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Hiệu quả của các phương pháp xử lý dữ liệu mất cân bằng trong chấm điểm tín dụng: Trường hợp tại các Ngân hàng thương mại Việt Nam

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Nội dung Text: Hiệu quả của các phương pháp xử lý dữ liệu mất cân bằng trong chấm điểm tín dụng: Trường hợp tại các Ngân hàng thương mại Việt Nam

- INTERNATIONAL CONFERENCE FOR YOUNG RESEARCHERS IN ECONOMICS & BUSINESS 2020

ICYREB 2020THE EFFECTIVENESS OF METHODS IN DEALING

WITH IMBALANCED DATA IN CREDIT SCORING: THE CASE

OF VIETNAM COMMERCIAL BANKS

HIỆU QUẢ CỦA CÁC PHƯƠNG PHÁP XỬ LÝ DỮ LIỆU

MẤT CÂN BẰNG TRONG CHẤM ĐIỂM TÍN DỤNG: TRƯỜNG HỢP

TẠI CÁC NGÂN HÀNG THƯƠNG MẠI VIỆT NAMNguyen Thi Lien, MS ; Nguyen Thi Thu Trang, MS; Nguyen Thi Dung

National Economics University

lientkt@neu.edu.vnAbstract

This article investigates the effectiveness of imbalanced data processing methods in the

problem of customer classification at commercial banks. This is a common issue in a customer

classification attempt, where observations of one class outnumber the remaining class. We apply

the methods widely used in the world including undersampling, oversampling, bothsamling tech-

niques, and SMOTE (Synthetic Minority Oversampling Technique) to deal with imbalances. The

logit model is applied to datasets that have been processed by these methods to classify customers.

Using 7501 transaction data from individual customers, the classification results using data

processed with these techniques all improve significantly compared to using untreated data. Be-

sides, the results also show that the most efficient method is SMOTE technique combined with

the logit model using variables transformed by Weight of Evidence (WOE).

Keywords: Bothsampling, credit scoring, oversampling, SMOTE, undersampling, WOE.Tóm tắt

Bài báo này nghiên cứu hiệu quả của phương pháp xử lý dữ liệu mất cân bằng trong bài

toán phân loại khách hàng tại các ngân hàng thương mại. Đây là một vấn đề phổ biến trong vấn

đề phân loại khách hàng, trong đó các quan sát của một lớp nhiều hơn lớp còn lại trong dữ liệu.

Chúng tôi áp dụng các phương pháp được sử dụng rộng rãi trên thế giới bao gồm kỹ thuật lấy

mẫu dưới (Undersampling), lấy mẫu quá mức (Oversampling), kỹ thuật lấy mẫu cả hai (booth-

sampling) và SMOTE (Synthetic Minority Oversampling Technique) để giải quyết vấn đề mất

cân bằng. Mô hình logit được áp dụng cho các tập dữ liệu đã được xử lý mất cân bằng để phân

loại khách hàng. Sử dụng 7501 dữ liệu giao dịch từ các khách hàng cá nhân, kết quả phân loại

sử dụng dữ liệu đã được xử lý mất cân bằng đều cải thiện đáng kể so với sử dụng dữ liệu không

được xử lý. Bên cạnh đó, kết quả cũng cho thấy phương pháp hiệu quả nhất là kỹ thuật SMOTE

kết hợp với mô hình logit sử dụng các biến đầu vào được chuyển đổi sang trọng số bằng chứng

(Weight of Evidence – WOE).

Từ khóa: Chấm điểm tín dụng, lấy mẫu cả hai, lấy mẫu quá mức, lấy mẫu dưới, SMOTE,

WOE1273

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ICYREB 20201. Introduction

In the case of classification algorithms, we aim to predict the false observations with the

highest accuracy. However, the accuracy of these algorithms can be influenced by the imbalanced

data (López, 2013), where observations of one class outnumber the other class(es) at least 5:1

for the binary case (He & Garcia, 2009). In order to improve the prediction accuracy and lower

the computation expense, a previous identification and quantification of the most relevant input

variables of the model is always highly advised. Imbalanced data also appears in some other

fields, such as fault detection (Gong & Qiao, 2012; Silva et al., 2006; Wei et al., 2013), toxin de-

tection (Harley et al., 2020), medical diagnosis (Ertekin et al., 2007) and customer churn predic-

tion (Bing Zhu et al., 2017). In medical area, consider the group of cancer patients as the positive

class and remaining persons are in the negative class, the difference about quantity between these

two groups in datasets is large and unequal in the number of observations between the groups.

