Journal of Volcanology and Geothermal Research 258 (2013) 203–214

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Journal of Volcanology and Geothermal Research

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Volcano clustering determination: Bivariate Gauss vs. Fisher kernels

Edgardo Cañón-Tapia ⁎CICESE, Earth Sciences Division, P.O. Box 434843, San Diego, CA 92143, USA

⁎ Tel.: +1 526461750500.E-mail address: ecanon@cicese.mx.

0377-0273/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.jvolgeores.2013.04.015

a b s t r a c t

a r t i c l e i n f o

Article history:Received 18 February 2013Accepted 23 April 2013Available online 3 May 2013

Keywords:Volcano clusteringKernel methodsVolcano distributionSpatial density estimation

Underlying many studies of volcano clustering is the implicit assumption that vent distribution can be stud-ied by using kernels originally devised for distribution in plane surfaces. Nevertheless, an important changein topology in the volcanic context is related to the distortion that is introduced when attempting to repre-sent features found on the surface of a sphere that are being projected into a plane. This work explores theextent to which different topologies of the kernel used to study the spatial distribution of vents can introducesignificant changes in the obtained density functions. To this end, a planar (Gauss) and a spherical (Fisher)kernels are mutually compared. The role of the smoothing factor in these two kernels is also explored withsome detail. The results indicate that the topology of the kernel is not extremely influential, and that eithertype of kernel can be used to characterize a plane or a spherical distribution with exactly the same detail(provided that a suitable smoothing factor is selected in each case). It is also shown that there is a limitationon the resolution of the Fisher kernel relative to the typical separation between data that can be accuratelydescribed, because data sets with separations lower than 500 km are considered as a single cluster usingthis method. In contrast, the Gauss kernel can provide adequate resolutions for vent distributions at awider range of separations. In addition, this study also shows that the numerical value of the smoothing fac-tor (or bandwidth) of both the Gauss and Fisher kernels has no unique nor direct relationship with the rele-vant separation among data. In order to establish the relevant distance, it is necessary to take intoconsideration the value of the respective smoothing factor together with a level of statistical significance atwhich the contributions to the probability density function will be analyzed. Based on such reference level,it is possible to create a hierarchy of clustering degrees that allows us to obtain significant information in ageologic (and particularly volcanic) context. To illustrate this aspect of the kernel method, two examplesusing volcanic fields along the Peninsula of Baja California and the American continent are reported.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Characterization of spatial patterns defined by the distribution ofvolcanic vents provides clues concerning the size and shape of themagma source, as well as about the geological structure and stress con-ditions of the crust at the time of activity. Most of the earlier studies ofthis type focused on the identification of possible vent alignmentsthought to reflect fracture patterns or other linear geologic structures(Lutz, 1986; Wadge and Cross, 1988; Connor, 1990). Since alignedvents are often found amidst a greater number of unaligned cones(Connor and Conway, 2000), the analysis of the spatial distribution ofvolcanic vents gradually evolved to include the identification not onlyof potential vent alignments, but also to provide a quantitative descrip-tion of vent clusters in a more general form. Over the years, severalstudies have shown that the required description of vent distributioncan be achieved with relative ease through non-parametric density es-timation tools (e.g., Connor and Hill, 1995; Lutz and Gutmann, 1995;

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Martin et al., 2004; Weller et al., 2006; Kiyosugi et al., 2010; Capelloet al., 2012; Connor et al., 2012). In particular, the kernel method(Silverman, 1986; Wand and Jones, 1999) has become increasinglypopular.

Despite its increasing popularity, however, there are some aspectsof the kernel method that might have not received proper attentionfrom the volcanological community. For example, all the kernelfunctions used to date to study the distribution of volcanic vents(Epanechnikov, nearest neighbor, Cauchy and Gauss) were originallydevised to provide a description of two-dimensional data found overa planar surface. Such inherent planar topology deserves furtherattention because, in the strictest sense, the data points that areaimed to be described by a kernel function in the volcanic contextdo not lie over a plane, but actually lie on a portion of a curved sur-face. The problem when using a plane-designed kernel function,therefore, might be similar to that found when making a map projec-tion, in the sense that a certain amount of distortion might be intro-duced by flattening the original curved surface. Since the generalrule is that the amount of distortion increases as the size of the regionto be projected increases in size, it is necessary to consider if there aresome maximum dimensions of a volcanic field that can be studied

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using plane-related kernel functions without introducing a noticeabledistortion on the resulting distribution. Although somewhat intui-tively it might seem that the use of a kernel devised to characterizea set of data found over a planar surface might be justified whenstudying vent distributions in volcanic fields that have relativelysmall extensions, the fact remains that at present we lack quantitativeevidence that can be used to establish whether this impression is jus-tified or not.

Other aspects of the kernel method that require further examina-tion are related to the physical significance of the parameter variouslycalled the bandwidth, kernel width, or smoothing factor (Silverman,1986; Wand and Jones, 1999). There is some evidence suggestingthat the selection of this parameter might have a stronger influenceon the results than the selection of a specific kernel function(Weller et al., 2006). Furthermore, the selection of an optimal band-width is a non-trivial subject that requires careful consideration inmost practical applications of the kernel method (Wand and Jones,1995). In this sense, the role played by the bandwidth parameter inany kernel function used to calculate the density of volcanic ventsdoes not seem to be an exception, and needs to be examined in detail.

The just described issues concerning the kernel method areaddressed in this paper by introducing into a volcanic context a ker-nel function that has a spherical native topology, making it a priorimore akin to the description of large portions of a planet. By compar-ing the distributions obtained by that spherical kernel function withthose obtained by using a representative plane-defined kernel func-tion, it is possible to establish in a quantitative form whether thereis any influence on the results that can be associated with the nativekernel topology. The analysis is made by first focusing on the generalcharacteristics of the kernel functions, and later by illustrating theiruse in the analysis of volcano distributions at two different scales.Along this process, the influence of the bandwidth is also exploredin detail.

