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  • View in gallery

    Demonstration of consensus clustering via categorical intersection illustrated in the spatial domain, depicting (a) an optimal intersection and (b) an intersection leading to the production of orphaned clusters.

  • View in gallery

    (Continued )

  • View in gallery

    Very simple example of overclustering and underclustering, depicted in a p = 2 dimensional geometric space defined by two attributes: X1 and X2. In (a), which illustrates overclustering at the higher clustering level, the 2-cluster solution would likely be stable to perturbations, while the specific object assignments in the 4-cluster partition would likely be more unstable (owing to perturbation-induced shifts in the intercluster boundaries). In (b), which illustrates underclustering at the lower clustering level, the 3-cluster solution composed of groups A, B, and C would be stable, but the partition at the 2-cluster level would likely be very sensitive to perturbations in the analysis strategy. In this example, such perturbations could arise from varying the clustering algorithm or the number of objects employed. In this paper, both of those are held constant, and it is the attribute selection and dimensionality that are varied in the bootstrap resampling. However, the basic concept is qualitatively similar.

  • View in gallery

    Adjusted Rand index summary statistics from the bootstrap resampling cluster sensitivity test, by clustering level for the (a) temperature and (b) precipitation with independent clusterings. Statistics include median, mean, and upper (Q3) and lower (Q1) quartiles, compiled for 10 resampling trials for each clustering level. See Table 3 for additional information.

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    The 4-cluster independent temperature solution used in the construction of the low-order consensus solution shown in Fig. 6. Shown are (a) the control regionalization and (b) the discrepancy count map resulting from bootstrap resampling perturbation analysis at the same clustering level. Locations of the cluster boundaries are superimposed on panel (b). The discrepancy count for a given climate division object reflects the number of times in the 10 perturbed trials that it was assigned to a different cluster than in the control partition shown in panel (a). See Table 3 for further information.

  • View in gallery

    The 6-cluster independent precipitation solution used in the construction of the low-order consensus solution shown in Fig. 6. Shown are (a) the control regionalization and (b) the discrepancy count map resulting from bootstrap resampling perturbation analysis. Locations of the cluster boundaries are superimposed on panel (b). The discrepancy count for a given climate division object reflects the number of times in the 10 perturbed trials that it was assigned to a different cluster than in the control partition shown in panel (a). See Table 3 for further information.

  • View in gallery

    Lower-order 14-cluster consensus outcome obtained by overlaying the temperature and precipitation solutions shown in Figs. 4 and 5. The gray shadings and patterns used in the temperature and precipitation clusterings are retained, producing a graphical approach to categorical intersection. This permits the reader to either focus upon the precipitation subzones of the temperature clusters (or vice versa) or to consider each combination of gray shading and pattern to be a separate grouping. Inset map depicts cluster boundaries, with pattern fill only employed for spatially discontinuous clusters. Three suspected orphan clusters have been reassigned, as discussed in the text and Table 4.

  • View in gallery

    Discrepancy count map, superimposed with cluster boundaries, for the (a) 7-cluster temperature and (b) 15-cluster precipitation independent clustering solutions. As with the discrepancy maps shown in Figs. 4 and 5, the count reflects the number of times a given climate division object was differently assigned in the 10 perturbed trials. See Table 3 for further information.

  • View in gallery

    Higher-order 26-cluster consensus outcome resulting from categorical intersection of a 7-cluster temperature solution with a 15-cluster precipitation solution. Regionalization shown has been adjusted to remove some suspected orphan clusters, as discussed in text and summarized in Table 5. Instead of using graphical overlay (as in the lower-order outcome in Fig. 6), patterns were selected to match those used in Fovell and Fovell’s (1993) 25-cluster reference solution as closely as possible (see their Fig. 7). The two 4-member clusters located in eastern Tennesse–western North Carolina and central North Carolina–eastern Virginia were judged to be especially unstable consensus clusters, and their reassignment could be justified.

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Consensus Clustering of U.S. Temperature and Precipitation Data

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  • 1 Department of Atmospheric Sciences, University of California, Los Angeles, Los Angeles, California
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Abstract

A “consensus clustering” strategy is applied to long-term temperature and precipitation time series data for the purpose of delineating climate zones of the conterminous United States in a “data-driven” (as opposed to “rule-driven”) fashion. Cluster analysis simplifies a dataset by arranging “objects” (here, climate divisions or stations) into a smaller number of relatively homogeneous groups or clusters on the basis of interobject dissimilarities computed using the identified “attributes” (here, temperature and precipitation measurements recorded for the objects). The results demonstrate the spatial scales associated with climatic variability and may suggest climatically justified ways in which the number of objects in a dataset may be reduced. Implicit in this work is the arguable contention that temperature and precipitation data are both necessary and sufficient for the delineation of climatic zones.

In prior work, the temperature and precipitation data were mixed during the computation of the interobject dissimilarities. This allowed the clusters to jointly reflect temperature and precipitation distinctions, but also had inherent problems relating to arbitrary attribute scaling and information redundancy that proved difficult to resolve. In the present approach, the temperature and precipitation data are clustered separately and then categorically intersected to forge consensus clusters. The consensus outcome may be viewed as having identified the temperature subzones of precipitation clusters (or vice versa) or as representing distinct groupings that are relatively homogeneous with respect to both attribute types simultaneously.

The dissimilarity measure employed herein is the Euclidean distance. As it employs only continuous time series data representing a single information type (temperature or precipitation), the consensus approach has the advantage of allowing an attractively simple interpretation of the total Euclidean distance between object pairs. The total squared distance may be subdivided into three components representing object dissimilarity with respect to temporal mean (level), seasonality (variability), and coseasonality (relative temporal phasing). Therefore, concerns about redundancy or arbitrary scaling problems are neutralized. This is seen as the chief advantage of consensus clustering.

The consensus strategy has several disadvantages. It is possible for two (or more) relatively general, undetailed clusterings to produce a very complex and fragmented clustering following categorical intersection. Further, the fact that the analyst chooses the clustering levels of the separate, contributing clusterings means that he or she has considerable freedom in fashioning the consensus outcome, which makes it difficult (if not impossible) to argue that true, “natural” clusters have been identified. The latter often applies to cluster analysis in general, however. It is believed that the consensus approach merits consideration owing to its advantages.

Two consensus outcomes are presented: a lower-order solution with 14 clusters and a higher-order solution with 26 clusters. The sensitivity of these clusterings to perturbations in the input data is assessed. The regionalizations are compared with those presented in prior work.

Corresponding author address: Prof. Robert G. Fovell, Dept. of Atmospheric Sciences, University of California, Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90095-1565.

Email: fovell@atmos.ucla.edu

Abstract

A “consensus clustering” strategy is applied to long-term temperature and precipitation time series data for the purpose of delineating climate zones of the conterminous United States in a “data-driven” (as opposed to “rule-driven”) fashion. Cluster analysis simplifies a dataset by arranging “objects” (here, climate divisions or stations) into a smaller number of relatively homogeneous groups or clusters on the basis of interobject dissimilarities computed using the identified “attributes” (here, temperature and precipitation measurements recorded for the objects). The results demonstrate the spatial scales associated with climatic variability and may suggest climatically justified ways in which the number of objects in a dataset may be reduced. Implicit in this work is the arguable contention that temperature and precipitation data are both necessary and sufficient for the delineation of climatic zones.

In prior work, the temperature and precipitation data were mixed during the computation of the interobject dissimilarities. This allowed the clusters to jointly reflect temperature and precipitation distinctions, but also had inherent problems relating to arbitrary attribute scaling and information redundancy that proved difficult to resolve. In the present approach, the temperature and precipitation data are clustered separately and then categorically intersected to forge consensus clusters. The consensus outcome may be viewed as having identified the temperature subzones of precipitation clusters (or vice versa) or as representing distinct groupings that are relatively homogeneous with respect to both attribute types simultaneously.

The dissimilarity measure employed herein is the Euclidean distance. As it employs only continuous time series data representing a single information type (temperature or precipitation), the consensus approach has the advantage of allowing an attractively simple interpretation of the total Euclidean distance between object pairs. The total squared distance may be subdivided into three components representing object dissimilarity with respect to temporal mean (level), seasonality (variability), and coseasonality (relative temporal phasing). Therefore, concerns about redundancy or arbitrary scaling problems are neutralized. This is seen as the chief advantage of consensus clustering.

The consensus strategy has several disadvantages. It is possible for two (or more) relatively general, undetailed clusterings to produce a very complex and fragmented clustering following categorical intersection. Further, the fact that the analyst chooses the clustering levels of the separate, contributing clusterings means that he or she has considerable freedom in fashioning the consensus outcome, which makes it difficult (if not impossible) to argue that true, “natural” clusters have been identified. The latter often applies to cluster analysis in general, however. It is believed that the consensus approach merits consideration owing to its advantages.

