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Data analysis techniques for the coastal zone
This article provides an introduction to several data analysis methods that are frequently used for the interpretation of morphological datasets.
Contents
General description
The aim of data analysis methods is generally to find a small number of shape functions or sinusoidal functions, or a small number of eigenvectors, that resolve with sufficient accuracy the spatial and temporal properties of the data. This data may relate to some of the forcings, like the waves, winds and currents, or to the bathymetry. An approximation of the data to about 80% to 85% may be sufficient for some applications and in such cases maybe 2 to 5 functions or eigenvectors may be chosen. However, it is generally preferably to be able to approximate the original data set with at least 90% (Gilmore and Lefranc, 2003)^{[1]}, especially when the objective is to find a set of variables embedded in the original dataset, as is the case for some chaotic techniques (described in more detailed below). Nonetheless, in coastal engineering it is common practice to approximate the data of interest with up to 5 functions or eigenvectors (see for example Rattan et al. 2005^{[2]}or Li et al., 2005^{[3]}) in order to simplify the analysis. Such methods are described in more detail below, following the reviews by Southgate et al. (2003) ^{[4]} and Larson et al. (2003)^{[5]}. Bulk statistics methods, discussed by Larson et al. (2003), are briefly summarized below. Then follows an analysis methods for beach level data. Finally some advanced linear and nonlinear data analysis methods are presented.
Bulk statistics methods
This method uses the statistical properties of a data time series (mean, range, variance, correlation, etc.) to characterize the behavior of a system. As such, the implementation of the method is very simple, and it has thus been extensively used in many fields, including coastal research. These methods have traditionally been applied to shortterm and longterm wave statistics, for instance. In shortterm wave analysis, a wave height may be analyzed directly, or after being decomposed in a sum of sinusoidal functions (that is, using a Fourier expansion), from which the moments of the data may be extracted. These methods allow, also, to calculate the properties of extreme events according to their probability of occurrence, and are thus very useful in coastal structure design (Larson et al., 2003^{[5]}). In relation to morphodynamics, statistical properties of the temporal and spatial evolution of different coastal features have been investigated, in particular as a preliminary step in studies when Principal Component Analyses are involved (discussed later).
Linear analysis of beach level data
The linear analysis of beach level data is demonstrated here using a set of beach profile measurements carried out at locations along the Lincolnshire coast (UK) by the National Rivers Authority (now the Environment Agency) and its predecessors between 1959 and 1991, as described in Sutherland et al. 2007^{[4]}. Locations backed by a seawall were chosen and a time series of beach levels at a set point in front of the seawall at Mablethorpe Convalescent Home are shown in Figure 1.
Use of trend line for prediction
Straight lines fitted to beach level time series give an indication of the rate of change of elevation and hence of erosion or accretion. The measured rates of change are often used to predict future beach levels by assuming that the bestfit rate from one period will be continued into the future. Alternatively, longterm shoreline change rates can be determined using linear regression on crossshore position versus time data.
Genz et al. (2007)^{[6]} reviewed methods of fitting trend lines, including using end point rates, the average of rates, ordinary least squares (including variations such as jackknifing, weighted least squares and reweighted least squares) and least absolute deviation (with and without weighting functions). Genz et al. recommended that weighted methods should be used if uncertainties are understood, but not otherwise. The ordinary least squares, reweighted least squares, jackknifing and least absolute deviation methods were preferred (with weighting, if appropriate). If the uncertainties are unknown or not quantified then the least absolute deviation methods is preferred.
The following question then arises: how useful is a bestfit linear trend as a predictor of future beach levels? In order to examine this, the thirty years of Lincolnshire data have been divided into sections: from 1960 to 1970, from 1970 to 1980, from 1980 to 1990 and from 1960 to 1990, for most of the stations. In each case a leastsquares bestfit straight line has been fitted to the data and the rates of change in elevation from the different periods are shown below:
 From 1960 to 1970 the rate of change was 17mm/year;
 From 1970 to 1980 the rate of change was 63mm/year;
 From 1980 to 1990 the rate of change was +47mm/year.
 From 1960 to 1990 the rate of change was 25mm/year.
