^{1}Electrical and Computer Engineering Department, King Abdul Aziz University, Saudi Arabia

^{2}Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Saudi Arabia

^{3}School of Electrical, Electronics and Computer Engineering, University of Western Australia, Australia

^{4}College of Engineering, Karachi Institute of Economics and Technology, Karachi, Pakistan

^{5}Laboratory of Biomechanics and Bioengineering (BMBI), Sorbonne Universities, University of Technology of Compiègne (UTC), Compiègne, France

^{6}Harvard John A Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, USA

^{7}Department of Chemical Engineering, Northeastern University, Boston, MA, USA

- *Corresponding Author:
- Adnan Memic

Center of Nanotechnology

King Abdul Aziz University, Saudi Arabia

**Tel:**617-373-4495

**E-Mail:**[email protected]

**Received Date**: May 09, 2017; **Accepted Date:** May 17, 2017; **Published Date**: May 25, 2017

**Citation: **Moinuddin M, Naseem I, Wasim A, et al. A Weighted Cosine RBF Neural Networks. J Mol Biol Biotech. 2017, 2:2.

Over the past decade, the field of bioinformatics and computer modeling applied to biological challenges has been growing exponentially. Many approaches have been developed that tackle applications ranging from protein classification, structure prediction to disease prediction. This work focuses on the design of a novel kernel for the Radial Basis Function Neural Networks (RBFNN). The proposed kernel is based on weighted cosine distance between the input vector and the center vectors associated with RBFNN. The weighting is introduced in: i) the inner product of input and neuron’s center, ii) norm of the input vector, and iii) norm of the center vector. We demonstrate how the weighting matrix can be chosen for different applications to optimize the performance of the WC-RBF. As case studies, we present the PDZ domain classification and the channel estimation problem with correlated inputs. We also design an adaptive technique to update the weighting matrix for an arbitrary data using the approach of steepest descent optimization. For the validation of our adaptive design, we present the problem of Leukemia disease prediction. Simulations are presented to validate the performance of the proposed RBFNN kernel in contrast to the conventional RBFNN kernels.

Radial basis function; Neural networks; RBF kernel; PDZ domain classification; Leukemia cancer prediction

Broomhead and Lowe were the first to introduce the Radial basis function neural network (RBFNN). Its concept was based on the Cover’s Theorem [1,2]. The RBFFNN is known as universal approximate due to its remarkable performance in the problem of function approximation [3-5]. Originally, the RBF networks were designed for data interpolation in a higher dimensional space [5]. However, its applications are in wide area of engineering and it has been used as an important tool for function approximation, prediction, estimation, and system control [4-9]. The main advantage of RBF compared to other algorithms based on neural networks is the simplicity of computation of its network parameters [5]. The RBF networks perform the complex nonlinear mapping of the data that enables a fast, linear, and robust learning mechanism without significant computational cost [2].

Some of the most commonly used basis kernels are: Gaussian kernels [2], multi-quadric kernels [2], inverse multi-quadric functions [2], thin-plate spline kernels [5], and cosine kernels [10] etc. However, the selection of a proper kernel is highly problemspecific. A usual design of an RBFNN involves the learning of the centers of the kernels, the widths of the kernels, and weights of the networks [2]. In this context, a huge amount of work is carried out in the literature [11-16].

The most commonly used RBF kernel is the Gaussian kernel which employs the Euclidian distances between the feature vectors and the centers of the kernels [2]. However, it is shown in [10] that there can be scenarios where cosine distances are more significant in separating the features compared to the Euclidean. Thus, the work in [10] developed an RBF kernel by employing a linear combination of Gaussian kernel and cosine distance based kernel. Later, a time varying combination of the two kernels is proposed in [17] to improve the overall performance. In this work, a weighted cosine RBF kernel is proposed in which the conventional cosine distance is weighted by a diagonal matrix. We show that the weighting matrix can either be application dependent or it can be made adaptive via steepest descent based optimization. For the first case, we present two different applications: (1) PDZ domain classification and (2) Channel estimation for correlated inputs. For the second case, we develop a mechanism to recursively update the weighting matrix for the weighting matrix via steepest descent minimization of mean square error (MSE) cost function. This adaptive weighting matrix based RBF kernel is used for Leukemia disease prediction problem.

