Optimizing Neural Networks for Academic Performance Classification Using Feature Selection and Resampling Approach

MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX ISSN: 1803-3814 (Printed), 2571-3701 (Online) https://doi.org/10.13164/mendel.2023.k.237 Detecting Outliers Using Modified Recursive PCA Algorithm For Dynamic Streaming Data Yasi Dani1,  , Agus Yodi Gunawan1,  ,Masayu Leylia Khodra2,4, Sapto Wahyu Indratno3,4 1Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10 Bandung, Indonesia 2School of Electrical Engineering and Informatics, Institut Teknologi Bandung, Jl. Ganesha 10 Bandung, Indonesia 3Department Statistics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10 Bandung, Indonesia 4Natural Language Processing and Big Data Analytics (U-CoE AI-VLB), University Center of Excellence on Artificial Intelligence for Vision, Institut Teknologi Bandung, Jl. Ganesha 10 Bandung, Indonesia [email protected]  , ayo[email protected]  , masa[email protected], [email protected] Abstract Outlier analysis has been widely studied and has produced many methods. However, there is still rare a method to detect outliers for dynamically streaming batch data (online learning). In the present research, a novel online algorithm to detect outliers in such dataset is proposed. Data points are proceeded by applying a modified recursive PCA to predict sequentially parameters of the model; eigenvalues and eigenvectors of the statistical detection model are recursively updated using approximate values by perturbation methods. More specifically, the recursive eigenstructure is obtained from the derivation of the covariance matrix using the first-order perturbation technique. The Mahalanobis distance is then used as an outlier score. Our algorithm performances are evaluated using some metrics, namely accuration, precision, recall, F1-score, AUC-PR, and the execution time. Results show that the proposed online outlier detection is computationally efficient in time and the algorithm’s performance effectiveness is comparable to that of the offline outlier detection algorithm via classical PCA. Keywords: Outlier, Online Learning, Recursive PCA, Eigendecomposition, Perturbation Method. Received: 27 October 2023 Accepted: 30 November 2023 Online: 07 December 2023 Published: 20 December 2023 1 Introduction An outlier is an observation that deviates so much from other observations that it raises the suspicion that it is produced by a different mechanism [15]. In detecting outliers, there are two outlier detection techniques, supervised outlier detection based on classification or regression and unsupervised outlier detection based on clustering. In the digital era, the introduction of a technology system is very important since we have entered an era, namely the era of big data. Big data is data on a large scale in terms of volume, intensity, and complexity that exceeds the capacity of standard analytical tools [27]. Emerson et al. [11] proposed that a data set will be considered large if it exceeds 20% of the RAM on a given machine and very large if it exceeds 50%, in which case even the simplest calculations will consume all the remaining RAM. Consequently, it would be time consuming to process the data using traditional methods. It is better to construct an appropriate algorithm so that the data can be managed properly and efficiently. Online learning, also known as incremental learning, is a machine learning method that builds a learnable model for effective classification in real-time detection [18]. While, offline learning is a traditional machine learning technique that requires large computational time and time to process all data. The model uses only the previously provided data (a set of historical data). It will then require manually updating the model on more recent emerging data and apply the resulting model whenever the normal system behavior changes. The online algorithm could use all available information without storing or revisiting individual data points [8,23]. Other previous studies developed several techniques in identifying outliers or anomalies. Bosman et al.