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Knit directory: polymeRID/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| html | f2ee83c | goergen95 | 2019-08-19 | Build site. |
| html | d960dc2 | goergen95 | 2019-08-19 | included calibration |
| html | b846f0b | goergen95 | 2019-08-19 | Build site. |
| Rmd | de84a71 | goergen95 | 2019-08-19 | large update for website |
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| Rmd | 6bef5e6 | goergen95 | 2019-08-14 | confusion matrix output in rf_exploration |
| html | 6bef5e6 | goergen95 | 2019-08-14 | confusion matrix output in rf_exploration |
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| Rmd | 293fd73 | goergen95 | 2019-08-14 | first step on svm_exploration |
| html | 293fd73 | goergen95 | 2019-08-14 | first step on svm_exploration |
Support Vector Machine (SVM) is a non-parametric classification method which was initially designed for binary classification problems and was developed in its current form by Boser, Guyon, and Vapnik (2004). A detailed overview of the SVM algorithm is found in Burges (1998). The principal idea behind SVM is to find an optimal hyperplane which separates two classes from each other by the largest possible margin. The algorithm is optimized by iteratively maximizing this margin. Only the closest observations to the margin are considered. These specific observations are also called support-vectors. Multidimensional data can be processed by mapping the data into a higher dimensional feature space through a specified mapping function. This function is called kernel function, and mainly four different groups are used: linear, polynomial, radial and sigmoid functions (Burges 1998). In this project only the radial basis function was used. Multi-class problems are addressed by calculating an optimal margin following the one class-against-all pattern and conducting a majority vote at the end of the calculations. SVM needs proper fine-tuning of its parameters. These are the regularization parameter C and the kernel width γ for radial basis functions. The regularization parameter is also called “penalty value”, as it is a constant giving penalty to misclassified observations. The optimal values might change if different representations of the data are presented to the algorithm.
Different techniques of data pre-processing might emphasize different features of the patterns to be learned by an algorithm. To grasp this, different transformations of the data were presented to the SVM algorithm. Additionally, the raw data signal was jittered to test which transformation might prove beneficial in delivering a high classification accuracy even in the presence of noise. To test this, we define a function which adds noise to the raw data and returns a list with the number of elements equal to the levels of noise applied.
addNoise = function(data, levels = c(0), category="class"){
data.return = list()
index = which(names(data) == category)
for (n in levels){
tmp = as.matrix(data[ , -index])
if (n == 0){
tmp = data
}else{
tmp = as.data.frame(jitter(tmp, n))
tmp[category] = data[category]
}
data.return[[paste("noise", n, sep="")]] = tmp
}
return(data.return)
}
data = read.csv(file = paste0(ref, "reference_database.csv"), header = TRUE)
noisy_data = addNoise(data, levels = c(0,10,100,250,500), category = "class")
# indivitual elements can be selected by using [[ and refering to the index or the name
head(noisy_data[["noise100"]])[1:3,1:3] wvn3992.63003826141 wvn3990.70123147964 wvn3988.77242469788
1 -0.010328834 0.01829306 0.001603137
2 0.005237827 0.01683373 -0.013921331
3 -0.011751581 0.01958805 0.002650597In another function which uses the noisy_data-object as input, specific data transformations are applied. These is normalization which centers and scales the input data, different forms of the Savitzkiy-Golay filter (Savitzky and Golay 1964), and first and second derivatives of the raw spectra. The function iterates through the elements in the noisy_data object and returns each specified transformation in a list element below the noise level. The exemplary function below applies the pre-processing for normalization, standard filtering and first derivative only. The implementation of the function used in the project can be found here.
