‘SongEvo’ package

Introduction

SongEvo simulates the cultural evolution of quantitative traits of bird song. SongEvo is an individual- (agent-) based model. SongEvo is spatially-explicit and can be parameterized with, and tested against, measured song data. Functions are available for model implementation, sensitivity analyses, parameter optimization, model validation, and hypothesis testing.

Overview of Functions

SongEvo implements the model
par.sens allows sensitivity analyses
par.opt allows parameter optimization
mod.val allows model validation
h.test allows hypothesis testing

Getting Started

Load and attach SongEvo package

library(SongEvo)

Functions

SongEvo implements the model par.sens allows sensitivity analyses par.opt allows parameter optimization mod.val allows model validation h.test allows hypothesis testing

Examples

EXAMPLE 1

Load the example data: song.data and global parameters

To explore the SongEvo package, we will use a database of songs from Nuttall’s white-crowned sparrow (Zonotrichia leucophrys nuttalli) recorded at three locations in 1969 and 2005.

data("song.data")

Examine global parameters. Global parameters describe our understanding of the system and may be measured or hypothesized. They are called “global” because they are used by many many functions and subroutines within functions. For descriptions of all adjustable parameters, see ?song.data.

data("glo.parms")
glo.parms$mortality.a.m <- glo.parms$mortality.a.f <- glo.parms$mortality.a
glo.parms$mortality.j.m <- glo.parms$mortality.j.f <- glo.parms$mortality.j
glo.parms$male.fledge.n.mean <- glo.parms$male.fledge.n.mean*2
glo.parms$male.fledge.n.sd <- glo.parms$male.fledge.n.sd*2
glo.parms <- glo.parms[!names(glo.parms) %in% c("mortality.a","mortality.j")]
str(glo.parms)
#> List of 17
#>  $ learning.error.d  : num 0
#>  $ learning.error.sd : num 430
#>  $ n.territories     : num 40
#>  $ lifespan          : num 2.08
#>  $ phys.lim.min      : num 1559
#>  $ phys.lim.max      : num 4364
#>  $ male.fledge.n.mean: num 2.7
#>  $ male.fledge.n.sd  : num 1
#>  $ disp.age          : num 2
#>  $ disp.distance.mean: num 110
#>  $ disp.distance.sd  : num 100
#>  $ terr.turnover     : num 0.5
#>  $ male.fledge.n     : num [1:40] 1 1 2 1 0 2 2 2 2 1 ...
#>  $ mortality.a.f     : num 0.468
#>  $ mortality.a.m     : num 0.468
#>  $ mortality.j.f     : num 0.5
#>  $ mortality.j.m     : num 0.5

Share global parameters with the global environment. We make these parameters available in the global environment so that we can access them with minimal code.

list2env(glo.parms, globalenv())
#> <environment: R_GlobalEnv>

Examine song data

Data include the population name (Bear Valley, PRBO, or Schooner), year of song recording (1969 or 2005), and the frequency bandwidth of the trill.

str(song.data)
#> 'data.frame':    89 obs. of  3 variables:
#>  $ Population: Factor w/ 3 levels "Bear Valley",..: 3 3 3 3 3 3 3 3 3 3 ...
#>  $ Year      : int  1969 1969 1969 1969 1969 1969 1969 1969 1969 1969 ...
#>  $ Trill.FBW : num  3261 2494 2806 2878 2758 ...

Simulate bird song evolution with `SongEvo()`

Define initial individuals

In this example, we use songs from individual birds recorded in one population (PRBO) in the year 1969, which we will call starting.trait.

starting.trait <- subset(song.data, Population=="PRBO" & Year==1969)$Trill.FBW

We want a starting population of 40 individuals, so we generate additional trait values to complement those from the existing 30 individuals. Then we create a data frame that includes a row for each individual; we add identification numbers, ages, and geographical coordinates for each individual.

starting.trait2 <- c(starting.trait, rnorm(n.territories-length(starting.trait), 
    mean=mean(starting.trait), sd=sd(starting.trait)))
init.inds <- data.frame(id = seq(1:n.territories), age = 2, trait = starting.trait2)
init.inds$x1 <-  round(runif(n.territories, min=-122.481858, max=-122.447270), digits=8)
init.inds$y1 <-  round(runif(n.territories, min=37.787768, max=37.805645), digits=8)

Specify and call the SongEvo model

SongEvo() includes several settings, which we specify before running the model. For this example, we run the model for 10 iterations, over 36 years (i.e. 1969–2005). When conducting research with SongEvo(), users will want to increase the number iterations (e.g. to 100 or 1000). Each timestep is one year in this model (i.e. individuals complete all components of the model in 1 year). We specify territory turnover rate here as an example of how to adjust parameter values. We could adjust any other parameter value here also. The learning method specifies that individuals integrate songs heard from adults within the specified integration distance (intigrate.dist, in kilometers). In this example, we do not includ a lifespan, so we assign it NA. In this example, we do not model competition for mates, so specify it as FALSE. Last, specify all as TRUE in order to save data for every single simulated individual because we will use those data later for mapping. If we do not need data for each individual, we set all to FALSE because the all.inds data.frame becomes very large!

iteration <- 10
years <- 36
timestep <- 1
terr.turnover <- 0.5
integrate.dist <- 0.1
lifespan <- NA
mate.comp <- FALSE
prin <- FALSE
all <- TRUE

Now we call SongEvo with our specifications and save it in an object called SongEvo1.

