Cheng Qian, Zhang Hongyi
(College of Chemistry and Environmental Science, Hebei University, Baoding 071002, China)

Abstract A 3-6-1 BPNN (back-propagation neural network) was used to link the relation between molecular structures and boiling points. The data sets were composed of 73 saturated aliphatic aldehydes and alkanones, with experimental boiling points ranging from 253.7 K to 631.2 K, and the number of carbon atoms ranging from 1 to 20. Each compound was characterized by three parameters obtained directly from its molecular structure information, namely functional group position index (P), carbon atom numbers (N) and methyl numbers (N_m), and the three parameters were together selected as inputs to the BPNN constructed in this paper. There was a good linear relationship between the predicted and experimental boiling points with regression coefficient of 0.9992, and the root mean square (RMS) error for the predicted boiling points was within 2.9 K. Through a comparison with multiple linear regression of experimental boiling points against functional group position index (P), carbon atom numbers (N) and methyl numbers (N_m), we found that the predicted boiling points by BPNN were superior than those predicted by multiple linear regression, which gave 11.6 K for the root mean square (RMS) error for the predicted boiling points. Compared with topological indices methods reported previously, the 3-6-1 BPNN proposed did not need additional knowledge or software package to calculate the complicated descriptors, and the number of selected parameters used for inputs was only 3 which was the smallest in all of the corresponding reports.
Keywords Quantitative structure-property relationship, Artificial Neural Network, Boiling point, Aliphatic aldehydes, Aliphatic alkanones.

1. INTRODUCTION
One of the most important purposes of applying mathematical, statistical and computer-based methods in chemistry is to gain the maximum information about the properties selected compounds by analyzing the chemical data. As a result, the interest in quantitative structure-property relationship (QSPR) studies has been increased substantially in these years.
    Although topological index as an important QSPR tool has been successfully used for prediction of all kinds of physical chemistry properties of organic compounds, including boiling points, reports about the prediction of boiling points for saturated aliphatic aldehydes and alkanones are still few, due to the presence of oxygen in their molecules resulting in more complicated calculation of their topological indices than that of saturated aliphatic chain hydrocarbons. Balaban et al. ^[1] predicted 200 carbonyl compounds contained 127 mono-and dialdehydes and –ketones and 73 esters using five topological indices. Toropov and Toropova ^[2] obtained one-variable models of the normal boiling points of carbonyl compounds by the nearest neighboring code (NNC). Toropov and co-workers ^[3] have used simplified molecular input line entry system (SMILES) to model normal boiling points of acyclic carbonyl compounds. Several works about boiling point of saturated aliphatic aldehydes and alkanones prediction have also been done using molecule (or atom) topological index by Lin Zhihua and co-workers ^[4], Chen Yan ^[5], Zhang Xiuli and co-workers ^{[6, 7]}, Feng Changjun, Yang Weihua ^[8], Wang Keqiang ^[9], and so on. However, the common shortcoming in these reports is that these models needs many topological indexes obtained through many difficult calculation steps. Therefore, it is necessary to find a new way for predicting the boiling points of saturated aliphatic aldehydes and alkanones.
    As a powerful chemometric technique, artificial neural network (ANNs) has been used in the study of QSPR ^{[10, 11]}. Among the numerous network architectures, the popular type for QSPR studies is the multilayer feed-forward network with back-propagation algorithm, usually called back-propagation neural network (BPNN). As far as we know, no BPNN method for predicting boiling points of saturated aliphatic aldehydes and alkanones was reported. In this paper, a 3-6-1 BPNN was used to link the relation between molecular structures and boiling points. The data sets were composed of 73 saturated aliphatic aldehydes and alkanones, with experimental boiling points ranging from 253.7 K to 631.2 K, and the number of carbon atoms ranging from 1 to 20. Each compound was characterized by three parameters obtained directly from its molecular structure information, namely functional group position index (P), carbon atom numbers (N) and methyl numbers (N_m), and the three parameters were together selected as inputs to the BPNN constructed in this paper.
    The results obtained by the 3-6-1 BPNN were validated, tested and compared with the results obtained either in previous reports or by multiple linear regressions.

