4.1 On data
In research on methodology to study spatial-temporal variability of atmospheric circulation, synthetic datasets play an essential role. The structure of such datasets is a very important factor since it constrains the results to a large extent. This seems to be of particular importance if one compares the two alternative perspectives of teleconnections, that is, the spatial patterns of temporal co-variability on one hand and the regime-based approach that sees teleconnections as typical anomaly patterns on the other.
In the two previous studies that directly compared the skill of SPCA and SOMs in extracting known patterns (modes) from synthetic datasets (Reusch et al. 2005; Liu et al. 2006), most of datasets were generated by repeating predefined patterns, which led to datasets consisting of a few distinct clusters. These clusters were represented well by SOMs in the latter study if there were at least as many SOM types as clusters; in the former study, considerably more types were necessary to avoid blending of clusters. SPCA, on the other hand, was in most cases unable to capture the spatial structure of the predefined modes, even though the data were strictly linear. Datasets generated in such a way seem beneficial to study methods and their ability to recognize such clusters; however, the application of SPCA on such datasets is inherently meaningless, since it is not designed to recognize clusters in realizations, linear or nonlinear, unlike TPCA (Compagnucci and Richman 2008). This, however, does not mean that SPCA is useless in the continuum perspective, since the principal components are typically projected on SOM arrays to discern the relevant types from those related to other teleconnections (e.g., Johnson et al. 2008; Rousi et al. 2020; Yuan et al. 2015).
Consequently, to obtain data that could be used to compare different perspectives on teleconnections and avoid biased results, we construct synthetic datasets in a way that can be understood as reversed SPCA—that is, constructing data from predefined orthonormal vectors (loadings) that carry the information on how grid points co-vary and times series of amplitudes randomly drawn from normal distributions that carry the variability itself. A resulting dataset forms a continuum of circulation fields rather than a few distinct clusters, which better approximates the structure of real circulation—see, e.g., Fig. 7 in Philipp et al. (2016). More importantly, datasets defined in such a way can still be successfully classified and do not limit the applicability of SOMs in any way, but they can also be linked to underlying modes of variability. Therefore, they make it possible to study both perspectives of teleconnections in parallel, in our case by means of visualizing both SOMs and the underlying modes of variability in one projection plane, for which the nonlinear Sammon mapping is utilized.
We did not add noise to the generated datasets as according to the results of Reusch et al. (2005) and Liu et al. (2006) we did not believe that doing so would be beneficial. Furthermore, our data is generated as white noise. Based on results in Part II of the study, where generated and real-world data are compared, we do not think that generating auto-correlated data would change results of the present study. Last, results by Liu et al. (2006; see Fig. 9 there) clearly show that linear methods blend those modes whose spatial patterns are correlated. Therefore, the three modes were defined such that they are mutually uncorrelated. We also defined the modes to be mutually equidistant, since one can hypothesize that SOMs may also tend to blend modes that are relatively close to one another, in terms of the distance metric.
All in all, even though our data are more complex than data used in previous similar studies, they are still relatively simplistic compared to real-world data. In Part II, additional datasets will be generated from modes extracted from reanalysis data to assess whether findings based on simplistic datasets hold in the real world.
4.2 On methodology
Beside the datasets, two other specific issues seem particularly important for our study due to its aim at methodological aspects. First, any results are constrained by the quality of SOMs, while the assessment of quality is to a large extent goal-specific and a result of choice of multiple parameters (e.g., Gibson et al. 2017). Previous studies provide only limited guidelines regarding these choices, and in many cases the parameters were selected randomly or not mentioned at all (Sheridan and Lee 2011). The second issue regards the application of Sammon mapping beyond a mere tool for validation, toward providing a visual link between data, modes and SOMs.
The choice of parameters was indicated in Fig. 2 and Sect. 2.2. The parameters were chosen experimentally based on an initial sensitivity study, with the exception of the algorithm type and grid topology, which were set to sequential and rectangular, respectively. Kohonen (2013) suggests that the batch algorithm be used instead of the sequential one, as it is faster and safer. However, it is not clear to what extent this recommendation relates to the typical utilization of SOMs in synoptic climatology consisting in defining relatively very small SOMs used for classification rather than for exploratory data projection. According to Jiang et al. (2015), the sequential algorithm was used by the majority of research in synoptic climatology.
