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NOIS_method.R
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NOIS_method.R
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##############################################################################
############## Näive Overfitting Index Selection (NOIS) ######################
## see https://doi.org/10.1016/j.isprsjprs.2017.09.012 ######################
## for more details about the method ######################
##############################################################################
library(MASS)
library(hsdar)
library(caret)
##############################################################################
############## simulate an uncorrelated predictors ##########################
##############################################################################
# generate data independent to the response variable
# Spectra_DF is a hyperspectra dataset
# For instance, the first colunm with the response variable (e.g. LAI or Chl) and several colunms with wavelenghts (e.g. from 400nm to 2100nm)
# The rows are the observations (e.g. n=100)
# It is a list with 3 different landscape or places
# read dataset
Spectra_DF <- read_csv("Spectra_DF.csv")
# in the case of time series or multiple landscapes
set.seed(1234)
GEdata000 <- data.frame(sapply(1:3, FUN = function(i)
data.frame((mvrnorm(nrow(Spectra_DF[[i]]),
colMeans(data.frame(Spectra_DF[[i]][-1]), # [-1] to exclude LAI
na.rm = FALSE, dims=1),
cov(data.frame(Spectra_DF[[i]][-1])) ))) ))
##############################################################################
# check the remain correlation between LAI and each wavelength
##############################################################################
CorMatrix <- data.frame(sapply(1:3, FUN = function(i)
data.frame(cor(data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]))) ))
CorYbandGE <- data.frame(sapply(1:3, FUN = function(i)
CorMatrix[[i]]$lai[-1]))
# maximum Rsquared between LAI and a wavelength
# based on the remain correlation a tolerance baseline can be adjusted
sapply(1:3, FUN=function(i) max(abs(CorYbandGE[[i]]))^2)
# plot correlation per wavelength
colour <- colorRamp(c("orange", "yellow"))
Corspectra <- sapply(1:3, FUN = function(i)
speclib(CorYbandGE[[i]], rep(401:2501)))
plot(Corspectra[[1]], FUN=1, ylim=c(-1,1),
col=rgb(colour(4/8),maxColorValue=255))
for (i in 2:3)
plot(Corspectra[[i]], FUN=i, new=FALSE,
col=rgb(colour(i/30), maxColorValue=255))
#############################################################################
# Tune Naive models and calculate the overfitting index NOIS
#############################################################################
# Tune the models increasing complexity
# 1 - example Partial Least Square regression (PLSR)
# 2 - example Support Vector Machine (SVM)
#############################################################################
nonControl <- trainControl(method = "none") # no cross-validation
set.seed(39)
kfold10 <- trainControl(method="repeatedcv", repeats=10) # K-fold cross-validation
#############################################################################
# 1 Partial Least Square regression (PLSR)
# Tune from 1 to 20 components (accumulated) without cross-validation
Plsfit1 <- c()
for (i in 1:3) {
Plsfit1[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 1), # 1 component
trControl = nonControl)} # no cross-validation
Plsfit2 <- c()
for (i in 1:3) {
Plsfit2[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 2),# 2 component
trControl = nonControl)}
Plsfit3 <- c()
for (i in 1:3) {
Plsfit3[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 3),
trControl = nonControl)}
Plsfit4 <- c()
for (i in 1:30) {
Plsfit4[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 4),
trControl = nonControl)}
Plsfit5 <- c()
for (i in 1:3) {
Plsfit5[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 5),
trControl = nonControl)}
Plsfit6 <- c()
for (i in 1:3) {
Plsfit6[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 6),
trControl = nonControl)}
Plsfit7 <- c()
for (i in 1:3) {
Plsfit7[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 7),
trControl = nonControl)}
Plsfit8 <- c()
for (i in 1:3) {
Plsfit8[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 8),
trControl = nonControl)}
Plsfit9 <- c()
for (i in 1:3) {
Plsfit9[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp=9),
trControl = nonControl)}
Plsfit10 <- c()
for (i in 1:30) {
Plsfit10[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 10),
trControl = nonControl)}
Plsfit11 <- c()
for (i in 1:3) {
Plsfit11[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 11),
trControl = nonControl)}
Plsfit12 <- c()
for (i in 1:3) {
Plsfit12[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 12),
trControl = nonControl)}
Plsfit13 <- c()
for (i in 1:3) {
Plsfit13[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 13),
trControl = nonControl)}
Plsfit14 <- c()
for (i in 1:3) {
Plsfit14[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 14),
trControl = nonControl)}
Plsfit15 <- c()
for (i in 1:3) {
Plsfit15[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 15),
trControl = nonControl)}
Plsfit16 <- c()
for (i in 1:3) {
Plsfit16[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 16),
trControl = nonControl)}
Plsfit17 <- c()
for (i in 1:) {
Plsfit17[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 17),
