Describing the spatial distribution of habitat trees based on line - - PowerPoint PPT Presentation

data reconstruction results model

Describing the spatial distribution of habitat trees based on line transect sampling and point pattern reconstruction

• H. Bäuerle
• A. Nothdurft

Forest Research Institute Baden-Württemberg

Workshop 03.-05.December 2009, University Lübeck

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

• utline

1

Data: Line transect samples as thinned point patterns

2

Method of reconstructing full point patterns in a model-independent approach

3

Evaluation of the reconstruction routine and analysis of the

• btained point patterns

4

Describing the spatial habitat tree distribution by fitting a Log-Gaussian Cox process model

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

empirical line transect sampling in the 378 ha study region

3463000 3463500 3464000 3464500 3465000 3465500 5425500 5426000 5426500 5427000 5427500 5428000 5428500

Natura 2000 area

• ●
• Figure: Observed habitat trees from

the 12 transects every 200 m

distance [m] detection function g(x) 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 1.2

detection probability using the half-normal distribution function

ˆ pi = exp −x2

i

2ˆ σ2

• point pattern reconstruction based on line transect sampling
• H. Bäuerle, A. Nothdurft

data reconstruction results model

estimation of local densities in segments

3463000 3463500 3464000 3464500 3465000 3465500 5425500 5426000 5426500 5427000 5427500 5428000 5428500

• ●
• Figure: Estimating the local density

in 1 ha segments local density ˆ Nj =

nj

• i=1

1/ˆ pji segments: j = 1, . . . , 199

• bserved trees:

i = 1, . . . , nj total number of habitat trees ˆ N = 199

j=1 ˆ

Nj

2·L·ω A

A: total area of the study site L: total length of the survey lines ω: = maximum horizontal distance

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction based on analysis of empirical cumulative distribution functions ECDFs

spherical contact distribution function:

Hs (r) = 1 − P (N (b (o, r)) = 0) for r ≥ 0

Probability that a disc b of radius r centered at any location o contains at least one tree.

nearest-neighbor distance distribution function:

D (r) = Po (N (b (o, r) / {o}) > 0) for r ≥ 0

Probability of finding the nearest neighbour of a tree within a disc b of radius r centered at any tree location o.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

1 The ECDFs for the original tree pattern are estimated. 2 New tree locations are generated to form the initial

configuration together with the original locations.

3 To improve this configuration one tree in the current pattern is

randomly generated or deleted.

4 For the new point pattern the ECDFs are estimated. 5 The contrast function Ct is determined and the new state is

accepted if Ct < Ct−1. Otherwise Ct is rejected and the point pattern is reset to the previous configuration.

6 Step 3-6 are repeated until the maximum number of iterations

10 × Q is achieved, with Q the initial number of trees in A.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

Step 1 The ECDFs for the original tree pattern are estimated.

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 r [m]

D ^

k(r)

500 1000 1500 0.0 0.2 0.4 0.6 0.8 1.0 r [m]

H ^

sk(r)

Figure: Distances were measured to the k = 1, . . . , 5 nearest trees, Dk(r) and Hs,k(r)

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

1 The ECDFs for the original tree pattern are estimated. 2 New tree locations are generated to form the initial

configuration together with the original locations.

3 To improve this configuration one tree in the current pattern is

randomly generated or deleted.

4 For the new point pattern the ECDFs are estimated. 5 The contrast function Ct is determined and the new state is

accepted if Ct < Ct−1. Otherwise Ct is rejected and the point pattern is reset to the previous configuration.

6 Step 3-6 are repeated until the maximum number of iterations

10 × Q is achieved, with Q the initial number of trees in A.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

Step 2 New tree locations are generated.

3463000 3463500 3464000 3464500 3465000 3465500 5425500 5426000 5426500 5427000 5427500 5428000 5428500

• Figure: Original habitat tree locations are fixed during the procedure.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

1 The ECDFs for the original tree pattern are estimated. 2 New tree locations are generated to form the initial

configuration together with the original locations.

3 To improve this configuration one tree in the current pattern is

randomly generated or deleted.

4 For the new point pattern the ECDFs are estimated. 5 The contrast function Ct is determined and the new state is

accepted if Ct < Ct−1. Otherwise Ct is rejected and the point pattern is reset to the previous configuration.

6 Step 3-6 are repeated until the maximum number of iterations

10 × Q is achieved, with Q the initial number of trees in A.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

Step 3 One tree in the current pattern is randomly generated or deleted.