The main contributions of this study are as follows:

1) In Section (Proposed Weighted Cosine Rbfnn (Wc-Rbfnn)), we propose a novel weighted cosine RBF kernel where the weighting is introduced in: i) the inner product of input and neuron’s center, ii) norm of the input vector, and iii) norm of the center vector.

2) In Section (Choice of Weighting Matrix for the Wc- Rbfnn), we show how the weighting matrix can be chosen differently for different applications in order to optimize the performance of the WC-RBFNN. For that purpose, we present two examples: the PDZ domain classification and the channel estimation problem with correlated inputs.

3) In Section (Design of an Adaptive Weight for Wc-Rbfnn), we develop an automatic mechanism to update the weighting matrix for an arbitrary data using the approach of steepest descent optimization. For the validation of our adaptive design, we present the problem of Leukemia disease prediction.

The paper is organized as follows: Following this introduction, we provide an overview of the conventional RBF kernels in Section (Overview of the Conventional Rbf Kernels). In Section (Proposed Weighted Cosine Rbfnn (Wc-Rbfnn)), we develop the proposed weighted cosine RBF kernel. The choice of weighting matrix for the WC-RBFNN is discussed in Section (Choice of Weighting Matrix for the Wc-Rbfnn). In Section (Design of an Adaptive Weight for Wc-Rbfnn), an adaptive strategy is developed to make the weighting matrix of the WC-RBFNN time varying. Simulation results are presented in Section (Simulation Results). Finally, the concluding remarks are provided in Section (Conclusion).

In this section, we provide an overview of existing RBFNNs which are conventionally used in the literature. The RBFNN transform the nonlinear classification to linear classification by mapping the data into a higher dimensional space using some nonlinear kernel. According to Covers theorem, translation of features from lower dimension to a higher dimension simplifies the classification task via linear separation [1].

The RBFNN consists of three layers: an input layer, a hidden
layer, and a linear output layer as shown in **Figure 1**. All inputs
are connected to each hidden neuron. The input vector is passed
through the hidden layer which consists of nonlinear mapping
function. The i^{th} neuron of the hidden layer employs a nonlinear
kernel (denoted by to compute distance between
the input vector x and its center vector ci. The nonlinear kernels
are of different types which can be categorized broadly into two:
Euclidian Distance based and Cosine Distance based. These are
discussed in the ensuing sub-sections.

**A. Euclidian distance based RBF kernels**

Most commonly used RBF kernels are as follows [2],

**Multiquadrics:**

Most commonly used RBF kernels are as follows [2],

**Multiquadrics:** (1)

**Inverse multiquadrics:** (2)

and **Gaussian: ** (3)

where Ƭ > 0, is a constant and β is spread parameter.

It can be seen from the above examples that the RBF kernels are usually function of the Euclidean distance between input and the center vectors. These distances are then mapped via some nonlinear functions such as Gaussian function given in (3). Here, the parameter β plays the role to adjust the sensitivity of the kernel. For example, its larger value will make the kernel less sensitive to a given input and vice versa.

**B. Cosine distance based RBF kernels**

One recent study [10] showed that there are scenarios where Euclidian distance becomes ineffective way of distinction. Thus, a cosine distance based RBF kernel was proposed in [10] which it evaluates the cosine distance between the input vector and the RBF’s center vector as follows:

(4)

where < xc_{i} > is showing the dot product of vectors x and c_{i}.