2. Bivariate Gauss and Fisher kernels

2.1. Definitions

Among the various kernels that have been used to study the spatialdistribution of volcanic vents, the bivariate Gauss kernel was selectedon this study as representative for the case of density functions with anative plane topology. This kernel reduces the intensity of the distribu-tion from amaximum centered at each data point, it has radial symme-try, and it has a distribution tail that is nonzero for all distances from thedata point (Weller et al., 2006). The spherical counterpart used in thiswork is the Fisher kernel, which can be considered to be the Gaussequivalent in terms of the statistical properties of the original distribu-tion functions. Unlike the Gauss kernel, however, the Fisher kernelwas specifically created to describe distributions of points possessingan inherent spherical topology (Fisher et al., 1993).

In symbols, the Gauss kernel is defined by:

f̂ x; yð Þ ¼ 12πNC2

g

XNi¼1

exp −12

di

Cg

" #2 !ð1Þ

where the point at which the density function is being estimated hascoordinates (x, y) on the plane, di is the distance between the estima-tion point and the ith volcano, and Cg is the smoothing parameter orbandwidth.

On the other hand, the Fisher kernel introduced by Diggle andFisher (1985), based on the distribution originally created by Fisher(1953), is given by:

f̂ Lat; Lonð Þ ¼ Cf

4πNsinh Cfð ÞXNi¼1

exp Cf cosθið Þ ð2Þ

where the estimation of a density is made at a point located over thesurface of a sphere that has coordinates (Lat, Lon), θi is the angulardistance between this point and the ith volcano, and Cf is the corre-sponding smoothing parameter.

Examination of Eqs. (1) and (2) reveals that the different topolo-gies native to each of those kernels is essentially defined by theform in which the distance between two points is introduced ineach case. The relevant distances are linear for the Gauss kernel(di), whereas they are calculated as angular distances for the Fishercase (θi). For this reason, di can take any value from zero to infinity,whereas θi can have values only between zero and 180°. Due to thischaracteristic of the Fisher kernel, it becomes easier to deal withcases that are at “extremes” of the longitude scale than it is to dealwith these cases on the Gauss kernel. For example, an added compu-tational effort might be required to determine that the distance be-tween two points located at the same latitude but longitudes 359°W and 1° W, respectively (or at longitudes 179° E and 179° W) isonly of two degrees if the Gauss kernel is used. Such a problem isnot found when the Fisher kernel is used.

Another formal difference between kernels is found relative to theform in which the smoothing parameter is introduced in each case. Inthe Gauss kernel (Eq. (1)) the smoothing parameter varies in a qua-dratic form both, inside and outside of the exponential, whereas inthe case of the Fisher kernel (Eq. (2)) the smoothing parameter variesfollowing a hyperbolic function outside the exponential. Althoughthose differences do not affect their statistical properties, they haveimportant implications concerning the form in which the smoothingfactor must be interpreted in each case, and impose a severe limita-tion on the Fisher kernel, as will be shown throughout this paper.

2.2. General comparison

In order to gain a general appreciation of the differences existingbetween the Gauss and Fisher kernels, it is convenient to makea direct comparison of the behavior of Eqs. (1) and (2). Althoughthere is a difference in the parameters that enter each of those equa-tions, it is possible to obtain a probability density function (PDF) atone arbitrary point in space, expressing it as a function of a commondistance-parameter. This is possible because geometrically, the lineardistances di of the Gauss kernel can be related to the angular dis-tances of the Fisher kernel through the equation:

d ¼ r θ ð3Þ

where r is the radius of a reference sphere (which can be equated tothe radius of the Earth), and θ, the angular distance between twopoints, is measured in radians. Thus, using Eq. (3) it is possible to cal-culate either a Gauss density distribution as a function of an equiva-lent angular distance (instead of the linear distance entering theoriginal definition), or alternatively, it is possible to calculate a Fisherdensity distribution as a function of an equivalent linear distance(instead of the original angular distance of the definition).

It is possible to facilitate the direct comparison of the PDFs of bothkernels evenmore by focusing our attention on the normalized valuesrelative to the maximum value of each PDF. Inspection of Eqs. (1) and(2) reveals that such a maximum value is obtained when the distancebetween points is set equal to zero, and such location of the maxi-mum density is obtained regardless of the value of the smoothing pa-rameter. Consequently, normalizing the PDFs obtained with Eqs. (1)and (2) is relatively straightforward. As for the smoothing parameterof both kernels, for the time being we can retain them as numbersthat enter the equations, and produce a series of PDFs with a rangeof values of those parameters. Fig. 1a and b show various normalizedPDFs of both kernels for a variety of the corresponding smoothingfactors.

Fig. 1. a) Normalized contribution to the probability density function (PDF) of a given setof data as a function of distance from the observation point for the Fisher (unornamented)and bivariate Gauss (asterisks) kernels. The numbers denote the value of the smoothingfactor that was used to calculate the corresponding curve (Cg — Gauss; Cf — Fisher ker-nels). The shadowed area to the left is expanded in the diagram 1b. c) Selected curvesshowing the absolute contributions to the PDFs as a function of smoothing factor. Lineswith similar symbols are identical if normalized. Upper group — Fisher kernel; lowergroup — Gauss kernel. (Triangles — Cg = 250, Cf = 700; * — Cg = 700, Cf = 85;+ — Cg = 1000, Cf = 45; o — Cg = 5000, Cf = 1.7.)

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The first key aspect that must be noted on those diagrams is thatthe numerical value of the smoothing factor has an inverted effecton the corresponding kernels. While an increase in the numerical

value of the smoothing factor in the Gauss kernel increases the contri-bution to the relative probability of a data located at given distancefrom the observation point, an increase in the numerical value ofthe smoothing factor on the Fisher kernel decreases its relative con-tribution. For this reason, the series of PDF curves of the Gauss kernelsare progressively farther from the vertical axes as the smoothing fac-tor increases, whereas in the Fisher kernel the series of PDFs are pro-gressively closer to the vertical axis of the figure as the correspondingsmoothing parameter increases. For this reason, it becomes evidentthat phrases such as “increasing the amount of smoothing” or “in-creasing the bandwidth” have the exact opposite implications,depending on which kernel is used.

