This document summarizes a research paper that proposes a new model called Quantum-Inspired Neuro-Evolutionary Computation (QINEA-BR) for optimizing the configuration of neural networks. QINEA-BR extends an existing quantum-inspired evolutionary algorithm (QIEA-BR) by adding a binary representation to allow optimization of categorical neural network parameters like input variables and hidden neuron count. The paper shows that QINEA-BR can successfully perform binary classification on a credit risk evaluation problem, outperforming other models. It divides the neural network parameters into binary and numerical parts to optimize using a hybrid binary-real chromosome representation in QIEA-BR.

1. Published in the World Congress on Nature and Biologically Inspired Computing (NaBIC'09), Coimbatore, India, Dec 09-11, 2009
A New Model for Credit Approval Problems: A Quantum-Inspired Neuro-
Evolutionary Algorithm with Binary-Real Representation
Anderson Guimarães de Pinho Marley Vellasco André Vargas Abs da Cruz
Department of Electrical Engineering, PUC-Rio
Rio de Janeiro, Brazil
agp@gmail.com marley@ele.puc-rio.br andrev@ele.puc-rio.br
Abstract—This paper presents a new model for neuro- topology and weights of a feed-forward neural network. This
evolutionary systems. It is a new quantum-inspired model is an extension of the QIEA- proposed by Cruz in
evolutionary algorithm with binary-real representation [6], which is a quantum-inspired evolutionary algorithm with
(QIEA-BR) for evolution of a neural network. The proposed real representation for numerical problems.
model is an extension of the QIEA- developed for numerical Cruz developed a quantum inspired evolutionary
optimization. The Quantum-Inspired Neuro-Evolutionary algorithm with a numerical representation to optimize the
Computation model (QINEA-BR) is able to completely weights of a neural network. On this paper, a binary
configure a feed-forward neural network in terms of selecting representation is added to the chromosome opening the
the relevant input variables, number of neurons in the hidden
possibility to optimize other types of variables that are
layer and all existent synaptic weights. QINEA-BR is evaluated
in a benchmark problem of financial credit evaluation. The
important for modeling a neural network, such as: which
results obtained demonstrate the effectiveness of this new attributes are relevant to be used on the input layer; how
model in comparison with other machine learning and many neurons to use on the hidden layer and which kind of
statistical models, providing good accuracy in separating good activation function to use on the hidden and output layers.
from bad customers. Such decisions are of categorical nature, and cannot be
efficiently represented as real numbers, thus leading to the
Keywords-quantum-inspired algorithms; genetic algorithms; use of a mix of representations into a single algorithm.
hybrid neuro-genetic systems;classification. This paper shows that the use of Quantum-Inspired
Evolutionary algorithms for training neural networks can be
I. INTRODUCTION used to successfully perform a binary classification. The
results presented here show a significant improvement when
A precise prediction of breach of contract has been the compared to other models.
objective of many companies in several segments. One of This paper is divided as follows: Section II presents
them, which has been extensively studied in the financial details of the new proposed quantum-inspired evolutionary
literature, is the credit default analysis. In this area, many algorithm, the QIEA-BR; Section III describes the
quantitative methods for creating models to separate bad and application of the proposed model to define the feed-forward
good customers have been explored [1], [2], [3], [14], [15], neural network configuration, the so-called QINEA-BR;
[16], [17], [18], [19], [20]. Section IV evaluates the proposed QINEA-BR model in a
Huang in [20] divides these methods in fields of the benchmark credit analysis application and compares its
science: such as discriminant analysis; logistic regression; results with other techniques; finally, Section V presents the
mathematical programming methods; recursive partitioning; conclusions and future work.
expert systems; neural networks; non-parametric methods of
smoothing; and models of time series.
Bose and Chen in [1] detailed some machine learning II. THE QIEA-BR ALGORITHM
techniques, such as: artificial neural networks; support vector The quantum-inspired genetic algorithm with binary-real
machines; genetic algorithms; genetic programming; representation (QIEA-BR) is a model where numerical and
evolutionary programming; and hybrid models of these binary parameters must be optimized.
techniques, with or without other statistical models. The binary representation part of the problem is based on
The use of hybrid models (neuro-genetic algorithms, for the concept of q-bits [7]. A q-bit can be in the "1" state, the
example), can help to overcome minor issues like neural "0" state, or in a superposition of both [4]. A state of a q-bit
networks overfitting problems and, thus, be more attractive can be represented as:
to solve complex problems with large volumes of data, such
as credit approval problems.   0  1 (1)
In this paper, a new model to evaluate credit approval in
financial problems is proposed. It is a Quantum-Inspired Where  and