The term in (4) represents the weighted norm of the vector x which can be evaluated using the following definition:

(5)

It is argued in [10] that the cosine distance based kernel is more suitable when the lengths of vectors are very close yet differ in their inclinations. Later this work was extended to a time varying convex combination of Euclidian distance based and cosine distance based kernels as follows [17]:

(6)

where α(n) is a time varying mixing parameter which is adapted by minimizing the mean square error cost function [17].

In this study, we propose a weighted cosine RBF kernel in which weight to the dot product and to the norm of the individual vectors are provided which result in the following kernel:

(7)

where Σ is a diagonal weight matrix with σ_{m} as its m^{th} diagonal
element, x^{H}Σc_{i} is the weighted correlation between vectors x and
ci such that:

(8)

where M is the length of the vectors and x_{m} and c_{i,m} are the m^{th} elements of x and c_{i}, respectively.

The proposed RBF kernel given in (7) is named as Weighted Cosine RBF (WC-RBFNN).

**^{1}The notation ()^{H} denotes conjugate transposition operation.
For the real input vectors, this is equivalent to taking transpose
only [18].

• The weighting matrix Σ is playing a role of giving specific importance to certain elements in the evaluation of cosine distance. In order to understand this further, we consider the weighted correlation between the two vectors x and ci defined in [8].

• It can be seen that the parameter σm is giving different
weights to the m^{th} term in this summation. Thus, if
the correlation between x and ci is dominant due to certain
terms, higher weights to these terms can increase their
impact on the overall summation which eventually results
in the enhancement of the cosine distance between the
two vectors. Hence, the output of those neurons will be
higher which are closer to specific input in terms of cosine
distance.

• Consider the two extreme scenarios:

a) If x = c_{i} (showing full positive correlation), the WCRBFNN’s
output for the ith neuron will be

(9)

b) If x = -c_{i} (showing full negative correlation), the WCRBFNN’s
output for the ith neuron will be

(10)

Thus, we conclude that the output of the WC-RBFNN’s neuron lies in the range [-1, +1] where extreme values show the full negative and full positive correlation cases, respectively.

There can be an obvious question as how to choose the diagonal weighting matrix. This choice mainly depends on the application and which way the proposed RBF kernel is utilized. Some of the examples of specific applications are discussed in the following sub-sections.

**A. Application in PDZ domain classification**

PDZs are structural domains that are contained in numerous otherwise unrelated proteins. In general, they consist of approximately 90 amino acids and in the human genome are considered to be the most common protein domain [19]. PDZ domains are mainly recognized for their mediator role in the assembly of receptors at the cellular membrane interface [19-24]. Although, there are several PDZ domain classes; but, the two most dominant recognition patterns are X-S/T-S-φ for Class I PDZ domains and X-φ-X-φ for Class II PDZ domains [25,26]. There are many works in the field of protein and PDZ domain analysis. One of the recent work in [27], topological predictions for integral membrane transport proteins as well as guides for the development of more reliable topological prediction programs for family-specific characteristics are provided. In another work by [28], the response surface methodology (RMS) and artificial neural network (ANN) modeling were applied to optimize medium components for spinosad production. The Genetic Algorithm (GA) is used for ANN’s input optimization. They conclude that the hybrid ANN/GA approach provides a viable alternative to the conventional RSM approach for the modeling and optimization of fermentation processes. Related to protein study, another work in [29] provides molecular modelling of CtHtrA protein active site structure identified putative S1S3 sub site residues I242, I265, and V266.

In our work, we focus on classifying the Class I PDZ domains (i.e.,
having X-S/T-S-φ pattern) and Class II PDZ domains (i.e., having
X-φ-X-φ pattern) which were reported in [25,26] using a novel
history weighted cosine RBF NN. Now, consider first the features
we have used in our classification task. The set X = A, B, C, D, E, F,
G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y is all possible amino acids (AAs). A protein sequence of length M, it has M-1 bigrams. For
example, ABCDEFG has the following bigrams: AB, BC, CD, DE,
EF, and FG. We computed the bigram normalized frequency of
amino acids in a PDZ sequence by counting the number of times
that bigram appears in the PDZ domain and dividing by M-1. To
design the required weighting matrix Σ, we first evaluate the
Bigram **(Figure 1)**.

