diff --git a/measles/measles.R b/measles/measles.R index bde0fb0..b64314b 100644 --- a/measles/measles.R +++ b/measles/measles.R @@ -182,11 +182,11 @@ library(doParallel) registerDoParallel() set.seed(998468235L,kind="L'Ecuyer") -mcopts <- list(preschedule=FALSE,set.seed=TRUE) foreach(i=1:4, .packages="pomp", - .options.multicore=mcopts) %dopar% { + .options.multicore=list(set.seed=TRUE) + ) %dopar% { pfilter(m1,Np=10000,params=theta) } -> pfs logmeanexp(sapply(pfs,logLik),se=TRUE) diff --git a/measles/measles.Rmd b/measles/measles.Rmd index fc51f7a..06399a4 100644 --- a/measles/measles.Rmd +++ b/measles/measles.Rmd @@ -996,11 +996,11 @@ library(doParallel) registerDoParallel() set.seed(998468235L,kind="L'Ecuyer") -mcopts <- list(preschedule=FALSE,set.seed=TRUE) foreach(i=1:4, .packages="pomp", - .options.multicore=mcopts) %dopar% { + .options.multicore=list(set.seed=TRUE) + ) %dopar% { pfilter(m1,Np=10000,params=theta) } -> pfs logmeanexp(sapply(pfs,logLik),se=TRUE) diff --git a/measles/measles.html b/measles/measles.html index c7386f4..c552224 100644 --- a/measles/measles.html +++ b/measles/measles.html @@ -825,11 +825,11 @@

Constructing the pomp object

registerDoParallel() set.seed(998468235L,kind="L'Ecuyer") -mcopts <- list(preschedule=FALSE,set.seed=TRUE) foreach(i=1:4, .packages="pomp", - .options.multicore=mcopts) %dopar% { + .options.multicore=list(set.seed=TRUE) + ) %dopar% { pfilter(m1,Np=10000,params=theta) } -> pfs logmeanexp(sapply(pfs,logLik),se=TRUE) diff --git a/mif/mif.R b/mif/mif.R index 7a1472a..0e11d9a 100644 --- a/mif/mif.R +++ b/mif/mif.R @@ -1,6 +1,7 @@ library(pomp) options(stringsAsFactors=FALSE) -stopifnot(packageVersion("pomp")>="1.4") +stopifnot(packageVersion("pomp")>="1.4.7") +set.seed(557976883) bsflu_data <- read.table("http://kingaa.github.io/short-course/stochsim/bsflu_data.txt") statenames <- c("S","I","R1","R2") paramnames <- c("Beta","mu_I","rho","mu_R1","mu_R2") @@ -67,8 +68,6 @@ library(foreach) library(doParallel) registerDoParallel() - -set.seed(2036049659,kind="L'Ecuyer") stew(file="pf.rda",{ t_pf <- system.time( pf <- foreach(i=1:10,.packages='pomp', diff --git a/mif/mif.Rmd b/mif/mif.Rmd index 0c5d52f..1561236 100644 --- a/mif/mif.Rmd +++ b/mif/mif.Rmd @@ -51,7 +51,8 @@ options(keep.source=TRUE,encoding="UTF-8") ```{r prelims,include=FALSE,purl=TRUE,cache=FALSE} library(pomp) options(stringsAsFactors=FALSE) -stopifnot(packageVersion("pomp")>="1.4") +stopifnot(packageVersion("pomp")>="1.4.7") +set.seed(557976883) ``` ---------------------------- @@ -383,7 +384,7 @@ Let's treat $\mu_{R_1}$ and $\mu_{R_2}$ as known, and fix these parameters at t We will estimate $\beta$, $\mu_I$, and $\rho$. It will be helpful to parallelize most of the computations. -Most machines nowadays have multiple cores and using this computational capacity involves only +Most machines nowadays have multiple cores and using this computational capacity involves only: i. the following lines of code to let R know you plan to use multiple processors; i. the parallel for loop provided by the **foreach** package; and @@ -394,14 +395,11 @@ library(foreach) library(doParallel) registerDoParallel() - -set.seed(2036049659,kind="L'Ecuyer") ``` The first two lines load the **foreach** and **doParallel** packages, the latter being a "backend" for the **foreach** package. The next line tells **foreach** that we will use the **doParallel** backend. By default, **R** will guess how many concurrent **R** processes to run on this single multicore machine. -The final line sets up a parallel random number generator and fixes its seed. ### Running a particle filter. diff --git a/mif/mif.html b/mif/mif.html index 14800dd..7c5da38 100644 --- a/mif/mif.html +++ b/mif/mif.html @@ -430,11 +430,11 @@