are complex numbers that determine
Neuro-Evolutionary Algorithm with binary-real the probability of observing the corresponding state, such
representation (QINEA-BR), which determines the final that:

2. Published in the World Congress on Nature and Biologically Inspired Computing (NaBIC'09), Coimbatore, India, Dec 09-11, 2009
2 2 ii) while t <= T
  1 (2) t = t+1
iii) generate classic population P(t) with mix
Thus, a q-bit, the smallest unit of information, can be representation, observing Q(t)
defined as a pair of numbers ( ,  ) as:
iv) evaluate P(t)
v) if t=1 then
B(t) = P(t)
  otherwise
  (3)
vi) P(t) = Classic recombination of P(t) and B(t-1)
 
vii) evaluate P(t)
viii) B(t) = Best individuals of [P(t) U B(t-1)]
where (2) applies. ix) updates the binary part of Q(t) with the best
individuals of B(t), using q-gate
The representation of the numerical (continuous) part of x) updates the real part of Q(t) with the best
the chromosome is performed as in [6], using a probability individuals of B(t), using quantum-crossover
density functions (PDF). For a simple PDF, a uniform end
distribution inside a defined interval, this numerical gene can end
be represented as a pair of parameters called center and q0
width of the gene. Thus, the center () and width () are all i) Each j in Q(t) is initialized with equal probabilities
the parameters that are needed to represent this function, as
follows: for all states. In
 q j b , all q-bits are equal to 1/ 2 .
0
 
In
 q j r , considering that all the weights of a network
0
  (4)
  could be optimized assuming values in the range of (-

2,2), and  would be equal to 0 and 2, respectively.
For a specific x  (   ,    ) , the number to be q0
optimized, where: Note that initially, a j is the linear superposition of all
  possible states, with equal probability of occurrence.
  
p( x) dx  1 , (5) ii) While t, the current generation, is less than the total
number of generations, QIEA-BR continues looping.
Thus, a quantum individual j, in an instant of time t, iii) The classical population P(t) is generated in compliance
representing mixed numeric and binary features, can be to the quantum states of individuals in Q(t). For each q-
defined as: bit, the algorithm generates a random number between 0
2



  
qtj   qtj qt  
b j r

and 1. If this number is between 0 and , then the
classic bit is generated with a value of 0; otherwise the
  t  t classical bit is 1. For the real classic gene, a number in
 tjk    tj1  tj 2  tjm  
 j1 j 2
...   ...   the interval (    ) to (    ) is randomly chosen.
 t t t   t  t t  
  j1  j 2  jk b  j1 j 2  jm r 
   
 (6) iv) Since the QIEA-BR model was developed for
classification problems, the evaluation of each
Thus, a population Q(t) in generation t, with n possible qt
individual j in P(t) considers the number of correctly
qt classified patterns, that is, correctly indicated states “1”
solutions j can be given as follows:
and “0”. Therefore, the evaluation function is calculated
 
t t t
by (8):
Q(t )  q1 , q2 ,..., qn (7)
( A j  D j  rc1/ c0 )
A detailed description of all steps of the QIEA-BR fj  (8)
(C j  rc1/ c0  B j )  
algorithm is provided below:
QIEA-BR Algorithm Aj D
Where, is the number of true-positive samples, j
start
B C
t=0 true-negative, j false-negative and j false-positive.
i) initializes quantum population Q(t) with mixed rc1/ c0
representation is the ratio between total of “1” and “0”, in the