Global History Matrix (BGHM) denoted as Gk for the kth class. In
order to compute BGHM, we first aligned our dataset separately
for Class I and Class II PDZ domains using CLUSTALX 2.0 [30]. The
concept of BGHM is illustrated in **Figure 1**, where we compute
the global frequencies of every existing bigram for a set of PDZ
domains sequences. Since, we are dealing with three classes of
PDZ; therefore, we have three BGHMs: G_{I} and G_{II}. Every Column in
G_{k} (k = I, II) indicates normalized frequencies of all possible bigrams
at that position in every PDZ domain. To ease the computational burden, we set the number of columns in G_{k} to the length of the
longest PDZ domain sequence for k^{th} class. Once the BGHM for
each class is evaluated, a history weight matrix (H) is computed
as shown in **Figure 2** for a given PDZ domain. To understand the
computation of the history matrix **H**, consider the example of
the PDZ domain ETRREIKLFKGPKGLGFSIAGGRRNQTKIIDGGA. In
this PDZ sequence, the bigram ”RR” appears 3^{rd}, 23^{rd}, and 35^{th} positions. Assuming that the corresponding elements in the
BGHM G_{k} has entries , respectively.
Thus, the element in the history matrix H corresponding to the
bigram RR is calculated by the summation of (as depicted by the **Figure 1**).

Finally, this history weight matrix **H** is used as the weighting
matrix for the RBF, i.e., **Σ=H** and the resulting RBF is named as
History weighted cosine RBFNN (HWC-RBFNN).

**B. Application in channel estimation for correlated inputs**

In this section, we show a a different domain of application for
the proposed WC-RBFNN, that is, the channel estimation using
the WC-RBFNN. Considering the problem of channel estimation
in the presence of correlated inputs x. If the channel to be
estimated is represented as an M×1 vector h, the output of the
channel at iteration n can be expressed as o(n)=xHh+v(n), where
v(n) is a zero mean i.i.d. noise sequence with variance σ^{2}_{v} and x_{i} is M×1 zero mean correlated input vector with correlation matrix
R_{x}, i.e., E[xx^{H}] =R_{x}. The goal of the RBF is to estimate the unknown
channel by minimizing the cost function

J(w)=E[e(n)^{2}]

where e(n) is the estimation error given by;

e(n)=d(n)-y(n) (11)

where d(n) and y(n) are the desired response and the output of the RBF, respectively.

The desired response, in the case of channel estimation, is d(n)=o(n)=xHh+v(n) while the output of the RBF can expressed as y(n)=φT(x)w(n) The estimation of the channel is obtained via recursive computation of the RBF weights w(n) via steepest descent optimization as follows:

w(n+1)=w(n)+μe*(n)φT(x) (12)

It is known fact from the theory of adaptive filtering that the
convergence speed of steepest descent optimization is severely
degraded by the input correlation [18]. For example, if the
adaptation is used by minimizing the mean-square-error cost
function, the step-size μ is bounded as [18], where
λ_{max} is the maximum eigenvalue of the input correlation matrix R_{x}.

Thus, higher the correlation among the inputs, slower is the learning speed for the adaptation. The same can be true for the case of RBF based channel estimation. In order to deal with this issue, it is proposed to set the weighting matrix Σ as follows:

(13)

Knowing the fact that the input correlation matrix is Hermitian,
we can factorize the Σ as^{2}

(14)

**^{2}The representation R^{H/2} is a short notation for (R^{1/2})^{H}.

By doing so, we can see that the output of the proposed RBF kernel for the ith neuron is transformed to

(15)

Now, if we define , we can write

As a result, the above expression can be reformulated as:

(16)

The correlation matrix of transformed vector can be evaluated as:

(17)

which proves that the input vector is transformed to a white vector with the choice of weighting matrix given in (14). Hence, the transformed output of the RBF given in (16) is now dealing with white input vectors which promise improvement in the convergence speed of the proposed RBFNN.

