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Introduction
The structure-activity relationship (SAR) and the
quantitative structure-activity relationship (QSAR) studies are
basically concerned with the correlation of structure with
activity. Many different statistical methods, such as the
multiple linear regression (MLR)[1] and partial least squares
(PLS) analysis[2] have been employed to SAR/QSAR
analyses. Several physicochemical descriptors, such as
hydrophobicity, topology, electronic parameters, and steric
effects, are usually used in QSAR studies. Recently, many
researches have been concerned with applying the support
vector machine (SVM) to investigate
SAR/QSAR[3_7]. The SVM, a statistical approach mentioned by
Vapnik[8] and developed in the last decade, has lately started to be applied in
this field. The potential of SVM for use in SAR/QSAR has
been discussed[9_15]. In this work, beta-adrenergic agonists
and antagonists (phenethyl-amines) are studied with the
application of SVM in the field of SAR and QSAR.
Beta-adrenergic agonists and antagonists are well known for their
enhanced effect on the growth hormone-releasing hormone
(GHRH)-induced growth hormone (GH) secretion in human
bodies. They are significant mediators on the GH response
to GHRH in anorexia nervosa[16_19]. As SVM is a robust tool
in the SAR/QSAR field, our objective is to study the
modeling of phenethyl-amines by SVM to classify
antagonists/agonists and predict the activity. To demonstrate the power
of SVM, computations were performed by the
cross-validation[20] and independent test, which are deemed the most
rigorous and objective test procedures in statistical
prediction. As the number of samples in this work is very
small, the independent test and a special cross-validation,
leave-one-out cross validation (LOOCV), were used to test
the models' generalization and
reliability[21].
Materials and methods
Support vector classification Support vector
classification (SVC) has been recently proposed as a very effective
method for solving pattern recognition problems. SVC is a
learning machine based on statistical learning theory. The
basic idea of applying SVM to pattern classification can be
stated briefly as follows[3,22]: suppose we are given a set of
samples, that is, a series of input vectors
xiÎRd (i=1, ..., N)
with corresponding labels (x1,
y1), ..., (xm,
ym), ..., yÎ{-1, +1};
where -1 and +1 are used to stand, respectively, for the 2
classes. The goal here is to construct 1 binary classifier or
derive 1 decision function from the available samples, which
has a small probability of misclassifying a future sample.
In other words, the goal is to seek an optimized linear division,
that is, construct a hyperplane,
wTx+b=0 that separates the 2
classes (this can be extended to multi-classes). Different
mappings construct different SVM. The mapping
xiÎRd (i=1,
..., N) is performed by a kernel function:
K(xi,
xj)=F(xi)F(x
j) (1)
which defines an inner product in the space H.
The decision function implemented by SVM can be
written as:
 (2)
where the coefficients αi are obtained by solving the
following convex quadratic programming problem:
Maximize (3)
subject to (4)
In Equation 4, C is a regularization parameter which controls
the trade-off between the margin and misclassification error.
These xj are called support vectors only if the
corresponding αi>0. Several typical kernel functions are:
K(xi,
xj)=(xi·
xj+1)d
(5)
K(xi,
xj)=exp(-γ || xi
_xj ||2 ) (6)
Equation 5 is the polynomial kernel function of degree
d which will revert to the linear function when
d=1. Equation 6 is the radial basic function (RBF) kernel with 1 parameter
γ.
The SVM training process always seeks a global
optimized solution and avoids over-fitting, so it has the ability to
deal with a large number of features. A complete description
to the theory of SVM for pattern recognition is given by
Vapnik[22].
Support vector regression
(SVR)[3,22] Support vector regression (VM) can be applied to regression by the
introduction of an alternative loss function and the results appear to
be very encouraging. For the case of regression
approxima-tion, suppose there is a given set of data
points (xi is input vector and is the desired value). SVR is based on
the structural risk minimization principle from the statistical
learning theory. Each instance is represented by a vector
x with molecular descriptors as its components. A kernel
function, K(xi,
xj), is used to map the vectors into a higher
dimensional feature space, and linear regression is then
conducted in this space. The optimal regression function can
be represented by:
 (7)
where l is the number of support vectors and the coefficients
αi, αi* and bias
b are determined by maximizing the following
Lagrangian expression:
 (8)
with the following constrains:
0¡Üαi ¡ÜC i=1, ..., l
0¡Üαi* ¡ÜC i=1, ..., l (9)
where l is the training set size and
C is a penalty for training errors.
