text:
\section{Introduction}
\label{sec:Intro}
 Knowledge Discovery and Data Mining (KDD) is an interdisciplinary area focusing on methodologies for extracting useful information from data. Classification is a fundamental tool in this domain. In the terminology of machine learning\cite{Ref1}, classification is considered an instance of supervised learning, i.e., learning where a training set of correctly identified observations is available. The task of the constructed classifier is to predict class labels given any input information such as financial crisis prediction\cite{Ref2}\cite{Ref3}. In this introductory section, we provide a brief overview of some well-developed classification algorithms to shed light on the complex issues of class imbalance.\\
       Decision trees can be used to visually and explicitly represent decisions and the decision making process. A typical tree consists of three components. The inner tree nodes represent  features with thresholds leading to the branching of edges eventually ended up in terminal nodes called leaves labeled with class decisions. In tree building, the decision tree model is built by recursively splitting the training data set based on a locally optimal criterion (typically information gain or gini impurity until certain stopping criteria are met. After building the decision tree, a tree pruning step is performed to prevent overfitting by reducing the size of the tree. In the presence of the class imbalance, more branches may need to be built in order to identify the minority class. CART is a well-known tree algorithm\cite{Ref4}.\\
       A random forest (RF) or random decision forest\cite{Ref5} is an ensemble learning method obtained by generating many decision trees and then combine their decisions. RF is a popular bagging algorithm which incorporates those trees\cite{Ref6}. Two types of randomness are built in the process. First, tree bagging is employed to select a bootstrap sample from original the original data for each decision tree. The bootstrap resampling method was introduced by Bradley Efrons\cite{Ref7}. It is implemented by sampling with replacement from an existing sample data independently with same size and conducting experiments among them. Second, a subset of features is randomly selected to generate the best split in each tree building process. The reason in this is to decrease the correlation between trees and reduce overfitting problem. \\
       A Support Vector Machine (SVM) introduced by Cortes is a discriminative classifier based on a separating hyperplane\cite{Ref8}.  Given training data, for two-group (i.e., bi-class) classifications, the algorithm outputs an optimal hyperplane where the margin is maximal. Akbani\cite{Ref9} illustrates that SVMs can be ineffective in determining the class boundary when the distribution is skewed.\\
       A neural network is a series of algorithms that aims to discover the underlying relationships in a set of training data through a process that mimics the way the human brain operates. It is a beautiful bio-inspired programming paradigm. However, some experimental results demonstrate that its performance when dealing with imbalanced data is not satisfactory. The principal reason is the minority class cannot be adequately weighted\cite{Ref10}.\\
       Besides these arguably the most fundamental classifiers, there are still many other classification algorithms which we will not be able to introduce exhaustively here. Our goal is modest as we aim to develop a better decision tree algorithm, and on top of which, a better random forest algorithm, and them compare them to these four original and fundamental algorithms: CART, Random Forest (RF), SVM, and Neural Network, plus the weighted version of CART and RF that in a way reflect the current state-of-the-art on combing class-imbalance. In the next section, we shall introduce several common solutions to the class-imbalance problem.

\section{Class imbalance problem}
\label{sec:2}
      A number of solutions to the class imbalance problem have been presented in the literature. Basically they can be divided into 3 broad categories, i.e., cost-sensitive learning, data level and algorithm level approaches.
\subsection{Cost-sensitive learning}
\label{sec:2.1}
       Cost-sensitive learning first introduced by Elkan\cite{Ref11} has shown to be an effective technique for incorporating the different misclassification costs into the classification process. The goal of this type of learning is to minimize the total cost of misclassification. The key point is to assign different values to the false positives and negatives. If we focus on identifying the minority which is the common case, the cost for labeling a positive example as negative can be higher than the cost for labeling a negative example as positive. Weighted Random Forest is the extended version that makes random forest more suitable for learning from extremely imbalanced data based on the idea of cost sensitive learning\cite{Ref12}\cite{Ref13}. Kim\cite{Ref14} proposed the weighted support vector machine which imposes weights to the loss term. Likewise, some other learning algorithms have their cost-sensitive correspondings, such as cost-sensitive decision tree, cost-sensitive neural network\cite{Ref15}\cite{Ref16}\cite{Ref17}\cite{Ref18}\cite{Ref19}, etc.
\subsection{Data level approaches}
\label{sec:2.2}
       The data-level approaches can also be interpreted as resampling techniques. That is, we can either undersample the majority class or oversample the minority class to attain the balanced distribution (the ratio of class distribution is 1:1).\\
    Undersampling can be defined as removing some observations of the majority class. Some algorithms have been applied to alleviate the skewness, e.g., random undersampling and cluster-based undersampling\cite{Ref20}. However an obvious drawback is that we are removing information that may be valuable which can then lead to an underfitting problem.\\
      Similarly, oversampling before splitting can achieve more minority observations. Well-known algorithms such as SMOTE\cite{Ref21} and ADASYN\cite{Ref22} have been widely implemented in dealing with imbalanced classification problems. However, it can cause overfitting and poor generalization.
\subsection{Algorithm level approaches}
\label{sec:2.3}
It is always a good rule of thumb to modify the training procedure of algorithms to boost its performance with uneven datasets. To develop an algorithmic solution, one needs in-depth knowledge of both the classifier learning algorithm and the application domain. Based on the receiver operating characteristics (ROC), our group had invented the ROC classification trees and the ROC random forest to identify the rare cases. Song suggested choosing the threshold with the largest harmonic mean of sensitivity and specificity as the node splitting criterion\cite{Ref23}\cite{Ref24}. Yan\cite{Ref25} extended the bi-class classification ROC Tree and ROC RF\cite{Ref23} to multiple classes. Our proposed algorithm PRC classification trees, and later the PRC random forest, are developed from a different perspective as will be introduced in details in the following chapter.
\subsection{Ensemble methods}
\label{sec:2.4}
       It is well known that through combining classifiers one can improve the prediction accuracy. It is effective to embed the data level approaches in boosting procedure which is one of the most popular combination techniques. Sun\cite{Ref26} introduced the cost items into the learning framework of AdaBoost for improving the classification of imbalanced data\cite{Ref27}. Boosting has been proven to be a general and effective method for improving the accuracy of a given learning algorithm\cite{Ref28}\cite{Ref29}. Nitesh\cite{Ref30} combined the SMOTE algorithm and the boosting procedure and WOTBoost\cite{Ref31} adjusted its oversampling strategy at each round of boosting to synthesize more targeted minority data samples. Similarly, RUSBoost alleviated class imbalance by introducing random undersampling techniques into the standard boosting procedure\cite{Ref32}. MEBoost mixed two different weak learners with boosting to improve the performance on imbalanced datasets\cite{Ref33}. So far, many learning methods have had their boosting counter parts, such as DataBoost\cite{Ref34}, EUSBoost\cite{Ref35}.

\subsection{Performance evaluation}
\label{sec:2.5}
Performance metric plays a key role in evaluating the classifier. Additionally, it is a guideline for the modeling process of classifiers. Accuracy is a traditional measure to evaluate the performance. It may be misleading, however, for a data set with unbalanced class distribution. Suppose we give every observation the prevalent class label, the accuracy can be high but the performance for the rare class is very poor. The following is a brief summary on some commonly used performance metrics.
\subsubsection{Confusion matrix}
\label{sec:2.5.1}
In the bi-class scenario, positive and negative class samples can be categorized into four groups based on the classification process. This can be denoted as the confusion matrix shown in Tabel 1.
\begin{table}
\centering
\caption{Confusion matrix}
\label{tab:1}
\begin{tabular}{lll}
\hline\noalign{\smallskip}
& Predicted as Positive & Predicted as Negative  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Actually Positive & True Positive (TP) &False Negative (FN) \\
Actually Negative &  False Positive (FP) & True Negative (TN) \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\\Several measures can be derived from this matrix:
\begin{align*}
  &True \, positive\, rate (Recall/Sensitivity): TPR=\frac{TP}{TP+FN}\tag{1}\\
  &False \, negative\, rate : FNR=\frac{FN}{TP+FN}\tag{2}\\
  &True\, negative\, rate (Specificity):TNR=\frac{TN}{TN+FP}\tag{3}\\
  &False\, positive\, rate :FPR=\frac{FP}{TN+FP}\tag{4}\\
  &Positive\, predictive\, value (Precision):PPV=\frac{TP}{TP+FP}\tag{5}\\
  &Negative\, predictive\, value :NPV=\frac{TN}{TN+FN}.\tag{6}\\
\end{align*}
\subsubsection{F-measure}
\label{sec:2.5.2}
For situation where the performance of positive class is the priority, two measures of the indicators described earlier are crucial – the true positive rate and the positive predictive value. Specifically, the true positive rate is defined as recall referring to the percentage of positive class samples that are correctly classified. Correspondingly, the positive predictive value is defined as precision denoting the percentage of predictive positive class samples that are correctly classified. \\
      F-measure is devised to integrate these two measures. In principle, it is the harmonic mean of precision and recall:
      \begin{equation}
F_{measure}=\frac{2}{recall^{-1}+precision^{-1}} .\tag{7}
\end{equation}
      \\In general, F-measure tends to be closer to the smaller one of the measures. Additionally, a high value indicates that both two measures are reasonably high and convincing.

