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Michael B. Richman

Abstract

A detailed analysis of obliquely rotated principal components as a map typing technique was performed. This type of transformation does not constrain orthogonality of the vectors, allowing the components or map types the freedom to better reflect the original data set. Meteorological map types of sea level pressure, with previously known configurations, were utilized to ascertain the advantages and disadvantages of unrotated, orthogonally rotated and obliquely rotated components. The consequences of rotating varying numbers of components also were explored to determine the robustness of the two classes of rotations. Map types of sea level pressure for June through August 1971–1975 were generated and utilized successfully to stratify precipitation events which occurred during project METROMEX.

The obliquely rotated principal components were shown to be 1) superior to other rotational methods in retrieving input fields of synoptic scale data and, 2) less likely to distort the map types when an incorrect number of components were rotated. The obliquely rotated components proved to be a powerful data reduction technique and exhibited potential as a forecast tool since the probability of a specific weather event could be associated with a particular weather pattern.

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John E. Walsh
and
Michael B. Richman

Abstract

No abstract available.

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John E. Walsh
and
Michael B. Richman

Abstract

Thirty-one years of monthly data are used to evaluate the seasonal dependence of the associations between large-scale temperature anomalies over the United States and the North Pacific Ocean. Both station (grid-point) values and empirical orthogonal functions of temperature are used in the correlative analysis.

The North Pacific sea surface temperature (SST) anomalies correlate most highly with temperature fluctuations over the southeastern and the far western states. Correlations with the SST anomalies have opposite signs in the two portions of the United States. The associations with the SST anomalies are independent of season only in the western states. The association involving the southeastern states is strongest during the winter and insignificant during the summer. The pattern of North Pacific SST that correlates most highly with the United States temperature is an east-west SST gradient between the West Coast and 35°N, 160°W. Statistically significant fractions of temperature variance over at least some areas of the United States are described by regression onto the SST anomalies in all seasons except spring. The results imply some seasonal predictability based on North Pacific SST patterns alone, although useful predictability appears to be confined primarily to the winter.

The pattern analysis shows that the failure to rotate the dominant eigenvectors can obscure the spatial and temporal interrelationships deduced from the two sets of data fields.

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Andrew E. Mercer
and
Michael B. Richman

Abstract

Three common synoptic storm tracks observed throughout the United States are the Alberta Clipper, the Colorado cyclone, and the East Coast storm. Numerous studies have been performed on individual storm tracks analyzing quasigeostrophic dynamics, stability, and moisture profiles in each. This study evaluated storms in each track to help diagnose patterns and magnitudes of the aforementioned quantities, documenting how they compare from track to track. Six diagnostic variables were computed to facilitate the comparison of the storm tracks: differential geostrophic absolute vorticity advection, temperature advection, Q-vector divergence, mean layer specific humidity, low-level stability, and midlevel stability. A dataset was compiled, consisting of 101 Alberta Clippers, 165 Colorado cyclones, and 159 East Coast cyclones and mean fields were generated for this comparison. Maxima and minima of the 25th and 75th percentiles were generated to diagnose magnitudes and patterns of strong versus weak cyclones and measure their similarities and differences to the mean patterns. Alberta Clippers were found to show the weakest magnitude of quasigeostrophic variables, while East Coast storms had the strongest magnitudes. Alberta Clippers maintained the lowest moisture content through their life cycle as well. However, East Coast storms were the most stable of the three tracks. Typically, correlations between storm tracks were high; suggesting that storm evolution is similar between tracks, in terms of the patterns of diagnostic variables measured. However, significant magnitude differences in the quasigeostrophic variables distinguished the storms in each track.

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Kimberly L. Elmore
and
Michael B. Richman

Abstract

Eigentechniques, in particular principal component analysis (PCA), have been widely used in meteorological analyses since the early 1950s. Traditionally, choices for the parent similarity matrix, which are diagonalized, have been limited to correlation, covariance, or, rarely, cross products. Whereas each matrix has unique characteristic benefits, all essentially identify parameters that vary together. Depending on what underlying structure the analyst wishes to reveal, similarity matrices can be employed, other than the aforementioned, to yield different results. In this work, a similarity matrix based upon Euclidean distance, commonly used in cluster analysis, is developed as a viable alternative. For PCA, Euclidean distance is converted into Euclidean similarity. Unlike the variance-based similarity matrices, a PCA performed using Euclidean similarity identifies parameters that are close to each other in a Euclidean distance sense. Rather than identifying parameters that change together, the resulting Euclidean similarity–based PCA identifies parameters that are close to each other, thereby providing a new similarity matrix choice. The concept used to create Euclidean similarity extends the utility of PCA by opening a wide range of similarity measures available to investigators, to be chosen based on what characteristic they wish to identify.

