PDM-ANN

PDM-ANN is based on the combined use of a backward feature selection method and an ANN [13, 15]. Initially, the full vector of input factors is applied as input to the ANN and PDM eliminates serially the factors that are less associated with the ANN output. The procedure is repeated until one factor remains in the feature vector. In the present study, a feed-forward ANN [35] was used, consisting of one input layer of input neurons equal in number to the applied factors, one hidden layer of adjustable number of neurons, and one output layer of one neuron. The hyperbolic tangent sigmoid and log sigmoid were used as activation functions in the hidden and output layer, respectively. Due to the use of hyperbolic tangent sigmoid function, the labels of categorical input variables were set to certain values in the range [-1.0, +1.0], e.g. the labels [1^st, 2^nd, 3^rd and 4^th class] of a 4-class variable were set to [-1.000, -0.333, +0.333, +1.000].The use of log sigmoid function in the output neuron resulted in output values in the range [0.0, +1.0]. A value of less than +0.5 corresponded to BMI ≤ 25 (C1), while a value greater or equal to +0.5 corresponded to BMI > 25 (C2). ANN was trained using the back-propagation algorithm with adaptive learning rate and momentum in order to control the ANN training procedure in terms of convergence rate [35]. Initial learning rate (lr) and momentum (mc) were set to lr = 0.01 and mc = 0.9, respectively, while several values for the number of hidden neurons (number of hidden neurons = 1, 2, 4, 6 and 8) were tested.

for the initial ANNs that use all N = 63 input factors. The procedure continues by deleting one factor from the total number of factors and constructing the 3-CV ANNs that use the remaining factors as input. In turn, each factor is deleted from the total number of factors and ANNs are constructed with the remaining ones. The 62-dimensional input factors set that yields the best average of mean accuracies $({\overline{A}}_{1(62)},{\overline{A}}_{2(62)},{\overline{A}}_{3(62)})$ in the 3-CV sets is assigned the fitness value F _{N = 62} (eq(1)) and is the one chosen at this dimensionality threshold. As the procedure goes on, an ANN that uses N inputs is derived from the one that uses N + 1 inputs by subtracting the least informative factor by means of mean accuracy in the 3-CV training and testing sets, and the set of N factors is assigned a fitness value F _N . This process is repeated until one factor remains. The best set of 3-CV ANNs and the corresponding input factors set are selected based on the values of fitness function F. The factors that are included in the selected set of factors are the most informative ones, while the ones left out are either redundant or do not affect the output.

GA-ANN

GA-ANN combines the evolutionary optimization method of GA [37] with an ANN classifier in order to obtain an optimal ANN architecture [38]. A three layer feed-forward ANN was again used, trained by the back-propagation algorithm with adaptive learning rate and momentum. The hyperbolic tangent sigmoid and log sigmoid were used as activation functions in the hidden and output layer, respectively, and the input values were used after their encoding in the range [-1.0, 1.0] as described for the PDM-ANN method. The GA was applied in order to optimize the ANN architecture, including the set of input factors to be used, and the fitness function, used by PDM-ANN, was utilized.

The whole GA-ANN process is the following: An initial generation of M = 100 random chromosomes is created within the first step of the GA. Each chromosome is a binary mask of lchrom = 76 binary digits (0 or 1). The lchrom digits of each chromosome encode the architecture of one ANN: i) the factors to be used as input in the training procedure are encoded by the first lf = 63 digits, ii) the number of hidden neurons by the next lh = 4 digits, iii) the range of the initial weights by the next lw = 3 digits, iv) the momentum term by the next lm = 4 bits, and v) the initial learning rate by the last ll = 2 bits. Three ANNs are constructed for each of the M chromosomes according to its content and tested within the 3-CV technique. The corresponding classification accuracies in 3-CV training and testing sets are measured and the fitness function value of each chromosome is calculated. Four genetic operators, i.e. selection, crossover, mutation, and election, are then applied to the initial generation of chromosomes. The selection operator uses the elitist selection method [37] and selects the chromosomes that will mate to produce the offsprings for the next generation. Random pairs of the selected chromosomes mate with probability P _c = 0.7 based on the two-point crossover operator [37] and bits within a chromosome are mutated (switched from 0 to 1 or vice-versa) with probability P _m = 0.01. In order to avoid that offsprings with a lower fitness value than their parents are included in the next generation, the election operator is finally used [38]. For the sake of low dimensionality of the selected subset of input factors, "penalty" function is applied to chromosomes that encode a subset exceeding a given dimensionality threshold T. Thus, these chromosomes are assigned a fitness value equal to 50% of the average population fitness. The whole procedure is repeated for N _G = 50 generations and results are stored. The algorithm yields to the optimal ANN classifier, which corresponds to the chromosome featured the maximum fitness value.