\section{Methodology }
\label{sec:3}
\subsection{ROC Curve and the area under the ROC curve (AUC)}
\label{sec:3.1}
ROC curve is a performance measurement for classification problem at various thresholds settings. The curve is plotted based on the ROC space with True positive rate (TPR) against the false positive rate (FPR) where TPR is on y-axis and FPR is on the x-axis. In the ROC space, a ROC curve is created by connecting all the pairs of TPR and FPR at each threshold for a specific classifier. It is a common method to calculate the area under the ROC curve (AUC) to compare the measurement of separability among different classifiers. The higher AUC refers to a stronger ability of distinguishing between classes.  There are several methods to perform a computational analysis of AUC estimators. The way we used in our analysis is to apply the trapezoidal estimators.  The estimation is based on an approximation of the entire area by summing up all subareas and the formula is:
\begin{equation}
   AUC_{trapezoid}=\frac{1}{2}\cdot\sum_{i=1}^{n}(f_{i+1}-f_{i})\cdot(t_{i+1}+t_{i})\tag{8}
\end{equation}
where f is the function of FPR and t is of TPR, the trajectory is partitioned in n-1 sections.
\subsection{PRC and the area under the precision-recall curve (AUPRC)}
\label{sec:3.2}
For the classification problem, the performance is typically defined with the confusion matrix generated by the associated classifier. It is possible to compute the recall (sensitivity), precision and specificity. Similar to ROC curves described in the previous chapter, we connect all the pair of measurements at each threshold to make the PRC plot. Precision-Recall curve plots Precision which is the fraction of observations with a positive predicted value that are truly positive versus Recall, measuring the fraction of the examples with positive labels that get a positive predicted result. \\
The perfect classifier will have a PRC that passes through the upper right corner (corresponding to 100\% precision and 100\% recall). In general, the closer a point is to that position, the better the test is. Figure 1 and 2 show PRC and ROC curve for a perfect test respectively. To compare two classifiers based on PRC, the reasonable measure is the area under precision-recall curve (AUPRC). The higher it is, the better the performance of the classifier is. If the area under ROC (AUC) is less than 0.5, representing the baseline value of random guessing, we should convert the negative predictions into positive ones to get a better result. For AUPRC, the corresponding baseline value is the ratio of positive cases in the distribution expressed as
\begin{equation}
Baseline \, Value =\frac{positive\,cases}{positive\,cases+negative\, cases} .\tag{9}
\end{equation}
\begin{figure*}
  \includegraphics[width=0.75\textwidth]{prcurve.eps}
\caption{PRC for a perfect test}
\label{fig:1}
\end{figure*}
\begin{figure*}
  \includegraphics[width=0.75\textwidth]{Rocurve.eps}
\caption{ROC curve for a perfect test}
\label{fig:2}
\end{figure*}
Algorithm 1 below shows the process of calculating the AUPRC and the computation\cite{Ref36} in the case of it less than the baseline value. For the estimation of AUPRC, we also applied the trapezoidal approach:
\begin{equation}
AUPRC_{trapezoid}=\frac{1}{2}\cdot\sum_{i=1}^{n}(r_{i+1}-r_{i})\cdot(p_{i+1}+p_{i})\tag{10}
\end{equation}
where r is the function of recall and p is of precision, the trajectory is partitioned in n-1 sections.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE Training data $(x_1,y_1 )$,\ldots,$(x_n,y_n )$, Feature set $f_i \in F$, target vector $y_i \in \{-1,+1\}$.
\ENSURE Area under precision-recall curve (AUPRC) of feature $f_i$.
\STATE \textbf{sort} data $(X[f_i ], Y)$ by feature
\STATE \textbf{set} $uniq\_ values_{f_i} \leftarrow sort (unique(f_i))$
\STATE \textbf{set} $total\_ positives \leftarrow length (which(Y==1))$
\STATE \textbf{set} $total\_ negatives \leftarrow length (which(Y==-1))$
\STATE \textbf{set} $PRC_{baseline} \leftarrow \frac{total\_ positives}{total\_ positives+total\_ negatives}$
\STATE \textbf{set} $recall\_ array \leftarrow rep(0,length(uniq\_ values_{f_i})$
\STATE \textbf{set} $precision\_ array \leftarrow rep(0,length(uniq\_ values_{f_i})$
\STATE \textbf{set} $AUPRC_{f_i} \leftarrow 0$
\FOR{$j=1$ to $length(uniq\_ values_{f_i})$}
\STATE $indice \leftarrow  which(X[f_i]\leq uniq\_ values_{f_i}[j])$
\STATE $recall\_ array \leftarrow \frac{length(which((X[f_i ],Y)[indice,"Y"]==1))}{total\_ positives}$
\STATE $precision\_ array \leftarrow \frac{length(which((X[f_i ],Y)[indice,"Y"]==1))}{length(indice)}$
\IF{$precision\_ array[j]< PRC_{baseline}$}
\STATE $recall\_ array[j] \leftarrow 1-recall\_ array[j]$
\STATE $precision\_ array[j] \leftarrow \frac{total\_ positives-length(which((X[f_i],Y)[indice,"Y"]==1))}{nrow(X[f_i],Y  )-length(indice)}$
\ENDIF
\IF{$j==1$}
\STATE  $AUPRC_{f_i} \leftarrow \frac{recall\_ array[j]\cdot(1+precision\_ array[j])}{2}$
\ELSE \STATE $AUPRC_{f_i} \leftarrow AUPRC_{f_i}+$\STATE$\frac{(recall\_ array[j]-recall\_ array[j-1])\cdot(precision\_ array[j]+precision\_ array[j-1])}{2}$
\ENDIF
\ENDFOR
\RETURN $AUPRC_{f_i}$
\STATE \textbf{end}
\end{algorithmic}
\caption{AUPRC\_calculation}
\label{al1}
\end{algorithm}
      The Precision-recall curve is usually a tortuous curve, fluctuating up and down. Consequently, PRC tends to intersect each other much more often than the ROC curve. The main difference between them is that the number of true-negative results is not used for constructing the PRC. Ekelund\cite{Ref37} illustrates that the precision-recall curves are not impacted by the addition of patients without disease which are the true-negative cases. Evaluation of classifiers via receiver operating characteristics (ROC) curve have been commonly used. However, for imbalanced data, the ROC may tend to give an optimistic view since the classifier is more likely to assign the observation as the majority class. In many real-life problems, we care more about the true positive cases, e.g., fraud detection and disease diagnosis. The Precision-Recall plot is more informative than the ROC Plot when evaluating binary classifiers on imbalanced datasets\cite{Ref38}. Therefore, we propose the classification trees based on maximizing the area under the Precision-Recall curve.
\subsection{Feature selection}
\label{sec:3.3}
Feature selection is deciding which variable to include in the model. The data features that are used to train tree-based models have a huge influence on the performance we can achieve. In the case of the PRC classification tree, it is done by Algorithm 2. It lets AUPRC decide which feature variable is useful to include in every split of tree node. Typically, the feature that yields the largest AUPRC value should be selected. In Algorithm 2, the function "AUPRC\_calculation" is implemented by Algorithm 1. The below pseudo code illustrates the process of generating the suitable feature with the largest AUPRC value.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE Training data $(x_1,y_1 )$,\ldots,$(x_n,y_n )$, Feature set $f_i \in F$, target vector $y_i \in \{-1,+1\}$.
\ENSURE Feature $f_i$ with the largest AUPRC.
\\ \textbf{Require}: AUPRC\_calculation.
\STATE \textbf{set} $Max_f  \leftarrow NULL$
\STATE \textbf{set} $Max_{AUPRC} \leftarrow 0$
\FOR{each feature $f_i \in F$}
\STATE $Temp_{AUPRC} \leftarrow AUPRC\_calculation(X[f_i],Y  )$
\IF{$Temp_{AUPRC}> Max_{AUPRC}$}
\STATE $Max_f \leftarrow f_i$
\STATE $Max_{AUPRC} \leftarrow Temp_{AUPRC}$
\ENDIF
\ENDFOR
\RETURN $Max_f, Max_{AUPRC}$
\STATE \textbf{end}
\end{algorithmic}
\caption{Feature\_Selection\_by\_AUPRC}
\label{al2}
\end{algorithm}
\subsection{Threshold selection}
\label{sec:3.4}
It is important to calculate the optimum threshold for each split in the PRC Tree. This could be done by selecting the largest F1-score in the chosen feature variable. The F-score is a measure of accuracy which is the harmonic mean of precision and recall.\\
          Note that the F-score does not take the true negatives into account. That's the reason that it is more effective than other measures in the classification for imbalanced data. Algorithm 3 illustrates the procedure of threshold selection. The basic goal is to choose the cutoff point with the largest F1 after comparing all the scores for the threshold set of the feature achieved by Algorithm 2.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE PRC matrix ($recall\_ array,precison\_ array$) of selected feature$f_i$, where $f_i \in F$; Threshold set $\Theta$.
\ENSURE Selected splitting threshold $\theta$.
\\ \textbf{Require}: function $HarmonicMean$.
\STATE \textbf{set} $Max_{F1}  \leftarrow 0$
\STATE \textbf{set} $Max_{\theta}  \leftarrow NULL$
\STATE \textbf{set} $uniq\_ split_{f_j} \leftarrow sort(unique(f_i))$
\FOR{{$j=1$ to $length(uniq\_ split_{f_j})$}}
\STATE $Temp_{F1} \leftarrow HarmonicMean(recall\_ array[j],precison\_ array[j])$
\IF{$Temp_{F1}> Max_{F1}$}
\STATE $Max_{F1} \leftarrow Temp_{F1}$
\STATE $Max_{\theta} \leftarrow uniq\_ split_{f_j}[j]$
\ENDIF
\ENDFOR
\RETURN $Max_{\theta}$
\STATE \textbf{end}
\end{algorithmic}
\caption{Threshold\_Selection\_by\_F1-score}
\label{al3}
\end{algorithm}
\subsection{PRC Tree algorithm}
\label{sec:3.5}
         The following Algorithm 4 is employed to generate the PRC Tree. The previous Algorithm 2 and 3 complete a node split process in each stage. Instead of measuring by traditional Gini impurity or information gain, the features and threshold are selected by AUPRC and F1-score respectively. Algorithm 4 "PRC\_Tree" integrates the previous algorithms to generate each split node until the corresponding stopping criteria is met. Each node has scores for different classes, measuring the percentage of each class in it. It is called nodescore in the algorithm. The corresponding nodelabel is the majority class of this node. By doing so, it could be easy to achieve the majority target label in the Terminal node. Below is the pseudo code to build PRC classification tree recursively. The prediction of a PRC Tree, $\mathcal{T}$, with $K$ terminal nodes and depth $L$, can  be written as
\begin{equation}
g(x_i;\hat{y},K,L)= \sum_{i=1}^{K}\hat{y_{k}}\mathbbm{1}_{\{x_i\in C_k(L)\}},\tag{11}
\end{equation}
where $C_k(L)$ is one of the $K$ partitions of the data. Each partition is a product of up to $L$ indicator function of the features which are selected by AUPRC algorithm.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE Training data $(x_1,y_1 )$,\ldots,$(x_n,y_n )$, Feature set $f_i \in F$, target vector $y_i \in \{-1,+1\}$; stopping criterion (maximum tree depth, minimum leaf size); $N_f$, the number of features used in each split.
\ENSURE PRC tree.
\\ \textbf{Require}: Feature\_Selection\_by\_AUPRC and Threshold\_Selection\_by\_F1-score
\STATE \textbf{Do} the nodescore and nodelabel for the root node.
\IF{the stopping criterion is met}
\STATE \textbf{return} PRC tree
\ELSE
\STATE sample $N_f$ features from the feature set $F$
\STATE \textbf{set} the selected features as $F^{'}$
\STATE $(Max_f, Max_{AUPRC}) \leftarrow Feature\_Selection\_by\_AUPRC(X,Y,F^{'})$
\STATE $Max_{\theta} \leftarrow Threshold\_Selection\_by\_F1-score(Max_f, Max_{AUPRC})$
\STATE $\{(X,Y)_{left},(X,Y)_{right}\} \leftarrow Node\_Split(Max_f, Max_{\theta})$
\STATE apply the function $PRC\_Tree\,(maximum tree depth\leftarrow maximum tree depth-1)$ to the subsets $\{(X,Y)_{left},(X,Y)_{right}\}$ recursively until resulting nodes are pure or the stopping criteria is met
\ENDIF
\RETURN PRC tree
\STATE \textbf{end}
\end{algorithmic}
\caption{PRC\_Tree}
\label{al4}
\end{algorithm}