SongEvo1 <- SongEvo(init.inds = init.inds, females = 1.0, iteration = iteration, steps = years,  
    timestep = timestep, n.territories = n.territories, terr.turnover = terr.turnover, 
    integrate.dist = integrate.dist, 
    learning.error.d = learning.error.d, learning.error.sd = learning.error.sd, 
    mortality.a.m = mortality.a.m, mortality.a.f = mortality.a.f,
    mortality.j.m = mortality.j.m, mortality.j.f = mortality.j.f, lifespan = lifespan, 
    phys.lim.min = phys.lim.min, phys.lim.max = phys.lim.max, 
    male.fledge.n.mean = male.fledge.n.mean, male.fledge.n.sd = male.fledge.n.sd, male.fledge.n = male.fledge.n,
    disp.age = disp.age, disp.distance.mean = disp.distance.mean, disp.distance.sd = disp.distance.sd, 
    mate.comp = mate.comp, prin = prin, all = TRUE)

Examine results from SongEvo model

The model required the following time to run on your computer:

SongEvo1$time
#>    user  system elapsed 
#>  21.756   0.023  21.781

Three main objects hold data regarding the SongEvo model. Additional objects are used temporarily within modules of the model.

First, currently alive individuals are stored in a data frame called “inds.” Values within “inds” are updated throughout each of the iterations of the model, and “inds” can be viewed after the model is completed.

head(SongEvo1$inds, min(5,nrow(SongEvo1$inds)))
#>                 coordinates   id age    trait        x1       y1
#> M1482  (-122.4754, 37.7904) 1482   8 2090.363 -122.4754 37.79040
#> M1511 (-122.4868, 37.79245) 1511   8 3174.851 -122.4868 37.79245
#> M1535 (-122.4515, 37.79301) 1535   7 3341.994 -122.4515 37.79301
#> M1547 (-122.4768, 37.79017) 1547   7 2880.297 -122.4768 37.79017
#> M1597  (-122.4811, 37.7908) 1597   6 2999.243 -122.4811 37.79080
#>       male.fledglings female.fledglings territory father sex fitness learn.dir
#> M1482               0                 0         0   1200   M       1         0
#> M1511               0                 0         0   1459   M       1         0
#> M1535               0                 0         0   1399   M       1         0
#> M1547               0                 0         0   1450   M       1         0
#> M1597               0                 0         0   1452   M       1         0
#>              x0       y0
#> M1482 -122.4775 37.79099
#> M1511 -122.4869 37.79170
#> M1535 -122.4524 37.79182
#> M1547 -122.4770 37.79012
#> M1597 -122.4786 37.79028

Second, an array (i.e. a multi-dimensional table) entitled “summary.results” includes population summary values for each time step (dimension 1) in each iteration (dimension 2) of the model. Population summary values are contained in five additional dimensions: population size for each time step of each iteration (“sample.n”), the population mean and variance of the song feature studied (“trait.pop.mean” and “trait.pop.variance”), with associated lower (“lci”) and upper (“uci”) confidence intervals.

dimnames(SongEvo1$summary.results)
#> $iteration
#>  [1] "iteration 1"  "iteration 2"  "iteration 3"  "iteration 4"  "iteration 5" 
#>  [6] "iteration 6"  "iteration 7"  "iteration 8"  "iteration 9"  "iteration 10"
#> 
#> $step
#>  [1] "1"  "2"  "3"  "4"  "5"  "6"  "7"  "8"  "9"  "10" "11" "12" "13" "14" "15"
#> [16] "16" "17" "18" "19" "20" "21" "22" "23" "24" "25" "26" "27" "28" "29" "30"
#> [31] "31" "32" "33" "34" "35" "36"
#> 
#> $feature
#> [1] "sample.n"           "trait.pop.mean"     "trait.pop.variance"
#> [4] "lci"                "uci"

Third, individual values may optionally be concatenated and saved to one data frame entitled “all.inds.” all.inds can become quite large, and is therefore only recommended if additional data analyses are desired.

head(SongEvo1$all.inds,  min(5,nrow(SongEvo1$all.inds)))
#>                   coordinates id age  trait        x1       y1 male.fledglings
#> I1.T1.1 (-122.4799, 37.79532)  1   2 4004.8 -122.4799 37.79532               1
#> I1.T1.2 (-122.4488, 37.79137)  2   2 3765.0 -122.4488 37.79137               1
#> I1.T1.3  (-122.4622, 37.7906)  3   2 3237.4 -122.4622 37.79060               2
#> I1.T1.4 (-122.4741, 37.80505)  4   2 3621.1 -122.4741 37.80505               0
#> I1.T1.5 (-122.4543, 37.79999)  5   2 3285.4 -122.4543 37.79999               0
#>         female.fledglings territory father sex fitness learn.dir x0 y0 timestep
#> I1.T1.1                 0         1      0   M       1         0  0  0        1
#> I1.T1.2                 0         1      0   M       1         0  0  0        1
#> I1.T1.3                 0         1      0   M       1         0  0  0        1
#> I1.T1.4                 1         1      0   M       1         0  0  0        1
#> I1.T1.5                 0         1      0   M       1         0  0  0        1
#>         iteration
#> I1.T1.1         1
#> I1.T1.2         1
#> I1.T1.3         1
#> I1.T1.4         1
#> I1.T1.5         1

Simulated population size

We see that the simulated population size remains relatively stable over the course of 36 years. This code uses the summary.results array.

plot(SongEvo1$summary.results[1, , "sample.n"], xlab="Year", ylab="Abundance", type="n", 
    xaxt="n", ylim=c(0, max(SongEvo1$summary.results[, , "sample.n"], na.rm=TRUE)))
axis(side=1, at=seq(0, 40, by=5), labels=seq(1970, 2010, by=5))
    for(p in 1:iteration){
        lines(SongEvo1$summary.results[p, , "sample.n"], col="light gray")
        }
n.mean <- apply(SongEvo1$summary.results[, , "sample.n"], 2, mean, na.rm=TRUE)
lines(n.mean, col="red")

#Plot 95% quantiles
quant.means <- apply (SongEvo1$summary.results[, , "sample.n"], MARGIN=2, quantile, 
    probs=c(0.975, 0.025), R=600, na.rm=TRUE)
lines(quant.means[1,], col="red", lty=2)
lines(quant.means[2,], col="red", lty=2)