A second key aspect to be noted in the diagrams of Fig. 1a and b isrelated to the fact that it is not possible to establish a simple relation-ship between the numerical value of the smoothing factor used toproduce any of the curves, and the numerical values of the horizontalaxis. For example in the case of the Gauss kernel a smoothing factorequal to 15 leads to a curve that is noticeably above the horizontalaxis at separation distances that go from zero to 50 km, and asmoothing factor equal to 1000 has non-negligible contributions tothe PDF at distances between zero and 3000 km. Similarly, for theFisher kernel, a smoothing factor of 50 leads to a PDF that is notice-ably above the horizontal axis at separations between zero and slight-ly less than 3000 km, whereas a smoothing factor of 700 yields nonnegligible contributions to the relative probability if the separationbetween points is anywhere from zero to 750 km. These examples in-dicate that the physical interpretation of the numerical value of thebandwidth in these two kernels is not straightforward, as it is inother kernels (e.g. Epanechnikov), where there is a direct correlationbetween bandwidth and the distance between data and/or observa-tion points. Consequently, the numerical value of this parameter(for neither the Gauss nor the Fisher kernels) should not be associat-ed with any specific distance typical of the distribution. Nevertheless,based on the curves of Fig. 1a and b, it is possible to associate a specif-ic numerical value when referring to “relevant” or “significant” dis-tances, as is discussed in more detail in Section 3.2.

Yet a third key aspect to be noted from the curves of Fig. 1a and b,is the fact that it is possible to produce identical curves with both ker-nels given an appropriate numerical value of the respective smooth-ing factors. This is illustrated by the specific case of the curves thatare marked by a Gauss bandwidth of 250 and a Fisher bandwidth of700, and by the curves accompanied by two smoothing parameters(the one in parentheses corresponding to the numerical value of theFisher smoothing factor, and the other to the corresponding factorof the Gauss kernel). This finding clearly highlights the fact that de-spite their different mathematical definitions, both kernels studiedhere will lead to the same assessment of density estimations, provid-ed that a suitable value of their respective smoothing parameters isselected. Since the identical curves can be produced to cover almostall the range of relevant distances, it can be concluded that there isno particular limit that can be identified to mark the non-distortionsize associated to the plane-like kernel. As will be shown below, how-ever, there is a numerical practical limit that is found in the case ofthe Fisher kernel that limits its uses in the description of small scaledistributions.

It must be recalled at this point that the curves shown in Fig. 1aand b have been normalized relative to the maximum value of eachPDF. In contrast with the curves of those figures, the diagram inFig. 1c shows the actual (non-normalized) values of representativePDFs of both kernels. As can be observed, Fig. 1c has a vertical axiswith a logarithmic scale, necessary to accommodate the wide rangeof maximum values of all the curves shown. Also, it must be notedthat the lower limit of the vertical axis of this figure has been truncat-ed (i.e., the lower limit is not zero), in order to visualize the main partof the curves. If the whole range of values had been included, thecurves would have collapsed into a single line close to the horizontal

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axis, even if the logarithmic scale continued to be used to display thedata. This figure therefore shows that the actual numerical values as-sociated with each PDF have a strong dependence on both the numer-ical value of the smoothing parameter and on the type of kernel usedto calculate the PDF. Actually, the difference in numerical valuesmight be so large, that it is illustrative to examine the difference inmaximum values of a given PDF by making reference to the numbersprovided in Table 1, rather than representing them in graphical form.As inspection of this table reveals, the maximum values of the variousdensity curves include variations of six orders of magnitude in thecase of the Gauss kernel, whereas the maximum density calculatedwith the Fisher kernel changes only through 3 orders of magnitudefor the range of smoothing parameters included in the table. It is im-portant to note, however, that a smoothing parameter larger than 700leads to computational errors very easily, because the hyperbolicfunction entering the definition of this kernel increases its value ex-tremely rapidly beyond this value. Consequently, this particularvalue of the smoothing parameter of the Fisher kernel constitutes apractical limit that reduces its applicability. The extent to which thisreduction might become important in practical terms, is discussedin more detail in Section 4.

Detailed examination of the curves in Fig. 1c indicates that it ispossible to identify two groups. Those in the lower group wereobtained with the Gauss kernel, and those in the upper group wereobtained with the Fisher kernel. The curves marked with the samesymbol on each group would be identical if the normalized valueswere shown. The relevant point to be noted in this diagram is thatsince the maximum probability contributed by any data to the finalPDF is a function of the smoothing factor used in the calculations, itis not justified to make a direct numerical comparison of resultseven if the same kernel is used with two different smoothing factors.For example, indicating that the density of vents in a particular loca-tion is clearly more significant than the density obtained in anotherplace with the same kernel but different smoothing parameter isnot justified, since it could have occurred that in fact the differencesin computed densities reflect only the value of the smoothing factornot reflecting a real difference relative to the maximum density of agiven zone. Nevertheless, if reference is made to relative values ofthe maximum density obtained in each case with a constant smooth-ing factor (i.e., using the normalized PDFs), the comparison of resultsmight be meaningful even if those results were obtained with differ-ent kernels (Note that “different kernels” applies only for the Fisherand Gauss kernels studied here, and is not extensive to the other ker-nels that have been used to study volcano distributions). Actually, thischaracteristic of the kernel method allows us to make a detailed ex-ploration of the data at various scales, as will be discussed next.

3. Method resolution and cluster dimensions

3.1. Determination of an optimal smoothing factor

A key to fully take advantage of the characteristics of kernelmethods is to have a clear understanding of the form in which thesmoothing parameter influences the calculations. In simple terms,the effect of the smoothing factor is similar to the effect produced ina histogram of a unidimensional variable when different bin-widthsare selected (Fig. 2). A too-narrow bin-width tends to yield abumpy distribution in which no tendency might be clearly observed,and even spurious peaks (due entirely to an inadequate number of

Table 1Values of the maximum density that can be achieved as a function of the smoothing param

Kernel type Smoothing factor 1 5

Gauss 0.1592 0.0064Fisher 0.1841 0.7958

observations) may be produced (Fig. 2b). In contrast, a too-widebin-width may result in a density distribution where all the detailhas been lost (Fig. 2c). For this reason, a good balance must exist be-tween the number of observations and bin-width in order to revealrelevant characteristics of the distribution that are neither biased byinsufficient sampling nor extreme averaging that precludes the iden-tification of secondary modes.