Other examples of rare events include software defects (Rodriguez et al., 2014), cancer gene ex-

pressions (Wu et al., 2012), credit card transactions fraudulent (Kundu et al., 2009), fraud detec-

tion in telecommunication (Augustin, 2012), and natural disaster events (Kim et al., 2016). The

most prevalent class is called the majority class, while the rarest class is called the minority class

(Huang et al., 2016).

Several techniques to process imbalanced data are developed and applied on the aspect of

data while others concentrate on the algorithmic level. To address the imbalanced problem, four

categories in the applications include the preprocessing strategies, cost-sensitive learning meth-

ods, adaptation of machine learning techniques and combination of the previous three approaches.

The preprocessing strategies include resampling techniques and/or variable importance analysis.

Resampling algorithms are presented by oversampling and undersampling methods. Resampling

techniques are used to rebalance the sample space for an imbalanced dataset to alleviate the effect

of the skewed class distribution in the learning process. Resampling methods are more versatile

because they are independent of the selected classifier (López et al., 2013). Oversampling meth-

ods are created the synthetic minority samples are randomly duplicating the minority samples.

Some research (Chawla et al., 2002; Estabrooks et al., 2004; and Tahir et al., 2009) introduced

the random undersampling in managing the imbalanced dataset. The undersampling method drops

non-default observations to oversee the data imbalance. However, those methods have limitations

such as overfitting or loosing essential information. The SMOTE (Synthetic Minority Oversam-

pling Technique) method increases instances for minority class based on KNN (K Nearest Neigh-

bor) algorithm introduced by Batista et al. (2004). The added observations have data properties

that are close to the original observations, reducing the level of imbalance in the data set. Based

on the original SMOTE method, several newly developed algorithms include SMOTE-N (Syn-

thetic Minority Oversampling Technique Nominal), SMOTE-NC (Synthetic Minority Oversam-

pling Technique Nominal Continuous), MWMOTE (Majority Weighted Minority Oversampling

Technique). Sukarna Barua et al. (2012) introduced two methods, Borderline-SMOTE 1 and Bor-

derline-SMOTE 2, which only alter the observations at the border of the sample to reduce the

misclassification rate. The SMOTEBoost method (Chawla et al., 2003) improved the SMOTE

method by incorporating the AdaBoost M2 algorithm. The EasyEnsemble method (proposed by

Liu et al., 2009) has increased the efficiency of the undersampling algorithm by adding useful1274

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ICYREB 2020information to the clustering method to reduce the noise level of the generated sample. Besides,

the BalanceCascade method (Han et al., 2009) provides a method to guide the deletion of obser-

vations rather than the random deletion method. Besides, bothsampling is a combination of the

above methods (Dubey et al., 2014).

The traditional classification method for predicting the probability of default of a loan is

logistic regression (Hosmer et al., 2000; Maalouf and Trafalis, 2011; Shu et al., 2014; Maratea et

al., 2014); neural networks and decision trees (Quinlan, 1998; Sarkar et al., 2016). Other methods

are applied including gradient boosting, least square support vector machines (Jin et al., 2012).

These research models focus on statistical orientation or artificial intelligence (AI).

Input variable analysis of these algorithms has gained attention in many practical applica-

tions (Ferretti et al., 2016) due to the complexity of interactions among variables on large datasets.

Variable analysis is a crucial task to improve the model interpretability, reduce the computational

cost, optimize the data storage, and provide a smaller number of relevant input variables. Several

approaches have been considered, which can be grouped input variables into new categories or

values without losing explanatory prediction power. Principal Component Analysis (PCA) is only

suitable for data that are normally distributed, near-standard, or linearly related features (Jolliffe

and Cadima, 2016). Other analysis constructs a model base on the Weight of Evidence (WOE)

and Information Value (IV) in clustering algorithms (Polykretis and Chalkias, 2018).

In Vietnam, the data processing algorithm by the undersampling method of reducing ran-

dom elements on the data boundary is introduced in medical data (Phuong et al., 2015). Research

on SMOTE techniques (Lien et al., 2018) in machine learning techniques in credit card fraud de-

tection show that the algorithm is suitable for controlling credit card fraud detection in Vietnam.

