## Script to calculate IPM transition matrices from IBM population vector.
## This is to compare the covariance matrix from the IBM and the IPM approx.
rm(list=ls()) # Clear the workspace
####
#### Load libraries
####
library(communitySynchrony)
library(synchrony)
library(boot)
library(mvtnorm)
library(msm)
library(MASS)
library(statmod)
####
#### Set global parameters
####
sppList=c("ARTR","HECO","POSE","PSSP")
bigM=c(50,75,50,75) # Set matrix dimension for each species
maxSize=c(3000,202,260,225) # In cm^2: PSSP=225 HECO=202 POSE=260 ARTR=3000
A=10000 # Area of 100cm x 100cm quadrat
Nspp=length(sppList)
tlimit=2
constant=T
Nyrs=22
NoOverlap.Inter=F
####
#### Read in regression parameters, subset for Idaho and format
####
do_site <- "Idaho"
Gpars_all <- readRDS("../results/growth_params_list.RDS")[[do_site]]
Spars_all <- readRDS("../results/surv_params_list.RDS")[[do_site]]
Rpars_all <- readRDS("../results/recruit_parameters.RDS")[[do_site]]
spp_list <- sppList <- names(Gpars_all)
Nyrs <- nrow(Gpars_all[[1]])
Nspp <- length(spp_list)
site_path <- paste("../data/", do_site, sep="")
Gpars <- format_growth_params(do_site = do_site, species_list = spp_list,
Nyrs = Nyrs, Gdata_species = Gpars_all)
Spars <- format_survival_params(do_site = do_site, species_list = spp_list,
Nyrs = Nyrs, Sdata_species = Spars_all)
Rpars <- format_recruitment_params(do_site = do_site, species_list = spp_list,
Nyrs = Nyrs, Rdata_species = Rpars_all,
path_to_site_data = site_path)
if(constant==T){
#turn off random year effects
Rpars$intcpt.yr=matrix(Rpars$intcpt.mu,Nyrs,Nspp,byrow=T)
Gpars$intcpt.yr[]=0;Gpars$slope.yr[]=0
Spars$intcpt.yr[]=0;Spars$slope.yr[]=0
}
####
#### Vital rate functions
####
S=function(u,W,Spars,doYear,doSpp){
mu=Spars$intcpt[doSpp]+Spars$intcpt.yr[doYear,doSpp]+
(Spars$slope[doSpp]+Spars$slope.yr[doYear,doSpp])*u+
W%*%(Spars$nb[doSpp,])
return(inv.logit(mu))
}
G=function(v,u,W,Gpars,doYear,doSpp){
mu=Gpars$intcpt[doSpp]+Gpars$intcpt.yr[doYear,doSpp]+(Gpars$slope[doSpp]+Gpars$slope.yr[doYear,doSpp])*u+
W%*%(Gpars$nb[doSpp,])
sigma2=Gpars$sigma2.a[doSpp]*exp(Gpars$sigma2.b[doSpp]*mu)
out=dnorm(v,mu,sqrt(sigma2))
out
}
#number of recruits per area produced
# cover is stored in absolute area (cm^2)
get.rpa=function(Rpars,cover,doYear){
# cover is in m^2 per m^2; convert to % scale:
cover2=cover*100
# calculate recruits
Nspp=length(cover)
mu=rep(NA,Nspp)
for(i in 1:Nspp){
mu[i]=cover2[i]*exp(Rpars$intcpt.yr[doYear,i]+sqrt(cover2)%*%Rpars$dd[i,])
}
if(sum(is.na(mu))>0) browser() # stop for errors
rpa=mu/(cover*A) # convert from number recruits to recruits per cm^2
return(rpa)
}
# Fecundity function, expected number of recruits of size y produced by a size x individual
# The size distribution of recruits is on the log scale
f=function(v,u,Rpars,rpa,doSpp) {
nRecruits = rpa[doSpp]*exp(u)
#probability of producing a seedling of size v
tmp=dnorm(v,Rpars$sizeMean[doSpp],sqrt(Rpars$sizeVar[doSpp]))/(1-pnorm(-1.61,Rpars$sizeMean[doSpp],sqrt(Rpars$sizeVar[doSpp])))
#number recruits of each size
f=nRecruits*tmp
return(f)
}
####
#### Read in IBM population vector list; keep iteration 50 as initial pop vector
####
nt.save <- readRDS("../results/nt_popvecBIG_ibm.RDS")
nt <- list()
nt[[1]] <- as.numeric(nt.save[[2]][2:length(nt.save[[2]])])
nt[[1]][] <- 0
for(i in 1:length(nt.save)) nt[[i+1]] <- as.numeric(nt.save[[i]][2:length(nt.save[[i]])])
####
#### Simulation length, Matrix size and initial vectors
####
v=v.r=b.r=expv=Cr=WmatG=WmatS=list(length(sppList))
h=r.L=r.U=Ctot=numeric(length(sppList))
for(i in 1:Nspp){
# minimum (0.9*minimum size from data) and maximum sizes (1.1*maximum size from data)
L=log(0.2)
