Data Analysis and Related Applications 3 E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 393

Veröffentlichungsjahr: 2024

Beschreibung

The book is a collective work by a number of leading scientists, analysts, engineers, mathematicians and statisticians who have been working at the forefront of data analysis and related applications, arising from data science, operations research, engineering, machine learning or statistics. The chapters of this collaborative work represent a cross-section of current research interests in the above scientific areas. The collected material has been divided into appropriate sections to provide the reader with both theoretical and applied information on data analysis methods, models and techniques, along with appropriate applications. The published data analysis methodology includes the updated state-of-the-art rapidly developed theory and applications of data expansion, both of which go through outstanding changes nowadays. New approaches are expected to deliver and have been developed, including Artificial Intelligence.

Leseprobe

Big Data, Artificial Intelligence and Data Analysis Set

coordinated byJacques Janssen

Volume 11

Data Analysis and Related Applications 3

Theory and Practice – New Approaches

Edited by

Yiannis Dimotikalis

Christos H. Skiadas

First published 2024 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-962-4

PART 1Data Analysis Classification

1Walking into the Digital Era: Comparison of the European Union Countries in the Last Decade

This study allows us to understand how European Union countries are more or less prepared for confronting the digital era. In particular, we aim to understand whether the least developed countries are close or fairly distant from the others. For this study, data for some variables associated with digital skills, digital economy and digital society have been collected from the Eurostat database during the period 2010–2020. To analyze the data, we applied a double principal component analysis.

These results allow us to identify the differences and similarities between countries and indicators from 2010–2020, and, more precisely, to study the countries’ evolution trends and the evolution of the relations between the different indicators.

1.1. Introduction

This study allows us to understand how the countries in the European Union are more or less prepared for confronting the digital era. People and societies have to deal with these new challenges not only to gain knowledge but also to promote growth, development and opportunities for everyone. In particular, it is interesting to see whether the least developed countries are close or relatively distant from the others in this route, in order to develop new strategies and to overcome detected fragilities.

Table 1.1.Variables considered in the study grouped by topic

G1: Activity and employment status of individualsV1 – % employed persons in the universe of active persons with ICT educationV2 – % males in the universe of employed persons with ICT educationV3 – % persons with tertiary education among employed persons with ICT educationV4 – % persons aged 15–34 among the employed persons with ICT education

G2: Internet at households with population aged 16–74 yearsV5 – % households with Internet accessV6 – % households with broadband Internet connection

G3: Internet use by individualsV7 – % individuals with daily Internet access in the last three months before the surveyV8 – % individuals with Internet use in the last three months before the surveyV9 – % individuals with Internet use in the last 12 months before the surveyV10 – % individuals with Internet use more than a year ago before the surveyV11 – % individuals who have never used Internet before the survey

G4: Online purchase by individualsV12 – % individuals with last online purchase in the last three months before the surveyV13 – % individuals with last online purchase in the last 12 months before the survey

G5: Reasons of the individuals for using InternetV14 – % individuals using Internet for finding information about goods and servicesV15 – % individuals using Internet for looking for a job or sending a job applicationV16 – % individuals using Internet for selling goods and services

G6: Enterprises use for selling (having e-commerce)V17 – % enterprises with e-commerce sales of at least 1% turnover

G7: Enterprises that provided training to develop/upgradeICT skills, excluding enterprises of the financial sectorV18 – % small enterprises provided training to ICT/IT specialistsV19 – % medium enterprises provided training to ICT/IT specialistsV20 – % SMEs provided training to ICT/IT specialistsV21 – % large enterprises provided training to ICT/IT specialistsV22 – % small enterprises provided training to other employed personsV23 – % medium enterprises provided training to other employed personsV24 – % SMEs provided training to other employed personsV25 – % large enterprises provided training to other employed personsV26 – % small enterprises provided training to their personnelV27 – % medium enterprises provided training to their personnelV28 – % SMEs provided training to their personnelV29 – % large enterprises provided training to their personnel

1.1.1. The dataset

For this study, a large number of indicators associated with digital skills, digital economy and digital society (29 variables) have been considered, as well as all European countries for which we have data available (30 individuals). The data for these indicators were collected from the Eurostat database during the period 2010–2020.