\subsection{PRC random forest algorithm}
\label{sec:3.6}
Random forests are an ensemble learning method for classification by constructing a multiple of classification trees. We can build our PRC random forest by treating the PRC tree as the base classifier. The PRC random forest has competitive predictive performance and provides a reliable feature importance estimate. Algorithm 5 "PRC\_Random\_Forest" states how to build the forests based on the PRC tree. The parameter $N_t$ is used to decide the number of trees to form the "forest". As discussed before, the number of features $N_f$ for each node split is randomly chosen from the feature set F. It can decrease the prediction error of the model by doing so. Below is the procedure of our "forests".

\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE Training data $(x_1,y_1 )$,\ldots,$(x_n,y_n )$, Feature set $f_i \in F$, target vector $y_i \in \{-1,+1\}$; Number of trees $N_t$; Number of features for each node split $N_f$.
\ENSURE PRC random forest $\Re$.
\\ \textbf{Require}: PRC\_Tree
\STATE \textbf{Set} $\Re \leftarrow NULL$.
\FOR{{$j=1$ to $N_t$}}
\STATE generate bootstrap sample $(X,Y)^j$
\STATE for each node split, generate $F^{'}$ by randomly choosing $N_f$ features from $F$
\STATE $prc\_tree_j \leftarrow PRC\_Tree((X,Y)^j,F^{'})$
\STATE append $prc\_tree_j\, to\, \Re$
\ENDFOR
\RETURN PRC random forest $\Re$
\STATE \textbf{end}
\end{algorithmic}
\caption{PRC\_Random\_Forest}
\label{al5}
\end{algorithm}

\subsection{PRC-ROC Classification Trees }
\label{sec:3.7}
     Each evaluation approach has its own merits and demerits. As an aside, it is comprehensive to combine the PRC and ROC classification trees in meaningful and optimal ways. The following chapters will explore the method based on PRC and ROC curves in more detail.
\subsubsection{Feature selection}
\label{sec:3.7.1}
The one valuable feature of the RF is that we can use the out-of-bag (OOB) samples to make objective performance judgements based on the training data only. The OOB error is an error estimation technique often used to evaluate the accuracy of a random forest. We will use the weighted average of both areas to do the feature selection. Based on the training data and OOB accuracy score, we can tune the parameter to find the optimal weighted parameter for a given data. The parameter can be expressed as
\begin{equation}
a=\frac{(1-OOB\_error_{PRC RF})}{(1-OOB\_error_{PRC RF}+1-OOB\_error_{ROC RF} )}.\tag{12}
\end{equation}
Hence, the weighted average of both areas becomes
\begin{equation}
Area_{weighted}=a(AUPRC) + (1-a)(AUC), where\, 0\leq a\leq1.\tag{13}
\end{equation}
In the case of the PRC-ROC classification tree, it is done by Algorithm 7. Instead of AUPRC, it lets the weighted average of AUPRC and AUC decide which feature variable is useful to include in every split of tree node. The feature that yields the largest mean value should be selected in a similar fashion. Algorithm 6\cite{Ref25} shows how the function "AUC\_calculation" is implemented. Then we achieves the algorithm by integrating it to the previous feature selection Algorithm 2.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE Training data $(x_1,y_1 )$,\ldots,$(x_n,y_n )$, Feature set $f_i \in F$, target vector $y_i \in \{-1,+1\}$.
\ENSURE Area under curve (AUC) of feature $f_i$.
\STATE \textbf{sort} data $(X[f_i ], Y)$ by feature
\STATE \textbf{set} $uniq\_ values_{f_i} \leftarrow sort (unique(f_i))$
\STATE \textbf{set} $total\_ positives \leftarrow length (which(Y==1))$
\STATE \textbf{set} $total\_ negatives \leftarrow length (which(Y==-1))$
\STATE \textbf{set} $tp\_ array \leftarrow rep(0,length(uniq\_ values_{f_i})$
\STATE \textbf{set} $fp\_ array \leftarrow rep(0,length(uniq\_ values_{f_i})$
\STATE \textbf{set} $AUC_{f_i} \leftarrow 0$
\FOR{$j=1$ to $length(uniq\_ values_{f_i})$}
\STATE $indice \leftarrow  which(X[f_i]\leq uniq\_ values_{f_i}[j])$
\STATE $tp\_ array \leftarrow \frac{length(which((X[f_i ],Y)[indice,"Y"]==1))}{total\_ positives}$
\STATE $fp\_ array \leftarrow \frac{length(which((X[f_i ],Y)[indice,"Y"]==-1))}{total\_ negatives)}$
\IF{$j==1$}
\STATE  $AUC_{f_i} \leftarrow \frac{tp\_ array[j]*fp\_ array[j]}{2}$
\ELSE \STATE $AUC_{f_i} \leftarrow AUC_{f_i}+$
\STATE$\frac{(fp\_ array[j]-fp\_ array[j-1])\cdot(tp\_ array[j]+tp\_ array[j-1])}{2}$
\ENDIF
\ENDFOR
\IF{$AUC_{f_i}< 0.5$}
\STATE $AUC_{f_i} \leftarrow 1-AUC_{f_i}$
\ENDIF
\RETURN $AUC_{f_i}$
\STATE \textbf{end}
\end{algorithmic}
\caption{AUC\_calculation}
\label{al6}
\end{algorithm}

\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE Training data $(x_1,y_1 )$,\ldots,$(x_n,y_n )$, Feature set $f_i \in F$, target vector $y_i \in \{-1,+1\}$.
\ENSURE Feature $f_i$ with the largest weighted average of AUPRC and AUC.
\\ \textbf{Require}: AUPRC\_calculation, AUC\_calculation.
\STATE \textbf{set} $Max_f  \leftarrow NULL$
\STATE \textbf{set} $Max_{AUPRC\_AUC} \leftarrow 0$
\FOR{each feature $f_i \in F$}
\STATE $Temp_{AUPRC\_AUC} \leftarrow Area_{weighted}( AUPRC\_calculation(X[f_i],Y),$ \STATE $AUC\_calculation(X[f_i],Y))$
\IF{$Temp_{AUPRC\_AUC}> Max_{AUPRC\_AUC}$}
\STATE $Max_f \leftarrow f_i$
\STATE $Max_{AUPRC\_AUC} \leftarrow Temp_{AUPRC\_AUC}$
\ENDIF
\ENDFOR
\RETURN $Max_f, Max_{AUPRC\_AUC}$
\STATE \textbf{end}
\end{algorithmic}
\caption{Feature\_Selection\_by\_AUPRC\_AUC}
\label{al7}
\end{algorithm}