Two consensus outcomes are presented: a lower-order solution with 14 clusters and a higher-order solution with 26 clusters. The sensitivity of these clusterings to perturbations in the input data is assessed. The regionalizations are compared with those presented in prior work.

Corresponding author address: Prof. Robert G. Fovell, Dept. of Atmospheric Sciences, University of California, Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90095-1565.

Email: fovell@atmos.ucla.edu

1. Introduction

Many different climate classification systems and/or climatic regionalizations have been presented in the past. Though their motivations have varied, they have been useful for demonstrating climatic variability and/or suggesting how the degrees of freedom in complex datasets may be reduced in a climatically justified manner. Such efforts often began with the subjective identification of a set of measures deemed to be skillful in revealing climatic characteristics. Typically, this has included some form of temperature and/or precipitation data, although measures relating to moisture demand and supply, solar radiation, winds, soil characteristics, and other factors have also sometimes been employed.

Once selected, at least two very distinct approaches may be pursued. The first, encompassing the famous Koeppen (1923) and Thornthwaite (1931) systems, is the “rule-driven” strategy in which the climate types are first defined using the selected measures and then used to determine the classification of each locale for which the appropriate data are available, yielding the regionalization. A principal advantage of this approach is that it directly and quantitatively specifies the climate types and the boundaries among types. A disadvantage is that the classification rules are subjectively formulated and thus open to question.

The second approach avoids the direct specification of classification rules and instead constructs the climate regionalization in a “data-driven” fashion, often utilizing some form of cluster analysis [see Gong and Richman (1995) for a comprehensive review of clustering applications in the atmospheric sciences]. Some examples of data driven regionalizations are Gadgil and Joshi (1983) for the Indian subcontinent; Anyadyke (1987) for West Africa; Ronberg and Wang (1987) for China; Crutcher (1960), Steiner (1965), and Fovell and Fovell (1993, hereafter FF) for the conterminous United States; and Bunkers et al. (1996) for a portion of the U.S. northern Plains.

As in the rule-driven schemes, the a priori selection of a set of relevant measures is required for a clustering-based regionalization. Although the cited studies vary widely in this regard, they share the basic idea that locales with similar characteristics relative to the selected measures should have roughly similar climates, thereby allowing “climate zones” to be identified by the manner in which the locales are found to group together. For this reason, clusters with a high degree of internal homogeneity with respect to the selected measures are usually desired. A disadvantage of this approach is that the regionalization obtained may not be reducible to a simple set of classification rules, though it may perhaps assist in the formulation of those rules.

In this paper, the cluster analysis–based regionalization of the conterminous United States presented in FF is reconsidered, utilizing a simpler, and in some senses superior, methodology. In this introduction, the motivation for this reconsideration, as well as background information relevant to it, is presented.

Consider an n × p dataset X, where n represents the number of objects (spatial locales herein) for which p measurements (often termed variables or attributes) are recorded. Cluster analysis may be used to partition the n objects into a smaller number of categories or clusters on the basis of the distance between object pairs, thereby constructing a new classification or grouping variable and/or accomplishing a reduction in the number of objects in the dataset. Distance usually has a geometric interpretation and is derived from the set of p attributes that provide the means of distinguishing among the objects.

The attributes can take a wide variety of forms. For example, they may be p temporal samples of one data type, such as temperature or precipitation measures. For example, Gadgil and Joshi (1983) constructed separate regionalizations from monthly minimum temperature, precipitation, and moisture index time series data. More commonly, the attributes have represented combinations of disparate data types (e.g., temperature, precipitation, solar radiation, wind speed, and humidity measurements, etc.), reflecting the idea that climate is a multivariate phenomenon. This approach was pursued by Steiner (1965) and Anyadyke (1987), among others. Fovell and Fovell (1993) elected to employ a hybrid approach, combining time series of two different data types, temperature and precipitation, as did Bunkers et al. (1996).

Once selected, the attributes jointly define an amorphous p-dimensional geometric space in which the objects are distributed, with the width of the space along each attribute axis reflecting the attribute’s variance. The most popular dissimilarity measure is the Euclidean distance (ED), the shortest distance between two points (objects) within the p-dimensional space. Application of the Euclidean metric upon X creates an n × n symmetric distance matrix D. In the common parlance (cf. Arabie and Hubert 1992), the ED takes X, a “two-way, two-mode” (objects by attributes) matrix, and “destructively” converts it into the “two-way, one-mode” matrix D (objects by objects).

The distance matrix D is then input to a clustering algorithm, of which many forms and types exist. Many of the more commonly employed schemes strive to fashion “hard” (nonoverlapping), geometrically “compact” clusters in a “hierarchical” fashion. That is, the clusters are constructed such that each object belongs to only one cluster, objects within each cluster tend to be more similar to each other than to objects assigned to other clusters (such similarity being derived from the EDs), and the analysis proceeds stepwise, with the clustering generated at each step being dependent upon the results obtained at previous steps. In particular, an “agglomerative hierarchical” algorithm takes an initial solution of n clusters (each having one member) and successively fuses a cluster pair at each step until a terminal, all-inclusive cluster is created, thereby establishing a set of n nonindependent clustering levels, each representing a potential solution.

The result is sometimes displayed as a dendogram, a treelike structure that graphically portrays the successive fusions (“branches”) down to the terminal “root.” One or more clustering levels, representing different compromises between generality and detail, are then selected. The “number of clusters” problem remains an important, unresolved issue (Everitt 1979), but several proposed “stopping rules” appear to have some merit in guiding the clustering level selection (e.g., Milligan and Cooper 1985). A regionalization is generated when a clustering partition forged in geometric space is mapped into the spatial domain. Regionalizations often display considerable spatial coherence owing to the presence of spatial correlations among the objects.

Many different hierarchical methods have been proposed, and alternative (though less commonly used) approaches also exist. Some methods have been specifically designed to avoid geometric cluster shape biases, although a bias toward geometric compactness is often desirable as it strives to maximize within-cluster homogeneity. Partitioning algorithms construct each clustering level solution independently through optimization of an objective function. Fuzzy schemes further allow objects to participate in more than one cluster, thereby avoiding one of hard clustering’s most unrealistic assumptions. With hard clustering, it must be recognized that while two clusters may well be quite distinct with regard to their mean properties (averaged across their respective memberships), objects in the clusters that are located relatively close to an intercluster boundary in geometric space may not be very different from each other. This can apply to the clusters’ spatial boundaries as well, owing to object spatial correlation. As a result, even small alterations in the analysis strategy (including changing the clustering algorithm) may cause shifts in the specific spatial locations of the boundaries, even if the overall character of the regionalization is unchanged. Thus, the importance of the boundaries themselves should not be exaggerated.

In the FF study, the popular group average linkage hard hierarchical cluster analysis algorithm was applied to National Climatic Data Center (NCDC) climate division data in an attempt to construct regional climate zones in the conterminous United States. For each climate division object, monthly values for precipitation accumulation and average daily mean temperature were obtained spanning a 50-yr period. These data were recast into 50-yr means for each month, creating p = 24 attributes (12 each for temperature and precipitation). The group average linkage method has been widely used (e.g., Wolter 1987; Davis 1991; see also Gong and Richman 1995), often for its tendency to forge geometrically compact clusters (cf. Kalkstein et al. 1987).

Numerous decisions about how the initially selected attributes were handled and combined prior to clustering had to be made, decisions that could powerfully influence the resulting subdivisions. Since the ED destroys the attribute dimensions of the dataset, it is extraordinarily important to consider how the attributes participate in the interobject distance computations. This raises the complex issue of attribute scaling or weighting. Owing to the ED’s geometric interpretation, changing an attribute’s variance can profoundly alter its relative contribution to the total interobject distances (Sokal and Sneath 1963). Yet, variance reflects an arbitrary measurement scale, and this becomes a severe problem when the attributes represent a collection of disparate data types having incommensurable scales. In such cases, standardization of the attributes to common unit variance is often performed (see Milligan and Cooper 1988). This procedure, however, cannot detect the presence of, or compensate for, irrelevant or redundant attributes that can obscure and/or distort the clustering solution (FF).