trControl = nonControl)}
Plsfit18 <- c()
for (i in 1:3) {
Plsfit18[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 18),
trControl = nonControl)}
Plsfit19 <- c()
for (i in 1:3) {
Plsfit19[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 19),
trControl = nonControl)}
Plsfit20 <- c()
for (i in 1:3) {
Plsfit20[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneGrid = expand.grid(ncomp = 20),
trControl = nonControl)}
# With cross-validation and up 20 to component
Plsfitcv <- c()
for (i in 1:3) {
Plsfitcv[[i]] <- train(lai~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "pls", # package wrapped
type= "Regression",
tuneLength = 20, # number of components tested
trControl = kfold10 )} # with cross-validation
# Original data With cross-validation and up to 20 component
Pls000original <- c()
for (i in 1:3) {
Pls000original[[i]] <- train(lai~.,
data=data.frame(Spectra_DF[[i]]), #originaldata
method = "pls", # package wrapped
type= "Regression",
tuneLength = 20, #number of components tested
trControl = kfold10 )} # with cross-validation
###################################################################################################################
# calculating RMSE for the Naive Models - PLSR
###################################################################################################################
RMSE.000 <- t(data.frame(
"1"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit1[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"2"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit2[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"3"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit3[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"4"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit4[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"5"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit5[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"6"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit6[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"7"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit7[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"8"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit8[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"9"=sapply(1:3, FUN=function(i) postResample(predict(Plsfit9[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"10"=sapply(1:, FUN=function(i) postResample(predict(Plsfit10[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"11"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit11[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"12"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit12[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"13"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit13[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"14"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit14[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"15"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit15[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"16"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit16[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"17"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit17[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"18"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit18[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"19"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit19[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"20"=sapply(1:30, FUN=function(i) postResample(predict(Plsfit20[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',]))
###################################################################################################
########################## RMSE with the mean as prediction ########################################
###################################################################################################
RMSELAIav <- data.frame(RMSEy=sapply(1:30, FUN=function(i)
(postResample(rep(mean(unlist(Spectra_DF[[i]]$lai)), 100),
unlist(Spectra_DF[[i]]$lai))['RMSE'])))# 100 sample size
############################################################################
# Naive overfitting index for the 20 components
NOI000 = data.frame(sapply(1:30, FUN=function(i)
data.frame(NOI=1-(RMSE000[,i]/RMSELAIav[i,]))))
#############################################################################
## plot NOIS method
plot(rep(1:20), NOI000[[1]], type="b", col="green", new=F,
ylim=c(-0.1,1), xaxt='n', yaxt='n', cex.lab=1.3,
main="NOIS Index - Partial Least Squares (PLSR)",
xlab = "Ncomp", ylab="overfitting index (NOIS)")
axis(side = 1, at=1:20, cex.axis = 1.1)
axis(side = 2, at=c(0,.05,.1,.2,.3,.4,0.5,.6,.7,.8,.9,1),las=2,cex.axis=1.1)
abline(h=0.05, col="red", lwd=2)
abline(h=0.0, col="grey", lty="dotted", lwd=2)
for (i in 1:3) {
lines(rep(1:20), NOI000[[i]], type="b", col="green", new=F)
points(Plsfitcv[[i]]$bestTune$ncomp,
NOI000[Plsfitcv[[i]]$bestTune$ncomp,i], type="p",
col="black", new=F)
points(Pls000original[[i]]$bestTune$ncomp,
NOI000[Pls000original[[i]]$bestTune$ncomp,i],
type="p", col="red", new=F)}
###########################################################################
###############################################
############## Tune SVM #######################
###############################################
svmfit1 <- c()
for (i in 1:3) {
