1. Introduction
The West Florida Shelf (WFS) is a broad, gently sloping continental margin influenced by the Gulf of Mexico Loop Current system located seaward of the shelf break (Molinari et al. 1977; Huh et al. 1981; Paluszkiewicz et al. 1983; He and Weisberg 2003; Weisberg and He 2003) and by local wind and buoyancy forcing, including the fresh water of the Mississippi River generally found at midshelf in spring and summer (Gilbes et al. 1996; He and Weisberg 2002). The shelf circulation is dynamically linked to its varying water properties, and particularly to temperature, which exerts a primary control on density. The close relationship between the shelf water temperature variability and the variability of net surface heat flux and ocean circulation are reported in recent studies (He and Weisberg 2002, 2003; Weisberg and He 2003; Liu and Weisberg 2005a). Thus, a description of the characteristic patterns of SST variability adds to our understanding of the shelf circulation and its air–sea interactions (Weisberg et al. 2004).
a. Self-organizing map and its applications in meteorology and oceanography
Techniques for pattern detection in large oceanographic datasets are becoming increasingly important as datasets grow in size and complexity. The self-organizing map (SOM), an artificial neural network based on unsupervised learning, is an effective software tool of feature extraction (Kohonen 1982, 2001). It provides a nonlinear cluster analysis, mapping high-dimensional data onto a (usually) 2D output space while preserving the topological relationships between the input data. As a tool of pattern recognition and classification, the SOM analysis is in widespread use across a number of disciplines (Kaski et al. 1998; Oja et al. 2003). Since its first use in climate research by Hewitson and Crane (1994), extensive applications of the SOM technique have been found in the meteorological community (Malmgren and Winter 1999; Cavazos 1999, 2000; Ambroise et al. 2000; Cavazos et al. 2002; Hewitson and Crane 2002; Hsu et al. 2002; Hong et al. 2004). Recently, the SOM analysis has also been applied in oceanography. For example, Ainsworth (1999) and Ainsworth and Jones (1999) used this method to improve chlorophyll estimates from satellite data. Silulwane et al. (2001) and Richardson et al. (2002) used it to identify characteristic chlorophyll profiles in the ocean, and Hardman-Mountford et al. (2003) applied this method to altimeter data. Ultsch and Röske (2002) used it to predict sea level. The SOM technique was also used to extract SST and wind patterns from satellite data (Richardson et al. 2003; Risien et al. 2004), and to detect ocean current spatial patterns from moored velocity time series (Liu and Weisberg 2005b).
b. Growing hierarchical self-organizing map
Despite its wide applications, SOM analysis has inherent deficiencies. First, it uses a static network architecture w.r.t. the number and arrangement of neural nodes that have to be defined prior to the start of training. Second, hierarchical relations between the input data are difficult to detect in the map display. To address both issues within one framework, a neural network model of the growing hierarchical self-organizing map (GHSOM) was recently introduced (Dittenbach et al. 2002; Rauber et al. 2002; Dittenbach 2003; Pampalk et al. 2004). The GHSOM is composed of independent SOMs, each of which is allowed to grow in size during the training process until a quality criterion regarding data representation is met. This growth process is further continued to form a layered architecture such that hierarchical relations between input data are further detailed at lower layers of the neural network. To our knowledge, the GHSOM method has not yet been applied to meteorological or oceanographic research.
c. The SOM and GHSOM MATLAB toolboxes
Most of the above referenced SOM applications in meteorology and oceanography are based on a software package SOM_PAK 3.1 or earlier versions (Kohonen et al. 1995) written in C language. More recently the use of MATLAB, as a high-level programming language with graphics and visualization support, has facilitated an efficient, customizable SOM implementation (Vesanto et al. 2000). The SOM MATLAB Toolbox utilizes MATLAB structures, making it convenient to tailor the code for specific user needs. The SOM Toolbox version 2.0 can be downloaded from a Web site of the Helsinki University of Technology, Finland: http://www.cis.hut.fi/projects/somtoolbox/. The GHSOM Toolbox, developed jointly by the University of Aberdeen and Vienna University of Technology, can be downloaded at http://www.oefai.at/elias.pampalk/ghsom/.
d. Goal of this paper
A 5-yr set of daily SST composite maps on the WFS is analyzed using the SOM and the GHSOM Toolboxes. The purposes are twofold: 1) to demonstrate the usefulness of the GHSOM in feature extraction, and 2) to describe the characteristic SST patterns on the WFS and their temporal variations.