The variables are grouped by topic (groups G1–G7), and are listed in Table 1.1. When we refer to ICT (Information and Communication Technology) Education, in the general sense, it means providing users with a diverse set of technological tools, definitions and resources to create, store, communicate, manage and optimize information. The database aggregates the ICT education levels into two categories: upper-secondary and post-secondary non-tertiary (levels 3 and 4), and tertiary (levels 5–8). Concerning employed persons with ICT education, they are divided by two classes of age: 15–34 and 35–74. The enterprises are classified accordingly to the number of employees and self-employed persons as follows: small (10–49), medium (50–249), SMEs (10–249) and large (250 or more).

Of course, it would be interesting to include other indicators such as the percentage of the ICT sector in GPD, the R&D personnel in the ICT sector as a percentage of total R&D personnel or the business expenditure on R&D in the ICT sector as a percentage of total R&D expenditure, for instance. However, we do not have information for all countries and years during the period.

The countries (individuals) considered in this study are presented in Table 1.2. We also further analyzed six data tables, corresponding to the years 2010, 2012, 2014, 2016, 2018 and 2020, each one with the same countries and variables. As the United Kingdom was not part of the EU in 2020, we considered the values obtained in 2019, instead of 2020, so we could still include it in the analysis.

Table 1.2.Countries considered in the study

Countries

Countries

Countries

EU27 – European Union 27

FR – France

AT – Austria

BE – Belgium

HR – Croatia

PL – Poland

BG – Bulgaria

IT – Italy

PT – Portugal

CZ – Czechia

CY – Cyprus

RO – Romania

DK – Denmark

LV – Latvia

SI – Slovenia

DE – Germany

LT – Lithuania

SK – Slovakia

EE – Estonia

LU – Luxembourg

FI – Finland

IE – Ireland

HU – Hungary

SE – Sweden

GR – Greece

MT – Malta

UK – United Kingdom

ES – Spain

NL – Netherlands

NO – Norway

1.1.2. Preliminary analysis of the data

Before proceeding with a multivariate data analysis, we carried out an exploratory analysis in order to gain insight into the data. In particular, we represented the boxplots by year and for all of the variables (not presented here) and among other conclusions, it is worth mentioning here the huge number of outliers observed in the data, and reported in Table 1.3. The countries with an * are severe outliers. Some countries appear as outliers with respect to some variables during several years of the period 2010–2020, such as GR, RO and BG: GR is a severe lower outlier in variable V1 (the percentage of employed persons in the universe of active persons with ICT education is small comparatively to the corresponding percentage in the other countries); RO and BG are lower outliers in variables V21, V25 and V29 (the percentage of large enterprises providing training in ICT skills to their personnel, specialists or other employed persons is small when compared to what happens in other countries). We also mention, for instance, DK and FI, who appear during some years of the period as upper outliers in variable V15 (with a large percentage of individuals using the Internet to look for a job or sending a job application compared to other countries), and countries such as BE and FI, who in 2020 were upper outliers in variables V19, V23 and V27 (i.e. they have a large percentage of medium enterprises providing ICT training when compared with other countries). Other details can be found in Table 1.3.

Table 1.3.Variables that present several outliers along the period

Year

Variables and associated outliers

2010

Lower V1(GR), V3(IT), V21(RO), V29(RO)

2012

Lower V1(ES,GR*), V21(RO), V29(RO)Upper V10(CZ)

2014

Lower V1(PT,GR*), V7(RO), V21(RO), V29(RO)Upper V10(BG)

2016

Lower V1(GR*), V2(CY), V7(RO), V21(RO),V25(BG,RO), V29(BG,RO)Upper V15(DK), V17(IE)

2018

Lower V1(GR*,IT,PT), V3(PT), V21(RO),V25(BG,RO), V29(RO)Upper V26(NO)

2020

Lower V1(GR*,ES), V2(DK), V14(IT,RO),V19(RO), V21(RO), V23(BG,RO),V25(BG,RO), V27(BG,RO), V29(RO)Upper V10(BG,IT), V15(DK,FI), V19(BE),V23(FI),V27(FI)

After an introduction, aiming to present this study and detailed descriptions of the dataset, as well as a preliminary data analysis, this chapter is organized as follows. Section 1.2 gives a small description of the double principal component analysis method, the statistical methodology applied in this work, and also includes part of the results obtained from the application of this method to the data. The representation and interpretation of the trajectories are provided in section 1.3, and section 1.4 concludes the chapter with some final conclusions.