In FF’s study, the vexing difficulty involved redundancy because it was assumed from the outset that the attributes selected were indeed relevant to climate specification. The attributes, especially the temperature measures, tended to be highly intercorrelated. This raised the possibility that some of the temperature attributes represented redundant information that would allow temperature as a data type to dominate the computed dissimilarities and clustering outcomes unless the problem were addressed somehow. Although FF clearly demonstrated that redundancy is not synonymous with correlation, it remains that if two identical attributes are present in a dataset, the repeated attribute’s influence on the interobject distances is magnified without bringing anything demonstrably new to the analysis. Indeed, in such a case, the redundant attribute’s variance is unintentionally doubled, undermining any scaling decisions previously made (Fovell 1992) and distorting the shape of the geometric space. Further, removal of the redundant attribute could have a first-order effect on the clustering outcome because of the effect on scaling. This made attribute selection a very tricky issue.

Fovell and Fovell (1993) attempted to compensate for redundancy by applying principal components analysis (PCA) to the p = 24 initially selected attributes. As discussed in detail by FF (section 4c), however, PCA itself raises difficult scaling and truncation issues, forcing the conclusion that it is a “troublesome and inadequate” technique not ideally suited to the detection of redundant variance. Much of the problem stems from the fact that PCA may be considered a variance reapportioning technique. Danger exists because important decisions (such as where to truncate the analysis) are usually handled by using some function of the total aggregate attribute variance, which includes the redundant variance one seeks to remove.

The present reconsideration is motivated in large part by the difficulties faced in this portion of FF’s methodology. One of the alternate strategies that FF suggested, a form of consensus clustering accomplished through categorical intersection, is investigated herein.1 We still seek to construct climate zones that represent relatively homogeneous (i.e., geometrically compact) areas of temperature and precipitation but proceed without directly mixing the two disparate data types. Instead, two independent clustering solutions will be formed: one each from temperature and precipitation time series data. Since neither is expected to represent climate variability in isolation, the two solutions have to be merged in some fashion. This will be accomplished via categorical intersection of the two solutions, yielding the joint or “consensus” outcome. The result may be interpreted as highlighting the precipitation subzones of temperature regions (or vice versa) or as representing a set of distinct and separate climate zones. If the clusters in the independent solutions are relatively homogeneous with respect to their particular data type, then the consensus clusters should be relatively homogeneous with respect to both types jointly. In theory, any number of independently constructed clusterings could be used to construct a consensus solution.

As will be discussed in section 2b, the principal advantage of consensus intersection clustering is believed to be the very simple and straightforward interpretation of the interobject Euclidean distances it allows, one that effectively neutralizes concerns about redundancy bias. Another advantage is that it affords the analyst the freedom to choose the level of detail desired in each independent clustering. For example, a relatively general temperature and relatively detailed precipitation clustering, when intersected, emphasizes the precipitation component of the consensus outcome. This freedom, however, has a downside. It would be very difficult to support a claim that the consensus clusters are in any sense natural clusters lurking within the data. This is often highly problematic anyway. It also vastly increases the total number of possible consensus outcomes, thus making it difficult to choose among them.

Consensus clusters may differ from those forged from the mixing of different information sources with respect to distinctiveness and stability. With regard to the former, suppose four cluster temperature and five cluster precipitation solutions are constructed. In the present dataset, these solutions would be rather general, having already sacrificed a great deal of detail. The consensus outcome could have as many as 4 × 5 = 20 clusters, apparently a much more detailed solution, although some potential clusters may in fact be empty sets. These clusters may potentially be less distinctive that the same number of clusters formed using the mixed temperature and precipitation data, owing to the information loss prior to intersection. On the other hand, the mixed data clusters could be affected by “overclustering,” which can sometimes occur as more and more clusters are retained. In overclustering, a parent cluster, formed at a relatively low clustering level, is more stable to analysis perturbations than the subclusters that comprise it at higher clustering levels. In utilizing less detailed independent clusterings, the consensus clusters may be more stable to perturbations.

As the clustering level of the independent solutions and/or the number of independent solutions increases, the number of possible clusters in the consensus outcome quickly escalates. This could result in a fragmented solution containing a large number of “orphans,” small membership clusters of suspect validity residing in the boundary zones (in categorical and physical space) between larger groupings. Given the aforementioned sensitivity of hard clustering analyses with respect to cluster boundaries, removal of these orphaned clusters is indicated. To achieve a reasonable consensus outcome, it may be necessary to employ relatively low-order independent solutions, despite the loss of detail.

It is implicitly assumed in this study that temperature and precipitation accumulation information is both necessary and sufficient to discriminate among climate zones. As pointed out in FF, other data types, possibly including combinations of temperature and precipitation (such as moisture demand versus supply), could be considered. This remains a topic for future work, as the principal intent of this study is to explore an alternative to FF’s analysis procedure.

The structure of this paper is as follows. The data are introduced in section 2a, and the consensus clustering approach is justified in sections 2b and 2c through an examination of the squared Euclidean distance and its constituent components. Section 2d discusses the specific clustering algorithm employed to construct the independent clusterings. In section 3, lower-order and higher-order consensus outcomes are constructed and compared to FF’s regionalizations. The stability of the independent clustering solutions is also assessed. Section 4 presents the concluding discussion.

2. Methodology and justification

a. Data source and preprocessing

The NCDC dataset consists of 344 climate divisions (CDs) which cover the conterminous 48 United States (see FF’s Fig. 1). As in FF, one especially troublesome CD, located in northwestern South Carolina, was excised from the analysis, leaving a total of n = 343 objects. For each object, 51-yr time series of monthly mean temperature and precipitation accumulation (total p = 612 samples or, more generically, attributes), spanning the period 1931–81, were constructed. As the precipitation data are far from normally distributed, a square root transform was applied. Richman and Lamb (1985) found this transform improved normality in their 7-day precipitation dataset, even more so than in their 3-day data. Still more dramatic improvement was found in these data, probably because the sampling interval (1 month) was still longer. Fovell and Fovell (1993) did not apply this transformation, which does exert some effect on the outcome (see section 2b). No other scaling alterations were deemed necessary. Note here that the present study essentially employs the same data as used in FF, although the way in which these data will be processed, arranged, and combined herein is distinctly different.

b. The Euclidean distance and its components

Interobject Euclidean distance matrices, dimensioned n × n, were computed independently for the two data types: temperature and precipitation. Let xi represent the 1 × p row vector representing the monthly time series (either temperature or precipitation) for object i, with xik being the kth element from that series. Then, the squared Euclidean distance d2ij between objects i and j may be written as
i1520-0442-10-6-1405-e1
In view of the presence of significant low-order autocorrelation within each time series and FF’s concern about redundancy, it is proper to consider whether the total dimensionality of the geometric space should be reduced in some fashion. Since data of different types are not being combined, however, redundancy is not believed to be a concern in the present analysis. Using some algebra (see appendix), the total squared interobject distance (1) can be written as the sum of three components:
d2ijd2ijmd2ijsd2ijcs
where the components are defined as
i1520-0442-10-6-1405-e3
In the above, i is the temporal mean for object i (computed over the p months), Si is the object’s temporal sample standard deviation, and rij is the Pearson linear correlation coefficient between the two objects. The common factor p may be removed without effect. It is seen that each component shares the scaling of the total distance, being the square of the data type’s measurement scale (here, °C2 or cm). Some information (including terminology) pertinent to this discussion may be found in Cronbach and Gleiser (1953).

When the input data are continuous time series from a single source (data type), the three ED components are easily interpreted as mean (level), seasonality (variability), and coseasonality (temporal phase) component distances, respectively. The first two components utilize the series integrated properties of mean and standard deviation. The former differentiates objects with respect to their series means, while the latter gauges dissimilarity with regard to how the objects vary in time about their respective means: a measure of seasonality. Note that each of these properties can be expressed as a single number (i.e., one mean or one standard deviation) per object.

The third component, coseasonality distance, has been written in terms of the linear correlation coefficient for illustrative purposes. If each object time series is standardized to common mean and variance, the first two components naturally disappear and the coseasonality distance (and total squared distance d2ij) reduces to the correlation distance [d2ij]corr. Correlation distance responds to the relative phasing of the object’s temporal variations, reaching a maximum value when the two series are 180° out of phase (i.e., rij = −1). The coseasonality distance, then, may be viewed as a weighted phase distance. Note that, unlike the component distances (3), which are unbounded, the correlation distance component cannot exceed 4p. Also, note that neither coseasonality nor correlation distance can be reduced to a single datum per object.

Employing attributes of but a single data type at a time has two principal advantages. First, because the attributes all naturally share the same measurement units, problems arising from incommensurable scaling are largely avoided.2 When the attributes represent an aggregation of different (possibly intercorrelated) data types (e.g., temperature, precipitation, solar radiation, cloud cover fraction, etc.); however, the interpretation of the distance components is less clear. Even if commonality of scaling is enforced somehow upon the disparate data types (such as through standardization), one might be uncomfortable with employing the computed total EDs after realizing it is composed of statistics (means, standard deviations, correlation coefficients) computed across those different types.