The smoothing factor in kernel methods plays a role similar to theselection of bin-widths in histograms of unidimensional distributions.It is therefore through this parameter that it is possible to take ad-vantage of the potential that kernel methods have to explore theprobable occurrence of multimodality in one distribution at variousresolutions. Nevertheless, the selection of the smoothing parameterremains to be a matter of preference of individual workers, and onlyrough guidelines can be presented.

From a general point of view, there are two main alternative ap-proaches that have been followed to select a smoothing factor. Oneapproach is to rely on an automated procedure in which the datathemselves are used to select an optimal smoothing factor that yieldswhat is considered to be the best density distribution (Silverman,1986). Although this procedure may be advantageous in some re-spects, it assumes that a series of conditions are satisfied both bythe kernel function itself, and by the density distribution that isunder examination (Chiu, 1991). Unfortunately, since in general wecannot be sure that the data collection satisfies those conditions apriori, despite its apparent objectivity, this procedure introducessome degree of bias by making implicit assumptions concerning thedistribution that is under examination. Although on some occasionsthis procedure might be justified, it also might be undesirable inmany circumstances. For this reason, sometimes it is advisable toadopt an alternative approach to explore the data distribution with-out introducing any special assumptions. This alternative approachconsists of repeatedly estimating the spatial density distribution,each time using different values of the smoothing factor (alwayskeeping its value uniform across the whole extent of the area beingsurveyed during each calculation). The most significant, or optimal,density distribution can be selected among all of the distributionsproduced, based on some criteria, either statistical, or better yet,physical in nature.

A third approach that can be adopted, is to force a change in thevalue of the smoothing factor depending on the relative degree ofclustering found in the surroundings of each observation point. Thechange in value of the smoothing factor in this case is determinedby an iterative pre-exploration of the local database. Such a pre-exploration is used to establish a “typical” separation between datapoints in an area relatively closer to the observation point that alwaysremains smaller than the total area under investigation (Weller et al.,2006). In this case, the optimal density distribution is obtainedthrough the combination of various smoothing factors applied indifferent parts of the observation area. In principle, this approachyields more detail when larger concentrations are found, and lessdetail when the data points are more widely spaced, therefore beingmore efficient computationally than the other two methods. Thisapproach can prove to be advantageous in some kernels that assign auniform weight to data lying closer to the observation point, whileassigning a null contribution to data outside this range (e.g., theEpanechnikov kernel Connor, 1990). Nevertheless, if the contributionto the calculated PDF at a given point is always different from zero,and actually displays a strong dependence on the actual distance

eter for the Gauss-bivariate and Fisher kernels.

50 250 500 700

0.0001 2.6 × 10−6 6.4 × 10−7 3.3 × 10−7

7.9577 39.7887 79.5775 111.4085

Fig. 2. Unidimensional influence of the smoothing parameter in the calculated PDF,redrawn from Silverman (1986). The window width in this figure is equivalent tothe relevant distance mentioned in the text.

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between the data and the observation point (as in the case of the bivar-iate Gauss and Fisher kernels), the use of different smoothing factors indifferent parts of the area to be explored may not be entirely recom-mendable because it includes a certain bias on the interpretation of re-sults that might be related to biases in the available sampling ratherthan reflecting real variations of the spatial distribution as a whole. Inany case, as already mentioned, the approach followed to select asmoothing factor remains to be a personal decision of each worker,and even might be different on a case by case basis.

In this work it is considered that since the first and third ap-proaches just described entail a certain bias concerning the character-istics of the actual distribution, it seems a better alternative to adoptthe second approach as a general exploratory tool. An example ofthe utility of the third approach using a similar data base is providedelsewhere (Germa et al., submitted for publication).

3.2. Smoothing factor, relevant distance and grid density

To better appreciate the influence that the smoothing parameterhas on the resolution of the kernel methods, it is convenient tofocus attention on a simple situation in which there are very fewdata, and also relatively few evaluation points. Although in principlethe PDF of either Eq. (1) or (2) provide a continuous description ofprobability over the whole extent of the area being surveyed, in prac-tice we need to evaluate the PDF at a finite number of points that forma grid of some sort. Intuitively, it would be expected that the denserthe grid the more accuracy in the results will be obtained. As illustrat-ed by the several cases shown in Fig. 3, however, this is not necessar-ily true.

In Fig. 3, a series of situations are displayed, all of which involvethe same set of data (stars) and two different grids of observationpoints (large and small dots). The shadowed circles enclose the areathat makes a contribution to the PDF that is calculated at the observa-tion point lying at its center. The decreasing intensity in the shadowof those circles, represents the general form of the curves of Fig. 1aand b. In principle, and based on the definition of the Gauss and Fisher

kernels, those circles should extend to infinity, with an ever vanishingshadow, because the contribution of any data very far to the observa-tion point never is equal to zero. Nevertheless, we can take advantageof the normalized curves of Fig. 1a and b to make an operational def-inition of the radius of those circles as the distance at which the con-tribution to the PDF is “significant”. The meaning of the word“significant” in this context depends on the exact value of thesmoothing parameter, and on an arbitrarily defined threshold aswill be further discussed below. Nevertheless, for the time being, itcan be considered that the circles of Fig. 3 mark the limit betweenthe area where contributions are significant and the rest of the areaof study where the contributions to the calculated PDF are negligible.For facility, those circles will be referred in the following as the “rele-vant distance”.