The network input for a node i is given by:

where j represents nodes in the previous layer, w_ji is the weight associated with the connection from node j to node i, x_j is the output of node j, n is the number of the output, and q_i is a bias term or threshold value of node i responsible for accommodating nonzero offsets in the data. The output of node i is determined by the transfer function and the net input of the node. It is given by:

where y_i is the output of the node i, and f( ) is the transfer function which we can choose.

    The BPNN was used in our work. The structure of this kind of neural network is shown in Fig. 2. The first layer is the input layer with one node for each variable or feature of the data. The last layer is the output layer consisting of one node for each variable to be investigated. In between there are a series of one or more hidden layer(s). A node in hidden layer(s) can receive data from any nodes of the anterior layer, process the data, and output a signal.

Fig. 2 A fully connected multilayer feedforward back-propagation network

All data were standardized by using the following equation:

where x_i is the original values for functional group position index (P), carbon atom numbers (N), methyl numbers (N_m) and boiling points, n is the number of the data, and a_i was the result of the standardization.

    To finish the task of predicting boiling points of saturated aliphatic aldehydes and alkanones, the following procedures were done. First, we should choose the appropriate hidden layer nodes. Second, we compared the effect of different learning algorithm using in the network. Third, the prediction was done. Finally, we estimated the predicted results, and compared the BPNN method with other methods such as MLR method and topological index method previously reported.

3. RESULTS AND DISCUSSION
The neural network methodology has several empirically determined parameters. For example, the number of hidden layer nodes, the number of training epochs or the convergence criterion, the learning rate and momentum term, the initialization of the network, and so on. After confirming the input and target of the network, network optimization has to be done.
    The training course will be ended when the mean square error (MSE) values for training and testing sets simultaneously reaches the minimum. In the whole training process, the MSE values for training and testing sets can be calculated, and through monitoring the tendency of MSE we can determine if stopping the training process. Generally, the MSE value for training set will constantly decrease with evolution of training epochs. Similarly, the MSE value for testing set also decreases with evolution of training epochs in the begining stage. But if continuing the training epochs, the MSE for testing set will on the contrary increase, resulting in inferior prediction results for testing set. To prevent the overtraining phenomenon, it is necessary to monitor simultaneously the MSE for both training and testing sets.
    Early stopping was used in the optimizing training process. In ANN, early stopping ^{[15, 16]} is a pretty powerful and typical form using cross-validation which is the widely used method to avoid the overtraining (or over-fitting) phenomenon of neural networks. Early stopping means that the time stopping training proceeds is controlled by the minimum of errors of the validation (or testing) sets other than the minimum of errors of the training set. In general, the data set is divided into training, validation and test sets, while in the case of a small data set, the test set can substitute the validation set in cross-validation. So we divided the studied data of 73 groups into two sets, training set and testing set.
3.1 BPNN model confirmation
There were 3 nodes in the input layer and 1 node in the output layer. Our first aim was to determine the optimal number of hidden layer nodes. A series of neural networks with different numbers of hidden layer nodes were trained. The number of hidden layer nodes varied from 3 to 8. According to its generalization ability on the testing set, we calculated the mean square error (MSE) on different numbers of the hidden layer nodes. MSE is computed with the following equation:

where d_i is the desired output (the experimental boiling point) in testing set, o_i is the actual output in testing set, and n is the number of compounds in the testing set. The lower the value of MSE, the better the network model. To see the transformation trend intuitively, a curve of MSE versus the number of hidden layer nodes was plotted (as shown in Fig. 3). Fig. 3 shows that the best number of hidden layer nodes is 6. So a 3-6-1 BPNN model was selected for further studies.

Fig. 3 Hidden node numbers vs mean square error (MSE) on testing set of the saturated aliphatic aldehydes and alkanones

*GDBP: gradient descent back-propagation; GDABP: gradient descent with adaptive learning rate (lr) back-propagation; GDMBP: gradient descent with momentum back-propagation; GDXBP: gradient descent with momentum & adaptive lr back-propagation; L-MBP: L-M back-propagation.