The second, and more important, advantage is that redundancy concerns are eliminated. In FF’s situation, inclusion of additional correlated attributes intensified the redundancy bias problem and exerted a first-order effect on the clustering outcome. On the other hand, when temperature and precipitation data are handled separately, the inclusion of additional attributes (now representing temporal samples) is seen to have a beneficial effect on the outcome. The series integrated statistics upon which the ED components rely (temporal means, standard deviations, and correlation coefficients) may potentially be stabilized as the sample size increases.

Still, it is clear that climate cannot be represented by either data type alone. If the data types are not going to be combined prior to the computation of the interobject distances, then the separate temperature and precipitation clusterings have to be merged in some fashion. In the present strategy, the separate, independent clusterings are treated as categories that, upon intersection, create new subcategories or consensus clusters. The consensus clustering strategy is motivated primarily by the desire to exploit this scale-independent and redundancy-free interpretation of the Euclidean distance.

c. Relative contributions of the distance components

Ultimately, both temperature and precipitation clusterings were conducted using the total squared interobject distance (1), thereby retaining all three distance components. However, though the components are measured in the same units, it does not follow that they participate equally in determining the total distance. Nor are they equally valuable or applicable in any given study. It is thus useful to examine the relative contributions of the components to the total interobject distances, even when the total distance is employed. Doing so provides important information regarding the bases on which the clusters are formed and can assist in the labeling of the clusters.

The relevance of each distance component depends on the goals of the analysis, which may dictate how the object time series are processed prior to the computation of distances. One might be concerned with the construction of local forecast zones, for example. Clusters would then represent a collection of objects or stations that could be considered a single locale for forecasting purposes, accomplishing a reduction in the total number of objects. In that application, it may be judged prudent to standardize the objects to eliminate level and seasonality distinctions and cluster objects solely on the basis of correlation (unweighted phase) distance.

The present application is concerned with the construction of climate clusters, however, and so the mean/level and seasonality components ostensibly appear to be useful information. Two objects that differ markedly with respect to average temperature or precipitation accumulation may indeed be judged as being climatically distinct. Even two objects with comparable levels but significantly different seasonal amplitudes about those levels could be distinguished on the basis of their seasonality distance. Indeed, if these two components alone were used to judge dissimilarity, the geometric space collapses to a two-dimensional space defined by object temporal mean and standard deviation when the squared Euclidean distance is employed, simplifying the analysis and interpretation of the clusters considerably.

The value of the coseasonality component to a climate clustering may be less immediately obvious. It was found, however, that exclusion of coseasonality/phase distance permits several clusterings that do not subjectively appear to be climatically meaningful, especially in the precipitation solution. Two objects that happen to share identical temporal means and variances would still be distinguishable if the phasing of their temporal variations were markedly different. As an example, consider two CDs, one located on the West Coast where precipitation has a winter maximum, and the other in the Midwest with a summer maximum. If these locales happened to be roughly similar with regard to the series integrated properties of temporal mean and variability, they might be clustered together if the relative phasing of their temporal variations were not taken into account. It is arguable whether or not the fact that a given locale may be “winter wet” or “summer wet” is by itself a valid climatic indicator; this perhaps suggests the utility of measures like moisture demand and supply.

It turns out that the temperature and precipitation data types differ greatly with regard to the relative contributions of the three distance components to the total interobject Euclidean distance. This is illustrated in Table 1. For a given CD and data type, there are n − 1 = 342 total squared distances to be computed, each of which represents the sum of the three component distances. The fractional contribution of each component to each total distance was determined and then averaged over all of the n − 1 distances. This was done for each CD. Then, the fractions obtained were averaged over all 343 CDs, yielding the data shown in the table. Because the data domain includes significant topography, the analysis was also performed in a subdomain consisting of the 169 CDs bounded between the Rocky and Appalachian Mountains. Those statistics are also reported in Table 1.

Although no explicit information about spatial separations is employed in the clustering, some spatial cohesiveness in the regionalization is expected, owing to spatial correlation among the objects. Thus, some degree of correlation between Euclidean distance and spatial distance would be expected. However, we believe that if a particular distance component’s contribution is very highly predictable from spatial distance alone, it brings no useful information to a climate clustering. Clustering of spatial separation information would result in the construction of a large number of clusters that would be nearly circular when spatially mapped (subject to the constraints of the clustering algorithm employed; see FF’s section 4b) and have no skill in discriminating among climate zones. Table 2 summarizes the average linear correlations between spatial distance and four distance components (including correlation distance), computed using the same strategy employed in Table 1.

Temperature distances are greatly dominated by mean level dissimilarity, especially in the 169 CD central region where it contributes an average of 82% of the total distance (Table 1). Except for CDs along the West Coast, locales tend to vary together temporally, with even seasonal amplitudes dwarfed by level differences. The relatively small contribution of the coseasonality/phase distance is itself highly predictable from spatial separation (Table 2), especially in the central region where topographic variation is not sufficiently large to mask this effect. This suggests that the coseasonality (and correlation) distance should not be employed in isolation for a climate clustering. Although including coseasonality tended to make the temperature clusters slightly less distinctive, we subjectively decided to retain it because it encouraged a somewhat more spatially cohesive regionalization, reducing the spatial fragmentation of the consensus solution slightly.

In the precipitation data, coseasonality/phase distance is the most important component (Table 1). In contrast to the temperature situation, coseasonality was not highly predictable from spatial distance, even in the central zone (Table 2). Spatial plots of this component (not shown) were spatially cohesive but usually elliptically shaped, indicating the presence of information beyond mere physical separation. The almost insignificant contribution of seasonality distance was not anticipated, but there was no reason to remove it. We did consider employing correlation distance alone in the precipitation clustering, but ultimately decided that the mean/level distance component was too large and relevant to this study to ignore.

d. Clustering algorithms considered in this study

As in FF, the hierarchical group average linkage hard clustering algorithm was chosen for this study, owing to its tendency to form compact clusters and its relative lack of cluster membership size bias. Two other approaches, both partitioning strategies, were also considered: the k-means method (e.g., Spaeth 1980), which also yields hard clusters, and a fuzzy (overlapping) clustering scheme discussed in Kaufman and Rousseeuw (1990). In the consensus clustering strategy, categorical intersection is much easier to accomplish when the component clusterings are nonoverlapping, but a fuzzy solution may be “hardened” by assigning each object to the cluster in which its membership coefficient is largest. It is possible that a hardened fuzzy clustering may be superior to a solution generated by a hard algorithm because the fuzzy clustering upon which it was based was made without the restrictions imposed by hardness.

Despite their inherent advantages, however, both schemes were found to result in seemingly less distinctive clusterings than group average linkage. Both partitioning algorithms, especially the fuzzy approach, evinced pronounced bias toward constructing clusters with roughly uniform membership sizes. In their section 4b, FF discussed the ramifications of this bias. On the precipitation data, in fact, the fuzzy clustering algorithm was unable to find any distinctive groupings at all, a very disappointing (and apparently illogical) result. Partitioning methods in general are somewhat sensitive to the manner in which they are initialized. Variations of the initial partition (including initializing the fuzzy clustering with the group average linkage solution), however, only marginally improved the results. These alternate solutions will not be discussed herein.

e. Categorical intersection and orphaned clusters

As noted in the introduction, one potential disadvantage of consensus clustering through categorical intersection is the possibility that orphaned clusters might be produced. This term is used to denote consensus clusters with relatively few members (most commonly single-member isolated clusters) that form between the boundaries of larger clusters but do not appear to be justifiably separate entities. Although the consensus intersection that may result in these orphaned clusters is conducted in categorical space, a simplified example cast in the spatial domain may facilitate the discussion. Owing to the tendency for objects to exhibit a high degree of spatial cohesiveness, orphaned clusters tend to be readily identifiable in physical space.

Figure 1a illustrates the outcome of what is considered to be an optimal categorical intersection of two independent temperature and precipitation clusterings, each of which had 2 clusters. This example is an idealization of what generally occurs in the eastern third of the United States. In the temperature clustering, the spatial domain is divided into northern and southern segments (labeled “cool” and “warm,” respectively), while the precipitation solution differentiates between dry western locales and wet eastern stations. In categorical as well as physical space, the consensus intersection establishes 4 clusters, representing all possible combinations of the temperature and precipitation classes. The spatial subdivision seen in the consensus solution is neat, although again it is cautioned that the boundaries may not actually be as distinct in geometric space as the graphic appears to suggest in physical space.

