2.1 Synthetic Data

In the study, multiple synthetic data sets with predefined—and thus known—structure are used to train SOMs. The data sets are composed of circulation fields, which are linear combinations of three modes of circulation variability that describe variations in zonal (Z), meridional (M), and circular (C) flow. The three modes are defined as mutually orthogonal and of unit length to ensure that the contribution of individual modes to the total variability can be fully controlled by generated indices (amplitudes). To avoid calculation with very small numbers and to improve the readability of the resulting generated fields, all three patterns are scaled up by the same factor, the size of which is not relevant, but explains the larger isoline intervals in the spatial patterns of modes (Figure 1). The patterns can be interpreted as anomalies of an unspecified circulation variable that represent the modes in their positive phase and intensity of +1.0.

Each data set comprises 1,000 circulation fields; each field consisting of 20 × 20 grid points is a linear combination of the three patterns, whereby each mode is multiplied by a scalar index. The index therefore represents the intensity (amplitude) of a particular mode at a particular time instant. For each mode, the indices are drawn from normal distributions, the mean of which is zero and variance is a predefined fraction (explained variance hereafter) of the total variance σ² = 1. A total of 12 combinations of explained variances (EV), and thus data sets, are predefined (Table 1) such that . For example, in the data set “Z70-M20-C10,” 70% of the total variability is in the zonal flow, while the contribution of the meridional and circular flow is 20% and 10%, respectively. We also define several simple “bivariate” data sets in which the third mode is “turned off” (e.g., Z70-M30). Data sets in which other than the zonal mode was the strongest (e.g., Z30-M70) were not included in the study as they did not lead to any new findings. Note that the whole process can be understood as reverse PCA whereby a circulation data set is reconstructed from PCs by a linear superposition of orthonormal eigenvectors of a correlation matrix (i.e., PC loadings, or spatial patterns of modes) multiplied by time series of amplitudes (or, PC scores).

2.2 Training and Assessment of Self-Organizing Maps

Here, we present the methodology used for training, selecting and evaluating SOMs. A schematic diagram illustrating the methodology is shown in Figure 2. We discuss the choices of parameters and their effects on the results in Section 4.2. For a more detailed description of SOMs, we refer the reader to Kohonen (2001) and Sheridan and Lee (2011). Note that we use the term type instead of node in the paper to avoid any confusion with the term mode. We use the same term also when referring to SOM patterns to avoid any confusion with the patterns of modes.

The training of a SOM is an unsupervised and iterative process that does not require intervention from the researcher. However, several parameters affect the SOM's properties, some of which require subjective decisions to be made beforehand. To initialize the SOM array, n starting vectors (in our case, generated circulation fields) are randomly selected from a data set; n is also the resultant number of types. Subsequently, the fields are projected onto the array in multiple iterations, and the position of vectors that are close (in terms of Euclidean distance) to the field is adjusted. The adjustment depends on three parameters that have to be specified in advance. First, the neighborhood radius (referred to as radius for short) determines the size of the affected area around the projected data point that is adjusted. Second, the neighborhood function defines the adjustment method, and third, the learning rate specifies how much the vectors within the affected area are attracted to the projected field. The Gaussian neighborhood function was chosen over the commonly used bubble function, as it produced SOMs with somewhat better topology. The remaining two parameters decrease in value in each iteration step, allowing the overall topological structure of the data set to be captured first before fine-tuning the resulting classification over later iteration steps. Two more parameters relate to the SOM architecture: the algorithm type (sequential vs. batch) and form of array (rectangular vs. hexagonal). Kohonen (2001) suggests that batch algorithm and hexagonal array be used, but most applications of SOMs in climatology appear to rely on the other two options. To make our study comparable, we respect the common practice in the field with our selection.

