2.3 Selection of modeling approach for QSTR model development
Since, the experimental toxicity data considered here exhibited complex nonlinearities, it was appropriate to select nonlinear modeling approach for establishing the QSTR models for predicting the toxicities of pesticides in HB. Accordingly, we selected the nonlinear PNN and GRNN modeling approaches for development of QSTR models for qualitative (toxicity classes) and quantitative (endpoint toxicity values) prediction of toxicities of structurally diverse chemical pesticides in honey bee using a set of simple molecular descriptors. PNN and GRNN approaches have successfully been applied in computational toxicology earlier (Singh ...view middle of the document...
The node computes the distance d(s,x) from the test vector x to the training samples and output the value of the kernel function. The outcome of each of the layer one cell is added separately for the different classes by the connections to the output cells with weight one.
2.3.2 GRNN-QSTR model
GRNN uses nonlinear Gaussian kernel regression and estimates any arbitrary function between the input and output vectors, drawing the function estimate directly from the training data and provide the optimal estimation of continuous variables implementing the statistical concepts of conditional probability (Du et al., 2008). It is a four-layered NN (Fig. 2b) consisting of an input layer, a pattern layer, a summation layer, and an output layer, and does not require any iterative training procedure to converge to the desired solution. The input layer provides input vector x to pattern layer, which consists of neurons for each training datum or for each cluster center. In this layer, the weighted squared Euclidean distance is calculated according to; D_i^2= (x-x^i )^(T ) (x-x^i), where xi refers to ith training vector. Any new input applied to network is first subtracted from pattern layer neuron values, then according to the distance function either squares or absolute values of subtracts are summed and applied to activation function. Results are transferred to summation layer. Neurons in summation layer add dot product of pattern layer outputs and weights. At output layer, multiplication of pattern layer outputs and training data output Y values (Yf(x)K) are divided by weighted outputs of the pattern layer (f(x)K) to estimate desired Y, the conditional mean of Y given x, as (Specht, 1991);
Y(x)= (∫_(-∞)^∞▒Yf(x,Y)dY)/(∫_(-∞)^∞▒f(x,Y)dY) (2)
where K is a constant associated with Parzen window and f (x,Y) represents the known joint continuous probability density function (PDF) of a vector random variable, x, and a scalar random variable, Y. Since, the density, f (x,Y) is not known and it is usually estimated from a sample of observations of x and Y, the probability estimator f(x,Y) is based on a sample value xi and Yi of the random variables x and Y, as;.
f ̂(x,Y)= 1/(〖n(2π)〗^█(((p-1))/2@ ) σ^((p+1)) ) ∑_(i=1)^n▒exp〖[-((x-x^i )^(T ) (x-x^i))/〖2σ〗^2 〗 ] (3)
where n is the number of sample observations, p is the dimension of the vector variable x, σ is the smoothing parameter. The probability estimate, f(x,Y) assigns a sample probability of width σ for each sample xi and Yi, and the probability estimate is the sum of these sample probabilities. Now, supplementing the joint probability evaluation in Eq (3) into the conditional mean Eq (2) yields,
Y(x)= (∑_(j=1)^p▒〖y_j exp[(-D_j)/〖2σ〗^2 ] 〗)/(∑_(j=1)^p▒exp[(-D_j)/〖2σ〗^2 ] ) (4)
Here, Eq. (4) presents the exponentially weighted sum of each training pattern according to the smoothing parameter, σ and its Euclidean distance to the unknown pattern x. It is...