Leave one out cross-validation (LOOCV) During the
process of LOOCV, both the training dataset and testing
dataset are actually open and a sample will in turn move from
one to the other. Each sample in the dataset is in turn singled
out as a tested sample and all the rule-parameters are
calculated based on the remaining samples. In other words, each
sample is predicted by the rule parameters derived using all
the other samples except the one which is being
predicted[20_21].
Sensitivity analysis Sensitivity analysis (SA) is the study
of how the variation in the attributes of a model can be
apportioned, qualitatively or quantitatively, to different
targets of variation. A mathematical model is defined by a
series of equations, input factors, attributes, and variables
aimed to characterize the process being investigated. Good
modeling requires an evaluation of the confidence in the
model, possibly assessing the uncertainties related to the
modeling process and with the outcome of the model itself.
SA offers a valid method for characterizing the uncertainty
associated with a model. The model is run for a set of sample
points for the attributes of concern or with straightforward
changes in the model structure. This approach is often used
to investigate how the target changes significantly in
relation to the changes of attributes. The application of this
approach is straightforward, and it has been widely employed
in nonlinear modeling[23,24]
Data sets and descriptors The compounds with various
substituents (X, Y, R, R1, and
R2)[25] are shown in Figure 1.
The structural details and the activity class of 32 chosen
phenethyl-amine compounds are given in Table 1.
Seventeen of these are antagonists (class 1) and fifteen are
agonists (class 2). The data set is divided into the training set
and the testing set.
The substituents in the phenyl group were described by
the fragment constant value of Tekker
(f)[26], electronic parameters of Taft
(s)[27], STERIMOL parameters (L and B) and
the electronic parameters of Hammett
(ES)[28], plus the calculated
pKa of the amino group.
Owing to the redundancy of some parameters, the
selection of descriptors for a SAR/QSAR study is necessary, but
not straightforward[29]. However, the selection of
descriptors will contribute a lot to construct the actual model.
Recently, promising results have been reported on the
problem of feature selection[29_31]. Generally, the presence of
irrelevant or redundant features can cause the model to lose
sight of the broad picture that is essential for generalization
beyond the training set. In particular, this problem is
compounded when the number of observations is also relatively
small. The main goal of feature selection is to select a subset
of d features from the given set of
D measurements (d<D) without significantly degrading the performance of the
mathematical model. In this work, the global searching method
was applied to the selection based on descriptors which
were used in Wold's study[25]. Prediction accuracy was
calculated with different combinations of the descriptors. Based
on the calculation results, we will choose the descriptors
with the best prediction accuracy. Therefore, 4 selected
descriptors (pKa, fR2, sR2, and S-p) were found suitable to
construct the SVC model; 3 selected descriptors
(pKa, fR2, and EsR2) were found suitable to construct the SVR model
for the activities of agonists, and 4 selected descriptors (fPh,
fR2, fR1, and B4) were found suitable to construct the SVR
model for the activities of antagonists. Further
combinations of descriptors may exist which may be useful for the
SAR/QSAR study of the data set used here, but the selected
subset proved to be appropriate for a successful prediction
of activities, therefore we did not look for further sets.
Results
Classification model-based SVC of the antagonist and
agonist
Selection of kernel function and the capacity
parameter C It is well known that similar to other multivariate
statistical models, the performance of SVC is related to
dependent and independent variables, as well as the
combination of parameters used in a model. In the computation of
SVC, the capacity parameter C (or regularization parameter)
and the kernel type used in modeling must be selected.
In this study, the suitable capacity parameter
C and the kernel function type in building the model were selected
based on the LOOCV method. The accuracy of prediction
(PA) in LOOCV was employed as a criterion in selection.
PA can be computed as follows:
 (10)
where NT is the number of samples in the whole data set and
NC is the number of samples whose classes are correctly
predicted in LOOCV.
