Department of Primary Industries and Fisheries and Agricultural Production Systems Research Unit, Toowoomba, Queensland, Australia

National Wildlife Research Center, 4101 Laporte Ave, Fort Collins, CO, USA

National Wildlife Research Center, USDA/APHIS/ADC, Hawaii Field Station, P.O. Box 10880, Hilo, Hawaii 96721, USA

Abstract

Background

Plotless density estimators are those that are based on distance measures rather than counts per unit area (quadrats or plots) to estimate the density of some usually stationary event, e.g. burrow openings, damage to plant stems, etc. These estimators typically use distance measures between events and from random points to events to derive an estimate of density. The error and bias of these estimators for the various spatial patterns found in nature have been examined using simulated populations only. In this study we investigated eight plotless density estimators to determine which were robust across a wide range of data sets from fully mapped field sites. They covered a wide range of situations including animal damage to rice and corn, nest locations, active rodent burrows and distribution of plants. Monte Carlo simulations were applied to sample the data sets, and in all cases the error of the estimate (measured as relative root mean square error) was reduced with increasing sample size. The method of calculation and ease of use in the field were also used to judge the usefulness of the estimator. Estimators were evaluated in their original published forms, although the variable area transect (VAT) and ordered distance methods have been the subjects of optimization studies.

Results

An estimator that was a compound of three basic distance estimators was found to be robust across all spatial patterns for sample sizes of 25 or greater. The same field methodology can be used either with the basic distance formula or the formula used with the Kendall-Moran estimator in which case a reduction in error may be gained for sample sizes less than 25, however, there is no improvement for larger sample sizes. The variable area transect (VAT) method performed moderately well, is easy to use in the field, and its calculations easy to undertake.

Conclusion

Plotless density estimators can provide an estimate of density in situations where it would not be practical to layout a plot or quadrat and can in many cases reduce the workload in the field.

Background

Plotless density estimators are those that based on distance measures rather than counts per unit area (quadrats or plots) to estimate the density of some fixed event, e.g. burrow openings, damage to plant stems, etc. Plotless density estimators can provide an estimate of density in situations where it would not be practical to layout a plot or quadrat, e.g. difficult terrain, crops, situations where a low impact is required. These techniques make certain assumptions about the spatial distribution of the event that in the worst case assume that the event is randomly distributed, a situation that occurs infrequently in nature. Other techniques permit greater degrees of non-randomness. It is important therefore to understand when a certain plotless density estimator is robust to departures from non-randomness.

An evaluation of which plotless density estimator (PDE) is suitable for a given field situation requires examination of fully enumerated field populations and is ideally suited to computer simulation. Inferences about PDEs using simulated populations

Methods

Estimation Methods Used

We selected the eight best estimators from the 24 evaluated by ^{th }random point to the first, second or third closest individual; from the closest individual to the first or second nearest neighbor and; the distance from a transect baseline of width w, to the g^{th }event such that all g events are within the transect. Estimators used in this study (Table

Summary of estimators used, their formulae and main reference.

Estimator

Formula^{§}

Reference

**Basic Distance (BD) estimators**

Compound of CI, NN & 2NN (BDAV3)

_{(1)i}/^{2})

[1]

_{(1)i}/^{2}

_{(2)i}/^{2}

BDAV3 = (BDCI + BDNN + BD2N)/3

**Kendall-Moran (KM) estimators**

CI and NN Search areas pooled (KMP)

_{i }+ _{i})] - 1}/∑ _{i}

[5,6]

CI, NN and 2NN search areas pooled (KM2P)

_{i }+ _{i }+ _{i})] - 1}/∑ _{i}

[5]

**Ordered Distance (OD) estimators**

Second Closest Individual (OD2C)

_{(2)i})^{2}

[7,8]

Third closest Individual (OD3C)

_{(3)i})^{2}

**Angle-Order (AO) estimators**

Second closest individual in each quadrant (AO2Q)

[7,8]

Third closest individual in each quadrant (AO3Q)

[7,8]

**Variable Area Transect (VAT)**

Variable Area Transect

_{i})

[9]

**Quadrat (QUAD)**

Quadrat

_{i}/_{i}_{i}

[17]

