Tornado Risk Analysis: Is Dixie Alley an Extension of Tornado Alley?

In general, kernel density estimation is a non- Comments on “Tornado parametric method of estimating the underlying Risk Analysis: Is Dixie probability density function (PDF) of a finite dataset. This is done by choosing a weighting function, or Alley an Extension of kernel, a measure of an area subject to the weighting function (this measure is often called the kernel Tornado Alley?” bandwidth or simply bandwidth) and then applying the weighting function to the finite dataset over the

—Pa t r i c k T. Ma r s h prescribed area. A kernel can be any function K(u) School of Meteorology, that is nonnegative, real valued, integrable, and satis- University of Oklahoma, fies (Wilks 2006) Norman, Oklahoma Cooperative Institute for Mesoscale Meteorological Studies, ∫ K(u) du = 1 (1) Norman, Oklahoma and NOAA/National Severe Storms Laboratory, Norman, Oklahoma K(–u) = K(u), "u. (2)

—Ha r o l d E. Br o o k s Several common kernels exist and include those NOAA/National Severe Storms Laboratory, Norman, Oklahoma chosen by BDK03 and DMCA11. BDK03 used a Gaussian kernel with a bandwidth of 120 km; DMCA11, however, used an Epanechnikov kernel ixon et al. (2011, hereafter DMCA11) present followed by a uniform kernel, both with a bandwidth a tornado risk analysis that found parts of the of 25 miles (40.25 km). Although different kernels D southeast United States are the most tornado will result in different probability density functions, prone in the nation, instead of “Oklahoma, the state DMCA11 state that the bandwidth, not the kernel, “is previously thought to be the maximum for tornado much more important for density calculations as it activity (Schaefer et al. 1986; Brooks et al. 2003).” determines the size of the surrounding area that will Because both Brooks et al. (2003, hereafter BDK03) influence the value at any given point.” Although this and DMCA11 employ kernel density estimation to is true, it is misleading. Specific bandwidth values achieve their depictions of tornado risk, a natural cannot be directly compared across different kernels question that arises is why these analyses are so dif- (Marron and Nolan 1988). Herein lie the major different. DMCA11 attempt to explain these differences ferences between BDK03 and DMCA11. as a consequence of their focus on tornado path length Before considering the impact of the difference instead of tornado frequency. Although there is no choices of bandwidth, let us examine the impact of question that the focus on slightly different underlying kernel choice in one dimension. In the case of an datasets affects the resulting analyses, the differences Epanechnikov kernel, the one-dimensional repre- between the tornado path length and tornado fre- sentation is quency datasets are small enough that this explanation inadequately explains the differences between BDK03 (3) and DMCA11. This comment offers a different explanation as to why the differences in the two different where h is the bandwidth and studies exist, one that focuses on the differences in how the kernel density estimation was conducted.

DOI:10.1175 / BAMS - D -11- 0 020 0.1 is the specific indicator function generalized as

