An artificial neural network (ANN) is an interconnected group of simple processing elements that can reproduce complex patterns in data. The basic model function, fx),

Figure 5 An example of a simple artificial neural network. The predictor x is transformed into a three-dimensional vector by the set of functions h, which is then transformed into a two-dimensional vector by the set of functions g, which is finally transformed into function f. In this case, the components of individual layers are independent of each other (e.g., the two components of g are independent given their input h). This enables parallel implementation of network computations.

Figure 5 An example of a simple artificial neural network. The predictor x is transformed into a three-dimensional vector by the set of functions h, which is then transformed into a two-dimensional vector by the set of functions g, which is finally transformed into function f. In this case, the components of individual layers are independent of each other (e.g., the two components of g are independent given their input h). This enables parallel implementation of network computations.

that we have previously used to relate the response variable y to the predictor x is now defined to consist of a number of other functions, g(x), which can further be defined in terms of additional functions, h(x). This can be graphically represented as a network structure (Figure 5), with arrows depicting the relations between functions (alternatively, between latent variables with values determined by those functions).

Networks such as that shown in Figure 5 are referred to as 'feedforward networks', because they consist of a directed acyclic graph. Networks with cycles are called 'recurrent networks'. Generally, such apparent cycles actually represent a temporal sequence in which the function takes on a different value or form at each point in time.

The advantage of ANNs is that they can be used to generate a functional model that can produce nearly any possible behavior. Further, this process can be automated so that the model 'learns' the appropriate functions from data. Usually, a process called 'supervised learning' is employed in which example pairs of x and y are used to choose functions from among an allowed class that lead to predictions which best match the observations. This process is called 'training' and occurs by minimizing a specified error function, which is often taken to be the same sum of squared errors that is used in regression analysis. In 'unsupervised learning', the error function is selected by the algorithm itself according to the data type and problem setting.

With appropriate selection of an error function, one can use formal statistical methods to estimate the predictive uncertainty of a trained neural network model. For example, if the sum of squared errors was employed in the training process, the mean squared error that results from applying the model to a separate validation data set can be used as a reasonable estimate of predictive variance.

A major disadvantage of ANNs for application to ecological modeling is that they must be viewed as

'black box' models. That is, the combination of functions that is produced by the model is often so complex that interpretation in terms of ecologically meaningful processes is usually impossible.

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