In Fig. 1b, a situation akin to what actually was found to transpire in northern Florida is depicted. Both temperature and precipitation increase toward the south, but the boundaries between the cool/warm and dry/wet clusters do not precisely coincide. As the boundaries themselves are subject to uncertainty owing to the unrealistic cluster hardness constraint, their importance should not be exaggerated. Upon intersection, however, small clusters (with respect to spatial and geometric extent, as well as cluster membership size) are created in the boundary zone between the two principal clusters. These small clusters would be judged to be orphaned clusters, and the stations located in them should be reassigned among the principal clusters. While reassignment could be subjectively performed, this issue isn’t particularly serious as the solutions are not actually radically altered in the process. It is noted that a similar result could occur from the intersection of two clusterings made from the same data but employing different clustering algorithms. In that case, it is logical to avoid exaggerating what would in effect be relatively small discrepancies between the clustering solutions.

3. Consensus clustering results

a. Selection of the independent temperature and precipitation solutions

For each dataset, temperature and precipitation, average linkage clustering solutions representing the 2–20 clustering levels, inclusive, were constructed, for a total of 38 “control” partitions. As in FF, the Calinski and Harabasz (1974) and Duda and Hart (1973) tests are employed to assist in the essentially subjective task of selecting among these candidate partitions. These criteria are related to the standard F test and will tend to highlight clustering levels in which the among cluster variation is large relative to the within cluster variation (see FF for further information). Thus, these criteria are consistent with the present goal of constructing clusters that maximize both distinctiveness and internal homogeneity in geometric space. Both tests performed very well in Milligan and Cooper’s (1985) admittedly limited and idealized evaluation of cluster stopping rules.

In the temperature clustering, both tests suggested the 2, 4, 10, and 13 clustering levels, while the Duda–Hart test also pointed to the 7-cluster solution. For precipitation, the 3-, 6-, and 15- cluster partitions were highlighted by both tests. To keep the initial consensus clustering solution of reasonable size, the 4-cluster temperature and 6-cluster precipitation solutions were selected for what will be termed the “lower-order consensus solution.” The higher-order consensus solution will utilize the 7-cluster temperature and 15-cluster precipitation partitions. The chosen clustering levels are subjectively believed to represent different, but adequate, compromises between generality and detail.

b. Assessment of cluster stability

As noted earlier, the specific locations of the cluster boundaries, whether considered in geometric or physical space, may be sensitive to perturbations in the analysis strategy, especially owing to the unrealistic constraint of cluster hardness. Thus, it is useful to attempt to assess the stability of the regionalizations with regard to cluster boundary placement. In addition, the objects could conceivably suffer from overclustering or underclustering, depending on the clustering level chosen (Fig. 2 presents a very simplified example). In the former case, it would be expected that the stability of the partitions to analysis perturbations would generally increase as fewer clusters are retained. That is, the principal cluster could be far more stable than the subclusters it could be further partitioned into. In the latter case, stability would likely degrade as progressively more disparate clusters are forced to merge, with small perturbations possibly forcing the creation of entirely different fusions. Thus, it is also useful to attempt to ascertain whether a given clustering level is appropriate for the data.

Following Gong and Richman (1995), “bootstrap resampling” was employed as a stability assessment tool. In this strategy, fictional temperature and precipitation datasets were constructed by randomly sampling p′ = 500 attributes with replacement from the original p attributes. For each data type (temperature and precipitation), 10 trials were conducted, resulting in a total of 20 new “perturbed” datasets. As in Gong and Richman (1985), 10 trials were deemed to be adequate for this task. These data were cluster analyzed in the same manner as the original datasets. Clustering solutions from 2 to 20 clusters, inclusive, were constructed, yielding a total of 380 perturbed clustering solutions. Other, including smaller, values of p′ were also investigated, but these merely served to confirm the fundamental results of this test.

The perturbed solutions must be compared somehow with the control partitions made using the entire set of attributes. This is a potentially formidable problem faced in many clustering studies. Here, two somewhat related approaches were pursued. In the first, for each data type and at each clustering level, a contingency table was constructed using each perturbed partition and its corresponding control. This resulted in a set of 380 such k × k tables, where k is the number of clusters in both the perturbed and control partitions. The generally accepted procedure for quantitatively comparing two partitions is the Hubert and Arabie (1985) adjusted Rand index (ARI), a chance-adjusted measure of association. The ARI is a normalized index bounded between zero (chance association) and unity (perfect association). In a comparison of five competing measures Milligan and Cooper (1986) concluded that the Hubert–Arabie ARI was the “index of choice” for partition comparison.

For each data type and at each clustering level, the ARI values for each of the 10 trials was computed and summary statistics were compiled. Figure 3 shows how those summary statistics varied with clustering level for the temperature and precipitation clusterings. Table 3 presents detailed information for the clusterings selected for consensus intersection.

In this specific application, the usefulness of these summary statistics, and the ARI in general, is limited by several important factors. In the above analysis, a k-cluster perturbed solution was only compared to the k-cluster control partition, even though it could have been more strongly comparable with the control solution at a slightly different clustering level. (FF discusses this in their section 4.) In these cases, the perturbed solutions could be forging nearly comparable clusters as the control solution, but in a somewhat different order within the hierarchical clustering. This would make the perturbed and control clustering solutions at a given clustering level appear less comparable than they actually are. This difficulty becomes more pronounced as the number of clusters retained in the solutions increases.

Conversely, at the higher clustering levels, the number of raw discrepancies—CDs assigned to different clusters in the control and perturbed solutions at the same clustering level—tended to escalate, a factor not particularly well captured by the chance-adjusted ARI. For large k, the k × k contingency table between the control partition and a perturbed solution could have a lot of empty cells (and thus the association could appear far better than chance) even if the two clustering solutions are very different. This is why the variation among the trials tended to decrease with increasing clustering level in Fig. 3. For small k, on the other hand, the presence of a few discrepancies can result in very low ARI values, even when the differences between the partitions appear to be climatologically irrelevant. This occurred, for example, at the 3-precipitation-cluster level in Fig. 3b.

This may argue for an association measure based instead on the raw number of discrepancies, which means removing the adjustment for chance in the ARI. By itself, however, a discrepancy count could be deceptive, as a small, climatologically unimportant, shift in the boundary of a particular cluster could result in a relatively large number of CDs shifting in cluster assignment, leading to a high discrepancy count that obscures the true measure of association between the two partitions. Unfortunately, there can be some degree of subjectivity in the determination of which cells in the contingency table represent matches and which represent discrepancies. If the partitions are highly comparable, however, discrepancy identification is obvious.

For this application, it is believed that cluster stability can be best assessed and demonstrated by presenting spatial mappings of discrepancy counts. This represents an enhanced version of the method employed in FF. For each of the four designated independent clusterings, the ten perturbed trials were examined and the discrepancies were recorded. Each CD object, then, has a discrepancy count that ranges between 0 and 10 (the number of trials) for each independent clustering. Superimposing the cluster boundaries on the spatial mapping of the discrepancy count then provides both a qualitative and quantitative measure of the stability of the constructed clusters and illustrates the uncertainty regarding the specific placements of cluster boundaries in the physical domain.

c. A lower-order consensus intersection

As noted above, the 4-cluster temperature and 6-cluster precipitation partitions were chosen for the low-order consensus clustering solution. For temperature, the four clusters had membership sizes of 154, 91, 91, and 7 members; for precipitation, membership sizes were 151, 88, 72, 20, 6, and 6. As a test, the coseasonality component was removed from the temperature distances. This clustering, which will not be considered further, also contained a prominent 4-cluster solution with (unsurprisingly) more unevenly sized memberships of 198, 93, 46, and 6 objects.

The independent temperature and precipitation solutions are shown in Figs. 4 and 5, respectively, and are summarized in Table 4. It is clear that both solutions are very general. The four temperature clusters (Fig. 4a) reflect the overwhelming dominance of mean/level distance in the temperature EDs already revealed in section 2c. Thus, the labels used in Table 4 (coolest, cooler, warmer, and warmest) were assigned on the basis of cluster average mean annual temperature. These easily computed cluster averages might be used to quantify the temperature climate types represented by the clusters, or to suggest classification bounds for rule-driven schemes like the Koeppen approach. Indeed, east of the Rocky Mountains, the clustering is reminiscent of the Koeppen partition, though it is unsurprising that the data- and rule-driven assignments do not precisely match. In the Koeppen scheme, these locales are judged to receive sufficient precipitation and are thus differentiated solely on the basis of temperature information, though the temperatures of the extreme months are used in place of annual mean data.