The “Kohonen” R package (Wehrens & Buydens, 2007; Wehrens & Kruisselbrink, 2018) was used to train SOMs. For each of the 12 data sets, SOMs of 13 different shapes (number of rows × number of columns) of arrays were calculated: 2 × 3, 2 × 4, 3 × 3, 3 × 4, 3 × 5, 4 × 4, 4 × 5, 4 × 6, 5 × 5, 5 × 6, 5 × 7, 6 × 6, and 6 × 7, the resulting classifications ranging from six (2 × 3 SOMs) to forty-two (6 × 7) types. The number of iterations was set to 1,000 as a higher value did not lead to better SOMs, likely due to the relatively simple structure of data. The initial value of the learning rate was set to between 0.04 (smallest SOMs) and 0.12 (largest SOMs) and decrease linearly to 0.01 over the 1,000 iterations. The radius parameter was initialized as the 66th percentile of type-to-type Euclidean distances, calculated from coordinates (i.e., row and column) of types within the SOM array, and was set to linearly decrease to a specific terminal radius. To account for the sensitivity of SOMs to the terminal radius, described by Jiang et al. (2015) and Gibson et al. (2017) and also apparent in our preliminary analyses, 16 values of this parameter (0.0–1.5 at a step of 0.1) were tested for SOMs up to 4 × 4 types and 10 values (0.9–1.8 at a step of 0.1) for larger SOMs, which are computationally more expensive. Note that radius less than 1.0 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 instance, radius set to linearly decrease from 3 to 0 means that the last third of iterations do not modify neighbors. Lastly, to account for the sensitivity of SOMs to random initialization, 20 variants of SOMs were trained for each data set—SOM shape—terminal radius combination, resulting in a total of 37,440 SOMs.

Two kinds of quality metrics are typically used to assess SOMs—quantization error and topological error (Kohonen, 2001). While the quantization error shows the degree of data separation into classes and can be therefore understood as a measure of quality of a classification, the topological error expresses the degree to which the organization of types within the rectangular SOM array captures the structure of high-dimensional classified data. Here, quantization error is calculated as the Euclidean distance between each field and its closest type, averaged over all 1,000 classified fields. The topological error represents the percentage of fields whose two closest types do not lie next to each other in the SOM array. From the trained SOMs, one SOM was chosen for each data set and SOM shape by selecting the SOM with the smallest quantization error among the SOMs in whose projections onto the Sammon space folding of arrays did not occur. In case multiple SOMs met these conditions, the one with the smallest topological error was selected. A total of 156 SOMs were retained for the study.

2.3 Sammon Mapping

Sammon mapping is a non-linear dimension reduction method that maps multivariate data points (here, 400-dimensional fields) to lower- (here, two-) dimensional space such that the data structure is approximately preserved; that is, inter-point distances (typically, Euclidean) in the original and the projection Sammon space correspond (Sammon, 1969). This is achieved by iteratively looking for a solution that minimizes the error function

(1)

where denotes the Euclidean distance between ith and jth fields in the original space and d_ij denotes their Euclidean distance in the projection space.

Sammon maps were trained using the function “sammon” of the R package MASS (Venables & Ripley, 2002). To find a solution close to the global optimum, the function utilizes classical multidimensional scaling (principal coordinates analysis) to obtain the initial configuration of the Sammon map. A reader is referred to Lee and Verleysen (2007, p. 258) for more information on initialization and optimization of Sammon mapping and Stryhal and Plavcová (2023) for an up-to-date review of its applications in climatology.

The purpose of Sammon mapping in the study is twofold. First, it is used as an additional quality measure to assess the topological structure of trained SOMs. In Figure 3, Sammon map of one data set is shown together with one SOM and its projection onto the Sammon map. Note that in the Sammon map in Figure 3a, relative to the orthogonal array of types in Figure 3b, the array is slightly deformed such that the distance between individual SOM types approximates the Euclidean distance between the spatial patterns of the types in the original space. A SOM whose Sammon projection is significantly deformed (e.g., when folding of arrays occurs and type-to-type links cross each other) does not capture data topology.

Second, we utilize Sammon mapping as a means of visualizing relationships between the generated data sets, representation of these data sets by SOMs and the underlying modes. The following approach is used to project SOMs and modes on Sammon maps: The projection of a SOM pattern p onto a Sammon map is obtained by finding the point in the Sammon space that minimizes the error F:

(2)

where (d_ip) denotes the Euclidean distance between the ith circulation field and type p in the original (projection) space. Note that F is almost identical to E, except that it does not integrate over all pairs of data points. Equation 2 is also used to project the modes onto Sammon maps. However, since the modes are not defined as anomaly patterns but rather as functions from which anomaly patterns are generated, we multiply each of the patterns in Figure 1 by several scalar indices (−3.0 to +3.0 at the step of +0.5). As explained in Section 2.1, we can understand the modes in terms of PCA, that is, as orthogonal vectors that form a 3-dimensional feature space. By multiplying the patterns of modes, we obtain an arbitrary coordinate system that represents this space and that can be projected onto Sammon maps.