If the relevant distance is too small relative to the separation of ob-servation points, we might face a situation in which some data do notmake an important contribution to any place where the PDF is beingevaluated (Fig. 3a), which is equivalent to saying that some of thedata are being left out from the analysis. Although increasing thelength of the relevant distance might result in an increase of thedata that are included in the analysis, as shown by Fig. 3b, whethera given relevant distance results in a large number of observationpoints with significant contributions depends not only on the size ofthe relevant distance but on the combination of this distance withthe separation of the points forming the observation grid. In particu-lar, if only the grid of large circles is used to evaluate the PDF, it isclear that in both Fig. 3a and b there are data that are being left outof the analysis. If the grid of small points is used instead, it is possibleto find observation points including significant contributions fromsome of the data in both situations (the yellow points in Fig. 3a andb), although the number of those points is larger in the latter case. Ac-tually, if the grid of large points is used to evaluate the PDF, the omis-sion of some data would still take place even if the relevant distance isincreased to the dimensions shown in Fig. 3c and d. Nevertheless, asmoothing parameter of this size might start to capture some of thecharacteristics of the distribution even in a widely spaced grid asshown in Fig. 3c, where the lower-left observation point is likely tohave a much larger density estimate than the other three points onthe grid of large points due to the relative clustering of the threestars enclosed by the relevant distance associated to the lower left ob-servation point. If the grid of small points is used instead, a more ac-curate density distribution would be obtained where the observationpoints near (but not exactly at) the lower left corner of the study areahave the largest density estimates (red circles), those at the center ofthe studied region have intermediate density estimates (orangecircles) and some of the observation points in the periphery of thestudy area have either small (yellow circles) or null (black circles)density estimates. Evidently, in this case, some of the fine structureof the data distribution is being captured, in particular the asymmetryof the distribution towards the lower left corner of the study area. Ifthe size of the relevant distance is increased even further, the asym-metry of the distribution will be captured even if the grid of large cir-cles is the one used for the study (Fig. 3e). Nevertheless, if the finergrid is used instead, some of the resolution would start to be lost,since every observation point in this case would have a large densityestimate (color omitted to allow observation of the effect experiencedin the grid of large points). If the relevant distance is still further in-creased, as in Fig. 3f, all resolution of the distribution of data enclosedwithin the observation cell defined by the four observation pointslocated at the corners of the grid is also entirely lost.

The diagrams of Fig. 3e and d have a particular interest withregard to guidelines that can be useful when interpreting density dis-tributions obtained in several situations of practical interest. For ex-ample, as is shown by the diagram of Fig. 3e, in order to avoidmissing any data from the resulting PDFs using the grid of large cir-cles, relevant distances of at least half the diagonal of a cell formed

Fig. 3. Schematic relationship among grid points (circles), actual data (stars) and a relevant distance (shadowed circles) controlled by the smoothing factor in kernel methods. Thecolor of the circles represents the relative density calculated at that observation point (red closer to the maximum, yellow, low density). See text for details.

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by four observation points is required. In this case, the relative contri-butions of each of the data to each of the observation points will bedifferent, and therefore the uneven distribution of data within the ob-servation cell might be detected. On the other hand, if the grid isdenser, and the separation between observation points is more orless of the same magnitude than the average separation betweendata, a relevant separation equal to the diagonal of a single observa-tion cell can provide more details concerning the fine structure ofthe distribution. Thus, as these two diagrams show, it is difficult todefine an “optimal” value of the smoothing parameter unless a previ-ous knowledge concerning the density of the actual observationpoints and the distribution between actual data has been achieved.Nevertheless, a sequential exploration of density contours with afixed grid separation can be useful to identify trends in the data distri-bution at various scales of clustering. Once such knowledge has beenestablished, a denser grid can be used to characterize the data if sorequired.

Although the examples shown in Fig. 3 are a crude approximationto a real calculation made with either Eq. (1) or (2), this figure illus-trates basic aspects of the kernel methods examined here that areconvenient to appreciate. In particular, the figure serves to illustratethe fact that different scales of clustering can be identified if a suiteof PDFs is calculated each using a different value of the smoothing

factor (and a fixed grid separation). Fig. 3 also illustrates that a toodense grid may turn out to be useless if the smoothing parameterleads to a circle with a radius larger than the dimensions of the obser-vation cell. Finally, the diagrams in Fig. 3 show that the resolution ofthe method depends on the correct combination of grid separationand relevant distance, inasmuch as it depends on the typical separa-tion of the individual data. In practical terms this implies that thereis no single value of the smoothing factor that can be identified as athreshold of resolution valid for the accurate description of a givendistribution, and that a too dense grid does not necessarily guaranteelarger resolution of the method. Most likely, some trial and error willbe desirable to explore the best combination suitable for the charac-terization of specific situations.

4. Discussion

4.1. Kernel method resolution as a function of the relevant distance(smoothing factor)

Despite their formal differences, and inherently different topology,the previous sections show that the Gauss and Fisher kernels assignrelative densities based on the same set of qualitative rules, andthat therefore both kernels can be used to obtain exactly the same

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general description of volcanic distribution regardless of the extent ofthe area being surveyed. The only condition that is required to thisend, is to select suitable values of the corresponding smoothingparameter.

To fully appreciate the spatial scales associated to each kernel, andin particular of the correct interpretation of the smoothing factor inthese cases, it is convenient to take a closer look to the curvesdisplayed in Fig. 1a and b, in the context of the diagrams of Fig. 3. Inpractical terms, the relevant distance indicated by the radius of thecircles on Fig. 3 can be numerically determined by using the curvesof Fig. 1a and b in combination with a convenient threshold value.For example, it might be considered that a “negligible” contributiontakes place beyond the distance at which the normalized curves ofFig. 1a and b have a value lower than 90% of the maximum relativeweight. Alternatively, in some other instances it might be more con-venient to set the threshold value to 10%, 1%, or even smaller. Whencombining these thresholds with a given value of a particularsmoothing factor, the relevant distances will have a unique and welldetermined value, and therefore any ambiguity associated with itsdetermination will be removed. For example, threshold values of90%, 10% and 1% of the maximum density of a PDF calculated with asmoothing factor of 700 for the Fisher kernel (or 250 for the Gausscase), lead to relevant distances of 100, 500 and 750 km, respectively.If a smoothing factor equal to 50 is used on the Fisher kernel, the rel-evant distances (for the same threshold factors) become 400, 1950and 2750 km, but if a smoothing factor of 5 is used with the Gausskernel, the relevant distances would be 2, 11 and 15 km. Each ofthose relevant distances provide an indication of the size of the clus-ters that can be detected (and equally important, of the inter-clusterdistance) at the selected significance level. Consequently, a widerange of relevant distances can be explored by simply changing thesmoothing parameter and selecting a specific significance level.

