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#' Fit Matching-Centrality Model
#'
#' Fit a model that contains both a trait-matching and a centrality term based on Rohr et al. (2016)
#'
#' @param list Network List
#' @param N_runs Number of different start points for k2 and lambda to try. The best (maximum likelihood)
#' half will be used to construct the probability matrix
#' @param maxit Default = 10'000
#' @param method Passed to optim, default = 'Nelder-Mead'
#' @param ExtraSettings Other control settings to pass to optim()
#'
#' @return Network list with added 'B_par',the best fitting parameters, 'M_ProbsMatrix', the probability matrix
#'
#'
#'@references Rohr, R.P., Naisbit, R.E., Mazza, C. & Bersier, L.-F. (2016). Matching-centrality
#' decomposition and the forecasting of new links in networks. Proc. R. Soc. B Biol. Sci., 283, 20152702
#' @export
#'
FitBothMandC <- function(list,N_runs=10, maxit = 10000, method='Nelder-Mead', ExtraSettings=NULL){
A <- list$obs>0
N1 <- dim(A)[1]
N2 <- dim(A)[2]
N= N1+N2
CCA<- vegan::cca(A)
StartPointM<- c(scale(CCA$rowsum), scale(CCA$colsum))
ListOfFits<-map(1:N_runs, Optimiser, A=A, N_p=N, fixedSt_P=StartPointM, N_unif_P=2, func= Both_LogLikFunc, maxit=maxit, method=method, ExtraSettings=ExtraSettings)
# Best_B<- PerfectBest(Fits_B)
Best_Fits<-which(rank(map_dbl(ListOfFits, 'value')) >= max(1,N_runs/2) ) # Find Best
ProbMatrix<- matrix(0,dim(A)[1], dim(A)[2] )
for( i in Best_Fits){
p = ListOfFits[[i]]$par
best_m1V = rep(p[1:N1],N2)
best_m2V = rep(p[(N1+1):N],each=N1)
best_c1V = rep(p[(N+1):(N+N1)],N2)
best_c2V = rep(p[(N+N1+1):(N+N)],each=N1)
best_lambda = abs(p[N+N+2])
best_kb = p[N+N+1]
Both_Probs = boot::inv.logit(-best_lambda*((best_m1V - best_m2V)^2) + best_c1V + best_c2V+ best_kb)
Both_ProbsMatrix = matrix(signif(Both_Probs,6), ncol=N2)
ProbMatrix <-ProbMatrix+Both_ProbsMatrix
}
list$A<-A
list$N1<- N1
list$N2<- N2
list$N <- N
Best_i<-which.min(map_dbl(ListOfFits, 'value'))
Best_Pars <- ListOfFits[[Best_i]]$p
Best_Pars[N+N+2] = abs(Best_Pars[N+N+2])# Make lambda absolute
names(Best_Pars) <- c(paste0('M1_', 1:N1),
paste0('M2_', 1:N2),
paste0('C1_', 1:N1),
paste0('C2_', 1:N2),
'B_k2', 'B_lam' )
list$B_par<- Best_Pars
list$B_ProbsMat<- ProbMatrix
return(list)
}
Both_LogLikFunc <- function(p,A){
N1 <- dim(A)[1]
N2 <- dim(A)[2]
N= N1+N2
m1V = rep(p[1:N1],N2)
m2V = rep(p[(N1+1):N],each=N1)
c1V = rep(p[(N+1):(N+N1)],N2)
c2V = rep(p[(N+N1+1):(N+N)],each=N1)
kb = p[N+N+1] # intercept
lambda = abs(p[N+N+2])# lambda to scale diffs
Probs = boot::inv.logit(-lambda*((m1V - m2V)^2) + c1V + c2V+ kb)
a_ij_V<-as.vector(A)
## Prior trait distribution pulls towards middle:
## Cauchy distribution with scale =2
TraitPriors <- dcauchy(c(m1V,m2V),scale = 2, log=TRUE)
LogLiks = a_ij_V*log(Probs) + (1- a_ij_V)*(log(1-Probs) )
NegLogLik = -sum(LogLiks) - sum(TraitPriors)
return(NegLogLik)
}
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