1.2. Double principal component analysis: a brief description

To analyze the data, we applied a double principal component analysis (DPCA), a method of multivariate data analysis introduced in Bouroche (1975) to analyze three-way data with quantitative variables. This method can be modified to allow the analysis of categorical data. See, for instance, Lera et al. (2006).

DPCA is an extension of the principal component analysis (PCA) method, and allows us to jointly analyze several data tables with information collected on the same variables and individuals, in several instants of time, for instance. In this study, we have six data tables, corresponding to the years 2010, 2012, 2014, 2016, 2018 and 2020, all with the same countries and variables.

The application of the DPCA method comprises four steps: (1) interstructure, i.e. the analysis of the global evolution of the data along the period of time; (2) analysis of the clouds of individuals; (3) intrastructure, i.e. the choice of the best common space to represent the individuals and the variables; and (4) representation and interpretation of the trajectories of the individuals and variables.

From Figure 1.1, we can observe that the years appear in chronological order along the first axis. As can be expected, there are big changes between the periods of 2010 and 2020.

Figure 1.1.Representation of the interstructure

The analysis of individuals’ clouds and, in particular, the percentage of explained inertia when they are projected in a subspace of a smaller dimension, lead us to select the plan of the first two principal axes obtained from the PCA of the year 2014 to represent the individuals and variables for all years. This system of principal axes minimizes the average loss of information, i.e. it globally maximizes the percentage of explained inertia. The projection of the countries and variables in this plan globally explains 76.6% of the total inertia. If we consider three axes, the percentage of the explained inertia increases to 81%, but the balance between the gain and the increase of complexity to interpret the trajectories lead us to choose only two axes. The results corresponding to the last step of the method are presented in the next section.

Figure 1.2.Variables’ trajectories

1.3. Representation of the trajectories

The representation and interpretation of the trajectories is an important step in DPCA. It allows us to identify the differences and similarities between individuals (countries), and analyze the evolution of the correlations between the different variables during the 2010–2020 period. These trajectories are presented in Figures 1.2 and 1.3. All of the graphics have the same scale, i.e. the range in the first axis is the same, as well as the range in the second axis, to have comparable graphics. We have highlighted the year 2010 in red and the year 2020 in blue.

Looking at Figure 1.2, among the variables that present closed trajectories (i.e. they are variables with correlation values with the other variables stable along the period), we highlight the variables V18, V20, V25 and V27, related to the percentage of small or medium enterprises providing training in ICT skills to specialists or to their personnel employees or large enterprises who provide training to other employed persons. All of the other variables have extended trajectories, i.e. they are variables with relevant changes in their correlations with the others along the period. Among them, we highlight the variables V1, V2, V3 and V4 from group 1 (Activity and employment status of individuals), the variable V10 – % of individuals with Internet more than a year ago, the variables V15 and V16 from group 5 (Reasons for using Internet) and the variable V17 – % enterprises with e-commerce having received orders online, at least 1%.

Figure 1.3.Countries’ trajectories

As shown in Figure 1.3, in general, most of the countries present large or extended trajectories, at least along one of the axes, and in particular, we highlight the trajectories of BG, ES, FR, CY, NL, AT, PL and NO. With very closed trajectories, we highlight EU27 and BE. The countries PT and GR also exhibit some instability in their trajectories.

For the interpretation of the trajectories, we only considered the countries that contributed the most to the axes, with their trajectories represented together in Figures 1.4–1.5, and the variables correlated more so with them.

Concerning the first axis, which explains 65.3% of the total inertia, almost all of the variables have high correlations with the axis in all years, except V1, V2 and V3. Therefore, we only highlight the variables with correlations above 0.8 in absolute value. The first axis opposes the group of countries NO, FI, DK, SE, DE and BE that have large values in the variables V5–V7, V9, V12–V14 and V19–V29 and small values in variable V11, to the group of countries RO, BG, IT, GR, LT and PL that present large values in the variable V11 and small values in the others.

Figure 1.4.Interpretation of trajectories in the first axis (65.3% inertia)

The second axis only explains 11.3% of the total inertia, and few variables have significant correlations with this axis. Thus, we highlight the variables with correlations above 0.5 in absolute value. The second axis opposes the group of countries PT, HR, FI, CY, AT and BE, all of which have high values in the variables V24, V26 and V28 (years 2010, 2012 and 2014), V23–V24 (years 2010 and 2012) and small values in variable V1 (years 2010, 2012, 2014 and 2016), compared to the group of countries NL, LU, EE, LT and LV, all of which present high values in variable V1 and small values in the others.