Currently, commercial banks in Vietnam have many difficulties in building internal risk

management models according to Basel II capital standards (issued by the Basel Committee on

Banking Supervision (BIS,2004)) and Circular No.41/2016/TT-NHNN (issued by the State Bank

of Vietnam). Specific difficulties occur in the process of data collection, data processing, selection

and evaluation of the effectiveness of the internal risk management model. More seriously, ac-

cording to VietstockFinance (2020), the bad debt ratio of Vietnamese commercial banks is all

lower 3.42% in 2019. As a result, the imbalanced datasets effect on the efficiency of the good

and bad classification algorithms at Vietnamese commercial banks. This means that the number

of non-performing loans is much less than that of performing loans in the datasets. These imbal-

anced datasets can lead to biased prediction in favor of the major group in the classification al-

gorithms (Ganganwar, 2012).

. Reality shows the need to find algorithms for imbalanced data processing and debt clas-

sification models appropriate to the context in Vietnam. Earlier studies yet investigate the effec-

tiveness of variable analysis based on grouped by WOE after managing imbalanced processed

datasets. This article aims to oversee an imbalanced credit dataset by oversampling, undersam-

pling, bothsampling and SMOTE algorithms in the Vietnam bank context. After that, we apply

this handled imbalanced data into the logistic regression with input variables transformed by

WOE to evaluate the importance of these algorithms and select an efficient credit scoring method

at commercial banks in Vietnam.1275

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ICYREB 20202. Credit scoring method at commercial banks

The credit scoring process at commercial banks has a number of steps. The first step is the

data processing. One of the challenging problems when applying regression models in practice

is the quality of the data. The data processing step also consumes a great deal of time, accounting

for nearly 80% of the total time. The data processing includes segmenting, sampling, data parti-

tioning, processing missing data, and outliers. The next step is the scorecard construction process.

After data cleaning, the variables will be grouped by binning. The new value for each group is

WOE (Weight of Evidence). WOE presents the predictive power of an independent variable in

relation to the dependent variable.Distribution of good is the percentage of good customers in a particular group.

Distribution of bad is the percentage of bad customers in a particular group.

The WOE approach is suitable for the logistic regression model thanks to the convenience

of scoring and no need to deal with missing data. Binning is to reduce the number of attributes

because if original variables are used in the regression model, a great deal of dummy variables

will be created. Also, binning variables is useful, especially for variables whose relationship to

the dependent variable is nonlinear. This technique helps deal with nonlinear effects in a linear

model. The next step is to remove some variables before running the model. They are the variables

which has the poor ability to differentiate between good and bad accounts. Eliminating them be-

fore running the model could improves the model quality and shortens processing time afterwards.

The index of value information – IV is used to remove weak variables (Siddiqi, 2012). The IV is

calculated using the following formula:

Criteria for evaluating the variable predictiveness according to IV:Table 1. The assessment of IV

IV Assessment

< 0.02 Not useful for prediction

0.02 – 0.1 Weak predictive power

0.1 – 0.3 Medium predictive power

– 0.5 Strong predictive power

>0.5 Suspicious predictive power(Source: Siddiqi, 2012)

1276

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ICYREB 2020From the IV, some variables that are not useful for prediction or have weak predictive

power are immediately removed from the model. For remaining variables, it is necessary to com-

bine with expert opinion to achieve the goal of dropping unnecessary variables but not losing

good factors.

Logit regression method

The dependent variable is the default status of the customer, symbolized by D (D = 0 if the

customer pays on time, D = 1 if the customer pays late). Modelling the relationship between cus-

tomer characteristics and repayment ability through the logit function as follows:The maximum likelihood method is applied to estimate the coefficients β, thereby calcu-

lating the default probability for each specific customer.

Criteria for selecting variables for the logit model are statistical significance and expert

judgement. At the same time, operational efficiency evaluation includes considering resources

mobilized for data collection, considering whether the use of variables is consistent with the pro-

visions of law or not.

3. Algorithms for handling imbalanced datasets

Oversampling

The common method of increasing instances in the group of bad accounts is randomly re-

peating the data in this group. Take a set of randomly selected minority examples in the minority

group, then augment the size of this group by replicating the selected examples and adding them

to it. The result is that total examples in the bad group will be increased and the class distribution

becomes more balanced accordingly.