The training was initialized using randomly selected data points. We did not choose the alternative initialization from leading two principal components to assure that the relative orientation of SOMs and predefined modes in the Sammon space was not biased by this step. Since random initialization was quite sensitive to initial conditions, 20 variants of SOM were calculated for each combination of the remaining parameters. Furthermore, the commonly used Euclidean distance was chosen as the metric of similarity, as other metrics provided by the package (e.g., Manhattan, sum-of-squares) did not lead to higher quality SOMs. Last, we kept the number of training iterations fairly low (1,000), and the training was done in a single phase. Increasing the number of iterations and/or splitting the training into two phases of “rough” and “fine” ordering, as suggested by Kohonen (2013), did not lead to qualitatively different SOMs, probably due to the combination of small arrays and small datasets.
The package provides two neighborhood functions: bubble and Gaussian. Given the limited options provided by the algorithm, we did not test other functions such as the Epanechikov function that was found superior by Liu et al. (2006), although Jiang et al. (2015) found no notable difference between Epanechikov and Gaussian functions for SOMs optimized for classification. Both available functions led to SOMs with good topology; however, each of them required a specific range of neighborhood radii. In general, the Gaussian function led to SOMs less sensitive to the radius, notably to its lower terminal value. According to Wehrens and Kruisselbrink (2018) the Gaussian neighborhood always adjusts all types to some extent, even if the terminal radius is close or equal to the threshold of 1.0. For the bubble neighborhood, iterations calculated at the threshold account only for the winning type, which turns the process into k-means (Kohonen 2001). Regardless the function, in our case such iterations typically led to a marked change in quantization (decrease) and topological (decrease) errors and to dissipation of SOM topology. Gibson et al (2017) suggest training two sets of SOMs that differ in the inclusion of “k-means” iterations, arguing that the two sets may complement each other, one providing good topology, the other good clustering. Since SOM topology itself was the subject of our study, SOMs were only allowed to include a “k-means” phase if no folding of SOMs was detected in the Sammon projection.
Note that radius less than 1.0 mentioned in the study has no physical meaning and is specific to the “Kohonen” R package; there may be differences in other available software (Gibson et al. 2017). It allows for specifying the fraction of iterations with no neighbor modification: for example, radius set to linearly decrease from 3 to 0 means that the last one third of training does not modify neighbors.
We experimentally found that the effect of the learning rate and the initial radius was relatively small, once obvious errors were removed. In a few cases, which were discarded, an insufficient initial radius led to poor initial ordering of types in larger arrays. Additionally, too large learning rate in later iterations broke the topology of SOMs. These findings are in line with other studies (e.g., Sheridan and Lee 2011; Gibson et al. 2017).
The last point regards the use of Sammon mapping. Strictly speaking, Sammon mapping adds a step into the analysis that could be considered redundant or even introducing new uncertainty due to the black-box-like character of the projection that utilizes the nonlinear relations among all data points to define the projection plane. However, the idealized rectangular array of a SOM limits the study of its topological structure to such an extent that the benefits of Sammon mapping outweigh the negatives. Furthermore, utilization of a data projection technique allows one to see individual data points instead of SOM type frequencies only, which makes it possible to show how the linear modes of variability project on data in a nonlinear continuum framework.
4.3 On using SOMs to study teleconnections
Given the high number of combinations retained for the analysis (13 SOMs for each of the 12 datasets), only a relatively small subset of results could be chosen for presentation that, nevertheless, well represents the range of results. While the number of SOM types (size of array) does not seem to play that important a role in SOM topology (contrary to its strong link to explained variation), the shape of the array (ratio of the number of columns and rows) as well as the structure of data are important factors.
The projection of two leading modes of variability on SOM array diagonals, a topology pattern suggested in earlier studies (e.g., Reusch et al. 2007), seems quite rare. An alignment of SOM diagonals with two leading modes—which results in mirrored patterns of positive and negative phases of modes in opposite corners—was detected only in square SOMs (in about 50% of cases). On the other hand, in about 60% of strongly elongated SOMs (2×4, 3×5, 4×6, and 5×7) another topology pattern was observed, in which the two leading modes are approximately aligned with the center row and column of the array, the diagonals then representing their combinations. Consequently, SOMs with an odd number of rows/columns show the spatial structure of modes somewhat better, especially those with small arrays. However, most of the remaining cases, and also most large SOMs and SOMs of datasets in which two (or all three) modes have approximately the same strength, cannot be described with either of the two idealized topology patterns, the projection and identification of modes becoming less straightforward.