Since the GHSOM method is new to the meteorological and oceanographic community, a brief discussion on the philosophy behind the SOM and GHSOM techniques is given in section 2. The SST dataset is described in section 3. Applications of linear, EOF, and nonlinear, GHSOM methods are described in sections 4 and 5, respectively. The results are discussed and summarized in section 6.
2. The SOM and GHSOM
3. Data
The monthly mean SST patterns, computed over the entire 5-yr analysis period, show a seasonal variation (Fig. 5). An across-shelf SST gradient is found in all the winter months, but it is not obvious in the summer months. A spring cold tongue structure that is prominent in April and May is consistent with previous literature (e.g., Weisberg et al. 1996; He and Weisberg 2002). These SST features will be used for comparison with those derived from linear EOF and nonlinear GHSOM analyses.
4. EOF patterns of the SST
5. GHSOM mapping of the SST
In this section, the GHSOM is performed on the original SST data and the SST anomaly data, T̂(x, t) and T̃(x, t), respectively.
a. GHSOM analysis of the original SST data
The 5-year-long daily SST data are used as input to the GHSOM without any preconditioning. In the application of the GHSOM Toolbox, all the parameters are set to the default values except τ1 and τ2, the breadth- and depth-controlling parameters. Different (τ1, τ2) values are used to test the GHSOM performance (see Table 1). Generally, when smaller (τ1, τ2) values are chosen there are more nodes, that is, larger SOM arrays, in the output. A large SOM array identifies a large number of patterns and reveals more detailed structure within the data, whereas a small SOM array identifies fewer, more generalized patterns. We chose the case of (τ1 = 0.6, τ2 = 0.06) to analyze simply because the results have two layers and the SOM arrays are large enough to represent characteristic SST features and small enough to be visualized.
The layer-1 GHSOM enables a nonlinear classification of the 5-year-long daily SST on the WFS into four categories, as shown in the 2 × 2 SOM array in Fig. 7. Each unit explains a particular set of SST characteristics. Unit 1 reveals a typical low SST pattern (16°C < SST < 25°C) in which the isotherms are approximately aligned with the isobaths, with the coldest water centered around Florida Big Bend region and with the warmest water seaward of the shelf break associated with the Loop Current. Unit 4 reveals a high SST pattern (SST > 28°C), with no obvious horizontal temperature gradient. Both units 2 and 3 are transitional patterns between the units 1 and 4 extremes.
For each of the 5-year-long daily SST maps, a best-matching unit (BMU) can be identified. The BMU is defined to have the smallest weighted distance from the input data. Time series of the BMU (given by its number, 1–4) show obvious seasonal fluctuations (Fig. 8). Unit 1 is best matched in winter, while unit 4 is best matched in summer. Unit 2 is best matched in spring and early winter, and unit 3 in early summer and autumn. The cycle of units 1 → 2 → 3 → 4 → 3 → 2 → 1 therefore takes place in a year. To quantify the representation of each unit (1–4), the frequency of occurrence is computed by summing the hits of that unit and dividing by the total record length. The relative frequency of occurrence of each unit is shown in the upper-right corner of each map in Fig. 7. For example, units 1 and 4 represent 26.3% and 33.7% of all the SST data, respectively. A monthly climatology of the frequency of occurrence during the 5 years (Fig. 9) shows that the first pattern appears mostly in January–March, the second in April, November, and December, the third in May and October, and the fourth in June–September.