In this section, we aim to design the mechanism to make the
weighting matrix Σ time varying. To do so, we employ the
steepest descent approach to minimize the mean-squareserror
(MSE) cost function, i.e., J(Σ)=e^{2}(n), where e(n)=d(n)-y(n) is
the error at the output for the nth iteration. We consider more
general scenario in which every neuron has different weighting
matrix. Thus, the steepest descent recursion for the weighting
matrix of i^{th} neuron is given by

(18)

where represents the gradient of cost function J(Σ_{i})
w.r.t to the weighting matrix Σ_{i}.

Since, Σi is a diagonal matrix such that diag we can evaluate the as

(19)

whose m^{th} diagonal element can be evaluated as

(20)

where w_{m}(n) is the m^{th} element in the weight vector w(n) and is defined as

(21)

which is found to be

(22)

Thus, the adaptive weight matrix of the ith neuron can be computed via Equations (18), (19), (20), (21) and (22). Hence, the resulting WC-RBFNN utilizes an adaptive weighting matrix and it is abbreviated as AWC-RBFNN.

In this section, we present three examples to investigate the performance of the two proposed variants of the WC-RBFNN in previous sections. These examples are discussed in detail in the ensuing subsections.

**A. PDZ domain classification**

In order to validate the performance of our algorithm, we
perform a comparative study using holdout cross validation and
compare the performance of the proposed WC-RBFNN with the
conventional variants of the RBFNN, that is, Eucledian Distance
based RBF (ED-RBFNN), Cosine Distance based RBF (CDRBFNN).
For the WC-RBFNN, we have used the two types of weighting
matrix: One proposed in Section (Choice of Weighting Matrix
for the Wc-Rbfnn- Application in PDZ Domain Classification),
i.e., History Weighted WC-RBFNN or simply HWC-RBFNN and
the other proposed in Section (Design of an Adaptive Weight for
Wc-Rbfnn), i.e., AWC-RBFNN with adaptive weighting matrix. The
data set of the PDZ domains we used are taken from the work in
[25,26] which has two classes: Class I with 45 PDZ domains and
Class II with 20 PDZ domains. This data set is divided into two
sets: training and testing. In each run, we used training set to
train the weights for NN and the testing set is used to evaluate
the classifiers accuracy. Since Area Under Curve (AUC) is more
discriminating measure than accuracy, we used AUC to compare
the effectiveness of the two algorithms. Since our feature vectors
are high dimensional the ED based classifier suffers with the
challenge of dimensionality which is also evident in the ROC curve
plots^{3} presented in **Figure 2**, that 3The ROC curves are the curves
for true positive rate (TPR) vs false positive rate (FPR) shows the
comparative ROC curve for the four RBF based classifiers: ED-RBF,
CD-RBF, HWC-RBF, and AWC-RBF. The issue of dimensionality
was so critical that the conventional ED-RBF could not manage
it, yielding a poor AUC 0.543. The CD RBF performs better with
AUC of 0.759. The AUC of 0.842 for the adaptive weighting based
AWC-RBF shows its better performance compared to the ED-RBf
and the CD-RBF. However, the HWC-RBF achieves the highest
AUC of 0.913. This shows that the proper selection of weighting
matrix can improve the overall performance and it can perform
even better than the adaptive weighting matrix design **(Figure 2)**.

**^{3}The ROC curves are the curves for true positive rate (TPR) vs
false positive rate (FPR).

**B. Channel estimation in the presence of correlated inputs**

Next, we investigate the proposed RBFNN for a channel estimation problem with an unknown channel selected as h=[0.227, 0.460, 0.688, 0.460, 0.227]T. The RBFNN used in this experiment has 10 numbers of neurons at the hidden layer. The input to the RBFNN and unknown channel is an M dimensional correlated complex Gaussian input which is generated with the following correlation matrix:

where 0 < αc < 1 is the factor that controls the correlation between the regressor elements.