Load Hmisc package for plotting functions.

library("Hmisc")

Simulated trait values

We see that the mean trait values per iteration varied widely, though mean trait values over all iterations remained relatively stable. This code uses the summary.results array.

plot(SongEvo1$summary.results[1, , "trait.pop.mean"], xlab="Year", ylab="Bandwidth (Hz)", 
    xaxt="n", type="n", xlim=c(-0.5, 36), 
    ylim=c(min(SongEvo1$summary.results[, , "trait.pop.mean"], na.rm=TRUE), 
    max(SongEvo1$summary.results[, , "trait.pop.mean"], na.rm=TRUE)))
    for(p in 1:iteration){
        lines(SongEvo1$summary.results[p, , "trait.pop.mean"], col="light gray")
        }
freq.mean <- apply(SongEvo1$summary.results[, , "trait.pop.mean"], 2, mean, na.rm=TRUE)
lines(freq.mean, col="blue")
axis(side=1, at=seq(0, 35, by=5), labels=seq(1970, 2005, by=5))#, tcl=-0.25, mgp=c(2,0.5,0))

#Plot 95% quantiles
quant.means <- apply (SongEvo1$summary.results[, , "trait.pop.mean"], MARGIN=2, quantile, 
    probs=c(0.95, 0.05), R=600, na.rm=TRUE)
lines(quant.means[1,], col="blue", lty=2)
lines(quant.means[2,], col="blue", lty=2)

#plot mean and CI for historic songs.  
 #plot original song values
library("boot")
sample.mean <- function(d, x) {
    mean(d[x])
}
boot_hist <- boot(starting.trait, statistic=sample.mean, R=100)#, strata=mn.res$iteration)  
ci.hist <- boot.ci(boot_hist, conf=0.95, type="basic")
low <- ci.hist$basic[4]
high <- ci.hist$basic[5]
points(0, mean(starting.trait), pch=20, cex=0.6, col="black")
errbar(x=0, y=mean(starting.trait), high, low, add=TRUE)
 #text and arrows
text(x=5, y=2720, labels="Historical songs", pos=1)
arrows(x0=5, y0=2750, x1=0.4, y1=mean(starting.trait), length=0.1)

Trait variance

We see that variance for each iteration per year increased in the first few years and then stabilized. This code uses the summary.results array.

 #plot variance for each iteration per year
plot(SongEvo1$summary.results[1, , "trait.pop.variance"], xlab="Year", 
    ylab="Bandwidth Variance (Hz)", type="n", xaxt="n", 
    ylim=c(min(SongEvo1$summary.results[, , "trait.pop.variance"], na.rm=TRUE), 
    max(SongEvo1$summary.results[, , "trait.pop.variance"], na.rm=TRUE)))
axis(side=1, at=seq(0, 40, by=5), labels=seq(1970, 2010, by=5))
    for(p in 1:iteration){
        lines(SongEvo1$summary.results[p, , "trait.pop.variance"], col="light gray")
        }
n.mean <- apply(SongEvo1$summary.results[, , "trait.pop.variance"], 2, mean, na.rm=TRUE)
lines(n.mean, col="green")

#Plot 95% quantiles
quant.means <- apply (SongEvo1$summary.results[, , "trait.pop.variance"], MARGIN=2, quantile, 
    probs=c(0.975, 0.025), R=600, na.rm=TRUE)
lines(quant.means[1,], col="green", lty=2)
lines(quant.means[2,], col="green", lty=2)

Maps

The simulation results include geographical coordinates and are in a standard spatial data format, thus allowing calculation of a wide variety of spatial statistics.

Load packages for making maps.

library("sp")
library("reshape2")
library("lattice")

Convert data frame from long to wide format. This is necessary for making a multi-panel plot.

all.inds1 <- subset(SongEvo1$all.inds, SongEvo1$all.inds$iteration==1)
w <- dcast(as.data.frame(all.inds1), id ~ timestep, value.var="trait", fill=0)
all.inds1w <- merge(all.inds1, w, by="id")
years.SongEvo1 <- (dim(w)[2]-1 )
names(all.inds1w@data)[-(1:length(all.inds1@data))] <-paste("Ts", 1:(dim(w)[2]-1 ), sep="")

Create a function to generate a continuous color palette–we will use the palette in the next call to make color ramp to represent the trait value.

rbPal <- colorRampPalette(c('blue','red')) #Create a function to generate a continuous color palette

Plot maps, including a separate panel for each timestep (each of 36 years). Our example shows that individuals move across the landscape and that regional dialects evolve and move. The x-axis is longitude, the y-axis is latitude, and the color ramp indicates trill bandwidth in Hz.

spplot(all.inds1w[,-c(1:ncol(all.inds1))], as.table=TRUE, 
    cuts=c(0, seq(from=1500, to=4500, by=10)), ylab="", 
    col.regions=c("transparent", rbPal(1000)), 
    #cuts specifies that the first level (e.g. <1500) is transparent.
colorkey=list(
    right=list(
          fun=draw.colorkey,
          args=list( 
                key=list(
                at=seq(1500, 4500, 10),
                col=rbPal(1000),
                labels=list(
                at=c(1500, 2000, 2500, 3000, 3500, 4000, 4500),
                labels=c("1500", "2000", "2500", "3000", "3500", "4000", "4500")
                )
                )
                )
            )
    )
)

In addition, you can plot simpler multi-panel maps that do not take advantage of the spatial data class.

 #Lattice plot (not as a spatial frame)
it1 <- subset(SongEvo1$all.inds, iteration==1)
rbPal <- colorRampPalette(c('blue','red')) #Create a function to generate a continuous color palette
it1$Col <- rbPal(10)[as.numeric(cut(it1$trait, breaks = 10))]
xyplot(it1$y1~it1$x1 | it1$timestep, groups=it1$trait, asp="iso", col=it1$Col, 
    xlab="Longitude", ylab="Latitude")

Test model sensitivity with `par.sens()`

This function allows testing the sensitivity of SongEvo to different parameter values.