Testing the codes.

geom_line()+ facet_grid(variable~.)+ theme_classic() -

+

Before engaging in iterated filtering, it is a good idea to check that the basic particle filter is working since iterated filtering builds on this technique. The simulations above check the rprocess and rmeasure codes; the particle filter depends on the rprocess and dmeasure codes and so is a check of the latter.

pf <- pfilter(bsflu,params=params,Np=1000)
plot(pf)
-

+

These plots show the data along with the effective sample size of the particle filter (ess) and the log likelihood of each observation conditional on the preceding ones (cond.logLik).

@@ -444,7 +444,7 @@

Setting up the estimation problem.

##     mu_R1     mu_R2 
 ## 0.3324675 0.5541126

We will estimate \(\beta\), \(\mu_I\), and \(\rho\).

-

It will be helpful to parallelize most of the computations. Most machines nowadays have multiple cores and using this computational capacity involves only

+

It will be helpful to parallelize most of the computations. Most machines nowadays have multiple cores and using this computational capacity involves only:

  1. the following lines of code to let R know you plan to use multiple processors;
  2. the parallel for loop provided by the foreach package; and
  3. @@ -453,10 +453,8 @@

    Setting up the estimation problem.

    library(foreach)
     library(doParallel)
     
    -registerDoParallel()
    -
    -set.seed(2036049659,kind="L'Ecuyer")
    -

    The first two lines load the foreach and doParallel packages, the latter being a “backend” for the foreach package. The next line tells foreach that we will use the doParallel backend. By default, R will guess how many concurrent R processes to run on this single multicore machine. The final line sets up a parallel random number generator and fixes its seed.

    +registerDoParallel() +

    The first two lines load the foreach and doParallel packages, the latter being a “backend” for the foreach package. The next line tells foreach that we will use the doParallel backend. By default, R will guess how many concurrent R processes to run on this single multicore machine.

Running a particle filter.