This random method simply replicates a portion of minority class in order to increase the

weights of those examples. Because the replacement process is totally random, this method re-

creates some existing examples in the original minority class. Therefore, its main drawback is

that the overfitting phenomenon can occur. This method is the most fundamental among over-

sampling techniques. Many other common oversampling algorithms used in the real world are

developed based on this method (Peng Jun Huang, 2015).

Undersampling

Undersampling method randomly reduces observations in the good group. The downside

of this technique is that it is possible to eliminate useful observations from the majority group.

The common technique applied in undersampling is the weighted sampling. It is described

as follows: Calculate the weighted Euclidean distance of each negative instance from each of the

positive instances. All features are weighted by its Fishers discriminant score (F1 measure), which

measures the overlapping per attribute. After that, for each positive sample, sort negative instances

in ascending order of distance from the positive sample. Finally, for each positive instance, select1277

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ICYREB 2020a user-defined number of negative instances. The user-defined number indicates the desired ratio

of negative samples to positive samples. At this stage, special care is taken to avoid repetitive se-

lection of negative samples. If a particular negative sample has been already selected, the next

available negative sample is selected (Fernández, 2018).

Bothsampling

Due to the fact that oversampling results in too many bad observations and undersampling

can cause loss of the original data, the combination of these two methods can be chosen in the

hope of fixing those problems. However, it may not yield good results for the study either because

the method simultaneously multiply bad observations and reduce the random good ones.

SMOTE (Synthetic Minority Over-sampling Technique)

The SMOTE algorithm carries out an oversampling approach, but the key difference is that

it introduces synthetic examples but not replicate existing instances. To create these new data

points, the KNN algorithm is used for the minority group, clustering the data into K different

groups. Then, between two or more observations of the same group, it is needy to create more

observations. For this reason, the procedure is said to be focused on the “feature space” rather

than on the “data space”. For example, an xi positive instance is selected as basis to create new

synthetic data points. Based on a distance metric, several NNs of the same class (points xi1 to

xi4) are chosen from the training set. Finally, a randomized interpolation is carried out in order

to obtain new instances r1 to r4 (Fernández, 2018).

The SMOTE method works in “feature spaces” but not “data spaces”. It treats nominal and

continuous attributes in different ways. In the closest neighbor determination calculations, it uses

Euclidean distances for the continuum property and the Value Distance Metric for the nominal

property.

For continuum attributes:

hGet the difference between the feature vector (the minority class pattern) and one of its

k closest neighbors (the minority class sample).

hMultiply this difference with a random number between 0 and 1.

hAdding this difference to the object value of the original object vector constitutes a new

feature vector.

For nominal attributes:

hTake the ratio of majority groups between the feature vector under consideration and the

k nearest neighbors. In the case of a tie, take it at random.

hAssign that value to an additional new case for the minority class.

hUsing SMOTE creates additional regions for the minority, thereby allowing the classi-

fication methods to predict more minorities.

SMOTEboost algorithm

The SMOTEboost algorithm is the integration of the SMOTE algorithm into the standard1278

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ICYREB 2020boosting process. Thus, it gains the benefits of both boosting and SMOTE. This algorithm applies

to some asymmetric data at the medium and high levels, improves the prediction results for the

minority and improves the F-value.

Boosting helps improve the predictability of classifiers by recalculating the weights of mis-

classified observations. SMOTE only improves the classifier of minority cases. By attaching

SMOTE to boosting, it promotes boosting that focuses more on minority cases than in the ma-

jority.

SMOTEboost implicitly increases the weight of false negatives in the distribution because

the SMOTE algorithm increases the number of minority observations. So, in the next iteration,

SMOTEboost could create a wider decision area for the minority. SMOTEboost combines

SMOTE’s Recall-improving power and boosting’s Precision-improving power. Altogether it im-

proves the F-value or .

4. Validation

In the classification problem, to evaluate the performance of the model, two methods are

used including Confusion matrix and AUROC curve.

