Nonparametric estimation of directional highest density regions

Animal orientation example. Behavioral plasticity is considered by biologists as a feature of adaptation to changing beach environments. In particular, orientation is an adaptation characteristic that can not be modified by a single factor. Nonetheless, experts found some regularities in the orientation of sandhoppers and other animals from beach environments by changing one factor at a time under other controlled conditions.

For instance, the orientation of two sandhoppers species (Talitrus saltator and Talorchestia brito) is analyzed in Scapini et al. (2002). Both species are shown in Fig. 1. Bottom pictures can be found in Dekker (1978). Comparing the two species through regresion procedures, Scapini et al. (2002) conclude that Talitrus saltator showed more differentiated orientations, depending on the time of day, period of the year and sex, with respect to Talorchestia brito. Moreover, it seems that Talitrus saltator shows a higher flexibility (variation) of orientation than Talorchestia brito under the same environmental conditions, supporting the hypothesis that the former has a higher level of terrestrialization. As an illustration, Fig. 2 (left panel) shows the 36 orientation points (slightly jittered) corresponding to males of the specie Talitrus saltator measurements during the noon in April. It also contains the 77 angles (slightly jittered) when the measures are taken in October (Fig. 2, right panel). Differences in the distribution on the circle of these two samples can be easily observed. Therefore, the month of the year seems to play a significant role in sandhoppers behavior. In particular, two clusters for October measurements can be detected around the angle \(\pi \) but they are not present for the April sample. Similar comments could be done for the situation registered around the angles \(\pi /2\). HDRs reconstruction (with low probability content) would allow to determine the biggest modes of the distribution, and then, its clusters. Therefore, HDRs can be seen as a useful alternative to analyze sandhoppers orientation.

Earthquakes occurrences. The European-Mediterranean Seismological Centre (EMSC)² is a non-governmental and non-profit organisation that has been established in 1975 at the request of the European Seismological Commission. Since the European-Mediterranean region has suffered several destructive earthquakes, there was a need for a scientific organisation to be in charge of the determination, as quickly as possible (within one hour of the earthquake occurrence), of the characteristics of such earthquakes. These predictions are based on the seismological data received from more than 65 national seismological agencies, mostly in the Euro-Med region. Figure 3 (left) shows the geographical coordinates (red points), downloaded from EMSC website, of a total of 272 medium and strong world earthquakes registered between 1\(^{th}\) October 2004 and 9\(^{th}\) April 2020. The magnitude of all these events is at least 2.5 degrees on the Richter scale. Of course, these planar points correspond to spherical coordinates on Earth. Due to the important damages that earthquakes cause, cluster detection of HDRs could be also useful to identify, from a real dataset, where earthquakes are specially likely. This information is key for decision-making, for example, to update construction codes guaranteeing a better building seismic-resistance. An interactive representation of the sphere can be seen in Appendix D.

This paper is organized as follows. Section 2 contains some background ideas on directional level set estimation including some discussion on error measurements and some existing consistency results in the directional setting that will be really useful to extent the definition of HDRs for directional data in Sect. 3. There, plug-in estimators and the corresponding confidence regions are also established. Concretely, we consider the plug-in methods based on a well-known directional kernel density estimator, which requires a smoothing parameter (bandwidth) for its practical implementation. An appropriate bootstrap bandwidth selector, the first one specifically designed for directional HDRs estimation, is also introduced in this section. Section 4 presents an extensive simulation study illustrating the performance of the resulting plug-in reconstruction for the HDRs (for circular and spherical domains) considering the new bandwidth selector. These results are compared with those obtained with directional smoothing parameters not specifically designed for HDR estimation. In Sect. 5, the proposed methodology is applied to analyze the two real data examples presented in the Introduction. Finally, some conclusions and ideas for further research are presented in Sect. 6. Appendix A and the supplementary material that completes this work include further information on the datasets. Appendix B specifies the parameters taken for the construction of the spherical densities in the simulation study. Appendix C contains some additional results of simulations. Appendix D collects the description of the bandwidth selectors considered in the simulation study. All the methods presented in this paper, along with the real data examples, are accessible in HDiR package.