createTrainingSet = function(data, category = "class",
SGpara = list(p=3,w=11), lag = 15){
data.return = list()
for (noise in names(data)){
tmp = as.data.frame(data[[noise]])
classes = tmp[,category]
tmp = tmp[!names(tmp) %in% category]
# original data
data.return[[noise]][["raw"]] = as.data.frame(data[[noise]])
# normalised data
data_norm = preprocess(tmp, type="norm")
data_norm[category] = classes
data.return[[noise]][["norm"]] = data_norm
# SG-filtered data
data_sg = preprocess(tmp, type="sg", SGpara = SGpara)
data_sg[category] = classes
data.return[[noise]][["sg"]] = data_sg
# first derivative of original data
data_rawd1 = preprocess(tmp, type="raw.d1", lag = lag)
data_rawd1[category] = classes
data.return[[noise]][["raw.d1"]] = data_rawd1
}
return(data.return)
}
# applying the function
test_dataset = createTrainingSet(noisy_data, category = "class")
# individual transformations at a certain noise level can be accessed with [[
head(test_dataset[["noise500"]][["raw.d1"]])[1:3,1:3] wvn3963.69793653488 wvn3961.76912975311 wvn3959.84032297134
1 0.08295627 0.02234302 0.17399975
2 0.04253842 0.07918397 0.03849980
3 -0.05579848 0.11670539 0.02503375The database of Primpke et al. (2018) currently shows 1863 variables for each observation. Most of these data points do not hold relevant information to distinguish between different types of particles. To shorten the computation time, one can use dimensionality reduction techniques such as principal component analysis (PCA). PCA has already been used to transform spectral data of microplastic in marine ecosystems (Jung et al. 2018; Lorenzo-Navarro et al. 2018). PCA basically takes the input data for a given number of observations and performs a orthogonal transformation to derive uncorrelated principal components from the possibly correlated variables. Both redundancies in the data as well as the presence of noise can be accounted for this way. PCA has previously been successfully applied to FTIR-spectroscopy data (Hori and Sugiyama 2003; Nieuwoudt et al. 2004; Mueller et al. 2013; Ami, Mereghetti, and Maria 2013; Fu, Toyoda, and Ihara 2014). Simultaneously, the number of variables can be significantly reduced by applying PCA and thus speeding up the training process. Below we apply a PCA to the raw data as an example only.
library(factoextra)
tmp = test_dataset[["noise0"]][["raw"]]
pca = prcomp(tmp[ ,-1864]) # omitting class variable
var_info = factoextra::get_eigenvalue(pca)
# setting a threshold of 99% explained variance
threshold = 99
thresInd = which(var_info$cumulative.variance.percent>=threshold)[1]
pca_data = pca$x[,1:thresInd]We can use the index variable thresInd we just defined to take a look upon all the principal components which explain 99% of the variance of the data.
| eigenvalue | variance.percent | cumulative.variance.percent | |
|---|---|---|---|
| Dim.1 | 1.5618496 | 57.7479523 | 57.74795 |
| Dim.2 | 0.3875503 | 14.3293173 | 72.07727 |
| Dim.3 | 0.2226548 | 8.2324559 | 80.30973 |
| Dim.4 | 0.1823539 | 6.7423696 | 87.05209 |
| Dim.5 | 0.0897978 | 3.3201919 | 90.37229 |
| Dim.6 | 0.0538395 | 1.9906665 | 92.36295 |
| Dim.7 | 0.0443676 | 1.6404525 | 94.00341 |
| Dim.8 | 0.0324243 | 1.1988595 | 95.20227 |
| Dim.9 | 0.0297937 | 1.1015939 | 96.30386 |
| Dim.10 | 0.0215566 | 0.7970366 | 97.10090 |
| Dim.11 | 0.0173647 | 0.6420450 | 97.74294 |
| Dim.12 | 0.0145994 | 0.5397977 | 98.28274 |
| Dim.13 | 0.0113917 | 0.4211964 | 98.70394 |
| Dim.14 | 0.0069264 | 0.2560972 | 98.96003 |
| Dim.15 | 0.0042735 | 0.1580104 | 99.11804 |
We effectively reduced the number of variables from 1683 to 15 which still bears 99% of the variance we find in the original data. When it comes to machine learning, however, it is important to realize that this new data set is not fit to be used in a training process. If we now randomly split the observations into a training and testing set, we effectively mix up these two sets because information of the testing set has already influenced the outcome of the PCA. Therefore, the data set needs to be split before applying the PCA. The analysis is done on the training data only and then the same orthogonal transformations is applied to the testing data. This way it can be ensured that the test set is truly independent of the training process.
