CHAPTER 5 MODELING METHODOLOGY
5.5 USE OF ANN FOR PARAMETRIC STUDIES
5.5.1 ASSESSMENT OF CONTRIBUTION OF INPUT VARIABLES
Convinced by the predictive quality of artificial neural network model in the present work, studies on their explanatory capacity were carried out. Influence of each variable in terms of its contribution to the output was first found out. Partial derivative method (PaD) was used to analyze the contribution of inputs. Algorithms were obtained from the study done by Gevrey et.al (2003). However in ecology relationships are the results of multivariate and nonlinear conditions and phenomena are rarely due to a simple cause or to a unique perturbation. Hence modification of PaD method, which uses the second order derivative suggested by Gevrey et.al (2006), was used to analyze the contribution of all possible pair-wise combinations of input variables, taking into account two-way interaction between variables.
For a network with ni inputs, one hidden layer with nh neurones, and one output, the partial derivatives of the output yj with respect to input xj (with j_/1, . . ., N and N the total number of observations) are:
) )) , (
* ) , ( 1 (
* ) , (
* ) , 1 ( ( (
*
1
1 1
1
∑
2=
−
=
nhnh j
ji
D w nh y nh j y nh j w nh i
d
Where Dj is the derivative of the output neuron with respect to its input and is given by, D =y2 (1-y2) and
y2 = response vector of the given data set.
y1 (nh, j) is the response of the hth hidden neuron, w2 and w1 are the weights between the output neuron and hth hidden neuron, and between the ith input neuron and the hth hidden neuron.
The relative contribution of the ANN output to the data set with respect to an input is calculated by a sum of the squared partial derivatives obtained per input variable:
SSDi = 2
1
) ( ji
N
j
∑
d=
One SSD (Sum of Square Derivatives) value is obtained per input variable. The SSD values allow classification of the variables according to their increasing contribution to the output variable in the model. The input variable that has the highest SSD value is the variable, which influences the output variable most (Gevrey et.al 2003).
By considering two-way interactions, the PaD2 algorithm uses the computation of the partial derivatives of the ANN output with respect to the two inputs. Two types of results can be obtained. The first one is a profile of the output variation for small changes of two input variables, and the second is the relative contribution of a paired-input variable to the network output perturbations. For a network with ni inputs, one hidden layer with nh neurons, and one output, the partial derivatives of the output yi with respect to inputs xij and xi+1 j (with j=1, . . . N and N the total number of observations and with xi
and xi+1 the two inputs which constitute the interaction studied) are:
dddjjj =
( )
))) (
* 2 1 (
* )) ( 1 (
* ) (
* ) , 1 (
* ) 1 , (
* ) , ( (
)) 1 , (
* )) ( 1 (
* ) (
* , 1 ( (
* ) 2 1 ((
*
1 1
1 2
1 1
1
1 1
1 1
2 2
nh y nh
y nh
y nh w i
nh w i nh w
i nh w nh y nh
y nh w y
D
nh
nh
nh
nh
−
− +
+ +
−
−
∑
∑
=
=
When D is the derivative of the output neuron with respect to its input, an is given by D =y2 (1-y2) and
y2 = response vector of the given data set.
y1(nh) the response of the hth hidden neuron, w2 (1,nh) the weight between the output neuron and hth hidden neuron, and w1 (nh, i) and w1 (nh, i+1) are, respectively, the weights between the first studied input neuron and the hth hidden neuron and between the second studied neuron and the hth hiddenneuron.
The relative contribution of the ANN output to the data set with respect to a pair of input is the sum of the squared partial derivatives obtained per pair of input variables:
ssd = 2
1
)
∑
N (djThe value of sum of square derivatives (SSD) is obtained per pair of input variables. The SSD values allow the classification of the interactions according to their increasing contribution to the output variable in the model. The pair of input variables that has the highest SSD value is the pair that influences the output most (Gevrey et.al (2006).
These calculations were done by writing the code in MATLAB environment by calling out the text file containing the weights and bias matrices of the optimized neural network.
5.5.2 VARIABLE OPTIMIZATION USING CONTOUR PLOTS
In this study contours were drawn by keeping the values of two parameters fixed (lower values in one trial and the higher values in the next trial) and giving some increment for other two parameters so as to cover the whole domain used for the study.
Contour plots provide some clue for the decision makers about the range of input parameters to be maintained so that the output parameter can be maintained with in the safe or the desirable level. Maximum value was selected such that beyond which the system no longer can maintain the dissolved oxygen in the acceptable level. Four sets of contour plots were prepared.
1. Contour plots for DO concentrations vs TP and BOD, when (a) Conductivity = 50 µS/cm and alkalinity = 25 mg/L, (b) Conductivity = 300 µS /L and alkalinity =150 mg/L
2. Contour plots for DO concentration vs conductivity and alkalinity when (a) BOD = 0 mg/L, TP = 0.03 mg/L (b) BOD =10 mg/L, TP= 1 mg/L.
3. Contour plots of DO concentration vs conductivity and BOD when
(a) alkalinity = 25 mg/L, TP = 0.04 mg/L. (b) alkalinity = 180 mg/L. TP = 1 mg/L.
4. Contour plots of DO concentration Vs conductivity and TP when
(a) BOD = 0 mg/L, alkalinity = 25 mg/L, (b). BOD = 8 mg/L, alkalinity = 200 mg/L.
After obtaining the contour plots for some specific values, original data sets with output values above the desirable level were selected to obtain contour plots. This was done to have a comparison and to check the reliability of the contour plots in parametric studies.