svmfit1[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.00005)))}
svmfit2 <- c()
for (i in 1:3) {
svmfit2[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.0001)))}
svmfit3 <- c()
for (i in 1:30) {
svmfit3[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.00025)))}
svmfit4 <- c()
for (i in 1:3) {
svmfit4[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.0005)))}
svmfit5 <- c()
for (i in 1:3) {
svmfit5[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.001)))}
svmfit6 <- c()
for (i in 1:3) {
svmfit6[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.0025)))}
svmfit7 <- c()
for (i in 1:3) {
svmfit7[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.005)))}
svmfit8 <- c()
for (i in 1:3) {
svmfit8[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.01)))}
svmfit9 <- c()
for (i in 1:3) {
svmfit9[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.05)))}
svmfit10 <- c()
for (i in 1:3) {
svmfit10[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.1)))}
svmfit11 <- c()
for (i in 1:3) {
svmfit11[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = nonControl,
tuneGrid = expand.grid(cost=c(0.25)))}
# with cross-validation and cost from .00005 to .25
costgrid <- expand.grid(cost=c(0.00005,0.0001,0.00025,0.0005, 0.001,
0.0025, 0.005, 0.01, 0.05, 0.1, 0.25))
svmfitcv <- c()
for (i in 1:3) {
svmfitcv[[i]] <- train(lai ~.,
data=data.frame(lai=Spectra_DF[[i]]$lai, GEdata000[[i]]),
method = "svmLinear2",
trControl = kfold10,
tuneGrid = costgrid)}
# original data with cross-validation and cost from .00005 to .25
svm000original <- c()
for (i in 1:3) {
svm000original[[i]] <- train(lai ~.,
data=data.frame(Spectra_DF[[i]]),
method = "svmLinear2",
trControl = kfold10,
tuneGrid = costgrid)}
###################################################################################################################
# RMSE for the Naive Models - SVMR
###################################################################################################################
RMSEsvm <- t(data.frame(
"0.000005"=sapply(1:3, FUN=function(i) postResample(predict(svmfit1[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.00001"=sapply(1:3, FUN=function(i) postResample(predict(svmfit2[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.00005"=sapply(1:3, FUN=function(i) postResample(predict(svmfit3[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.0001"=sapply(1:3, FUN=function(i) postResample(predict(svmfit4[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.00025"=sapply(1:3, FUN=function(i) postResample(predict(svmfit5[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.0005"=sapply(1:3, FUN=function(i) postResample(predict(svmfit6[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.001"=sapply(1:3, FUN=function(i) postResample(predict(svmfit7[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.0025"=sapply(1:3, FUN=function(i) postResample(predict(svmfit8[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.005"=sapply(1:3, FUN=function(i) postResample(predict(svmfit9[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.01"=sapply(1:3, FUN=function(i) postResample(predict(svmfit10[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',],
"0.05"=sapply(1:3, FUN=function(i) postResample(predict(svmfit11[[i]]), unlist(Spectra_DF[[i]]$lai)))['RMSE',]))
############################################################################
# Naive overfitting index for the 11 levels of cost
############################################################################
NOI000svm = data.frame(sapply(1:3, FUN=function(i)
data.frame(NOI=1-(RMSEsvm[,i]/RMSELAIav[i,]))))
#############################################################################
costnames<-c(0.00005,0.0001,0.00025,0.0005, 0.001,
0.0025, 0.005, 0.01, 0.05, 0.1, 0.25)
cost <- sapply(1:3, FUN = function(i) svmfitcv[[i]]$bestTune$cost)
cost[cost==0.00005] <- 1
cost[cost==0.0001] <- 2
cost[cost==0.00025] <- 3
cost[cost==0.0005] <- 4
cost[cost==0.001] <- 5
cost[cost==0.0025] <- 6
cost[cost==0.005] <- 7
cost[cost==0.01] <- 8
cost[cost==0.05] <- 9
cost[cost==0.1] <- 10
cost[cost==0.25] <- 11
cost2 <- sapply(1:3, FUN = function(i) svm000original[[i]]$bestTune$cost)
cost2[cost2==0.00005] <- 1
cost2[cost2==0.0001] <- 2
cost2[cost2==0.00025] <- 3
cost2[cost2==0.0005] <- 4
cost2[cost2==0.001] <- 5
cost2[cost2==0.0025] <- 6
cost2[cost2==0.005] <- 7
cost2[cost2==0.01] <- 8
cost2[cost2==0.05] <- 9
cost2[cost2==0.1] <- 10
cost2[cost2==0.25] <- 11
##########################################################################
###### plot NOIS method - SVM ############################################
##########################################################################
plot(rep(1:11),NOI000svm[[1]], type="b", col="green", new=F,
ylim=c(-0.1,1), xaxt='n', yaxt='n', cex.lab=1.3,
main="NOIS Index - Support Vector Machine (SVM)",
xlab = "Cost", ylab="overfitting index (NOIS)")
axis(side = 1, at=1:11, cex.axis = 1, labels=costnames)
axis(side = 2, at=c(0,.05,.1,.2,.3,.4,0.5,.6,.7,.8,.9,1),las=2,cex.axis=1.1)
abline(h=0.05, col="red", lwd=2) # tolerance baseline
abline(h=0.0, col="grey", lty="dotted", lwd=2)
for (i in 1:3) {
lines(rep(1:11), NOI000svm[[i]], type="b", col="green", new=F)
points(cost[i], NOI000svm[cost[i],i], # complexity tuned generated data model
type="p", col="black", new=F)
points(cost2[i], NOI000svm[cost2[i],i],# complexity tuned original data
type="p", col="red", new=F)}