We apply a 10-fold cross-validation which is repeated five times. The following code takes a complete data set as input, applies a splitting function from the caret package and then builds the PCA upon the the training set and finally applies the same transformation to the testing set. Here, it is only applied for the raw data. We also randomly split the data to a 50% training and a 50% testing set.
folds = 10
repeats = 5
split_percentage = 0.5
threshold = 99
tmp = test_dataset[["noise0"]][["raw"]]
set.seed(42) # ensure reproducibility
fold_index = lapply(1:repeats, caret::createDataPartition, y=tmp$class,
times = folds, p = split_percentage)
fold_index = do.call(c, fold_index)
pcaData = list()
for (rep in 1:repeats){
rep_index = fold_index[(rep*folds-folds+1):(rep*folds)] # jumps to the correct number of folds forward in each repeat
pcadata_fold = lapply(1:folds,function(x){
# splitting for current fold
training = tmp[unlist(rep_index[x]),]
validation = tmp[-unlist(rep_index[x]),]
# keep response
responseTrain = training$class
responseVal = validation$class
# apply PCA
pca = prcomp(training[,1:1863])
varInfo = factoextra::get_eigenvalue(pca)
thresInd = which(varInfo$cumulative.variance.percent >= threshold)[1]
pca_training = pca$x[ ,1:thresInd]
pca_validation = predict(pca, validation)[ ,1:thresInd]
training = as.data.frame(pca_training)
training$response = responseTrain
validation = as.data.frame(pca_validation)
validation$response = responseVal
foldtmp = list(training, validation)
names(foldtmp) = c("training","validation")
return(foldtmp)
})
names(pcadata_fold) = paste("fold", 1:folds, sep ="")
pcaData[[paste0("repeat",rep)]] = pcadata_fold
}We now have a list object with the number of elements equivalent to the repeats. Below the level of repeats individual folds can be accessed. There the splitted data set can be accessed by referring to "training" and "testing".
pcaData[["repeat5"]][["fold10"]][["training"]][1:3,1:3] PC1 PC2 PC3
1 -0.4779164 -0.7786770 -0.02761266
2 -0.4869471 -0.7181236 -0.02492981
4 -0.4661699 -0.7463479 -0.11424932pcaData[["repeat5"]][["fold10"]][["validation"]][1:3,1:3] PC1 PC2 PC3
3 -0.5240744 -0.5931111 0.05961383
5 -0.4718366 -0.6995376 -0.09273887
7 -0.5255215 -0.5928214 0.05184969summary(pcaData[["repeat5"]][["fold10"]][["training"]]$response)FIBRE FUR HDPE LDPE PA PE PES PET PP PS PUR WOOD
14 12 5 6 7 4 8 5 6 4 4 2 summary(pcaData[["repeat5"]][["fold10"]][["validation"]]$response)FIBRE FUR HDPE LDPE PA PE PES PET PP PS PUR WOOD
13 11 5 5 7 4 7 4 6 3 3 2 For the SVM algorithm a simple search pattern for the optimal configuration for the regularization parameter C and the kernel width γ was implemented. We did this by applying a search grid for the parameters and calculating each possible combination. Note that due to limits in computation capacities we restricted the search to 25 combinations only. The code below calculates a model for each possible combination, evaluating its capacity to correctly classify the training data and then chooses the optimal model to evaluate the validation data.
