Pielou’s index measures the spacing of the points by examining the size of the empty spaces in the plot+. Pielou has shown that the probability density function of the random variable z = 2 . n . (Y for random spatial patterns is given by the x2 distribution with 2. n degrees of freedom. It can be shown that (Y has a probability density function given by f(a) = n” . Ly’n-ll . exp c-n . cy)/T’(n 1,

(5)

E(a) = 1.

(61

with mean and variance Var (a) = l/n.

(71

Here I’(n 1= (n - l)! is the gamma function evaluated at n. For random spacing, then, (Y has an expected value of 1. If the point spatial pattern is regular, (Y << 1, and, conversely, if the spatial pattern is aggregated, (Y >> 1. Confidence intervals for (Y are available for random spacing. We shall assume a sample size of n = 200 for Pielou’s index throughout the remainder of this paper. A hypothesis of randomness is then acceptable with 95% confidence if 0.865 5 cu 5 l-142, and with 99% confidence if 0.825 5 Q 5 1.189. 4. Derivation

of Distributions

for Fraser’s

Indicators

Significant correlation exists between Fraser’s spatial indicators and Pielou’s index of non-randomness. Regressing Pielou’s index on Fraser’s i The FORTRAN IV programs PTEST and ALPHA (Stauffer, 1976, 19776) which calculate Pielou’s index for two-dimensional spatial patterns and a generalized Pielou’s index for one-, two-, and three-dimensional spatial patterns, respectively, are available upon request from the author. $ Fraser points out that this is not the only way to conceive of “holes” in the spatial pattern. For instance, Delaunay triangles, defined by triples of points whose circumscribed circle contains no other points in the spatial pattern, furnish another notion of holes.

FRASER’S

SPATIAL

517

INDICATORS

indicators and using the known distribution for Pielou’s index, distributions and confidence intervals will be derived for Fraser’s spatial indicators. The correlations and regressions were established using data generated by the Tree and Stand Simulator (TASS) which grows each tree in a forest of Douglas fir (Mitchell, 1975; Fraser & Errico, 1978). The development of seven spacing regimes of identical initial density (1210 trees/acre) was simulated from age O-182 years: (i) regular spacing-trees 6’ apart in rows separated by 6’; (ii) regular spacing-trees 4’ apart in rows separated by 9’; (iii) regular spacing-trees 3’ apart in rows separated by 12’; (iv) random spacing; (v) aggregated spacing, 20% clumped toward cluster centers; (vi) aggregated spacing, 40% clumped toward cluster centers; and, (vii) aggregated spacing, 60% clumped toward cluster centers. The aggregation was simulated using Newnham’s (1968) Cluster Center Method. Natural tree mortality due to crown overtopping incorporated in the model reduced the density of the stand and the variation in the tree spacing. Pielou’s index (200 samples) and Fraser’s indicators were calculated for 16 separate ages over 182 years. Hence, data was generated for a variety of distributions covering a spectrum of “realistic” spatial pattern. Figures 2-4 lists the correlation and regression statistics. The regressions yield the linear relationships a =1.72510. a =0.69911. CY=0*93916.

Fl+O.23846, F2+0.40883, F3+0.30625.

(8)

(9) (10)

Distributions, mean values, and confidence intervals for F,, Fz, and F3 can then be derived from those known for (Y as follows. If (Y = A . F + B where A, B are constants and F is a Fraser indicator, then F = (a - B)/A. Let f(a) be the probability density function for (Y given by (5) and let g(F) be the probability density function for F. Then

g(F) = f[a WI . db )/d(F) =A.n".(A.F+B)'"-"

.exp[-n.

(A.F+B)]/I'(n).

(11)

The mean values for F are given by

E(F)=E[(a-B)/A] =[E(a)-B]/A.

(12)

518

H. B.STAUFFER O-1200 1.470-I

0.2067 I

o-2933 I

04667 I

03aco I

05533 I

064 3 I I +

I.324

I

t 1.179-

I I

+ +

I.033 o-887-

I I I It* I ;I; Hi’, I

+

+ +++ I + I I

+2 *+ I ,21 I

1251+*144 'fl:;;: : 4 2 t*l21 I +* I *t*2 I

0741-

I

0596 -

II

+ +I

0.450 , 01200

I 02067

FIG. 2. Correlation Fraser’s

F, spatial

I 02933

I 0.3600

I 04667

I 05533

I 0.6400

and regression statistics for Pielou’s index of non-randomness indicator. (Data generated by TASS. 1 Scattergram of (Y vs. F,.

(a J and

Legend: n = (n)O, 1,. + * Mean

,9)

and standard

where where where deviation

(x, y) occurs n times (x, y^) occurs (x, y) and (x, 9) coincide. of a and RI : mean 0.81611

0.33486

R Correlation between level): 0.92024. R-square

value:

S.d.

0.14507

~1 and Fr (correlation

0.07738. coefficients

bO.24187

are significant

at the l”/,

R2 = 0.84684

CX:

standard error = 0.05703 F-value = 608.20 F-probability cc 1% coefficient constant Fl

0.23846 1.72510

F-value

98.43 608.20

F-prob cc 1%

<<1Yo

s.d.

0.02404 0.06995.

FRASER’S oQoo I.470

0223 I

-’

SPATIAL 0447

0670 I

I

519

INDICATORS 0893 I

HI7 I

I

I.34 I I

+ I

. 2 +.

+

i

I I 0.450

,

+

oooo

I 0233

1 0670

I 0447

I 0893

I l-117

1 I34

)

FIG. 3. Correlation and regression statistics for Pielou’s index of non-randomness Fraser’s F2 spatial indicator. (Data generated by TASS.) Scattergram of a vs. Fz.