diff --git a/parest/parest.R b/parest/parest.R index 76951e9..071a788 100644 --- a/parest/parest.R +++ b/parest/parest.R @@ -1,3 +1,6 @@ +library(pomp) +stopifnot(packageVersion("pomp")>"1.4.5") +set.seed(1173489184) f <- seq(0,1,length=100) R0 <- -log(1-f)/f plot(f~R0,type='l',xlab=expression(R[0]),ylab="fraction infected",bty='l') @@ -161,64 +164,8 @@ plot(I~time,data=out,type='l',log='y') plot(R~time,data=out,type='l',log='y') plot(I~S,data=out,log='xy',pch='.',cex=0.5) par(op) -rain <- read.csv(paste0(baseurl,"data/dacca_rainfall.csv")) -rain$time <- with(rain,year+(month-1)/12) - -plot(rainfall~time,data=rain,type='l') - -interpol <- with(rain,approxfun(time,rainfall,rule=2,method='constant')) - -data.frame(time=seq(from=1920,to=1930,by=1/365)) -> smoothed.rain -smoothed.rain$rainfall <- sapply(smoothed.rain$time,interpol) - -plot(rainfall~time,data=rain,col='black',cex=2,pch=16,log='') -lines(rainfall~time,data=smoothed.rain,col='red') - - -rain.sir.model <- function (t, x, params) { - a <- params["a"] - b <- params["b"] - mu <- params["mu"] - gamma <- params["gamma"] - R <- interpol(t) - beta <- a*R/(b+R) - dSdt <- mu*(1-x[1])-beta*x[1]*x[2] - dIdt <- beta*x[1]*x[2]-(mu+gamma)*x[2] - dRdt <- gamma*x[2]-mu*x[3] - list(c(dSdt,dIdt,dRdt)) -} - -params <- c(a=500,b=50,mu=1/50,gamma=365/13) -xstart <- c(S=0.07,I=0.00039,R=0.92961) -times <- seq(from=1920,to=1930,by=7/365) -out <- as.data.frame( - ode( - func=rain.sir.model, - y=xstart, - times=times, - parms=params - ) - ) - -op <- par(fig=c(0,1,0,1),mfrow=c(2,2), - mar=c(3,3,1,1),mgp=c(2,1,0)) -plot(S~time,data=out,type='l',log='y') -plot(I~time,data=out,type='l',log='y') -plot(R~time,data=out,type='l',log='y') -plot(I~S,data=out,log='xy',pch='.',cex=1) -par(op) - - -fit <- loess(log1p(rainfall)~time,data=rain,span=0.05) - -data.frame(time=seq(from=1920,to=1930,by=1/365)) -> smoothed.rain -smoothed.rain$rainfall <- expm1(predict(fit,newdata=smoothed.rain)) - -plot(rainfall~time,data=rain,col='black',cex=2,pch=16,log='') -lines(rainfall~time,data=smoothed.rain,col='red') - -url <- paste0(baseurl,"data/niamey.csv") -niamey <- read.csv(url,comment.char="#") +# niamey <- read.csv("http://kingaa.github.io/parest/niamey.csv",comment.char="#") +niamey <- read.csv("niamey.csv",comment.char="#") plot(measles~biweek,data=niamey,type='n') lines(measles~biweek,data=subset(niamey,community=="A"),col=1) lines(measles~biweek,data=subset(niamey,community=="B"),col=2) @@ -286,7 +233,7 @@ f <- function (beta, X.0) { grid <- expand.grid(beta=seq(from=100,to=300,length=50), X.0=seq(from=4000,to=20000,length=50)) grid$SSE <- with(grid,mapply(f,beta,X.0)) -require(lattice) +library(lattice) contourplot(sqrt(SSE)~beta+X.0,data=grid,cuts=30) ## ?optim dat <- subset(niamey,community=="A") diff --git a/pfilter/pfilter.R b/pfilter/pfilter.R index 3f1e0d2..79879c1 100644 --- a/pfilter/pfilter.R +++ b/pfilter/pfilter.R @@ -6,7 +6,7 @@ library(grid) library(plyr) library(reshape2) options(stringsAsFactors=FALSE) -set.seed(594709947L) +set.seed(594709947) base_url <- "http://kingaa.github.io/short-course/" url <- paste0(base_url,"stochsim/bsflu_data.txt") bsflu <- read.table(url) diff --git a/pfilter/pfilter.Rmd b/pfilter/pfilter.Rmd index 2c29ecd..8081865 100644 --- a/pfilter/pfilter.Rmd +++ b/pfilter/pfilter.Rmd @@ -69,7 +69,7 @@ library(grid) library(plyr) library(reshape2) options(stringsAsFactors=FALSE) -set.seed(594709947L) +set.seed(594709947) ``` ## Objectives diff --git a/pfilter/pfilter.html b/pfilter/pfilter.html index c4006e0..d3ebb01 100644 --- a/pfilter/pfilter.html +++ b/pfilter/pfilter.html @@ -326,7 +326,7 @@

The graph of the likelihood function: The likelihood surface

theta } -> p

Note that we’ve used the foreach package with the parallel backend (doParallel) to parallelize these computations. To ensure that we have high-quality random numbers in each parallel R session, we use a parallel random number generator (kind="L'Ecuyer", .options.multicore=list(set.seed=TRUE)).