3.1 Links Between SOM Parameters, SOM Quality, and Data Structure

In Figure 4, quantization and topological errors and their dependence on terminal radius, SOM shape and data set structure are shown for selected cases. Not surprisingly, SOMs with larger arrays have a lower quantization error, as they better explain data variations. The error is also smaller in data sets with two modes and highest in data sets having three modes of similar variance, or very little structure. Notably, a significant drop in the error occurs when the terminal radius decreases below 1.0, but there is no further reduction beyond this threshold. For future studies, experimenting with the terminal radius close to the threshold of 1.0 should, therefore, suffice. However, this may not be the case if large SOMs were combined with data having little structure. Here, for example, terminal radius of 1.4–1.6 was the minimum that led to SOMs with reasonable topological structure for arrays of 30–42 types combined with data sets of little structure (not shown here).

The relationship between SOM parameters and the topological error is somewhat less clear, especially for terminal radius values above 1.0. The relationship can be indifferent, directly proportional, and indirectly proportional, depending on the particular data set—SOM shape combination. However, for almost all 156 combinations tested, a sharp increase in error can be observed when the terminal radius drops below 1.0, except for a few cases with simple data sets. The relatively high spread of the error, particularly when the terminal radius is set under 1.0, underscores the sensitivity of SOMs to initialization and, thus, the importance of calculating multiple variants of SOMs to avoid the risk of achieving only suboptimal local optima of classification.

Although SOMs trained with identical parameters may have nearly identical quantization error, they typically have a wide range of topological errors depending on initialization. For instance, we provide a detailed analysis of a specific case (Z50-M40-C10 data set, 3 × 4 array, and radius of 0.7) for which three distinct topological structures appear. The case is indicated by the red box in Figure 4b and the blue stars highlight the three local optima in the topological error, which are shown in more detail in Figure 5. Note that the SOM in Figure 5a has a markedly better topology than the other two.

3.2 SOM Topology and the Modes of Variability

Here, we present how SOM types are organized within an array according to their spatial similarity to modes. Sammon mapping is utilized to visualize the topological structure of both data and SOM arrays, and Pearson's pattern correlation is used to measure the spatial similarity between patterns of types and modes. Additionally, several spatial patterns representative of various intensities and phases of the modes are projected onto Sammon maps to obtain a generalized non-linear projection of the underlying linear modes.

3.2.1 Two Modes of Variability

The generated circulation fields are clearly organized in the Sammon map according to their spatial similarity to patterns of modes. In all three simple data sets (Z90-M10, Z70-M30, Z50-M50), projections of the two modes are orthogonal to each other. In Figure 6, this is apparent both indirectly from correlations between data and modes (small colored dots) and directly from projections of idealized patterns representing the modes onto the Sammon maps (large colored dots). Furthermore, projections of these idealized patterns are linear. In the two data sets of unequal strength of modes, the modes project along the two axes of the Sammon space (Figures 6a and 6b). In the special case of their equal strength (Figure 6c), the modes remain orthogonal after projection, but do not project on the horizontal and vertical axes of the Sammon map. This is likely due to the fact that the horizontal axis is by default oriented in the direction of maximum explained variance, which is along the leading mode. This direction is unstable for data of spherical shape, that is, when two leading modes explain the same share of variance. However, this does not affect our analysis, as relationships between projected objects do not change if the Sammon map is linearly transformed (e.g., rotated).

In Figure 6, three SOMs (4 × 4, 4 × 5, and 4 × 6) are projected onto each of the three Sammon maps (data sets) to illustrate the link between their topology and projection of modes. The diagonals of the square 4 × 4 SOMs project on the axes of the Sammon space in the case of Z90-M10 and Z70-M30 data sets, that is, opposite corners of SOM arrays depict opposite phases of modes in their rather extreme state. In the case of Z50-M50 (Figure 6c), however, the modes strongly project on the (theoretical) center row and column of the array, with the corner types representing fields with about equal contribution of both modes.