The data above indicates that 10year averages provide little predictive capability for estimating the change in elevation for the next 10years, let alone for the planning horizon that might need to be considered for a coastal engineering scheme. Few of the 10year averages are close to the 30year average.
A prediction horizon is defined as the average length of time over which a prediction (here an extrapolated trend) produces a better level of prediction of future beach levels than a simple baseline prediction. Sutherland et al. (2007)^{[4]} devised a method of determining the prediction horizon for an extrapolated trend using the Brier Skill Score (Sutherland et al., 2004^{[7]}). Here the baseline prediction was that future beach levels would be the same as the average of the measured levels used to define the trend. A 10 year trend was found to have a prediction horizon of 4 years at Mablethorpe Convalescent Home (Fig. 2). Similar values have been found at other sites in Lincolnshire.
Gaussian distribution of residuals
The good news is that distribution of residual (i.e. detrended) beach levels seems to follow the common assumption that it is Gaussian, or normal distribution, as shown for the Mablethorpe data in Fig. 2.
Linear and nonlinear analysis of datasets
Wavelets
The wavelet technique is similar to a Fourier analysis approach, where the signal is approximated by some basis functions, which in wavelet analysis are simply wavelet functions. The drawback of Fourier analysis or the more general harmonic analysis, in which data are represented by a superposition of sinusoidal terms, is the assumption of cyclicity beyond the spatial or temporal range of the dataset. This assumption may be justified if dominant processes have a linear or weakly nonlinear character (see, for example, the article Stability processes), but in practice many morphological features and processes are influenced by highly nonlinear perturbations both in space (e.g., presence of geological sedimentary structures) and in time (e.g., occurrence of extreme storms). In this case no good representation of the dataset can be obtained with a limited number of sinusoidal functions. Wavelets, on the contrary, can represent highly nonlinear behavior and do not assume any cyclicity, as they are localized in space and in time (Burrus et al., 1998^{[8]}, see also [1]). Time resolution is achieved with wavelets by using a scalable modulated window that is shifted along the signal. Hence, generally a small number of wavelets is needed to reconstruct a function with sufficient accuracy. An important property of wavelets is that their mean is zero and their average squared norm is unity. A very wellknown example of a wavelet is the Mexican hat and the Morlet wavelet, see Fig. 3. These examples are examples of mother wavelets, which may be dilated and transformed to form the basis. The first wavelet function was developed by Haar (1910).^{[9]} Wavelets have traditionally been used in data analysis to increase the signaltonoise ratio, and also to compress the data to only a few wavelet functions.
Wavelets were first used in coastal morphodynamics by Sarah Little et al. (1993) ^{[3]} to analyze large scale (of the order of 100 to 1000 kms) bathymetric evolution offshore the Hawaian islands; the wavelets the authors adopted for this analysis were Daubechies wavelets, a family of discrete orthogonal wavelets introduced by I. Daubechies (1988)^{[10]}. Thanks to the wavelet scale analysis and application of a wavelet transform, the authors were able to discover a small, lowfrequency topographic feature of around 200 kms in length, whose details suggest it is a slowspreading rift. After this pioneering work, other topography identification investigations have followed (eg. Little et al., 1996^{[11]}). More recently, Li et al. (2005)^{[3]} analyzed nearshore beach profile variability in Duck, North Carolina (USA); the space scales in this case were, instead, of the order of 0.1 kms. The objective of the study was to analyze both time and space variability of the bathymetry. Thus, the authors chose Daubechies’ wavelets as a base and adopt an adapted maximum overlap discrete wavelet transform (AMODWT), as both are very suitable for decomposition of signals with strong space and time variations.
Li et al. (2005)^{[3]} studied in detail a bathymetry profile that has been thoroughly surveyed since 1981. They identified the variance across the profile as nonstationary, with largest variations in the sandbank region; this region occurs between 100 and 400500 m offshore. Within this region, the 128256 m spatial scale contained most of the information, and did make the largest contribution to the variance for all the months surveyed. The authors suggested this is because highenergy waves would affect the bathymetry from the surf zone to deep water, that is for distances of the order of 100 meters. However, why highenergy waves, rather than the more ubiquitous moderate wave conditions, should have a larger effect on the wavelet decomposition is unclear. It is worth noting that the largest variations of the 128m occurred in the sandbar region, indicating this is the region where the morphology evolves the most, which is to be expected. Contrary to the spatial scales, the temporal wavelets contributed differently to the total variance depending on the month considered and the position along the profile. However, it may be pointed out that the two temporal wavelets that span from 3264 and 64128 months, respectively, contain most of the variance. Contributions of lower order appeared as large peaks in the profiles, indicating they are mostly eventrelated, rather than part of the average trend. This is highlighted by the authors with several examples. This work proves wavelets are a useful technique in signal decomposition and have great potential in coastal research.
