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C. Karul . S. Soyupak
13.3.1
Artificial Neural Network Approach
An algorithmic approach for developing artificial neural network models for
estimating chlorophyll-a concentration in lakes and reservoirs have been discussed
in detail previously by Karul et al. (1999b; 2000). A summary of previously
established approach is given below:
13.3.1.1
Training Method
A three layer feed-forward neural network model was used. Fig. 13.1 shows the
adopted neural network topology for the estimation of output parameters for
Keban Dam Reservoir as an example. A tangent-sigmoid transfer function was
selected between the input layer and the hidden layer, and a linear transfer
function was selected between the hidden layer and the output layer. The Neural
Network Toolbox of MatLab by Mathworks Co. (Demuth and Beale 1998) was
used during the study.
There are many variations of the backpropogation algorithm and the simplest
implementation of it updates the network weight and bias values in the direction in
wh ich the performance function decreases most rapidly, i.e. the negative of the
gradient. One iteration of the backpropagation algorithm is given by Equation
(13.1).
(13.1)
where x k is the vector of weights and biases at the k
th iteration; ~ is the learning
rate at the k
th iteration; gk is the gradient at the k
th iteration.
The Levenberg-Marquardt variation of the backpropagation algorithm was
employed in calculation of all neural network weights. The Levenberg-Marquardt
algorithm converges faster than other back propagation algorithms and is probably
best when there are neurons up to a few hundred. Hagan and Menhaj (1994) give
detailed information on the utilization of the Levenberg-Marquardt algorithm and
a summary for its implementation is included in Demuth and Beale (1998). The
algorithm is:
(13.2)
where x k is the vector of weights and biases at the k
th iteration; J is the Jacobian
matrix, which contains first derivatives of the network errors with respect to
weights and biases; e is the vector of network errors; I is the identity matrix and Il
is a scalar.
C. Karul . S. Soyupak
13.3.1
Artificial Neural Network Approach
An algorithmic approach for developing artificial neural network models for
estimating chlorophyll-a concentration in lakes and reservoirs have been discussed
in detail previously by Karul et al. (1999b; 2000). A summary of previously
established approach is given below:
13.3.1.1
Training Method
A three layer feed-forward neural network model was used. Fig. 13.1 shows the
adopted neural network topology for the estimation of output parameters for
Keban Dam Reservoir as an example. A tangent-sigmoid transfer function was
selected between the input layer and the hidden layer, and a linear transfer
function was selected between the hidden layer and the output layer. The Neural
Network Toolbox of MatLab by Mathworks Co. (Demuth and Beale 1998) was
used during the study.
There are many variations of the backpropogation algorithm and the simplest
implementation of it updates the network weight and bias values in the direction in
wh ich the performance function decreases most rapidly, i.e. the negative of the
gradient. One iteration of the backpropagation algorithm is given by Equation
(13.1).
(13.1)
where x k is the vector of weights and biases at the k
th iteration; ~ is the learning
rate at the k
th iteration; gk is the gradient at the k
th iteration.
The Levenberg-Marquardt variation of the backpropagation algorithm was
employed in calculation of all neural network weights. The Levenberg-Marquardt
algorithm converges faster than other back propagation algorithms and is probably
best when there are neurons up to a few hundred. Hagan and Menhaj (1994) give
detailed information on the utilization of the Levenberg-Marquardt algorithm and
a summary for its implementation is included in Demuth and Beale (1998). The
algorithm is:
(13.2)
where x k is the vector of weights and biases at the k
th iteration; J is the Jacobian
matrix, which contains first derivatives of the network errors with respect to
weights and biases; e is the vector of network errors; I is the identity matrix and Il
is a scalar.