33.

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Page 127

(3.27)

in which n denotes the iteration number, τ is a constant for neurone i, vj is generated in terms of a monotonically increasing non-linear mapping function f(uj), typically a sigmoid function (see Equation (3.2) and Figure 3.1 for details), and λ is a parameter which scales the input to function f(u) as: . Also, where f−1( ) is the inverse function of the input-output relationship f( ). The right hand side of Equation (3.27) specifies the amount of change in ui following the change in time n.

Each state of the Hopfield network has an associated energy function, which is used to indicate the degree of error. The higher the energy, the larger the error, and the aim of the network operations is to minimise the energy. The energy minimisation concept is used again in Chapter 7. At each iteration, the change of network state is directed towards a search for a lower energy state until convergence is reached. Hopfield (1984) describes two energy functions for the discrete and continuous cases. In the discrete version, energy E is expressed as:

(3.28)

while in the continuous version, E is defined as:

(3.29)

For very high values of λ, which are typical of Hopfield networks, the last term in Equation (3.29) is near zero and can be neglected. Equation (3.29) then becomes equivalent to Equation (3.28).

3.4.3 Network convergence

The successive updating of the network state is directed towards the search for a lower energy state. Eventually the network will converge at a point at which energy is a minimum. This may be a local or global minimum. The following paragraphs illustrate how the network is seeking a lower energy value during the successive changes of network state.

In the discrete version of the Hopfield model, each neurone outputs either 0 or 1. If there is a change of state for neurone i from time n to n+1, the difference in energy, denoted by ΔE, due to the ith neurone can be written as (refer to Equation (3.28)):

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Classification Methods for Remotely Sensed Data
Classification Methods for Remotely Sensed Data, Second Edition
ISBN: 1420090720
EAN: 2147483647
Year: 2001
Pages: 354

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