The discrepancy map for the temperature regionalization (Fig. 4b) shows that discrepancies between the control and perturbed clusterings mainly resided along the clusters’ spatial boundaries, illustrating the uncertainty regarding specific placements of boundaries when hard clustering is employed. For example, as the dataset was perturbed, the boundary between the coolest and cooler clusters in the eastern United States tended to wander somewhat. Two of the perturbed datasets (trials 3 and 8 in Table 3) resulted in radically different partitionings in the mountainous west (Wyoming, Montana, etc.) at the 4-cluster level; these divisions joined the cooler cluster instead, weak evidence for possible underclustering.3 In contrast, none of the 91 CDs in the warmer cluster, extending from southern California eastward to Georgia, shifted in cluster assignment at all through the perturbation trials.

The precipitation regionalization (Fig. 5a) identifies three areally extensive regions (the mountainous west, Midwest, and southeast–east coast) and three smaller zones (Florida, a general West Coast grouping, and the Pacific Northwest), the latter three identifying the most distinctive CDs in the dataset. Labels were assigned on the basis of cluster average mean annual precipitation accumulation, along with phasing reflecting the coseasonality distance component, again owing to the dominance of those two particular ED components (section 2c). However, as the important coseasonality distance is not reducible to a single cluster average (taken over all of the cluster’s members), this makes it very difficult to derive classification rules from the precipitation climate clusters.

The discrepancy map for the precipitation solution (Fig. 5b) again shows that shifts in cluster assignments in the perturbed datasets occurred most frequently along the cluster boundaries. As was the case with temperature, two of the perturbed datasets (again, trials 3 and 8 in Table 3) resulted in radically different clustering solutions (this time in the Great Plains area), again representing weak evidence of underclustering.4 The most severe problem appeared in the Idaho region, which clustered with northern California in the control solution but rarely did so in the perturbed datasets; these CDs fused with the inland west region instead. Owing to the high number of discrepancies, and the fact that they themselves tend to cluster spatially, it is apparent that this portion of the control solution is very suspect. However, revision of the control solution was deemed unnecessary because the northern California and Idaho regions did become separated after the consensus intersection was performed anyway. Therefore, altering the precipitation solution actually would not affect the resulting consensus clusters. In other words, the intersection has tended to mitigate some of the instability present in this component clustering.

Table 4 also documents the result of the categorical intersection of the two solutions shown in Figs. 4a and 5a. Figure 6 overlays the graphical symbols employed in the independent solutions, permitting the reader to view the temperature subzones of the precipitation clusters (or vice versa) if desired. Intersection creates 24 possible temperature/precipitation subtypes, of which 7 are empty and 3 are single-member isolated clusters. The three isolates were judged to have been orphaned members and were subjectively reassigned (see Table 4), yielding a 14-cluster consensus solution. Reassignment of the northeast Florida CD was considered particularly well justified. On the basis of the discrepancy maps, this CD was reassigned to the principal southeastern cluster that was forged by categorical intersection.

The consensus result may be compared to FF’s 14-cluster solution (their Fig. 6), which was obtained with mixed temperature and precipitation information. While the overall level of detail appears similar, the consensus solution is less complex along the West Coast and more detailed in the mountainous interior west. The principal California cluster has also claimed the Puget Sound lowlands region in Washington State, as these areas were not distinguishable with respect to either temperature or precipitation at the present level of detail. The southeast clusters in the two regionalizations are very similar, while some other clusters (such as the one in central Texas and Oklahoma) appear qualitatively similar though shifted in space. Two points to consider are that 1) the consensus strategy has resulted in a different effective combination of temperature and precipitation information, and 2) as noted in FF, hard algorithms in particular will be especially sensitive to any procedural alterations in areas where the cluster boundaries are not very well defined anyway. This was revealed in the discrepancy maps.

One advantage of the consensus approach is that the analyst can see from the independent, contributing clusterings which data type encouraged which particular cluster fusion or division. For example, the consensus and FF regionalizations both divide the eastern third of the United States from northern Maine to southern Georgia into three groupings: northern New England, an east central zone, and a southeast zone. As these CDs all fall into a single precipitation cluster, the distinction is purely on the basis of temperature (especially mean temperature owing to its dominance in the total temperature distance). It is the temperature contribution that causes the southern section of Texas to stand apart from the rest of the state in both regionalizations, while precipitation distinguishes much of the Pacific Northwest coast (apart from the Puget Sound area) from the balance of the West Coast in the consensus outcome. The present strategy allows this to be easily visualized.

d. A higher-order solution

The 15-cluster precipitation and 7-cluster temperature clusterings were chosen for the higher-order consensus outcome. Although the independent clusterings themselves are not shown, the cluster boundaries are again superimposed on the spatial discrepancy maps presented in Fig. 7. The temperature clustering had cluster sizes of 142, 91, 51, 40, 10, 7, and 2 members (see Table 5). The two warmer clusters from the 4-cluster solution were unchanged, while the cooler clusters were further subdivided. The coolest cluster (shown in white in Fig. 4a) was divided into western and eastern segments at the Montana–North Dakota border. Recall that it was this segment of the 4-cluster solution’s coolest cluster that was least stable in the perturbed analyses. In the other cooler-than-average grouping (shown in light grey), the West Coast CDs were not only separated from the remaining members but also further partitioned into two zones that meet at the latitude of San Francisco.

The 15-cluster precipitation solution had membership sizes of 86, 68, 47, 41, 31, 24, 14, 9, 6, 5, 4, 4, and 2 clusters, along with two isolates. The large eastern cluster of the 6-cluster regionalization (Fig. 5b) was divided into three smaller zones located in the northeastern, southeastern, and southwestern sections of the original cluster, along with a small, narrow coastal grouping incorporating southern Louisiana and Mississippi, as well as an isolated member in southeast Texas. The southern part of Texas was separated from the long, narrow centrally located cluster in the 6-cluster solution. The extensive interior western zone was split into western and eastern segments roughly along the front range of the Rocky Mountains. The Florida Keys separated from the remaining southern Florida CDs, becoming an isolate. Only the Pacific Northwest cluster was unaltered.

Both independent clusterings were found to be more sensitive to perturbations than their more general counterparts employed in the low-order consensus solution. The least stable region in the 7-cluster temperature solution is located in a belt of CDs starting in Nevada and extending through Utah, northern Arizona, and New Mexico (Fig. 7a). In both the 4- and 7-cluster control solutions, these CDs joined with the cooler cluster extending eastward from Kansas. In the perturbed solutions, however, these locales sometimes were merged with the Montana/Wyoming cluster located to the north, while in other trials they joined with the cluster residing in the northeastern United States. This same area was determined to be more stable in the 4-cluster perturbation tests, so it is possible the 7-cluster solution has overclustered at least this portion of the spatial domain. As was the case in the 4-cluster solution, the principal southern (warmer) cluster stretching from California eastward to Georgia was found to be extremely stable.

The discrepancy map for the precipitation clustering (Fig. 7b) shows that the most sensitive area by far is located in the mountainous west. As this same area was found to be more stable in 6-cluster precipitation solution, this may again indicate that this region has been overclustered. Overclustering is also suggested in the southeast quadrant. The principal cluster spanning this region at lower clustering levels was far more stable to analysis perturbations than the subclusters of which it was composed.5

The consensus clustering generated upon categorical intersection of the two contributing solutions is necessarily more complex. Yet, of the 105 possible consensus clusters, only 33 were nonempty and 8 of those were isolates (Table 5). Subjective inspection, guided by the discrepancy maps and the lower-order consensus solution, led to the reassignment of seven of the eight isolates, yielding 26 clusters total. Reassignment was permitted to alter an object’s temperature or precipitation classification, but not both. Some of the isolates were clearly orphaned clusters, such as the CD in western Kentucky that resided at the spatial meeting point of 3 consensus clusters. Reassignment reduced the total number of precipitation categories to 14 (Table 5).6

The consensus regionalization is shown in Fig. 8. Because of the greater complexity of this consensus solution, no attempt was made to show the temperature/precipitation subzones. Instead, patterns were selected to correspond as closely as possible to those used in FF’s 25-cluster solution (their Fig. 7) to facilitate comparison with that clustering.

