Spatial Modeling of grub density of the may beetle (forest cockchafer: Melolontha hippocastani) in the Hessischen Ried area

Transcription

1 Spatial Modeling of grub density of the may beetle (forest cockchafer: Melolontha hippocastani) in the Hessischen Ried area Matthias Schmidt und Rainer Hurling

2 Motivation Since several decades the forests in the Hessisch Ried area are subject to high abiotic and biotic stress factors. Subsequently in some areas the stands show an extensive decay. Temperature veg = 16,6 C, Precipitation veg = 371 mm, Index of aridity = 38.4 Extensive drawdown of ground water, fragmentation of forests, emission of pollutants (e.g. kerosin by the nearby Frankfurt airport, nitrogen emission) Outbreaks of may beetle, gipsy moth, bark beetle

3 Motivation

4 Motivation Biology: The may beetle lives usually 3 years as grub below ground (including moulting), at the end of summer of the third year it cocoons and the adult beetle hatches out from the ground in May of the 4 years. An extensively mature grub takes place, followed by copulation and death of the males, several egg depositions in the ground and death of the females. There are several tribes in the Rhine valley that follow their own cycle. Spatial prognosis of grub density (E 3 ) to support decision making in practical forestry concerning regeneration planning (planting, harrowing), (airborne) operations of forest protection applying chemical and biological pesticides against the adult beetle. Tests of hypothesis: Effect of distance to ground water table? Effect of soil type? Effect of forest stand type?..on grub density. Quantifying the spatial pattern of grub density. Method: Generalized additive regression models (mgcv, gamlss)

5 Data base: grub sampling design

6 Data base: grub inventory 2009 Grubs E 3 /m² >10 Samples= % Min. 1 st Quartil Median Mean 3 rd Quartil Max. Distance to GWT 2007 [m] Clay thickness [%] Number of samples with massive clay layer = 55

7 Data base: grub inventory 2009 The grub densities are count data Count data usually show certain properties: integer values, positive data range, significant skewness to the right, high proportion of zeros Standard regression models are not appropriate Distribution assumption: Poisson, zero-inflated Poisson, negative binomial

8 Generalized additive models g( i )= f 1 (DWT i )+f 2 (CTH i )+f 3 (east i,north i ) [1] GD i Grub density at sample plot i [n/m²] GD i x i ~ Poisson( i ) with E(y i x i ) = i and Var (GD i x i ) = i ; GD i = 0, 1, 2,. DWT i : Regionalized distance to GWT in October 2007 at sample plot i [m] CTH i : Regionalized clay thickness at sample plot i [%] east i, north i : Easting and northing of sample plot i using the Gauß-Krüger-System referring to the 3. meridian f 1, f 2 : 1-dimensional smoothing functions (penalized thin plate basis regressions splines) f 3 : 2-dimensional smoothing function (penalized thin plate basis regressions spline) Distribution assumptions: Poisson, zero inflated Poisson, negative binomial library(mgcv) Wood library(gamlss) Rigby & Stasinopoulos library(aer) Kleiber and Zeileis

9 Poisson distribution Model AIC Dispersion Parameter g( i )= f 1 (DWT i )+f 2 (CTH i ) [2.0] *** g( i )= f 1 (DWT i )+f 2 (CTH i )+f 3 (east i,north i ), edf for f 3 (east i,north i )= [2.1] *** edf for f 3 (east i,north i )= [2.2] *** zero-inflated Poisson distribution Modell AIC Mixture Parameter g 1 ( i )= f 11 (DWT i )+f 21 (CTH i ) [3.0] g 2 ( i ) = g 1 ( i )= f 11 (DWT i )+f 21 (CTH i )+f 31 (east i,north i ) edf for f 31 (east i,north i ) = [3.1] g 2 ( i )= f 12 (DWT i ) negative binomial distribution Model AIC Dispersion Parameter g 1 ( i )= f 11 (DWT i )+f 21 (CTH i ) [4.0] g 2 1/ ) = g 1 ( i )= f 11 (DWT i )+f 21 (CTH i )+f 31 (east i,north i ), edf for f 3 (east i,north i )= [4.1] g 2 1/ ) = edf for f 3 (east i,north i )= [4.2] g 2 1/ ) = edf for f 3 (east i,north i )= [4.3] g 2 1/ ) = 1.734

