Requirements

All analyses require the R software (v3.4) for data simulation, processing, and summarizing model results; and the Stan software for Hamiltonian Monte Carlo (HMC) simulation.

We begin by installing the tvvarss package (if necessary) and then loading it.

#if (require(devtools)) {
#    devtools::install_github("nwfsc-timeseries/tvvarss")
    library("tvvarss")
#}

Simulating data

The tvvarss package includes the the function simTVVAR() to simulate the process component of a TVVARSS model (i.e., it does not add observation error). The primary input to simTVVAR() is the [n x n] matrix (or [n x n x 1] array) Bt, which specifies the intra- and inter-species interactions. If Bt is a matrix, then it is used to specify the initial conditions for \(\mathbf{B}_t\).

the food web topology. Specifically, interactions are expressed as the effect of column on row; the diagonals indicate the strength of density-dependence. All elements of the matrix corresponding to no interaction are set to 0.

tvvarss is designed to work with symbolic representations within B0, based on the following character codes:

"dd" for density-dependence (this is implied in TVVARSS models)
"td" for top-down
"bu" for bottom-up
"cf" for competitive/facilitative

We show four different examples of simulated food web topologies.

Ex 1: Linear food chain

For the first example, we model 4 tropic levels stacked in a linear food chain from primary producers PP at the bottom to tertiary consumers TC at the top.

lvls <- c("TC","SC","PC","PP")
nn <- length(lvls)
cat(paste0(paste0(lvls[1:(nn-1)],"\n|\n",collapse = ""),lvls[nn]))

## TC
## |
## SC
## |
## PC
## |
## PP

Here is the topology in a matrix form that simTVVAR() will understand.

## initial conditions for B_t
B0_ex1 <- matrix(list(0),nn,nn)
dimnames(B0_ex1) <- list(lvls,lvls)
## diagonal elements = density-dependence
diag(B0_ex1) <- rep("dd",4)
for(i in 1:(nn-1)) {
  B0_ex1[i,i+1] <- "bu"
  B0_ex1[i+1,i] <- "td"
}
## inspect B0
B0_ex1

##    TC   SC   PC   PP  
## TC "dd" "bu" 0    0   
## SC "td" "dd" "bu" 0   
## PC 0    "td" "dd" "bu"
## PP 0    0    "td" "dd"

We can now simulate a TVVAR process based on this expression of the food web topology. In addition to the matrix specifying the topology, the simulator needs to know the length of time series and some information about the variances of the process errors for \(\mathbf{B}_t\) and the states \(\mathbf{x}_t\). For this example, we will assume IID errors.

Here is a 30-unit, simulated TVVAR process.

TT <- 35
## simulate
set.seed(666)
ex1 <- simTVVAR(Bt = NULL, topo = B0_ex1, TT = TT,
                var_QX = seq(4)/20, cov_QX = 0,
                var_QB = 0.10, cov_QB = 0)
## plot states
clr <- c("purple","darkred","blue","darkgreen")
par(mai=c(0.9,0.9,0,0), omi=c(0.1,0.1,0.1,1.5))
matplot(t(ex1$states), type="l", lty="solid", lwd=2, xpd=NA,
        col=clr, ylab="Log density")
legend("right", legend=lvls, lty="solid", lwd=2, bty="n",
       col=clr, inset=-0.2, xpd=NA, cex=0.9)

Now we simulate many TVVAR processes of this form and inspect them.

## number of simulations
ee <- 10
## list for results
ex1 <- vector("list",ee)
for(i in 1:ee) {
  ex1[[i]] <- simTVVAR(Bt = NULL, topo = B0_ex1, TT = TT,
                       var_QX = seq(4)/20, cov_QX = 0,
                       var_QB = 0.10, cov_QB = 0)
  par(mai=c(0.6,0.9,0.3,0), omi=c(0.1,0.1,0.1,1.5))
  matplot(t(ex1[[i]]$states), type="l", lty="solid", lwd=2, xpd=NA,
          col=clr, main=paste0("Simulation ",i), ylab="Log density")
  legend("right", legend=lvls, lty="solid", lwd=2, bty="n",
         col=clr, inset=-0.2, xpd=NA, cex=0.9)
}

Clearly some of the simulated processes must have diagonal values in B are close to 1, which combine with some of the off-diagonal elements to create unrealistic, boom-bust population dynamics. Let’s develop a screening process to toss those out.