\subsubsection{Threshold selection}
\label{sec:3.7.2}
To calculate the optimum threshold for each split in the PRC-ROC Tree, we select the largest harmonic mean of three performance metrics, i.e., recall, precision and specificity in the chosen feature variable. The F-score in this case becomes the harmonic mean of precision, recall and specificity:
      \begin{equation}
F_{3}=\frac{3}{recall^{-1}+precision^{-1}+specificity^{-1}} .\tag{14}
\end{equation}
          Algorithm 8 illustrates the procedure of threshold selection. The main goal is to choose the cutoff point with the largest harmonic mean after comparing all the scores for the threshold set of the feature achieved by feature selection.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE PRC\_AUC matrix ($recall\_ array, precison\_ array, fp\_ array$) of selected feature$f_i$, where $f_i \in F$; Threshold set $\Theta$.
\ENSURE Selected splitting threshold $\theta$.
\\ \textbf{Require}: function $HarmonicMean$.
\STATE \textbf{set} $Max_{F3}  \leftarrow 0$
\STATE \textbf{set} $Max_{\theta}  \leftarrow NULL$
\STATE \textbf{set} $uniq\_ split_{f_j} \leftarrow sort(unique(f_i))$
\FOR{{$j=1$ to $length(uniq\_ split_{f_j})$}}
\STATE $Temp_{F3} \leftarrow HarmonicMean(recall\_ array[j],precison\_ array[j],fp\_ array[j])$
\IF{$Temp_{F3}> Max_{F3}$}
\STATE $Max_{F3} \leftarrow Temp_{F3}$
\STATE $Max_{\theta} \leftarrow uniq\_ split_{f_j}[j]$
\ENDIF
\ENDFOR
\RETURN $Max_{\theta}$
\STATE \textbf{end}
\end{algorithmic}
\caption{Threshold\_Selection\_by\_F3-score}
\label{al8}
\end{algorithm}

\subsubsection{PRC-ROC Tree algorithm}
\label{sec:3.7.3}
         The following Algorithm 9 is employed to generate the PRC-ROC Tree. The following algorithm "PRC-ROC\_Tree" integrates the previous algorithms to generate each split node until the corresponding stopping criteria is met. Each node has scores for different classes, measuring the percentage of each class in it. It is called nodescore in the algorithm. The corresponding nodelabel is the majority class of this node. By doing so, it could be easy to achieve the majority target label in the Terminal node. Below is the pseudo code to build the classification tree recursively. The prediction of a PRC-ROC Tree, $\mathcal{T}$, with $K$ terminal nodes and depth $L$, can also be written as
\begin{equation}
g(x_i;\hat{y},K,L)= \sum_{i=1}^{K}\hat{y_{k}}\mathbbm{1}_{\{x_i\in C_k(L)\}},\tag{15}
\end{equation}
where $C_k(L)$ is one of the $K$ partitions of the data. Each partition is a product of up to $L$ indicator function of the features which are selected by AUPRC\_AUC algorithm.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE Training data $(x_1,y_1 )$,\ldots,$(x_n,y_n )$, Feature set $f_i \in F$, target vector $y_i \in \{-1,+1\}$; stopping criterion (maximum tree depth, minimum leaf size); $N_f$, the number of features used in each split.
\ENSURE PRC-ROC tree.
\\ \textbf{Require}: Feature\_Selection\_by\_AUPRC\_AUC and Threshold\_Selection\_by\_F3-score
\STATE \textbf{Do} the nodescore and nodelabel for the root node.
\IF{the stopping criterion is met}
\STATE \textbf{return} PRC-ROC tree
\ELSE
\STATE sample $N_f$ features from the feature set $F$
\STATE \textbf{set} the selected features as $F^{'}$
\STATE $(Max_f, Max_{AUPRC\_AUC}) \leftarrow Feature\_Selection\_by\_AUPRC\_AUC(X,Y.F^{'})$
\STATE $Max_{\theta} \leftarrow Threshold\_Selection\_by\_F3-score(Max_f, Max_{AUPRC\_AUC})$
\STATE $\{(X,Y)_{left},(X,Y)_{right}\} \leftarrow Node\_Split(Max_f, Max_{\theta})$
\STATE apply the function $PRC-ROC\_Tree\,(maximum tree depth\leftarrow maximum tree depth-1)$ to the subsets $\{(X,Y)_{left},(X,Y)_{right}\}$ recursively until resulting nodes are pure or the stopping criteria is met
\ENDIF
\RETURN PRC-ROC tree
\STATE \textbf{end}
\end{algorithmic}
\caption{PRC-ROC\_Tree}
\label{al9}
\end{algorithm}

\subsubsection{PRC-ROC random forest algorithm}
\label{sec:3.7.4}
    We can build our PRC-ROC random forest by treating the PRC-ROC tree as the base classifier in a similar way. The PRC-ROC random forest has competitive predictive performance and provides a reliable feature importance estimate. Algorithm 10 "PRC-ROC\_Random\_Forest" states how to build the forests based on the PRC-ROC tree. The parameter $N_t$ is used to decide the number of trees to form the "forest". As discussed before, the number of features $N_f$ for each node split is randomly chosen from the feature set $F$. It can decrease the prediction error of the model by doing so.
\begin{algorithm}[H]
\begin{algorithmic}[1]
\REQUIRE Training data $(x_1,y_1 )$,\ldots,$(x_n,y_n )$, Feature set $f_i \in F$, target vector $y_i \in \{-1,+1\}$; Number of trees $N_t$; Number of features for each node split $N_f$.
\ENSURE PRC-ROC random forest $\Re$.
\\ \textbf{Require}: PRC-ROC\_Tree
\STATE \textbf{Set} $\Re \leftarrow NULL$.
\FOR{{$j=1$ to $N_t$}}
\STATE generate bootstrap sample $(X,Y)^j$
\STATE for each node split, generate $F^{'}$ by randomly choosing $N_f$ features from $F$
\STATE $prc\_tree_j \leftarrow PRC-ROC\_Tree((X,Y)^j,F^{'})$
\STATE append $prc\_tree_j\, to\, \Re$
\ENDFOR
\RETURN PRC-ROC random forest $\Re$
\STATE \textbf{end}
\end{algorithmic}
\caption{PRC-ROC\_Random\_Forest}
\label{al10}
\end{algorithm}

\subsection{Complexity of algorithm}
\label{sec:3.8}
Let us consider a dataset with input size $n$. Time complexity is the measurement of the speed of the algorithm when performing for the input data. Random Forest is the ensemble model of decision trees. The situation is same to our proposed algorithms. For each tree, it will take $O(N_f\ast nlog(n))$, where $N_f$ is the number of features for each node split. For the random forest algorithm, the complexity would be $O(N_t\ast N_f\ast nlog(n))$ and $N_t$ is the number of the trees.

\section{Experimental Studies }
\label{sec:4}
        The experimental studies fall into two parts. Experiments in the first part are designed for simulation study. The goal is to evaluate the performance on classifying the imbalanced simulated data of binary classes. Experiments in the second part focus on several real-world data sets.\\
        The following step after implementing our proposed algorithm is to find out how effective the model is based on the metrics and datasets. Five performance metrics as follows are used to make evaluation between different algorithms in our analysis. Accuracy is the number of correct predictions made by the model over all kinds of predictions made. Specificity is a measure that tells us what proportion of the majority class, are predicted correctly. Precision shows us what proportion of the predicted minority class which is actually positive. Recall or sensitivity tells us the proportion of cases that actually belong to the minority class is identified by the algorithm as positive cases. F1-score, measuring both precision and recall, is the harmonic mean of them.
\subsection{Simulation study}
\label{sec:4.1}

          In the simulation experimental setting, the number of simulated data sets for classifying binary cases is set to 5,000 each. The feature variables for each class are generated from multivariate Gaussian distribution with specific mean and standard deviation. For our simulation, we divide the class-imbalance extent into 3 categories, namely "Mild", "Moderate" and "Extreme", according to the proportion of minority class. In general, the data set whose minority proportion is between 20\% and 40\% should be placed in the mild category. The criterion for moderate category is from 1\% to 20\% and the extreme case is when the proportion is less than 1\%. Similarly, the dimension of features set to be 5 is considered as low feature category and 20 for the higher dimension. Furthermore, the noisy data may also make it difficult to learn the rare class. In the real world, data contains various types of errors, either random or systematic. Random errors are often referred to as noise. To complete the simulation setting, we also give the scenarios with noisy features for comprehensive comparison. Overall, we have two categories (i.e., easy-to-classify and hard-to-classify) in the simulated data and the following sections will give more detail about setting.\\
    For the simulation study, we compared the performance of our proposed algorithms with that of CART, weighted CART and ROC Tree. The CART and weighted CART are implemented by functions in R package "rpart"\cite{Ref39}. Furthermore, we also compared the performance with random forest algorithms such as standard random forest and weighted version. Random forest (RF) is fit by the function in package "randomForest"\cite{Ref40} and weighted Random Forest (Weighted RF) is implemented in package "wsrf"\cite{Ref13}.
\subsubsection{Easy to classify, mild imbalanced data set with low feature dimension}
\label{sec:4.1.1}
  In this scenario, the degree of class-imbalance is set to be mild, i.e., 30\% of the simulated data points belong to the minority class. Five features are generated randomly from multivariate Gaussian distribution with specifies standard deviation 1. The means of the features for the majority and minority class are set to 0 and 3 respectively.
  Figure 3 shows the distribution of the data with 2 features. Table 2 shows the performance metrics of all compared algorithms and it is divided in two. The first block is for single classification tree and the other is for random forest. The best results are denoted in bold respectively. Both PRC algorithms have better performance in F1 score and accuracy than ROC. The PRC is slightly better than CART. In addition, the PRC-ROC tree achieves the best performance in all aspects in the first block. PRC-ROC RF yields the highest recall, accuracy and F1-score among the random forest algorithms.
\begin{figure*}[h]
\centering
\includegraphics[width=0.75\textwidth]{easy.eps}
\caption{Scatterplot of data in scenario 1 (Mild imbalanced extent with low feature dimension) for easy-to-classify setting }
\label{fig:3}
\end{figure*}

\begin{table}[h]
\centering
\caption{Summary of classification results under scenario 1 (Mild imbalanced extent with low feature dimension)}
\label{tab:2}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Algorithms&Recall&Specificity&Precision&Accuracy&F1 Score  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
CART&	0.9782	&0.9933	&0.9846	&0.9887 &0.9814\\
Weighted CART&	0.9760&	0.9722&	0.9391&	0.9733&	0.9572\\
ROC Tree&	0.9214&	\textbf{0.9942}&0.9860&	0.9720&	0.9526\\
PRC Tree&	0.9891&	0.9904&	0.9784&	0.9900&	0.9837\\
PRC-ROC Tree&\textbf{0.9934}&\textbf{0.9952}&\textbf{0.9891}&\textbf{0.9947}&\textbf{0.9912}\\
\cdashline{1-6}[0.8pt/2pt]
RF&	0.9476	&\textbf{0.9981}&\textbf{0.9954}	&0.9827 &0.9709\\
Weighted RF&	0.9934&	0.9923&	0.9827&	0.9927&	0.9880\\
ROC RF&	0.9476&	0.9923&0.9819&	0.9787&	0.9644\\
PRC RF&	\textbf{1.0000}&	0.9904&	0.9786&	0.9933&	0.9892\\
PRC-ROC RF&\textbf{1.0000}&0.9914&0.9807&\textbf{0.9940}&\textbf{0.9903}\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

