APPLICATIONS IV: ALGAE POPULATION ESTIMATION

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Figure 13.1 Modular decomposition of the implemented system.

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A block diagram of the system is presented in Figure 13.1, showing both the training and runtime stages of the system. During the training stage, 7 unreduced datasets (one per alga species) of 11 conditional attributes each are obtained from water samples. The datasets are reduced with FRFS to obtain 7 datasets with (on average) 7 conditional attributes each. These are then provided to the classi er, which induces 7 models (one for each species of alga). During runtime, the water samples are analyzed to obtain only 7 (on average) of the original 11 conditional attributes, as per the reduct set chosen by FRFS. This simpli es, speeds up, and reduces the costs associated with the data-gathering stage. In running experimental simulations, these new 7-attribute datasets are used by classi ers to provide the system s user with estimations of the 7 algae populations.

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13.1.2 Predictors

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To facilitate a careful experiment-based analysis of the present work, ve predictors were used to estimate the algae populations [382]: standard linear regression, a backpropagation neural network (BPNN), M5Prime, Pace regression, and a support vector-based system called SMOreg. The following brie y introduces these methods with details omitted (readers can refer to the respective references given). The linear regression model [92] is applicable for numeric classi cation and prediction provided that the relationship between the input attributes and the

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EXPERIMENTATION

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output attribute is almost linear. The relation is then assumed to be a linear function of some parameters, the task being to estimate these parameters given training data. The estimates are often accomplished by the method of least squares, which consists of nding the values that minimize the sum of squares of the residuals. Once the parameters are established, the function can be used to estimate the output values for unseen data. BPNNs [36] consist of a network of nodes arranged in several layers - the input, hidden, and output layers. Input and output layers buffer the input/output for the model, respectively. The hidden layer(s) provide a means for representing input relations. The network is trained by repeatedly presenting it with (labeled) training data and backpropagating any resulting errors in classi cation through it, adjusting weights between nodes in the process. This weight modi cation is achieved via the gradient of error curve. M5Prime is a rational reconstruction of Quinlan s M5 model tree inducer [375]. While decision trees were designed for assigning nominal categories, this representation can be extended to numeric prediction by modifying the leaf nodes of the tree to contain a numeric value that is the average of all the dataset s values that the leaf applies to. Projection adjustment by contribution estimation (Pace) regression [376] is a recent approach to tting linear models, based on competing models. Pace regression improves on classical ordinary least squares regression by evaluating the effect of each variable and using a clustering analysis to improve the statistical basis for estimating their contribution to the overall regression. SMOreg is a sequential minimal optimization algorithm for training a support vector regression using polynomial or radial basis function kernels [272, 346]. It reduces support vector machine training down to a series of smaller quadratic programming subproblems that have an analytical solution. This has been shown to be very ef cient for prediction problems using linear support vector machines and/or sparse data sets.

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13.2 EXPERIMENTATION

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For each of the 7 algae datasets, 10-fold cross-validation [353] was used to estimate the predictor s performance. The experimental results are given as two types of graph: root mean squared error (RMSE) and mean absolute error (MAE). The mean absolute error is computed by summing the absolute difference between the actual and predicted target value for each instance and then taking the average. The root mean squared error is determined by summing the squared differences between actual and predicted values, and taking the square root of the average. Both quantities are given for each predictor over the 7 datasets.

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