Confusion Matrix

In these problems, it is common to define the more critical data class as the Positive class

(P-positive), the other one called the Negative class (N-negative). In a good-bad classification

problem, bad means Positive, and good means Negative. From there, we define True Positive

(TP), False Positive (FP), True Negative (TN), False Negative (FN) to create confusion matrix

not normalized as follows:

Table 2. Confusion matrixPredicted “bad” Predicted “good”

Actual “bad” True Positives (TP) False Negatives (FN)

Actual “good” False Positives (FP) True Negatives (TN)

Evaluation criteria from the confusion matrix as following:

Table 3. Evaluation criteria

Predicted “bad” Predicted “good”

Actual “bad” TPR = TP/(TP+FN) FNR = FN/(TP+FN)

Actual “good” FPR = FP/(FP+TN) TNR = TN/(FP+TN)

FPR (False Positive Rate) is also known as a false forecast rate, FNR (False Negative Rate)

is also known as omission rate.1279

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ICYREB 2020For the classification problem where the data sets of the classes are hugely different from

each other, there is an efficiency measure commonly used as the Precision-Recall.

Precision = TP/(TP+FP)

Recall = TP/(TP+FN) = TPR

High precision means the accuracy of the bad observations found is high. Precision = 1 or

FP = 0 means that all observations predicted to be bad are true as bad. However, this does not

guarantee to find out all bad observations.

A high recall means a high True Positive Rate, meaning a low rate of omitting bad obser-

vations. Recall = 1 or FN = 0 means finding all bad observations. However, it is unlikely that all

bad prediction observations are correct.

Therefore, a good classification model is one that has both Precision and Recall as high as

possible, as close to 1 as possible. To measure the quality of the classifier based on both Precision

and Recall we use the F1 score.Choosing β higher than 1 means to value Recall over Precision and vice versa, β less

than 1 means that Precision is more important than Recall. Two commonly used values of β

are 2 and 0.5.

AUROC curve

The values of FPR and TPR change when the good or bad threshold changes. When the

threshold decreases, both FPR and TPR increase, which means more false statements than omis-

sions. Conversely, when the threshold increases, both the FPR and TPR decrease, which means

more omission than false statements.(Source: Sarang Narkhede, 2018)

Figure 1. Receiver Operating Characteristic curve or ROC curve

1280

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ICYREB 2020For each threshold, there is one pair of value (FPR, TPR). Standing for the points (FPR,

TPR) on the graph, when changing the threshold from 0 to 1, we get a line called the Receiver

Operating Characteristic curve or ROC curve.

Plot the graph with the horizontal axis FPR (some graphs denoted by 1 – Specificity) and

the vertical axis as TPR (some graphs denoted as Sensitivity).

Based on the ROC curve, one can show whether a model is useful or not. An efficient

model has low FPR and high TPR, there exists a point on the ROC curve that is close to the point

with the coordinate (0, 1) on the graph (upper left corner). The closer the curve is, the more effi-

cient the model is.

There is another parameter used to evaluate a model called Area Under the Curve or

AUROC, which is the area under the ROC curve. This value is a positive number less than or

equal to 1. The larger the value, the better the model. The meaning of AUROC criteria

Table 4. Assessment of model by AUROCAUROC Assessment of model

> 0.90 Excellent

0.80 – 0.90 Good

0.70 – 0.80 Fair

0.60 – 0.70 Poor

0.50 – 0.60 Fail(Source: D’Agostino et al., 2013)

Gini coefficient

The Gini coefficient is calculated according to the formula (Schechtman, 2016) of

Gini = 2* AUROC – 1

This coefficient is also used in evaluating the relevance or significance level of models. Its

measured values are between 0 and 1, where a score of 1 means that the model is 100% accurate

in predicting the outcome. The closer the Gini is to one, the better model is. On the other hand,

a Gini score equal to 0 means the model is entirely inaccurate.

2. Results of experimental research

The article uses credit dataset at a commercial bank in Vietnam, including 7052 observa-

tions with 456 bad observations. With the bad rate of 7.13%, the data surveyed is imbalanced be-

tween the two groups of good observations and the number of bad ones. The data relating to the

information of the customers is encrypted. After outliers processing, imbalanced data is handled

by methods of oversampling, undersampling, bothsampling and SMOTE. After that, the processed

datasets are divided into two datasets including training set (70%) and testing set (30%).