As in the lower-order consensus regionalization, greater simplicity in the northwestern quadrant means enhanced complexity in the remainder of the conterminous United States at the same clustering level. Also, comparison of the consensus outcome with FF’s 25-cluster solution again illustrates how a regionalization may change when the effective mixture of the contributing information is altered. Temperature information appears to play a greater role in the consensus outcome than in FF’s regionalization. Because of this, the consensus clusters are more spatially fragmented in the mountainous west; the reason for this may be seen even in the 4-cluster temperature map (Fig. 4a). The southeast, on the other hand, is less fragmented. The FF 25-cluster regionalization produced a spatially discontinuous cluster joining western Texas and the middle section of the east coast but excluding most of the CDs located in between. That combination was encouraged by the precipitation component, as a local maximum in winter precipitation resided in the area in between that cluster’s fragments (see the second principal component scores displayed in FF’s Fig. 4b). It may be impossible to objectively judge which of the two solutions is the superior regionalization, but since the present strategy allows for a more controlled combination of the contributing information types, the consensus solution deserves consideration.

e. Comment on cluster stability and the order of consensus intersection

In the introduction, the large information loss that may occur if the component clusterings employed in a consensus solution are too general (i.e., too few clusters retained) was identified as a possible disadvantage of the consensus clustering strategy. For k, l small, the consensus outcome could still generate a lot of clusters (up to k × l in number) that may be potentially less distinctive than the clusters forged from mixed data at the same clustering level q. However, for these data, the results discussed above suggest this possible disadvantage may be mitigated by the fact that the more general component clusterings tended to be substantially more stable to analysis perturbations. In addition, some of the instability present in the component solutions did not affect the consensus outcome at all anyway, owing to the nature of the categorical intersection. Thus, it may actually be advantageous to construct relatively detailed solutions from more general component clusterings in the case where overclustering is a substantial problem.

4. Discussion and conclusions

In this paper, Fovell and Fovell’s (1993) problem of using cluster analysis to construct “climate clusters” in the conterminous United States was revisited with a simpler, different, and in some ways superior, methodology applied. These climate clusters are useful for demonstrating the spatial scales of climatic variability and also for suggesting avenues for reducing the number of stations in a dataset in a climatically justified manner. Both studies shared the same basic motivations: that climate can be defined from temperature and precipitation time series data alone, and that climatic similarity can be inferred from the manner in which locales group together in a cluster analysis employing those data.

These studies represent the data-driven as opposed to the rule-driven approach to climate regionalization or classification. An example of the latter is the famous Koeppen (1923) classification, in which locales are assigned to predefined climate types on the basis of externally specified classification criteria (which also employed temperature and precipitation data). One advantage of the data-driven strategy is that it is useful in instances where the classification rules are unknown or arguable. One disadvantage of this approach is that its regionalization may not be easily reducible to a set of systematic classification rules, and thus, unlike the rule-driven strategy, it may not satisfy the desire or need to quantify climate type on the basis of statistical properties or mathematical formulae.

In Fovell and Fovell (1993), the time series were transformed into a total of 24 long-term monthly averages (12 each for temperature and precipitation). These variables or attributes were highly intercorrelated, however, which raised concerns about redundant information. Although the correlation coefficient is a very imperfect measure of attribute redundancy, it remains that if two attributes are perfectly correlated, one merely magnifies the scale of the other. Therefore, the attributes were further manipulated using principal component analysis. Difficult truncation and scaling issues had to be confronted before an interobject Euclidean distance matrix could be constructed and a clustering algorithm applied to it, however, and the regionalizations obtained were very sensitive to how those issues were resolved.

A distinctly different approach was taken in this work. The main problem in the Fovell and Fovell (1993) strategy is that different types of information (specifically, temperature and precipitation data), measured in arbitrary and incommensurable units, were being mixed prior to the computation of interobject distances. In the present work, independent clusterings from temperature and precipitation time series data were constructed and then combined through categorical intersection to produce a new, consensus outcome. Dealing with one information type at a time in large measure neutralizes not only concerns about scaling (which is well known to be of paramount importance in clustering) but also information redundancy.

When the data employed represent continuous time series drawn from a single source or data type, the squared Euclidean distance between any two objects may be considered as the sum of three components that reflect differences in the series’ temporal means (levels), variabilities (seasonal amplitudes) and covariabilities (phase differences or coseasonalities), respectively. Since the attributes do not represent a collection of disparate data types, it does not matter if they, as temporal observations, are not independent. Indeed, in this situation, it may be desirable to obtain very long time series, as the mean, standard deviation and correlation coefficient statistics upon which the Euclidean distance components depend might be more accurately and stably calculated. This is a major advantage of the consensus approach.

In some situations, it might be advisable to adjust the data to remove one or two of the distance components, although the total combined distance was employed in this work. It turned out, however, that the temperature distances were overwhelmingly dominated by interobject differences with respect to long-term annual average temperature, while the precipitation distances were significantly influenced by the covariability component. The latter component frustrates the reduction of a precipitation clustering into a set of classification rules, but it is felt that the realism of a climate clustering solution would suffer without it.

A consensus outcome is obtained when k-cluster temperature and l-cluster precipitation component clusterings are categorically intersected, yielding a solution with as many as k × l consensus clusters, although many potential clusters may be empty sets. The clustering levels for the contributing solutions (which need not be limited to two, as herein) are determined separately. This is both an advantage and a disadvantage. The user may independently select the desired level of detail for the contributing clusterings, but this makes the outcome essentially subjective and lacking in uniqueness. In order to avoid an excessively muddled consensus solution, it may be necessary to employ relatively general independent clusterings in which a large amount of detail has already been sacrificed. This is a potential disadvantage of the consensus strategy.

On the other hand, it was found herein that the more general component clusterings also tended to be more stable to analysis perturbations and that consensus intersection tended to mitigate some of the instability present in the component clusterings. Thus it is quite possible that at a given clustering level (say q clusters), a q-cluster consensus solution constructed from more general and stable component clusterings (e.g., k < q, l < q) may be more stable than the q-cluster solution obtained when the data types were combined. Further work would be necessary to establish the general applicability of this result.

Also, it was shown that categorical intersection can result in the formation of unrealistic orphan clusters (usually containing a single member), requiring essentially subjective reassignment. In this study, orphan reassignment caused little alteration to the consensus solution, and thus the problem was certainly not serious.

Using the group average linkage hierarchical clustering algorithm, two consensus solutions were constructed and examined, a lower-order partition using relatively general contributing clusterings (containing 4 temperature and 6 precipitation clusters) and a higher-order solution designed to preserve a greater amount of detail (with 7 temperature and 15 precipitation clusters). The sensitivity of the independent clusterings to analysis perturbations was assessed using bootstrap resampling. The clusterings employed in the lower-order consensus result were judged to evince very weak evidence of underclustering (i.e., retention of too few clusters), while strong evidence of overclustering was found in those used in the higher-order consensus solution. After reassignment of suspected orphan clusters, the former resulted in a 14-cluster consensus outcome while the latter produced 26 groupings. These outcomes were compared to Fovell and Fovell’s (1993) 14- and 25-cluster regionalizations, respectively.

In general, the consensus outcomes appeared to be acceptable and consistent to some degree with the solutions produced using the Fovell and Fovell (1993) approach. Discrepancies between the clusterings of the two approaches could be related to how the temperature and precipitation data types were mixed. Other clustering algorithms were examined for this study but were found to not work as well as group average linkage with these data.

Fovell and Fovell (1993) concluded by stating several ways their study could be improved, and many of those recommendations remain to be explored. For example, this study implicitly assumes that temperature and precipitation data themselves are sufficient to construct a reasonable climatic clustering. It may be that different data, perhaps reflecting moisture demand and supply, would produce an objectively superior outcome with either the consensus strategy or the Fovell and Fovell (1993) approach. Also, the consensus strategy should be applied to other datasets to more fully assess its advantages and shortcomings. It is believed that the present results argue for the general validity of the consensus clustering approach.

Acknowledgments

This work was supported by the Academic Senate of the University of California, Los Angeles, and by the National Science Foundation under Grant ATM-9421847.

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APPENDIX

Derivation of the Euclidean Distance Components

Equation (1) shows how the Euclidean distance d2ij between two objects i and j is computed using two time series x of length p. Expansion of the squared term on the right-hand side yields
i1520-0442-10-6-1405-ea1
Dividing the two time series into mean (designated using overbars) and deviation (indicated with primes) quantities produces, after further expansion,
i1520-0442-10-6-1405-ea2
Distribution of the summation through the right-hand side eliminates the four terms that are products of mean and deviation values (since the sum of deviations is naturally zero). Using conventional definitions for the sample variance Si and Pearson product moment correlation coefficient rij, the following expressions are created:
i1520-0442-10-6-1405-ea3
which yield, after substitution into (A2),
i1520-0442-10-6-1405-ea6
Further simplifying the first term on the right and completing the square involving the remaining three terms results in
d2ijpij2pSiSjpSiSjrij
The three terms on the right-hand side are the mean (level), seasonality (variability), and coseasonality (phase) distance components previously given in (3a). In the preceding, the sample estimate of the population standard deviation (si) may have been used in place of the sample standard deviation Si; this would cause p in the last two terms to be replaced by (p − 1). This has no effect on the computed components or their interpretations. For p large, as herein, p/(p − 1) ∼ 1 anyway, so the distinction is negligible.

Fig. 1.
Fig. 1.

Demonstration of consensus clustering via categorical intersection illustrated in the spatial domain, depicting (a) an optimal intersection and (b) an intersection leading to the production of orphaned clusters.

