A Wavelet-based Data Analysis to Credit Scoring

Abstract

Nowadays, the dramatic growth in consumer credit has made ineffective the methods based on the human intervention, aimed to assess the potential solvency of loan applicants. For this reason, the development of approaches able to automate this operation represents today an active and important research area named Credit Scoring. In such scenario it should be noted how the design of effective approaches represents an hard challenge, due to a series of well-known problems, such as, for instance, the data imbalance, the data heterogeneity, and the cold start. The Centroid wavelet-based approach proposed in this paper faces these issues by moving the data analysis from its canonical domain to a new time-frequency one, where this operation is performed through three different metrics of similarity. Its main objective is to achieve a better characterization of the loan applicants on the basis of the information previously gathered by the Credit Scoring system. The performed experiments demonstrate how such approach outperforms the state-of-the-art solutions.

Full text

A Wavelet-based Data Analysis to Credit Scoring
Roberto Saia, Salvatore Carta, Gianni Fenu
Department of Mathematics and Computer Science
University of Cagliari, Via Ospedale 72 - 09124 Cagliari, Italy
Email: {roberto.saia, salvatore, fenu}@unica.it
Abstract—Nowadays, the dramatic growth in consumer
credit has made ineffective the methods based on the human
intervention, aimed to assess the potential solvency of loan
applicants. For this reason, the development of approaches
able to automate this operation represents today an active
and important research area named Credit Scoring. In such
scenario it should be noted how the design of effective
approaches represents an hard challenge, due to a series of
well-known problems, such as, for instance, the data
imbalance, the data heterogeneity, and the cold start. The
Centroid wavelet-based approach proposed in this paper
faces these issues by moving the data analysis from its
canonical domain to a new time-frequency one, where this
operation is performed through three different metrics of
similarity. Its main objective is to achieve a better
characterization of the loan applicants on the basis of the
information previously gathered by the Credit Scoring
system. The performed experiments demonstrate how such
approach outperforms the state-of-the-art solutions.
Index Terms—business intelligence, credit scoring, pattern
mining, data processing, wavelets, classifications, metrics
I. INTRODUCTION
The approaches of Credit Scoring are aimed to
evaluate the user reliability in several contexts such as,
for instance, those related to the loan applications (from
now on named as instances). They cover a more and
more crucial role in this our age dominated by the
consumer credit, since the amount of money lost by the
financial operators due to loans fully or partially not
repaid depends on their effectiveness.
A Credit Scoring approach works by classifying each
new instance as reliable or unreliable by exploiting an
evaluation model defined on the basis of the previous
instances. They can be considered as a series of statistical
methods aimed to evaluate the probability that a new
instance will lead to a fully or partially non-repayment of
a loan [1]. The definition of effective Credit Scoring
approaches is not an easy task, due to a series of well-
known problems.
The idea around which this paper revolves is mainly
based on the the exploitation of the Discrete Wavelet
Transformation (DWT) [2], which is used in order to
move the data analysis in a non-canonical time-frequency
domain. In such domain the analysis is performed
through three different metrics of similarity, which are
aimed to better characterize the classes of data involved
in the Credit Scoring processes . The contributions given
by this paper are as follows:
definition of the time series to be use as input of
the DWT process, defined on the basis of the
previous instances;
conversion of the instance time series into the
frequency-time domain by using the DWT
process;
formalization of the Centroid Wavelet-based
Approach (CWA) algorithm able to classify the
new instances.
The paper is organized as follows: Section II provides
an overview of the Credit Scoring scenario; Section III
introduces the formal notation adopted in this paper;
Section IV describes our approach; Section V provides
details about the performed experiments and their results;
Section VI draws certain conclusions and points to some
further directions for research.
II. BACKGROUND AND RELATED WORK
An ideal Credit Scoring system should be able to
evaluate each new instance correctly, by classifying it as
reliable or unreliable on the basis of the information
available in the previous instances.
Literature proposes a considerable number of
classification techniques aimed to perform such task [3],
as well as many studies focused on the evaluation of their
performance [4], on the optimal tuning of their
parameters [5], and on the most suitable metrics of
evaluation [6]. On the basis of the results offered by the
Credit Scoring techniques is possible to predict when an
application (e.g., for a loan) potentially leads towards a
risk of partial or total non-repayment [7].
Regardless of the adopted technique, there are a
number of problems that complicate such tasks. The
imbalanced class distribution of data is the most
important of them and it happens because the previous
instances, collected by a Credit Scoring system to train its
evaluation model, are composed by a big number of
reliable cases, compared to the number of unreliable
ones. It leads towards a reduction of the Credit Scoring
techniques effectiveness [8].
Another problem to face is the data heterogeneity
which in literature is described as the incompatibility
among similar features resulting in the same data being
represented differently in different datasets [9].
The cold start problem instead happens when the
previous instances are not representative for both classes
of information (reliable and unreliable), preventing the
definition of an effective evaluation model [10].
The proposed approach is mainly based on the
Discrete Wavelet Transformation (DWT) process [11].
Such process exploits the wavelets, a task that in
literature is usually performed in order to reduce the size
or the noise of data (e.g., in the image compression and
filtering tasks). The wavelets are mathematical functions