CI – closest individual, NN – nearest neighbor, 2NN – second nearest neighbor. R_{(1)i }= the distance from the i^{th }sample point to the CI; R_{(2)i }= the distance from the i^{th }sample point to the second CI; R_{(3)i }= the distance from the i^{th }sample point to the third CI; R_{(3)ij }= the distance from the i^{th }sample point to the third CI for the j^{th }quadrant; H_{(1)i }= the distance from the ith CI to its NN; H_{(2)i }= the distance from the NN at the i^{th }random point; p_{i}, n_{i}, m_{i }= the number of Cis, NNs and 2^{nd }NNs respectively, B_{i }= the total search area at the i^{th }sample point for the CI and NN combined; C_{i}, = the total search area at the i^{th }sample point for the CI, its NN, and the second NN combine; N = the sample size (number of random sample points used to gather distance information); w_{i }= width of quadrat; w = width of transect; l_{i }= length of quadrat.

Complete results from all simulations.

Click here for file

- Basic distance estimators

- Kendall-Moran estimators

Schematic representation of how KM2P and BDAV3 are implemented in the field

**Schematic representation of how KM2P and BDAV3 are implemented in the field**. Shading shows the search area less intersection used in the calculation of KM2P. R – the random sample point CI – closest individual; NN – nearest neighbor; 2NN – second nearest neighbor, R_{(1)i }= the distance from the i^{th }sample point to the CI; H_{(1)i }= the distance from the i^{th }CI to its NN; H_{(2)i }= the distance from the NN at the i^{th }random point.

- Ordered distance and angle order methods

The

- variable area transect method

Simulation Study Design and Data Sets

Eight plotless density estimators were examined in the present study using 5000 Monte Carlo simulations, Table

Natural data sets

Seventeen data sets (Table ^{-2 }(bee-eater nest sites) to 19.3 m^{-2 }(damaged sugar). A boundary strip of 10% of the length and width of the extent of the population of points was used to remove the bias associated with sampling close to the edge of the study area.

Description of data sets used and density of the event.

Data Set

Description

n

Dimensions (m)

Density (m^{-2})

Bee eater

Bee eater nest sites

64

41.5*24

0.06

Corn 1

Rat damage to corn in the Philippines for three different fields

2406

89.25*103.2

0.26

Corn 2

1596

86.25*121.6

0.15

Corn 3

1342

99.2*96.75

0.14

PG 92

Active pocket gopher burrows – 1992

132

28.5*22

0.21

PG 93

Active pocket gopher burrows – 1993

136

32.6*22.5

0.19

Rice 1

Rat damage to rice in the Philippines for five different fields

1678

63.5*12.25

2.16

Rice 2

177

7.31*16.66

1.45

Rice 3

3105

17.8*19.8

8.81

Rice 4

262

18.36*8.16

1.75

Rice 5

275

21.08*7.99

1.63

Sugar 1

Rat damage in sugarcane, Mauna Kea Agribusiness fields, Hawaii, USA

921

7.99*5.96

19.34

Sugar 2

199

7.77*5.94

4.31

Sugar 3

689

7.98*5.98

14.44

Sugar 4

174

7.48*6.52

3.57

Waterfowl

Alaskan waterfowl nests

497

26.3*5.9

3.20

Xanth

Distribution of grass trees (

748

25*50

0.60

For ground or cliff nesting birds the density of nest sites provide important information on the number of breeding females or pairs. Two data sets were used with densities of 0.06 (bee eater) and 3.2 m^{-2 }(Alaskan waterfowl nests).

Burrowing species such as gophers and rabbits can be monitored through the presence of active burrows. Two data sets of a population of pocket gophers measured in two successive years were used to demonstrate the application of PDE as a suitable method for monitoring populations.

The use of PDEs for monitoring damage to crops was done using corn and rice in the Philippines, and sugar cane in Hawaii.