An example how a signal can be decomposed into different spectral bands using orthogonal wavelets was given by Różyński and Reeve (2005) ^{[12]}. Fig. 4, left panel, shows the time series of water level at a Baltic Sea coastal segment in Poland featuring a stormy event on 27th Oct. 2002, sampled at the rate of 0.5 Hz. Next, Fig. 4, right panel, presents the results of wavelet decomposition with the orthogonal Haar wavelet for frequency bands covering wind waves (4 – 8 s), swell (816 s) and infragravity waves (16256 s). We can clearly see the growth of wave height of all those components during the buildup of storm.
Consequently, Fig. 4, right panel, demonstrates residual spectral bands, which are roughly the same throughout the storm. The patterns belonging to each spectral band are called ‘details’ in wavelet terminology, whereas the pattern in Fig. 5 contains elements with periods above 4096 s i.e. the ultraslow varying trend, featuring the storm surge. This pattern is called ‘smooth representation’ in wavelet terminology. We can see that the storm was decomposed into spectrally disjoint, orthogonal patterns, allowing for their detailed, individual examination; in wavelet terminology the smooth representation and details together are all parts of multiresolution analysis of a time series. Jagged trajectories of the smooth representation and details describing lowperiod variability originate from the use of Haar wavelet; more advanced nearly asymmetric dbN (Daubechies^{[10]}) or nearly symmetric coifN wavelets (also constructed by I. Daubechies) do not produce such artefacts.
Principal Component Analysis (PCA)
Principal Component Analysis is a data analysis method that is intended to identify the most important patterns in large data sets, i.e. the patterns that represent the most important variance in the data. The PCA method has first been developed by Pearson (1901)^{[13]} and Hotelling (1933)^{[14]}.
We consider a dataset [math]\; h_{i,t}, \; i = 1, .., n; \; t = 1, .., n_t \;[/math] consisting of a number of [math]n_t[/math] successive observations of a set of [math]n[/math] variables. The most important patterns in the dataset are represented by a limited number of principal components, each representing a pattern of variation in the variables of a certain order of magnitude. The principal component of the first order represents the largest pattern of variation, the principal component of the second order represents the second largest pattern of variation after the first order variation is subtracted, and so on. The essential difference between PCA and data analysis methods such as decomposition according to Fourier or wavelet components is that in this case the principal components are not based on predetermined shape functions, but on shape functions that are derived from the data set itself. This allows the dataset to be rather well reproduced with only a small number of principal components; in practice, two or three principal components are often sufficient.
In PCA the shape functions are not continuous; they are formed by a set of normalized weight vectors [math]\; \bf e^{(p)} \;[/math] of order [math]\; p = 1,2, .. \;[/math] with each the same number of dimensions [math]n[/math] as the set of variables. The elements of these weight vectors are indicated by [math]\; e^{(p)}_i \;[/math] and have the property [math]\; \sum_{i = 1}^n [e^{(p)}_i]^2 = 1 \;[/math].