\subsubsection{Easy to classify, moderate imbalanced data set with higher feature dimension}
In this scenario, the class-imbalance extent becomes moderate. The number of features changes from 5 to 15. Table 3 shows the performance metrics of all the algorithms and the best results are denoted in bold correspondingly. The ROC Tree is perfect in specificity and precision, but the performance of recall is not ideal. We can conclude that the PRC still outperforms both CART in this scenario. Furthermore, PRC-ROC is better than PRC in the five-dimensional evaluation. In the second block, the weighted RF work better than other algorithms under this situation.
\begin{table}[h]
\centering
\caption{Summary of classification results under scenario 2 (Moderate imbalanced extent with higher feature dimension)}
\label{tab:7}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Algorithms&Recall&Specificity&Precision&Accuracy&F1 Score  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
CART&	0.9556	&0.9963	&0.9627	&0.9927   &0.9591\\
Weighted CART&	0.9259&	0.9905&	0.9058&	0.9847&	0.9157\\
ROC Tree&	0.4667&	\textbf{1.0000}&\textbf{1.0000}&0.9520&	0.6364\\
PRC Tree&0.9704	&0.9985	&0.9850	&0.9960  &0.9776\\
PRC-ROC Tree&\textbf{0.9778}&0.9993&0.9925&	\textbf{0.9973}&\textbf{0.9851}\\
\cdashline{1-6}[0.8pt/2pt]
RF&0.9556&0.9832&	0.8487&	0.9807&0.8990\\
Weighted RF&0.9259&\textbf{1.0000}&\textbf{1.0000}&\textbf{0.9933}&\textbf{0.9615}\\
ROC RF&	\textbf{1.0000}&0.9875&	0.8882&	0.9887&0.9408\\
PRC RF&	\textbf{1.0000}&0.9685&	0.7584&	0.9713&0.8626\\
PRC-ROC RF&\textbf{1.0000}&0.9817&	0.8437&	0.9833&0.9152\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

\subsubsection{Easy to classify, extreme imbalanced data set with low feature dimension}
Under this scenario, only 1\% of the simulated data points belong to the minority class. The parameter setting of features remains the same as scenario 1. Table 4 shows the performance metrics. The PRC achieves a higher number in recall than the weighted one. Both PRC methods apparently outclass than CART and ROC especially in recall. PRC-ROC obtains larger values in specificity, precision and F1-score than other methods. Although the weighted RF achieves the best specificity and precision, PRC-ROC RF yields the highest recall, accuracy and F1-score among the random forest algorithms.
\begin{table}[h]
\centering
\caption{Summary of classification results under scenario 3 (Extreme imbalanced extent with low feature dimension)}
\label{tab:10}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Algorithms&Recall&Specificity&Precision&Accuracy&F1 Score  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
CART&	0.5833	&0.9980	&0.7000	&0.9947& 0.6363\\
Weighted CART&0.8333&0.9919&0.4545&	0.9907&	0.5882\\
ROC Tree&	0.3333&	0.9980&0.5714&0.9927&	0.4210\\
PRC Tree&\textbf{0.9167}&0.9973&	0.7333&	0.9967&	0.8148\\
PRC-ROC Tree&0.7500&\textbf{0.9993}&\textbf{0.9000}&\textbf{0.9973}&\textbf{0.8182}\\
\cdashline{1-6}[0.8pt/2pt]
RF&	0.7500&0.9993&0.9000&\textbf{0.9973}&0.8182\\
Weighted RF&0.5833&\textbf{1.0000}&\textbf{1.0000}&0.9967&0.7368\\
ROC RF&	0.8333&0.9973&0.7143&0.9960&0.7692\\
PRC RF&	\textbf{1.0000}&0.9966&	0.7059&	0.9967&0.8276\\
PRC-ROC RF&\textbf{1.0000}&0.9973&	0.7500&	\textbf{0.9973}&\textbf{0.8571}\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

\subsubsection{Hard to classify, mild imbalanced data set with higher feature dimension and noise}
In this hard-to-classify scenario, the degree of class-imbalance is set to be mild, i.e., 30\% of the simulated data points belong to the minority class. Five features are generated randomly from multivariate Gaussian distribution with specifies standard deviation 1. The means of the features for the majority and minority class are set to 0 and -1 respectively. Figure 4 shows the distribution of the data with 2 features. We can see many overlaps compared to the previous setting.
The setting for this scenario is with ten normal features and five noise features. We change their means to 1 to treat them as noisy features. Table 5 summarizes the performance metrics of the algorithms mentioned and the best performance is denoted in the same way. The results obviously indicate that in this specific case PRC-ROC surpasses other tree-algorithms in every metrics except the recall value. For the comparison of the random forest methods, PRC-ROC RF outperforms others.
\begin{figure*}[h]
\centering
  \includegraphics[width=0.75\textwidth]{hard.eps}
\caption{Scatterplot of data in scenario 4 (Mild imbalanced extent with low feature dimension) for hard-to-classify setting}
\label{fig:4}
\end{figure*}
\begin{table}[ht]
\centering
\caption{Summary of classification results under scenario 4 (Mild imbalanced extent with higher feature dimension and noise)}
\label{tab:17}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Algorithms&Recall&Specificity&Precision&Accuracy&F1 Score  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
CART&	0.6372&	0.9141&0.7554&	0.8327 &0.6913\\
Weighted CART&\textbf{0.7846}&	0.8036&0.6245&	0.7980&	0.6955\\
ROC Tree&	0.6689&	0.8687&0.6797&	0.8100&	0.6743\\
PRC Tree&	0.7347&	0.8820&0.7216&	0.8387&	0.7281\\
PRC-ROC Tree&0.7143&\textbf{0.9056}&\textbf{0.7590}&\textbf{0.8493}&\textbf{0.7360}\\
\cdashline{1-6}[0.8pt/2pt]
RF&	0.5011&0.8971&0.6697&0.7807&0.5733\\
Weighted RF&0.7279&	0.9452&0.8470&	0.8813&	0.7829\\
ROC RF&	0.9297&	0.8291&0.6937&	0.8587&	0.7945\\
PRC RF&\textbf{0.9751}	&0.8263&0.7003&0.8700&0.8152\\
PRC-ROC RF&0.9478&\textbf{0.8565}&\textbf{0.7333}&\textbf{0.8833}&\textbf{0.8269}\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

\subsubsection{Hard to classify, extreme imbalanced data set with higher feature dimension and noise}
 In this extreme data set, the features' distribution is the same as scenario 4. Table 6 summarizes the comparison of all mentioned methods and the best results are denoted in bold. The PRC-ROC tree attains the best performance in the last four dimensions and makes huge improvement in F1 score. The ROC RF achieves the highest F1-score in the second block.\\
\begin{table}[h]
\centering
\caption{Summary of classification results under scenario 5 (Extreme imbalanced extent with higher feature dimension and noise)}
\label{tab:25}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Algorithms&Recall&Specificity&Precision&Accuracy&F1 Score  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
CART&	0.2000&	0.9966&	0.2857&	0.9913&	0.2353\\
Weighted CART&	\textbf{0.6000}&	0.9403&	0.0632&	0.9380&	0.1144\\
ROC Tree&	0.1000&	0.9919&0.0769&	0.9860&	0.0869\\
PRC Tree&	0.2000&0.9987&0.5000&	0.9933&0.2857\\
PRC-ROC Tree&0.2000&\textbf{0.9993}&\textbf{0.6667}&\textbf{0.9940}&\textbf{0.3077}\\
\cdashline{1-6}[0.8pt/2pt]
RF&	0.1000&	0.9933&	0.0909&	0.9873&	0.0952\\
Weighted RF&0.1000&	\textbf{0.9960}&\textbf{0.1429}&\textbf{0.9900}&0.1177\\
ROC RF&\textbf{0.7000}	&	0.9322&0.0648&	0.9307&	0.1186\\
PRC RF&	0.6000&0.9483&0.0723&0.9460&0.1290\\
PRC-ROC RF&0.6000&0.9523&0.0779&0.9500&\textbf{0.1381}\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

To sum up, the comparisons under most of those cases illustrate that two PRC trees achieve better performance than the both CART trees when the data set is imbalanced in either easy-to-differentiate or hard-to-differentiate case. Especially in the case of the extreme data set, the PRC tree is more powerful to identify the minority class and achieves higher score in precision and recall than traditional CART. The ROC and PRC Tree have different focuses when dealing with imbalanced data.The PRC Tree is better than the ROC Tree when the main goal is to identify the rare cases and achieve the satisfactory performance in F1 score. Moreover,PRC-ROC is slightly better than PRC in our 5-dimensional evaluation in most cases since it inherits the advantage of both classification trees. For the second block, the two PRC RFs also show some advantages in classifying the minority cases. The following section moves on to the hard-to-classify setting. The next section focuses on the evaluation in the real-world data set.