Figure 2 shows the curve of PA versus the capacity
para-meter C (its value from 1 to 200) with different kernel
functions (linear and polynomial radial basis functions) by using
SVC LOOCV. It can be seen in Figure 2 that
PA gets maximal value employing the RBF kernel function with the capacity
parameter C³146. Gamma (g) is another parameter of SVC
which will affect the prediction capacity when using the RBF
kernel function. Hence, we calculated the prediction
accuracy under different C and g (for RBF). Figure 3 shows the
PA versus C (C=1_200, step=1) and
g (g=0.1_3, step=0.1) with the RBF kernel function. From Figure 3 we can see that
although g affected the prediction accuracy, the maximal
value of PA is 91.67%. In this case,
g was set as default
value 1. So the SVC model with best performance could be
expected by using the RBF kernel function with the capacity
parameter C=146, g=1.
Modeling of SVC According to the results obtained
from above section, the optimal model of SVC for
discriminating antagonists from agonists is as follows, using RBF
kernel function with the capacity parameter
C=146, g=1:
 (11)
where x is a vector (pattern of sample) with unknown activity
to be discriminated and xi is one of the support vectors.
According to Equation 11, the compounds could be
discriminated as antagonists if
g(x)³0 and the accuracy of classification
(CA) is 95.83%.
CA can be defined as follows:
 (12)
where NT is the number of samples in the whole data set and
NC is the number of samples which are discriminated correctly.
Figure 4 shows the effect of classification with the trained
SVC model. It demonstrates that only 1 sample is classified
incorrectly.
Results of LOOCV for classification The predictive
ability was evaluated by the accuracy of prediction
(PA) in LOOCV.
Figure 5 shows the effect of prediction validated by the
LOOCV method. In Figure 5, there are also only 2 samples
predicted incorrectly, and the accuracy of prediction
(PA) is 91.67%. Additionally, to compare with the prediction
generalization ability of SVC method, the Fisher method, the
K-nearest neighbor (KNN, K=5), and the artificial neural
network (ANN), with 3 hidden nodes and Sigmoid
transformation function were also applied in this work, with the
prediction accuracies were 75%, 83.33%, and 79.17% in the LOOCV
test, respectively.
Results of independent test for
classification To validate the generalization and reliability of the classification
model (Equation 11), the independent test set was employed
in this study. The accuracy of prediction is 100%. The
Fisher method, the KNN, K=5, and the ANN were also
applied in this work, and the prediction accuracies were
87.5%, 75%, and 75% in the independent test, respectively.
Predicting the activity of
phenethyl-amines SVM models relate not only to the task of distinguishing between
pairs of molecules based on SVC, but the task of predicting
actual biological activity based on SVR. Just like the SVC
method, the performance of SVR in predicting the activity is
also related to dependent and independent variables, as well
as the combination of parameters used in a model. In the
computation of SVR, the capacity parameter
C, the epsilon insensitive loss function (e), the gamma
(g), and the kernel type used in modeling must all be selected. In this
computa-tion, the least root mean square error (RMSE) in LOOCV was
employed as the criterion to obtain the appropriate kernel
function and the optimal capacity parameter
C, e, and g. RMSE is defined as follows:
where ei is the experimental value of sample
i, pi is the predicted value of sample
i in the LOOCV of SVR, and n is the number of the total samples. In general, the smaller the value
of RMSE obtained, the better predictive ability expected.
As the mechanisms of antagonists' and agonists'
activity are different, their QSAR models would be built
respec-tively.
Optimal SVR model for antagonists To obtain the
suitable modeling parameters, the RMSE was calculated
under different parameters (e, C, and
g) and different kernel functions (including the linear kernel function, the
polynomial kernel function, and the RBF kernel function) by using
the LOOCV of SVR. Figure 6A_6E shows the effects of
RMSE with different kernel functions and different
parameters. Figure 6A illustrates RMSE versus
C (C=0.1_200, step=0.1) and e (e=0.01_0.3, step=0.01) with the linear
kernel function. Figure 6B illustrates RMSE versus
C (C=
0.1_200, step=0.1) and e (e=0.01_0.3, step=0.01) with the
polynomial kernel function. Figure 6C illustrates RMSE
versus C (C=0.1_200, step=0.1) and e
(e=0.01_0.3, step=0.01) with the RBF kernel function. Figure 6D illustrates RMSE
versus g (g=0.1_3, step=0.1) and e
(e=0.01_0.3, step=0.01) with the RBF kernel function. Figure 6E illustrates RMSE
versus C (C=0.1_200, step=0.1) and
g (g=0.1_3.0, step=0.1) with the RBF kernel function. After comparing the above
plots, the author found that the RBF kernel function is better
than both the linear function and poly function in the
building model. Hence, according to Figure 6, the optimal SVR
model for antagonists can be presented as follows, with the
RBF kernel function (C=0.5, e=0.1, and
g=1.4):
ACT=Σ(αi -
αi*)×exp[-1.400×||
x - xi*
||2]+0.471 (13)
where (αi -
αi*) is the Lagrange coefficient corresponding to
the support vector. Figure 7 illustrates the relationship of
predicted activities and the experimental activities of agonists,
with R=0.943.