The dataset can be described by a matrix [math]\; \bf H_0 \;[/math] with elements [math]\; h_{i,t}  \overline h_i \;[/math], where [math]\; \overline h_i \;[/math] is the average over all [math]\; n_t \;[/math] observations. The principal component of order [math]\; p [/math] is represented by the [math]n_t[/math]dimensional vector [math]\; \bf h^{(p)} \;[/math] with elements
[math]\; h^{(p)}_t = \sum_{i = 1}^n \; (h_{i,t}  \overline{h_i}) \; e^{(p)}_i . \quad \quad (1)[/math]
The first order principal components [math]\; h^{(1)}_t [/math] describes the greatest pattern of variation in the data set, corresponding to the maximum of [math]\; \sum_{t=1}^{n_t} [h^{(1)}_t]^2 \;[/math]. The corresponding weight vector [math]\; \bf e^{(1)} [/math] can be determined by using the following result from mathematical theory: The weight vector [math]\; \bf e^{(1)} [/math] is given by the eigenvector with the highest eigenvalue of the matrix [math]\; \bf C_0 = H_0^T H_0 [/math] with elements
[math]\; c_{i,j}=\sum_{t = 1}^{n_t} (h_{i,t}\overline{h_i})(h_{j,t}  \overline{h_j}) . \quad \quad (2)[/math]
Routine mathematical procedures can be used to determine this eigenvector. The first order approximation to the original dataset is given by
[math]h^{(1)}_t = \sum_{i=1 }^n \; (h_{i,t}  \overline{h_i}) \; e_i^{(1)}. \quad \quad (3)[/math]
As previously stated, the secondorder principal component [math]\;\bf h^{(2)} [/math] can be determined by the same procedure after the firstorder principal component has been subtracted from the data set. A more detailed explanation of the mathematical background of the PCA data analysis methods is given in the Wikipedia article PCA.
Empirical Orthogonal Functions (EOF)
The application of Principal Component Analysis to the analysis of morphological datasets is generally termed EOF, Empirical Orthogonal Functions. EOF methods have been used with success to analyze nearshore beach topography, as will be described below. However, the technique may not be appropriate for studies of bar dynamics as eigenfunctions are fixed in space but bars, on the contrary, are wavelike patterns that travel in time. Extended EOFs and Complex Principal Component Analysis, both modifications of EOFs, do not have such shortcoming; however, they rely on timelagged data, and thus the data needs to be sampled at constant time intervals. This is not usual in coastal applications, as noted by Larson et al. (2003), but may be achieved via data interpolation.
Larson et al. (2003)^{[5]} cite three papers (Hayden et al. 1975 ^{[15]}, Winant et al., 1975 ^{[16]} and Aubrey et al., 1979^{[17]}) as pioneering applications of EOFs in coastal morphology, in particular for beach profile behavior; these researchers, as Larson et al. (2003) point out, observed the lower order EOF modes could be related to particular coastal features, i.e. the mean profile, bars and berms, and lowtide terraces to the first, second and third order modes, respectively. Therefore, these studies also constitute first attempts of coastal characterization via EOFs. More recently, the EOF method together with a moving window model were used by Wijnberg and Terwindt (1995) ^{[18]} to divide the Dutch coast into regions according to their characteristic patterns of behavior. They analyzed 115 kms of Dutch coast via 14 thousand near crossshore transects at generally 250 m longshore intervals. These regions vary from 5 to 42 kms in size, each characterized mainly by what the authors define as ‘secondary’ features, that is features diverging from the mean profile such as mounds or sandbars (this example and those mentioned below are such that the mean has not been removed from the data). The authors observed subdecadal shifts of shoreline positions and speculate this could be related to sandbar dynamics. Larson et al. (2003) applied the same technique of Wijnberg and Terwindt (1995) to nearshore topography in a Dutch and a German coastal area. For the Dutch coastal site, the modes were related to the coastal features, with similar results as in Aubrey (1979) except that third EOF was shifted 90 degrees in phase with respect to the second and was also related to the bar system. For the German site, the technique was applied to study beach nourishment effects on topography evolution at a beach resort that has suffered from severe erosion in the past (Dette and Newe, 1997).^{[19]} In this case the first EOF indicated an increase in mean elevation. Similarly to other EOF analysis at other sites infilled sites, rapid changes occur at the beginning and were then followed by gradual adjustment to an equilibrium. In general the process takes one year if beach nourishment is nearshore, or considerably longer if the beach nourishment is at the berm, as Larson et al. (2003) observed.
Mathematical procedure
Some key mathematical details of the EOF analysis method are presented below. In sum, it should be underlined that the EOF, as well as the SSA, EEOF and MSSA methods, are variants of the same PCA methodology, based on the covariance structure of a studied system.
