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In this chapter we deal with applications to situations different from the ones analyzed in the previous chapters in order to show the wide range of applications of the artificial neural network (ANN). Therefore, we apply the methods discussed in the previous chapters to the case of buoys moored off the the California coast, the postprocessing of temperatures for a certain gauge station in the south of Italy and to forecasting precipitation. In the first section we describe the structure of the network and of the optimal error chosen for the data reconstruction of the California coast. In the second section we show the application of this NN algorithm, proposed in Chapter 4 and applied in Chapter 7, and how it is able to model the SWH measured by the NOAA (National Oceanic and Atmospheric Association) National Data Buoy Center (NDBC-USA) near the California coast. In this application the NN uses as input the data of either one or more correlated data series from buoys located near San Francisco. The reconstruction of this time series is proposed by using different input data. [ABSTRACT FROM AUTHOR]

In this chapter we discuss how we selected a model for describing the probability distribution of the extreme events of sea waves and how the model has been made to suit the structure of the data. We also discuss the model's consequences, such as the evaluation of an m-year return level. Also, we check whether the time series obtained by means of artificial neural network (ANN) reconstruction has the same extreme events probability distribution as the original one. We further discuss the definition of a special ANN method dedicated to the reconstruction of single selected extreme or large events. [ABSTRACT FROM AUTHOR]

In this chapter we describe other algorithms as a possible alternative to artificial neural network (ANN) method for solving the reconstruction problem. Many algorithms, unlike ANN and simply NN, have been used for solving analogous problems. We selected two algorithms: the approximation operators which are a different version of ANN, already studied and explained in detail in Chapter 5, and the classical autoregressive integrated moving average (ARIMA) models widely used in the framework of time-series analysis. We apply both of them to our problem and we show with some examples that the ANN models have a much better performance. [ABSTRACT FROM AUTHOR]

In this chapter we will show the application of the algorithms and methods explained in Chapter 4 to the time series of sea level (SL) and sea wave height (SWH) measurements. As specified in Chapter 2, SL is the height of the tide, and SWH is the significant wave height. The phenomenologies of the two time series are different and each has its own problems. [ABSTRACT FROM AUTHOR]

The statistics of waves is important in understanding the forces acting on the sea shore and for determining its evolution. Interaction among waves and winds is crucial for wave motion. Knowledge of the probability of occurrence of extreme events is necessary for designing secure structures in the sea environment. Extreme-value theory provides powerful tools to evaluate the probability of extreme events. In this chapter our aim is to collect several contributions to the theory of extreme events in order to make a self-contained exposition. We present a selection of the papers which seem best suited to our procedures, aims and tastes ([55], [56], [16], [19], [8], [29], [17]). Theorems are outlined without giving the proofs, which can be found in the quoted literature; we prefer to underline their importance for operations on the data. The chapter is divided in two parts. The first describes the method for deriving the distribution of the maxima in the case of independent random variables from the statistics of the exceedances of the time series over a certain threshold. This method is called POT (peak over threshold) and will be used in Chapter 9 to show the results for sea measurements. Section 6.1 also gives the fundamentals of the theory. The hypothesis of independent random variables is very restrictive and obliges the researcher to extract subsequences of i.i.d. variables from stationary processes, getting too few data in the case of long-range correlations. In the second part we deal with theorems and useful results for weakly dependent data. There is an introduction to each section with exact mathematical statements where the ideas are explained in simple and more intuitive terms. [ABSTRACT FROM AUTHOR]

In this chapter we discuss and show some results for the use of the neural network (NN) as a complete set of functions. The fact that the combination of the sigmoidal function corresponding to an NN can approximate any function is a simple consequence of the Stone-Weierstrass theorem and so such an approach is a convincing one. Furthermore, in the case of approximation theory the synaptic weights are given by some a priori estimates and in many cases could be directly evaluated from the data. This approach has, as a drawback, more errors than the NN constructed using the procedures described in the previous chapter. [ABSTRACT FROM AUTHOR]