Specify and call `par.sens()`

Here we test the sensitivity of the Acquire a Territory submodel to variation in territory turnover rates, ranging from 0.8–1.2 times the published rate (40–60% of territories turned over). The call for the par.sens function has a format similar to SongEvo. The user specifies the parameter to test and the range of values for that parameter. The function currently allows examination of only one parameter at a time and requires at least two iterations.

parm <- "terr.turnover"
par.range = seq(from=0.4, to=0.6, by=0.025)
sens.results <- NULL

Now we call the par.sens function with our specifications.

extra_parms <- list(init.inds = init.inds, 
                    females = 1,  # New in SongEvo v2
                    timestep = 1, 
                    n.territories = nrow(init.inds),
                    integrate.dist = 0.1,
                    lifespan = NA, 
                    terr.turnover = 0.5, 
                    mate.comp = FALSE, 
                    prin = FALSE,
                    all = TRUE,
                    # New in SongEvo v2
                    selectivity = 3,
                    content.bias = FALSE,
                    n.content.bias.loc = "all",
                    content.bias.loc = FALSE,
                    content.bias.loc.ranges = FALSE,
                    affected.traits = FALSE,
                    conformity.bias = FALSE,
                    prestige.bias=FALSE,
                    learn.m="default",
                    learn.f="default",
                    learning.error.d=0,
                    learning.error.sd=200)
global_parms_key <- which(!names(glo.parms) %in% names(extra_parms))
extra_parms[names(glo.parms[global_parms_key])]=glo.parms[global_parms_key]
par.sens1 <- par.sens(parm = parm, par.range = par.range, 
                      iteration = iteration, steps = years, mate.comp = FALSE, 
                      fixed_parms=extra_parms[names(extra_parms)!=parm], all = TRUE)
#> [1] "terr.turnover =  0.4"
#> [1] "terr.turnover =  0.425"
#> [1] "terr.turnover =  0.45"
#> [1] "terr.turnover =  0.475"
#> [1] "terr.turnover =  0.5"
#> [1] "terr.turnover =  0.525"
#> [1] "terr.turnover =  0.55"
#> [1] "terr.turnover =  0.575"
#> [1] "terr.turnover =  0.6"

Examine par.sens results

Examine results objects, which include two arrays:

The first array, sens.results, contains the SongEvo model results for each parameter. It has the following dimensions:

dimnames(par.sens1$sens.results)
#> [[1]]
#>  [1] "iteration 1"  "iteration 2"  "iteration 3"  "iteration 4"  "iteration 5" 
#>  [6] "iteration 6"  "iteration 7"  "iteration 8"  "iteration 9"  "iteration 10"
#> 
#> [[2]]
#>  [1] "1"  "2"  "3"  "4"  "5"  "6"  "7"  "8"  "9"  "10" "11" "12" "13" "14" "15"
#> [16] "16" "17" "18" "19" "20" "21" "22" "23" "24" "25" "26" "27" "28" "29" "30"
#> [31] "31" "32" "33" "34" "35" "36"
#> 
#> [[3]]
#> [1] "sample.n"           "trait.pop.mean"     "trait.pop.variance"
#> [4] "lci"                "uci"               
#> 
#> [[4]]
#> [1] "par.val 0.4"   "par.val 0.425" "par.val 0.45"  "par.val 0.475"
#> [5] "par.val 0.5"   "par.val 0.525" "par.val 0.55"  "par.val 0.575"
#> [9] "par.val 0.6"

The second array, sens.results.diff contains the quantile range of trait values across iterations within a parameter value. It has the following dimensions:

dimnames(par.sens1$sens.results.diff)
#> [[1]]
#> [1] "par.val 0.4"   "par.val 0.425" "par.val 0.45"  "par.val 0.475"
#> [5] "par.val 0.5"   "par.val 0.525" "par.val 0.55"  "par.val 0.575"
#> [9] "par.val 0.6"  
#> 
#> [[2]]
#>  [1] "Quantile diff 1"  "Quantile diff 2"  "Quantile diff 3"  "Quantile diff 4" 
#>  [5] "Quantile diff 5"  "Quantile diff 6"  "Quantile diff 7"  "Quantile diff 8" 
#>  [9] "Quantile diff 9"  "Quantile diff 10" "Quantile diff 11" "Quantile diff 12"
#> [13] "Quantile diff 13" "Quantile diff 14" "Quantile diff 15" "Quantile diff 16"
#> [17] "Quantile diff 17" "Quantile diff 18" "Quantile diff 19" "Quantile diff 20"
#> [21] "Quantile diff 21" "Quantile diff 22" "Quantile diff 23" "Quantile diff 24"
#> [25] "Quantile diff 25" "Quantile diff 26" "Quantile diff 27" "Quantile diff 28"
#> [29] "Quantile diff 29" "Quantile diff 30" "Quantile diff 31" "Quantile diff 32"
#> [33] "Quantile diff 33" "Quantile diff 34" "Quantile diff 35" "Quantile diff 36"

To assess sensitivity of SongEvo to a range of parameter values, plot the range in trait quantiles per year by the parameter value. We see that territory turnover values of 0.4–0.6 provided means and quantile ranges of trill bandwidths that are similar to those obtained with the published estimate of 0.5, indicating that the Acquire a Territory submodel is robust to realistic variation in those parameter values.

In the figure, solid gray and black lines show the quantile range of song frequency per year over all iterations as parameterized with the published territory turnover rate (0.5; thick black line) and a range of values from 0.4 to 0.6 (in steps of 0.05, light to dark gray). Orange lines show the mean and 2.5th and 97.5th quantiles of all quantile ranges.