[5] identified anomalies in sensor systems with parameter estimation using the recursive least square (RLS) method, Zangeneh-Nejad et al. [31] studied anomalies with the DIA (Detection, Identification, and Adaptation) algorithm after estimating parameters using the RLS method. Later, Schifano et al. [25] and Wang et al. [29] detected outliers with standardized predictive residuals and to test for outliers in the n-th data where after estimating the parameters with the Bayesian framework method. Hoeltgebaum et al. [17] identified anomalies using The Hall-Buckley Eggleson (HBE) method after estimating parameters using the LASSO method. Then, Ippel et al. [19] estimated the parameters recursively with 237 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX stochastic gradient descent. And Majdoubi et al. [22] recursively estimated the parameters based on the recursive least squares method. Thuy et al. [28] provided a method consisting a deep neural network and heuristic algorithms combined with LR to boost the accuracy of attack detections in an intrusion detection system (IDS). Fieri and Suhartono [12] studied two types of soft voting models, namely machine learning-based and deep learning-based to detect offensive language a Twitter dataset and to improve the performance of the soft voting classifier method. According to Jolliffe [20], Principal Component Analysis (PCA), is one of the oldest and most wellknown multivariate analysis techniques. This PCA method is applied primarily to reduce the dimensions of the dataset by projecting each data point into the first few principal components to obtain data of less dimension while retaining as much as possible of the variation present in the dataset [3,2]. The Mahalanobis distance is equal to the Euclidean distance between data point and in such a transformed (axisrotated) dataset after dividing each of transformed coordinate values by the standard deviation of that direction. Thus, PCA can also be used to calculate the Mahalanobis distance [1]. In this research, we propose a novel online outlier detection algorithm to identify outliers with Mahalanobis distance using modified recursive PCA where outliers are identified as soon as a new data record appears in dataset. To be explicit, recusive eigenstructures are calculated from the covariance matrix using the first-order perturbation technique. For outlier detection score the Mahalanobis distance is applied. We then evaluate the effectiveness and efficiency of the algorithm performance evaluation using some metrics, i.e. accuration, precision, recall, F1-score, AUC-PR, and the execution time. We apply this online outlier detection algorithm on synthetic dataset. The remainder of the paper is organized as follows: Section 2gives recursive formulas for the sample mean and covariance matrix. It also presents approximate perturbation methods for recursive PCA, our proposed online detection algorithm, and algorithm performance evaluation using some metrics. In Section 3, we apply online and offline algorithm approaches to synthetic dataset and discuss the Mahalanobis score, and performance analysis for both approaches. Conclusions are written in Section 4. 2 Materials and Methods In this section, we build an online outlier detection algorithm consisting of the online parameter estimation formula via modified recursive PCA and identifying the outliers in each new data arrive using the Mahalanobis distance as the outlier score. 2.1 Parameter Estimation via Modified Recursive PCA We first derive a recursive mean formula. Given vector data x x x1,x x x2, ...,x x xn+1 ∈ ℜp. A recursive mean formula is calculated as µ µ µn+1 =1 n+ 1 n+1 X i=1 x x xi =1 n+ 1 n X i=1 x x xi+1 n+ 1x x xn+1 =n n+ 1µ µ µn+1 n+ 1x x xn+1.(1) For the covariance matrix, a recursive formula is given by the following C C Cn+1 =1 n+ 1 n+1 X i=1 (x x xi−µ µ µn+1)(x x xi−µ µ µn+1)T =1 n+ 1 n+1 X i=1 x x xix x xT i−µ µ µn+1µ µ µT n+1 =n n+ 1(1 n n X i=1 x x xix x xT i−µ µ µnµ µ µT n) +n (n+ 1)2(x x xn+1 −µ µ µn)(x x xn+1 −µ µ µn)T =n n+ 1C C Cn+n (n+ 1)2(x x xn+1 −µ µ µn)(x x xn+1 −µ µ µn)T =C C Cn+1 n+ 1[n n+ 1(x x xn+1 −µ µ µn)(x x xn+1 −µ µ µn)T −C C Cn].(2) In Eq. (2) we have substituted µ µ µn+1 by Eq. (1). Let C C Cnbe the covariance matrix at step n-th whose size is p×p. Assume that at (n+ 1)-th, C C Cnchanges slightly by matrix B B B. This can be written as C C Cn+1 =C C Cn+εB B B, (3) with small parameter ε=1 n+1 . In other words, C C Cnis perturbed by B B B. Now, an eigen equation for C C Cn+1 is given by C C Cn+1v v vn+1 =λn+1v v vn+1 ⇒(C C Cn+εB B B)v v vn+1 =λn+1v v vn+1, (4) where (λn+1,v v vn+1) is the eigenpair of C C Cn+1. Assuming that there are pdistinct eigenvalues. We denote the j-th eigenpair of C C Cnby (λj,n, v v vj,n) with j= 1,2,3, ..., p. The first order asymptotic approximation for (λj,n+1,v v vj,n+1) can be written as λj,n+1(ε)≈λj,n +ελ1j,n,(5) v v vj,n+1(ε)≈v v vj,n +εv v v1j,n,(6) with λ1j,n =< B B Bv v vj,n,v v vj,n > < v v vj,n,v v vj,n >(7) 238 