In this chapter we explain some fundamental concepts used in neural networks (NNs) with special regard to the ones applied to forecasts and data analysis. The structure of the back-propagation NN is shown in connection with its use for time-series analysis. The concepts of learning and training errors are explained in some detail and the main types of learning algorithms are exhibited. We also give some exact estimates of the probability that the learning error differs from the training error by more than a small quantity and a priori bounds of the learning error using extreme-value theory for the simple perceptron. Since NNs are also analyzed as a universal approximation of functions, we expose some useful properties of NNs treated in this way in Chapter 5. These topics do not cover all the literature dealing with NN application and theory, but they are good examples of the present situation. Heuristic remarks and rigorously proven statements are also distinguished using mathematical language for exactly proven facts and ordinary language for conjectures and hypotheses. This type of presentation suits the general aim of our book: we want to start from the very beginning of the theory to arrive at the very end of some applications of NNs. In the literature, especially in the applications, there is often confusion between proven facts and conjectures. [ABSTRACT FROM AUTHOR]

In this chapter we explain the main ideas and open problems of the model of wind waves. It is important to include this presentation in our book since we will compare the results of neural network (NN) reconstruction with those of the wave amplitude model (WAM) model. This comparison is done to check the order of magnitude of the significant wave height (SWH) reconstructed by means of the NN. Moreover, an understanding of this chapter is useful to obtain a good comprehension of the sea phenomena such as the sea evolution, the interaction among the sea and the wind and the formulation of the problem of dissipation forces. [ABSTRACT FROM AUTHOR]

The characteristics of wave motion are important for understanding the evolution of a coast. In the first three sections of this chapter we give the definitions of the quantities describing waves and a description of the current instruments and methodologies for their measurement. We describe the network of buoys used for attaining the significant wave height (SWH) time series analyzed in this book. We use a similar approach for tides: some of the theory and the corresponding network of buoys are described. [ABSTRACT FROM AUTHOR]

There are some important reasons for writing this book about neural networks (NNs) even though many books and articles on this topic have already been published. One reason is that we deal with the application of NNs as an algorithm for data analysis in the field of sea phenomena. This is quite a new field of application for the NN; few papers have yet been published using this approach. NNs have a double significance: one is that they provide very flexible and adaptable algorithms to handle data analysis and the other is that some versions of these algorithms can be used to describe the behavior of real neurons. The first meaning of NN is the one which is dealt with in this book. We do not deal with all the possible applications to the complex world of time series, which is impossible to do in the space of one book (it would require an entire encyclopedia), but only with the applications of NNs to time series of sea levels and other related important variables. [ABSTRACT FROM AUTHOR]

We have analyzed and explained many topics in this book. They differ very much but are all connected with the sea events and the algorithm selected for developing the analysis. The choice among the different algorithms has not been simple; we think that we have solved it in the optimal way, according to our taste and interests. The first principle used for collecting the various chapters has been to bring together all the theoretical and experimental facts concerning the sea time series and the phenomenology of the waves. Thus, the first three chapters have been devoted to an exposition of the main phenomenology (Chapters 1, 2) of the sea events as well as historical information. The measuring techniques are discussed and displayed in some detail so that the reader can feel the complexity of the process of collecting the data and also the practical and historical motivations which led to the development of such instruments. The waves and tides are distinguished and analyzed in detail. Chapter 3 covers the theoretical model currently used for forecasting sea waves. We discussed briefly the problems of the construction of the WAM which is used in Europe for the prediction of waves. We emphasize the fact that the actual derivation of this model disregards the free boundary nature of the problem. In Chapter 7 we compared the results of NN reconstruction with the output of the astronomical model for a check of the results and with the output of the WAM to check the order of magnitudes of the NN outputs. Chapters 4 and 5 give the main aspects and fundamentals of the theory and practice of NNs. [ABSTRACT FROM AUTHOR]