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50 training examples were necessary to have reasonable learning of connection weights Since we had only six ef cient cases (ef ciency of 100%) and a minimum of 50 hospitals for training were desirable, we selected the top 50 hospitals (sorted in descending order of their relative ef ciency) in our ef cient set The word ef cient in the context of DEA means DMUs with ef ciency of 100% Since the ef cient set in our cases doesn t contain all DMUs with 100% ef ciency, we use quotation marks to identify that the word ef cient has a little different meaning than the word ef cient when used in the context of DEA The same line of reasoning applies for the word inef cient (ef ciency < 100%) when used in the context of DEA and inef cient when used to represent the 50 hospitals that had the lowest ef ciency rating The performance of ef cient and inef cient hospitals was then tested on the set of 75 hospitals in the test set Among the design issues were the following: 1 Normalization of Dependent Variable Because we were using the logistic activation function f x 1= 1 e x , it can be shown that lim f x 1 and lim f x 0 We normalized our dependent variable, the number of employees (y), so that y 2 (0,1), where 0 stands for zero employees and 09 was the largest number of employees in the sample 2 Training Sample Size Our initial decision on a training set sample size of 100 and a test set of 75 was determined on the heuristic of the training set size being !10 times the number of independent variables Because we had ve independent variables, we selected the training set sample size of over 50 3 Learning, Generalizability, and Overlearning Issues The network convergence criteria (stopping criteria) and learning rate determine how well and how quickly a network learns A lower learning rate increases the time it takes for the network to converge, but it does nd a better solution The learning rate was set to 008 The convergence criteria were set as follows: IF (|Actual_Output - Predicted_Output| 01 for all examples) OR Training iterations ! Maximum iterations THEN Convergence = Yes ELSE Convergence = No We selected the above-mentioned convergence criteria to account for the high variability of the dependent variable A more strict convergence criterion was possible; however, an issue arose regarding over tting the network on training data Evidence in the literature shows that over tting minimizes the sumof-square error in the training set at the expense of the performance on the test set We believed that the above-mentioned convergence criteria can make the
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network learning more generalizable It is important to note that convergence (stopping) criteria should not be confused with the procedure used to learn the weights A standard back-propagation algorithm was used to learn the weights The back-propagation algorithm uses minimization of the sum of squares as the optimization criteria The maximum number of iterations was set to 5,000 4 Network Structural Issues The network structure that we chose for our study was similar to the one shown in Figure 33 Our networks had ve input nodes and one output node We had a three-layer (of nodes) network for modeling a nonlinear relationship between the independent variables and the dependent variable The number of hidden nodes was twice the number of input nodes 1, which is a more common heuristic for smaller sample sizes In the case of larger sample sizes, a higher number of hidden nodes are recommended For our research, we tried two different sets of hidden nodes, 11 and 6, respectively 5 Input and Weight Noise One way to develop a robust neural network model is to add some noise in its input and weight nodes while the network is training Adding a random input noise makes the ANN less sensitive to changes in input values The weight noise shakes the network and sometimes causes it to jump out from a gradient direction that leads to a local minimum In one of our experiments, we added input and weight noise during the network training phase The objective of our two experiments can be summarized as follows: in our rst experiment, the objective was to identify the best ANN con guration and use this con guration in our second experiment The objective of the second experiment was to use the proposed DEA-based training data selection to test if there are any performance differences when ef cient versus inef cient training data are used Based on the design considerations, we conducted four different tests by varying the structural design and noise parameters in a two-layer network Tables 32 and 33 illustrate the results of different structural designs for our four tests for network training on 100 cases and testing the trained network on 75 unseen cases, respectively The rst two tests represent the hidden number of nodes of 11 and 6 in a two-layer
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TABLE 32 Performance of Four Different Structural Designs During the Training Phase Number of Hidden Nodes 11 11 6 6 Input Noise 0 008 0 008 Weight Noise 0 001 0 001 RMS Error 0037 005 0044 0041 Prediction Accuracy 97% 96% 97% 98%
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