that work by decomposing the input data into different
frequencies at different scales. The input data of a DWT
process is usually a time series [12], a sequence of values
obtained by measuring the variations during the time of a
specific type of data (e.g., voltage, temperature, etc.). The
output is a new representation of data in a frequency-time
domain (data representation in terms of both frequency
and time). In our case, the time series used as input of the
DWT process are the values assumed by the instance
features.
The frequency-time domain offers us some interesting
advantages, the most important of them is the Multi-
Resolution Analysis operates by DWT that allows us to
observe the data at different levels of resolutions [13],
with the possibility of obtaining an approximated or
detailed vision on them. Our approach exploits this in
order to have a better characterization of the reliable and
unreliable instances.
The Equation 1 shows the formalization of the
Continuous Wavelet Transform (CWT), where Ψ(t) is the
mother wavelet (i.e., a continuous function in both the
time and frequency domain) and * denotes the complex
conjugate.



Considering that, for several reasons (e.g., the
computational load), it is not possible to perform a data
analysis by using all the wavelet coefficients, a common
approach is to use a discrete subset of the upper half-
plane, so we can be able to rebuild the original data by
using the corresponding wavelet coefficients. Such
discrete subset is composed by all the points
, where , and after this operation
we can formalize the child wavelets (Equation 2) .
A data compression that allows us to have a data
overview (approximated data view) is related to the use
of small scales, as this is equivalent to using high
frequencies (because the scale is given by the formula
1/frequency). In a opposite way, a data expansion that
allows us to observe the data changing (detailed data
view) is related to the use of large scales, as it is
equivalent to using low frequencies.
A number of functions can be used as mother wavelet
(e.g., Haar, Daubechies, Symlets, Meyer, Coiflets, etc),
but for the objectives of this paper we take into account
only the Haar [13] one. It is a sequence of rescaled
square-shaped functions which represent a wavelet
family. They are based on the mother Ψ and father φ
(scaling) functions shown in Equation 3.
The proposed Centroid Wavelet-based Approach
compares the instances in the new time-frequency
domain through three metrics of similarity, which are
aimed to evaluate different aspects of the instances.
III. FORMAL NOTATION
Given a set of classified instances ,
and a set of features that compose
each instance i, we denote as the subset of
reliable instances, as the subset of unreliable
ones, and as C={reliable,unreliable} the set of possible
instance classifications. It should be noted that an
instance can belong only to one class . We also
denote as a set of unclassified
instances and as these instances
after the classification process, thus . Finally,
we denote as and
, respectively, input and output of
the DWT process.
IV. PROPOSED APPROACH
The proposed approach has been implemented through
the three steps listed and explained in the following:
Input Data Definition: definition of the time
series to use in the DWT process, made by using
the sequence of values assumed by each single
feature of an instance;
Output Data Generation: generation of the new
data representation in the frequency-time domain,
by processing the time series through the DWT
process;
Instance Classification: formalization of an
algorithm based on our Centroid Wavelet-based
Approach (CWA), able to classify a new instance
as reliable or unreliable on the basis of three
different metrics of similarity.
A. Input Data Definition
The first step is aimed to prepare the time series to use
as input in the DWT process.
It is performed by using the sequence of value
assumed by the features of an instance, then we
consider the features of all instances, i.e., subset of
previous reliable instances , subset of previous
unreliable instances , and set of instances to
evaluate in .