AMERICAN METEOROLOGICAL SOCIETY march 2012 | 405 Unauthenticated | Downloaded 10/01/21 01:13 PM UTC and falls off symmetrically to 0 as |x| approaches (4) infinity (Fig. 1). The smaller amplitude of the Gaussian PDF is The one-dimensional uniform kernel is a consequence of the Gaussian kernel’s PDF being nonzero everywhere, whereas the Epanechnikov (5) PDF is nonzero only when |x| ≤ h and the combined Epanechnikov–uniform kernel PDF is nonzero only when |x| ≤ 2h. Using the results of Marron and and the one-dimensional Gaussian kernel is Nolan (1988), it can be shown that, when comparing Epanechnikov and Gaussian kernels, the (6) Epanechnikov kernel must be 2.2138 times larger than the Gaussian bandwidth to achieve a similar Applying the Epanechnikov kernel to a single point, response function. It is apparent that the resulting (0, 1), results in a PDF maximized at x = 0 with a value one-dimensional PDFs are different between kernels of 0.75. This PDF falls off symmetrically to 0 for |x| < h for the same bandwidth. However, DMCA11 used and is 0 for |x| ≥ h. Applying the uniform kernel to a bandwidth that was one-third the size of BDK03. the resulting PDF further reduces the maximum of This choice of a smaller bandwidth acts to amplify the PDF to slightly larger than 0.5 and broadens the the differences between the Gaussian and combined PDF such that the PDF symmetrically decreases to 0 Epanechnikov–uniform PDFs (Fig. 1). for |x| < 2h. Applying a Gaussian kernel to the same As striking as the effects of the different kernels initial point results in a PDF that is maximized at and bandwidths are in one dimension, they are mag- x = 0 with a value of nified in two dimensions. To illustrate this, consider a tornado with pathlength zero at location (0, 0). If we apply the two-dimensional Epanechnikov kernel with the DMCA11 bandwidth, approximately 40 km, and examine the output on a 5-km grid, we find that the maximum of the PDF is 0.0117, and nonzero probabilities are found out to a radius of 40 km (Fig. 2a). Application of a uniform distribution further smoothes the PDF, resulting in a maximum of the PDF of 0.0060 and nonzero probabilities out to a radius of 80 km (Fig. 2b). Contrast the Epanechnikov and combined Epanechnikov–Gaussian probabilities with those generated from a Gaussian distribution of identical bandwidth (Fig. 2c). The maximum of the PDF is 0.0025, but nonzero probabilities extend to infinity in both the x and y directions. The maximum Gaussian probability is 4.68 times smaller than the Epanechnikov kernel and 2.4 times smaller than the combined Epanechnikov–uniform kernel. As was the case in one dimension, when comparing the Gaussian kernel with a bandwidth of 120 km to a combined Epanechnikov–uniform kernel with a bandwidth of 40 km, the differences are Fi g . 1. Comparison of one-dimensional kernel density estimation for Epanechnikov kernel (solid blue line), combined magnified further. The BDK03 kernel re- Epanechnikov–uniform kernel (dashed blue line), and Gaussian sults in a maximum PDF value of 0.0003 kernel (solid green line) with bandwidth equal to 1. A Gaussian (Fig. 2d), whereas the DMCA kernel is kernel with bandwidth of 3 is shown in the dashed green line. 0.0060.

406 | march 2012 Unauthenticated | Downloaded 10/01/21 01:13 PM UTC Comparing the radii at which the different kernels produce probabilities that are 5% of the maximum PDF value offers a cursory comparison of the kernel’s two-dimensional spatial extents. Using the previously described two-dimensional sce- nario, the Epanechnikov radius is approximately 38 km, whereas the Epanechnikov–uniform combined kernel is 64 km. The Gaussian kernels have much larger radii. The Gaussian kernel with the same bandwidth as the Epanechnikov and Epanechnikov–uniform kernels has a radius of approximately 97 km, Fi g . 2. Comparison of two-dimensional kernel density estimation for (a) and the Gaussian kernel Epanechnikov kernel, (b) combined Epanechnikov–uniform kernel, and (c) used in BDK03 has a Gaussian kernel, all with bandwidth of 40 km. (d) A Gaussian kernel with band- radius of approximately width 120 km is depicted. All kernel density estimations are presented on a 293 km. common grid with grid length of 5 km. The practical impli- cation of this for tornado risk analysis is highlighted NOAA–University of Oklahoma Cooperative Agreement with the following thought experiment. The area of NA17RJ1227, U.S. Department of Commerce. the United States east of the Rocky Mountains (where climatologically most tornadoes occur) is approximately 6.4 million square kilometers. If we look at REFERENCES the area enclosed by 95% of the BDK03 smoother, an Brooks, H. E., C. A. Doswell, and M. P. Kay, 2003: area of roughly 270,000 km2 (4.2% of United States) Climatological estimates of local daily tornado is captured. However, 95% of the Epanechnikov– probability for the United States. Wea. Forecasting, uniform combined smoother of DMCA11 is roughly 18, 626–640. 12,000 km2 (0.19%). Furthermore, if we assume that Dixon, P., A. Mercer, J. Choi, and J. Allen, 2011: Tornado each of the approximately 1,300 tornadoes per year risk analysis: Is Dixie Alley an extension of Tornado is uniformly distributed, a point on a grid (any grid) Alley? Bull. Amer. Meteor. Soc., 92, 433–441. using the BDK03 smoother will use information from Doswell, C. A., 2007: Small sample size and data qual- approximately 55 tornadoes per year. The DMCA11 ity issues illustrated using tornado occurrence data. smoother uses information from approximately 2 Electron. J. Severe Storms Meteor., 2, 1–16. tornadoes per year. As a result, the greater detail Marron, J., and D. Nolan, 1988: Canonical kernels for seen in DMCA11 arises from the choice of a much density estimation. Stat. Probab. Lett., 7, 195–199. smaller kernel and a corresponding smaller effective Schaefer, J., D. Kelly, and R. Abbey, 1986: A minimum sample size (Doswell 2007) is used to estimate the assumption tornado-hazard probability model. true distribution. J. Appl. Meteor., 25, 1934–1945. Wilks, D. S., 2006: Statistical Methods in the Atmospheric ACKNOWLEDGMENTS. Funding was provided by Sciences. 2nd ed. International Geophysics Series, NOAA/Office of Oceanic and Atmospheric Research under Vol. 91, Elsevier, 627 pp.