U=log(maxSize[i])*1.1
# boundary points b and mesh points y. Note: b chops up the size interval (L-U) into bigM-equal-sized portions.
b = L+c(0:bigM[i])*(U-L)/bigM[i]
# v calculates the middle of each n-equal-sized portion.
v[[i]] = 0.5*(b[1:bigM[i]]+b[2:(bigM[i]+1)])
# step size for midpoint rule. (see equations 4 and 5 in Ellner and Rees (2006) Am Nat.)
h[i] = v[[i]][2]-v[[i]][1]
# variables for Wr approximation
b.r[[i]]=sqrt(exp(b)/pi)
v.r[[i]]=sqrt(exp(v[[i]])/pi)
expv[[i]]=exp(v[[i]])
r.L[i] = sqrt(exp(L)/pi)
r.U[i] = sqrt(exp(U)/pi)
WmatG[[i]]=matrix(NA,length(v.r[[i]]),Nspp) # storage of size-specific W values for each focal species
WmatS[[i]]=matrix(NA,length(v.r[[i]]),Nspp)
} # next species
tmp=range(v.r)
size.range=seq(tmp[1],tmp[2],length=50) # range across all possible sizes
####
#### Utility functions
####
get_pairs <- function(X, pop_vector){
pairs <- expand.grid(X, X)
# pairs$tag <- pairs[,1] - pairs[,2]
pairs$multi <- pairs[,1]*pairs[,2]*pop_vector
return(pairs$multi)
}
get_cov <- function(K){
test <- apply(K, MARGIN = 2, FUN = "get_pairs",
pop_vector=(nt[[doSpp]]))
mat_dim <- sqrt(dim(test)[1])
test <- as.data.frame(test)
test$tag <- rep(c(1:mat_dim), each=mat_dim)
cov_str <- matrix(ncol=mat_dim, nrow=mat_dim)
for(do_i in 1:mat_dim){
tmp <- subset(test, tag==do_i) #subset out the focal i
rmtmp <- which(colnames(tmp)=="tag") #get rid of id column
# Sum over k columns
cov_str[do_i,] <- (-h[doSpp]^2) * apply(tmp[,-rmtmp], MARGIN = 2, FUN = "sum")
}
diag(cov_str) <- 1
return(cov_str)
}
GenerateMultivariatePoisson<-function(pD, samples, R, lambda){
normal_mu=rep(0, pD)
normal = mvrnorm(samples, normal_mu, R)
pois = normal
p=pnorm(normal)
for (s in 1:pD){pois[s]=qpois(p[s], lambda[s])}
return(pois)
}
make.R.values=function(v,u, #state variables
Rpars,rpa,doYear,doSpp){
f(v,u,Rpars,rpa,doSpp)
}
make.P.values <- function(v,u,muWG,muWS, #state variables
Gpars,Spars,doYear,doSpp){ #growth arguments
S(u,muWS,Spars,doYear,doSpp)*G(v,u,muWG,Gpars,doYear,doSpp)
}
make.P.matrix <- function(v,muWG,muWS,Gpars,Spars,doYear,doSpp) {
muWG=expandW(v,v,muWG)
muWS=expandW(v,v,muWS)
P.matrix=outer(v,v,make.P.values,muWG,muWS,Gpars,Spars,doYear,doSpp)
return(h[doSpp]*P.matrix)
}
make.R.matrix=function(v,Rpars,rpa,doYear,doSpp) {
R.matrix=outer(v,v,make.R.values,Rpars,rpa,doYear,doSpp)
return(h[doSpp]*R.matrix)
}
# Function to format the W matrix for the outer product
expandW=function(v,u,W){