10 Validation / quantile residuals Assumption: the observations y 1,,y n are independently distributed with probability density function (pdf) f(y i, i, ). F(y i, i, ) denotes the corresponding (cumulative) distribution function (cdf). the quantile residual r i corresponding to observation y i is defined as: r i =F - F( ; ˆ, ˆ ) y i i with F and F - denoting the distribution and quantile function of the standard normal distribution. r i = ~N(0,1), if f(y i, i, ) is indeed the correct model for the observations Dunn and Smyth, 1996

11 Validation quantile residuals Poisson distribution negativ binomial distribution zipoisson distribution negativ binomial distribution

12 Validation quantile residuals Draw a random sample from the fitted negative binomial model of 10 times the original sample size applying the original predictors and fitting a Poisson and a negative binomial model to the data. negativ binomial distribution Poisson distribution

13 Results model effects

14 Model predictions: Effect of distance to GWT

15 Effect of the spatial trend function

16 Conclusions Generalized additive regression models (~negative binomial distribution) are a well suited approach for modeling count data of may beetle grubs. The most important model effect on grub density results from geographic location. The causal predictors distance to GWT and clay thickness show significant effects on grub density also. If distance to GWT exceeds more than 4 m and clay thickness is less than 40% no effects are observed. For validation purposes quantile residuals should be used. In 2013 the next grub inventory will take place on the same grid and the model will be parameterized again. The potential changes in the spatial pattern of grub density over time and the effects of applied pesticides will be tested. An additional inventory of pest-damages should be used to quantify the relation between E 3 /m³ and corresponding damages.

17 Chemical and biological operations of forest protection both require a good information base! Busch (1865)

18 Simulation der Effekte unterschiedlicher Grundwasserbewirtschaftungspläne auf die Engerlingsdichte

19 Simulation der Effekte unterschiedlicher Grundwasserbewirtschaftungspläne auf die Engerlingsdichte E 3 /m²

20 Model predictions: Effect of clay thickness

21 Poisson distribution g i ) = i = x i = + 1 x i1 + + k x ik or i = exp( i ) = exp(x i ) = exp( ) exp( 1 x i1 ) exp( k x ik ) mit g(.) : Verknüpfungsfunktion: natürlicher Logarithmus und y i x i ~ Poisson( i ) mit E(y i x i ) = i and Var (y i x i ) = i ; y i = 0, 1, 2,. Wahrscheinlichkeitsdichtefunktion der Poissonverteilung: f(y; ) = y e - y=0, 1, 2,.. y!

22 zero inflated Poisson distribution g 1 i ) = i = x i = + 11 x i1 + + k1 x ik mit g 1 (.) : Verknüpfungsfunktion: natürlicher Logarithmus g 2 ( i ) = i = x i = + 12 x i1 + + k2 x ik und g 2 (.) : Verknüpfungsfunktion: logistisch und y i x i ~ ZIP( i, i ) mit E(y i x i ) = (1- ) i und Var(y i x i ) = i (1- ) (1+ i ) [3a] [3b] y i = 0, 1, 2,. Wahrscheinlichkeitsdichtefunktion der Zero-inflated Poissonverteilung: - e -, y = 0 f(y;, ) = y - e y! - y, y = 1,2,.

23 negativ binomial distribution g 1 i ) = i = x i = + 11 x i1 + + k1 x ik mit g 1 (.) : Verknüpfungsfunktion: natürlicher Logarithmus und y i x i ~ Poisson( i ) with i ~ Gamma(shape=, scale=1/ ) = a Gamma(Mittelwert=1, Varianz=1/ ) und y i x i ~ NegBin( i, ) mit E(y i x i )= i und Var(y i x i ) = i i ; y i = 0, 1, 2,. Wahrscheinlichkeitsdichtefunktion der negativ Binomialverteilung: f(y,, ) = ( y) y ( ) y! ( ) y, mit Gammafunktion. y = 0, 1, 2,. g 2 1/ ) = i = x i = + 12 x i1 + + k2 x ik mit g 2 (.) : Verknüpfungdfunktion: natürlicher Logarithmus b