Figure 1. Evaluation Criterion
B. Output Data Generation
In the second step we perform the DWT process by
using as input the time series related to the and
sets.
Our Credit Scoring approach exploits two wavelet
properties. The first one is the Dimensionality reduction:
the DWT process reduces the dimensionality of a time
series by performing an orthonormal transformation that
allows us to recover the original data. This can be
exploited in order to reduce the computational load. The
second one is the Multiresolution analysis: the DWT
process allows us to analyze the data by using an
approximated or detailed point of view. This can be
exploited in order to define a model of representation of
the instances less or more affected by data variation.
The proposed approach exploits these two properties
by transforming the original instances data through the
Haar wavelet, preferring an approximation coefficients
N/2 in order to obtain a more stable model able to face
the data heterogeneity issue. The result of each instance
transformation is the set O.
C. Instance Classification
Each unevaluated instance is classified after a
comparison process performed between it and all the
instances in the training set (i.e., the reliable ones in the
subset I+ and the unreliable ones in the subset I-). Such
comparison process is performed in terms of the three
metrics detailed described later, i.e., the Cosine Similarity
(Ө), the Features Weighted Sequence Similarity (Φ), and
the Normalized Magnitude Similarity (μ). Premising that
the radius (ρ) is a value experimentally defined in
Section V, we classify a new instance by adopting the
following criteria:
first we define a centroid + (see Figure 1), using
as coordinates of the x and y axes, respectively,
max(Ө)-ρ and max(Ф)-ρ, where max(Ө) stands for
the maximum value of the Cosine Similarity and
max(Ф) stands for the maximum value of the
Features Weighted Sequence Similarity, both
calculated between the instance to evaluate and all
the instances in the training set;
the classification process is based on the type and
magnitude of the instances bounded by the
circular area of radius ρ, centered in + (the
circular area of Figure 1, where the size of the
cases represents the Normalized Magnitude
Similarity μ of the instance);
a new instance is classified as reliable if the
weight (in terms of μ) of the selected reliable
instances within the radius ρ is greater than that of
the unreliable ones, otherwise it is classified as
unreliable.
Figure 1 shows a case when the instance under
evaluation is classified as unreliable since the sum of the
weight of the three unreliable instances within the radius
is greater than that of the other two reliable ones. By
following a prudential criterion, we classify the new
instances as unreliable when the sum of the weight of the
reliable ones within the radius is equal than that of the
unreliable ones.
1) Metrics
This section describes the three metrics used in our
approach.
Cosine Similarity: The Cosine Similarity (Ө) metric is
able to measure the similarity between two vectors v1 and
v2 with size larger than zero. More formally, given two
vectors v1 and v2 of attributes, it is represented using a
dot product and magnitude as shown in the Equation 4.
We normalized the result in a range [0,1], where 0
indicates two completely different vectors and 1 two
equal vectors, and the intermediate values indicate
different levels of similarity between the two vectors.

Features Weighted Sequence Similarity: The
Features Weighted Sequence Similarity (Ф) is not a
canonical metric, since it was defined in the context of
this paper in order to evaluate the similarity between
instances in terms of the weighted sequence of the
features that composed them. The idea behind this metric
is that similar instances present similar sequences of
features, i.e., if we sort the features of two instances on
the basis of their values, the sequences of their indexes
will be similar in terms of cosine similarity Ө. More
formally, given two instances i(1) and i(2) we calculate Φ as
shown in Equation 5, where TS(1) and TS(2) are the time
series of the instances to compare and the function idx
gives us the sorted time series TS in terms of former
element indexes (i.e., the indexes of the TS elements
before sorting).

Normalized Magnitude Similarity: The Normalized
Magnitude Similarity (μ) is also a non-canonical metric

defined in the context of this paper with the aim to
evaluate the difference in terms of magnitude between
two instances i(1) and i(2).
It is measured by taking into account their DWT
outputs O(1) and O(2). It is calculated as shown in
Equation 6, where max(Δ) is the maximum value
assumed by Δ in the context of all the instances in the
training set I.