training = pcaData[["repeat1"]][["fold1"]][["training"]]
validation =pcaData[["repeat1"]][["fold1"]][["validation"]]
x_train = training[ ,1:ncol(training)-1]
y_train = training$response
x_test = validation[ ,1:ncol(validation)-1]
y_test = validation$response
tuneGrid = expand.grid(gamma =seq(0.1,1,0.2),cost = seq(1,5,1) )
accuracy = c()
models = list()
for ( i in 1:nrow(tuneGrid)){
model = e1071::svm(x = x_train, y = y_train,
kernel = "radial",
gamma = tuneGrid$gamma[i],
cost = tuneGrid$cost[i])
pred = predict(model, x_train)
conf = caret::confusionMatrix(pred, y_train)
accuracy = c(accuracy, conf$overall["Kappa"])
models[[i]] = model
}
best_model = models[[which(accuracy == max(accuracy))[1]]]
prediction = predict(best_model, x_test)
confMat = caret::confusionMatrix(prediction, y_test)
print(confMat$table) Reference
Prediction FIBRE FUR HDPE LDPE PA PE PES PET PP PS PUR WOOD
FIBRE 12 0 0 0 0 0 5 2 0 0 0 2
FUR 1 11 0 0 0 0 0 0 0 0 0 0
HDPE 0 0 3 0 0 0 0 0 0 0 0 0
LDPE 0 0 0 5 0 2 0 0 0 0 0 0
PA 0 0 0 0 7 0 0 0 0 0 0 0
PE 0 0 1 0 0 2 0 0 0 0 0 0
PES 0 0 0 0 0 0 2 1 0 0 0 0
PET 0 0 1 0 0 0 0 1 0 2 0 0
PP 0 0 0 0 0 0 0 0 6 0 0 0
PS 0 0 0 0 0 0 0 0 0 1 0 0
PUR 0 0 0 0 0 0 0 0 0 0 3 0
WOOD 0 0 0 0 0 0 0 0 0 0 0 0The process of splitting the data set into training and testing was automated by putting the above code in a function which can be found here. Finally, this function was applied to the different pre-procesesing levels, as discussed before.
source("code/functions.R")
wavenumbers = readRDS(paste0(ref,"wavenumbers.rds"))
# add noise to data
noisyData = addNoise(data,levels = c(0,10,100,250,500), category = "class")
# preprocessing
testDataset = createTrainingSet(noisyData, category = "class",
SGpara = list(p=3, w=11), lag=15,
type = c("raw", "norm", "sg", "sg.d1", "sg.d2",
"sg.norm", "sg.norm.d1", "sg.norm.d2",
"raw.d1", "raw.d2", "norm.d1", "norm.d2"))
types = names(testDataset[[1]])
levels = lapply(names(testDataset), function(x){
rep(x, length(types))
})
levels = unlist(levels)
results = data.frame(level=levels, type = types, kappa = rep(0,length(levels)))
for (level in unique(levels)){
for (type in types){
print(paste0("Level: ",level," Type: ",type))
tmpData = testDataset[[level]][[type]]
tmpData[which(wavenumbers<=2420 & wavenumbers>=2200)] = 0 # setting C02 window to 0
tmpModel = pcaCV(tmpData, folds = 10, repeats = 5, threshold = 99, metric = "Kappa", p=0.5, method="svm")
saveRDS(tmpModel,file = paste0(output,"svm/model_",level,"_",type,"_",round(tmpModel[[1]],2),".rds"))
results[which(results$level==level & results$type==type),"kappa"] = as.numeric(tmpModel[[1]])
print(results)
}
}
saveRDS(results, paste0(output,"svm/exploration.rds"))The plot below shows the Kappa scores the algorithm achieved during training for different representations of the data at increasing noise levels.
