ANEXO III

Neste anexo são mostrados dois artigos, que foram apresentados em eventos relacionados ao tema em estudo, decorrentes da pesquisa. O primeiro artigo, "Classifying Chromosomes Using Radial Basis Function Networks", foi apresentado no First International Congress of Industrial Engineering and XV National Congress of Production Engineering - ENEGEP, em São Carlos - SP, no período de 03 à 07 de Setembro de 1995. Este artigo se encontra nos anais do Congresso, no volume 2, páginas 1137 à 1141. O segundo artigo, "Using Radial Basis Function Networks to Classify Human Chromosomes", foi apresentado no II Simpósio Brasileiro de Redes Neurais - II SRRN, também em São Carlos - SP, no período de 18 à 20 de Outubro de 1995. Este artigo também pode ser encontrado nos anais do Simpósio, nas páginas 133 à 138.

Abstract: This paper presents an approach to the automatic classification of G-banded human metaphase chromosomes using Radial Basis Function Networks (RBFN). The features chosen to represent the chromosome were length, centromere position, and density. These features are the input to the network that classifies the chromosomes according to the Denver group (7 outputs).

Different topologies and free parameters of the network were tested confirming that with the use of this approach classification performance and accuracy of the RBFN compares favorably with a well-developed parametric classifier and multi-layer perceptrons (MLP), used in previous studies.

We now are investigating the use of this approach for the classification of 24 classes of chromosomes (1 - 22 pairs of "outosomes" and two sex chromosomes). We intend to use the output of the Denver classification as input to the next processing stage, together with the features from banding pattern.

Keywords: Artificial Neural Network, RBFN, Automated chromosome classification.

1 - INTRODUCTION

The microscopic analysis of human metaphase chromosomes has as its objective to classify the chromosomes (karyotyping) to evaluate their structural integrity. The microscopic study is delayed and requires the extreme experience of a professional, making it a highly skilled work. The inspection of chromosomes is an essential procedure in many fields of investigation such as detecting genetic abnormality, damage due to environmental factors, cancer diagnosis, or pre-natal diagnosis. For clinical purposes, karyotyping is required in which chromosomes must be assigned to one of 24 classes (Paris conference in 1971). The karyotyping requires great ability of the cytogenecist; this task contains substantial elements of a tedious and repetitive nature, resulting in some interest in recent years in developing automated karyotyping systems [1, 2, 13, 14, 16, 17, 21].

To form a karyotyping of the 46 chromosomes in a normal human cell, some features of chromosomes are needed. The length, centromere position and band pattern along the longitudinal axis have been used for more than 20 years [2]. Piper selected 28 available features, and says that it is impractical to make a exhaustive search for the optimal feature subset [16]. Various classification methods have been applied to such band transition sequences [8], weighted density distributions [14], structural band descriptions [4], Markov networks [3,5] and more recently neural network [1, 19].

Neural network classifiers have been shown to be highly adaptable and capable of generalizing on classes based on training data [6, 11]. RBFN have been applied in classification tasks where classical pattern recognition methods and MLP, usually trained with backpropagation, have not been applied or have been unsuccessful [7].

In the following sections a description of the data base used and the RBFN designare are given.

2 - CHROMOSOME DATABASE AND FEATURES

Three data bases of annotated measurements from G-banded chromosomes, originating in Copenhagen, Edinburgh and Philadelphia, have been used in this study. In each data set correct classification into one of the 24 human chromosome classes was provided by experienced cytogeneticists to permit evaluation of the approach. These three data sets have been largely used in previous classification studies [1]. Details of these data are given in Table 1.

The first data base was collected at Rigshospitalet, Copenhagen in 1975-1978 and consists of 180 G-banded peripheral blood metaphase cells. These chromosomes were carefully measured by densitometry of photographic negatives from selected cells of high quality. The second data base obtained at the MRC, Edinburgh in 1984, contains 125 human male peripheral blood cells. The third data base was obtained at the Jefferson Medical College, Philadelphia, in 1987 and contains 130 cells.

The different preparation techniques and digitization result in obvious differences in chromosome morphology and banding in the three data bases. So, a standardization or normalization is needed to minimize the length and gray level variation between cells (caused mainly for illumination and staining). The features used in our study were length, centromere position, and density. The density was taken from the average points in the longitudinal axis of the chromosome and normalized after that.