In general, with an increasing elongation of SOM arrays, its center row rather than a diagonal tends to be aligned with the leading mode, but a wide range of results was obtained in different data sets. The leading mode may strongly project on two opposite corners (as in 4 × 5 and 4 × 6 SOMs of Z90-M10 data), on only one corner (4 × 5 SOM of Z70-M30 data) or none at all (4 × 6 SOM of Z70-M30 data). The weaker mode does not project most strongly on the remaining two corners but rather on the types next to and second next to the corners in 4 × 5 and 4 × 6 SOMs, respectively. The corner types themselves represent more complex fields with non-negligible contribution of both modes. The results obtained for SOMs of other shapes are similar to those presented above: Modes project well on the diagonals (center row and column in the case of Z50-M50) of square SOM arrays regardless of their size, while only the leading mode occasionally projects on a diagonal in oblong SOMs.

3.2.2 Three Modes of Variability of Different Strength

Four data sets composed of three modes of different strength were analyzed in the study (Z70-M20-C10, Z60-M30-C10, Z50-M40-C10, and Z50-M30-C20). Relative to the bivariate data sets, shifting 10%–20% of total variability to the third-order circular mode does not markedly affect the projection of the first two modes onto the Sammon map: the zonal (meridional) modes project on the horizontal (vertical) axes of the projection plane (Figure 7). When projecting the circular mode on the Sammon map, its stronger realizations (i.e., ±1 × C and beyond) appear along the vertical axis (i.e., orthogonally to the leading mode, i.e., more or less parallel to the second-order mode). However, fields with such strong contribution of circular flow are generated only very rarely in these four data sets, the maximum index (amplitude) of the mode being—in absolute values—1.23 (C10 data set) and 1.49 (C20 data set), while the 95th percentile is approximately 0.6. On the contrary, the ±0.5 × C patterns project in all four cases very close to the center of the Sammon map. In general, fields highly correlated with the circular mode are scattered across the whole Sammon map, but their highest density is close to the center. In the center, fields with only weak circulation (“turned-off” zonal and meridional modes) occur, and therefore, even a small contribution of the circular mode leads to relatively high pattern correlation between its pattern and data fields.

The orientation of SOM arrays relative to modes and the shape of the arrays also resemble those obtained for bivariate data sets: the two leading modes project onto diagonals in all square SOMs except those of the Z50-M40-C10 data set. On the other hand, the two leading modes project along the central row and column in most SOMs that are strongly elongated (i.e., those with two more columns than rows), the corner types then comprising linear combinations of the two leading modes. The orientation of SOMs with one more column than rows varies between the typical orientation of square and strongly elongated SOMs, depending on data set and SOM parameters (number of types, terminal radius). The corner types of these slightly elongated SOMs can be either linear combinations of two or three modes or strong projections of one particular mode. In the latter case, patterns in opposite corners typically are not mirror images of each other (see Figures 8a and 8b); such a situation is further referred to as “non-mirrored patterns.”

Consistent with the projection of the ±0.5 × C patterns close to the center of Sammon maps, also types resembling the pattern of this mode occur close to the center of SOM arrays. Note, however, that the proximity to the center is in terms of the Sammon map rather than SOM array coordinates. This is apparent in types [1,3] in Figure 8b, and [4,3] and [4,4] in Figure 8c, which—despite their location at the edge of SOM arrays—lie relatively close to the center of their respective Sammon maps, due to a rather deformed shape of the SOM array in the former case (Figure 7b) and rather small array (consequence of a higher terminal radius necessary to produce a SOM with good topological structure) in the latter (Figure 7c). However, the occurrence of types resembling the circular mode in the center does not mean that other types do not show signs of circular flow; rather the opposite holds, which is apparent from the spatial patterns of types (Figure 8).

3.2.3 Three Modes of Variability of Equal Strength

One example of equal strength of modes was presented above, regarding the bivariate Z50-M50 data set. Additional five data sets were defined in which the first- and second-order modes (Z40-M40-C20), the second- and third-order modes (Z60-M20-C20, Z50-M25-C25, and Z40-M30-C30), or all three modes (Z33-M33-C33) have equal strength.