It can be observed that with higher noise level the Kappa score is reduced significantly. All data transformations yield to very low accuracies when the noise increases. In the absence of significant noise, however, the simple Savitzkiy-Golay filter, the raw data, as well as the first and second order derivatives yield to a kappa score of about 0.75. Looking at this more mathematically by calculating the average slopes of the Kappa scores reveals the transformations with the most stable results. Note that we only take the kappa score at noise level 0 and 10 to calculate the average slope, since higher noise levels yielded to unsatisfactory accuracies.
noise0 = results[results$level == "noise0", ]
noise10 = results[results$level == "noise10", ]
slopes = noise10$kappa - noise0$kappa
types = unique(results$type)
df = data.frame(type = types, slope = slopes, row.names = NULL)
df = df[order(-slopes),]| type | slope | |
|---|---|---|
| 3 | sg | -0.0007272 |
| 1 | raw | -0.0011139 |
| 4 | sg.d1 | -0.0148920 |
| 10 | raw.d2 | -0.0161989 |
| 9 | raw.d1 | -0.0167638 |
| 5 | sg.d2 | -0.0642191 |
| 7 | sg.norm.d1 | -0.2300057 |
| 8 | sg.norm.d2 | -0.2324091 |
| 2 | norm | -0.5853342 |
| 6 | sg.norm | -0.6006121 |
This confirms that the average decrease in Kappa score is the lowest for the Savitzkiy-Golay filtered data followed by the raw data. Then, the first derivative of the filtered data achieves the next lowest slope, but it has to be noted that the overall kappa score level of this data transformation is lower than for the derivatives of the unfiltered data.
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sessionInfo()R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Linux Mint 19.1
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=de_DE.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=de_DE.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] plotly_4.9.0 knitr_1.24
[3] factoextra_1.0.5 tensorflow_1.14.0
[5] abind_1.4-5 e1071_1.7-2
[7] keras_2.2.4.1 workflowr_1.4.0.9001
[9] baseline_1.2-1 gridExtra_2.3
[11] stringr_1.4.0 prospectr_0.1.3
[13] RcppArmadillo_0.9.600.4.0 openxlsx_4.1.0.1
[15] magrittr_1.5 ggplot2_3.2.0
[17] reshape2_1.4.3 dplyr_0.8.3
loaded via a namespace (and not attached):
[1] httr_1.4.1 tidyr_0.8.3 viridisLite_0.3.0
[4] jsonlite_1.6 splines_3.6.1 foreach_1.4.7
[7] prodlim_2018.04.18 shiny_1.3.2 assertthat_0.2.1
[10] stats4_3.6.1 highr_0.8 yaml_2.2.0
[13] ggrepel_0.8.1 ipred_0.9-9 pillar_1.4.2
[16] backports_1.1.4 lattice_0.20-38 glue_1.3.1
[19] reticulate_1.13 digest_0.6.20 promises_1.0.1
[22] colorspace_1.4-1 recipes_0.1.6 httpuv_1.5.1
[25] htmltools_0.3.6 Matrix_1.2-17 plyr_1.8.4
[28] timeDate_3043.102 pkgconfig_2.0.2 SparseM_1.77
[31] caret_6.0-84 xtable_1.8-4 purrr_0.3.2
[34] scales_1.0.0 whisker_0.3-2 later_0.8.0
[37] gower_0.2.1 lava_1.6.5 git2r_0.26.1
[40] tibble_2.1.3 generics_0.0.2 withr_2.1.2
[43] nnet_7.3-12 lazyeval_0.2.2 mime_0.7
[46] survival_2.44-1.1 crayon_1.3.4 evaluate_0.14
[49] fs_1.3.1 nlme_3.1-140 MASS_7.3-51.4
[52] class_7.3-15 tools_3.6.1 data.table_1.12.2
[55] munsell_0.5.0 zip_2.0.3 compiler_3.6.1
[58] rlang_0.4.0 grid_3.6.1 iterators_1.0.12
[61] htmlwidgets_1.3 crosstalk_1.0.0 labeling_0.3
[64] base64enc_0.1-3 rmarkdown_1.14 gtable_0.3.0
[67] ModelMetrics_1.2.2 codetools_0.2-16 R6_2.4.0
[70] tfruns_1.4 lubridate_1.7.4 zeallot_0.1.0
[73] rprojroot_1.3-2 stringi_1.4.3 Rcpp_1.0.2
[76] rpart_4.1-15 tidyselect_0.2.5 xfun_0.8