Not all units in the first layer grow to the same depth in the GHSOM hierarchy. Only units 1, 2, and 4 are further expanded in a second-layer map. The second-layer GHSOM grown from unit 1 (winter SST patterns) of the first-layer GHSOM (GHSOM 2–1) is shown in Fig. 10. Different features of the cold coastal water and the warm Loop Current water are classified into the 3 × 4 SOM array. Unit 2 of this 3 × 4 s layer has the highest frequency of occurrence (12.9%), showing a typical Loop Current feature seaward of the shelf break. Note that the relative frequency of occurrence here is referred to the total number of hits of unit 1 in the first layer only; that is, the frequency of occurrences for each of the sublayer SOMs sum to 100%. The second-layer GHSOM grown from unit 2 (spring and early winter SST patterns) of the first-layer GHSOM (GHSOM 2–2) is shown in Fig. 11. Specifically, the upper three units (1, 3, and 5) represent spring patterns, while the lower units (2, 4, and 6) represent early winter patterns, and this is evident in the time series of the BMUs (Fig. 12). The general sequence of the SST variation is units 1 → 3 → 5 for the spring evolution and units 6 → 4 → 2 for the early winter evolution. The spring cold tongue structure may be identified in units 3 and 5.
The second layer GHSOM grown from unit 4 (summer SST patterns) of the first-layer GHSOM (GHSOM 2–4, a 2 × 5 array) is shown in Fig. 13. The peak summer SST patterns are shown in the rhs of the SOM array (units 7∼10), while the early and late summer SST patterns are arranged in the lhs (units 1∼6). The evolution of the summer SST patterns from early to late summer stages in each year is illustrated in Fig. 14. The general characteristic of summer SST is uniformly high temperature. Thus, it is difficult to divide the coastal and the Loop Current waters based on summer SST.
b. GHSOM analysis of the SST anomaly with the temporal mean map removed
Here the SST anomaly T̂(x, t) is used as input to the GHSOM analysis. Similar to that in section 5a, a set of controlling parameters are used to run the GHSOM model, and when τ1 = 0.2 and τ2 = 0.02 the GHSOM has only one layer (24 units). This case is chosen to demonstrate the capability of the GHSOM to function as a basic SOM. A 4 × 6 SOM array of 5-year-long daily SST anomalies is shown in Fig. 15. Similar SST patterns are located adjacent to one another in the SOM mapping, while dissimilar patterns are at the opposite extremes of the SOM space. There is a continuum of change across the SOM array, with the typical summer and winter SST anomaly patterns at the lower-right and upper-left hand corners, respectively, and an annual cycle is obvious in the BMU time series (Fig. 16). For either peak winter (January ∼ February) or peak summer (July ∼ August) patterns, the SST anomaly is smaller on the ocean side and larger on the shelf with the largest SST anomaly near the coast of the Florida Big Bend. This result is the same as that from the first EOF mode (Fig. 6). The spring cold tongue structure may be identified only in unit 5, while the warm tongue structure may be identified in many units (10, 14, and 16).
c. GHSOM analysis of the SST anomaly with both the temporal mean map and spatial mean time series removed
Here the SST anomaly T̃(x, t) is used in the GHSOM model. By removing the time series of spatial mean SST, the strong seasonal variation is reduced, while the relative spatial structure is not altered; that is, the horizontal SST gradient is not changed. Similar to that in section 5a, a set of controlling parameters are used to run the GHSOM model. We present the results of τ1 = 0.8 and τ2 = 0.08 for the same reason as in section 5a.
The first layer GHSOM is still a 2 × 2 array representing four categories of the SST anomaly patterns on the WFS (Fig. 17). Unit 1 reveals a wide warm tongue structure on the shelf, mostly appearing in November as a fall transition (Fig. 18); in contrast, unit 4 reveals a spring cold tongue structure on the shelf (peaking in April). Unit 2 reveals a pattern with a coastal and deep ocean contrast in summer (from June to September). On the other hand, unit 3 shows the reverse of the unit 2 pattern in winter (from January to February). Units 1∼4 represent 16.3%, 43.6%, 24.4%, and 15.7% of the SST anomaly maps, respectively. Generally, the SST anomaly patterns revealed by the GHSOM may be compared with the first two mode EOFs. Units 2 and 3 resemble the two extremes of the first mode eigenvector with positive and negative weights, respectively; and units 1 and 4 may be ascribed to the second mode EOF with negative and positive weights, respectively. However, the amplitudes of the winter SST anomalies (unit 3) are larger than those of the summer SST anomalies (unit 2). Also, the shapes of the cold and warm tongues are different as shown in units 4 and 1, respectively. These asymmetric phenomena are not identified in the lowest mode EOF results. We note that all the four units may be further expanded in a subsequent layer to reveal more detailed structures, but these are not pursued here.