The remaining data set is from a coastal sand island, north of Brisbane, Australia. Grass trees,

Simulated data sets

Five data sets whose spatial characteristics were predetermined were also included for comparison. The artificial data sets (where n is the number of individuals, ^{-2}) had distributions that were Poisson (n = 100,

Statistics

The relative root mean square error (RRMSE) was used as the basis of comparisons between the different PDEs _{est }is the estimated density and

In addition, relative bias (RBIAS) shows the bias relative to the true density and the direction of that bias such that:

The R index,

R index, standard error of expected mean, s, and z statistic [13] for the data sets used. When the pattern is entirely random R = 1, if the events are uniform then R > 1 (R = 2.149 for a perfect hexagonal uniform distribution) and conversely when the population of events is clumped R < 1 (R approaches 0 for maximally clumped distribution). The z test statistic considers the null hypothesis that the spatial distribution is random. Data sets comparable to those generated in [1] in italics.

Dataset

R

s

z

Pattern

2.15

0.003

76.53

Uniform

Rice 3

1.41

0.002

39.57

Uniform

PG 92

1.36

0.067

6.16

Uniform

Bee-eater

1.34

0.086

8.17

Uniform

PG 93

1.3

0.077

4.9

Uniform

1.25

0.003

17.34

Uniform

Rice 1

1.08

0.004

6.12

Uniform

Rice 5

1.04

0.016

0.99

Random

0.98

0.003

-1.1

Random

Rice 4

0.94

0.013

-1.59

Random

Xanth

0.89

0.017

-4.3

Clumped

Corn 1

0.88

0.014

-8.69

Clumped

Rice 2

0.88

0.019

-2.43

Clumped

Sugar 1

0.83

0.002

-8.15

Clumped

0.8

0.003

-14.79

Clumped

0.8

0.002

-16.34

Clumped

Sugar 3

0.76

0.003

-9.45

Clumped

Corn 2

0.75

0.039

-11.12

Clumped

Sugar 4

0.7

0.011

-6.57

Clumped

Waterfowl

0.65

0.007

-12.6

Clumped

Sugar 2

0.64

0.013

-7.32

Clumped

Corn 3

0.47

0.043

-21.97

Clumped

where R_{O }is the average observed nearest neighbor distance, r_{i }is the nearest neighbor distance to the i^{th }sample point and n is number of nearest neighbor distances measured;

where R_{E }is the expected nearest neighbor distance for a random pattern of events;

R was calculated for the complete data set less a 10% buffer. When the pattern is entirely random R = 1, if the events are uniform then R > 1 (R = 2.149 for a perfect hexagonal uniform distribution) and conversely when the population of events is clumped R < 1 (R approaches 0 for maximally clumped distributions). The z test statistic was calculated that measured the difference between the observed and expected values of R, i.e. it considers a null hypothesis that the spatial distribution is random.

where se is the standard error of R_{E}

A Spearman (rank) correlation coefficient was calculated between the log of (_{est }for AO3Q, BDAV3, KM2P and VAT across all natural data sets.

Results and Discussion

Interpretation of the performance of estimators based on relative root mean square error (RRMSE) (Table

Mean relative root mean square error for 10, 25, 50 and 100 samples/simulation for each density estimator and each spatial pattern for the natural data sets (see Table 3)

Sample size

Sample size

RRMSE

Rank

Estimator

10

25

50

100

10

25

50

100

Uniform (n = 5)

AO2Q

0.306

0.266

0.247

0.238

7

8

8

8

AO3Q

0.280

0.247

0.232

0.224

6

7

7

7

BDAV3

0.254

0.202

0.182

0.173

3

5

5

5

KM2P

0.247

0.199

0.177

0.166

1

3

3

4

KMP

0.256

0.201

0.177

0.165

4

4

4

3

OD2C

0.307

0.229

0.202

0.188

8

6

6

6

OD3C

0.251

0.182

0.157

0.143

2

1

1

1

VAT

0.258

0.194

0.167

0.148

5

2

2

2

Poisson (n = 2)

AO2Q

0.392

0.353

0.341

0.333

8

8

8

8

AO3Q

0.345

0.316

0.307

0.302

7

7

7

7

BDAV3

0.270

0.157

0.114

0.091

4

3

3

3

KM2P

0.232

0.149

0.111

0.088

1

1

2

2

KMP

0.288

0.193

0.153

0.130

5

5

5

5

OD2C

0.304

0.199

0.159

0.131

6

6

6

6

OD3C

0.253

0.160

0.123

0.098

2

4

4

4

VAT

0.256

0.154

0.107

0.077

3

2

1

1

Clumped (n = 10)