With each generated dataset, run the logistic model with the original variables and WOE

grouping variables for credit scoring. The model evaluation results are as follows:1281

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ICYREB 2020Table 5. Results of logistic models

Model Logistic Method AUROC Gini Precision Recall F_score

with

[1] Original Original 0.5521 0.1042 0.008 0.999 0.023

variables

[2] Original Oversam- 0.6271 0.254 0.581 0.499 0.308

variables pling

[3] Undersam- 0.6089 0.3166 0.628 0.784 0.448

pling

[4] Bothsam- 0.6296 0.2646 0.595 0.627 0.372

pling

[5] SMOTE 0.6105 0.221 0.628 0.580 0.353[6] (WOE) Oversam- 0.7051 0.4196 0.656 0.639 0.385

grouping pling

variables

[7] Undersam- 0.6228 0.244 0.599 0.634 0.376

pling

[8] Bothsam- 0.6978 0.3424 0.636 0.618 0.373

pling

[9] SMOTE 0.8208 0.642 0.757 0.742 0.447(Source: author’s calculation)

Using the original imbalanced dataset, the logistic model has the AUROC of 0.5521 and

the G coefficient of 0.1042. This result shows that the model is not able to distinguish between

good and bad observation. The low precision and high Recall coefficients (0.008 and 0.999, re-

spectively) means the accuracy of the bad observations found is low. As a result, predict results

are biased towards the majority group.

After processing with oversampling, undersampling, bothsampling, and SMOTE methods,

the validation results of the logit regression models with the unbalanced processed data set were

improved compared to the model [1]. The results show that the imbalanced handling method in-

creases the efficiency of the classification methods. The AUROC of these logistic models increase

greater than 0.5, showing that these models are capable of differentiation. Besides, the under-

sampling method shows better results than the other three methods because the coefficients of

Gini, Precision, Recall, F_score are all higher. However, the Gini coefficient is in the range of

0.2 – 0.3, showing that the classification ability of these models is still quite weak.1282

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ICYREB 2020To improve the differentiation of the model, we use a combination of logistic regression

variable analysis. Validation results of the model with unbalanced processed data sets showed a

marked improvement in the evaluation criteria of classification ability. The AUROC coefficient

results show that using binning WOE variables has improved the ability to classify good and bad

customers. Also, the SMOTE methods using the logistic model [9] combined WOE grouping

variables are significantly improved by the high AUROC (0.821) and Gini coefficients (0.624).

This model has the highest recall of 0.757 and precision coefficients of 0.742, compared to [6],

[7], and [8] models. This conclusion is also true for the F_ score coefficient.

6. Conclusion

Imbalanced data problem occurs in some different fields and reduces the efficiency of the

classification regression. Several algorithms are introduced to handle this problem in preprocess-

ing data steps. Various methods are introduced in this study. Each method has both advantages

and drawbacks. The bank must consider rationality, stability, and strength, and complexity when

working with each method. The oversampling algorithm is easy to implement. However, the data

size increases, but observations are repeated from the original observations, so in some cases,

the oversampling method shows ineffective classification. The undersampling technique that ran-

domly reduces the data in the Good observation group is also easy to perform, in turn, it can

eliminate useful observations in the majority group. Overcoming the disadvantages of oversam-

pling, the SMOTE method complements the minority, creating a completely new advantage

through artificial methods to show more optimal results in most classification cases.

We have tested the effectiveness of the oversampling, undersampling, bothsampling and

SMOTE algorithms with specific data in Vietnam. Based on the AUROC, Gini, recall, precision,

and F-score coefficients; the results show that imbalanced datasets have a negative effect on the

ability to classify good and bad customers. Furthermore, the undersampling technique is more

efficient when used in the logit model with the original variables. To improve performance of

the classification regression, when combining variable analysis grouped by WOE, the SMOTE

algorithm shows outstanding efficiency compared to the mentioned others.

Appendix: The Figures represent the AUROCFigure 2. AUROC Figure 3. AUROC Figure 4. AUROC

of the logistic model with of the logistic model with of the logistic model with

original variables oversampling data undersampling data1283

- INTERNATIONAL CONFERENCE FOR YOUNG RESEARCHERS IN ECONOMICS & BUSINESS 2020

ICYREB 2020Figure 5. AUROC Figure 6. AUROC Figure 7. AUROC

of the logistic model with of logistic model with of Logistic model clustering

bothsampling data SMOTE data variables by WOE with

oversampling dataFigure 8. AUROC Figure 9. AUROC Figure 10. AUROC

of Logistic model clustering of Logistic model clustering of Logistic model clustering

variables by WOE with variables by WOE with variables by WOE with

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