An important step in projecting future climate change impacts on extremes involves quantifying the underlying probability distribution functions (PDFs) of climate variables. However, doing so can prove challenging when multiple models and large domains are considered. Here an approach to PDF quantification using k-means clustering is considered. A standard clustering algorithm (with k = 5 clusters) is applied to 33 years of daily January surface temperature from two state-of-the-art reanalysis products, the North American Regional Reanalysis and the Modern Era Retrospective Analysis for Research and Applications. The resulting cluster assignments yield spatially coherent patterns that can be broadly related to distinct climate regimes over North America, e.g., low variability over the tropical oceans or temperature advection across stronger or weaker gradients. This technique has the potential to be a useful and intuitive tool for evaluation of model-simulated PDF structure and could provide insight into projections of future changes in temperature.
 Although global warming impacts on climate are often framed in terms of mean change, the potential changes in extremes arguably pose a greater concern for societal or ecosystem adaptive capacity [Kharin et al., 2007; Trenberth et al., 2007; IPCC, 2012]. Nonlinear relationships between changing means and extremes suggest that even small changes in the former can be associated with large changes in the latter [Griffiths et al., 2005]. Quantifying vulnerability to and risks associated with extreme climatic events—and more critically, projecting how such risks may change in the future—requires detailed knowledge of the underlying probability distribution functions (PDFs) of important climate variables. Furthermore, in order to constrain uncertainties in model simulations of future climate extremes, standardized metrics are required to evaluate model fidelity.
 Because climate change may alter multiple moments of the PDF of a climate variable [Hannachi, 2006], evaluation metrics should give insight on multiple aspects of PDF structure. For example, Donat and Alexander  present evidence of increasing variance in the Tropics since the mid-twentieth century as well as a tendency toward more positive skewness. Additionally, there can be considerable spatial variation or dependence on small-scale processes in these PDFs [Easterling et al., 2000; Diffenbaugh et al., 2005]. While some theoretical guidance exists for relating the governing dynamics of a system to its PDF characteristics [e.g., Bourlioux and Majda, 2002; Sura and Sardeshmukh, 2008; Neelin et al., 2010; Stechmann and Neelin, 2011], further effort is needed to apply theoretical understanding to climate variables in observations and models.
 Analyzing surface temperature (Ts) PDFs from the Global Surface Summary of the Day product, Ruff and Neelin  documented non-Gaussian, often asymmetric long tails occurring over a wide range of geographic and climatic settings. They further noted how the details of the PDF tails significantly impact the estimation of threshold exceedances. For example, under a warming-induced shift of the distribution, locations with high-side Gaussian tails would experience a greater increase in a given warm threshold exceedance relative to locations with a fat (e.g., exponential) tail.
 The size and scope of currently available observational and model data products present practical challenges for generalizing the diagnosis, interpretation, validation, and intercomparison of PDF characteristics. Consequently, evaluation and comparison of large observational and model data sets require the development of flexible yet standardized and readily applicable diagnostics to facilitate model evaluation, interpretation, and development. To that end, we present results of a cluster analysis applied to Ts PDFs obtained from two reanalysis products. Our results demonstrate that a few stable PDF categories can be obtained and related to some readily understood aspects of different climatic regimes.
2 Data Sets and Methodology
 The North American Regional Reanalysis (NARR) [Mesinger et al., 2006] is a high-resolution reanalysis product covering North America. This data set is derived from a data assimilation scheme with near-surface observations ingested hourly and atmospheric profiles of temperature, winds, and moisture from rawinsondes and dropsondes ingested every 3 h. The native NARR data are available on a Lambert conformal grid (3-hourly, approximately 32 km).
 Additionally, Ts data are analyzed from the Modern Era Retrospective Analysis for Research and Applications (MERRA), developed by NASA's Global Modeling and Assimilation Office and disseminated by the Goddard Earth Sciences Data and Information Services Center [Rienecker et al., 2011]. MERRA assimilates observations from multiple sources including weather stations and balloons, satellite data, ships, buoys, and aircraft. This data product is defined on a global, regular uniform grid at a spatial resolution of 0.5° latitude × 0.67° longitude, which is coarser than NARR but finer than other widely used reanalysis products. Because of the different grid nests for NARR and MERRA, the latter covers more of the Earth at lower latitudes than the NARR domain. While other reanalysis products cover this domain, these two were chosen because the relatively high resolution allows for analysis of regional scale phenomena.
