It is found that the combination of models has a prediction error slightly worse compared to a simple PLS model using physical chemical data. This is not so bad. On the one hand, PLS is very much a predictive method, so, it should be performing better. On the other hand, PLS does not take curvature into account. This means that the PLS model has a handicap for some of the details. It would seem that these two balance out. Finally, the separate models can be interpreted better than a PLS model.

## Model fit

The model uses all the separate parts which were defined previously in this blog. For ease of use this is all placed in a small function. A second function is used to generate predictions. The first figure shows the predictions using all data. In addition the ideal line (y=x) is added.

The normal predictions are not very impressive. Especially, the least liked products are not found least liked at all. Part of the problem may be in an intermediate step. CJuiciness, which is the juiciness observed by the consumers, is not well predicted. As it was earlier discovered that CJuiciness is the main explaining factor for liking, this may explain part of the lack of prediction power.

## Model prediction capability using cross validation

Cross validation predictions are given in the second figure. The sub figures make very clear that the model does not do a good job of predicting new data. Again, Juiciness is a big part of the problem, now both the sensory observed juiciness and the consumer observed juiciness.

### Prediction error

For completeness, the prediction error is calculated. The Root Mean Squared Error of Prediction is 0.170. As a comparison, a PLS model of liking from physical chemical data has a slightly better prediction error. This makes the model defined at least a competitor to PLS. It also shows that predicting the liking poses a difficult problem.

Data: X dimension: 17 4

Y dimension: 17 1

Fit method: kernelpls

Number of components considered: 4

VALIDATION: RMSEP

Cross-validated using 17 leave-one-out segments.

(Intercept) 1 comps 2 comps 3 comps 4 comps

CV 0.174 0.1626 0.1617 0.2029 0.2007

adjCV 0.174 0.1615 0.1610 0.2005 0.1984

TRAINING: % variance explained

1 comps 2 comps 3 comps 4 comps

X 34.74 80.63 87.17 100.00

CLiking 33.80 36.71 40.89 40.96

### R code

library(xlsReadWrite)

library(gam)

library(plyr)

datain <- read.xls('condensed.xls')

datain <- datain[-grep('bag|net',datain$Products,ignore.case=TRUE),]

datain$week <- sapply(strsplit(as.character(datain$Product),'_'),

function(x) x[[2]])

dataval <- datain

vars <- names(dataval)[-1]

for (descriptor in vars) {

dataval[,descriptor] <- as.numeric(gsub('[[:alpha:]]','',

dataval[,descriptor]))

}

# all fits in a model

FitCombinedModel <- function(dataval) {

# link instrument to sensory

MJuicsSEL3 <- gam(SJuiciness ~ AInstrumental.firmness + s(AJuice.release,3) +

s(ATitratable.acidity,3) ,data=dataval)

MSweetSEL <- gam(SSweetness ~ AInstrumental.firmness + s(AJuice.release,3) +

ASoluble.solids + ATitratable.acidity ,data=dataval)

MCrispSEL <- gam(SCrispness ~ s(AInstrumental.firmness,3) + AJuice.release +

ASoluble.solids + ATitratable.acidity ,data=dataval)

MMealiSEL <- gam(SMealiness ~ s(AInstrumental.firmness,3) + AJuice.release +

ASoluble.solids + ATitratable.acidity ,data=dataval)

# link sensory to consumer experience

MCJuic <- lm(CJuiciness ~ SJuiciness + SSweetness,data=dataval)

MCSweet <- lm(CSweetness ~ SSweetness + SCrispness ,data=dataval)

# link consumer experience to liking

Mliking <- lm(CLiking ~ CJuiciness+CSweetness + I(CJuiciness^2) ,data=dataval)

return(list(MJuicsSEL3=MJuicsSEL3,MSweetSEL=MSweetSEL,MCrispSEL=MCrispSEL,

MMealiSEL=MMealiSEL,MCJuic=MCJuic,MCSweet=MCSweet,Mliking=Mliking ))

}

PredictCombinedModel <- function(modellist,newdata) {

preddata <- data.frame(SJuiciness = predict(modellist$MJuicsSEL3,newdata),

SSweetness = predict(modellist$MSweetSEL,newdata),

SCrispness = predict(modellist$MCrispSEL,newdata),

SMealiness = predict(modellist$MMealiSEL,newdata))

preddata$CJuiciness <- predict(modellist$MCJuic,preddata)

preddata$CSweetness <- predict(modellist$MCSweet,preddata)

preddata$CLiking <- predict(modellist$Mliking,preddata)

preddata

}

FCM <- FitCombinedModel(dataval)

datain.nomiss <- dataval[-1,]

predictions <- PredictCombinedModel(FCM,datain.nomiss)

par(mfrow=c(2,3))

m_ply(c('SJuiciness','SSweetness','SCrispness','CJuiciness',

'CSweetness','CLiking'),function(x) {

plot(y=predictions[,x],x=datain.nomiss[,x],ylab=paste('predicted',x),

xlab=paste('observed',x))

abline(a=0,b=1)

})

#cross validation section

lopred <- mdply(1:nrow(datain.nomiss),function(x) {

datain.nomisslo <- datain.nomiss[-x,]

lomodel <- FitCombinedModel(datain.nomisslo)

PredictCombinedModel(lomodel,datain.nomiss[x,])

})

#lopred <- do.call(rbind,lopred)

par(mfrow=c(2,3))

m_ply(c('SJuiciness','SSweetness','SCrispness','CJuiciness',

'CSweetness','CLiking'),function(x) {

plot(y=lopred[,x],x=datain.nomiss[,x],

ylab=paste('cv predicted',x),xlab=paste('observed',x))

abline(a=0,b=1)

})

# root mean squared error of prediction

sqrt(mean((lopred$CLiking-datain.nomiss$CLiking)^2))

#pls as benchmark

library(pls)

plsmodel <- plsr(CLiking ~ AInstrumental.firmness + AJuice.release + ASoluble.solids + ATitratable.acidity,data=datain.nomiss,

scale=TRUE,validation='LOO' )

summary(plsmodel)

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