## Sunday, October 4, 2015

### Predicting Titanic deaths on Kaggle VII: More Stan

Two weeks ago I used STAN to create predictions after just throwing in all independent variables. This week I aim to refine the STAN model. For this it is convenient to use the loo package (Efficient Leave-One-Out Cross-Validation and WAIC for Bayesian Models). See also the paper by Aki Vehtari, Andrew Gelman and Jonah Gabry.
Since the package does the heavy lifting, it only remains to wrap it a function so I can quickly compare some models. A potential next step is to automate some more, but I did not do that pending current results.

### Data

Same as last time.

### Model

To keep the model similar as last time, I need to get a full design matrix for each independent variable in the model. So I made a mechanism which takes a model formulation and creates both the design matrix and a bunch of indices to keep track which column corresponds to which part of the model. To be specific, terms contains 1 to nterm if the corresponding column in xmat is from variable 1 (intercept) to the last variable. This sits in the function des.matrix.
The generated quantities block is purely for the LOO statistic.
It is preferred to compile the model only once, hence fit1 is calculated beforehand. Having done that preparation, MySmodel  is a function which does model fitting, LOO statistic and output it all in one step. In this function I can just drop in the formula and get something usable as output, so I can easily examine a bunch of models. It seemed to me that forward selection was a suitable way to examine the model space. I know it is not ideal, but at this point I mainly want to know if this actually will function.

### Results

To my surprise, Title was the parameter which gave the best predictions. I had expected sex to play that role.
Survived ~ Title -445.4972 16.46314
Computed from 4000 by 891 log-likelihood matrix

Estimate   SE
elpd_loo   -445.5 16.5
p_loo         4.1  0.2
looic       891.0 32.9

All Pareto k estimates OK (k < 0.5)
The next variable was passenger class

Survived ~ Title + Pclass -395.5926 17.42705
Unfortunately after adding a few independent variables things gave only minor improvements. This os not because of anything faulty, I made a classical mechanism to leave 10% out and predict the remainder. Those results were similar, but took more time and showed more run to run variation in the results. The only true advantage was that it gave results on the same scale as previous cross validations.
I expanded the model formula to about 10 terms. At that point, the expected prediction error decreased so slow that I decided on an eight term model. (Title + Pclass + sibsp + Title:Pclass + Embarked + oe + Title:sibsp + parch). The functions myPmodel and mySpred refit the model and perform the actual predictions. The result was a disappointing 0.78 on Kaggle. A minor improvement on the previous STAN result, but boosting is still better.

### Code

library(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())
library(loo)

# read and combine
train\$status <- 'train'
test\$status <- 'test'
test\$Survived <- NA
tt <- rbind(test,train)

# generate variables
tt\$Embarked[tt\$Embarked==''] <- 'S'
tt\$Embarked <- factor(tt\$Embarked)
tt\$Pclass <- factor(tt\$Pclass)
tt\$Survived <- factor(tt\$Survived)
tt\$age <- tt\$Age
tt\$age[is.na(tt\$age)] <- 999
tt\$age <- cut(tt\$age,c(0,2,5,9,12,15,21,55,65,100,1000))

tt\$Title <- sapply(tt\$Name,function(x) strsplit(as.character(x),'[.,]')[])
tt\$Title <- gsub(' ','',tt\$Title)
tt\$Title[tt\$Title=='Dr' & tt\$Sex=='female'] <- 'Miss'
tt\$Title[tt\$Title %in% c('Capt','Col','Don','Sir','Jonkheer','Major','Rev','Dr')] <- 'Mr'
tt\$Title[tt\$Title %in% c('Lady','Ms','theCountess','Mlle','Mme','Ms','Dona')] <- 'Miss'
tt\$Title <- factor(tt\$Title)
# changed cabin character
tt\$cabchar <- substr(tt\$Cabin,1,1)
tt\$cabchar[tt\$cabchar %in% c('F','G','T')] <- 'X';
tt\$cabchar <- factor(tt\$cabchar)
tt\$ncabin <- nchar(as.character(tt\$Cabin))
tt\$cn <- as.numeric(gsub('[[:space:][:alpha:]]','',tt\$Cabin))
tt\$oe <- factor(ifelse(!is.na(tt\$cn),tt\$cn%%2,-1))
tt\$Fare[is.na(tt\$Fare)]<- median(tt\$Fare,na.rm=TRUE)
tt\$ticket <- sub('[[:digit:]]+\$','',tt\$Ticket)
tt\$ticket <- toupper(gsub('(\\.)|( )|(/)','',tt\$ticket))
tt\$ticket[tt\$ticket %in% c('A2','A4','AQ3','AQ4','AS')] <- 'An'
tt\$ticket[tt\$ticket %in% c('SCA3','SCA4','SCAH','SC','SCAHBASLE','SCOW')] <- 'SC'
tt\$ticket[tt\$ticket %in% c('CASOTON','SOTONO2','SOTONOQ')] <- 'SOTON'
tt\$ticket[tt\$ticket %in% c('STONO2','STONOQ')] <- 'STON'
tt\$ticket[tt\$ticket %in% c('C')] <- 'CA'
tt\$ticket[tt\$ticket %in% c('SOC','SOP','SOPP')] <- 'SOP'
tt\$ticket[tt\$ticket %in% c('SWPP','WC','WEP')] <- 'W'
tt\$ticket[tt\$ticket %in% c('FA','FC','FCC')] <- 'F'
tt\$ticket[tt\$ticket %in% c('PP','PPP','LINE','LP','SP')] <- 'PPPP'
tt\$ticket <- factor(tt\$ticket)
tt\$fare <- cut(tt\$Fare,breaks=c(min(tt\$Fare)-1,quantile(tt\$Fare,seq(.2,.8,.2)),max(tt\$Fare)+1))
tt\$sibsp=factor(c(1:4,rep(4,6))[tt\$SibSp+1])
tt\$parch=factor(c(1:4,rep(4,6))[tt\$Parch+1])