if(dim(W)[1]!=length(u)) stop("Check size of W")
Nspp=dim(W)[2]
W=as.vector(W)
W=matrix(W,length(W),ncol=length(v))
W=as.vector(t(W))
W=matrix(W,nrow=length(u)*length(v),ncol=Nspp)
return(W)
}
# Function to calculate size-dependent crowding, assuming no overlap
wrijG=function(r,i,j){
return(2*pi*integrate(function(z) z*exp(-alphaG[i,j]*(z^2))*Cr[[j]](z-r),r,r+r.U[j])$value+
pi*Ctot[j]*exp(-alphaG[i,j]*((r+r.U[j])^2))/alphaG[i,j]);
}
WrijG=Vectorize(wrijG,vectorize.args="r")
wrijS=function(r,i,j){
return(2*pi*integrate(function(z) z*exp(-alphaS[i,j]*(z^2))*Cr[[j]](z-r),r,r+r.U[j])$value+
pi*Ctot[j]*exp(-alphaS[i,j]*((r+r.U[j])^2))/alphaS[i,j]);
}
WrijS=Vectorize(wrijS,vectorize.args="r")
# Function to sum total cover of each species
sumCover=function(v,nt,h,A){
out=lapply(1:Nspp,function(i,v,nt,h,A) h[i]*sum(nt[[i]]*exp(v[[i]]))/A,v=v,nt=nt,h=h,A=A)
return(unlist(out))
}
# Function to sum total density of each species
sumN=function(nt,h){
out=lapply(1:Nspp,function(i,nt,h) h[i]*sum(nt[[i]]),nt=nt,h=h)
return(unlist(out))
}
# Function to calculate size variance of each species
varN=function(v,nt,h,Xbar,N){
out=lapply(1:Nspp,function(i,v,nt,h,Xbar,N) h[i]*sum((exp(v[[i]]-Xbar[i])^2)*nt[[i]])/N[i],v=v,nt=nt,h=h,Xbar=Xbar,N=N)
return(unlist(out))
}
####
#### Calculate the equilibrium areas.
####
## initial population density vector
new.nt=nt
# set up matrix to record cover
covSave = matrix(NA,tlimit,Nspp)
covSave[1,]=sumCover(v,nt,h,A)
# set up list to store size distributions
sizeSave=list(NULL)
for(i in 1:Nspp){
sizeSave[[i]]=matrix(NA,length(v[[i]]),(tlimit))
sizeSave[[i]][,1]=nt[[i]]/sum(nt[[i]])
}
# initial densities
Nsave=matrix(NA,tlimit,Nspp)
Nsave[1,]=sumN(nt,h)
yrSave=rep(NA,tlimit)
####
#### Run simulation
####
covmat <- list()
for (i in 2:(tlimit)){
#draw from observed year effects
allYrs=c(1:Nyrs)
doYear=sample(allYrs,1)
yrSave[i]=doYear
#get recruits per area
cover=covSave[i-1,]; N=Nsave[i-1,]
rpa=get.rpa(Rpars,cover,doYear)
#calculate size-specific crowding
alphaG=Gpars$alpha
alphaS=Spars$alpha
if(NoOverlap.Inter==F){#T: heterospecific genets cannot overlap; F: overlap allowed
for(ii in 1:Nspp){
# first do all overlap W's
Xbar=cover*A/N # multiply by A to get cover back in cm^2
varX=varN(v,nt,h,Xbar,N)
muWG = pi*Xbar*N/(A*alphaG[ii,])
muWS = pi*Xbar*N/(A*alphaS[ii,])
muWG[is.na(muWG)]=0
muWS[is.na(muWS)]=0
WmatG[[ii]]=matrix(muWG,nrow=length(v[[ii]]),ncol=Nspp,byrow=T)