AO2Q

0.390

0.321

0.293

0.277

2

2

3

3

AO3Q

0.362

0.307

0.284

0.271

1

1

1

2

BDAV3

0.461

0.331

0.287

0.263

6

3

2

1

KM2P

0.424

0.374

0.361

0.354

3

4

4

4

KMP

0.468

0.427

0.413

0.406

7

7

6

6

OD2C

0.491

0.466

0.459

0.455

8

8

8

8

OD3C

0.448

0.426

0.420

0.417

5

6

7

7

VAT

0.439

0.414

0.407

0.403

4

5

5

5

All (n = 17)

AO2Q

0.368

0.311

0.287

0.274

4

4

4

4

AO3Q

0.338

0.292

0.273

0.263

1

2

2

2

BDAV3

0.380

0.273

0.236

0.216

6

1

1

1

KM2P

0.346

0.297

0.279

0.269

2

3

3

3

KMP

0.387

0.335

0.316

0.305

7

7

7

7

OD2C

0.417

0.367

0.350

0.340

8

8

8

8

OD3C

0.369

0.325

0.311

0.301

5

6

6

6

VAT

0.366

0.321

0.303

0.291

3

5

5

5

Mean relative bias for 10, 25, 50 and 100 samples/simulation for each density estimator for each spatial pattern (see Table 3)

Sample size

Sample size

RBIAS

Rank

Estimator

10

25

50

100

10

25

50

100

Uniform (n = 5)

AO2Q

0.222

0.225

0.225

0.225

8

8

8

8

AO3Q

0.205

0.207

0.207

0.207

7

7

7

7

BDAV3

-0.095

-0.136

-0.147

-0.154

4

4

5

6

KM2P

-0.136

-0.145

-0.148

-0.150

6

6

6

5

KMP

-0.130

-0.141

-0.145

-0.148

5

5

4

4

OD2C

-0.051

-0.077

-0.091

-0.097

1

1

2

2

OD3C

-0.070

-0.091

-0.101

-0.106

2

3

3

3

VAT

-0.074

-0.080

-0.081

-0.083

3

2

1

1

Poisson (n = 2)

AO2Q

0.324

0.324

0.326

0.325

8

8

8

8

AO3Q

0.295

0.295

0.296

0.296

7

7

7

7

BDAV3

0.073

0.014

-0.002

-0.009

6

2

2

2

KM2P

-0.037

-0.050

-0.052

-0.052

3

4

4

3

KMP

-0.070

-0.089

-0.092

-0.094

5

6

6

6

OD2C

-0.045

-0.065

-0.070

-0.071

4

5

5

5

OD3C

-0.031

-0.047

-0.051

-0.052

2

3

3

4

VAT

0.019

0.005

0.000

-0.003

1

1

1

1

Clumped (n = 10)

AO2Q

-0.079

-0.082

-0.081

-0.080

2

3

3

3

AO3Q

-0.063

-0.065

-0.065

-0.064

1

2

2

2

BDAV3

0.080

-0.008

-0.036

-0.049

3

1

1

1

KM2P

-0.319

-0.325

-0.333

-0.337

4

4

4

4

KMP

-0.350

-0.377

-0.386

-0.391

5

5

5

5

OD2C

-0.410

-0.435

-0.444

-0.447

8

8

8

8

OD3C

-0.376

-0.399

-0.407

-0.410

7

7

7

7

VAT

-0.365

-0.387

-0.393

-0.396

6

6

6

6

All (n = 17)

AO2Q

0.055

0.054

0.055

0.055

2

2

1

1

AO3Q

0.056

0.056

0.056

0.056

3

3

2

2

BDAV3

0.033

-0.039

-0.061

-0.072

1

1

3

3

KM2P

-0.227

-0.241

-0.246

-0.249

4

4

4

4

KMP

-0.254

-0.276

-0.282

-0.286

7

7

7

7

OD2C

-0.266

-0.290

-0.300

-0.304

8

8

8

8

OD3C

-0.248

-0.270

-0.278

-0.281

6

6

6

6

VAT

-0.236

-0.253

-0.257

-0.260

5

5

5

5

An ideal estimator is one that is robust across many spatial patterns, i.e. RRMSE and RBIAS are low, and where the amount of fieldwork required can be minimized or at least be undertaken efficiently. Basic distance estimators were largely dismissed by

Schematic representation of how AO3Q is implemented in the field

**Schematic representation of how AO3Q is implemented in the field**. The order of the quadrants is arbitrary. In practice much time is spent deciding which is the third closest individual and into which quadrant an individual lies. R_{(3)ij }= the distance from the i^{th }sample point to the third CI for the j^{th }quadrant.