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Dani-g2igHX,gD etecting Outliers Using Modified Recursive PCA Algorithm For Dynamic Streaming Data v v v1j,n = p X k=j < v v vk,n,B B Bv v vj,n > (λj,n −λk,n)v v vk,n +βv v vj,n,(8) where βis an arbitrary constant and v v vj,n is an orthonormal vector. Detailed derivation of Eq. (7) and (8) can be found in [16]. We now apply Eq. (4) to Eq. (2) with B B B= n n+1 (x x xn+1−µ µ µn)(x x xn+1−µ µ µn)T−C C Cnand ε=1 n+1 . The recursive eigenvalue and eigenvector estimation formula can now be obtained respectively as λj,n+1 ≈λj,n +1 n+ 1(n n+ 1ϕ2 j,n −λj,n),(9) v v vj,n+1 ≈v v vj,n +1 (n+ 1)(X k=j n n+1 ϕk,nϕj,n λj,n −λk,n v v vk,n +βv v vj,n), (10) where ϕj,n = (x x xn+1−µ µ µn)Tv v vj,n,ϕk,n =v v vT k,n(x x xn+1−µ µ µn), and βis an arbitrary constant. 2.2 Outlier Detection Method The Mahalanobis distance measures the number of standard deviations that an observation is from the mean of a distribution, introduced by Prasanta Chandra Mahalanobis in 1930 [21]. Then the Mahalanobis distance as the outlier score of a data point x x xcan be defined by score(x x x) = p X j=1 |(x x x−µ µ µ)·v v vj|2 λj (11) where µ µ µis the centroid of the data, λis the eigenvalue, and v v vis the eigenvector. For our problem at hand the eigenstructure is given by Eq. (9) and (10). An outlier has to satisfy the following condition score(x x x)>qχ2 d,1−α(12) where χ2 d,1−αis chi-square distribution with degrees of freedom dand (1−α)% the significance level (For more information concerning Eq. (12), see Aggarwal [1]) 2.3 The Proposed Online Outlier Detection Algorithm Algorithm 1describes the procedure of detecting outliers using the Mahalanobis distance via modified recursive PCA. We describe how these parameters are updated and how outliers are identified with every step of new data. 2.4 Performance Evaluation In the field of machine learning and computing, evaluating performance of a classification algorithm is very important [24,14]. In binary classification, the input Algorithm 1: Mini-batch Outlier Detection Algorithm via Modified Recursive PCA. Input: a data matrix X X X∈ ℜn×p 1Initialization: 2µ µ µn←1 nPn i=1 x x xi 3C C Cn←1 nPn i=1(x x xi−µ µ µn)(x x xi−µ µ µn)T 4λj,n and v v vj,n of C C Cnwith j= 1,2,3, ..., p 5for n+ 1 to final do 6ϕj,n ←(x x xn+1 −µ µ µn)Tv v vj,n 7λj,n+1 ←λj,n +1 n+1 (n n+1 ϕ2 j,n +−λj,n) 8ϕk,n ←(x x xn+1 −µ µ µn)Tv v vk,n 9v v vj,n+1 ←v v vj,n +1 (n+1) Pk=j n n+1 ϕk,nϕj,n λj,n−λk,n v v vk,n 10 v v vj,n+1 ←v v vj,n+1 ||v v vj,n+1|| 11 µ µ µn+1 ←n n+1µ µ µn+1 n+1x x xn+1 12 score(x x xn+1)←Pp j=1 |(x x xn+1−µ µ µk+1)·v v vj,n+1|2 λj,n+1 13 if score(x x xn+1)>qχ2 d,1−αthen 14 outlier detected 15 record outlier with label 1 16 else 17 record inlier with label 0 18 end if 19 end if 20 end for Output: binary classification data is grouped into one of two classes. In measuring the performance of an algorithm that is often used in machine learning, especially the classification model, it creates a confusion matrix [4]. This research performs a binary classification, so the results of the confusion matrix are two classes. The confusion matrix aims to compare the classification results of an algorithm with the truth classification results [13,9].The representation of the confusion matrix is a matrix table with four combinations of predicted values and the actual value where the table can be seen in Table 1. We define the following loss function I:Y×Y→ {T P, T N, F P, F N}. Let y∈ {i0, i1}be the prediction where i0= inlier and i1= outlier. The mapping of the Ifunction as follows [7] •If y=i1and ˆy=i1, then I(y, ˆy) = T P ; •If y=i0and ˆy=i0, then I(y, ˆy) = T N; •If y=i0and ˆy=i1, then I(y, ˆy) = F P ; •If y=i1and ˆy=i0, then I(y, ˆy) = F N. Next, when the prediction result is a real number, a threshold value of tis needed to distinguish positive and negative classes [10]. Furthermore, we use the results of the confusion matrix table to evaluate the performance of the machine learning algorithm for making predictions, namely by calculating the values of accuracy, precision, recall/sensitivity, F1-score. To add a measure to evaluate 239 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Table 1: Confusion Matrix. Predicted Values Positive Negative Actual Values Positive T P F N Negative F P T N the performance of the algorithm, we also use AUC-PR that stands for area under the precision-recall curve [32,30]. The range of AUC-PR values is between 0 and 1. Generally, the higher the AUC-PR score, the better a classifier performs for the given task. Next, we also calculate the execution time of the algorithm to evaluate the efficiency of the algorithm. 