Perform Principal Component Analysis - Management Assignment Help

Assignment Task:

Activity 1: Principal Component Analysis In this first activity, you are asked to: 
1. Perform Principal Component Analysis (PCA) on the Stamps data in the 9-dimensional space of the numerical predictors ( PB_Predictors ), and show the Proportion of Variance Explained (PVE) for each of the nine resulting principal components. Plot the accumulated sum of PVE for the first components, as a function of , and discuss the result: 
(a) How many components do we need to explain 90% or more of the total variance? 
(b) How much of the total variance is explained by the first three components? 
2. Do some research by yourself on how to render 3D plots in R, and then plot a 3D scatter-plot of the Stamps data as represented by the first three principal components computed in the previous item ( x = PC1 , y = PC2 , and z = PC3 ). You can use, for example, the function scatter3D() from the package plot3D . Use the class labels ( PB_class ) to plot inliers and outliers in different colours (for example, inliers in black and outliers in red). Make sure you produce multiple plots from different angles (at least three). Recalling that the class labels would not be available in a practical application of unsupervised outlier detection, do the outliers (forged stamps) look easy to detect in an unsupervised way, assuming that the 3D visualisation of the data via PCA is a reasonable representation of the data in full space? How about in a supervised way? Why? Justify your answers. 
Activity 2: Unsupervised outlier detection In this second activity, you are asked to perform unsupervised outlier detection on the Stamps data in the 9dimensional space of the numerical predictors ( PB_Predictors ), using KNN Outlier with different values of the parameter  (at least the following three: ). For each , produce the same 3D PCA visualisation of the data as in Activity 1 (PCA), but rather than using the class labels to colour the points, use instead the resulting KNN Outlier Scores as a continuous, diverging colour scale. Then, for each , produce a second plot where the top-31 outliers according to the KNN Outlier Scores are shown in red, while the other points are shown in black. Do these plots give you any insights on the values of  that look more or less appropriate from an unsupervised perspective (ignoring the class labels)? Justify your answer. 
Activity 3: Supervised anomaly detection In this third activity you are asked to: 
1. Perform supervised classification of the Stamps data, using a KNN classifier with the same values of as used in Activity 2 (unsupervised outlier detection). For each classifier (that is, each value of ), compute the Area Under the Curve ROC (AUC-ROC) in a Leave-One-Out Cross-Validation (LOOCV) scheme. 
2. Compare the resulting (supervised) KNN classification performance for each value of , against the classification performance obtained in an unsupervised way by the KNN Outlier method with the same value of . Notice that, if we rescale the KNN Outlier Scores (obtained in Activity 2 (unsupervised outlier detection)) into the  interval, these scores can be interpreted as outlier probabilities, which can then be compared with the class labels (ground truth) in PB_class to compute an AUC-ROC value. This way, for each value of , the AUC-ROC of the supervised KNN classifier can be compared with the m m k k = 5,25,100 k k k k k k k [0,1] k
AUC-ROC of KNN Outlier as an unsupervised classifier. Compare the performances of the supervised versus unsupervised classifiers and discuss the results. For example, recalling that the supervised method makes use of the class labels, whereas the unsupervised method doesn’t, what can you conclude considering there are applications where class labels are not available?

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