Figure: Generation of a tree Figure: Deletion of a tree

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

1 The ECDFs for the original tree pattern are estimated. 2 New tree locations are generated to form the initial

configuration together with the original locations.

3 To improve this configuration one tree in the current pattern is

randomly generated or deleted.

4 For the new point pattern the ECDFs are estimated. 5 The contrast function Ct is determined and the new state is

accepted if Ct < Ct−1. Otherwise Ct is rejected and the point pattern is reset to the previous configuration.

6 Step 3-6 are repeated until the maximum number of iterations

10 × Q is achieved, with Q the initial number of trees in A.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

Step 4 For the new point pattern the ECDFs are estimated.

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 r [m]

D ^

k(r)

500 1000 1500 0.0 0.2 0.4 0.6 0.8 1.0 r [m]

H ^

sk(r)

Figure: ECDFs for the new point pattern.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

1 The ECDFs for the original tree pattern are estimated. 2 New tree locations are generated to form the initial

configuration together with the original locations.

3 To improve this configuration one tree in the current pattern is

randomly generated or deleted.

4 For the new point pattern the ECDFs are estimated. 5 The contrast function Ct is determined and the new state is

accepted if Ct < Ct−1. Otherwise Ct is rejected and the point pattern is reset to the previous configuration.

6 Step 3-6 are repeated until the maximum number of iterations

10 × Q is achieved, with Q the initial number of trees in A.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

Step 5 Contrast function Ct is determined.

Ct =

5

• k=1

nD

• u
• ˆ

Dk (uζ|t) − ˆ Dk (uζ) 2 +

5

• k=1

nH

• v
• ˆ

Hs,k (vδ|t) − ˆ Hs,k (vδ) 2

whereas:

ˆ Dk (uζ) and ˆ Hs,k (vδ) obtained from the original line transect samples ˆ Dk (uζ|t) and ˆ Hs,k (vδ|t) derived after iteration t of the

• ptimisation procedure

total number of distance classes: nD for ˆ Dk and nH for ˆ Hs,k grouped distance classes: ζ = 10 m and δ = 5 m

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

Step 5 Contrast function Ct is determined.

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 r [m]

D ^

k(r)

500 1000 1500 0.0 0.2 0.4 0.6 0.8 1.0 r [m]

H ^

sk(r)

Figure: ECDFs after the initial step of the algorithm

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

1 The ECDFs for the original tree pattern are estimated. 2 New tree locations are generated to form the initial

configuration together with the original locations.

3 To improve this configuration one tree in the current pattern is

randomly generated or deleted.

4 For the new point pattern the ECDFs are estimated. 5 The contrast function Ct is determined and the new state is

accepted if Ct < Ct−1. Otherwise Ct is rejected and the point pattern is reset to the previous configuration.

6 Step 3-6 are repeated until the maximum number of iterations

10 × Q is achieved, with Q the initial number of trees in A.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

reconstruction routine: an iterative optimisation procedure

Step 6 Step 3-6 are repeated until the maximum number of iterations 10 × Q is achieved.

200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 r [m]

D ^

k(r)

500 1000 1500 0.0 0.2 0.4 0.6 0.8 1.0 r [m]

H ^

sk(r)

Figure: Contrast measure after the maximum number of iterations

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

special features in the reconstruction routine

Special Point 1 Implementing the varying tree density

3463000 3463500 3464000 3464500 3465000 3465500 5425500 5426000 5426500 5427000 5427500 5428000 5428500

• ●
• Figure: Stretched segments cover an

area of 2 ha

• 20

40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 p(adding)

Dj ^

(a)

• 20

40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 p(deletion)

Dj ^

(b)

Figure: Nomination-probability relative to the local point density

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

special features in the reconstruction routine

Special Point 2 Estimating the ECDFs

just to remember...

distance [m] detection function g(x) 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Figure: Detection function

3464535 3464545 3464555 3464565 5426400 5426600 5426800 5427000 5427200

• Figure: Transforming the tree locations to the

transect and measure the “waiting” distances

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

special features in the reconstruction routine

Special Point 3 Thinning the point pattern:

intension of the scan is to provide estimates of the ECDFs similar to those surveyed on the transects

3463000 3463500 3464000 3464500 3465000 3465500 5425500 5426000 5426500 5427000 5427500 5428000 5428500

• Figure: Dense scanning of the study

region

3463000 3463500 3464000 3464500 3465000 3465500 5425500 5426000 5426500 5427000 5427500 5428000 5428500

• Figure: Random sampling with

respect to allocated empirical detection probabilities

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

full mapped 35 ha forest stand of a national park

4390900 4391100 4391300 5661400 5661600 5661800 5662000

national park

• ●
• ●
• ●
• ●
• complete habitat tree pattern

500 resamples for line transect surveys with randomly shifted transect locations distance between lines: 150 m 500 point pattern reconstructions

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

pair correlation function to evaluate the reconstruction routine & to describe the spatial distribution of habitat trees

5 10 15 20 25 30 5 10 15 r[m]

g(r)

Figure: PCF of the 500 reconstructed point patterns in the national park. True PCF and 95%-envelopes from the resampling.