 #plot of range in trait quantiles by year by parameter value
plot(1:years, par.sens1$sens.results.diff[1,], ylim=c(min(par.sens1$sens.results.diff, 
    na.rm=TRUE), max(par.sens1$sens.results.diff, na.rm=TRUE)), type="l", 
    ylab="Quantile range (Hz)", xlab="Year", col="transparent", xaxt="n")
axis(side=1, at=seq(0, 35, by=5), labels=seq(1970, 2005, by=5))

  #Make a continuous color ramp from gray to black
grbkPal <- colorRampPalette(c('gray','black'))
  
  #Plot a line for each parameter value
for(i in 1:length(par.range)){
lines(1:years, par.sens1$sens.results.diff[i,], type="l", 
    col=grbkPal(length(par.range))[i])
}

  #Plot values from published parameter values
lines(1:years, par.sens1$sens.results.diff[2,], type="l", col="black", lwd=4)

  #Calculate and plot mean and quantiles
quant.mean <- apply(par.sens1$sens.results.diff, 2, mean, na.rm=TRUE)
lines(quant.mean, col="orange")

#Plot 95% quantiles (which are similar to credible intervals)
  #95% quantiles of population means (narrower)
quant.means <- apply (par.sens1$sens.results.diff, MARGIN=2, quantile, 
    probs=c(0.975, 0.025), R=600, na.rm=TRUE)
lines(quant.means[1,], col="orange", lty=2)
lines(quant.means[2,], col="orange", lty=2)

Optimize parameter values with `par.opt()`

This function follows par.sens to help users optimize values for imperfectly known parameters for SongEvo. The goals are to maximize accuracy and precision of model prediction. Accuracy is quantified by three different approaches: i) the mean of absolute residuals of the predicted population mean values in relation to target data (e.g. observed or hypothetical values (smaller absolute residuals indicate a more accurate model)), ii) the difference between the bootstrapped mean of predicted population means and the mean of the target data, and iii) the proportion of simulated population trait means that fall within (i.e. are “contained by”) the confidence intervals of the target data (a higher proportion indicates greater accuracy). Precision is measured with the residuals of the predicted population variance to the variance of target data (smaller residuals indicate a more precise model).

Prepare current song values

target.data <- subset(song.data, Population=="PRBO" & Year==2005)$Trill.FBW

Specify and call `par.opt()`

Users specify the timestep (“ts”) at which to compare simulated trait values to target trait data (“target.data”) and save the results in an object (called par.opt1 here).

ts <- years
par.opt1 <- par.opt(sens.results=par.sens1$sens.results, ts=ts, 
    target.data=target.data, par.range=par.range)

Examine results objects (residuals and target match).

par.opt1$Residuals
#> , , Residuals of mean
#> 
#>               Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5
#> par.val 0.4     337.67248    42.14224    618.3192   295.42507   296.43127
#> par.val 0.425    14.88322    64.83654    157.5851   220.03256   505.43352
#> par.val 0.45     98.19680   426.78861    130.5791    37.26350    91.81459
#> par.val 0.475   480.51629   151.77558    329.3065   172.13522   173.00116
#> par.val 0.5      38.88184   235.34810    174.1565   476.72616   371.41632
#> par.val 0.525   246.14716    85.87224     33.6524   176.26693   438.59028
#> par.val 0.55    205.79261   171.11830    386.3405   305.85898   350.03035
#> par.val 0.575   166.16689   237.88605    123.9382    12.36809    32.29153
#> par.val 0.6     324.83059   240.07137    211.8161   233.80812   262.50084
#>               Iteration 6 Iteration 7 Iteration 8 Iteration 9 Iteration 10
#> par.val 0.4     136.33191   434.04084   363.43588    337.6249     103.5518
#> par.val 0.425   401.41800   190.58312   365.07502    105.6916     259.4278
#> par.val 0.45     94.77140   394.94548   237.53560    393.6234     130.3846
#> par.val 0.475   184.07631   411.56752   205.27598    284.1895     279.7190
#> par.val 0.5      71.92619    61.22911    89.23354    257.2702     288.6791
#> par.val 0.525   107.86898   246.29488   514.83296    150.7413     348.6759
#> par.val 0.55    105.74324   282.10859   286.41502    105.3514     230.7253
#> par.val 0.575   353.74372   332.41011   316.94364    134.5768     204.4446
#> par.val 0.6     523.07773   265.56457   242.19849    309.3421     471.0858
#> 
#> , , Residuals of variance
#> 
#>               Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5
#> par.val 0.4    15713.1977   13839.242   11639.100   2908.3894   21015.808
#> par.val 0.425  12241.3845   13840.693   10582.837   7101.4268    1225.971
#> par.val 0.45   10123.3088    5596.804   20580.338   3074.3697   11074.651
#> par.val 0.475   3697.0456   16446.050    1924.619    572.1751    2866.875
#> par.val 0.5      746.2376   19626.442    6360.826  16660.5471     241.952
#> par.val 0.525  20974.3797   23365.129    3747.057  18754.1502   11432.909
#> par.val 0.55    7211.3114    9319.934   12130.865  10441.0221   18145.817
#> par.val 0.575  15631.7062   18728.480    8317.070  10932.7747    7733.102
#> par.val 0.6     1187.0508   12701.042    2140.042  11966.6007    4146.222
#>               Iteration 6 Iteration 7 Iteration 8 Iteration 9 Iteration 10
#> par.val 0.4     12436.211   20752.072   4328.3979   14323.847     3715.786
#> par.val 0.425   24174.412   25259.289   5496.4018    5140.107     4913.227
#> par.val 0.45    21283.045    9472.799    231.4417    8591.068     6376.929
#> par.val 0.475   17411.252    5637.427  14698.5215    5431.506     6768.395
#> par.val 0.5      3234.145   20065.146  19035.1251    5962.384    10092.814
#> par.val 0.525   21616.796   12381.724  13612.8361   12271.126     2340.125
#> par.val 0.55     2492.179   13210.131    102.5839   12249.199    26210.343
#> par.val 0.575    8169.606    4321.632   1697.6651    7776.691    19964.172
#> par.val 0.6      1881.575    6018.208   2966.9014    7593.890     8458.041
par.opt1$Target.match
#>               Difference in means Proportion contained
#> par.val 0.4              296.4976                  0.1
#> par.val 0.425            228.4967                  0.2
#> par.val 0.45             165.5880                  0.1
#> par.val 0.475            267.1563                  0.0
#> par.val 0.5              198.7103                  0.3
#> par.val 0.525            228.1638                  0.2
#> par.val 0.55             242.9484                  0.0
#> par.val 0.575            189.0033                  0.2
#> par.val 0.6              308.4296                  0.0