Figure 1.5.Interpretation of trajectories in the second axis (11.3% inertia)

1.4. Some final conclusions

The results of this study allow us to identify some existing differences or similarities between groups of European countries during the period 2010–2020 under analysis, concerning ICT skills and employment, Internet usage and digital development.

As expected, this study confirms our perception that the countries of the north and central Europe are very prepared for confronting the challenges of the digital age, since they have already greatly invested in providing training in ICT skills to their populations and how everyone is comfortable using the Internet for a variety of purposes: communication, purchase, selling, finding, and so on. Among these countries, the most similar are the following subgroups of countries: (DE and SE), (NO and DK), FI and BE.

On the other hand, the countries of the East and South of Europe are less prepared, having made little investment in ICT education and, in most of them, a large part of the population has never used the Internet. Among these countries, the closest are the subgroups (PL and LT), (IT and GR), BG and RO.

We also observe that in countries such as PT, AT, CY, HR, BE and FI, where the percentage of employed persons with ICT education is yet small, a large percentage of small and medium enterprises provide training to their personnel or to other employed persons to develop or upgrade ICT skills. The opposite happens, for instance, in the countries LT, LV, EE, LU and NL.

1.5. Acknowledgements

This work was supported by the National Funds through the Portuguese funding agency, FCT – Fundação para a Ciência e a Tecnologia, within the framework of the projects UIDB/00006/2020 (CEAUL) and LA/P/0063/2020 (INESC TEC).

1.6. References

Bouroche, J.M. (1975). Analyse des donnés ternaires : la double analyse en composantes principales. Third-cycle PhD Thesis, Université de Paris VI, Paris.

Lera, L., Pérez, R., Bouquet, A. (2006). El doble análisis em componentes principales para datos categóricos y su applicación en un estudio de migración.

Revista Columbiana de Estadística

, 29(1), 17–34.

Note

Chapter written by Fernanda Otilia FIGUEIREDO and Adelaide FIGUEIREDO.

2Multivariate Kernel Discrimination Applied to Bank Loan Classification

The purpose of this chapter is to apply a kernel discriminant analysis to classify bank loans and determine which loans are at risk of default. This study begins by introducing the concept of kernel density estimation, which is a widely used non-parametric technique to obtain an estimate for the probability density function. This procedure is based on two main parameters: the kernel function and the bandwidth, the latter being the crucial parameter. The multivariate kernel density estimator is later applied to discriminant analysis to obtain kernel discrimination. This is a method that classifies observations into a predetermined number of distinct and disjointed classes. Finally, we apply multivariate kernel discriminant analysis to a sample of bank loans to determine which loans can be classified as defaulted. This model can help predict the likelihood that future loans may default.

2.1. Introduction

As the name implies, credit risk analysis is used to assess the likelihood of a borrower’s repayment failure and the loss caused to the financer when default occurs. This concept greatly affects the long-term success of any bank or financial institution. In Malta, the non-performing loan ratio stood at 3% in December 2019 (Central Bank of Malta). This ratio reflects the country’s credit quality of loans. Banks need to determine the probability of non-performing loans of companies to decrease the prospect of weakened liquidity. The aim of this study is to create a model that predicts whether a company will be able to pay its outstanding loan, based on several variables. To obtain such a model, we apply the multivariate kernel discriminant analysis. This technique uses kernel density estimation, as well as non-parametric discriminant analysis.

Rosenblatt (1956), Whittle (1958), Parzen (1962) and Watson and Leadbetter (1963) studied and presented a number of results regarding univariate kernel density estimation. Moreover, Cacoullos (1966) extended Parzen’s results to the multivariate case. Loftsgaarden and Quensenberry (1965) and Epanechnikov (1969) also applied multivariate non-parametric kernel density estimation.

The quality of multivariate kernel density estimation depends mainly on the choice of the bandwidth matrices. The univariate plug-in bandwidth estimator provided by Sheather and Jones (1991) was extended to the multivariate case by Wand and Jones (1993). Moreover, a number of authors, including Chacón and Duong (2010, 2012) and Horová et al. (2013), also contributed to the field of optimal bandwidth selection by applying a number of novel techniques.