To illustrate this aspect of the kernel methods, it is convenient toexamine an artificial example in which two clusters are located atthree different distances from each other (Fig. 4). The average dimen-sions of each cluster are approximately the same, but their internalstructure (i.e., the relative distances between data inside each cluster)is slightly different. Since we are interested in examining the influ-ence of the relevant distance and significance level, the internal struc-ture of each cluster is kept constant in all the examples. In contrast,the relative distance between clusters is adjusted to take three values(each illustrated in one column of the figure). In the first case, theinter-cluster distance is larger than the average dimensions of theclusters, in the second case (middle column) it is of the same order,and on the third case (rightmost column) it is zero. Actually, in thethird case there is some overlapping of a few of the data on their com-mon boundary.

The density contours for each of these three cluster configurationswere calculated by using four different values of the smoothing pa-rameter. Those values were selected using the curves of Fig. 1a, takingas a guide the known inter-cluster distance of each of the three con-figurations, so that in three cases (the three lowermost rows) the rel-evant distance at a significance level of 20% was approximately equalto the inter-cluster separation. The fourth value of the smoothing pa-rameter (used to generate the diagrams of the first row) was selectedso that the relevant distance in this case was larger than the largestinter-cluster distance used in the examples.

As can be appreciated in the figure, when the relevant distance islarger than the inter-cluster separation (even if slightly), the densitycontours define a continuous figure that acts as an envelope to thedata. Depending on the exact distances involved, all the data can becontained within the curve marking the 40% maximum density(Fig. 4a), or in some cases a few data can be left out of the curve mark-ing the 10% maximum density (Fig. 4k and l). Nevertheless, in allcases the lowest density contour shown gives a general idea of theoverall shape of the total distribution defined by both clusters. If

the relevant distance is equal to the inter-cluster distance (Fig. 4band g), the contours split into two concentric zones, each aroundone of the clusters. The maximum densities achieved in each cluster,however, are different, as could be expected due to their differentdegrees of packing. It should be noted that the identification of twoseparate clusters, with a relative variation of maximum density, alsocould have been achieved using a larger relevant distance, althoughin this case it would have been necessary to focus on the contoursof higher density that are not contiguous (e.g., 70% in Fig. 4a or 30%in Fig. 4f), instead of focusing attention on the lowermost densitycontour shown.

As the relevant distance starts to take values that are smaller thanthe inter-cluster separation, the contours are reduced in size, leavingsome data outside the lowermost density contour shown, but delin-eating the general boundaries of each cluster. Furthermore, some de-tails of the internal structure of each cluster start to be revealed inthe form of small zones of high density, that may be identified assub-clusters lying inside specific parts of the original cluster. Notethat such internal structure continues to be identifiable by the higherdensity contours when the two clusters are slightly overlapping, evenif in this case the lower density contours remain unified.

The examples shown in Fig. 4 illustrate the fact that a progressiveexploration of a given collection of data using kernel methods canprovide insights concerning the distribution at different scales of res-olution. Whether the fine structure inside each cluster is truly signif-icant or not, or alternatively, whether the overall shape of the twoclusters is more significant in a geologic context than the identifica-tion of each of the two clusters as separate entities, requires a differ-ent source of information that cannot be extracted from the analysisof the density contours alone. In any case, the examples in this sectionshow that kernel methods are well suited to establish a hierarchy ofsignificant clusters that can be identified by paying attention to theform in which different contour levels evolve as the relevant distancechanges during the exploration of PDFs.

4.2. Practical uses of Gauss and Fisher kernels

To illustrate the potential that the adoption of a sequential explo-ration of density contours calculated with constant smoothing factorshas from the point of view of volcanic processes, in this section twoexamples of volcano distributions involving two contrasting spatialscales are explored with some detail. One of those distributions is de-fined by the vents forming six different volcanic fields located alongthe Peninsula of Baja California. The second is defined by the locationof volcanoes listed in the world catalog along the Latin American con-tinent. These two distributions were selected because they share acommon geometric characteristic (elongated overall distributionwith various sub-clusters) despite their clear difference in scale, andbesides in both cases there are different zones of volcano clusteringthat have been documented previously. Consequently, the generalbehavior of both kernels and of the sequential use of smoothing pa-rameter values illustrates the same basic principles in both situations.

4.2.1. Intermediate scale volcanic distributionPost-subduction volcanism produced several volcanic fields along

the Peninsula of Baja California, each field containing several tens ofvolcanic vents (Negrete-Aranda and Cañón-Tapia, 2008). A detailedstudy of the distribution of such fields, and of vents within eachfield using an adaptive approach related to the numerical value ofthe smoothing factor has been provided by Germa et al. (submittedfor publication). In this paper, the same database containing the loca-tion of all the volcanic vents located within six volcanic fields in BajaCalifornia is used to illustrate the utility of calculating a sequence ofPDFs using different values of the smoothing factor. Fig. 5a showsthe location of the volcanic centers associated to these fields, andthe grid used to study their vent distribution. The concentric pattern

Fig. 4. Evolution of the density contours as a function of a decrease in relevant distances (rows) shown for three different inter-cluster distances (columns). The relative magnitudeof the relevant distance (R.d.) and the inter-cluster separation (I.C.S.) is indicated on the right corner of each diagram. See text for details.

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of colors corresponds to the densities obtained when a smoothingfactor equal to 250 is used on the Gauss kernel. The same PDF isshown in the map of Fig. 5b where it is observed that despite theelongated distribution of the volcanic centers the probability densityhas a circular nature, suggestive of a concentric array of volcanoesroughly located at the middle of the Peninsula. An entirely equivalentdistribution would be obtained if the Fisher kernel was used with asmoothing factor of 700. Due to the numerical limitations foundwhen the Fisher kernel is considered, it is impossible to obtain moredetail of the characteristics of the distribution in this case by usingthis kernel. Nevertheless, if the Gauss kernel is used, a lot more infor-mation can be obtained. For this reason, the following focuses on theuse of the Gauss kernel. If a smoothing factor of 50 is used, instead ofthe previous value of 250, the density contours acquire an elongatedstructure dominated by a larger concentration (almost circular) ofvolcanic edifices at its center. Further reducing the smoothing factorto 30 yields density contours that indicate the existence of two loca-tions with higher density towards the northern extreme of the distri-bution, and a continuous tail of lower density towards its southernextreme. A smoothing factor of 15 yields two separate regions of vol-canism, each with its own zone of relative high density, and welldefined boundaries, although of a very distinctive size. The concentra-tion at the north is much smaller than that at the south, and it has anearly circular structure. In contrast, the region at the south has arather elongated shape, with its maximum located closer to thenorthern extreme. Further reduction of the smoothing parameterto a value of 5, yields six individual regions that correspond to the

individual volcanic fields of Jaraguay, San Borja, Vizcaino, Santa Claraand San Ignacio–San José de Gracia. As shown in Fig. 5f, the maximumdensity achieved at each of these independent fields is not equal acrossregions, being slightly larger in the two northern fields than on theremaining four.