Optimal SVR model for agonists In this section, the
data set of agonists was investigated by the same method
used for antagonists. Figure 8A_8E shows the effects of
RMSE with different kernel functions and different
para-meters.
According to Figure 8, the optimal SVR model for
agonists can be presented as follows, with the RBF kernel
function (C=1.9, e=0.01, and g=1.4):
ACT=Σ(αi -
αi*)×exp[-1.400×||
x - xi*
||2]+0.223 (14)
where (αi -
αi*) is the Lagrange coefficient corresponding to
the support vector. Figure 9 illustrates the relationship of
predicted activities and the experimental activities of
antagonists, with R=0.949.
Results of LOOCV of SVR In order to further confirm
our findings, we investigated the relationships between
experimental and predicted activity values. The detailed
experimental values and calculated values are given in Table 2.
These relationships are shown in Figure 10A,10B, which
illustrates the effect of prediction with the aforementioned
model (Equations 13 and 14) validated by the LOOCV method.
Figure 10 suggests that the predicted activities are in
agreement with experimental activities for agonists and antagonists,
with correlation (R) 0.831 and 0.865, respectively.
MLR, PLS regression (PLSR), and ANN were also utilized
to predict the activities of phenethyl-amine compounds with
special consideration of their generalization abilities in the
LOOCV test compared to SVR. The calculated results are
given in Table 2. The RMSE in the LOOCV test using SVR,
MLR, PLS, and ANN, respectively, are listed in Table 3. It
can be found that the generalization ability of SVR is
superior to the other methods which are often used in QSAR
studies in the LOOCV test.
Results of independent SVR test As a demonstration
of practical application, predictions were also conducted for
the independent data set based on the models derived from
the training data set. MLR, PLSR, and ANN were also
utilized to predict the activities of phenethyl-amine compounds
with special consideration of their generalization abilities in
the independent test compared to SVR. The calculated
results are given in Table 4. Table 5 lists the RMSE in the
independent test using SVR, MLR, PLS, and ANN,
respec-tively. It can be found that the generalization ability of SVR
is superior to the other methods which are often used in
QSAR studies.
Discussion
Parameters It is important to set the adjustable
parameters of SVR to obtain the model with the better, or at least
sufficient, predictive capability. Parameter
C is an important parameter because of its possible effects on the trade-off
between maximizing the margin and minimizing the training
error. If the value of C is too small, then insufficient stress
may be placed on fitting the training data. If the value of
C is too large, then the algorithm may over-fit the training data.
From Figures 6 and 8, we can see that if C >100 or
C<0.01, the value of RMSE will be very large, which means that the
prediction accuracy of activity will be very poor. The optimal
value of e is another important parameter. e prevents the
entire training set meeting boundary conditions, and so
allows for the possibility of sparsity in the dual formulation's
solution. Hence, choosing the appropriate value of
e is critical from theory, but it is hard to find an appropriate value of
e as there is no rule as to how to optimize it. In this study, we
predict the value of RMSE while the range of e is 0.001_0.31,
and we found that if the value is smaller, the result will be
better. However, it will take more time when the value of
e is very small. The optimal value of g also plays a significant
role in this study. g controls the amplitude of the Gaussian
function, therefore controls the generalization ability of SVM.
In this study, the prediction accuracy of activity changed
greatly when g ranged from 0.1 to 3; when the value of
g was out of this range, the result was poor. The parameters should
be optimized together with the kernel functions type adopted
in SVR modeling.