AMERICAN METEOROLOGICAL SOCIETY march 2012 | 407 Unauthenticated | Downloaded 10/01/21 01:13 PM UTC as the radius of the 95th percentile), so we expanded Reply to “Comments on our effective kernel radius to 310 km (a 155-km ‘Tornado Risk Analysis: Is Epanechnikov kernel and an additional uniform smoothing radius of 155 km). Our original effective Dixie Alley an Extension kernel radius is 80.50 km (a 40.25-km Epanechnikov kernel and an additional uniform smoothing radius of Tornado Alley?’” of 40.25 km). The spatial patterns, especially maxima and minima, remain mostly unchanged with

—P. Gr a d y Di x o n variations in kernel function (Figs. 1a,b), but there Department of Geosciences, are strong differences when comparing results based Mississippi State University, only on tornado points or tornado paths (Figs. 1b,d; Mississippi State, Mississippi Figs. 2a,c; Figs. 2b,d). Correlations of grid values for differing methods yield an r2 of 0.63 for path —An d r e w E. Me r c e r Department of Geosciences, densities based on different kernel bandwidths Mississippi State University, (Figs. 1c,d) and 0.49 when comparing point density Mississippi State, Mississippi to path density (Figs. 1b,d) with the same kernel bandwidth. The exact magnitude at many locations may vary with different kernel functions, which is why it is usually unwise to compare results based n their comment, Marsh and Brooks (2011, here- on different kernels, but the spatial patterns should after MB11) have provided a clear explanation be less affected. I of kernel density estimation (KDE). The visual The original maps presented here make use of all comparisons of multiple kernel types (MB11) help to tornado events, whereas BDK03 use “tornado days” reduce confusion by readers while aiding with repeat- as counted within a fixed grid. DMCA11 made use ability for future studies. However, as explained in our of tornado days by selecting the longest path within original paper (Dixon et al. 2011, hereafter DMCA11), 40.25 km (25 miles) of every event. The reason we the kernel function (i.e., the shape of the probability did not employ tornado days for this reply is that we density function) is much less important than the could not justify any particular method to select rep- bandwidth (i.e., the kernel radius). This claim is resentative events when using only tornado points, supported by several previously published works on and our starting grid is fine enough that tornado geospatial analyses (Silverman 1986; Schabenberger days for each grid cell would not effectively reduce and Gotway 2005; de Smith et al. 2007; Xie and Yan the population bias that is known to affect tornado 2008; Poulos 2010). A kernel function and bandwidth records. should be chosen with the data and the scientific To ensure that the differences in results displayed question in mind. Therefore, as MB11 argue, it does by the two papers in question are not due simply to not make sense to strictly compare magnitudes differences in study periods, we applied our kernel that were calculated using different kernels. This function to point and path data from the same period does not mean that the choice of kernel function or (1950–2007) as DMCA11 (Fig. 2). Although magni- bandwidth can be expected to completely change the tudes may increase with an increased bandwidth, character of the original data. In this paper, we show the general spatial patterns are different only when that the differences in results reported by DMCA11 comparing point density and path density. There and Brooks et al. (2003, hereafter BDK03) are due are some noticeable differences between the study largely to the use of tornado path data and point data, periods, but the point data still result in maxima respectively. restricted mostly to the southern Great Plains and Using tornado data from the same period northeastern Colorado, whereas the path data yield (1980–99) as BDK03, we show that almost identical maxima across the central Great Plains and the results are achieved with different kernel functions Deep South. so long as the kernel bandwidths are similar (Fig. 1). We are not arguing that any particular method MB11 explain that BDK03 employ an effective kernel (points or paths, Gaussian or Epanechnikov, or radius of approximately 308 km (293 km is reported large or small bandwidth) is superior to the others. Regardless of the methods, the original data should be recognizable in the patterns and conclusions DOI:10.1175 / BAMS - D -11- 0 0219.1 and the results should be reproducible. DMCA11