train <- tt[tt\$status=='train',]
test <- tt[tt\$status=='test',]

#end of preparation and data reading

options(width=90)

#################4444444444444444444##############################

des.matrix <- function(formula,data) {
form2 <- strsplit(as.character(formula),'~',fixed=TRUE)
resp <- form2[[length(form2)]]
form3 <- strsplit(resp,'+',fixed=TRUE)[]
la <- lapply(form3,function(x)
model.matrix(as.formula(paste('~' , x, '-1' )),data) )
nterm <- c(1,sapply(la,ncol))
terms <- rep(1:length(nterm),nterm)
ntrain <- nrow(data)
mat <- do.call(cbind,la)
mat <- cbind(rep(1,ntrain),mat)
np <- ncol(mat)
list(
survived = c(0,1)[data\$Survived],
np=np,
ntrain=nrow(data),
terms=terms,
nterm=max(terms),
tx=mat)
}

datain <- des.matrix(~ Sex+Pclass,data=train)

parameters=c('std','f','log_lik')
my_code <- '
data {
int<lower=0> ntrain;
int survived[ntrain];
int<lower=1> np;
int<lower=1> nterm;
int terms[np];
matrix <lower=0,upper=1> [ntrain,np]  tx;
}
parameters {
vector[np]  f;
real <lower=0> std[nterm];
real <lower=0> stdhyp;
}
model {
stdhyp ~ normal(0,2);
std ~ normal(0,stdhyp);
for (i in 1:np) {
f[i] ~ normal(0,std[terms[i]]);
}
survived ~ bernoulli_logit(tx*f);
}
generated quantities {
vector [ntrain] log_lik;
for (i in 1:ntrain) {
log_lik[i] <- bernoulli_logit_log(survived[i], tx[i]*f);
}
}
'

fit1 <- stan(model_code = my_code,
data = datain,
pars=parameters,
iter = 1000,
chains = 4,
open_progress=FALSE)
#fit1

#log_lik1 <- extract_log_lik(fit1)
#loo1 <- loo(log_lik1)
#print(loo1, digits = 3)

print.mySmodel <- function(x) {
print(x\$loo1)
cat('\n')
invisible(x)
}

mySmodel <- function(formula,data) {
datain <- des.matrix(formula,data)

fitx <-
stan(model_code = my_code,
data = datain,
pars=parameters,
fit=fit1,
iter = 2000,
chains = parallel::detectCores(),
open_progress=FALSE)
log_lik1 <- extract_log_lik(fitx)
loo1 <- loo(log_lik1)
ll <-  list(myform=formula,fitx=fitx,loo1=loo1)
class(ll) <- 'mySmodel'
cat(format(formula),loo1\$elpd_loo,loo1\$se_elpd_loo,'\n')
ll
}
mySmodel(Survived ~
Title ,
data=train)

################# prediction functions

myPmodel <- function(formula,data) {
datain <- des.matrix(formula,data)

fitx <-
stan(model_code = my_code,
data = datain,
pars=parameters,
fit=fit1,
iter = 2000,
chains = parallel::detectCores(),
open_progress=FALSE)
list(myform=formula,fitx=fitx)
}

PredM <- myPmodel(~ Title + Pclass + sibsp + Title:Pclass + Embarked + oe + Title:sibsp + parch
,data=train)

mySpred <- function(mymodel,newdata) {
pfit <- as.matrix(mymodel\$fitx)
fmat <- pfit[,grep('^f\\[',colnames(pfit))]
px <- des.matrix(mymodel\$myform,data=newdata)\$tx
mylpred <- tcrossprod(fmat,px)
mpred <- apply(mylpred,2,function(x) mean(x))
pred <- as.numeric(gtools::inv.logit(mpred)>.5)
factor(pred)
}
preds <- mySpred(PredM,test)

out <- data.frame(
PassengerId=test\$PassengerId,
Survived=as.numeric(as.character(preds)),
row.names=NULL)
write.csv(x=out,
file='stanqua.csv',
row.names=FALSE,
quote=FALSE)

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