Hidden neuron

In neural network design, the above parameters have no precise number/answers because it is dependent on the particular application. However, the question is worth addressing. In general, the more patterns and the fewer hidden neurons to be used, the better the network. It should be realized that there is a subtle relationship between the number of patterns and the number of hidden layer neurons. Having too few patterns or too many hidden neurons can cause the network to memorize. When memorization occurs, the network would perform well during training, but tests poorly with a new data set. 

Minimum number of hidden neurons [number of patterns]... 

During the selection of the number of hidden layer neurons, the desired tolerance should also be considered. In general, a tight tolerance requires that the selected network be trained with fewer hidden neurons. As mentioned earlier, cross-validation during training can be used to monitor the error progression, which subsequently serves as a guideline in the selection of the hidden layer neurons. 

To be a bit more precise, let us use the index a to label the states of the visible neurons, the index fl to label the states of the hidden neurons, and the combined index a(3 to label the complete states of the whole system. Then the probability Pa of finding the visible neurons in state a is given by... 

As we mentioned above, however, linearly inseparable problems such as the XOR-problem can be solved by adding one or more hidden layers to the perceptron. Figure 10.9, for example, shows a solution to the XOR-problem using a perceptron that has one hidden layer added to it. The numbers appearing by the links are the values of the synaptic weights. The numbers inside the circles (which represent the hidden and output neurons) are the required thresholds r. Notice that the hidden neuron takes no direct input but acts as just another input to the output neuron. Notice also that since the hidden neuron s threshold is set at r = 1.5, it does not fire unless both inputs are equal to 1. Table 10.3 summarizes the perceptron s output. 

Given pattern p, the hidden neuron receives a weighted input equal to... 

Aqueous solubility is selected to demonstrate the E-state application in QSPR studies. Huuskonen et al. modeled the aqueous solubihty of 734 diverse organic compounds with multiple linear regression (MLR) and artificial neural network (ANN) approaches [27]. The set of structural descriptors comprised 31 E-state atomic indices, and three indicator variables for pyridine, ahphatic hydrocarbons and aromatic hydrocarbons, respectively. The dataset of734 chemicals was divided into a training set ( =675), a vahdation set (n=38) and a test set (n=21). A comparison of the MLR results (training, r =0.94, s=0.58 vahdation r =0.84, s=0.67 test, r =0.80, s=0.87) and the ANN results (training, r =0.96, s=0.51 vahdation r =0.85, s=0.62 tesL r =0.84, s=0.75) indicates a smah improvement for the neural network model with five hidden neurons. These QSPR models may be used for a fast and rehable computahon of the aqueous solubihty for diverse orgarhc compounds. 

Fig. 6.18. Schematic representation of a multilayer perceptron with two input neurons, three hidden neurons (with sigmoid transfer functions), and two output neurons (with sigmoid transfer functions, too)...

With Eq. (6.126) and a GAUSsian activation function the output of the hidden neurons (the RBF design matrix) becomes... 

Figure 5.13 Layers of units and connection links in an artificial neuronal network. Ij-ij.- input neurons, h-hj hidden neurons, b, b exit and output bias neurons, w,2hi weight of transmission, ij-h, x.-Xji input process variables and y output...

Variations of this simple architecture exist. One such case is the so-called jump network, where there are connections directly from inputs to outputs, in addition to the hidden neuron connections. Another... 

Multi-layer feedforward networks contain an input layer connected to one or more layers of hidden neurons (hidden units) and an output layer (Figure 3.5(b)). The hidden units internally transform the data representation to extract higher-order statistics. The input signals are applied to the neurons in the first hidden layer, the output signals of that layer are used as inputs to the next layer, and so on for the rest of the network. The output signals of the neurons in the output layer reflect the overall response of the network to the activation pattern supplied by the source nodes in the input layer. This type of network is especially useful for pattern association (i.e., mapping input vectors to output vectors). 

Hidden neurons communicate only with other neurons. They are part of the large internal pattern that determines a solution to the problem. The information that is passed from one processing element to another is continued within a set of weights. Some of the interconnections are strengthened and some are weakened, so that a neural network will output a more corrected answer. The activation of a neuron is defined as the sum of the weighted input signals to that neuron ... 

The FFBP distinguishes itself by the presence of one or more hidden layers, whose computation nodes are correspondingly called hidden neurons of hidden units. The function of hidden neurons is to intervene between the external input and the network output in some useful manner. By adding one or more hidden layers, the network is enabled to extract higher order statistics. In a rather loose sense, the network acquires a global perspective despite its local connectivity due to the extra set of synaptic connections and the extra dimension of NN interconnections (Hagan and Menhaj, 1994). Figure 1 depicts die structure of a FFBP neural network. 

In parallel to the SUBSTRUCT analysis, a three-layered artificial neural network was trained to classify CNS-i- and CNS- compounds. As mentioned previously, for any classification the descriptor selection is a cmcial step. Chose and Crippen published a compilation of 120 different descriptors, which were used to calculate AlogP values as weU as drug-likeness [53, 54]. Here, 92 of the 120 descriptors and the same datasets for training and tests as for the SUBSTRUCT algorithm were used. The network consisted of 92 input neurons, five hidden neurons, and one output neuron. 

Cross-validation estimates model robustness and predictivity to avoid overfitting in QSAR [27]. In 3D-QSAR models, PLS and NN model complexity are established by testing the significance of adding a new dimension to the current QSAR, i.e., a PLS component or a hidden neuron, respectively. The optimal number of PLS components or hidden neurons is usually chosen from the analysis with the highest q2 (cross-validated r2) value, Eq. (3). The most popular cross-validation technique is leave-one-out (LOO), where each compound is left out of the model once and only once, yielding reproducible results. An extremely fast LOO method, SAMPLS [42], which evaluates the covariance matrix only, allows the end user to rapidly estimate the robustness of 3D-QSAR models. Randomly repeated cross-validation rounds using leave 20% out (L5G), or leave 50% out (L2G), are routinely used to check internal... 

Fig. 9.18. Comparison between the expected results and those obtained with the WNN with 3 hidden neurons and 3 output neurons. The graphs correspond to the three species under study. The dashed line corresponds to ideality (y=x) and the sohd line is the regression of the comparison data. Plots at top correspond to training and plots at bottom to testing.

Using the original piincijjal compwnent scores as inputs, the best architecture consist of a three layer network with 23 input neurons, 10 neurons in the hidden layer and one neuron in the output layer. Considering RPC scores as inputs, the best architectures were achieved with almost the same number of hidden neurons. The hidden neurons consist of 9 and 10 neurons respectively. Training was carried out for a maximum 10000 iterations. Selection of the network was performed at maximum correlation coefficient (R) and 95% confidence limit. 

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