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Michael B. Richman
and
Peter J. Lamb

Abstract

This paper presents the results of climatic pattern analyses of three- and seven-day summer (May–August) rainfall totals for the central United States. A range of eigenvectorial methods is applied to 1949–80 data for a regularly spaced network of 402 stations that extends from the Rocky to the Appalachian Mountains and from the Gulf Coast to the Canadian border. The major objectives are to quantitatively assess the sensitivity of eigenvectorial results to several parameters that have hitherto been the subject of considerable qualitative concern, and to identify the potential applications of those results.

The entire domain variance fractions cumulatively explained by a) the first 10 correlation-based unrotated Principal Components (PCs) and b) the 10 orthogonally rotated (VARIMAX criterion) PCs derived from them are identical for the same data. They vary between 35–47 percent depending on the data time scale and form, being higher for seven- than three-day totals and further enhanced when those totals are square-root (especially) and log10 transformed. The (highly contrasting) sets of unrotated and VARIMAX PC spatial loading patterns are invariant with respect to data time scale and form. They receive strong statistical support from analyses performed on subsets of the data, their covariance- and cross-products-based equivalents, counterpart common factor patterns, and (for VARIMAX) an obliquely rotated (Hanis–Kaiser Case II B′B criterion) PC analysis. The unrotated PC loading patterns very closely resemble the set that Buell claimed would tend to characterize a domain of the present rectangular shape, irrespective of the meteorological parameter treated. They receive little physical support from analyses performed separately for subareas of the domain or from comparison with the interstation correlation matrix from which they are derived. The VARIMAX PC loading patterns, in contrast, derive strong physical support from those verifications. Each of these patterns emphasizes a relatively strong anomaly in a different part of the domain; they collectively yield a regionalization of the domain into 10 subareas within which three- and seven-day summer rainfall tends to be spatially coherent. The regionalization is suggested to be of considerable potential utility for crop-yield modeling, short-range weather prediction, and research into climatic variation and change.

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Michael B. Richman
and
Xiaofeng Gong

Abstract

When applying eigenanalysis, one decision analysts make is the determination of what magnitude an eigenvector coefficient (e.g., principal component (PC) loading) must achieve to be considered as physically important. Such coefficients can be displayed on maps or in a time series or tables to gain a fuller understanding of a large array of multivariate data. Previously, such a decision on what value of loading designates a useful signal (hereafter called the loading “cutoff”) for each eigenvector has been purely subjective. The importance of selecting such a cutoff is apparent since those loading elements in the range of zero to the cutoff are ignored in the interpretation and naming of PCs since only the absolute values of loadings greater than the cutoff are physically analyzed. This research sets out to objectify the problem of best identifying the cutoff by application of matching between known correlation/covariance structures and their corresponding eigenpatterns, as this cutoff point (known as the hyperplane width) is varied.

A Monte Carlo framework is used to resample at five sample sizes. Fourteen different hyperplane cutoff widths are tested, bootstrap resampled 50 times to obtain stable results. The key findings are that the location of an optimal hyperplane cutoff width (one which maximized the information content match between the eigenvector and the parent dispersion matrix from which it was derived) is a well-behaved unimodal function. On an individual eigenvector, this enables the unique determination of a hyperplane cutoff value to be used to separate those loadings that best reflect the relationships from those that do not. The effects of sample size on the matching accuracy are dramatic as the values for all solutions (i.e., unrotated, rotated) rose steadily from 25 through 250 observations and then weakly thereafter. The specific matching coefficients are useful to assess the penalties incurred when one analyzes eigenvector coefficients of a lower absolute value than the cutoff (termed coefficient in the hyperplane) or, alternatively, chooses not to analyze coefficients that contain useful physical signal outside of the hyperplane. Therefore, this study enables the analyst to make the best use of the information available in their PCs to shed light on complicated data structures.