20 40 60 80 2 4 6 8 r[m]

g(r)

Figure: PCF of the 500 reconstructed point patterns in the Natura 2000 area. Mean PCF and 95%-envelopes from the resampling.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

Fitting a Log-Gaussian Cox process model to the reconstructed point patterns

A two-stage random mechanism:

1 non-negative random intensity process

Λ (s) = exp {Z (s)} where {Z (s)} is a Gaussian random field

2 conditional on this an inhomogeneous Poisson process with

intensity function λ (s) is formed λ = exp

• µ + 1

2σ2

• mean

µ; variance σ covariance function c (r) = σ2exp−r/α

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

Simulation results for fitting a LGC process model to the reconstructed point patterns

mean std 2.5% 97.5% N P ˆ σ2 0.9 1.5 0.00000003 7.5 ˆ µ

• 7.6

0.7

• 10.5
• 7.1

λ 0.0008 0.0002 0.0005 0.001 N 2000 ˆ σ2 0.9 0.1 0.7 1.1 ˆ µ

• 8.7

0.1

• 8.8
• 8.6

λ 0.0003 0.000002 0.0002 0.0002

Table: A maximum radius of 50 m was defined to fit the LGC process model.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

Evaluation of the fitted Log-Gaussian Cox process model

5 10 15 20 25 30 2 4 6 8 10 12 14

r[m]

g (r)

mean fit

• riginal

CI fit (0.025, 0.975)

Figure: PCF for the fitted Log-Gaussian Cox process model in the national park.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

References

Illian, J., Penttinen, A., Stoyan, H., & Stoyan, D. (2008): Statistical analysis and modelling of spatial point patterns. John Wiley & Sons. Ripley, B.D. (1987): Stochastic simulation. John Wiley & Sons, Inc. New York, NY, USA. Hastings, W. 1970. Monte Carlo sampling methods using Markov chains and their application. Biometrika, 57, 97-109. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. & Teller, E. 1953. Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087-1092. Pommerening, A., & Stoyan, D. (2008): Reconstructing spatial tree point patterns from nearest neighbour summary statistics measured in small subwindows. Canadian Journal of Forest Research, 38(5), 1110–1122. Tscheschel, A., & Stoyan, D. (2006): Statistical reconstruction of random point patterns. Computational Statistics and Data Analysis, 51(2), 859–871. point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

K-function (KF) to evaluate the reconstruction routine & to describe the spatial distribution of habitat trees

5 10 15 20 25 30 1000 2000 3000 4000 5000 r[m]

K(r)

Figure: KF of the reconstructed point patterns in the national park

10 20 30 40 50 2000 4000 6000 8000 10000 12000 r[m]

K(r) (c)

Figure: KF of the reconstructed point patterns in the Natura 2000 area

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

histogram of the 500 reconstructed point patterns in the Natura 2000 area

trees per hectare

frequency 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 20 40 60 80 100 120 sample reconstruction

(a) total number of trees

frequency 910 920 930 940 950 960 970 20 40 60 80 100 120 140

(b)

920 930 940 950 960 0.0 0.2 0.4 0.6 0.8 1.0

• distr. total number of trees

cumulative frequency

(c)

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

characterisation of the implemented improvements-only algorithm according to Tscheschel & Stoyan, 2006

improvements-only algorithm is closely related to the general birth-and-death algorithm (Ripley, 1987) and the Metropolis-Hastings algorithm (Metropolis, 1953; Hastings, 1970) improvements-only algorithm does not require any assumption

• f the point process

values of the estimated summary characteristics serve as model parameters modified simulated annealing algorithm, only states where Ct < Ct−1 are accepted

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft

data reconstruction results model

example of a point pattern for the whole study region

3463000 3463500 3464000 3464500 3465000 3465500 5425500 5426000 5426500 5427000 5427500 5428000 5428500

• ●
• ●
• Figure: Reconstructed point pattern with varying local habitat tree

density.

point pattern reconstruction based on line transect sampling

• H. Bäuerle, A. Nothdurft