\subsection{Real-world data set experiment}
\label{sec:4.2}
The PRC and PRC-ROC Tree have shown their advantage in classifying the simulated data set which is severely skewed. In this section we choose several datasets from Kaggle competition and UCI Machine Learning Repository to testify the feasibility of PRC random forests. We will compare the performance with standard classification algorithms such as random forest, SVM and neural network. In addition, weighted CART and ROC random forest are all taken into account. Random forest is fit by the function in package "randomForest"\cite{Ref40}. SVM fits in package "e1071"\cite{Ref41} and the package "neuralnet"\cite{Ref42} is for realizing neural network. Weighted Random Forest (Weighted RF) is implemented in package "wsrf"\cite{Ref13}.\\
The Table 7 summarizes the basic information of the three selected data sets. The first two data sets are about financial distress prediction and default of clients. The remaining one is about breast cancer diagnosis. For the credit data, this study employed a binary variable, default payment (Yes = 1, No = 0), as the dependent variable. Besides, this data set used 23 features as the independent variables. Similarly, the target variable in the second data is denoted by "Financial Distress" if it is greater than -0.50 the company should be considered as healthy (0). Otherwise, it would be regarded as financially distressed (1). We have 83 explanatory variables for it. The binary outcome of interest for the cancer data set that contains 30 attributes is whether a tumor is benign (0) or malignant (1). \\
\begin{table}[h]
\centering
\caption{Summary of selected real-world datasets}
\label{tab:26}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lllll}
\hline\noalign{\smallskip}
Dataset Name&	Data Sources&	Number of Observations&	Proportion of Minority Class&	Number of Features  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Default of Credit Card Clients&	UCI&	30000&	3.7\%&	23\\
Financial Distress Prediction&	Kaggle&	3672&	22\%&	83\\
Breast Cancer Wisconsin&	UCI&	569&	37\%&	30\\
\noalign{\smallskip}\hline
\end{tabular}%
}
\end{table}
    The out-of-bag error is an error estimation technique often used to evaluate the accuracy of a random forest. We split the data into training and testing and use the OOB to estimate the error rate of the training sample. The following table shows the out-of-bag error of all the corresponding random forests and the minimum values are colored in red. Then we still use the separate testing data to further validate our RF model built with the training data and the results are given in the last three tables. \\

\begin{table}[h]
\centering
\caption{Summary of Out-of-bag error rate}
\label{tab:27}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lllll}
\hline\noalign{\smallskip}
Dataset Name&	RF&	Weighted RF&	ROC RF&	PRC RF\\
\noalign{\smallskip}\hline\noalign{\smallskip}
Default of Credit Card Clients&	\textcolor{red}{0.1900}&	0.2300	&0.2446&	0.1921\\
Financial Distress Prediction&	0.0414&	0.0500&	0.0345&\textcolor{red}{0.0000}\\
Breast Cancer Wisconsin	&0.1000&0.0912&	0.0714&	\textcolor{red}{0.0455}\\
\noalign{\smallskip}\hline
\end{tabular}%
}
\end{table}

\begin{table}[h]
\centering
\caption{Summary of performance results for Default of Credit Card Clients Data}
\label{tab:28}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Algorithms&Recall&Specificity&Precision&Accuracy&F1 Score\\
\noalign{\smallskip}\hline\noalign{\smallskip}
SVM	&0.3343&	\textbf{0.9628}&	0.7204&	0.8229&	0.4567\\
\textcolor{blue}{Neural Network}&	0.3947&	0.9441&	0.6692&	0.8218&	0.4965\\
\cdashline{1-6}[0.8pt/2pt]
CART&	0.3378	&0.9627	& 0.7218	&0.8236 &0.4602\\
Weighted CART&	0.5788&	0.8253&	0.4870&	0.7704&	0.5289\\
ROC Tree&	\textbf{0.7061}&	0.5330&	0.3022&	0.5716&	0.4233\\
PRC Tree&	0.4187&	0.9200&	0.5997&	0.8083&	0.4931\\
PRC-ROC Tree &	0.3353&	0.9634&	\textbf{0.7241}&	0.8236&	0.4584\\
RF&	0.3493&	0.9608	&0.7187&	0.7839&	0.4701\\
Weighted RF&	0.3558&	0.9570&	0.7032&	0.8231&	0.4968\\
ROC RF&	0.6307&	0.5850&	0.3033&	0.5952&	0.4096\\
\textcolor{green}{PRC RF}&	0.5744&	0.8439&	0.5132&	\textbf{0.8247}&	\textbf{0.5432}\\
PRC-ROC RF &	0.5813&	0.8340&	0.5009&	0.7778&	0.5381\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

\begin{table}[h]
\centering
\caption{Summary of performance results for Financial Distress Prediction Data}
\label{tab:29}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Algorithms&Recall&Specificity&Precision&Accuracy&F1 Score\\
\noalign{\smallskip}\hline\noalign{\smallskip}
SVM	&0.0000	&\textbf{1.0000}	&\textbf{NaN}&	0.9600&	\textbf{NaN}\\
\textcolor{blue}{Neural Network}&	0.3100&	0.9774&	0.3514&	0.9528&	0.3294\\
\cdashline{1-6}[0.8pt/2pt]
CART&	0.1951	&0.9830	& 0.3077	&0.9537 &0.2388\\
Weighted CART&	0.8049&	0.8727&	0.1964&	0.8702&	0.3158\\
ROC Tree&	0.0730&	0.9906&	0.2308&	0.9564&	0.1110\\
\textcolor{green}{PRC Tree} &	0.3902&	0.9802&	0.4324&	0.9583&	\textbf{0.4102}\\
PRC-ROC Tree &	0.2439&	0.9934&	0.5882&	\textbf{0.9655}&	0.3448\\
RF&	0.0244&	0.9981&	0.3333	&0.9619	&0.0455\\
Weighted RF&	0.1951&	0.9953&	\textbf{0.6154}&	\textbf{0.9655}&	0.2963\\
ROC RF&	\textbf{0.9512}&	0.8021&	0.1566&	0.8076&	0.2689\\
PRC RF&	0.9024&	0.8633&	0.2033&	0.8648&	0.3318\\
PRC-ROC RF &0.9268&	0.8681&	0.2135&	0.8702&	0.3471\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

\begin{table}[h]
\centering
\caption{Summary of performance results for Breast Cancer Data}
\label{tab:30}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Algorithms&Recall&Specificity&Precision&Accuracy&F1 Score\\
\noalign{\smallskip}\hline\noalign{\smallskip}
\textcolor{blue}{SVM}	&0.9672&	0.9909&	0.9833	&0.9616	&\textbf{0.9751}\\
\textcolor{blue}{Neural Network}&	0.9672&	0.9909&	0.9833&	0.9616&	\textbf{0.9751}\\
\cdashline{1-6}[0.8pt/2pt]
CART&	0.9016	&0.9909	& 0.9821	&0.9591 &0.9401\\
Weighted CART&	0.9344&	0.9818&	0.9661&	0.9649&	0.9500\\
ROC Tree&	0.8197&	0.9909&	0.9804&	0.9298&	0.8929\\
PRC Tree &0.9180&	0.9455&	0.9032&	0.9357&	0.9105\\
PRC-ROC Tree &0.9344&	0.9818&	0.9661&	0.9649&	0.9476\\
RF&		0.8689&	0.9909	&0.9814&	0.9473&	0.9217\\
Weighted RF&	0.9180	&\textbf{1.0000}	&\textbf{1.0000}&	0.9707&	0.9572\\
ROC RF&	0.8361&	0.9099&	0.9808	&0.9357	&0.9027\\
PRC RF&	\textbf{0.9836}	&0.9636	&0.9375	&0.9708	&0.9600\\
\textcolor{green}{PRC-ROC RF}&	\textbf{0.9836}&	0.9727&	0.9524&	\textbf{0.9766}&	0.9677\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

Then we experiment with real-world data sets and summarize the performance comparison in Tables 9 to 11. The best performance is denoted in bold. Furthermore, we use green to highlight the best tree model and blue to color the best non-tree model. The results are quite consistent with the out-of-bag error rate table. In general, the proposed algorithm can achieve higher recall and F1-score than traditional random forest. PRC random forest achieves the largest accuracy, F1-score and PRC-ROC attains the highest precision in the first case. Despite random forest achieving a slightly higher specificity, the recall and F1-score are much lower than PRC random forest and PRC-ROC RF. This is even more confirmed in the Financial Distress Prediction Data. In this distress data, SVM tends to classify all the cases as the majority class. Weighted RF did a good job in the performance of precision and accuracy. However, PRC random forest and PRC-ROC RF achieve good performance in F1 score on average. To conclude, PRC Random Forest is superior to other algorithms especially traditional random forest in identifying the minority class which is exactly the goal of imbalanced classification. In the last two instances, the PRC-ROC Random Forest has higher accuracy and F1-score compared to other algorithms and PRC RF's behavior is close to it.