WmatS[[ii]]=matrix(muWS,nrow=length(v[[ii]]),ncol=Nspp,byrow=T)
# now do conspecific no overlap W
Ctot[ii]=h[ii]*sum(expv[[ii]]*nt[[ii]])
Cr[[ii]]=splinefun(b.r[[ii]],h[ii]*c(0,cumsum(expv[[ii]]*nt[[ii]])),method="natural")
WmatG[[ii]][,ii]=WrijG(v.r[[ii]],ii,ii)/A
WmatS[[ii]][,ii]=WrijS(v.r[[ii]],ii,ii)/A
}
}else{
for(ii in 1:Nspp){
Ctot[ii]=h[ii]*sum(expv[[ii]]*nt[[ii]])
Cr[[ii]]=splinefun(b.r[[ii]],h[ii]*c(0,cumsum(expv[[ii]]*nt[[ii]])),method="natural")
}
for(jj in 1:Nspp){
WfunG=splinefun(size.range,WrijG(size.range,jj,jj))
WfunS=splinefun(size.range,WrijS(size.range,jj,jj))
for(ii in 1:Nspp) {
WmatG[[ii]][,jj]=WfunG(v.r[[ii]])/A
WmatS[[ii]][,jj]=WfunS(v.r[[ii]])/A
}
}
} # end NoOverlap if
for(doSpp in 1:Nspp){
if(cover[doSpp]>0){
# make kernels and project
P.matrix <- make.P.matrix(v[[doSpp]],WmatG[[doSpp]],WmatS[[doSpp]],Gpars,Spars,doYear,doSpp)
R.matrix <- make.R.matrix(v[[doSpp]],Rpars,rpa,doYear,doSpp)
newK <- P.matrix+R.matrix
covmat[[doSpp]] <- get_cov(K=P.matrix)
pCont <- GenerateMultivariatePoisson(pD = length(nt[[doSpp]]),
samples = 1,
R = covmat[[doSpp]],
lambda = P.matrix%*%nt[[doSpp]])
rCont <- rpois(length(nt[[doSpp]]),R.matrix%*%nt[[doSpp]])
new.nt[[doSpp]] <- pCont+rCont
# sizeSave[[doSpp]][,i]=new.nt[[doSpp]]/sum(new.nt[[doSpp]])
}
} # next species
# nt=new.nt
# covSave[i,]=sumCover(v,nt,h,A) # store the cover as cm^2/cm^2
# Nsave[i,]=sumN(nt,h)
#
# print(i)
# flush.console()
#
# if(sum(is.na(nt))>0) browser()
} # next time step
####
#### Read in IBM covariance matrix for comparison
####
ibm_covmat <- readRDS("../results/ibm_covmat.RDS")
for(i in 1:3) ibm_covmat[[i]][which(is.na(ibm_covmat[[i]])==TRUE)] <-0
for(i in 1:3) ibm_covmat[[i]] <- ibm_covmat[[i]][2:(nrow(ibm_covmat[[i]])),2:(nrow(ibm_covmat[[i]]))]
# pdf("../results/ipm_ibm_covmatdiffs.pdf")
par(mfrow=c(1,3))
for(i in 1:3){
matdiff <- ibm_covmat[[i]] - covmat[[i+1]]
hist(matdiff, main=sppList[i+1])
}
# library(gplots)
# for(i in 1:3){
# matdiff <- ibm_covmat[[i]] - covmat[[i+1]]
# heatmap.2(matdiff,Rowv=FALSE, Colv=FALSE, dendrogram="none",main=sppList[i+1],
# col=cm.colors(20), tracecol="#303030", trace="none",
# notecol="black", notecex=0.8, keysize = 1.5, margins=c(5, 5))
# }
# dev.off()
par(mfrow=c(1,1))
plot(ibm_covmat[[1]], covmat[[2]], ylim=c(-0.2,0.2), xlim=c(-0.2,0.2), xlab="Truth", ylab="Prediction")
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