Sensitivity analysis (SA) In this paper, 2 QSAR models
of phenethyl-amines are attained with highly accurate
predictions. However, the aforementioned models are
nonlinear, and it is hard to analyze the relationship between
attributes and activity. Hence, we used the sensitivity
analysis to study how the attributes affect the target.
SA of QSAR model of agonists From Figure 11A, we
can see that when the values of fR2 and ESR1 are fixed, the
value of activity varies with the increase of
pKa, and a maximum value of activity exists when the value of
pKa is around 9.4. From Figure 11B, we can see that when the values of
pKa and ESR1 are fixed, the value of activity will decrease
with the increase of fR2. From Figure 11C, we can see that
when the values of pKa and fR2 are fixed, the value of
activity will decrease with the increase of ESR1. According to
these findings, we can conclude that decreasing the value of
ESR1 and fR2 and choosing the value of pKa around 9.4 will
result in higher activity of agonists.
SA of QSAR model of antagonists From Figure 12A,
we can see that when the values of fR2, fR1, and fPh are
fixed, the value of activity will increase with the increase of
B4. From Figure 12B, we can see that when the values of B4,
fR1, and fPh are fixed, the activity varies with the increase of
fR2, and a maximum value of activity exists when the value of
fR2 is around 0.95. From Figure 12C, we can see that when
the values of B4, fR2, and fPh are fixed, the value of activity
will increase with the increase of fR1. From Figure 12D, we
can see that when the values of B4, fR1, and fR2 are fixed, the
value of activity will decrease with the increase of fPh.
According to these findings, we can conclude that there if we
want to get higher activity of antagonists, maybe we can
increase the value of B4 and fR1, decrease the value of fPh,
and choose a proper value of pKa around 0.95.
Advantages and disadvantages of SVM Compared with
other algorithms used in chemometrics, SVM has
outstanding advantages: it can be used for both classification (SVC)
and regression (SVR); it is suitable for both linear and
nonlinear problems; it has special generalization ability,
especially for problems of small sample size; and it has no local
minimum problem. As a newly-proposed algorithm, SVM
has a bright future as a powerful tool for chemistry and
related fields owing to these advantages. The investigations
presented here show that the SVM technology is a robust
and highly accurate, intelligent classification and regression
technique which can be successfully applied to deriving
statistical models with good statistical qualities and good
predictive capabilities within areas, well suited to SAR/QSAR
analyses of drug research.
Although SVM outperforms PLS, MLR, and ANN techniques in this study, different machine-learning algorithms
would have their own advantages and disadvantages in
different data sets. In fact, the different approaches may
provide complementary information from different point of views,
and sometimes the combination of different methods may
result in better results than the single one for special data
sets. Despite the advantage of SVM, adjusting the
parameters is hard work and affects the application of SVM.
However, no general guidelines are available to select these
parameters. The approach we used in this study is
time-consuming and there are possibly still better parameters than
the ones we employed. Hence, developing an efficiency
method to adjust the parameters is important in future work.
In this work, the SAR and QSAR analyses based on the
SVM method for 32 phenethyl-amines was studied, and 1
SAR and 2 QSAR models were developed. The predictive
powers of these models were verified with the LOOCV test
and independent test methods. For the LOOCV test, the
accuracy of the classification using Equation 11 with LOOCV
was 91.67%. The RMSE for antagonists using Equation 13
was 0.5881, and the RMSE for agonists using Equation 14
was 0.4779. For the independent test, the accuracy of the
classification using Equation 11 with LOOCV was 100%. The
RMSE for antagonists using Equation 13 was 0.3727, and
the RMSE for agonists using Equation 14 was 0.1435.
Taking the promising results made above into account, it could
be concluded that the SVM method could be employed to
SAR/QSAR modeling with much improved quality and predictability.
Appendix
The calculations were implemented on a 1830 MHz
Centrino Duo computer (Dell, Xiamen, China), running the
Windows XP operating system (Microsoft, Redmond, USA).
All the learning input data were range-scaled to (0, 1) in this
work. The SVM software package named ChemSVM
including SVC/SVR, was programmed in our
lab[32_34].The validation of the software has been tested in some applications in
chemistry and chemical
technology[33,34].
Acknowledgement
The authors wish to thank Dr Beke TAMÁS for reading
this manuscript and improving its presentation.
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