Plot results of `par.opt()`

Accuracy of `par.opt()`

Difference in means.

plot(par.range, par.opt1$Target.match[,1], type="l", xlab="Parameter range", 
    ylab="Difference in means (Hz)")

Plot proportion contained.

plot(par.range, par.opt1$Prop.contained, type="l", xlab="Parameter range", 
    ylab="Proportion contained")

Calculate and plot mean and quantiles of residuals of mean trait values.

res.mean.means <- apply(par.opt1$Residuals[, , 1], MARGIN=1, mean, na.rm=TRUE)
res.mean.quants <- apply (par.opt1$Residuals[, , 1], MARGIN=1, quantile, 
    probs=c(0.975, 0.025), R=600, na.rm=TRUE)
plot(par.range, res.mean.means, col="orange", ylim=c(min(par.opt1$Residuals[,,1], 
    na.rm=TRUE), max(par.opt1$Residuals[,,1], na.rm=TRUE)), type="b", 
    xlab="Parameter value (territory turnover rate)", 
    ylab="Residual of trait mean (trill bandwidth, Hz)")
points(par.range, res.mean.quants[1,], col="orange")
points(par.range, res.mean.quants[2,], col="orange")
lines(par.range, res.mean.quants[1,], col="orange", lty=2)
lines(par.range, res.mean.quants[2,], col="orange", lty=2)

Precision of `par.opt()`

#Calculate and plot mean and quantiles of residuals of variance of trait values
res.var.mean <- apply(par.opt1$Residuals[, , 2], MARGIN=1, mean, na.rm=TRUE)
res.var.quants <- apply (par.opt1$Residuals[, , 2], MARGIN=1, quantile, 
    probs=c(0.975, 0.025), R=600, na.rm=TRUE)
plot(par.range, res.var.mean, col="purple", 
    ylim=c(min(par.opt1$Residuals[,,2], na.rm=TRUE), 
    max(par.opt1$Residuals[,,2], na.rm=TRUE)), type="b", 
    xlab="Parameter value (territory turnover rate)", 
    ylab="Residual of trait variance (trill bandwidth, Hz)")
points(par.range, res.var.quants[1,], col="purple")
points(par.range, res.var.quants[2,], col="purple")
lines(par.range, res.var.quants[1,], col="purple", lty=2)
lines(par.range, res.var.quants[2,], col="purple", lty=2)

Visual inspection of accuracy and precision of `par.opt()`: plot trait values for range of parameters

par(mfcol=c(3,2),
    mar=c(2.1, 2.1, 0.1, 0.1),
    cex=0.8)
for(i in 1:length(par.range)){
plot(par.sens1$sens.results[ , , "trait.pop.mean", ], xlab="Year", ylab="Bandwidth (Hz)", 
    xaxt="n", type="n", xlim=c(-0.5, years), 
    ylim=c(min(par.sens1$sens.results[ , , "trait.pop.mean", ], na.rm=TRUE), 
    max(par.sens1$sens.results[ , , "trait.pop.mean", ], na.rm=TRUE)))
    for(p in 1:iteration){
        lines(par.sens1$sens.results[p, , "trait.pop.mean", i], col="light gray")
        }
freq.mean <- apply(par.sens1$sens.results[, , "trait.pop.mean", i], 2, mean, na.rm=TRUE)
lines(freq.mean, col="blue")
axis(side=1, at=seq(0, 35, by=5), labels=seq(1970, 2005, by=5))

#Plot 95% quantiles
quant.means <- apply (par.sens1$sens.results[, , "trait.pop.mean", i], MARGIN=2, quantile, 
    probs=c(0.95, 0.05), R=600, na.rm=TRUE)
lines(quant.means[1,], col="blue", lty=2)
lines(quant.means[2,], col="blue", lty=2)

#plot mean and CI for historic songs.  
 #plot original song values
library("boot")
sample.mean <- function(d, x) {
    mean(d[x])
}
boot_hist <- boot(starting.trait, statistic=sample.mean, R=100)#, strata=mn.res$iteration)  
ci.hist <- boot.ci(boot_hist, conf=0.95, type="basic")
low <- ci.hist$basic[4]
high <- ci.hist$basic[5]
points(0, mean(starting.trait), pch=20, cex=0.6, col="black")
library("Hmisc")
errbar(x=0, y=mean(starting.trait), high, low, add=TRUE)
 
 #plot current song values
library("boot")
sample.mean <- function(d, x) {
    mean(d[x])
}
boot_curr <- boot(target.data, statistic=sample.mean, R=100)#, strata=mn.res$iteration) 
ci.curr <- boot.ci(boot_curr, conf=0.95, type="basic")
low <- ci.curr$basic[4]
high <- ci.curr$basic[5]
points(years, mean(target.data), pch=20, cex=0.6, col="black")
library("Hmisc")
errbar(x=years, y=mean(target.data), high, low, add=TRUE)

  #plot panel title
text(x=3, y=max(par.sens1$sens.results[ , , "trait.pop.mean", ], na.rm=TRUE)-100, 
    labels=paste("Par = ", par.range[i], sep=""))  
}

Model validation with `mod.val()`

This function allows users to assess the validity of the specified model by testing model performance with a population different from the population used to build the model. The user first runs SongEvo with initial trait values from the validation population. mod.val() uses the summary.results array from SongEvo, along with target values from a specified timestep, to calculate the same three measures of accuracy and one measure of precision that are calculated in par.opt.