Both the Sammon map and projection of modes in the case of Z40-M40-C20 data (not shown) resemble those of Z50-M40-C10 (Figure 7b), except the leading two modes are not aligned with the Sammon map axes, similarly to Z50-M50 (Figure 6c). The three cases with equal strength of the meridional and circular modes are presented in Figure 9, including Sammon maps of 4 × 4, 4 × 5, and 4 × 6 SOMs—the spatial patterns of the latter are shown in Figures 8d–8f. In all three cases, the leading mode is approximately aligned with the horizontal axis of the Sammon map. However, despite the equal strength of the two remaining modes, the three cases differ in how these two modes project onto the Sammon map of data as well as onto the SOMs.

In the case of Z60-M20-C20 (Figure 9a), both modes project onto the vertical axis of the Sammon map, such that their positive (negative) values occur mostly above (below) the horizontal axis. A similar distribution is also apparent in patterns of SOM types (Figure 8d): excluding the clearly zonal types, the upper two rows tend to capture southerly (M⁺) and cyclonic (C⁺) flow, and their combinations, while northerly (M⁻) and anticyclonic (C⁻) flow prevails in the bottom two rows. Consequently, the SOM selectively and clearly underrepresents concurrence of phases M⁺C⁻ and M⁻C⁺ (in general, types with circular features in the right-hand side of their patterns). Additionally, the two types that project onto the vertical axis of the Sammon map ([1,3] and [4,4]) have non-mirrored patterns, the former resembling M⁺, while the latter being a linear combination of M⁻ and C⁻. This illustrates the fact that the position of types within an array and their distances from the center of its Sammon map are functions of Euclidean distance. As a consequence, the distances do not necessarily convey the information on the organization of anomalies within the patterns, at least not to the same degree as in the case of pattern correlation.

In the case of the Z50-M25-C25 data set (Figure 9b), the three modes project onto the Sammon map similarly to Z50-M30-C20 (Figure 7c), with the zonal and meridional modes aligned with the two axes and the circular mode scattered across the map with increased density of high correlations in the center of the map. The SOM is in this case deformed such that its upper left corner (see type [1,1] in Figure 8e) and the center of the bottom row (type [4,4]) lie close to this area of increased data density and represent both phases of the circular mode. On the other hand, strong projections of the meridional mode occur in types that lie farthest from the horizontal axis in the Sammon map (types [1,3] and [4,6]). Thus, while two corners of the SOM that lie close to the horizontal axis show opposite phases of the leading mode, the remaining two corners show one phase of one of the remaining two modes.

The Sammon map of the Z40-M30-C30 data set (Figure 9c) reveals that in this case it is the circular mode that projects orthogonally to the leading zonal mode. The projection of the meridional mode on the vertical axis is somewhat less clear than in Z60-M20-C20, but still apparent; note, however, that it does not have a clear representation among the types (Figure 8f). Nonetheless, the northerly (southerly) component of flow does project on most types in the upper (bottom) half of the SOM. Given the alignment of the zonal and circular modes with the central row and column of SOMs (in the case of this particular data set), it matters whether the number of rows and columns of a SOM is odd or even; the odd number causing the center row/column to clearly represent the spectrum of realizations of one particular mode, the remaining types being linear combinations of (typically) all three modes (not shown). This particular alignment occurred in Z50-M40-C10, Z50-M30-C20, and Z40-M30-C30 and also in most of small strongly elongated (2 × 4, 3 × 5) SOMs.

4.1 On Data

Synthetic data sets are crucial in studying the ability of methods to capture atmospheric circulation and its structure. However, the definition of such data sets as well as the structures emphasized in them can vary considerably. These differences arise mainly from variations in how the studied feature, in this case, the mode of variability or teleconnection, is conceptualized.

In two previous studies that directly compared the skill of PCA and SOMs in extracting known patterns, or modes, from synthetic data sets (Liu et al., 2006; Reusch et al., 2005), most of the data sets were generated by repeating a few predefined patterns. This resulted in data consisting of a few distinct clusters, which is suitable for studying the ability of methods to classify spatial fields based on similarities in their patterns. Since SOMs represent a topologically ordered nonlinear classification, their application to such a task seems straightforward. However, applying PCA to this task is a misconception. Compagnucci and Richman (2008) showed that for PCA to produce a classification, the similarity matrix to be decomposed must express relations (e.g., pattern correlations) between individual fields (so-called T-mode PCA) and not spatial correlations, which is the case for the commonly used S-mode PCA. The reader is referred to chapter 6 of Richman (1986) for more information on modes of decomposition of data by PCA. To our knowledge, nevertheless, a comparison of SOMs with T-mode PCA has not been done yet.