3 Results and Discussion Our experiments are performed on several synthetic datasets. Our synthetic datasets have same number of data points that are created with the scikit-learn module “make classification” algorithm [26,6].We generate synthetic datasets to simulate a two-class classification problem with 1000 samples (100 samples of the train set and 900 samples of the test set) with different number of features (p= 2,3,5,10) randomly drawn following a standard Normal distribution, i.e N(0,1) and one binary dependent variable y∈Y={0,1}being the class of the data points, the distribution of the two classes defined by Yis imbalanced, i.e. the proportion of observations for which y= 0 was 99% and y= 1 was 1% with no redundant variables in the dataset. In the following subsections, we compare the Mahalanobis distance results as outlier scores and evaluate the algorithm performance via modified recursive PCA (online algorithm) and classical PCA (offline algorithm) with respect to ground-truth outlier information. In simulating each test data point in the offline outlier detection algorithm, all training data and previous test data are still used. 3.1 Mahalanobis Distance Analysis as Outlier Score In this section, we show plots of the Mahalanobis score difference on the y-axis obtained from the results of the online and offline algorithm against the test dataset on the x-axis. The scores were obtained from the results of subtracting the Mahalanobis score using the modified recursive PCA method with the classical PCA method. All plots display the difference in Mahalanobis scores converging to zero for some features although there is a slight spread across some test points. In particular, at 2, 3, and 5 features (Fig. 1to 3) that the difference in Mahalanobis score values is close to zero. Meanwhile, the 10 features in Fig. 4show that the Mahalanobis score is slightly different from zero, but still converges around zero. Figure 1: Mahalanobis Score Difference for 2 Features. Figure 2: Mahalanobis Score Difference for 3 Features. Figure 3: Mahalanobis Score Difference for 5 Features. Figure 4: Mahalanobis Score Difference for 10 Features. 240 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Dani-g2igHX,gD etecting Outliers Using Modified Recursive PCA Algorithm For Dynamic Streaming Data Figure 5: Plot Test Data with Two Features on Ground-Truth. Figure 6: Plot Test Data with Two Features on Offline Algorithm Prediction Values. 3.2 Performance Analysis of Online and Offline Algorithm In this section, we show the representation of outlier detection results on test data with two features using online and offline algorithms compared to groundtruth. The outliers are shown in yellow color and the inliers are shown in blue color. The ground-truth of the test set used is shown in Fig 5and the predicted results of the two algorithms on the test data are shown in Fig. 6and 7. The number of outliers on the groundtruth data is 14 as shown in Table 2. Meanwhile, the number of outliers predicted by the two algorithms is 10 as shown in Table 2. For more details about the outlier detection results with two features, it can be seen in Table 2. And in this section, we also show the confusion matrix of the results of the online and offline outlier detecFigure 7: Plot Test Data with Two Features on Online Algorithm Prediction Values. tion algorithms respect to ground-truth of all test data (Table 3to 6). The results of the confusion matrix of the two algorithms are exactly the same for 2 and 3 features where both algorithms correctly predict a total of 10 outliers and correctly predict a total of 886 inliers. While the results of the confusion matrix are different for 5 and 10 features. For the 5 features, the offline algorithm correctly predicts a total of 9 outliers and correctly predicts a total of 887 inliers, while the online algorithm correctly predicts a total of 10 outliers and correctly predicts a total of 886 inliers. For 10 features, the offline algorithm correctly predicts a total of 9 outliers and correctly predicts a total of 886 inliers, while the online algorithm correctly predicts a total of 7 outliers and correctly predicts a total of 886 inliers. Table 7represents the comparison of the performance results of the online and offline outlier detection algorithms on the given test data. The accuracy and precision values of the two algorithms are very satisfying since the values are more than 0.90, then the accuracy of the two algorithms is exactly