-

+


Exercise: Likelihood slice

diff --git a/polio/global_search.rda b/polio/global_search.rda index af6e2fb..0d2707f 100644 Binary files a/polio/global_search.rda and b/polio/global_search.rda differ diff --git a/polio/local_search.rda b/polio/local_search.rda index 9dd7103..520a39e 100644 Binary files a/polio/local_search.rda and b/polio/local_search.rda differ diff --git a/polio/polio.R b/polio/polio.R index f4a8a29..4f73e49 100644 --- a/polio/polio.R +++ b/polio/polio.R @@ -189,6 +189,7 @@ pf1 %>% ## ----local_search-------------------------------------------------------- stew(file="local_search.rda",{ + w1 <- getDoParWorkers() t1 <- system.time({ m1 <- foreach(i=1:90, .packages='pomp', .combine=rbind, @@ -250,6 +251,7 @@ polio_box <- rbind( ## ----global_search------------------------------------------------------- stew(file="global_search.rda",{ + w2 <- getDoParWorkers() t2 <- system.time({ m2 <- foreach(i=1:400,.packages='pomp',.combine=rbind, .options.multicore=list(set.seed=TRUE) @@ -310,37 +312,37 @@ library(reshape2) library(magrittr) bake(file="profile_rho.rds",{ - params %>% - subset(loglik>max(loglik)-20, - select=-c(loglik,loglik.se,rho)) %>% - melt(id=NULL) %>% - daply(~variable,function(x)range(x$value)) -> box - - starts <- profileDesign(rho=seq(0.01,0.025,length=30), - lower=box[,1],upper=box[,2], - nprof=10) - - foreach(start=iter(starts,"row"), - .combine=rbind, - .options.multicore=list(set.seed=TRUE), - .options.mpi=list(seed=290860873,chunkSize=1) - ) %dopar% { - mf <- mif2(polio, - start=unlist(start), - Np=2000, - Nmif=300, - cooling.type="geometric", - cooling.fraction.50=0.5, - transform=TRUE, - rw.sd=rw.sd( - b1=0.02, b2=0.02, b3=0.02, b4=0.02, b5=0.02, b6=0.02, - psi=0.02, tau=0.02, sigma_dem=0.02, sigma_env=0.02, - IO_0=ivp(0.2), SO_0=ivp(0.2) - )) - mf <- mif2(mf,Np=5000,Nmif=100,cooling.fraction.50=0.1) - ll <- logmeanexp(replicate(10,logLik(pfilter(mf,Np=5000))),se=TRUE) - data.frame(as.list(coef(mf)),loglik=ll[1],loglik.se=ll[2]) - } + params %>% + subset(loglik>max(loglik)-20, + select=-c(loglik,loglik.se,rho)) %>% + melt(id=NULL) %>% + daply(~variable,function(x)range(x$value)) -> box + + starts <- profileDesign(rho=seq(0.01,0.025,length=30), + lower=box[,1],upper=box[,2], + nprof=10) + + foreach(start=iter(starts,"row"), + .combine=rbind, + .options.multicore=list(set.seed=TRUE), + .options.mpi=list(seed=290860873,chunkSize=1) + ) %dopar% { + mf <- mif2(polio, + start=unlist(start), + Np=2000, + Nmif=300, + cooling.type="geometric", + cooling.fraction.50=0.5, + transform=TRUE, + rw.sd=rw.sd( + b1=0.02, b2=0.02, b3=0.02, b4=0.02, b5=0.02, b6=0.02, + psi=0.02, tau=0.02, sigma_dem=0.02, sigma_env=0.02, + IO_0=ivp(0.2), SO_0=ivp(0.2) + )) + mf <- mif2(mf,Np=5000,Nmif=100,cooling.fraction.50=0.1) + ll <- logmeanexp(replicate(10,logLik(pfilter(mf,Np=5000))),se=TRUE) + data.frame(as.list(coef(mf)),loglik=ll[1],loglik.se=ll[2]) + } }) -> m3 ## ----save_profile_rho---------------------------------------------------- @@ -375,7 +377,7 @@ params %>% bake(file="sims.rds",seed=398906785, simulate(polio,nsim=2000,as.data.frame=TRUE,include.data=TRUE) - ) -> sims +) -> sims ddply(sims,~sim,summarize,zeros=sum(cases==0)) -> num_zeros num_zeros %>% @@ -413,7 +415,7 @@ print(pl2,vp=viewport(x=0.8,y=0.75,height=0.4,width=0.2)) sims %>% subset(sim!="data") %>% summarize(imports=coef(polio,"psi")*mean(SO+SB1+SB2+SB3+SB4+SB5+SB6)/12 - ) -> imports + ) -> imports ## ----check_class_m3,eval=F----------------------------------------------- ## class(m3) diff --git a/polio/polio.Rmd b/polio/polio.Rmd index ecb150f..a3d01ac 100644 --- a/polio/polio.Rmd +++ b/polio/polio.Rmd @@ -64,7 +64,7 @@ set.seed(5996485L) 1. Show how partially observed Markov process (POMP) methods can be used to understand transmission dynamics of polio. 2. More broadly, discuss the use of POMP methods for compartmental models for biological systems having age structure, seasonality and other covariates. 