6. Discussion and summary
As a data analysis method, the EOF conveniently orders patterns of variability on the basis of variance. However, as a linear method, it may be a suboptimal way of spanning a data space if the system is nonlinear. The nonlinear SOM orders patterns of variability on the basis of topology rather than the variance. A major strength of the SOM is that the underlying patterns in a dataset can be visualized in the same form as the original data. Thus, if input data are SST images, then the outputs are SST patterns, not SST anomaly patterns. This is an advantage over the EOF, in which the temporal mean field is removed prior to the analysis. As the SOM output patterns resemble the input format, their qualitative interpretation may be easier than that from the EOF. Also, the SOM is not subject to the symmetry bias of a given EOF mode, such that the SOM patterns may be more realistic than the EOF patterns. As shown in section 5c, the asymmetric SST anomaly patterns of winter and summer and of the cold and warm tongues revealed by the SOM cannot be identified in the individual EOF patterns. Another advantage of the SOM analysis is that the algorithm is robust in handling missing data, without a priori estimation. Thus, the SOM method can be used to explore incomplete datasets. Moreover, the SOM can be used as a data interpolation technique, estimating missing data from input data that are similar (Hewitson and Crane 2002).
The major advantages of the GHSOM model over the standard SOM are the following. First, the overall training time is reduced since only a necessary number of units are developed to organize the data at a certain level of detail. Second, the GHSOM uncovers the hierarchical structure of the data, allowing the user to understand and analyze a large amount of data in an exploratory way. Each SOM array in the hierarchy explains a particular set of characteristics of the data. This makes the GHSOM analysis an excellent tool for feature extraction and classification. Third, the size of the SOM array does not have to be specified subjectively before hand; the GHSOM automatically expands in a three-dimensional structure.
Here we used the GHSOM method to extract characteristic patterns of SST variability on the WFS from a time series of daily SST maps that span the 5-year interval 1998∼2002. Four characteristic SST patterns are extracted in the first-layer GHSOM array: characteristic winter and summer patterns, and two transitional patterns. Three of these are further expanded in a second layer, yielding more pattern evolution details. The results show that a seasonal cycle dominates the SST variability on the shelf. Winter SST is characterized by cold water (16°C < SST < 25°C) with the isotherms aligned approximately along the isobaths and with the coldest water centered within the Florida Big Bend region and with the warmest water located seaward of shelf break in association with the Loop Current. In contrast, summer SST is characterized by horizontally uniform, warm water (SST > 28°C) making it difficult to discern shelf from Loop Current waters. The spring transition includes a midshelf cold tongue.
When the GHSOM analysis is performed on the SST anomaly data (with both the temporal mean map and the time series of spatial mean values removed), four characteristic SST anomaly patterns are also obtained in the first GHSOM layer, representing the SST anomaly patterns in the four seasons. The winter SST anomaly pattern shows the cooling effect of shoaling isobaths on shelf and the warming influence of advection by the Loop Current, while the summer pattern reveals the warming effect of shoaling isobaths on shelf relative to the Loop Current. The spring pattern shows a midshelf cold tongue, while the fall pattern shows a warm tongue on the shelf. These seasonal patterns, whether extracted from the original data or the anomaly fields, exhibit asymmetries that are not readily apparent in the complementary EOF analysis.
Acknowledgments
Support was provided by the Office of Naval Research under Grant N00014-98-1-0158 for observations and modeling of the west Florida continental shelf circulation and Grant N00014-02-1-0972 for the Southeast Atlantic Coastal Ocean Observing System.
REFERENCES
Ainsworth, E. J., 1999: Visualization of ocean colour and temperature from multispectral imagery captured by the Japanese ADEOS satellite. J. Visual, 2 , 195–204.
Ainsworth, E. J., and Jones S. F. , 1999: Radiance spectra classification from the ocean color and temperature scanner on ADEOS. IEEE Trans. Geosci. Remote Sens, 37 , 1645–1656.