In this paper, we defined our data sets such that they mimic real circulation, which consists of a continuum of circulation fields rather than a few well separated clusters (Philipp et al., 2016). In contrast to previous studies, the modes are not defined as anomaly fields recurring more or less unaltered in synthetic data sets, but rather represent various statistical links across the spatial domain, linear combinations of which create individual circulation fields. In other words, the generation of data can be understood as a reversed PCA. Such data do not limit the applicability of SOMs in any way, and because they are generated from orthogonal modes, they also make it possible to study how these modes project on SOM arrays, including lower-order modes, which has not been attempted before.

4.2 On Methodology

Beside the data sets, two other specific issues seem particularly important for our study due to its aim at methodological aspects. First, any conclusions are constrained by the quality of trained 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 & 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 training was initialized using randomly selected data points. We did not choose the alternative “linear” 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 SOM topology appeared to be sensitive to initial conditions, 20 variants of SOM were calculated for each combination of the remaining parameters to avoid locally optimal solutions. Experimentally we found that training additional variants did not lead to markedly better SOMs, and we conclude that the variants retained for the study are very close to their respective global optima. Rousi et al. (2015) arrived at the same conclusion after training 10 randomly initialized variants of SOMs of Euro-Atlantic circulation.

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 data sets.

The R 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, which 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 training into k-means clustering (Kohonen, 2001). In our case, including such iterations typically led to a marked change in quantization (decrease) and topological (decrease) errors and to dissipation of SOM topology, regardless of the neighborhood function. 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.

We experimentally found that the effects of the learning rate and the initial radius on SOM quality were relatively small, if chosen reasonably. 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 previous studies (e.g., Gibson et al., 2017; Sheridan & Lee, 2011).

The last point regards the use of Sammon mapping. Strictly speaking, Sammon mapping adds a step into the analysis that could introduce new uncertainty. It evaluates nonlinear relations among all data points to define a two-dimensional projection plane, but stresses an accurate representation of local topology. It represents only one of many possible frameworks, in which the results could be presented. However, the highly idealized rectangular array of SOMs limits the study of its topological structure to such an extent that the benefits of Sammon mapping seem to considerably outweigh its limitations. 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. An interested reader is referred to Stryhal and Plavcová (2023) for more information on Sammon mapping.

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 data sets), 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) is an important factor. Multiple local optima in SOM topology are often found for oblong SOMs of data that have equal or nearly equal strength of two modes (zonal and meridional, or meridional and circular). This suggests that exploratory data projection by PCA prior to SOM training may aid selection of SOM parameters, such as initialization and shape of array. One such PCA application can be found in Lee (2017), who chose a square SOM array after finding that the leading two PCs accounted for nearly equal variance in their data.

A characteristic type of SOM's behavior, or topology pattern, consisting in a projection of two leading modes of variability on the diagonals of a SOM array, which was suggested in earlier studies (Lennard & Hegerl, 2015; 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 data sets 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 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. Fields very strongly correlated with the circular mode occur only when the amplitude of the two leading modes is close to zero (the modes are close to their neutral phase). Without the contribution of the leading modes, fields with only relatively weak anomalies are generated, which project 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 Figures 1 and 6 of their study). While the leading two principal components, which explained 39% and 13% of variability, projected relatively strongly on all the four corners of the SOM, the third-order principal component (11%) did not have a clear representation within the 5 × 6 SOM array. Despite similarities between the pattern and several types in the center of the array (e.g., type 16, Figure 1), the two phases of this pattern 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 two combinations of phases being underrepresented. Each of these three topology patterns 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 data set—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 & Livezey, 1987; Casado & Pastor, 2012; Cortesi et al., 2019; Hynčica & 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. We would also like to emphasize that the primary goal and strength of SOMs do not lie in explicitly identifying leading modes of circulation variability, but rather their various combinations represented by recurrent circulation fields. Whether SOMs provide high-quality classifications that outperform other methods, such as k-means, is a different aspect that was not evaluated in this study. However, this does not diminish its importance.

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 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 data sets, 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 data sets 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.