3. Published in the World Congress on Nature and Biologically Inspired Computing (NaBIC'09), Coimbatore, India, Dec 09-11, 2009
training sample. This ratio is used to avoid problems of  
the quantum individual jn . Then, the update of jn
specialization in unbalanced databases. Finally,  is a occurs as follows:
small parameter to avoid division by zero when the
problem is fully separable by the optimized model (i.e.  jn'   jn  ( g jn   jn )* random (10)
Cj B
and j are equal to zero). where random is a random number between 0 and 1,
generated for each quantum center, that determines the
v) In the first generation, the population of best individuals 
founded in B(1) is the population observed P(1). speed of the update of jn in the direction of g jn . The
second parameter of the numerical gene, the pulse
vi) If it is not the first generation, the recombination occurs
- as in traditional genetic algorithms - between B(t-1), width  jn , is updated in a similar way, by calculating
the population of the best individuals in the previous the total height of B(t) among all individuals, given
generation, and P(t). In all experiments carried out in by max( g jn )  min(g jn ) . So,  jn is updated by:
this work the uniform crossover operator has been
applied, acting differently if gene is real or binary and  jn'   jn  ((max( g jn )  min( g jn ))   jn )* random (11)
applied in a pair of genes of an individual of B(t-1) and
another individual of P(t). Probability of crossover was This type of update for numerical representation on
specified by the user. No mutation operators have been quantum algorithms is inspired on the work of Cruz [6].
employed. Since quantum algorithms have good
potential of exploration and exploitation simultaneously,
classic mutation operation is no longer motivated on this III. QIEA-BR MODEL APPLIED TO NEURO-EVOLUTION
study, because we want to obtain potential results The main objective of the QIEA-BR is to apply it to
proved essentially by the quantum algorithm and neuro-evolution, that is, to completely configure a feed-
operators. forward neural network, with one hidden layer, to binary
classification models. With this objective in mind, the
vii) Evaluate the new population P(t)', resultant of the following parameters must be defined by the QIEA-BR
recombination of P(t) and B(t-1), applying (8). model:
a. Which variables, among the available ones, are
viii) The new population in B(t) is given by the best relevant to be used as inputs for the neural network?
individuals from the union of P(t) with B(t-1), b. How many neurons must be used in the single
respecting the size of B(t) population defined by the hidden layer?
user. c. What kind of activation function must be applied
qt throughout the network? Sigmoid logistic or
ix) The q-bit of a quantum individual j is updated by the hyperbolic?
q-gate rotation operators [7], [11]. Initially, one classic The above parameters will be represented as binary genes
individual from B(t) and another from Q(t) are selected in the hybrid chromosome representation of the QIEA-BR
randomly. Each q-bit is updated in the direction of the model. The main reason that we choose between logistic and
individual from B(t), increasing or decreasing the hyperbolic activation function is that according to Haykin
probability of a state "0" or "1". A new q-bit can be [23], page 40, these function are the most common used
obtained as follows: when constructing artificial neural networks.
Other important neural network configuration
 cos( )  sin( )    j1 
t parameters, which will be represented as numerical genes in
    (9) the QIEA-BR chromosome, are the following:
 sin( ) cos( )    tj1 
  d. Synaptic weights of the single hidden layer;
e. Synaptic weights of the output layer;
where  is the angle of rotation and should be f. Threshold value, at the output neuron, that separates
assigned considering the type of problem by the user. the two output classes.
Depending on the intention of increase or decrease the Therefore, to optimize all these parameters, the proposed
probability, the sine terms (positive and negative) are QIEA-BR model, described in the previous section, is
exchanged. applied, with the following parameters:
 nh: Maximum number of neurons in the hidden
x) To update the numeric genes of the quantum individual layer;
Q(t), the same classic individual B(t), used in the  numQuantum: number of quantum individuals;
previous step, is employed. Consider the n-th numeric  numClassic: number of classic individuals;
g  numGeneration: Number of generations;
gene of a classic individual as jn , and the n-th gene of  C- Crossover: classical crossover rate;