Ambroise, C., Seze G. , Badran F. , and Thiria S. , 2000: Hierarchical clustering of self-organizing maps for cloud classification. Neurocomputing, 30 , 47–52.
Cavazos, T., 1999: Large-scale circulation anomalies conducive to extreme events and simulation of daily rainfall in northeastern Mexico and southeastern Texas. J. Climate, 12 , 1506–1523.
Cavazos, T., 2000: Using self-organizing maps to investigate extreme climate events: An application to wintertime precipitation in the Balkans. J. Climate, 13 , 1718–1732.
Cavazos, T., Comrie A. C. , and Liverman D. M. , 2002: Intraseasonal variability associated with wet monsoons in southeast Arizona. J. Climate, 15 , 2477–2490.
Chu, P. C., Lu S. H. , and Chen Y. , 1997a: Temporal and spatial variabilities of the South China Sea surface temperature anomaly. J. Geophys. Res, 102 , 20937–20955.
Chu, P. C., Tseng H. C. , Chang C. P. , and Chen J. M. , 1997b: South China Sea warm pool detected from the Navy's Master Oceanographic Observational Data Set (MOODS). J. Geophys. Res, 102 , 15761–15771.
Dittenbach, M., 2003: The growing hierarchical self-organizing map: Uncovering hierarchical structure in data. J. Austrian Soc. Artif. Intell, 22 , 3. 25–28.
Dittenbach, M., Rauber A. , and Merkl D. , 2002: Uncovering the hierarchical structure in data using the Growing Hierarchical Self-Organizing Map. Neurocomputing, 48 , 199–216.
Espinosa-Carreon, T. L., Strub P. T. , Beier E. , Ocampo-Torres F. , and Gaxiola-Castro G. , 2004: Seasonal and interannual variability of satellite-derived chlorophyll pigment, surface height, and temperature off Baja California. J. Geophys. Res, 109 .C03039, doi:10.1029/2003JC002105.
Gilbes, F., Tomas C. , Walsh J. J. , and Muller-Karger F. E. , 1996: An episodic chlorophyll plume on the West Florida Shelf. Cont. Shelf Res, 16 , 1201–1224.
Hardman-Mountford, N. J., Richardson A. J. , Boyer D. C. , Kreiner A. , and Boyer H. J. , 2003: Relating sardine recruitment in the Northern Benguela to satellite-derived sea surface height using a neural network pattern recognition approach. Progress in Oceanography, Vol. 59, Pergamon, 241–255.
He, R., and Weisberg R. H. , 2002: West Florida shelf circulation and temperature budget for the 1999 spring transition. Cont. Shelf Res, 22 , 719–748.
He, R., and Weisberg R. H. , 2003: West Florida shelf circulation and temperature budget for the 1998 fall transition. Cont. Shelf Res, 23 , 777–800.
He, R., Weisberg R. H. , Zhang H. , Muller-Karger F. , and Helber R. W. , 2003: A cloud-free, satellite-derived, sea surface temperature analysis for the West Florida Shelf. Geophys. Res. Lett, 30 .1811, doi:10.1029/2003GL017673.
Hewitson, B. C., and Crane R. G. , 1994: Neural Nets: Applications in Geography. Kluwer Academic, 208 pp.
Hewitson, B. C., and Crane R. G. , 2002: Self-organizing maps: Applications to synoptic climatology. Climate Res, 22 , 13–26.
Hong, Y., Hsu K. , Sorooshian S. , and Gao X. , 2004: Precipitation estimation from remotely sensed imagery using an artificial neural network cloud classification system. J. Appl. Meteor, 43 , 1834–1853.
Hotelling, H., 1933: Analysis of a complex of statistical variables into principal components. J. Educ. Psych, 24 , 417–441.
Hsu, K., Gupta H. V. , Gao X. , Sorooshian S. , and Imam B. , 2002: SOLO—An artificial neural network suitable for hydrologic modeling and analysis. Water Resour. Res, 38 .1302, doi:10. 1029/2001WR000795.
Huh, O. K., Wiseman W. J. , and Rouse L. J. , 1981: Intrusion of loop current waters onto the West Florida continental shelf. J. Geophys. Res, 86 , 4186–4192.