Absolute relative bias (i.e. regardless of sign) for the AO and BD estimators was an order of magnitude smaller than the others for clumped data sets. However, AO estimators showed higher positive bias for Poisson data sets compared to the near zero for the others. In uniform data sets the OD and VAT estimators showed a RBIAS close to zero.

BDAV3 and KM2P use the same field methodology, however, data processing is much simpler for BD than for KM estimators. These estimators use information from the closest individual, distance to its nearest neighbor and the second nearest neighbor and that may help to explain why they are robust across all spatial patterns studied here, compared to estimators such as AO that rely on information derived from the closest individual.

Whereas the calculation for KM2P looks deceptively simple (Table

For uniform patterns the OD3C, VAT or KM2P methods were the most suitable, however, the method of searching in VAT is the simplest to implement. The fieldwork required for BDAV3 and KM2P are the same and although BDAV3 is much easier to calculate it is less able to cope with uniform data sets. The selection of the required sample size should be undertaken on a case-by-case basis using a pilot study. Accuracy will be improved with larger sample sizes and the techniques used to minimize the variance through stratified sampling, randomization, etc. should be employed.

The VAT method would seem the most straightforward to utilize in most field situations, and under optimized sampling constraints the method holds promise for row crops

Correlation between mean density estimate against known density for all data sets

**Correlation between mean density estimate against known density for all data sets**. Line shows complete agreement between known and estimated density. Spearman's correlation coefficient shown in parentheses. Symbols denote spatial pattern of data set: Uniform – filled circle, Poisson – filled triangle, Clumped – open circle.

Furthermore, the present study aimed to examine PDE methods as originally presented, without attempting to improve performance through optimizing procedures. Thus we examined VAT sampling using g = 3. The number of individuals for which to search has been optimized with substantial improvements in estimation quality for g ≥ 5

When damage is the event to be estimated and is caused by an animal that invades a crop or forestry coup it is usual to find the damage along the edge. Figures

Examples of diversity of spatial patterns found

**Examples of diversity of spatial patterns found**. (a) uniform distribution of pocket gopher burrows; (b) aggregated nesting pattern of waterfowl; (c) random pattern of rodent damage in rice; (d) highly clumped damage within a cornfield.

Typically the data sets of damage were clumped, however, random and uniform patterns were also found for data sets that mapped the distribution of burrows or nest sites. It is a characteristic of field data that the spatial pattern can vary within the study area. This was demonstrated by recalculating the R index for regions within the Corn 2 data set (Figure

Subsets within the highly clumped Corn 2 data set showing random and uniform patterns, see Table 6

Subsets within the highly clumped Corn 2 data set showing random and uniform patterns, see Table 6.

R index, standard error of expected mean, s, and z statistic [13] for subsets within Corn 2 see Figure 5.

Dataset

R

s

z

Pattern

Region 1

1.1

0.053

1.5

Random

Region 2

0.92

0.173

-1.23

Random

Region 3

1.21

0.047

3.21

Uniform

Conclusion

Plotless density estimators can provide an estimate of density in situations where it would not be practical to layout a plot or quadrat and can in many cases reduce the workload in the field.

Authors' contributions

NAW ran the simulations and with RME and HWK drafted and finalised the manuscript. RTS developed the original fortran code. All authors read and approved the final manuscript.

Acknowledgements

The authors wish to thank L. F Pank, R M Anthony and E Benigo for providing some of the field data sets and R K Schumacher and P Hallgren for their helpful comments on an earlier draft of the manuscript. The authors wish to thank the three anonymous referees for their comments and suggestions. This work was originally supported by the Queensland University of Technology.