4.2.2. Large scale volcanic distributionIn Fig. 6, a similar progression of volcano groupings is shown, but at a

different scale of separation between volcanoes and of the size of thearea surveyed. In this case, the volcano distribution corresponds to thevolcanoes of Mexico, Central and South America listed in the world cat-alog of volcanoes (Siebert and Simkin, 2002). The density contours inthis case were obtained using the Fisher kernel (except in Fig. 6f),with the corresponding smoothing factors as indicated in each case. Atsmoothing factors of 10 (Fig. 6b), the relevant distance used to calculatethe PDF at each observation point is very large, and therefore theresulting density contours cover a large proportion of the Earth surface,including many locations in which there is no active volcanism at all.Nevertheless, even with such large values of relevant distances, thedensity contours reflect some of the characteristics of the distributionsince the general character of the distribution is such that contourlines are elongated roughly following the west coast of the continent.Also, it is noted that the density contours reveal a larger concentrationof volcanoes shifted towards the north of the surveyed area. When thesmoothing factor is increased to 175, the resulting density contours de-crease in size, covering an area that is more evidently related to the lo-cation of the actual volcanoes. More importantly, the distributions

Fig. 5. Maps showing a progression in the form of the density contours associated to volcanic vents forming six volcanic fields on the Peninsula of Baja California. a) Close-up of thesurveyed area showing the actual vents (triangles) and the grid used for the calculation of PDFs. b) Regional map showing the extension of the surveyed area and the same contoursshown in a for reference. c) to f) Maps showing the obtained density contours by selecting the smoothing parameter indicated on the right corner of each diagram for thebivariate Gauss kernel. The color bar shown in a) is valid to describe the contours in the rest of the maps. Volcanic fields: J — Jaraguay, SB — San Borja, SC — Santa Clara, SI —San Ignacio, V — Vizcaino.

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obtained are such that twomain distinctive clusters have been defined,each of which contains two separate regions of relative maximumdensity. One of the main clusters corresponds to the South and CentralAndean ranges, and another includes the Northern Andes, theGalapagos, Central America and the Trans-Mexican Volcanic Belt, orTMVB. A third cluster of relatively smaller density, but independent ofthe other two, is defined towards the northern part ofMexico, includingthe Peninsula of Baja California. When the smoothing factor is set to

400, theGalapagos volcanoes and theNorthern Andes become indepen-dent clusters, and althoughmore detailed structure can be observed be-tween the Southern and Central Andes, these two volcanic regionsremain as part of a single cluster. In order to identify these two partsof the Andean volcanoes as separate chains, it is necessary to set asmoothing factor of 700. Interestingly, at this value of the smoothingfactor the volcanoes of Central America and the TMVB continue to bepart of a single cluster, although themaximum density is clearly related

Fig. 6.Maps showing a progression in the form of the density contours associated to volcanic vents along the Latin-American continent. a) Actual vents (triangles). b) to e)Maps showingthe obtained density contours by selecting the smoothing parameter in the Fisher kernel indicated on the bottom of each diagram. f) Results obtained by using the Gauss kernel with theindicated smoothing parameter. Volcanic regions: NM— Northwest Mexico, TMVB— Trans Mexican Volcanic Belt, CAm— Central America, G — Galapagos, NA— Northern Andes, CA—

Central Andes, SA — Southern Andes.

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to the Central America volcanoes. Actually, the density of volcanoes inCentral America achieves a larger maximum value than that achievedin the Southern Andes, and this in turn is larger than that achievedat the other independent clusters. The contours obtained with theGauss kernel and a smoothing factor of 250 are shown in Fig. 6f for com-parison. As expected from the curves of density contribution shown inFig. 1, the resulting density contours are almost identical for bothkernels.

4.3. Geologic implications

As for the geological implications of the use of both methods, it isimportant to note that previous studies have shown that the spatialdistribution of volcanic vents has fractal characteristics, displayingsome degree of self-similar clustering (Connor, 1990; Mazzarini andD'Orazio, 2003; Mazzarini et al., 2008). This characteristic of vent dis-tributions has been investigated in further detail by Mazzarini (2007),who applied an agglomerative hierarchical method in which two

neighboring observations are successively joined to form one cluster,until all the observations are part of a single grouping. An importantpart of this process requires the calculation of new coordinates forthe centroid of each cluster formed, because those coordinates areused as the new “observations” that will be merged in the followingstep. Although this method has some appeal for the unbiased deter-mination of a hierarchy of clusters, it might yield centroids of clustersthat are located in a section of the area under study that is entirelydevoid of any actual observations (Mazzarini et al., 2010). Additional-ly, the method used by those authors might result in some closeddensity contours that could be interpreted as being independentclusters that nevertheless are not associated with the coordinatesof a centroid, as is the case in the analysis made by Mazzarini et al.(2010) on the Michoacán–Guanajuato volcanic field (specificallytheir Fig. 5a). To some extent, the apparent disconnection betweenthe location of local clusters as revealed by a contour plot of relativevent density, and the location of the calculated centroids mightprove to be a disadvantage because it is difficult to assign a direct

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physical significance of melt production to a centroid that is locatedaway from the actual vents. Similarly, a further justification for theexclusion of a cluster that is evident on the contours yet has no spe-cific centroid associated with it might be required.

The problem just described is not found when the density con-tours are drawn by using a progressive succession of smoothing fac-tors with either the bivariate Gauss or Fisher kernels.