2) Algorithm
The classification Algorithm 1 takes as input the set of
previous instances I, an instance to evaluate, and
the radius value ρ, returning as output a boolean value
class that indicates the classification of the instance
(i.e., true=reliable or false=unreliable).
Algorithm 1. Classification Algorithm
V. EXPERIMENTS
Our approach has been developed in Java, by using the
Jwave1 library to perform the Discrete Wavelet
Transformations and the WEKA2 library to implement
the state-of-the-art competitor used to evaluate its
performance (i.e., Random Forests).
A. Datasets
The real-world datasets used to evaluate our approach
are freely downloadable at the UCI Repository of
Machine Learning Databases3. They represent three
benchmarks in the Credit Scoring field, allowing us to
evaluate our approach in different data scenarios, both for
number of instances and for class balancing.
German Credit Data (GCD): This dataset contains
the classification of people as reliable or unreliable in
terms of credit risks. We used the numerical version,
which is composed by 1,000 instances: 700 classified as
reliable (70.0%) and 300 classified as unreliable (30.0%).
1https://github.com/cscheiblich/JWave/
2http://www.cs.waikato.ac.nz/ml/weka/
3ftp://ftp.ics.uci.edu/pub/machine-learning-databases/statlog
Each instance is defined by 24 features and a binary class
variable (reliable or unreliable).
Australian Credit Approval (ACA): This dataset
contains credit card applications classified as reliable or
unreliable on the basis of their final outcome. We used
the numerical version, which is composed by 690
instances: 307 classified as reliable (44.5%) and 383
classified as unreliable (55.5%). Each instance is defined
by 14 features and a binary class variable (reliable or
unreliable).
Japanese Credit Screening (JCS): This dataset
contains a number of people instances classified as
reliable (granted credit) or unreliable (non granted
credit). We used the numerical version, which is
composed by 125 instances: 85 classified as reliable
(68.0%) and 40 classified as unreliable (32.0%).
Each instance is defined by 16 features and a binary
class variable (reliable or unreliable).
B. Strategy
All the experiments have been performed by adopting
the k-fold cross-validation criterion, with k=10, in order
to reduce the impact of data dependency, improving the
value of the obtained results. In more detail, each dataset
has been divided in k subsets, and each k subset has been
used as test set, while the other k-1 subsets have been
used as training set. The result is given by the average of
all the obtained results.
The final results have been verified for the existence of
a statistical difference between them, by using the
independent-samples two-tailed Student's t-tests
(p<0.05).
Given that the Algorithm 1 needs as input a radius
value ρ, it has been experimentally calculated for each
dataset by testing a large range of values in the context of
the training set I, choosing as final value the one that
leads towards the best performance in terms of F-
measure. The results indicate as optimal ρ values 0.967
for the GCD dataset, 0.849 for the ACA dataset, and
0.789 for the JCS dataset, since these values maximize
the F-measure.
C. Competitor
Although the literature indicates Random Forest as one
of the most performing approach for the Credit Scoring
[4], we have however carried out a preliminary study
aimed to compare the AUC performance of ten binary
classification approaches (i.e., Naive Bayes, Logistic
Regression, Multilayer Perceptron, Random Tree,
Decision Tree, Logic Boost, SGD, Voted Perceptron, K-
nearest, and Random Forests), adopting the same cross-
validation criterion used for the other experiments. The
results indicate that Random Forest outperforms all the
other approaches (GCD=0.79, ACA=0.93, JCS=0.97), so
it is the only one we will be confronting with.
We have also performed a series of experiments in
order to optimize the Random Forest performance. After
we configured as unlimited the maximum depth of the
tree parameter, we tested different values of the number
of randomly chosen attributes (nrca) parameter. In order
to avoid the overfitting problem, the tuning process has