In the EOF variant the system matrix is composed of covariances representing fluctuations of a studied parameter/quantity at two points of the domain at the same moments in time, ignoring the time lags between consecutive measurements. For example, when seabed is sampled [math]n_t[/math] times, the measurements usually consist of [math]n_y[/math] crossshore profiles with [math]n_x[/math] sampling points each of them. The resulting lag0 covariance matrix [math]\mathbf C_0[/math] with elements [math]c_{i,j}[/math] has [math]n*n[/math] terms, where [math]n = n_x *n_y [/math], such that:
[math]c_{i,j} = \sum_{t=1}^{n_t} \; (h_{i,t}  \overline{h_i})\;(h_{j,t}\overline{h_j}) , \quad \quad (4)[/math]
with [math]i,j = 1, .... , n[/math]. Note that this is identical to Eq. (2). For [math]i=j[/math] (main diagonal) the terms represent variances; overlined quantities represent average seabed positions at locations [math]i[/math] and [math]j[/math] respectively. Eigenvectors [math]\mathbf e^{(p)}[/math] of this matrix are scaled to unit length. When plotted in the real physical space (e.g. the studied seabed domain), the terms most largely departing from zero represent areas of high variability. They represent the spatial side of EOF decomposition. The matrix [math]\mathbf C_0[/math] has [math]n_t[/math] positive eigenvalues [math]\lambda_p[/math], so [math]n_t[/math] determines the rank of [math]\mathbf C_0[/math]. It is so, because, the number of measurements [math]n_t[/math] is usually much smaller than the number of spatial ponts [math]n [/math]. The eigenvalues sum up to the total signal variance. i.e. the sum of the terms along the main diagonal of [math]\mathbf C_0[/math]. They show the distribution of variance among eigenvectors, thus indicating their importance. From a practical point of view several key modes normally contain at least 95% of the total variance, so the most important features of a studied system can be identified.
The temporal side of EOF decomposition (principal components) can be obtained using the following sums:
[math]h^{(p)}_t=\sum_{i=1}^{n } \;( h_{i,t}  \overline{h_i}) \; e^{(p)}_i . \quad \quad (5)[/math]
[math] h^{(p)}_t [/math] here denotes the element of [math]p[/math]th time series, known as the [math]p[/math]th principal component [math]\mathbf h^{(p)}[/math], at [math]t[/math]th moment in time ([math]t=1, ...., n_t[/math]), associated with [math]p[/math]th eigenvector [math]\mathbf e^{(p)}[/math]. Very importantly, its variance is equal to [math]\lambda_p[/math]. In sum, each pair [math]\mathbf h^{(p)}[/math] and [math]\mathbf e^{(p)}[/math] provides information on spatiotemporal evolution of [math]p[/math]th EOF mode.
Singular Spectrum Analysis (SSA)
A particular modification of PCA, namely Singular spectrum analysis (SSA), has been used to identify chaotic properties of a system, that is, to determine the number (embedding dimension) of independent variables that are needed to describe the system, and the properties of the attractors in such system. SSA was extensively discussed by Southgate et al. (2003), and the main points raised by the authors are summarized here. Firstly, in the case of SSA the data matrix has in its columns not all the measured time series at all times, but the data at successive equitemporal lags, up to the maximum shift needed for a full system’s description. The number of columns of the data matrix defined as such is called the embedding dimension, d, and the SSA will not resolve periods longer than that corresponding to d. It is of interest to note that the SSA technique is used not only for chaotic characterization studies, but also for noise reduction, data detrending, oscillatory characterization, or forecasting. Example applications to coastal morphology, given by Southgate et al. (2003), relate to longterm shoreline evolution. However, in general this technique has not been applied to coastal research, but rather to climatology (e.g. Ghil et al., 2002 ^{[20]}).
EOF can be generalized to Extended EOF (EEOF) for data sets containing many spatial points and few time realizations. For data sets where the number of spatial points is less than the number of realizations SSA can be generalized to Multichannel SSA (MSSA). An example of application of the MSSA method was presented by Różyński (2005) ^{[21]}, who analyzed shoreline variations from 1983 until 1999, sampled monthly at 27 equally spanned transects, covering a stretch of 2,600 m of an open sea coastal segment in Poland, transects 2911 and 0310, see Fig. 6. The study revealed three important patterns representing shoreline standing waves. Upon locations of their nodes, the wavelengths of those standing waves could be directly evaluated. Next, the magnitudes of the variation of antinodes were used for the assessment of their amplitudes. Finally, the periods were established by determination of time needed by antinodes to evolve from maximum seaward to maximum landward position. The largest standing wave had the wavelength of 1,500 m with amplitudes ranging between 4 and 20 m about the mean shoreline position at a given transect and the corresponding period of more than 32 years, see Fig. 7. No firm explanation for the existence of that wave could be provided.