408 | march 2012 Unauthenticated | Downloaded 10/01/21 01:13 PM UTC Fi g . 1. A comparison of (a) tornado frequency maps created by Brooks et al. (2003) and (b)–(d) those created using the kernel function of DMCA11. All maps are based on 1980–99 tornado reports. Differences in the maps are attributed to the original data being (a),(b) tornado starting points or (c),(d) tornado path data and to kernel bandwidths of (a),(b),(d) ~310 km versus (c) 80 km. Contouring style and colors were chosen to be consistent with (a) Brooks et al. (2003).

Fi g . 2. A comparison of tornado frequency maps based on data for the period 1950–2007. Differences in the maps are attributed to the original data being (a),(b) tornado starting points or (c),(d) tornado path data and to kernel bandwidths of (b),(d) 310 km versus (a),(c) 80 km. Contouring style and colors were chosen to be consistent with (c) DMCA11.

AMERICAN METEOROLOGICAL SOCIETY march 2012 | 409 Unauthenticated | Downloaded 10/01/21 01:13 PM UTC made use of path data because their goal was to as- to Principles, Techniques and Software Tools. 3rd ed. sess tornado risk to humans and property. We see Troubador, 516 pp. much value in the methods of BDK03, especially to Dixon, P. G., A. E. Mercer, J. Choi, and J. S. Allen, 2011: atmospheric science researchers, because predicting Tornado risk analysis: Is Dixie Alley an extension tornado occurrence requires knowledge about where of Tornado Alley? Bull. Amer. Meteor. Soc., 92, they form most frequently. The larger bandwidth of 433–441. BDK03 is helpful because it produces a product that Marsh, P. T., and H. E. Brooks, 2011: Comments on is less affected by individual events and is relatively “Tornado risk analysis: Is Dixie Alley an extension easy to interpret. The small bandwidth of DMCA11 of Tornado Alley?” Bull. Amer. Meteor. Soc., 93, is based on a threat radius that is commonly com- 405–407. municated by the Storm Prediction Center (“within Poulos, H., 2010: Spatially explicit mapping of hurri- 25 miles of a point”), and it allows local areas to be cane risk in New England, USA using ArcGIS. Nat. highlighted (due to relative maxima or minima) for Hazards, 54, 1015–1023. future study. Schabenberger, O., and C. A. Gotway, 2005: Statistical Methods for Spatial Data Analysis. Chapman and Hall/CRC, 488 pp. REFERENCES Silverman, B. W., 1986: Density Estimation for Statistics Brooks, H. E., C. A. Doswell, and M. P. Kay, 2003: and Data Analysis. Chapman and Hall, 175 pp. Climatological estimates of local daily tornado Xie, Z., and J. Yan, 2008: Kernel density estimation probability for the United States. Wea. Forecasting, of traffic accidents in a network space. Comput. 18, 626–640. Environ. Urban Syst., 32, 396–406. de Smith, M. J., M. F. Goodchild, and P. A. Longley, 2007: Geospatial Analysis: A Comprehensive Guide

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