The GA-ANN method was applied for dimensionality thresholds T = 30 and T = Inf, the latter corresponding to the case where no penalty function is applied and the method searches for the optimal set of factors with no limitation in terms of dimensionality.

The finally selected predictive models of PDM-ANN and GA-ANN were comparatively assessed using the metrics of accuracy as well as, sensitivity, i.e. the discriminative power regarding positive cases (BMI > 25), specificity, i.e. the discriminative power regarding negative cases (BMI ≤ 25), and area under Receiver Operating Characteristics (ROC) curve. ROC curves were generated by thresholding the output neuron in the range [0, 1], with a step equal to 0.02, and estimating the true positive (sensitivity) and false positive (1-specificity) rates for each threshold. All metrics were calculated for all 3-CV sets and mean values were computed separately for the training and testing sets.

provides an unbiased estimate of the statistical significance of the obtained value of Q in the 3-CV testing set. The whole procedure was repeated for each 3-CV testing set and the mean value of p was calculated.

PDM-ANN Results

The PDM-ANN method was firstly applied in order to identify the most important factors among the aforementioned genetic variations, nutrient intake measurements and gender that affect BMI when used as a two class output variable (C1 vs. C2). The obtained F _N values per number of selected factors N = 63, 62, ..., 1 are presented in Figure 1. Results show that the mean accuracy obtained in the 3-CV training and testing sets when using N factors was kept in the range 77.7%-79.6% for N = 63,..,32 and it started to decrease for N < 32. It continued to decrease up to N = 1, where a mean accuracy equal to 62.3% was obtained. The fact that the factors subtracted by the PDM up to the threshold N = 32 did not affect the F _N values indicates that these certain factors either do not have an important impact to BMI measurement or are redundant with other factors kept in the input variables set. The one by one subtraction of factors after dimensionality N = 32 caused a progressive reduction to F _N values and it is inferred that the factors subtracted after this dimensionality threshold have a high information content towards BMI measurement. Thus, the subset of 32 factors was considered as optimal for the classification of subjects according to BMI measurement in terms of obtained mean accuracy and number of used factors.

GA-ANN Results

GA-ANN was next applied in order to construct an optimal ANN in terms of input factors, architecture and training parameters (number of hidden neurons, momentum and learning rate of the ANN), which leads to a maximum classification accuracy. The procedure was followed both for T = Inf and T = 30 and results for the fitness value and mean accuracies in 3-CV training and testing sets were stored while GA evolved. Figure 2 presents the mean fitness value of all chromosomes obtained in each generation. Results show that while GA evolved, the fitness function improved in average and thus better ANN architectures for discriminating subjects in terms of BMI were obtained. This is more clear in the case T = 30 where the mean fitness value improved from 58.4% to 74.24%. After GA-ANN run for 50 generations, the ANN that corresponds to the chromosome of the best fitness value was selected as the optimal one. When no dimensionality threshold was applied (T = Inf) the best ANN (fitness = 79.18%) was obtained during the 47^th generation and used 32 out of 63 factors as inputs. In case of dimensionality threshold T = 30, the best ANN (fitness = 79.38%) was obtained again during the 47^th generation and was fed by a selected subset of 25 factors. Thus, the application of dimensionality threshold led to an input set of lower dimensionality, as compared to the case of no dimensionality threshold, provided though the same high fitness value. Table 2 presents the subsets of factors fed as inputs to the best ANNs for T = Inf and T = 30. The selected subsets included gender and factors related to nutrition and genetic variations, thus showing the multifactorial contribution of the studied factors to BMI.

Comparative Assessment and Discussion

The ANN obtained by GA-ANN (T = 30) was the best performing architecture in the 3-CV testing sets in terms of mean accuracy (61.46%), mean specificity (48.63%), mean area under ROC curve (0.608), and second best performing in terms of mean sensitivity (69.80%) after the one obtained by GA-ANN (T = Inf). It was also in favor of the specific architecture that it was fed by the most parsimonious set of factors obtained by the applied methods with a dimensionality equal to 25. It is, thus, shown here that the stochastic feature selection within the GA-ANN yielded better results than the serial backward feature selection within PDM-ANN.