 To construct PDFs, daily January Ts data are first deseasonalized by removing the daily climatology over the 33 year (1979–2011) period; long-term linear trends are also removed for individual grid points. Only January is considered here for demonstration purposes. Anomalies are computed so that all grid points have a mean of 0, allowing for systematic comparison of PDFs across the domain. Anomalies for all 1023 days are sorted into bins of 0.5 K width using d = 152 bins at all grid points and normalized by the total number of days. While this results in many grid points having multiple bins with zero counts, d = 152 was necessary to span the range of temperature anomalies at all grid points. Next, k-means cluster analysis is applied to group the PDFs. Clustering is performed on the log of probability (log10[bin count(i)/1023] where i = 1, 2,…, 152) to increase the weight of distribution tails in the clustering. In other words, the clustering algorithm seeks k sets among these vectors of length d of the log of probabilities, over the data set of n spatial points, that minimizes the within-cluster sum of squares of the distance in the d-dimensional space.
 Here, k = 5 clusters is used for demonstration purposes. While the choice of k = 5 clusters is arbitrary, in a simple sensitivity analysis in which the number of clusters was varied from three to eight, we found the results for k = 5 to be straightforward to interpret physically. The optimal number of clusters and associated sensitivities will be explored in more detail in ongoing work that applies this methodology to evaluation of climate models.
3 Results and Discussion
 Figure 1 depicts the standard deviation (SD) and skewness of the daily Ts variability. In general, MERRA has smaller SD along the margins of sea ice (Labrador Sea, Bering Sea), while NARR has smaller values over western Canada and Alaska. The overall geographic pattern of skewness is very similar in both products. Moreover, both products are able to capture local features such as the band of positive skewness over the coastal waters adjacent to California and much of Baja California caused by offshore winds that induce strong positive temperature excursions, e.g., the Santa Ana winds in Southern California [Hughes and Hall, 2010]. Loikith and Broccoli  document a similar skewness structure using coarser-resolution gridded daily Ts observations. In a simple sensitivity analysis where the data were divided into two equal temporal intervals, the SD and skewness values did not change appreciably in most of the domain, suggesting that these patterns and values are stable with respect to time period at least over the late 20th and early 21st centuries.
 Although the spatial patterns of the second and third moment statistics in Figure 1 capture important features of daily Ts variability, PDF modality is not readily discernable in terms of a single moment. In this sense, it may be instructive to consider diagnostics of the overall shape of the PDFs, especially if such diagnostics are sufficiently limited, i.e., the number of shape categories is small. To this end, we apply the k-means cluster method. Figure 2 depicts the PDFs associated with each of the clusters, showing cluster-mean PDFs (thick lines) and ±1 SD (shading; calculated as the SD of all the points of the cluster within each temperature bin). Maps of the pointwise cluster assignments are in Figure 3. Here the colors plotted on the map correspond to the individual PDFs that comprise the mean PDFs in Figure 2, e.g., the red curve in Figure 2 is the mean of the PDFs for each red-shaded grid point in Figure 3. The mean SD and skewness values, computed as the average of all grid points within the cluster, are indicated in Figure 2. The clusters are numbered based on the mean SD of all grid points within the cluster from high (C1) to low (C5) SD values.
 The grid points falling into C1 consist largely of sub-Arctic regions and are characterized by high temperature variance, as evident by the wide PDF, and a relatively large spread within the cluster, reflecting significant local variations. This region matches the band of high SD in Ts (Figure 1) and includes the transition zone from predominantly negative skewness to the south and positive skewness to the north reflected in the symmetrical PDF. This region is subject to strong anomalous Ts advection associated with synoptic-scale weather events [Loikith and Broccoli, 2012] across a gradient such that either colder or warmer air masses can be advected into the region.
 C2 exhibits relatively high variance and encompasses the Arctic as well as the continental midlatitudes. The Arctic is an area of predominantly positive skewness, while a mixture of negative and positive skewness occurs over the continental midlatitudes (Figure 1). This combination is reflected in the symmetrical mean PDF. While the two regions described by this cluster have little in common climatologically, different mechanisms may allow for similar PDF characteristics, especially variance. The Arctic (C2) has lower variance than areas to the south (C1) since the region is among the coldest in the hemisphere, thus precluding outbreaks of extreme cold air (in an anomalous sense) that can occur at lower latitudes. The midlatitude region is within the main storm track but has lower variance compared with areas immediately to the north (C1) due in part to the modification of extreme cold air masses as they move equatorward.