Table 1 - Details of the three chromosome data sets used. [16]

3 - RADIAL BASIS FUNCTION NETWORK (RBFN)

There exists a variety of different ways that neural networks (NNs) can be used in pattern classification [6]. The backpropagation algorithm for training a MLP (supervised) could be seen as an application of a method of optimization known in statistics as stochastic approximation [6, 15]. We can visualize the design of a NN as an approximation problem to find the best result (curve-fitting) in a space of high dimension. Looking at the problem this way, learning is equivalent to finding a surface in a multi-dimensional space proportioned by the best adaptation of the parameters of training, with the criterion of "the best adaptation" measured by some statistical method [6]. Correspondingly, generalization is equivalent to using this multi-dimensional surface to interpolate the test data. This is the approach used in RBFN, which was initially introduced in the problem solution of real multi-variate interpolation.

Broomhead and Lowe, in 1989, were among the first to explore the use of RBFN in the NN field. The RBFN basically consists of an input layer, a hidden layer, and the output layer. Each node in the hidden layer employs radial basis functions to produce a localized output with respect to the input signals. The outputs are combinations of weighted inputs that are mapped by an activation function that is radially symmetric. Each activation function requires a "center" and a scale parameter. The most common radial basis function is the Gaussian function, so that given an input vector X, the output of a single node will be

(3.1)

where the function could be

(3.2)

The values of , j = [1,n], are used in the same manner as with "normal" probability densities to provide "dispersion" scales in each component direction.

Another common variation on the basis functions is to increase their functionality using the Mahalanobis distance in the Gaussian function. The above equation becomes:

(3.3)

where K^-1 is the inverse of the covariance matrix of X associated with hidden node C .

Given p-exemplar n-vectors, representing p-classes, the network can be initiated with knowledge of the centers (locations of the exemplars). If c_j represents the jth exemplar vector, then we can define the weight matrix C as follows:

C = [c₁ c₂ ... c_n] (3.4)

such that the weights in the hidden node j are the components of the "center" c_j. Thus, a hidden-layer node calculates the expression of Eq. (3.2).

The output layer is a weighted sum of the hidden-layer outputs. When presenting an input vector to the network, the network implements

(3.5)

where represents the vector of functional outputs from the hidden layer, and the corresponding center vector. Given some training data with desired responses, the output weights W can be found using the LMS interactively or non-interactively, like descent gradient and pseudo inverse techniques, respectively.

Learning in the hidden-layer is performed using an unsupervised method, typically a clustering algorithm, heuristic clustering algorithm, or supervised algorithm to find the cluster centers (hidden node C). The most common clustering algorithm used to train the hidden layer RBFN is the generalized Lloyd algorithm or the K-means clustering algorithm [7]. Some studies have also used supervised learning of locations of the centers and self-organizing learning of the centers [10, 22].

A simple way of choosing the scaling factors for the Gaussian functions is to set them equal to the average distance between all training data

(3.6)

where is the set of training patterns grouped with cluster center C_j, and M_j is the number of patterns in .

Another manner of choosing the parameters is to calculate the distances between the centers in each dimension and use some percentage of this distance for the scaling factor. In this way, the p-nearest neighbor algorithm has been used. Sometimes, to improve the radius of the Gaussian function, it is interesting to multiply this variance by a constant. The objective is to increase the radius and consequently the amplitude or range of the neuron [18].

4 - CLASSIFICATION RESULTS

After normalization of the data in each data set, half of data were used to train and the other half to test. Then, we further divided the training set into a subtraining set and a cross-validation set. Alternative values for the user-specified parameters were then tried while training on the subtraining set and testing on the cross-validation set. The best-performance parameter values were then employed to train a network on the full training set. The generalization performance of the resulting network is then measured on the test set. In a subsequent experiment the roles of the training data and testing data were interchanged.

A comparison was made between the best performance using our approach and the MLP approach used by Errington & Graham, shown in the Table 2. To train and test a whole data set on a SUN sparc 10 takes about 1 minute. The results have proved that the RBFN have excellent characteristics as classifiers and have an advantage in reduced training time. Finally, it is possible to continually perform the learning process with sequential incoming data, therefore, the network always improves its performance [19].

Table 2 - Comparison of error rates for the best Denver classifiers used.

5 - CONCLUSION

We have presented in this paper the application of radial basis function networks to the classification of human chromosomes in the Denver group. The results have been compared and have shown improvement the conventional techniques and MLP, trained with back propagation. We are studying the use of this approach in the classification of human chromosomes into 24 classes, with the objective to separate the chromosomes and to form karyotyping.

ACKNOWLEDGMENT

The authors would like to thank all those who helped with this research, specially Dr. Jim Piper at MRC Cytogenetics Unit in Edinburgh, Scotland, to let us use his chromosome databases. The authors also would like to thanks the Brazilian Research Council (CNPq) for support to conduct this research.

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