Kaski, S., Kangas J. , and Kohonen T. , 1998: Bibliography of Self-Organizing Map (SOM) papers: 1981–1997. Neural Comput. Surv, 1 , 102–350.
Kohonen, T., 1982: Self-organized information of topologically correct features maps. Biol. Cybernetics, 43 , 59–69.
Kohonen, T., 2001: Self-Organizing Maps. Springer Series in Information Sciences, Vol. 30, 3d ed., Springer-Verlag, 501 pp.
Kohonen, J., Hynninen J. , Kangas J. , and Laaksonen J. , 1995: SOM_PAK, the self-organizing map program, version 3.1. Helsinki University of Technology, Laboratory of Computer and Information Science, Finland, 27 pp. [Available online at http://www.cis.hut.fi/research/som_pak/.].
Lagerloef, G. S. E., and Bernstein R. L. , 1988: Empirical orthogonal function analysis of Advanced Very High Resolution Radiometer surface temperature patterns in Santa Barbara Channel. J. Geophys. Res, 93 , 6863–6873.
Liu, Y., and Weisberg R. H. , 2005a: Momentum balance diagnoses for the West Florida Shelf. Cont. Shelf Res, 25 , 2054–2074.
Liu, Y., and Weisberg R. H. , 2005b: Patterns of ocean current variability on the West Florida Shelf using the self-organizing map. J. Geophys. Res, 110 .C06003, doi:10.1029/2004JC002786.
Malmgren, B. A., and Winter A. , 1999: Climate zonation in Puerto Rico based on principal components analysis and an artificial neural network. J. Climate, 12 , 977–985.
Molinari, R. L., Baig S. , Behringer D. W. , Maul G. A. , and Legeckis R. , 1977: Winter intrusions of the loop current. Science, 198 , 505–506.
Oja, M., Kaski S. , and Kohonen T. , 2003: Bibliography of Self-Organizing Map (SOM) papers: 1998–2001 addendum. Neural Comput. Surv, 3 , 1–156.
Paluszkiewicz, T., Atkinson L. P. , Posmentier E. S. , and McClain C. R. , 1983: Observations of a loop current frontal eddy intrusion onto the West Florida Shelf. J. Geophys. Res, 88C , 9639–9651.
Pampalk, E., Widmer G. , and Chan A. , 2004: A new approach to hierarchical clustering and structuring of data with Self-Organizing Maps. Intell. Data Anal. J, 8 , 2. 131–149.
Rauber, A., Merkl D. , and Dittenbach M. , 2002: The Growing Hierarchical Self-Organizing Map: Exploratory analysis of high-dimensional data. IEEE Trans. Neural Networks, 13 , 1331–1341.
Richardson, A. J., Pfaff M. C. , Field J. G. , Silulwane N. F. , and Shillington F. A. , 2002: Identifying characteristic chlorophyll a profiles in the coastal domain using an artificial neural network. J. Plankton Res, 24 , 1289–1303.
Richardson, A. J., Risien C. , and Shillington F. A. , 2003: Using self-organizing maps to identify patterns in satellite imagery. Progress in Oceanography, Vol. 59, Pergamon, 223–239.
Richman, M. B., 1986: Rotation of principal components. J. Climatol, 6 , 293–335.
Risien, C. M., Reason C. J. C. , Shillington F. A. , and Chelton D. B. , 2004: Variability in satellite winds over the Benguela upwelling system during 1999–2000. J. Geophys. Res, 109C .C03010, doi:10.1029/2003JC001880.
Silulwane, N. F., Richardson A. J. , Shillington F. A. , and Mitchell-Innes B. A. , 2001: Identification and classification of vertical chlorophyll patterns in the Benguela upwelling system and Angola-Benguela Front using an artificial neural network. S. Afr. J. Mar. Sci, 23 , 37–51.
Ultsch, A., and Röske F. , 2002: Self-organizing feature maps predicting sea levels. Info. Sci, 144 , 91–125.