The examples of this paper show that by following this approach it ispossible to identify an evolution of density contours that ultimatelymight be related to significant changes in the long chain of events thatlead to the presence of volcanic activity. For example, differences inthe thermal structure of themantle wedge over time or spatially, varia-tions in composition of both themantle wedge and the subducting slab,changes in the fertility of the underlying rocks as the result of eventsof melting and depletion, and differences in the stress state of the over-lying plate are among the many factors that can contribute to thepresence or absence of a volcanic front in convergent margins (Stern,2002; Hall, 2012). Some of the clues concerning various of these param-eters that might be revealed by examination of the progression of con-tours such as those shown in Figs. 4 and 5 are examined next. As shownby the diagrams in Figs. 5 and 6, the different clusters always are definedin regions where the relative concentration of vents reaches a localmaximum, and therefore, the coordinates of the centroid (the middleof the cluster) will always bear a closer association with the actual ob-servations than they do in the method adopted by Mazzarini (2007).Furthermore, these clusters can be classified according to a thresholdof relative density that serves as a cut off value that not only identifiesindependent zones from a spatial point of view, but also assigns relativeweights to each cluster.

In the specific case of the progression observed for the distributionof volcanic vents in the Peninsula of Baja California, Fig. 5 suggeststhat at a regional scale there is a maximum density of volcanic centersalmost symmetrically disposed north and south of latitude 28° N(Fig. 5b and c). If attention is given to the concentration of vents with-in the boundaries of individual volcanic fields, however, there seemsto be a progression of larger to smaller density from north to south(Fig. 5e and f). In other words, although the total number of volcanicvents north and south of 28° N is more or less equilibrated, there aremore fields south of this latitude, and the vents within the fields tothe north are more clustered than the vents in the southern fields.Consequently, the various diagrams of Fig. 5 indicate that there is adifferent hierarchical distribution of the volcanic vents that can beidentified by using a progression of smoothing factors in the estima-tion of spatial PDFs. Such hierarchy might be related to aspects asso-ciated with the source of melts at depth as follows. The relationshipbetween the extent of volcanism in a given area and the correspond-ing dimensions of the melt source at depth has been investigatedin other volcanic fields around the world (Bernhard Spörli andEastwood, 1997; Kiyosugi et al., 2010). In these cases, individual clus-ters are formed at the places where magma supply is slightly larger,and can be related to zones of crustal weakness. Consequently, thegeneral shape of the contours found in Baja suggests a magma sourceregion resembling an elliptical dome from where a series of fingersemanate (Tamura et al., 2002). Each of these fingers is associatedwith hot zones marking preferential paths for rapid magma ascentonce the critical conditions for magma tapping are reached in themantle source, and define the boundaries between fields, that aremore marked on the northern part of the studied area.

A similar progression is observed in Fig. 6, although in this case thevents considered are defined by large volcanic centers located along aband that extends for a much larger total distance than in the case ofthe fields in the Peninsula of Baja California. At a hemispheric scalethe distribution is elongated and displays a larger concentration ofvolcanoes to the north, (Fig. 6b) but if attention is given to the con-centration of volcanoes within specific zones it is seen that the south-ern Andes and Central America are the two regions with larger

relative concentrations (Fig. 6e). In this case, the shape of themagma source also seems to follow the contours of the convergentmargin, but the identification of separate clusters might have a differ-ent origin than that suggested to control the formation of fields in thePeninsula of Baja California. In any case, while the segmentation inSouthern America is easily detected by the kernel methods, theTMVB–Central America region seems to define a continuous groupingof volcanism that contains two places of relative maximum density ofvolcanoes, but that nonetheless are not separated regions at the samesignificance level than the South American volcanic chain. Such dif-ference between both zones therefore might be reflecting a funda-mental difference in the slab-mantle-crustal structure underlyingthose two broad zones, but the difference between regions deservesa more detailed discussion beyond the purpose of the present work.

5. Conclusions

This paper has established that despite their differences in inherenttopology, there is no particular distortion of the distribution calculatedusing the Gauss and Fisher kernels, at any spatial scale, as long as thedistances between data and observation points are calculated in termsof Eq. (3). This equation eliminates the geometric problems associatedwith the distortion introduced by any map projection, and allowsboth kernel methods to identify spatial clusters with exactly the sameaccuracy, provided that a suitable value of the smoothing factor is se-lected. Perhaps the only advantage of the Fisher kernel is that due toits inherent spherical topology, it avoids any ambiguity that couldarise if a data set includes longitudes near the meridian of reference.On the other hand, the Fisher kernel has a limitation concerning its abil-ity to provide good resolution of spatial data that have separation dis-tances that are relatively small (b500 km). Vent distributions withtypical separations below this threshold need the Gauss kernel to be ac-curately described.

This work also has shown that the numerical value of the smooth-ing factor (or bandwidth) of both kernels has no unique nor direct re-lationship with the concept of a relevant separation among data, orbetween data and observation points. Actually, in order to establishthe radius of the relevant distance around a given observation point,it is necessary to take into consideration the value of the respectivesmoothing factor together with a reference level at which the contri-butions to the PDF will be considered to be significant.

Once a relevant distance at a particular significance level has beenestablished, both kernels have the potential to yield insights concerningthe formation of significant clusters at various hierarchical levels thatare easy to identify when density distributions are determined using arange of values of the smoothing parameter. Specifically, the results inFigs. 5 and 6 indicate that the progressive exploration of PDF contoursas a function of the smoothing parameter can provide an unbiasedtool to establish the boundaries of volcanic fields, or regions, by payingattention to the moment when a specific density contour breaks apartto form separate clusters. This procedure will also indicate which ofthose clusters has a larger relative concentration of volcanic centers,which in turn provides a form to classify different zones ranking themin terms of descending maximum relative concentration. This type ofinformation might in turn provide some clues that are worth exploringin more detail in a broader geologic or tectonic context.

Acknowledgments

The help provided by Ramón Mendoza-Borunda in several aspectsof this research is greatly appreciated. I also thank the two anony-mous reviewers for the comments they made that were instrumentalin clarifying several concepts, and in correcting an erroneous state-ment made in a previous version. This research was partially fundedby a CONACYT grant.

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