Multivariate kernel density estimation has been applied to discriminant analysis. The properties of this technique feature in Murphy and Maron (1986) and Silverman (1986). Over the past several years, kernel discriminant analysis has been applied to several fields. For example, Liberati et al. (2012) and Widiharih et al. (2018) both used kernel discriminant analysis to address the problems of credit risk analysis in banking and making efficient credit granting decisions. Some further applications of kernel discriminant analysis include Hand (1983) and Chen et al. (2005) in medicine, Martin (2011) in agriculture and Farida and Aidi (2016) in education.

This section concludes by giving a brief overview of the structure of this chapter. In section 2.2, we discuss multivariate kernel density estimation. In section 2.3, we outline the properties of kernel discriminant analysis. In section 2.4, we apply these techniques to a local dataset that contains not only the data of a number of local companies but also the status of their bank loan (whether the company is in default or otherwise). Further details concerning the said dataset are provided in section 2.4. As mentioned earlier, the main aim will be to create a model which uses kernel discriminant analysis, which can identify firms that are close to bankruptcy. In this section, we split the data into the training set and test set. We train the model using the former dataset and test the model performance using the latter dataset. Section 2.5 contains some concluding remarks and suggestions for future research.

2.2. Multivariate kernel density estimation

Let p denote the total number of variables in the dataset and let n denote the total number of companies in the dataset.

Let Xi be a random variable representing the ith variable. Moreover, let X be a random vector, where X = (X1, X2, …, Xp)T and xn = (xn1, xn2, …, xnp)T will be the vector that contains all of the observations of the nth company. Let be the n×p matrix which contains all the observations of the dataset, such that:

Multivariate kernel density estimation is composed of two main components: the kernel function, which will now be denoted by K, and the bandwidth, which will now be denoted by H. In a multivariate setting, the bandwidth is a p × p matrix, where p represents the dimension of the data, i.e. the number of variables being studied. The approach chosen for multivariate kernel density estimation highly depends on the type of bandwidth matrix considered. When H is taken to be diagonal, this is usually referred to as a constrained bandwidth matrix. Otherwise, it is referred to as an unconstrained bandwidth matrix. The diagonal bandwidth matrix is thus a special case of the unconstrained bandwidth matrix.

2.2.1. Kernel density estimator

For the multivariate case with a bandwidth matrix H, the kernel density estimator is defined as follows:

where n represents the sample size and KH(x) is said to be the scaled kernel, which is defined as:

where is the inverse of the square root of the determinant of the matrix H and is the inverse of the matrix square root. Therefore, the kernel density estimator can be written as:

where K is the multivariate kernel function. K is assumed to be the symmetric function. Furthermore, and , implying that K satisfies the properties of a multivariate probability density function. There are various types of functions that satisfy the above-mentioned properties, which include the standard multivariate normal kernel density function and the multivariate Epanechnikov kernel function. Similar to the univariate case, the choice of the kernel function is not crucial. However, the choice of the bandwidth matrix is extremely important.

It is important to note that the formula discussed above regarding the kernel density estimator is not affected by the choice of bandwidth matrix (constrained or unconstrained).

2.2.2. The optimal bandwidth

Finding the optimal bandwidth is crucial when performing kernel density estimation. This is because an incorrect bandwidth matrix will lead to either over fitting or under fitting. There are three ways to find the optimal bandwidth and these are: unbiased cross-validation (UCV), biased cross-validation (BCV) and smoothed cross-validation (SCV). These techniques can be applied to constrained and unconstrained bandwidths.

2.2.2.1. Unbiased cross-validation

Rudemo (Rosenblatt 1956) and Bowman (1984) discuss the method of UCV. This can be classified as a leave-one-out type cross-validation method and is defined as follows:

where .

Hence, the optimal bandwidth, denoted by , can be obtained as follows:

2.2.2.2. Biased cross-validation

Scott and Terrell (Rudemo 1982) discuss this method of cross-validation and refer to it as BCV and is defined as follows:

where . Moreover, ⊗ represents the Kronecker product.

Then, we obtain the optimal bandwidth :

2.2.2.3. Smoothed cross-validation

The last version of cross-validation discussed is SCV.

Hall et al. (1992) discuss this technique and propose the use of a pilot kernel function, which we denote by L, and a pilot bandwidth matrix, which we denote by G, such that L and G may be different from K and H, obtaining:

where , and * represents the convolution operator.

