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Traditionally, the classes in thematic maps have been treated as crisp sets, using classical set theory. In this formulation, map classes are assumed to be mutually exclusive and exhaustive. This approach limits the ability of thematic maps to represent the continuum of variation found in most landscapes. Substitution of fuzzy sets allows more exibility for treatment of map classes in the areas of accuracy assessment and area estimation. Accuracy assessment methods based on fuzzy sets allow consideration of the magnitude of errors and assessment of the frequency of ambiguity in map classes. An example of an accuracy assessment from a vegetation map of the Plumas National Forest illustrates the implementation of these methods. Area estimation based on fuzzy sets and using accuracy assessment data allows estimation of the area of classes as a function of levels of class membership. The fuzzy area estimation methods are an extension of previous methods presented by Card (1982) . One interesting result is that the sum of the areas of the classes in a map need not be unity. This approach allows a wider range of queries within a GIS.

Background: Malaria is the leading cause of morbidity and mortality in Ethiopia, accounting for over five million cases and thousands of deaths annually. The risks of morbidity and mortality associated with malaria are characterized by spatial and temporal variation across the country. This study examines the spatial and temporal patterns of malaria transmission at the local level and implements a risk mapping tool to aid in monitoring and disease control activities.

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During the last thirty years there has been much research effort in regional science devoted to modeling interactions over geographic space. Theoretical approaches for studying these phenomena have been modified considerably. This paper suggests a new modeling approach, based upon a general nested sigmoid neural network model. Its feasibility is illustrated in the context of modeling interregional telecommunication traffic in Austria, and its performance is evaluated in comparison with the classical regression approach of the gravity type. The application of this neural network approach may be viewed as a three-stage process. The first stage refers to the identification of an appropriate network from the family of two-layered feedforward networks with 3 input nodes, one layer of (sigmoidal) intermediate nodes and one (sigmoidal) output node (logistic activation function). There is no general procedure to address this problem. We solved this issue experimentally. The input-output dimensions have been chosen in order to make the comparison with the gravity model as close as possible. The second stage involves the estimation of the network parameters of the selected neural network model. This is performed via the adaptive setting of the network parameters (training, estimation) by means of the application of a least mean squared error goal and the error back propagating technique, a recursive learning procedure using a gradient search to minimize the error goal. Particular emphasis is laid on the sensitivity of the network performance to the choice of the initial network parameters, as well as on the problem of overfitting. The final stage of applying the neural network approach refers to the testing of the interregional teletraflic flows predicted. Prediction quality is analyzed by means of two performance measures, average relative variance and the coefficient of determination, as well as by the use of residual analysis. The analysis shows that the neural network model approach outperforms the classical regression approach to modeling telecommunication traffic in Austria.*The quality of this paper was substantially improved by the comments provided by three anonymous reviewers on the first submission of the paper; we gratefully appreciate their comments. The second author wishes to acknowledge the assistance and computational environment provided

-A new methodology for automatic mapping from Landsat Thematic Mapper (TM) and terrain data, based on the fuzzy ARTMAP neural network, is developed. System capabilities are tested on a challenging remote sensing classification problem, using spectral and terrain features for vegetation classification in the Cleveland National Forest. After training at the pixel level, system performance is tested at the stand level, using sites not seen during training. Results are compared to those of maximum likelihood classifiers, as well as back propagation neural networks and K Nearest Neighbor algorithms. ARTMAP dynamics are fast, stable, and scalable, overcoming common limitations of back propagation, which did not give satisfactory performance. Best results are obtained using a hybrid system based on a convex combination of fuzzy ARTMAP and maximum likelihood predictions. A prototype remote sensing example introduces each aspect of data processing and fuzzy ARTMAP classification. The example shows how the network automatically constructs a minimal number of recognition categories to meet accuracy criteria. A voting strategy improves prediction and assigns confidence estimates by training the system several times on different orderings of an input set.

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