The 2nd wave had the wavelength in the range of 1,0001,400 m, amplitudes of 10 m and the period of 8 years, see Fig. 8. Later studies, Różyński (2010) ^{[22]} and Różyński (2015) ^{[23]}, identified a coupling of this standing wave to variations of the winter index of North Atlantic Oscillation (for Dec., Jan., Feb. and Mar.), which contains a significant component with the period of 8 years and was found to be controlling winter wave climates and water levels in the Baltic Sea to a considerable degree.
The 3rd standing wave was found to be less regular; the estimated wavelength fell between 1,400 – 1,600 m, the amplitudes varied from 10 m at one antinode to only 6 m at the other, see Fig. 9. The processes responsible for the presence of this wave could not be identified in the absence of records of longterm hydrodynamic background.
Mathematical procedure
In the SSA method the spatial dimension is reduced to a single point: [math]n_x n_y = 1[/math], but the main focus is on temporal evolution of a studied quantity at that point. For this purpose, the lagged covariance matrix is built:
[math]\mathbf{C} = \begin{bmatrix} c(0) & c(1) & c(2) & \cdots & c(M_L1) \\ c(1) & c(0) & c(1) & \cdots & c(M_L2) \\ \vdots & \vdots & \vdots & & \vdots \\ c(M_L1) & c(M_L2) & c(M_L3) & \cdots & c(0) \end{bmatrix} [/math]
with the terms:
[math]c(j)=\frac{1}{n_t  j} \sum_{i=1}^{n_tj} \; (h_i  \overline h)\;(h_{i+j}  \overline h) . \quad \quad (6)[/math]
The parameter [math]M_L[/math] is called either window length or embedding dimension and determines the maximum covariance lag; a practical rule of thumb suggests that [math]M_L \le n_t / 3[/math]. The matrix [math]\mathbf C[/math] is symmetrical and has positive eigenvalues; if one or more eigenvalues are zero then the signal contains a deterministic component, represented by a perfect sine function. Formally, we can compute the eigenvectors [math]\mathbf E^{(p)}[/math] and principal components [math] \mathbf{h^{(p)}}[/math] of this matrix. The latter are derived from the formula:
[math]h^{(p)}_i=\sum_{j=1}^{M_L} \; (h_{i+j1}  \overline h) \; E^{(p)}_j[/math] for [math]1 \le I \le n_tM_L +1 . \quad \quad (7)[/math]
Thus, we have to take [math]M_L[/math] elements of the original series [math]\mathbf h[/math] from [math]i[/math]th to [math](i+M_L)[/math]th element, compute their products with the corresponding elements of the eigenvectors [math]\mathbf E^{(p)}[/math] of the matrix [math]\mathbf C[/math] and sum these products to obtain [math]i[/math]th element of the [math]p[/math]th principal component. Hence, the principal components are time series of the length [math]n_t – M_L[/math]. Importantly, despite being orthogonal, the prinicipal components are not correlated only at lag zero. It originates from the fact that [math]M_L[/math] consecutive elements of the original series are needed to compute one term of every principal component, so the correlation structure of the original series must be imprinted in principal components. Moreover, there may be up to [math]M_L[/math] subsets of the original time series containing the specific element [math]h_{i+j}[/math], so there may be up to [math]M_L[/math] different ways of reconstructing this element with principal components:
[math]h_{i+j+1}=\sum_{p=1}^{M_L} \; (h^{(p)}_i  \overline h) \; E^{(p)}_j . \quad \quad (8)[/math]
Thus, using principal components we do not obtain unique expansion of the original series. However, uniqueness can be established when we calculate the mean values of all possible ways of reconstructing the original signal:
[math]h^{(p)}_i=\frac{1}{M_L} \sum_{j=1}^{M_L} \; (h^{(p)}_{ij+1}  \overline h) \; E^{(p)}_j \quad \quad (9)[/math]
for [math]M_L \le i \le n_t – M_L +1[/math] at the middle part of the signal,
[math] h^{(p)}_i =\frac{1}{i} \sum_{j=1}^{i} \; (h^{(p)}_{ij+1} – \overline h) \; E^{(p)}_j \quad \quad (10)[/math]
for [math]1 \le i \le M_L 1[/math] at the beginning of the signal,
[math] h^{(p)}_i =\frac{1}{n_t i +1} \sum_{j=i  n_t + M_L}^{M_L} \; (h^{(p)}_{ij+1} – \overline h) \; E^{(p)}_j \quad \quad (11)[/math]
for [math]n_tM_L +2\le i \le n_t [/math] at the end of the signal.