 C3 encompasses the southwestern United States, northern Mexico, and the coastal waters of southern Alaska, the northern Gulf of Mexico, and the western Atlantic Ocean. Included in C3 are coastal regions of high temperature gradient on the West Coast and regions of high oceanic temperature gradient off the East Coast of the U.S. Comparing to Figure 1, C3 includes some ocean regions with relatively high variance as well as the southwestern portion of the continent which has relatively low variance for a continental region. The mean PDF also exhibits negative skewness, especially evident over coastal Alaska. Here, the negative skewness is likely caused by extreme cold outbreaks associated with advection from the continental interior combined with a limited warm tail associated with the moderating effect of the ocean. The region over the Atlantic has high storm frequency in the winter, which elevates the temperature SD relative to other marine regions.
 C4 and C5, describing the midlatitude oceans and Tropics, respectively, have the smallest variance of the five clusters, associated with a smaller temperature gradient in the Tropics and with the moderation of advective effects by ocean heat capacity over C4. A substantial part of C4 is also to the south of the main storm track. C5 is south of the storm track and experiences smaller effects by midlatitude synoptic-scale weather variability. The mean PDF of C4 (and C5 for NARR) is characterized by a long cold tail, likely reflecting the occasional incursion of cold air masses from across the temperature gradient on the midlatitude side. The relatively short warm tail likely reflects the small gradient toward warmer tropical temperatures; as such, it is not possible to strongly increase temperatures by warm advection.
 The approach described here yields a first view of regional distributions of PDFs; however, to emphasize differences in PDF shape, cluster analysis is applied to PDFs computed from anomalies normalized by their SD. If all the distributions were Gaussian, the normalization would tend to collapse them into a single cluster, so this approach can be anticipated to give a view of the prevalence of non-normality. Figure 4 shows an example in which three clusters are used to group PDFs of normalized temperature anomalies. The PDF cluster assignments reflect the higher-order moments of skewness and kurtosis. While skewness appears to be the most apparent characteristic for clustering, kurtosis is also influential with C1 having the highest kurtosis.
4 Summary and Conclusions
 Variations in Ts PDFs over a large geographic area encompassing North America and surrounding oceans are examined using simple k-means clustering. In both data sets, the cluster analysis yields stable, spatially coherent patterns that can be understood in terms of distinct Ts regimes, such as smaller variability over tropical oceans and larger variability over the high-latitude continental interior. The shape of the reconstructed PDF for each cluster, along with the geographical distribution of the clusters, fits well with physical interpretations in terms of temperature advection in the presence of a maintained background temperature gradient and advection by synoptic-scale events. In general, temperature variances appear to be the leading determinant in defining clusters. Skewness also affects some cluster assignments, suggesting that cluster-based approaches are useful for identifying regions with common PDF shape. By normalizing the temperature anomalies by their SD, it is possible to use cluster analysis to group PDFs based on higher moment statistics, providing important information for characterizing regional sensitivity of temperature extremes to future warming.
 Future work will focus on developing the cluster analysis approach outlined in this paper for categorizing PDF characteristics in regional climate model simulations for the purpose of evaluating model data against observations/reanalysis. While other methodologies exist for systematic PDF evaluation, the ability of this tool to be used over large domains or numbers of grid points makes it particularly versatile. For example, Perkins et al.  developed and applied a PDF skill score for model evaluation over Australia using relatively homogenous subregions. Their technique provides a concise and standardized way to evaluate models; however, the clustering method has the advantage that it works over large inhomogeneous domains. Furthermore, this approach may serve to identify regions where future changes in Ts or other climate variables are likely to be relatively homogeneous. As such, this method may provide a foundation for elucidating changes in future climate extremes.
 Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Part of this research was funded by NASA National Climate Assessment 11-NCA11-0028 and AIST AIST-QRS-12-0002 projects, and the NSF ExArch 1125798 (P.C.L., J.K., H.L., and D.E.W). Part of this research was funded by NOAA NA11OAR4310099 (J.D.N.) and New Jersey Agricultural Experiment Station Hatch grant NJ07102 (B.R.L.). We thank Joyce Meyerson for her assistance with figure visualization.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.