The optimal bandwidth is therefore obtained as follows:

These three cross-validation techniques will be applied to the dataset. Moreover, we will at first assume an unconstrained bandwidth and subsequently take a constrained bandwidth. The UCV, BCV and SCV formulas for the constrained bandwidth are similar to the unconstrained case described above. The results will then be compared and contrasted, and the best result will be highlighted. We now move on to define kernel discriminant analysis.

2.3. Kernel discriminant analysis

Fix and Hodges (1951) proposed kernel discriminant analysis as a non-parametric version of discriminant analysis. Let 1,2,…,m denote each distinct population, where m represents the number of available populations; and let fj denote the multivariate density function of the observations drawn from population j. Moreover, let nj denote the number of observations obtained from population j. Finally, let πj denote the prior probability that an observation belongs to population j.

Discriminant analysis uses Bayes theorem. If bi denotes the event that an individual belongs to the ith population, and a denotes the event that an individual has a particular vector of observations, then Bayes theorem states that the probability of event bi occurring, given that event a occurred is:

Therefore, by Bayes theorem, we have that the probability that an observation x belongs to the population j is defined as:

An individual whose set of observations are contained in vector xn+1 is allocated to population j if and only if πjfj(xn+1) = argmaxj∈{1,…,m}πjfj(xn+1). Hence, we deduce that an individual whose set of observations are contained in vector xn+1 is allocated to population j if and only if . We can define the log-odds ratio that the point x is classified to population j rather than to population i as follows:

Hence, the point x is allocated to population j if Lji > 0 for all i ≠ j. Duong (2007) suggests replacing the density functions fj discussed above with the kernel density estimator . This approach involves the estimation of the density function fj from a training dataset using kernel density estimation, and then substituting the density function with this kernel density function to obtain the kernel discriminant rule:

An individual whose set of observations are contained in vector xn+1 is allocated to population j given data X if and only if .

The techniques discussed in both this section and section 2.2 can now be applied to the dataset.

2.4. Applying kernel discriminant analysis to local data

2.4.1. Description of data

This dataset was originally divided into five distinct populations:

non-impaired;

impaired but not in default;

default because unlikely to pay;

default because of past due more than 90 days;

default because both unlikely to pay and past due more than 90.

Table 2.1.Detailed description of variables

Variable number and name

Description

1) Amount in arrears

Shows the aggregate amount of principal, interest and any other fee payment that is outstanding at the reference date.

2) Credit lines limit

Shows the outstanding amount plus any unutilized credit limit that can be drawn upon by the customer.

3) Utilized balance

Shows the amount availed of from the authorized amount.

4) Outstanding balance

Shows the amount outstanding on the utilized balance at the end of the reference date, excluding any accrued interest.

5) Collateral market value of residential property

Shows the market value of the residential property that backs the exposure as per the latest valuation.

6) Collateral extendible value of residential property

Shows the collateral for the residential property which can be considered for credit risk mitigation at the reporting date.

7) Collateral market value of commercial property

Shows the market value of the commercial property which backs the exposure as per the latest valuation.

8) Collateral extendible value of commercial property

Shows the collateral for the commercial property which can be considered for credit risk mitigation at the reporting date.

9) Collateral life policy surrender value

Shows the amount of money policy holders will receive if they exit the policy before maturity. In this case, this amount is taken by the creditor to be paid for the loan.

10) Collateral cash or quasi cash

Cash or instruments that are easily converted to cash and used as collateral.

11) Collateral extendible value of other

Shows the collateral for any other types of assets not previously mentioned which can be considered for credit risk mitigation at the reporting date.

12) Specific credit risk adjustment

Shows the amount of provision against credit risk.

13) Risk-weighted assets

Shows the risk-weighted exposure amounts.

14) Interest rate

Shows the interest rate that is calculated.

15) Days past due

Shows the total number of days past due as from day 1.

For simplicity, it was decided to reclassify the data into two classes: the first group was labeled as “non-defaulted” and includes those loans that fall under the non-impaired population; the second was labeled as “defaulted”, and includes all the remaining observations. Apart from the above-mentioned classification variable, the dataset contains 15 other quantitative variables. A detailed description of these variables can be found in Table 2.1. Each variable was given a number (which can be found in Table 2.1). This will be used to refer to a specific variable.

The dataset was divided into two sets: the training and test sets in the ratio of 70:30. When observing the training set, it was noted that 88.83% of the observations fall under the non-defaulted population. Such an imbalance in the dataset could lead to problems when using any classification technique. Indeed, the method used in this study is not immune to such problems.