The projection of the third-order mode on SOM arrays has so far received only little attention. Here, when the circular mode is weaker than the leading two modes, it projects strongly only on a few types close to the center of the array; but some curvature of isolines is apparent in most types. This is in line with how the mode projects on the Sammon map of data—fields relatively strongly correlated with the pattern of the mode may appear anywhere in the phase space, but in the center of the Sammon map such cases are more common, and refer to fields in which both leading modes are close to their neutral phase. As a result, such fields do not comprise marked overall anomalies and occur, therefore, close to the center of the Sammon space. A similar SOM structure can be seen in the results of Reusch et al. (2007) who analyzed North-Atlantic monthly mean SLP patterns (see Figs. 1 and 6 of their study). While the leading two PCs, which explained 39% and 13% of variability, projected relatively strongly on all the four corners of the SOM, the third-order PC (11%) did not have a clear representation within the 30-member SOM array. Despite similarities between the PC pattern and several types in the center of the array (e.g., type 16, Fig. 1), the two phases of this PC were not recognized as noteworthy regimes of North-Atlantic variability, owing to their overall weak projection on a SOM, at variance with the similar strength of the second- and third-order modes.
The identification of the second and third modes in SOM arrays becomes particularly challenging if their strength is identical. Several different topology patterns were found here, each pointing to a particular limitation of SOMs. First, one of the modes may project relatively weakly onto one or a few types (typically in the center of the SOM) as if its variance was smaller than that of the other mode—the mode thus being underrepresented. Second, parts of the SOM array not occupied by patterns similar to the leading mode (e.g., two opposite corners) may represent one (any) phase of one mode each, the remaining phases being underrepresented. Third, one part of a SOM array (e.g., upper half) may predominantly show one particular combination of phases (positive or negative) of the two modes, the other (e.g., lower) half showing a different combination of phases, but not necessarily opposite to the upper half for both modes, the remaining combinations of phases being underrepresented. Each of these three topology patterns—if forced—may have specific utility, except that the pattern itself is largely outside of researchers’ control due to the unsupervised nature of SOMs. In real data, the underrepresentation of specific combinations of phases of modes may be, but equally likely may not be, a consequence of these combinations occurring less frequently in the training dataset—here it was a result of only minor differences due to sampling. This illustrates the fact that one has to pay a considerable price for the topology preservation constraint of SOMs, relative to simple cluster analysis.
Last, it has to be stressed that in both the Euro-Atlantic and the Pacific-American regions at least four modes of variability are necessary to describe variations in winter circulation, summer circulation being even more complex (Barnston and Livezey 1987; Casado and Pastor 2012; Hynčica and Huth 2020). Therefore, selective and uncontrollable underrepresentation of important modes of variability and their interactions will be an even more important issue in SOMs of real data.
4.4 On using SOMs to study the nonlinearity of teleconnections
We clearly show that the topology of SOMs strongly depends on the structure of data. Even if the leading two modes of variability project along elements of a SOM array (such as its diagonals), marked irregularities can be identified in the spatial patterns of types if the data comprise more than the two modes of variability. One way these irregularities demonstrate themselves in the spatial patterns of SOM types is a presence of non-mirrored patterns in opposite types within a SOM array, for example in opposite corners. Reusch et al. (2007) pointed out that such non-mirrored patterns are a feature that distinguishes SOMs from linear analyses (such as PCA). However, one needs to remember that such apparent nonlinearities do not necessarily point to nonlinearity of the underlying modes of variability—non-mirrored patterns in (any) two opposite positions within a SOM array clearly prevail even if data are linear. In the case of our synthetic datasets, these differences may be due to sampling, or simply be artifacts of projecting multidimensional data onto a two-dimensional phase space. We showed that especially lower order modes can project on SOM arrays in various ways, which, nevertheless, all lead to some kind of selective underrepresentation of co-occurrences of some phases of modes. Our data can be understood bivariate or trivariate—as regardless of the number of grid points, or input variables, all variability can be linked to only two or three modes. The fact that nonlinearity appears so clearly in SOMs of such simplistic datasets suggests that it will occur in any complex (multimodal, noisy) real-world data regardless of how well such data may be approximated by linear models. The tendency of nonlinear methods to detect nonlinearity where there is none has already been documented (Christiansen 2005), and one should, therefore, not forget the strong predisposition of results by the character of the used research method.
4.5 Outlook
The topology of SOMs relative to underlying modes of variability seems to be rather sensitive to data sampling. Here, we focused only on the topology; however, the response of the frequency of occurrence of individual SOM types to changes in the underlying modes may also be of interest, especially regarding the ability of SOMs to capture inter-decadal changes in teleconnections and distinguishing them from sampling variability. This will be further analyzed in Part II of the study. Furthermore, the modes of variability predefined in the present study were highly idealized; generating synthetic datasets from real modes of variability—and potentially from a larger number of modes, adding noise to the generated data, and/or defining more complex nonlinear functions of data variability may represent other useful steps in studying the links between different perspectives of teleconnections. Last, other methods than SOMs might be utilized to study the continuum of teleconnections to assess whether they may improve on some of the limitations of SOMs discovered here.