the same, only the precision differs slightly on 5 features where the offline algorithm’s precision is slightly better. The recall and F1-score of the two algorithms are good enough where the values are in the range [0.50,0.83], there are only differences in 5 and 10 features where the recall and F1-score of the online algorithm are comparable to the offline algorithm. The AUC-PR of both algorithms is also satisfactory since the value more than 0.75, there are only differences in 5 and 10 features where the offline algorithm’s AUC-PR is slightly better. Then the execution time of the online algorithm is always faster than the offline algorithm for each feature. Overall, it can be seen that the effectiveness of the two algorithms decreases slightly as the features increase, the effectiveness of the online algorithm is comparable to the offline algorithm, and the efficiency of the online algorithm is the most outperforming. 4 Conclusions and Remarks A way to construct an online algorithm to identify outliers for data streams was discussed in this paper. This research applied a recursive schema strategy that predicts the iterative model to update the parameters in the model when new data appears and detects outliers. In other words, an iterative schema plays an important role for data streams. We constructed an online algorithm to identify outliers with Mahalanobis distance using modified recursive PCA. In conclusion, this work showed that in terms of effectiveness the performance of the online algorithm is comparable to that of the offline algorithm and in terms of efficiency the performance of the online algorithm outperforms the offline algorithm. We remark here that our proposed algorithm is still limited, it is quite appropriate only for the incoming data whose changes are not too large since we apply the perturbation method in our algorithm. Other things, 241 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Table 2: The Results of Outliers Detection. 2 Features Ground-Truth Outlier Predicted Values (nth test data) (Offline Algorithm) (Online Algorithm) 92th √ √ 114th - - 164th √ √ 238th √ √ 260th √ √ 326th - - 330th √ √ 397th √ √ 498th √ √ 516th √ √ 644th - - 710th √ √ 723th - - 785th √ √ Total Outliers 14 10 10 Table 3: Confusion Matrices Results of The Test Set for 2 Features. p= 2 Offline Algorithm Online Algorithm Predicted Values Predicted Values Positive (1) Negative (0) Positive (1) Negative (0) Actual Values Positive (1) 10 4 10 4 Negative (0) 0 886 0 886 Table 4: Confusion Matrices Results of The Test Set for 3 Features. p= 3 Offline Algorithm Online Algorithm Predicted Values Predicted Values Positive (1) Negative (0) Positive (1) Negative (0) Actual Values Positive (1) 10 4 10 4 Negative (0) 0 886 0 886 Table 5: Confusion Matrices Results of The Test Set for 5 Features. p= 5 Offline Algorithm Online Algorithm Predicted Values Predicted Values Positive (1) Negative (0) Positive (1) Negative (0) Actual Values Positive (1) 9 4 10 3 Negative (0) 0 887 1 886 Table 6: Confusion Matrices Results of The Test Set for 10 Features. p= 10 Offline Algorithm Online Algorithm Predicted Values Predicted Values Positive (1) Negative (0) Positive (1) Negative (0) Actual Values Positive (1) 9 5 7 7 Negative (0) 0 886 0 886 242 MENDEL — Soft Computing Journal, Volume 29, No.gk, .2+2K#2` 2023, Brno, Czech RepublicX Dani-g2igHX,gD etecting Outliers Using Modified Recursive PCA Algorithm For Dynamic Streaming Data Table 7: Performance Comparison of Online and Offline Outlier Detection Algorithms. The number of Features (p): 2 3 5 10 Accuracy Offline Algorithm 1.000 1.000 1.000 0.990 (100%) Online Algorithm 1.000 1.000 1.000 0.990 Precision Offline Algorithm 1.000 1.000 1.000 1.000 Online Algorithm 1.000 1.000 0.910 1.000 Recall Offline Algorithm 0.710 0.710 0.690 0.640 Online Algorithm 0.710 0.710 0.770 0.500 F1-score Offline Algorithm 0.830 0.830 0.820 0.780 Online Algorithm 0.830 0.830 0.830 0.670 AUC-PR Offline Algorithm 0.859 0.859 0.848 0.824 Online Algorithm 0.859 0.859 0.841 0.754 Time Offline Algorithm 0.461 0.480 1.082 1.574 (second) Online Algorithm 0.300 0.414 0.726 1.286 the present algorithm only applies for the data streams arriving one by one. Therefore, for the future work, an outlier detection algorithm involving the incoming data with mini-batch size issue can be extended. Acknowledgement: The authors sincerely wish to thank the journal editors and reviewers who critically reviewed the manuscript and made valuable suggestions for its improvement. 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