3. Get some practice maximizing the likelihood for such models. How does one set up a _global_ search for a maximum likelihood estimate (MLE)? - How does one test whether such a search has been successful? +How does one test whether such a search has been successful? ## Introduction @@ -402,6 +402,7 @@ We use the iterated filtering algorithm IF2 of @ionides15, which uses a random w We set a constant random walk standard deviation for each of the regular parameters and a larger constant for each of the initial value parameters. ```{r local_search} stew(file="local_search.rda",{ + w1 <- getDoParWorkers() t1 <- system.time({ m1 <- foreach(i=1:90, .packages='pomp', .combine=rbind, @@ -427,7 +428,7 @@ stew(file="local_search.rda",{ },seed=318817883,kind="L'Ecuyer") ``` -This investigation took `r round(t1["elapsed"]/60,1)` minutes on a machine with `r getDoParWorkers()` cores. +This investigation took `r round(t1["elapsed"]/60,1)` minutes on a machine with `r w1` cores. The maximum likelihood we obtain is `r with(subset(m1,loglik==max(loglik)),round(loglik,1))`. These repeated stochastic maximizations can also show us the geometry of the likelihood surface in a neighborhood of this point estimate: ```{r pairs_local,fig.width=6,fig.height=6} @@ -505,6 +506,7 @@ We then carry out a search identical to the local one except for the starting pa ```{r global_search} stew(file="global_search.rda",{ + w2 <- getDoParWorkers() t2 <- system.time({ m2 <- foreach(i=1:400,.packages='pomp',.combine=rbind, .options.multicore=list(set.seed=TRUE) @@ -533,7 +535,7 @@ write.csv(params,file="polio_params.csv",row.names=FALSE,na="") ``` This search gives a maximized likelihood estimate of `r with(m2,round(max(loglik),1))` with a standard error of `r with(subset(m2,loglik==max(loglik)),round(loglik.se,2))`. -It took about `r round(t2[3]/60)` mins on a `r getDoParWorkers()`-core machine. +It took about `r round(t2[3]/60)` mins on a `r w2`-core machine. This compares with the results of @Martinez-Bakker2015, who report a maximized likelihood of -794.3 with a standard error of 0.11. Plotting these diverse parameter estimates can help to give a feel for the global geometry of the likelihood surface. @@ -596,37 +598,37 @@ library(reshape2) library(magrittr) bake(file="profile_rho.rds",{ - params %>% - subset(loglik>max(loglik)-20, - select=-c(loglik,loglik.se,rho)) %>% - melt(id=NULL) %>% - daply(~variable,function(x)range(x$value)) -> box - - starts <- profileDesign(rho=seq(0.01,0.025,length=30), - lower=box[,1],upper=box[,2], - nprof=10) - - foreach(start=iter(starts,"row"), - .combine=rbind, - .options.multicore=list(set.seed=TRUE), - .options.mpi=list(seed=290860873,chunkSize=1) - ) %dopar% { - mf <- mif2(polio, - start=unlist(start), - Np=2000, - Nmif=300, - cooling.type="geometric", - cooling.fraction.50=0.5, - transform=TRUE, - rw.sd=rw.sd( - b1=0.02, b2=0.02, b3=0.02, b4=0.02, b5=0.02, b6=0.02, - psi=0.02, tau=0.02, sigma_dem=0.02, sigma_env=0.02, - IO_0=ivp(0.2), SO_0=ivp(0.2) - )) - mf <- mif2(mf,Np=5000,Nmif=100,cooling.fraction.50=0.1) - ll <- logmeanexp(replicate(10,logLik(pfilter(mf,Np=5000))),se=TRUE) - data.frame(as.list(coef(mf)),loglik=ll[1],loglik.se=ll[2]) - } + params %>% + subset(loglik>max(loglik)-20, + select=-c(loglik,loglik.se,rho)) %>% + melt(id=NULL) %>% + daply(~variable,function(x)range(x$value)) -> box + + starts <- profileDesign(rho=seq(0.01,0.025,length=30), + lower=box[,1],upper=box[,2], + nprof=10) + + foreach(start=iter(starts,"row"), + .combine=rbind, + .options.multicore=list(set.seed=TRUE), + .options.mpi=list(seed=290860873,chunkSize=1) + ) %dopar% { + mf <- mif2(polio, + start=unlist(start), + Np=2000, + Nmif=300, + cooling.type="geometric", + cooling.fraction.50=0.5, + transform=TRUE, + rw.sd=rw.sd( + b1=0.02, b2=0.02, b3=0.02, b4=0.02, b5=0.02, b6=0.02, + psi=0.02, tau=0.02, sigma_dem=0.02, sigma_env=0.02, + IO_0=ivp(0.2), SO_0=ivp(0.2) + )) + mf <- mif2(mf,Np=5000,Nmif=100,cooling.fraction.50=0.1) + ll <- logmeanexp(replicate(10,logLik(pfilter(mf,Np=5000))),se=TRUE) + data.frame(as.list(coef(mf)),loglik=ll[1],loglik.se=ll[2]) + } }) -> m3 ``` ```{r save_profile_rho} @@ -671,7 +673,7 @@ params %>% bake(file="sims.rds",seed=398906785, simulate(polio,nsim=2000,as.data.frame=TRUE,include.data=TRUE) - ) -> sims +) -> sims ddply(sims,~sim,summarize,zeros=sum(cases==0)) -> num_zeros num_zeros %>% @@ -709,7 +711,7 @@ print(pl2,vp=viewport(x=0.8,y=0.75,height=0.4,width=0.2)) sims %>% subset(sim!="data") %>% summarize(imports=coef(polio,"psi")*mean(SO+SB1+SB2+SB3+SB4+SB5+SB6)/12 - ) -> imports + ) -> imports ``` Since only `r with(subset(num_zeros,sim!="data"),round(sum(zerosModel implementation (L1 <- logmeanexp(sapply(pf1,logLik),se=TRUE))
##                        se 
 ## -820.4411300    0.2642793
-