We parameterized SongEvo with initial song data from Schooner Bay, CA in 1969, and then compared simulated data to target (i.e. observed) data in 2005.

Prepare initial song data for Schooner Bay.

starting.trait <- subset(song.data, Population=="Schooner" & Year==1969)$Trill.FBW
starting.trait2 <- c(starting.trait, rnorm(n.territories-length(starting.trait), 
    mean=mean(starting.trait), sd=sd(starting.trait)))

init.inds <- data.frame(id = seq(1:n.territories), age = 2, trait = starting.trait2)
init.inds$x1 <-  round(runif(n.territories, min=-122.481858, max=-122.447270), digits=8)
init.inds$y1 <-  round(runif(n.territories, min=37.787768, max=37.805645), digits=8)

Specify and call SongEvo() with validation data

iteration <- 10
years <- 36
timestep <- 1
terr.turnover <- 0.5

SongEvo2 <- SongEvo(init.inds = init.inds, females = 1.0, iteration = iteration, steps = years,  
    timestep = timestep, n.territories = n.territories, terr.turnover = terr.turnover, 
    integrate.dist = integrate.dist, 
    learning.error.d = learning.error.d, learning.error.sd = learning.error.sd, 
    mortality.a.m = mortality.a.m, mortality.a.f = mortality.a.f,
    mortality.j.m = mortality.j.m, mortality.j.f = mortality.j.f, lifespan = lifespan, 
    phys.lim.min = phys.lim.min, phys.lim.max = phys.lim.max, 
    male.fledge.n.mean = male.fledge.n.mean, male.fledge.n.sd = male.fledge.n.sd, male.fledge.n = male.fledge.n,
    disp.age = disp.age, disp.distance.mean = disp.distance.mean, disp.distance.sd = disp.distance.sd, 
    mate.comp = mate.comp, prin = prin, all = TRUE)

Specify and call mod.val()

ts <- 36
target.data <- subset(song.data, Population=="Schooner" & Year==2005)$Trill.FBW
mod.val1 <- mod.val(summary.results=SongEvo2$summary.results, ts=ts, 
    target.data=target.data)

Plot results from mod.val()

plot(SongEvo2$summary.results[1, , "trait.pop.mean"], xlab="Year", ylab="Bandwidth (Hz)", 
    xaxt="n", type="n", xlim=c(-0.5, 36.5), 
    ylim=c(min(SongEvo2$summary.results[, , "trait.pop.mean"], na.rm=TRUE), 
    max(SongEvo2$summary.results[, , "trait.pop.mean"], na.rm=TRUE)))
    for(p in 1:iteration){
        lines(SongEvo2$summary.results[p, , "trait.pop.mean"], col="light gray")
        }
freq.mean <- apply(SongEvo2$summary.results[, , "trait.pop.mean"], 2, mean, na.rm=TRUE)
lines(freq.mean, col="blue")
axis(side=1, at=seq(0, 35, by=5), labels=seq(1970, 2005, by=5))

#Plot 95% quantiles 
quant.means <- apply (SongEvo2$summary.results[, , "trait.pop.mean"], MARGIN=2, quantile, 
    probs=c(0.95, 0.05), R=600, na.rm=TRUE)
lines(quant.means[1,], col="blue", lty=2)
lines(quant.means[2,], col="blue", lty=2)

#plot mean and CI for historic songs.  
 #plot original song values
library("boot")
sample.mean <- function(d, x) {
    mean(d[x])
}
boot_hist <- boot(starting.trait, statistic=sample.mean, R=100)
ci.hist <- boot.ci(boot_hist, conf=0.95, type="basic")
low <- ci.hist$basic[4]
high <- ci.hist$basic[5]
points(0, mean(starting.trait), pch=20, cex=0.6, col="black")
library("Hmisc")
errbar(x=0, y=mean(starting.trait), high, low, add=TRUE)

 #text and arrows
text(x=5, y=2720, labels="Historical songs", pos=1)
arrows(x0=5, y0=2750, x1=0.4, y1=mean(starting.trait), length=0.1)

 #plot current song values
library("boot")
sample.mean <- function(d, x) {
    mean(d[x])
}
boot_curr <- boot(target.data, statistic=sample.mean, R=100)
ci.curr <- boot.ci(boot_curr, conf=0.95, type="basic")
low <- ci.curr$basic[4]
high <- ci.curr$basic[5]
points(years, mean(target.data), pch=20, cex=0.6, col="black")
library("Hmisc")
errbar(x=years, y=mean(target.data), high, low, add=TRUE)

 #text and arrows
text(x=25, y=3100, labels="Current songs", pos=3)
arrows(x0=25, y0=3300, x1=36, y1=mean(target.data), length=0.1)

The model did reasonably well predicting trait evolution in the validation population, suggesting that it is valid for our purposes: the mean bandwidth was abs(mean(target.data)-freq.mean)Hz from the observed values, ~21% of predicted population means fell within the 95% confidence intervals of the observed data, and residuals of means (~545 Hz) and variances (~415181 Hz) were similar to those produced by the training data set.

Hypothesis testing with `h.test()`

This function allows hypothesis testing with SongEvo. To test if measured songs from two time points evolved through mechanisms described in the model (e.g. drift or selection), users initialize the model with historical data, parameterize the model based on their understanding of the mechanisms, and test if subsequently observed or predicted data match the simulated data. The output data list includes two measures of accuracy: the proportion of observed points that fall within the confidence intervals of the simulated data and the residuals between simulated and observed population trait means. Precision is measured as the residuals between simulated and observed population trait variances. We tested the hypothesis that songs of Z. l. nuttalli in Bear Valley, CA evolved through cultural drift from 1969 to 2005.