In this experiment, the input vector has the same length of
that of the channel, that is, M=5. We set the SNR to 20 dB. We
compare the MSE performance of the conventional ED-RBFNN,
CD-RBFNN, the whitening based WC-RBFNN proposed in Section
(Choice of Weighting Matrix for the Wc-Rbfnn- Application in
Channel Estimation for Correlated Inputs) and the AWC-RBFNN
proposed in Section (Design of an Adaptive Weight for Wc-Rbfnn)
and the result is reported in **Figure 3**. It can be easily seen that
the proposed whitening based WC-RBF out performed all of its
counterparts in both convergence speed and the steady-state
error. This indicates that the knowledge of weighting matrix
for specific application can improve the performance and its
performance is even better than that of the AWC-RBF.

**C. Prediction of leukemia cancer**

In this section, we present application from the field of
bioinformatics, specifically, using gene microarray data as a tool
to predict cancer. Using the standard Leukemia ALL/AML data
[31], we aimed to develop a method of Leukemia prediction.
The data set we used was comprised of both training and testing
samples. More specifically, the bone marrow specimen consisted
of 38 training samples (27 ALL and 11 AML) and 34 testing
samples. These 34 test samples (20 ALL and 14 AML) are prepared
using different experimental conditions and were isolated from
either bone marrow (24) or by blood sample collection (10).
In total, the dataset consisted of 7129 target genes. Minimum
Redundancy and Maximum Relevance (mRMR) technique was
used a way to isolate the most significant gene products, as it is
well established in the field [32]. The mRMR approach yielded
the top five genes for further experiments **(Figure 3 and Table 1)**.
For the training phase, the MSE curves of different approaches
are shown in **Figure 4**.

**Table 1 **Results for the pattern classification problem.

Approach |
Training Accuracy | Testing Accuracy |
---|---|---|

Cosine kernel | 100.00% | 94.12% |

Euclideankernel | 100.00% | 58.82% |

WC-RBFNN | 100.00% | 97.06% |

The Euclidean kernel out performs the cosine kernel achieving
a minimum MSE of 0.1033. The cosine kernel attained an MSE
of 0.6934. The proposed method is able to undergo dynamic
adaptation of the diagonal weighting matrix achieving an MSE of
0.5480. The **Table 1** present the training accuracy of each of the
cases studied. One important note is that in all cases, the training
accuracy achieved is 100% for the training samples. However, a
better evaluation for predictive systems is to look at the outcome
of the testing stage, i.e., the unseen samples.

Even though, the Euclidean kernel initially achieved a high training
accuracy in the training stage, its performance in the testing stage
was relatively low at 58.82%. Therefore, the Euclidean kernel was
likely over trained on the training samples and therefore incurred
the problem of over fitting”. The kernel from this study using
the proposed dynamic approach outperformed the both the
conventional cosine and Euclidean kernels by a margin of 2.94%
and 38.24%, respectively **(Figure 4)**.

This work proposes a novel kernel for the Radial Basis Function Neural Networks (RBFNN). The proposed kernel is based on weighted cosine distance between the input vector and the center vectors associated with RBFNN. The weighting is introduced in:

i) the inner product of input and neuron’s center,

ii) norm of the input vector, and

iii) norm of the center vector.

We demonstrate how the weighting matrix can be chosen for different applications to optimize the performance of the WCRBF. As case studies, we present the PDZ domain classification and the channel estimation problem with correlated inputs. We also design an adaptive technique to update the weighting matrix for an arbitrary data using the approach of steepest descent optimization. For the validation of our adaptive design, we present the problem of Leukemia prediction. Simulations that were presented seem to validate the performance of the proposed RBFNN kernel in contrast to the conventional RBFNN kernels.

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under Grant No. (G/503/135/1436). The authors, therefore, acknowledge with thanks DSR for their technical and financial support.

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