Citation: Journal of Climate 10, 6; 10.1175/1520-0442(1997)010<1405:CCOUST>2.0.CO;2

Fig. 2.
Fig. 2.

Very simple example of overclustering and underclustering, depicted in a p = 2 dimensional geometric space defined by two attributes: X1 and X2. In (a), which illustrates overclustering at the higher clustering level, the 2-cluster solution would likely be stable to perturbations, while the specific object assignments in the 4-cluster partition would likely be more unstable (owing to perturbation-induced shifts in the intercluster boundaries). In (b), which illustrates underclustering at the lower clustering level, the 3-cluster solution composed of groups A, B, and C would be stable, but the partition at the 2-cluster level would likely be very sensitive to perturbations in the analysis strategy. In this example, such perturbations could arise from varying the clustering algorithm or the number of objects employed. In this paper, both of those are held constant, and it is the attribute selection and dimensionality that are varied in the bootstrap resampling. However, the basic concept is qualitatively similar.

Citation: Journal of Climate 10, 6; 10.1175/1520-0442(1997)010<1405:CCOUST>2.0.CO;2

Fig. 3.
Fig. 3.

Adjusted Rand index summary statistics from the bootstrap resampling cluster sensitivity test, by clustering level for the (a) temperature and (b) precipitation with independent clusterings. Statistics include median, mean, and upper (Q3) and lower (Q1) quartiles, compiled for 10 resampling trials for each clustering level. See Table 3 for additional information.

Citation: Journal of Climate 10, 6; 10.1175/1520-0442(1997)010<1405:CCOUST>2.0.CO;2

Fig. 4.
Fig. 4.

The 4-cluster independent temperature solution used in the construction of the low-order consensus solution shown in Fig. 6. Shown are (a) the control regionalization and (b) the discrepancy count map resulting from bootstrap resampling perturbation analysis at the same clustering level. Locations of the cluster boundaries are superimposed on panel (b). The discrepancy count for a given climate division object reflects the number of times in the 10 perturbed trials that it was assigned to a different cluster than in the control partition shown in panel (a). See Table 3 for further information.

Citation: Journal of Climate 10, 6; 10.1175/1520-0442(1997)010<1405:CCOUST>2.0.CO;2

Fig. 5.
Fig. 5.

The 6-cluster independent precipitation solution used in the construction of the low-order consensus solution shown in Fig. 6. Shown are (a) the control regionalization and (b) the discrepancy count map resulting from bootstrap resampling perturbation analysis. Locations of the cluster boundaries are superimposed on panel (b). The discrepancy count for a given climate division object reflects the number of times in the 10 perturbed trials that it was assigned to a different cluster than in the control partition shown in panel (a). See Table 3 for further information.

Citation: Journal of Climate 10, 6; 10.1175/1520-0442(1997)010<1405:CCOUST>2.0.CO;2

Fig. 6.
Fig. 6.

Lower-order 14-cluster consensus outcome obtained by overlaying the temperature and precipitation solutions shown in Figs. 4 and 5. The gray shadings and patterns used in the temperature and precipitation clusterings are retained, producing a graphical approach to categorical intersection. This permits the reader to either focus upon the precipitation subzones of the temperature clusters (or vice versa) or to consider each combination of gray shading and pattern to be a separate grouping. Inset map depicts cluster boundaries, with pattern fill only employed for spatially discontinuous clusters. Three suspected orphan clusters have been reassigned, as discussed in the text and Table 4.

Citation: Journal of Climate 10, 6; 10.1175/1520-0442(1997)010<1405:CCOUST>2.0.CO;2

Fig. 7.
Fig. 7.

Discrepancy count map, superimposed with cluster boundaries, for the (a) 7-cluster temperature and (b) 15-cluster precipitation independent clustering solutions. As with the discrepancy maps shown in Figs. 4 and 5, the count reflects the number of times a given climate division object was differently assigned in the 10 perturbed trials. See Table 3 for further information.

Citation: Journal of Climate 10, 6; 10.1175/1520-0442(1997)010<1405:CCOUST>2.0.CO;2

Fig. 8.
Fig. 8.

Higher-order 26-cluster consensus outcome resulting from categorical intersection of a 7-cluster temperature solution with a 15-cluster precipitation solution. Regionalization shown has been adjusted to remove some suspected orphan clusters, as discussed in text and summarized in Table 5. Instead of using graphical overlay (as in the lower-order outcome in Fig. 6), patterns were selected to match those used in Fovell and Fovell’s (1993) 25-cluster reference solution as closely as possible (see their Fig. 7). The two 4-member clusters located in eastern Tennesse–western North Carolina and central North Carolina–eastern Virginia were judged to be especially unstable consensus clusters, and their reassignment could be justified.

Citation: Journal of Climate 10, 6; 10.1175/1520-0442(1997)010<1405:CCOUST>2.0.CO;2

Table 1.

Contributions of the Euclidean distance components to the total squared interobject distance.

Table 1.
Table 2.

Average linear association between Euclidean distance components and spatial separation.

Table 2.
Table 3.

Results of bootstrap resampling perturbation analysis for selected independent (control) clustering solutions. All trials used p′ = 500 sampled attributes, as discussed in text. The ARI statistic is Hubert and Arabie’s (1985) adjusted Rand index. The discrepancy count reflects the number of climate division objects (of 343 total) that were differently assigned in a given perturbed trial than in the control solutions.

Table 3.
Table 4.

Lower-order consensus clustering solution.

Table 4.
Table 5.

Higher-order consensus solution (revised, showing modifications), constructed from the independent 7-cluster temperature and 15-cluster precipitation solutions.

Table 5.

1

A large part of the rapidly expanding literature on consensus clustering (cf. Day 1986; Barthelemy and Monjardet 1988; Arabie and Hubert 1992) concentrates on techniques for merging independently constructed dendograms from hierarchical clusterings into a compromise or best-fitting dendogram, rather than on accomplishing categorical intersection of independently constructed clustering solutions as performed herein. The consensus dendogram approach is especially useful in situations when it is desirable to find the compromise solution among a set of dendograms obtained by applying different clustering algorithms on the same set of attributes. As our independent dendograms are obtained from different attribute sets and could be expected to generate very different dendograms, we believe that the strategy of intersecting the clustering solutions is indicated. Our approach is consistent with the term consensus clustering (P. Arabie 1995, personal communication).

2

This statement strictly applies only when scaling alterations involve linear transformations, such as the conversion of temperature data from degrees Fahrenheit to degrees Celsius. In that case, although the computed ED values are changed, the clustering outcome is not. Nonlinear transformations, such as the square root transform applied to the precipitation data or the conversion to standardized scores, alters the shape of the geometric space and therefore can affect the clustering outcome.

3

At the 5-cluster level, these two trials were very similar to the counterpart control solution. In making the 4-cluster control solution, however, the CDs in Wyoming and Montana join with the northeast United States, while in these two particular trials, those CDs join with the West Coast CDs instead. Underclustering is suggested because most of this same area was found to be more stable at higher clustering levels (see Fig. 6 and section 3d). However, the relatively small discrepancy counts in this region suggest the control clustering as shown is not especially unstable or unjustifiable.

4

At a higher clustering level in both the perturbed and control solutions, a separate cluster resided in the region extending from northwest Texas northward to the Canadian border (see Fig. 6 and section 3d). In the control solution, this cluster fused with the balance of the interior west, while in those two particular trials it joined with the cluster immediately east of it. Again, owing to the relatively small number of discrepancies involved, this is considered only weak evidence of underclustering.

5

The discrepancies in the southeast quadrant occurred because, while half of the perturbed trials produced a very similar clustering to the control solution there, the other half (trials 2, 3, 6, 9, and 10 in Table 3) resulted in a radically different partitioning. The discrepancies among the control and perturbed solutions, however, decreased dramatically at lower clustering levels, providing strong evidence of overclustering in this quadrant. If the 14-cluster precipitation solution had been selected instead, virtually all of the discrepancies found in the states of Alabama, Georgia, and North and South Carolina would have been absent. The 14-cluster level was not indicated by the statistical tests in the control clustering; that is one reason why it was not selected.

6

Although we did not reassign consensus clusters with more than one member, it may have been appropriate to do so in at least two cases, those involving the 4-member clusters in eastern Tennessee–western North Carolina, and central North Carolina–eastern Virginia. If the 14-cluster precipitation solution had been selected instead, the only change that would have occurred in the consensus outcome is that these clusters would have vanished, the former grouping joining with the northeastern U.S. cluster and the latter merging with the CDs to the south. As noted earlier, this would eliminate many of the discrepancies found in the precipitation clustering in the southeast quadrant.

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