Prepare initial song data for Bear Valley.

starting.trait <- subset(song.data, Population=="Bear Valley" & Year==1969)$Trill.FBW
starting.trait2 <- c(starting.trait, rnorm(n.territories-length(starting.trait), 
    mean=mean(starting.trait), sd=sd(starting.trait)))

init.inds <- data.frame(id = seq(1:n.territories), age = 2, trait = starting.trait2)
init.inds$x1 <-  round(runif(n.territories, min=-122.481858, max=-122.447270), digits=8)
init.inds$y1 <-  round(runif(n.territories, min=37.787768, max=37.805645), digits=8)

Specify and call SongEvo() with test data

SongEvo3 <- SongEvo(init.inds = init.inds, females = 1.0, iteration = iteration, steps = years,  
    timestep = timestep, n.territories = n.territories, terr.turnover = terr.turnover, 
    integrate.dist = integrate.dist, 
    learning.error.d = learning.error.d, learning.error.sd = learning.error.sd, 
    mortality.a.m = mortality.a.m, mortality.a.f = mortality.a.f,
    mortality.j.m = mortality.j.m, mortality.j.f = mortality.j.f, lifespan = lifespan, 
    phys.lim.min = phys.lim.min, phys.lim.max = phys.lim.max, 
    male.fledge.n.mean = male.fledge.n.mean, male.fledge.n.sd = male.fledge.n.sd, male.fledge.n = male.fledge.n,
    disp.age = disp.age, disp.distance.mean = disp.distance.mean, disp.distance.sd = disp.distance.sd, 
    mate.comp = mate.comp, prin = prin, all = TRUE)

Specify and call h.test()

target.data <- subset(song.data, Population=="Bear Valley" & Year==2005)$Trill.FBW
h.test1 <- h.test(summary.results=SongEvo3$summary.results, ts=ts, 
    target.data=target.data)

The output data list includes two measures of accuracy: the proportion of observed points that fall within the confidence intervals of the simulated data and the residuals between simulated and observed population trait means. Precision is measured as the residuals between simulated and observed population trait variances.

Eighty percent of the observed data fell within the central 95% of the simulated values, providing support for the hypothesis that cultural drift as described in this model is sufficient to describe the evolution of trill frequency bandwidth in this population.

h.test1
#> $Residuals
#>              Residuals of mean Residuals of variance
#> Iteration 1           236.5796            133441.628
#> Iteration 2            94.7541             19757.392
#> Iteration 3           284.1701             13530.493
#> Iteration 4           157.2309              7559.532
#> Iteration 5           940.7100             59896.916
#> Iteration 6           295.9355             20705.733
#> Iteration 7           107.0326             35204.140
#> Iteration 8           358.8188             64719.731
#> Iteration 9           109.4810              8627.788
#> Iteration 10          465.8790             57366.316
#> 
#> $Prop.contained
#> [1] 0.4

We can plot simulated data in relation to measured data.

#Plot
plot(SongEvo3$summary.results[1, , "trait.pop.mean"], xlab="Year", ylab="Bandwidth (Hz)", 
    xaxt="n", type="n", xlim=c(-0.5, 35.5), 
    ylim=c(min(SongEvo3$summary.results[, , "trait.pop.mean"], na.rm=TRUE), 
    max(SongEvo3$summary.results[, , "trait.pop.mean"], na.rm=TRUE)))
    for(p in 1:iteration){
        lines(SongEvo3$summary.results[p, , "trait.pop.mean"], col="light gray")
        }
freq.mean <- apply(SongEvo3$summary.results[, , "trait.pop.mean"], 2, mean, na.rm=TRUE)
lines(freq.mean, col="blue")
axis(side=1, at=seq(0, 35, by=5), labels=seq(1970, 2005, by=5))#, tcl=-0.25, mgp=c(2,0.5,0))

#Plot 95% quantiles (which are similar to credible intervals)
quant.means <- apply (SongEvo3$summary.results[, , "trait.pop.mean"], MARGIN=2, quantile, 
    probs=c(0.95, 0.05), R=600, na.rm=TRUE)
lines(quant.means[1,], col="blue", lty=2)
lines(quant.means[2,], col="blue", lty=2)

 #plot original song values
library("boot")
sample.mean <- function(d, x) {
    mean(d[x])
}
boot_hist <- boot(starting.trait, statistic=sample.mean, R=100)#, strata=mn.res$iteration)  
ci.hist <- boot.ci(boot_hist, conf=0.95, type="basic")
low <- ci.hist$basic[4]
high <- ci.hist$basic[5]
points(0, mean(starting.trait), pch=20, cex=0.6, col="black")
library("Hmisc")
errbar(x=0, y=mean(starting.trait), high, low, add=TRUE)

 #plot current song values
points(rep(ts, length(target.data)), target.data)

library("boot")
sample.mean <- function(d, x) {
    mean(d[x])
}
boot_curr <- boot(target.data, statistic=sample.mean, R=100)#, strata=mn.res$iteration) 
ci.curr <- boot.ci(boot_curr, conf=0.95, type="basic")
low <- ci.curr$basic[4]
high <- ci.curr$basic[5]
points(years, mean(target.data), pch=20, cex=0.6, col="black")
library("Hmisc")
errbar(x=years, y=mean(target.data), high, low, add=TRUE)

 #text and arrows
text(x=11, y=2850, labels="Historical songs", pos=1)
arrows(x0=5, y0=2750, x1=0.4, y1=mean(starting.trait), length=0.1)
text(x=25, y=2900, labels="Current songs", pos=1)
arrows(x0=25, y0=2920, x1=years, y1=mean(target.data), length=0.1)