\section{Conclusion}
In this paper, we investigate the PRC classification tree for dealing with class imbalance problem. We also extend this to the general PRC-ROC tree and random forest by setting an additional weight parameter which selects the best weighted combination of the AUPRC and the AUC based on the training data, for each problem at hand. Four new tree-based algorithms (i.e., PRC Tree, PRC RF, PRC-ROC Tree and PRC-ROC Tree RF) are studied along with several existing related algorithms. Table 31 shows the F1-score performance of the proposed classifiers against other tree-based algorithms in all datasets. The results indicate that our methods are superior to other rivals for skewed classification problems.\\
Several research issues are needed for in-depth investigation. Our current PRC tree/forest and PRC-ROC tree/forest are all designed for binary classifications. We shall extend them to accommodate multiple-class classifications arising naturally from many practical scenarios. \\
In our experimental study, we do not consider the interaction term. Hence another related task is to uncover interaction effects in the framework of PRC classification tree. There are absolutely many other potential possibilities about classification of imbalanced data and it will continue to gain increasingly attention in academia and industry.
\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


text:
436 F. 2d 1363 - Consumers Union of United States Inc v. Veterans Administration
436 F2d 1363 Consumers Union of United States Inc v. Veterans Administration
CONSUMERS UNION OF UNITED STATES, INC., Plaintiff-Appellant,
VETERANS ADMINISTRATION, W. J. Driver, Administrator of
Veterans Affairs of the Veterans Administration, Francis E.
Blalock, Chief, Paperwork Management Division, Medical
Administrative Service, Department of Medicine and Surgery,
Veterans Administration, and H. M. Engle, M.D., Chief
Medical Director, Department of Medicine and Surgery,
Veterans Administration, Defendants-Appellees.
Nos. 165, 166, Dockets 33978, 34160.
United States Court of Appeals, Second Circuit.
Argued Nov. 9, 1970.
Decided Jan. 15, 1971.
Marvin M. Karpatkin, New York City (Karpatkin, Ohrenstein & Karpatkin, Michael N. Pollet, and Barry Satlow, New York City, on the brief), for plaintiff-appellant.
Robert V. Zener, Dept. of Justice, Washington, D.C. (William D. Ruckelshaus, Asst. Atty. Gen., Washington, D.C. and Whitney North Seymour, Jr., U.S. Atty., Southern District of New York, on the brief), for defendants-appellees.
Before LUMBARD, Chief Judge, and WATERMAN and ANDERSON, Circuit judges.
ANDERSON, Circuit Judge:
Consumers Union appeals from that part of the district court judgment denying its request for an injunction ordering the Veterans Administration (VA) to disclose comparative scores and the scoring schemes relating to tests conducted by the VA on hearing aids. The court did order the VA to disclose the raw test data. 301 F.Supp. 796 (S.D.N.Y.1969).
Since 1955, the VA has been engaged in a program of testing hearing aids for distribution to veterans. It sends invitations to bid to hearing aid manufacturers six months before the start of the contract year, and companies submitting bids are required to provide the VA with samples which are tested to determine their acoustical and electronic qualities. The raw test data is converted to a 'quality point score' for each sample by means of mathematical formulae termed the 'scoring scheme.'
The invitations issued by the VA to bid for the 1968 contract and those in prior years included a statement of policy that the results of the tests were primarily for VA use only;1 and, in accordance with this policy, the VA refused to disclose the scoring schemes and quality point scores to anyone. The raw scores were furnished to each manufacturer for his own model, and the testing methods, apparently, were revealed to anyone upon request.2
In September 1967, Consumers Union, a non-profit corporation which evaluates consumer products and publishes Consumers Reports, requested that the VA disclose the raw scores, scoring schemes and quality point scores for the 1965 and 1966 contract years pursuant to the Freedom of Information Act, 5 U.S.C. 552. The request was later amended to cover the 1968 contract year only. After the request was refused and all administrative remedies were exhausted, Consumers Union brought the present action on July 19, 1968 in the district court pursuant to 5 U.S.C. 552(a)(3),3 seeking injunctive relief ordering the VA to disclose the three classes of information for the 1968 contract year.
The district court granted Consumers Union's requested injunction with respect to the demand for the raw scores, but denied the request for the scoring schemes and quality index scores. It held that although the express terms of the Act permitted disclosure, the district court possessed discretion to deny relief where release of the information would result in harm to the public interest. Consumers Union appealed, arguing that the district court does not possess discretion to deny what the express terms of the Act permit and that such discretion, if possessed by the court, was improperly exercised. The VA cross-appealed, claiming that by ordering disclosure of the raw test data the district court improperly interpreted certain provisions of the Act.4
Subsequent to the district court decision, the VA amended its solicitation for the 1970 contract year to permit disclosure of all three types of information sought by Consumers Union, and the information was disseminated. This policy was made retroactive to cover contract year 1968, and Consumers Union was provided with all the information it sought in the district court action. The Government filed a motion to dismiss the appeal as moot which was denied by this court on February 27, 1970. Subsequently the VA withdrew its cross-appeal, and in its brief in connection with the appeal of Consumers Union5 renewed its argument that the appeal should be dismissed as moot.
The Government's claim of mootness is met first by Consumers Union's assertion that the doctrine of the law of the case precludes this court from reconsidering its prior determination that the appeal is not moot. See Eisen v. Carlisle & Jacquelin, 391 F.2d 555, 566-567 (2 Cir. 1968); 1B Moore's Federal Practice P0.404. Assuming this contention to be correct,6 nevertheless, the doctrine operates only to preclude a reconsideration of substantially similar, if not identical, issues. See 1B Moore's Federal Practice P0.404(1) at 404, P0.1404(4) at 453. The Government's motion to dismiss the appeal as moot was based on the fact that the relief requested had been supplied by the Government, thus an order by this court to disclose the information would be pointless. That motion was denied. The Government's position on its cross-appeal indicated it felt the disclosure was discretionary on its part and not required by the Freedom of Information Act, creating the distinct possibility that the dispute which generated the present action would recur. See e.g., Golden v. Zwickler, 394 U.S. 103, 89 S.Ct. 956, 22 L.Ed.2d 113 (1969); United States v. W. T. Grant Co., 345 U.S. 629, 73 S.Ct. 894, 97 L.Ed. 1303 (1953); Bowater Steamship Company v. Patterson, 303 F.2d 369, 371-372 (2 Cir.), cert. denied, 371 U.S. 860, 83 S.Ct. 116, 9 L.Ed.2d 98 (1962); United States v. Aluminum Co. of America, 148 F.2d 416, 448 (2 Cir. 1945). Moreover, the position of Consumers Union as a leading consumer-oriented investigator of products and its strong interest in communicating the results of tests on hearing aids7 suggested that any such future dispute would very likely involve Consumers Union, so that its requisite personal stake in the recurrence of the dispute was present. See Note, Mootness on Appeal in the Supreme Court, 83 Harv.L.Rev. 1672, 1682-85 (1970). Nevertheless, subsequent events have actually made a recurrence of the dispute unlikely, and the law of the case doctrine, therefore, is inapplicable.
Consumers Union repeatedly emphasized in its briefs and at oral argument that should the Government concede the illegality of its initial refusal to disclose the information and disclaim any intention of a repetition of the illegal activity, it would agree that the appeal was moot. At oral argument, the Government conceded that the results of the tests on hearing aids did not come within any of the exemptions to the Act and that no public interest rationale justified a withholding of the information. Although Consumers Union's concession is, of course, not dispositive, we think that the Government's is. It is therefore quite clear that the Government will not again assert that hearing aid test results fall outside the Act or should be concealed in the public interest. Under the circumstances injunctive relief ordering production of the quality point scores and scoring schemes is unnecessary. See Bowater Steamship Company v. Patterson, supra, 303 F.2d at 371-372.
The appeal is therefore dismissed as moot.
The invitation to bid for the 1968 contract year contained the following language:
The results of these tests and the evaluations based thereon are primarily for VA use only, without regard to any other governmental or private agency. It must be realized, however, that the (VA) may be required to release hearing aid test data, * * * to other agencies within the Federal Government structure. If this should occur, the (VA) would no longer retain control over release of the test data to the public.
The VA's policy of non-disclosure was modified in the 1969 solicitation to allow the VA, in its discretion, to release the raw scores
5 U.S.C. 552(a)(3) provides in pertinent part:
On complaint, the district court of the United States * * * has jurisdiction to enjoin the agency from withholding agency records and to order the production of any agency records improperly withheld from the complainant. In such a case the court shall determine the matter de novo and the burden is on the agency to sustain its action.
The Government argued that the formulae and quality point scores were exempted from disclosure pursuant to 552(b)(2), (3), (4), (5), (7), which provide exemptions from disclosure for certain categories of information
Since it has released to Consumers Union all the information sought in the appeal, the Government professes no interest in the disposition of the appeal, save for an interest in interpretations of law having continuing significance
To the extent that the law of moot cases is a jurisdictional consideration (see Sibron v. New York, 392 U.S. 40, 57, 88 S.Ct. 1889, 20 L.Ed.2d 917 (1968); Note, Mootness on Appeal in the Supreme Court, 83 Harv.L.Rev. 1672, 1673-74 n. 12 (1970)), we are not persuaded that the law of the case doctrine, based in the sound discretion of the court, would prevent a reconsideration of the earlier denial of the Government's motion. See 1B Moore's Federal Practice P0.404(1) at 405, P0.404(4) at 452. However, our disposition of the case makes it unnecessary to pass on this question
Consumers Reports had published findings on hearing aids, the data for which was presumably gained through its own research staff, in September 1950, January 1951 and January 1966 issues of the magazine

redpajama_set_name: RedPajamaCommonCrawl


text:
// Home Scots, Legal and Latin words /contact /site map /help
/ Problem Letters
/ Problem Words
/ Phonetic Spelling
/ Scots, Legal and Latin words
/ Sums of Money
/ Dates
/ Abbreviations
If you are sure you've identified the word correctly but do not know what it means it may be that the word is Scots. Access to a Scots dictionary is recommended. The Dictionary of the Scots Language is available free online www.dsl.ac.uk and combines three of the best used dictionaries. It also allows for fuzzy searches, which is very helpful for variations in spelling. For further details about this and other Scots dictionaries, including an online version, go to the Bibliography page.
Common Scots words occurring are umquhill = deceased , relict = widow, compeared = appeared before a court. Note also the Scots plural form -is, and the Scots past participle –it, which you may want to drop off a word when looking for a definition. Because different Scots accents pronounce words slightly differently it can sometimes be useful to change the vowels you are using to search the dictionary to find the right definition.
Is it a legal term?
The word might be legal jargon. If you are reading text from official documents such as contracts, bonds, testaments, sasines, or other public registers the document is likely to follow a particular pattern – this is known as diplomatic. There will be stock phrases occurring in a particular order to fulfil the legal obligations for which the document is created. This can be helpful when trying to identify a word as it shortned the list of possible words to choose from. There are a number of books which deal with Scots legal terms or diplomatic. There is a list of books which will help with legal jargon in the Bibliography.
Is it a Latin word?
In medieval times most records were written in Latin, which means that as time progresses into the 1500s and onwards there are relics of the use of Latin in which diminishes as the centuries progress. Latin will be used as legal jargon but also for commonly occurring phrases such as dates – septimo julii 1589, 7 July 1589; the commencement of a court sitting which usually begins curiae justiciarae; lists of people present – sederunt; indicating totals – summa and so on.
It is useful to have access to a Latin dictionary or a Latin word list, such as Eileen Gooder, Latin for Local History, London, 2nd. edition, 1978; and/or R. E. Latham, Revised Medieval Latin Word-list from British and Irish Sources, London, 1965.
If this hasn't solved the problem try another option:
Phonetic Spelling
Sums of Money
Return to the Problem Words page.