Vesanto, J., Himberg J. , Alhoniemi E. , and Parhankangas J. , 2000: SOM toolbox for Matlab 5. Helsinki University of Technology, Finland. [Available online at http://www.cis.hut.fi/projects/somtoolbox/.].
Weare, B. C., Navato A. R. , and Newell R. E. , 1976: Empirical orthogonal analysis of Pacific sea surface temperature. J. Phys. Oceanogr, 6 , 671–678.
Weisberg, R. H., and He R. , 2003: Local and deep-ocean forcing contributions to anomalous water properties on the West Florida Shelf. J. Geophys. Res, 108 .3184, doi:10.1029/2002JC001407.
Weisberg, R. H., Black B. , and Yang H. , 1996: Seasonal modulation of the west Florida shelf circulation. Geophys. Res. Lett, 23 , 2247–2250.
Weisberg, R. H., He R. , Kirkpatrick G. , Muller-Karger F. , and Walsh J. J. , 2004: Coastal ocean circulation influences on remotely sensed optical properties: A West Florida Shelf case study. Oceanography, 17 , 2. 68–75.
Illustration of how an SOM works. The data time series are rearranged in a 2D array such that the data at each time step are reshaped as a row vector. For each time step, the row vector is used to update the weight of the SOM via an unsupervised learning algorithm. This iterative process is called self-organizing. The outcome weight vectors of the SOM nodes are reshaped back into characteristic data patterns.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
An example of the hierarchical structure of the GHSOM. All of the four units in the first-layer SOM are expanded in the second layer. Only two units in one of the second-layer SOMs are further expanded in the third layer.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
A record-length mean SST map for the 5-yr period, 1998–2002, overlaid on the 20-, 50-, 100-, 200-, and 1000-m isobaths for the WFS analysis domain.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Time series of the spatial mean SST anomaly for this WFS analysis domain.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
SST monthly means on the WFS obtained by forming an average for each month over the 5-yr period 1998–2002.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Eigenfunctions and the associated temporal evolution functions for the first four EOF modes of the SST data. The percent of variance accounted for by each mode is indicated at the upper-right corner of each eigenfunction plot. The labels J, M, and S on the abscissa designate the first days of January, May, and September, respectively, here and in subsequent figures.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Layer 1 GHSOM (2 × 2) for the 5-yr-long daily SST data on the WFS. The frequency of occurrence of each pattern is also shown on each map.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Temporal evolution of the BMUs (labeled 1–4 on the ordinate) corresponding to the patterns in the layer 1 GHSOM of Fig. 7.
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Frequency of occurrence of the layer 1 GHSOM (Fig. 7) patterns as a function of month.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
GHSOM 2–1: The layer 2 SOM expanded from pattern 1 of the first-layer GHSOM. The relative frequency of occurrence of each pattern is shown in the upper-right corner of each map.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
GHSOM 2–2: The layer 2 SOM expanded from pattern 2 of the first-layer GHSOM. The relative frequency of occurrence of each pattern is shown in the upper-right corner of each map.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Temporal evolution of the BMUs for the GHSOM 2–2.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
GHSOM 2–4: The layer 2 SOM expanded from pattern 4 of the first-layer GHSOM. The relative frequency of occurrence of each pattern is shown in the upper-right corner of each map.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Temporal evolution of the BMUs for the GHSOM 2–4.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
A 4 × 6 SOM of the 5-yr-long daily SST anomalies. The data are preprocessed by removing the temporal mean map (Fig. 3). The relative frequency of occurrence of each pattern is shown in the upper-right corner of each map.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Temporal evolution of the BMUs for the 4 × 6 SOM in Fig. 15.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
The layer 1 GHSOM (2 × 2) of the 5-yr-long daily SST anomalies (τ1 = 0.8, τ2 = 0.08). The input data are preprocessed by removing both the temporal mean map and a time series of spatial mean values. The relative frequency of occurrence of each pattern is shown in the upper-right corner of each map.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Frequency of occurrence of the four characteristic maps in Fig. 17 as a function of month.
Citation: Journal of Atmospheric and Oceanic Technology 23, 2; 10.1175/JTECH1848.1
Total numbers of the SOM units in the GHSOM with different values of controlling parameters. It is the third case (shown in bold type) that is presented in this paper.