Notice that we set up a parallel random number generator (RNG). In particular, we use the L’Ecuyer RNG, which is recommended for use with doParallel. Note, too, that the replications are averaged using the logmeanexp function, which counteracts Jensen’s inequality. It is helpful to plot the effective sample size and conditional log likelihoods: These calculations took only 3 sec and produced a log likelihood estimate with a standard error of 0.26.

+

Notice that we set up a parallel random number generator (RNG). In particular, we use the L’Ecuyer RNG, which is recommended for use with doParallel. Note, too, that the replications are averaged using the logmeanexp function, which counteracts Jensen’s inequality. It is helpful to plot the effective sample size and conditional log likelihoods: These calculations took only 2 sec and produced a log likelihood estimate with a standard error of 0.26.

@@ -333,6 +333,7 @@

Parameter estimation

Likelihood maximization: A local approach

Now, let us see if we can improve on our initial guess. We use the iterated filtering algorithm IF2 of Ionides et al. (2015), which uses a random walk in parameter space to approach the MLE. We set a constant random walk standard deviation for each of the regular parameters and a larger constant for each of the initial value parameters.

stew(file="local_search.rda",{
+  w1 <- getDoParWorkers()
   t1 <- system.time({
     m1 <- foreach(i=1:90,
                   .packages='pomp', .combine=rbind,
@@ -356,7 +357,7 @@ 

Likelihood maximization: A local approach

} ) },seed=318817883,kind="L'Ecuyer")
-

This investigation took 4.2 minutes on a machine with 20 cores. The maximum likelihood we obtain is -796.5. These repeated stochastic maximizations can also show us the geometry of the likelihood surface in a neighborhood of this point estimate:

+

This investigation took 4.1 minutes on a machine with 20 cores. The maximum likelihood we obtain is -796.5. These repeated stochastic maximizations can also show us the geometry of the likelihood surface in a neighborhood of this point estimate:

pairs(~loglik+psi+rho+tau+sigma_dem+sigma_env,data=subset(m1,loglik>max(loglik)-20))

Because it is so useful to build up a view of the geometry of the likelihood surface, we save these likelihood estimates in a file for later use:

@@ -402,6 +403,7 @@

Global likelihood maximization: parameter estimation using randomized starti )

We then carry out a search identical to the local one except for the starting parameter values.

stew(file="global_search.rda",{
+  w2 <- getDoParWorkers()
   t2 <- system.time({
     m2 <- foreach(i=1:400,.packages='pomp',.combine=rbind,
                   .options.multicore=list(set.seed=TRUE)
@@ -427,7 +429,7 @@ 

Global likelihood maximization: parameter estimation using randomized starti library(plyr) params <- arrange(rbind(m1,m2[names(m1)]),-loglik) write.csv(params,file="polio_params.csv",row.names=FALSE,na="")

-

This search gives a maximized likelihood estimate of -794.9 with a standard error of 0.12. It took about 141 mins on a 20-core machine. This compares with the results of Martinez-Bakker et al. (2015), who report a maximized likelihood of -794.3 with a standard error of 0.11.

+

This search gives a maximized likelihood estimate of -794.9 with a standard error of 0.12. It took about 138 mins on a 20-core machine. This compares with the results of Martinez-Bakker et al. (2015), who report a maximized likelihood of -794.3 with a standard error of 0.11.

Plotting these diverse parameter estimates can help to give a feel for the global geometry of the likelihood surface.

pairs(~loglik+psi+rho+tau+sigma_dem+sigma_env,data=subset(m2,loglik>max(loglik)-20))

@@ -453,42 +455,42 @@

Profile likelihood

library(magrittr) bake(file="profile_rho.rds",{ - params %>% - subset(loglik>max(loglik)-20, - select=-c(loglik,loglik.se,rho)) %>% - melt(id=NULL) %>% - daply(~variable,function(x)range(x$value)) -> box - - starts <- profileDesign(rho=seq(0.01,0.025,length=30), - lower=box[,1],upper=box[,2], - nprof=10) - - foreach(start=iter(starts,"row"), - .combine=rbind, - .options.multicore=list(set.seed=TRUE), - .options.mpi=list(seed=290860873,chunkSize=1) - ) %dopar% { - mf <- mif2(polio, - start=unlist(start), - Np=2000, - Nmif=300, - cooling.type="geometric", - cooling.fraction.50=0.5, - transform=TRUE, - rw.sd=rw.sd( - b1=0.02, b2=0.02, b3=0.02, b4=0.02, b5=0.02, b6=0.02, - psi=0.02, tau=0.02, sigma_dem=0.02, sigma_env=0.02, - IO_0=ivp(0.2), SO_0=ivp(0.2) - )) - mf <- mif2(mf,Np=5000,Nmif=100,cooling.fraction.50=0.1) - ll <- logmeanexp(replicate(10,logLik(pfilter(mf,Np=5000))),se=TRUE) - data.frame(as.list(coef(mf)),loglik=ll[1],loglik.se=ll[2]) - } + params %>% + subset(loglik>max(loglik)-20, + select=-c(loglik,loglik.se,rho)) %>% + melt(id=NULL) %>% + daply(~variable,function(x)range(x$value)) -> box + + starts <- profileDesign(rho=seq(0.01,0.025,length=30), + lower=box[,1],upper=box[,2], + nprof=10) + + foreach(start=iter(starts,"row"), + .combine=rbind, + .options.multicore=list(set.seed=TRUE), + .options.mpi=list(seed=290860873,chunkSize=1) + ) %dopar% { + mf <- mif2(polio, + start=unlist(start), + Np=2000, + Nmif=300, + cooling.type="geometric", + cooling.fraction.50=0.5, + transform=TRUE, + rw.sd=rw.sd( + b1=0.02, b2=0.02, b3=0.02, b4=0.02, b5=0.02, b6=0.02, + psi=0.02, tau=0.02, sigma_dem=0.02, sigma_env=0.02, + IO_0=ivp(0.2), SO_0=ivp(0.2) + )) + mf <- mif2(mf,Np=5000,Nmif=100,cooling.fraction.50=0.1) + ll <- logmeanexp(replicate(10,logLik(pfilter(mf,Np=5000))),se=TRUE) + data.frame(as.list(coef(mf)),loglik=ll[1],loglik.se=ll[2]) + } }) -> m3
params <- arrange(rbind(params,m3[names(params)]),-loglik)
 write.csv(params,file="polio_params.csv",row.names=FALSE,na="")