Consider a random vector X taking values on a \(d-\)dimensional unit sphere \(S^{d-1}\) with density f. Given a level \(t>0\), the directional level set is defined as:The nature of different level sets is shown in Fig. 4, which represents \(G_f(t)\) in grey color for three different circular densities and three different values of the level t. The threshold t is represented through a dotted grey line. Note that, if large values of t are considered (bottom row in Fig. 4), \(G_f(t)\) coincides with the greatest modes. However, for small values of t, the level set \(G_f(t)\) is virtually equal to the support of the distribution.

It is important to noticed that, following (Hartigan 1975), we may also establish the concept of cluster in directional setting as the connected components of the level set \(G_f(t)\). With this view in mind, note that the density represented in the second row of Fig. 4 presents four connected components for all of the considered values for t, determining four population clusters.

Plug-in estimation is the most natural and common choice for reconstructing density level sets in the Euclidean space. A review of other existing estimation alternatives can be seen in Rodríguez-Casal and Saavedra-Nieves (2019). Plug-in methods are devised to reconstruct the level set in (1) aswhere \(g_n\) usually denotes the classical kernel estimator for Euclidean data (see Parzen 1962 and Rosenblatt 1956). This methodology, which has received considerable attention (see, for instance, Tsybakov 1997; Baíllo 2003; Mason and Polonik 2009; Rigollet and Vert 2009; Mammen and Polonik 2013; Polonik 2013 or Chen et al. 2017) can be easily generalized to the directional setting. Given a random sample \(\mathcal {X}_n=\{X_1,\ldots ,X_n\}\in S^{d-1}\) of the unknown directional density f, the corresponding level set \(G_f(t)\) in (2) can be reconstructed aswhere \(f_n\) denotes a nonparametric directional density estimator. Following the classical ideas for real-valued random variables, a kernel estimator on \(S^{d-1}\) is provided in Bai et al. (1989) (\(d>2\)) who also proved strong pointwise consistency, uniform consistency, and \(L_1-\)norm consistency of the estimator (see also Hall et al. 1987 and Klemelä 2000 for further results). Following Bai et al. (1989), from a random sample \(\mathcal {X}_n\) on a \(d-\)dimensional sphere the directional kernel density estimator at a point \(x\in S^{d-1}\) is defined aswhere \(1/h^2 > 0\) is concentration parameter and \(K_{vM}\) denotes the von Mises-Fisher kernel density (see Appendix B for explicit formulae). The consideration of a von Mises kernel in Eq. (4) is not the only option and it is particularly interesting to point out the use of a wrapped-normal kernel in the circular setting. In this case, Huckemann et al. (2016) proved that this kernel guarantees the monotonicity on the number of modes with respect to the smoothing parameter, something that also happens for the gaussian kernel in the linear case. It may be argued that such a kernel shoud be used in our problem. Nevertheless, it is computationally more expensive and our practical experience shows that results in practice are quite similar.

Note that the kernel estimator in (4) can be viewed as a mixture of von Mises-Fisher. Furthermore, the concentration parameter \(1/h^2\) plays an analogous role to the bandwidth in the Euclidean case. For small values of \(1/h^2\), the density estimator is oversmoothed. The opposite effect is obtained as \(1/h^2\) increases: with a large value of \(1/h^2\), the estimator is clearly undersmoothing the underlying target density. Hence, the choice of h is a crucial issue. For simplicity, in what follows, we refer to h as bandwidth parameter. Several approaches for selecting h in practice, in circular and even directional settings, have been proposed in the literature (see Appendix D). All the existing proposals aim to minimize some error criterion on the target density, but none of them is specifically designed focusing on the reconstruction of a directional level set.

Figure 5 shows three plug-in estimators \(\hat{G}_f(t)\) for models (black colour) and levels \(t_1\), \(t_2\) and \(t_3\) (dotted grey line) considered in Fig. 4. Kernel density estimators (grey color) in (4) have been determined from samples of size 250 considering the proposal in Oliveira et al. (2012) as bandwidth parameter.