## min log-density threshold
dens_min <- -5
## max log-density threshold
dens_max <- 7
ex1 <- vector("list",ee)
for(i in 1:ee) {
  tmp <- list(states=2*rep(dens_max,2))
  while(max(tmp$states) > dens_max | min(tmp$states) < dens_min) {
    tmp <- simTVVAR(Bt = NULL, topo = B0_ex1, TT = TT,
                    var_QX = seq(4)/20, cov_QX = 0,
                    var_QB = 0.10, cov_QB = 0)
  }
  ex1[[i]] <- tmp
  par(mai=c(0.6,0.9,0.3,0), omi=c(0.1,0.1,0.1,1.5))
  matplot(t(ex1[[i]]$states), type="l", lty="solid", lwd=2, xpd=NA,
          col=clr, main=paste0("Simulation ",i), ylab="Log density")
  legend("right", legend=lvls, lty="solid", lwd=2, bty="n",
         col=clr, inset=-0.2, xpd=NA, cex=0.9)
}

Ex 2: Grazers & plants

For the second example, we model 2 tropic levels with 2 different primary producers and 2 different consumers.

lvls <- c("C1","C2","P1","P2")
nn <- length(lvls)
cat(c("C1-C2\n| X |\nP1-P2"))

## C1-C2
## | X |
## P1-P2

Here is the topology in a matrix form that simTVVAR() will understand.

## initial conditions for B_t
B0_ex2 <- matrix(list(0),nn,nn)
dimnames(B0_ex2) <- list(lvls,lvls)
## diagonal elements = density-dependence
diag(B0_ex2) <- rep("dd",4)
B0_ex2[1:2,3:4] <- "bu"
B0_ex2[3:4,1:2] <- "td"
B0_ex2[1,2] <- B0_ex2[2,1] <- B0_ex2[3,4] <- B0_ex2[4,3] <- "cf"
## inspect B0
B0_ex2

##    C1   C2   P1   P2  
## C1 "dd" "cf" "bu" "bu"
## C2 "cf" "dd" "bu" "bu"
## P1 "td" "td" "dd" "cf"
## P2 "td" "td" "cf" "dd"

ee <- 5
## min log-density threshold
dens_min <- -5
## max log-density threshold
dens_max <- 7
ex2 <- vector("list",ee)
clr <- c("darkblue","steelblue","darkgreen","darkolivegreen4")
for(i in 1:ee) {
  tmp <- list(states=2*rep(dens_max,2))
  while(max(tmp$states) > dens_max | min(tmp$states) < dens_min) {
    tmp <- simTVVAR(Bt = NULL, topo = B0_ex2, TT = TT,
                    var_QX = seq(4)/20, cov_QX = 0,
                    var_QB = 0.10, cov_QB = 0)
  }
  ex2[[i]] <- tmp
  par(mai=c(0.6,0.9,0.3,0), omi=c(0.1,0.1,0.1,1.5))
  matplot(t(ex2[[i]]$states), type="l", lty="solid", lwd=2, xpd=NA,
          col=clr, main=paste0("Simulation ",i), ylab="Log density")
  legend("right", legend=lvls, lty="solid", lwd=2, bty="n",
         col=clr, inset=-0.2, xpd=NA, cex=0.9)
}

Introduction to using time varying vector autoregressive models (TVVARSS)

Eric Ward, Mark Scheuerell, Steve Katz

2021-10-04

Requirements

Simulating data

Ex 1: Linear food chain

Ex 2: Grazers & plants