text:
Lee Edmon McDuff passed away peacefully on December 22, 2020 surrounded by his devoted wife and family, of whom he was so proud. Lee, the 6th child of Joe E. and Rosa Lee Watson McDuff, was born February 24, 1925 in Ralls, Texas. He graduated from Ralls High School and Texas Tech University Class of 1949 with degrees in Petroleum Engineering and Geology.
Lee had a gift, a way of making the complex look simple. He could fix anything and knew that engineering was a perfect career choice; however, duty called Lee into serving his country in February, 1944 as a 2nd Lt. in the US Army Air Corps flying for the 405 Squadron, 371st Fighter Group. Lee flew many successful missions with "Shadrach", his trusty P-47, until he was shot down near Koblenz, Germany and held as a POW in Stalag-Luft I prison camp until his liberation in May 1945. His bravery and service to his country
earned him a Purple Heart and Bronze Star among other service awards.
Upon returning to Texas, Lee completed college and began a lifelong career as the 7th employee with newly formed Transcontinental Gas and Pipeline until his retirement as Vice President of Corporate Services. While working for Transco in St. Francisville, LA, Lee met the love of his life, Shirley Marie Navarre and they were married on March 25, 1951 in a double ceremony. Lee (known by close friends and family as Mac) and Shirley began their life in Houston after the birth of their first child. As their family grew, so did their love and devotion to each other. Aside from family, Lee had only one other passion and he practiced that at every opportunity as an avid golfer and long-time member of Pine Forest Country Club. Lee brought leadership and generosity to every endeavor, work, service or play throughout his life.
Lee was preceded in death by his parents, Joe E. and Rosa Lee Watson McDuff, his brothers, James Richard "Red" McDuff (Bonnie), William Clayborn "Dutch" McDuff (Gwen), his sisters, Rosie Jo McKay (Doug) and twins, Eros Caldwell (Elam), Eunice McDuff and daughter-in-law Carolyn McDuff.
Lee was surrounded by love during his life and is survived by his wife of 70 years, Shirley, and sons Gary McDuff (Mary Ann), Patrick McDuff, Michael McDuff, Timothy McDuff (Libby), and daughters Molly Levegood (Bill), Mary Lee House (Del) and Laura Karsten (Dan) and grandchildren Mary Levegood, Will Levegood, Jim Levegood, Ashley Pearson (Philip), Hailey House, Melanie House, Madeleine McDuff, Megan Karsten, Jon Karsten, Kelly Karsten, Claire Karsten and great grandchildren Rachel Pearson,
Sydney Pearson and William Lee Pearson (on the way). Lee will be missed beyond measure and his light and love will live on for generations.
"Do not let your hearts be troubled. Trust in God, trust in me." - John 14:1
Inurnment
Tue. Jan. 5, 2021
Houston National Cemetery
10410 Veterans Memorial Drive
Houston , TX 77038.
Lee had a gift, a way of making the complex look simple. He could fix anything and knew that engineering was a perfect career choice; however, duty called Lee into serving his country in February, 1944 as a 2nd Lt. in the US Army Air Corps flying for the 405 Squadron, 371st Fighter Group. Lee flew many successful missions with "Shadrach", his trusty P-47, until he was shot down near Koblenz, Germany and held as a POW in Stalag-Luft I prison camp until his liberation in May 1945. His bravery and service to his country earned him a Purple Heart and Bronze Star among other service awards.
Larry Dial
Tom Spencer
Lynn Harris
patrick mcnamara
Buddy Gomez
Thank you Lt. Mcduff for your dedication in helping save this country.


text:
Quotevine
Digitising Asset and Automotive Finance
Watch our introductory video!
The Quotevine Platform
ADA (Sales Enablement)
CORA (Enterprise Calculations)
Radia: Terms Management
Quotevine Knowledgebase
Broker Solutions
October 4, 2018 By Daniel Layne Leave a Comment
Electric vehicles – zapping assumed economics for asset finance
The rise of eco-conscious individuals has created a surge in demand for electric vehicles; according to the International Energy Agency, electric vehicle ownership will soar to 125 million by 2030.
Following the passing of the Automated and Electric Vehicle Act through the UK Parliament in July 2018, we can expect to see major steps forward in the infrastructure and consumer confidence that will support this trend, with new public charging points planned and a push for standardisation around reliability standards and payment mechanisms.
In addition to the widely discussed impact on the energy and transport infrastructure, this transport revolution will also bring new challenges and uncertainty for the auto finance sector.
The mix of governments' evolving subsidy strategies, shifting consumer demand patterns and new technology will up-end the traditional economics and pricing models associated with vehicle leasing, demanding new flexibility from traditional providers.
One of the key reasons for this shift is the change in engine type, which has a major impact on the residual vehicle valuation systems that are already in place.
Conventional depreciation models used to calculate the value of combustion engine vehicles after lease expiry will no longer apply – wear and tear of an electric motor is very different to that of an internal combustion engine.
A standard car is valued as a whole 'piece of metal', unlike an EV, which can be valued in components. Not only that, but the battery can make up 50% of the EV's value.
As electric vehicles continue to develop, we will see more and more technology integration. For example, over the air software updates and data connectivity between the vehicle and the home or smartphone.
Today's vehicles contain millions of lines of code, which can be updated and upgraded remotely with new and enhanced functions. With this much 'intelligence' being piped into cars after they are leased, traditional depreciation theory will be completely reversed, as software augments vehicle performance over time.
This trend is as relevant across the enterprise landscape as it is in a consumer context.
Data from the Corporate Vehicle Observatory shows that 30% of very large companies have already introduced some electric vehicles to their fleets, and 27% have adopted hybrids, with an intention to accelerate this movement in the coming years.
Meanwhile, research from Smart Energy GB shows that more than 8 million people in Britain are considering buying or leasing an electric vehicle in the next five years.
Tapping the data
To ensure appropriate pricing metrics and prevent against revenue losses in this 'new normal', auto financing businesses will need to understand how to better price electric vehicles.
The key to this lies in the ability to gain access to, source, and process vehicle and industry specific data in real-time.
Connected cars provide a significant opportunity for asset owners to dynamically and accurately price and update their offering based on actual vehicle ownership, rather than relying on estimations, guesswork and averages.
This becomes even more relevant in a corporate fleet context, when this is scaled up to encompass thousands of vehicles.
Data insight will allow the lease provider to transform their business relationship from transactional to value-add advisory, drawing on vehicle usage data to help enterprise fleet managers optimise their fleets to achieve cost savings, reduce carbon emissions in line with CSR targets, and improve driver safety; for example, by providing real-time notifications when a vehicle detects a fault, or uses machine learning to predict a future fault.
Calling for access
Rising levels of vehicle connectivity allow vehicle manufacturers to engage directly with customers, enabling them to maintain brand awareness and promote maintenance and other service offerings.
However, external service providers, including automotive finance companies, are currently at a significant commercial disadvantage as they are unable to access this data.
Action is being taken by stakeholders within the automotive finance industry, who are calling for greater levels of access to connected car data.
The European Commission's Gear 2030 report recognises the importance of direct, safe and secure access to a wide set of in-vehicle data in real-time for the provision of connected services.
We are on the cusp of some changes in the automotive sector that will shake the assumed foundations of so many parts of the value chain.
It is critical for the measures outlined in this document to come into play to ensure the advancement of the automotive finance industry.
With them in place, the automotive finance industry can present more innovative leasing solutions that are better matched to customer expectations, both retail and enterprise.
* Quotevine is a technology business that provides a suite of cloud-native SaaS solutions to digitise asset and automotive finance.
* Daniel Layne has 17 years experience designing multi-million pound systems for blue-chip automotive funders, providing technical consultancy and guiding projects from inception to delivery.
Article originally published on: https://www.assetfinanceinternational.com/index.php/people/thought-leaders/thought-leaders/17491-electric-vehicles-zapping-assumed-economics-for-asset-finance
Filed Under: Press
Quotevine Limited
The Merton Centre
45 St Peter's Street
MK40 2PN
info@quotevine.com
Quotevine is a technology business that provides a suite of SaaS solutions to digitise asset and automotive finance.
Quotevine Newsletter
© Quotevine Limited. Registered in England and Wales at Unit A, The Merton Centre, 45 St Peter's Street, Bedford, MK40 2PN. Company Number 07877335.
The Quotevine logo and Safer Than On-Site slogan are registered trademarks of Quotevine Limited. CAP is a registered trademark of CAP Automotive Limited.