4. Published in the World Congress on Nature and Biologically Inspired Computing (NaBIC'09), Coimbatore, India, Dec 09-11, 2009
  : parameter to update the binary part of the chromosome, only if its weights are active by the binary
quantum individuals; genes. If the binary genes are inactive, real genes are kept
 Q- Crossover: parameter to update the real part of unchangeable when applying q-crossover. On the other side,
the quantum individual. if binary genes are active, real genes could suffer updates by
Therefore, the final chromosome representation depends the crossover at least a number of “minGeneration”
on the maximum number of neurons defined by the user and controlled by the user.
the maximum number of possible variables in the input It is important to stress that all parameters presented
layer, so the length of the chromosome is fixed during the above should be adjusted according to the specific
evolutionary process. For example, in the case of a application problem.
maximum of 20 neurons in hidden layer and 30 available
input variables, the representation of an individual j, in an IV. EXPERIMENTAL RESULTS AND DISCUSSION
instant of time t, will contain 51 pairs of genes in the
quantum binary part: 30 to determine whether a variable in The proposed QIEA-BR model was evaluated in a
the input layer is active or not; 20 for activating the neurons benchmark application related to credit approval: the
in the hidden layer; and 1 gene to define the activation "Australian credit approval problem", which is available in
function that should be used in all layers (sigmoid or the UCI Machine Learning Repository [21]. As in any credit
hyperbolic). The numeric part of the chromosome analysis problem, this database provides a set of customers
representation contains, in this case, 621 pairs: 600 to define that are divided in good and bad payers. For confidential
the weight values for the synapses between input and hidden reasons, the meaning of the attributes is not provided by their
layers; 20 for the weights between hidden and output layers; administrators.
and 1 for the threshold value. This chromosome The database consists of 690 samples, with 307 (44.5%)
representation is provided below: composed of bad payers and 383 (55.5%) of good payers.
  
There are a total of 15 continuous and categorical attributes
qtj   qtj qt  

 b j r  (14 explanatory variables, and 1 that informs the class of the
costumer: good or bad). The database contained about 5% of
  t  t  tj 51    tj 52  tj 53  tj 672   (12) customers with missing values in at least one attribute, which
 j1 j 2     were treated by including the average and median (only one
 t t ... ...
t   t t t  
  j1  j 2  j 51 b  j 52  j 53  j 672 r 
    
for each attribute treated). Both categorical and numerical
attributes were pre-processed. Categorical attributes
(variable 5, 6 and 12) were transformed into the 1-of-N
It must be pointed out that the binary and numeric genes encoding. Variables 2, 3, 10, 13, and 14 were normalized by
are dependents. When a neuron is inactive in the classic mean and standard deviation.
individual, the crossover operator must no be applied to the The database was divided into 70% for training and 30%
weights associated to this neuron in the numeric part. for testing, in a 3-fold cross-validation process, thereby
Similarly, these weights can not be used to update the obtaining three sets of data for training and testing. We
parameters of the quantum individual. choose 3-fold for the cross-validation process, for
There are two additional parameters included in the comparisons with authors (see Lacerda page 178).
algorithm:
 minGeneration: minimum number of generations
that a neuron remains active before it can be A. Results
disabled by a crossover. This parameter is important
to avoid that a neuron is turned off before the The QINEA-BR model was implemented and tested in
evolutionary process had enough time to optimize Matlab. Varying the parameters presented in section III, we
its weights. could observe the results given by the evaluation function in
 updateGeneration: number of rounds that must be (8) from the best individual. After many tests, the parameters
executed before thee q-gate and q-crossover are presented in Table I were defined. Here we are going to
applied on the binary and numeric genes of the present just the final parameters adjusted. Sensitive analysis
quantum individual. This parameter controls the of how to control these parameters will be considered in
exploitation and exploration aspects of the futures works.
evolutionary process. That is, if a slower and After setting the parameters, each of the three training
gradual optimization process is desired, for a and test samples was subjected to evaluation by 3 neural
greater exploration of the search space, the higher networks, each developed independently. At the end, 9
his parameter should be. neural networks were obtained by the neuron-quantum
evolution. The results were evaluated by the percentage of
See that the binary part of a classic chromosome which wrong classified patterns (PWCP) and can be seen in Table
is responsible for enable or disable a neuron on the input and II.
hidden layers, is conditioned to the real part which As can be observed from Table II, among all the
determines the weights between neurons. And so, the experiments and samples, the model QINEA-BR showed an
algorithm must consider updating the real genes of a average PWCP of 15.0%, with a standard deviation of 2.9%.