been performed by using both the training and testing sets
and also in this case we adopted the same cross-validation
criterion used for the other experiments. The results
indicate as optimal nrca values 22 for the GCD dataset, 7
for the ACA dataset, and 15 for the JCS dataset, since
these values maximize the F-measure.
D. Results
The analysis of the results shown in Figure 2, where,
respectively, have been compared the performance of our
Centroid Wavelet-based Approach (CWA) to that of its
competitor Random Forests (RF), in terms of F-measure
and Area Under ROC Curve (AUC), leads towards the
following considerations.
Figure 2.a shows that our CWA approach outperforms
its competitor RF in terms of F-measure in the context of
all the datasets, regardless of their size and their level of
imbalance. This indicates its capability to classify the
instances correctly, both with regard to the number of all
performed classifications and to the number of the
classifications that should have been made. Such result
underlines two aspects, the first one related to the better
performance achieved by it, while the second one related
to the constancy of them. In fact, our approach
outperforms RF in the context of all the datasets and its
level of performance do not vary much (both in terms of
quality and range), differently from its competitor.
Figure 2.b shows that our CWA approach reaches AUC
performance similar (i.e., JCS dataset) or higher (i.e.,
GCD and ACA datasets) than that of its competitor RF,
regardless of the size of data and the level of imbalance
of them. The AUC metric measures the effectiveness of
the evaluation model and the results indicate that our
model gets higher performance than that of its competitor
in the context of all datasets. It should be also noted how
it obtains the best performance in a typical real-world
scenario (i.e., the GCD dataset) characterized by many
instances and a high degree of imbalance.
Additional considerations on the dimensionality
reduction: In our DWT approach we have exploited only
one of the two wavelet properties previously introduced
(i.e., the multiresolution analysis) in order to mitigate the
heterogeneity data issue through the pair-wise average
and directed distances operations made by the Haar
wavelet function. Now, we want to introduce the
possibility to exploit the second properties (i.e., the
dimensionality reduction) in order to reduce its
computational complexity without a significant
performance decay. Indeed, we can obtain a substantial
curtailment of the processed elements (i.e., |F|∙|I|) by
taking into account only the pair-wise average part of the
Haar wavelet function output (small differences in values
are to be attributed to the needed transformations to make
, with .
The results in terms of AUC performance show a
reduction of 48.0%, 46.0%, and 47.0% (respectively, in
the GCD, ACA, and JCS dataset), without detecting any
significant performance decay, getting even better results.
Figure 2. F-measure and AUC Performance
It should be added that we get a similar result by using
the directed distances part of the output instead of the
pair-wise average one.
VI. CONCLUSIONS AND FUTURE WORK
The Credit Scoring approach proposed in this paper is
based on a threefold assessment of similarity carried out
in the frequency-time domain offered by the Discrete
Wavelet Transformation process. The data analysis
performed in this non-canonical domain through three
metrics of similarity has led to a better capability to
characterize the user instances in their correct class of
destination (i.e., reliable or unreliable). Experimental
results proved the effectiveness of such approach in the
context of three datasets, where it outperforms its state-
of-the-art competitor in typical real-world scenarios
characterized by a considerable number of instances,
regardless of their data distribution.
Future work would be oriented to experiment
additional metrics of similarity, as well as other wavelet
functions in the Discrete Wavelet Transformation process,
with the aim to further improve the effectiveness of the
classification model. Another interesting future work
would be the experimentation of the proposed approach
in scenarios other than Credit Scoring.
ACKNOWLEDGMENT
This research is partially funded by Regione Sardegna
under project Next generation Open Mobile Apps
Development (NOMAD), Pacchetti Integrati di
Agevolazione (PIA) - Industria Artigianato e Servizi -
Annualità 2013.
REFERENCES
[1] W. E. Henley, “Statistical aspects of credit scoring.” Ph.D.
dissertation, Open University, 1994.
[2] P. Chaovalit, A. Gangopadhyay, G. Karabatis, and Z. Chen,
“Discrete wavelet transform-based time series analysis and
mining,” ACM Comput. Surv., vol. 43, no. 2, pp. 6:1–6:37, 2011.
[3] M. Doumpos and C. Zopounidis, “Credit scoring,” in Multicriteria
Analysis in Finance. Springer, 2014, pp. 43–59.
[4] S. Lessmann, B. Baesens, H. Seow, and L. C. Thomas,
“Benchmarking state-of-the-art classification algorithms for credit
scoring: An update of research,” European Journal of Operational
Research, vol. 247, no. 1, pp. 124–136, 2015.
[5] S. Ali and K. A. Smith, “On learning algorithm selection for
classification,” Appl. Soft Comput., vol. 6, no. 2, pp. 119–138,
2006.

[6] D. J. Hand, “Measuring classifier performance: a coherent
alternative to the area under the ROC curve,” Machine Learning,
vol. 77, no. 1, pp. 103–123, 2009.
[7] M. Siami, Z. Hajimohammadi et al., “Credit scoring in banks and
financial institutions via data mining techniques: A literature
review,” Journal of AI and Data Mining, vol. 1, no. 2, pp. 119–
129, 2013.
[8] N. Japkowicz and S. Stephen, “The class imbalance problem: A
systematic study,” Intell. Data Anal., vol. 6, no. 5, pp. 429–449,
2002.
[9] A. Chatterjee and A. Segev, “Data manipulation in heterogeneous
databases,” ACM SIGMOD Record, vol. 20, no. 4, pp. 64–68,
1991.
[10] J. Attenberg and F. J. Provost, “Inactive learning?: difficulties
employing active learning in practice,” SIGKDD Explorations,
vol. 12, no. 2, pp. 36–41, 2010.
[11] D. B. Percival and A. T. Walden, Wavelet methods for time series
analysis. Cambridge university press, 2006, vol. 4.
[12] R. Agrawal, C. Faloutsos, and A. N. Swami, “Efficient similarity
search in sequence databases,” in Foundations of Data
Organization and Algorithms, 4th International Conference,
FODO’93, Chicago, Illinois, USA, October 13-15, 1993,
Proceedings, ser. Lecture Notes in Computer Science, D. B.
Lomet, Ed., vol. 730. Springer, 1993, pp. 69–84.
[13] S. G. Mallat, “A theory for multiresolution signal decomposition:
the wavelet representation,” IEEE transactions on pattern analysis
and machine intelligence, vol. 11, no. 7, pp. 674–693, 1989.