There are [math]M_L[/math] quantities [math]\mathbf h^{(p)}[/math], which are termed reconstructed components and provide a unique expansion of the original signal. They are additive, but not orthogonal, so their variances are not cumulative. Therefore, a researcher should investigate not only single reconstructed components, but also their subsets in search for plausible interpretation of signal constituents. Traditional time series analysis techniques, mostly the Fourier analysis are used for this purpose. Usually, the entire useful information is contained in a few reconstructed components, so the analysis is not as tedious as might be suspected.
Finally, both the EEOF and MSSA methods provide unique expansions of the studied signals in time and space as well. Both methods are identical and differences in terminology are mostly practical; the term EEOF is used when [math]n_x n_y \gg n_t[/math], whereas MSSA is referred to when [math]n_t \gt n_x n_y [/math]. The resulting block system matrix is presented below:
[math]\mathbf{T} = \begin{bmatrix} T_{1,1} & T_{1,2} & \cdots & T_{1, n_x n_y} \\ T_{2,1} & T_{2,2} & \cdots & T_{2, n_x n_y}) \\ \vdots & \vdots & \ddots & \vdots \\ T_{n_x n_y, 1} & T_{n_x n_y, 2} & \cdots & T_{n_x. n_y , n_x n_y} \end{bmatrix} [/math]
The main diagonal contains autocovariance matrices of all [math]n_x n_y[/math] signals involved, the remaining (block) terms represent crosscovariances among them. Formally, this matrix can be manipulated analogously to previous description, so that [math]M_L[/math] reconstructed components are obtained. However, we should keep in mind that their interpretation can be difficult, because of the number of spatial points considered. Therefore, caution is recommended when applying these advanced techniques; such analysis should be preceded by ordinary EOF/SSA studies, depending on the problem studied.
Principal Oscillation Patterns and PIP
In a Principal Oscillation Pattern (POP) analysis the data is analyzed using patterns based on approximate forms of dynamical equations so may be used to identify changing patterns, such as standing waves and migrating waves (Larson et al, 2003)^{[5]}. POP is a linearized form of the more general Principal Interaction Pattern (PIP) analysis. A POP analysis using the longterm Dutch JARKUS dataset of crossshore beach profiles (Jansen, 1997^{[24]}) showed that POP systematically lost 4% to 8% more data than an EOF analysis. The prediction method was optimised using 8 POPs as adding more POPS included more of the noise. Różyński and Jansen (2002)^{[25]} applied POP analysis to 4 beach profiles at Lubiatowo (Poland) and recommended that an EOF analysis be carried out first.
Neural Networks
Background
Hodgkin and Huxley (1952)^{[26]} performed pioneering experimental studies on current propagation along the giant axon of a squid, consequently developing the first detailed mathematical model of neuron dynamics. The model is the first to include multiple ion channels and synaptic processes, as well as realistic neural geometry (Gerstner and Kistler, 2002 ^{[27]}). Following this work, several researchers started to study neurons as an interconnected system, developing the theory of Neural Networks. The term “perceptron” was introduced by Rosenblatt (1961)^{[28]} during this time to refer to an artificial neuron, rather than a natural one. However, according to Kingston (2003) ^{[29]}, the origin of artificial neurons may be traced back to McCulloch and Pitts (1943)^{[30]}.