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Xiaofeng Gong
and
Michael B. Richman

Abstract

Cluster analysis (CA) has been applied to geophysical research for over two decades although its popularity has increased dramatically over the past few years. To date, systematic methodological reviews have not appeared in geophysical literature. In this paper, after a review of a large number of applications on cluster analysis, an intercomparison of various cluster techniques was carried out on a well-studied dataset (7-day precipitation data from 1949 to 1987 in central and eastern North America). The cluster methods tested were single linkage, complete linkage, average linkage between groups, average linkage within a new group, Ward's method, k means, the nucleated agglomerative method, and the rotated principal component analysis. Three different dissimilarity measures (Euclidean distance, inverse correlation, and theta angle) and three initial partition methods were also tested on the hierarchical and nonhierarchical methods, respectively. Twenty-two of the 23 cluster algorithms yielded natural grouping solutions. Monte Carlo simulations were undertaken to examine the reliability of the cluster solutions. This was done by bootstrap resampling from the full dataset with four different sample size, then testing significance by the t test and the minimum significant difference test.

Results showed that nonhierarchical methods outperformed hierarchical methods. The rotated principal component methods were found to be the most accurate methods, the nucleated agglomerative method was found to be superior to all other hard cluster methods, and Ward's method performed best among the hierarchical methods. Single linkage always yielded “chaining” solutions and, therefore, had poor matches to the input data. Of the three distance measures tested, Euclidean distance appeared to generate slightly more accurate solutions compared with the inverse correlation. The theta angle was quite variable in its accuracy. Tests of the initial partition method revealed a sensitivity of k- means CA to the selection of the seed points. The spatial patterns of cluster analysis applied to the full dataset were found to differ for various CA methods, thereby creating some questions on how to interpret the resulting spatial regionalizations. Several methods were shown to incorrectly place geographically separated portions of the domain into a single cluster. The authors termed this type of result “aggregation error.” It was found to be most problematic at small sample sizes and more severe for specific distance measures. The choice of clustering technique and dissimilarity measure/initial partition may indeed significantly affect the results of cluster analysis. Cluster analysis accuracy was also found to be linearly to logarithmically related to the sample size. This relationship was statistically significant. Several methods, such as Ward's, k means, and the nucleated agglomerative were found to reach a higher level of accuracy at a lower sample size compared with other CA methods tested. The level of accuracy reached by the rotated principal component clustering compared with the other methods tested suggests that application of a hard and nonoverlapping clustering methodology to fuzzy and overlapping geophysical data results in a substantial degradation in the regionalizations presented.

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Michael C. Coniglio
,
David J. Stensrud
, and
Michael B. Richman

Abstract

This study identifies the common large-scale environments associated with the development of derecho- producing convective systems (DCSs) from a large number of events. Patterns are identified using statistical clustering of the 500-mb geopotential heights as guidance. The majority of the events (72%) fall into three main patterns that include a well-defined upstream trough (40%), a ridge (20%), and a zonal, low-amplitude flow (12%), which is identified as an additional warm-season pattern. Consequently, the environmental large-scale patterns idealized in past studies only depict a portion of the full spectrum of the possibilities associated with the development of DCSs.

In addition, statistics of derecho proximity-sounding parameters are presented relative to the derecho life cycle as well as relative to the forcing for upward motion. It is found that the environments ahead of maturing derechos tend to moisten at low levels while remaining relatively dry aloft. In addition, derechos tend to decay as they move into environments with less instability and smaller deep-layer shear. Low-level shear (instability) is found to be significantly higher (lower) for the more strongly forced events, while the low-level storm-relative inflow tends to be much deeper for the more weakly forced events. Furthermore, discrepancies are found in both low- level and deep-tropospheric shear parameters between observations and the shear profiles considered favorable for strong, long-lived convective systems in idealized simulations. This study highlights the need to examine DCS simulations within more realistic environments to help reconcile these disparities in observations and idealized models and to provide improved information to forecasters.

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John E. Walsh
,
Michael B. Richman
, and
David W. Allen

Abstract

Factor analysis and an orthogonal rotation to the varimax criterion are used to identify the synoptic-scale regions of the United States over which monthly precipitation amounts show the greatest spatial coherence. The regions are consistent with previously documented cyclone trajectories. The seasonal continuity of the patterns is seriously disrupted only in summer. Regional values of the Palmer Drought Index correlate most highly with the precipitation pattern amplitudes averaged over 13–18 months in the central United States and over 7–9 months along the East and West Coasts.

Associations between the regional precipitation and local 700 mb height parameters are strongest with the geostrophic wind components in the Ohio Valley and Great Lakes regions, and with geopotential height and vorticity in the Northern Plains. Sea level pressure anomalies over broad areas of the Atlantic and Pacific Oceans are associated with regional precipitation in the central and eastern United States, while the correlations with precipitation along the West Coast are somewhat more localized.

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