5. Published in the World Congress on Nature and Biologically Inspired Computing (NaBIC'09), Coimbatore, India, Dec 09-11, 2009
The results obtained by the QINEA-BR were also M odel PWCP Average Standart D.
compared with other models provided in Lacerda & NEIQ-BR 15,0% 2,9%
Carvalho in [14], for the same issue of Australian credit Average Other M odels 16,5% 3,4%
approval. This comparison is provided in Table III below.
M LP-Backprop 17,1% 1,8%
It can be observed that, on average, the QINEA-BR
model presents a lower PWCP average than the models Cascade correlation 18,0% 3,0%
estimated by Lacerda and Carvalho. However, the statistical Tower 14,7% 3,2%
t-test [22] for the difference between means leads to the Pyramid 16,9% 2,1%
acceptance of the equal means hypothesis by the level of 5% SVM 16,7% 2,6%
of confidence, i.e., the difference is not significant. RBF - Batch 16,7% 3,9%
Considering the confidence of 10%, the t-test is significant
RBF - DF 16,3% 2,5%
with p-value of 0,096. We should say that before we applied
the t-test to compare means, each sample was proved to be RBF - IO 17,8% 4,0%
normally distributed by the Kolmogorov-Smirnov test, where RBF - DFIO 16,7% 4,3%
null hypothesis was accepted [22]. RBF - IODF 17,3% 4,4%
Other algorithms from Carvalho and Lacerda apud Jones RBF - On line 16,9% 4,4%
and Quinlan, and their comparison with the QINEA-BR RBF - Optimal 15,9% 4,7%
model can be found in Table IV.
RBF - GA 14,0% 3,5%
TABLE I. PARAMETERS SET
nh 20 TABLE IV. COMPARISON WITH OTHER MODELS IN CARVALHO AND
numQuantum 2 LACERDA APUD JONES AND QUINLAN
numClassic 400 M odel PWCP Average
numGeneration 200 NEIQ-BR 15,0%
C-Crossover 0,95
Average of Other M odels 16,5%
 0,020*pi
Q-Crossover 0,95 C4.5 Rules 15,5%
minGeneration 3 C4.5 Trees 15,1%
updatesGeneration 10 Foil trad.1 17,8%
Foil trad.2 17,4%
Foil trad.3 17,0%
TABLE II. RESULTS OBTAINED FOR THE "AUSTRALIAN CREDIT
APPROVAL PROBLEM” Foil exd.1 18.,0%
Foil exd.2 16,4%
Samples PWCP
Experiment Foil exd.3 16,4%
1 2 3 Aver. Stand. D.
1 16,9% 13,5% 13,0% 14,5% 2,1% Again, the QINEA-BR model presented the lowest
2 18,4% 18,8% 11,1% 16,1% 4,3% PWCP average, but this difference is not proved by statistical
3 17,4% 13,5% 12,1% 14,3% 2,7% evidences.
Average 17,6% 15,3% 12,1% 15,0% -
Standart D. 0,7% 3,1% 1,0% - 2,9% V. CONCLUSIONS
This paper presented a new quantum-inspired
evolutionary computation model based on a hybrid binary
TABLE III. COMPARISON WITH OTHER MODELS PROVIDED IN and numeric representation, named QIEA-BR. The proposed
CARVALHO AND LACERDA [14] model was developed for a neuro-evolution application, and
tested in a benchmark application of credit approval.
The resultant neuro-evolution model provides the user
with a high degree of flexibility, avoiding the necessity to
perform variable selection and the specification of all neural
networks parameters, such as number of neurons in the
hidden layer and the threshold value used in the output layer
to define the final classification of the input pattern.
Although the difference in the percentage of wrong
classified patterns (PWCP) obtained by the QINEA-BR and
others models used in the literature has not been significant,
the results were quite surprising. It was shown that, on
average, the QINEA-BR model could replace but it is not
significant better than the others.

6. Published in the World Congress on Nature and Biologically Inspired Computing (NaBIC'09), Coimbatore, India, Dec 09-11, 2009
Many parameters, however, must be defined in the 2005, Springer Science and Business Media, Inc. Manufactured in
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