In Neural Networks, the propagation of the current along the main body of the cell and subsequent release of ions in the synapses constitutes a feedforward (or feedbackward) mechanism between perceptrons. This property has been extensively exploited in artificial Neural Network modeling. The release of ions and then retrieval at the synaptic end of the connected perceptron may be modeled as a sigmoidal function, as that shown inside the ellipse in Fig. 10. This sigmoidal function represents an excitation threshold density for the neuron population, that is an average population activity (other models have used a step function or a linear function to simulate the excitation threshold between perceptrons, but such representations have limited applicability). The number of inputs and the form of the transformation function may vary, and the precise form of the output may vary accordingly. Sigmoidal functions are important because they are nonlinear and continuous, but many other options are possible. Also, different inputs may be transformed using different functions, and may be transformed as well several times. The type of function, the number of functions used in each layer (or each transformation step), and the number of layers may be modified so that the perceptron may reproduce certain characteristics of the phenomenon under study. This can be to be able to reproduce a certain behavior of a group of cells, or simply to reproduce characteristics of some data that is being analyzed using a perceptron, regardless of the particular application. An introduction to the mathematical background is given in the Wikipedia article Artificial neural network.
Although Artificial Neural Networks models were originally developed to understand dynamics of brain cells, many applications exist now where artificial neural networks are used for analysis of data. Also, once the Neural Network parameters have been calibrated using the input variables, the model can then be used as a predictive tool. This is how Artificial Neural Networks have been applied to coastal morphodynamics.
Applications to coastal morphodynamics
Several applications to coastal morphodynamics using Neural Networks have been developed in the past. Here we will mention several examples described in Kingston (2003)^{[29]}, as well as a particular Neural Network model applied to the analysis of sandbank evolution. For the purposes of coastal modeling several input data may be considered. Of significance one may consider tidal measurements, shoreline evolution, bathymetry evolution, wave height, period and direction, to name but a few. Then the modeler needs to select the appropriate network parameters that reproduce most closely the observations. If the model needs many different layers, or different functions are each layer, then the processing power of the model is weak, but if the model is efficient with a small number of parameters then the processing power increases. It is also important to assess which input variables are the most relevant. Many applications relate to wave and tidal forecasting (Deo and Sridar Naidu, 1999^{[31]}and Tsai and Lee, 1999 ^{[32]}).
Concerning the sea floor evolution, several studies have concentrated on sandbank evolution, for instance to predict location and motion of sandbanks from video images. Fig. 11 is an example of video imaging at Egmond aan Zee, The Netherlands. A Neural Network technique is used to correct the quality of the image. Another application of Neural Networks concerns the dynamics of the sandbanks, that is, how they evolve with time. An example of this type of application is in Rattan, Ruessink and Hsieh (2005)^{[2]} used three different data analysis methods to study the behavior of three sandbank systems, one in Egmond aan Zee, a system in Hasaki, Japan, and the sandbank behavior in Duck Site, North Carolina. The three methods were based on Principal Component Analysis, but the most complex method included a Neural Network technique. The properties of each system were characterized according to how ell each of the methods reproduced the observed behavior. If, say, a simple linear model produced the results with the same accuracy as a nonlinear method, then the system was said to behave linearly. This method of system characterization has only started to be implemented and the conclusions it leads to are dependent on the properties of the methods themselves. Also, how closely these methods reproduce the actual behavior is unclear.
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tag; name "L" defined multiple times with different content  ↑ ^{4.0} ^{4.1} ^{4.2} Southgate, H. N., Wijnberg, K. M., Larson, M., Capobianco, M. and Jansen, H. 2003. Analysis of field data of coastal morphological evolution over yearly and decadal timescales. Part 2: Nonlinear techniques. Journal of Coastal Research 19: 776789. Cite error: Invalid
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tag; name "S" defined multiple times with different content  ↑ ^{5.0} ^{5.1} ^{5.2} ^{5.3} Larson, M., Capobianco, M., Jansen, H., Rozynski, G. N., Stive, M., Wijnberg, K. M. and Hulscher, S. 2003. Analysis and modeling of field data on coastal morphological evolution over yearly and decadal time scales. Part 1: Background and linear techniques. Journal of Coastal Research 19: 760775
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