Note that \(\rho\) is not perturbed in the IF iterations for the purposes of the profile calculation.

-

+

Simulation to investigate the fitted model: Local persistence

@@ -504,7 +506,7 @@

Simulation to investigate the fitted model: Local persistence

bake(file="sims.rds",seed=398906785, simulate(polio,nsim=2000,as.data.frame=TRUE,include.data=TRUE) - ) -> sims +) -> sims ddply(sims,~sim,summarize,zeros=sum(cases==0)) -> num_zeros num_zeros %>% @@ -538,13 +540,13 @@

Simulation to investigate the fitted model: Local persistence

print(pl1) print(pl2,vp=viewport(x=0.8,y=0.75,height=0.4,width=0.2)) -

+

sims %>%
   subset(sim!="data") %>%
   summarize(imports=coef(polio,"psi")*mean(SO+SB1+SB2+SB3+SB4+SB5+SB6)/12
-            ) -> imports
-

Since only 1.1% of the simulations had fewer zero-case months than did the data, the model does predict significantly more fadeouts than does the model. Months with no asyptomatic infections for the simulations were rare, on average 0.3 months per simulation. Months with fewer than 80 infections averaged 42.9 per simulation, which in the context of a reporting rate of 0.0121 can explain the absences of case reports.

-

For this model, the mean monthly infections due to importations (i.e., due to the term \(\psi\)) is 146.4. This does not give much opportunity for local elimination of poliovirus. One could profile over \(\psi\) to investigate how sensitive this conclusion is to values of \(\psi\) consistent with the data.

+ ) -> imports +

Since only 1.6% of the simulations had fewer zero-case months than did the data, the model does predict significantly more fadeouts than does the model. Months with no asyptomatic infections for the simulations were rare, on average 0.2 months per simulation. Months with fewer than 80 infections averaged 49.2 per simulation, which in the context of a reporting rate of 0.0141 can explain the absences of case reports.

+

For this model, the mean monthly infections due to importations (i.e., due to the term \(\psi\)) is 117.7. This does not give much opportunity for local elimination of poliovirus. One could profile over \(\psi\) to investigate how sensitive this conclusion is to values of \(\psi\) consistent with the data.

diff --git a/polio/polio_params.csv b/polio/polio_params.csv index b4d01bc..7097544 100644 --- a/polio/polio_params.csv +++ b/polio/polio_params.csv @@ -1,669 +1,669 @@ "b1","b2","b3","b4","b5","b6","psi","rho","tau","sigma_dem","sigma_env","SO_0","IO_0","delta","SB1_0","SB2_0","SB3_0","SB4_0","SB5_0","SB6_0","loglik","loglik.se" -3.63005754433999,-0.76328870790271,1.88872685344695,6.23435285829052,5.16689166383995,2.61946355210304,0.00286536343028887,0.0120689655172414,0.00123299947956615,0.0494414076846308,0.446140936926077,0.114924379257115,0.00169458663791264,0.0166666666666667,4069,4565,4410,4616,4305,4032,-794.760187258368,0.122902195199642 -3.93485355022026,-3.17784935774599,1.98814774006364,6.0002295517947,5.24189850326937,2.38536022344867,0.00241463092890594,0.0136206896551724,0.00170692977753935,0.0424549573024104,0.461960980727454,0.12587988060569,0.00105928534451406,0.0166666666666667,4069,4565,4410,4616,4305,4032,-794.88048753806,0.132587133690289 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