(a) and

Legend: n = (PIlO, 1,. . . , 9) + * Mean

and standard

Correlation between level): 0.93423. R-square

value:

where where where

(x, y) occurs n times (x, y*) occurs (x, y) and (x, 9) coincide.

deviation

“F (I and

of LY and

Fz:

mean O-81611 0.58258

F2 (correlation

s.d. o-14507 0.19386. coefficients

>0.24187

are significant

at the 1%

R2 = 0.87278

(I: standard error = 0.05198 F-value = 754.64 F-probability << 1%

F2: constant F2

coefficient 0.40883 0.69911

F-value 685.19 754.64

F-prob <
s.d. 0.01562 0.02545.

520

H. 0~140

0xX)

B.

0460

STAUFFER 0620

0780

0.940

Ho 0 1

:I-

I

I

I

I033 itt I I t+ I **II I I I I*** I \;u+:: I I

I

2431*** III+** 12,p;

t+ l t

t+ II I

122 21 21 I/” I

2212 II

I

+

I .: +

I

l I +

0.450

- , 0440

I 0300

I 0620

I 0460

I 0,780

I 0940

I Ilc 0

FIG. 4. Correlation and regression statistics for Pielou‘s index of non-randomness Fraser’s F3 spatial indicator. (Data generated by TASS.) Scattergram of (Y vs. F,.

[u, and

Legend: n={n\0,1,...,9) + * Mean

where(x,y)occursntimes where (x, 9) occurs where (x, y) and (x, $1 coincide.

and standard

deviation

of a and F3: mean 0.81611 0.54289

‘23

Correlation between level): 0.87010.

a and F3 (correlation

R-square

0.75707

value:

R’=

s.d. 0.14507

0.13440. coefficients

>0,24187

are significant

at the 1%

CK: standard error = 0.07 183 F-value = 342.81 F-probability cc 1% F3: constant 4

coefficient 0.30625 0.93916

F-value 116.60 342.81

F-prob c 1% cc 1 %

s.d. 0.02836 0.05072.

FRASER’S

SPATIAL

INDICATORS

521

The (1 -p) x 100% confidence limits (a, b) for (Y are similarly to [(a -B)/A, (b - B)/A] for F since a F(a) P/2

=

Jmmfb).

d(a)

=

JF(-m)

g(F).

transformed

d(F)

(a-EVA = J -co

g(F).

(13)

d(F),

and

J03 =J(b--Bb’A g(F). d(F).

p/2 = 6f(a).

d(a) = jFimi g(F).

d(F)

F(b)

(14)

Hence, using formulas (8)-(10) and the sample size it = 200 for Pielou’s index with mean E(cy) = 1,95% confidence intervals (a, 6) = (0*865,1*142) and 99% confidence intervals (a, 6) = (0.825, 1*189), distributions, mean values, and confidence intervals can be calculated for Fraser’s spatial indicators. The mean values and confidence intervals are listed in Table 1. TABLE Expected

values and confidence intervals indicators

Indicator FI Fl F2 F2 F3 F3

1

Confidence

a

(a, b) for Fraser’s Mean

spatial

b

95% 99%

0.36319 0*34000

0.44145 0.44145

0.52376 0.55101

95% 99%

0.65250 0.59529

0~84560 O-84560

1.04872 1.11595

95% 99%

0.59495 0.55236

0.73869 0.73869

0.88989 0.93994

Partial support for this research was provided by the Research Branch, British Columbia Ministry of Forests, Contract 65-3707, and the Natural Sciences and Engineering Research Council Canada, Grant A-3990. The author wishes to thank Ken Mitchell, Alan Fraser, Mik Kovats, and Darrell Errico, British Columbia Ministry of Forests, and Peter Belluce and Dale Rolfsen, University of British Columbia, for their helpful comments. Thanks also is extended to Ralph Schmidt, Director, Research Branch, British Columbia Ministry of Forests, and Colin Clark, Professor of Mathematics, University of British Columbia, for their encouragement and support.

522

H.

B.

STAUFFER

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CLARK, DAVID, FRASER, FRASER,

Statisticians, FRASER,

A.

Victoria,

Int. Union R. &

ERRICO,

For. Res. Oregon, D.

(1978).

Conf. Advisory Group of Forest Inst. Nat. Rech. Agr., Jouy-en-Josas, France, 277. Program Notes: FLDN. B.C. For. Serv.. Rex Br.,

B.C.

HOPKINS, B. (with an appendix by Skellam, J.R.) (1954). Ann. Bot. 18, 213. LLOYD, M. (1967). J. Anim. Ecol. 36, 1. MORISITA, M. (1959). Mem. Fuc. Sci. Kyushu U. Series E. (Biol.) 2, 215. MITCHELL, K. J. (1975). Forest Sci. Monograph 17. NEWNHAM, R. M. (1968). Dep. Forest. And Rural Deoel. Can.. Forest Manage.

Rep.

Inst. Inform.

FMR-X-10.

PIELOU, E. C. (1959). J. Ecol. 47,607. PIELOU, E. C. (1969). An Introduction to Mathematical Ecology. New York: Wiley. STAUFFER, H. B. (1976). PTEST. Dept. Fish. Environ., Can. For. Serv., Pac. For. Res. Cm., Victoria, B.C. Computing Services Program Library Documentation. STAUFFER, H. B. (1977a). Dep. Fish Environ., Can. For. Serv., Pac. For. Res. Cen., Victoria,

B.C.

BC-X-166.

STAUFFER,

Victoria, STAUFFER,

H. B. (19776).

B.C. Computing H. B. (1979).

ALPHA.

Dept. Fish. Environ.,

Services Program Library J. theor. Biol. 77, 19.

Can.

For. Serv., Pac. For. Res. Cen.,

Documentation.