For comparison reasons we used the linear discrimination analysis (LDA) method using the 38 continuous nutrition intake measurements in order to predict the status of the two class BMI output variable. Only continuous input variables can feed LDA and the factors corresponding to gender and genetic variations, available only as categorical variables, were discarded. The method provided mean values of accuracy, sensitivity and specificity equal to 58.2%, 58.3% and 58.1%, respectively, in the 3-CV testing sets and was outperformed by the selected models of PDM-ANN and GA-ANN in terms of accuracy and sensitivity. This is due to that PDM-ANN and GA-ANN can manipulate all factors (gender, nutrition measurements and genetic variations) after their transformation to categorical input factors, and simultaneously capture non-linear relationships. Even when LDA was fed with the sole factor selected by PDM-ANN (N = 1), i.e. cholesterol intake in food, it was outperformed by the corresponding PDM-ANN model in terms of mean accuracy (52.7% versus 60.2%).

It is important to discuss the factor selection results obtained here by the ANN-based methods. The studied factors may have a similar impact on the BMI trait, e.g. calories and cholesterol intake measurements, or intake of a vitamin in food and intake of a vitamin in supplement, and the selected ones may depend on the selection method used. Thus, the applied methods (PDM-ANN, GA-ANN (T = Inf), GA-ANN (T = 30) resulted to three overlapping subsets of factors suggested to contribute to BMI. Moreover, the selected subsets include common factors, e.g. the genetic variation of MTHFR C677T, Gender and Cholesterol-Intake in Food were suggested by all methods, while the genetic variations of CBS C699T, MTR A2756G and calories measurement were suggested by two out of three methods (CBS C699T by PDM-ANN and GA-ANN (T = Inf), MTR A2756G by GA-ANN (T = Inf and T = 30) and calories by PDM-ANN and GA-ANN (T = 30)). The consistency of these factors shows their strong impact. Nutrition related factors like calories and cholesterol intake have an obvious impact on BMI and are well known from daily life to trigger the onset of obesity. However, the current study showed the existence of other factors, i.e. intake of vitamins and genetic variations that complete the optimal sets of factors that can discriminate subjects with high or low BMI. It is worth noting that the polymorphisms of PPAR gamma 2 Pro12Ala, selected by GA-ANN (T = 30), and TNF alpha G308A, selected by PDM-ANN, have been related to obesity in [41] and [42], as well, respectively. Further research on the biochemical pathways in which the selected genes are involved could enlighten the way they solely affect BMI or how they interact with nutrition towards the complex BMI trait. Related examples include that the recommended intake of folic acid required to keep homocysteine (independent CVD risk factor) levels normal depends on the MTHFR gene variation, while the upper limits of saturated fat intake depend on the CETP, LPL and APOC3 genes.

The optimal ANN architecture yielded by GA-ANN has been integrated with a remotely located rule-based module that provides personalized information on required modifications of a subject's lifestyle habits in order to reduce the risk of complications related to the cardiovascular system. The proposed modifications are based on existing knowledge in the literature, regarding mostly the combined impact of genes and nutrition on cardiovascular health. The ANN architecture and the module for personalized advice provision communicate through web-services technology and have been integrated into a web-based platform [43].

Future work includes the use of MDR method [44] in order to reveal certain main effects or interactions within the factors selected here, and construct rules that describe these. Furthermore, we intend to use the hybrid ANN methods presented here in the analysis of other multifactorial CVD risk factors, e.g. hypertension and the fasting level of the measurements of TG and LDL-C, and include other factors that describe a subject's lifestyle, e.g. related with physical activity and smoking. Furthermore, the optimization of ANNs used for the analysis of multifactorial disease traits is an open research area in terms of both ANN architecture and ANN training. Our final scope is to construct an ensemble of predictors that estimate the risk of developing CVD and develop methods able to assess the reduction of CVD risk after certain lifestyle interventions. The resulting panel of ANNs could be used to derive an overall predictive risk score to CVD based on genetics and lifestyle. Furthermore, since the overall risk is based on genes and lifestyle, and lifestyle is modifiable, it would be possible to create lifestyle scenarios where any high risk is reduced by means of "reverse engineering".