For instance, for the sandhoppers example, Fig. 6 shows the plug-in estimators obtained for the two samples of sandhoppers represented in Fig. 2. It is possible to detect the largest modes of the two sample distributions corresponding to April and October samples. These results allow us to confirm the differences between the two populations. The largest cluster of April orientations is located around the angle \(7\pi /4\). However, the pattern observed for October registries is completely different. Although an only cluster is identified around the angle \(3\pi /2\), if the level t decreases slightly two additional groups can be detected around the angles \(3\pi /4\) and \(5\pi /4\), respectively.

Regarding the earthquakes illustration, Fig. 3 (right) shows the plug-in contour in blue obtained from the selected sample of world earthquakes considered. Chosing a convenient value of the level t, the greatest mode of sample distribution is identified in the Southeast of Europe. Countries such as Italy, Greece or Turkey (located withint this cluster) are the most affected areas in the recent past.

The level set \(\hat{G}_f(t_3)\) represented in Fig. 5 (third column) presents two connected components. However, Fig. 4 (right plot in the bottom row) shows that the theoretical level set \(G_f(t_3)\) has exactly three components. Therefore, the estimation error is considerably large. Distances between sets are the common criteria considered in set estimation to measure the discrepancies between the theoretical region to be estimated and the corresponding reconstruction. Of course, this is also applicable when the goal is to estimate level sets or HDRs.

The distance in measure \(d_{\mu }\) between two Borel sets A and B in \(\mathbb {R}^d\) is defined aswhere \(\mu \) denotes the Lebesgue measure and \(A\triangle B\), the symmetric difference of A and B calculated as \((A\cap B^c)\cup (A^c\cap B)\) with \(A^c\) representing the complementary of A. Consistency results for directional plug-in estimators have been already obtained in the literature for this distance. For the estimator established in (3) defined on \(S^2\), Cuevas et al. (2006) and Cholaquidis et al. (2020) check that \(\lim _{n\rightarrow \infty }d_\mu ( G_f(t), \hat{G}_f(t))=0, \text{ a.s. },\) and \(\lim _{n\rightarrow \infty }d_\mu (G_f(t),\hat{G}_f^{'}(t))=0, \text{ a.s. }\), where \(\hat{G}_f^{'}(t)=\{x\in S^{d-1}:f_n^{'}(x)\ge t\} \) and \(f_n^{'}\) denotes the kernel estimator (for manifolds with boundary) proposed in Berry and Sauer (2017). From the definition in (5), it easy to check that the distance in measure \(d_\mu \) does not penalize those level set estimators that have an isolated point as a connected component or any other set with null Lebesgue measure. Additionally, the undersmoothing caused by the choice of a small bandwidth value may provoque that the estimator \(\hat{G}_f(t)\) presents non-significant connected components with small Lebesgue measure. In this case, \(d_{\mu }\) would not be as effective as, for instance, the Hausdorff distance in detecting this situation.

Let us recall that, if A and B are now non-empty compact sets in \(\mathbb {R}^d\), the Hausdorff distance between A and B is established as followswhere \(\rho (\{x\},B)=\inf _{y\in B}\{\rho (x,y)\}\) being \(\rho (x,y)\) the distance between two points. Note that the definition of the Hausdorff distance is very general and depending on the selection of the distance \(\rho \), different error criteria emerge. Usually, \(\rho \) corresponds to the chordal distance (Euclidean distance in \(\mathbb {R}^d\), \(\rho _1\)).

The metric \(d_H\) is not completely successful in detecting differences in shape properties. In other words, two sets can be very close in Hausdorff distance and still show quite different shapes. This typically happens where the boundaries \(\partial A\) and \(\partial B\) are far apart, no matter the proximity of A and B. So a natural way to reinforce the notion of visual proximity between two sets provided by Hausdorff distance is to account also for the proximity of the respective boundaries. This error criterion has been also considered for establishing consistency results of several directional plug-in reconstructions. Cuevas et al. (2006) prove that \(\lim _{n\rightarrow \infty }d_H(\partial G_f(t),\partial \hat{G}_f(t))=0, \text{ a.s. },\) when the Hausdorff distance is defined from \(\rho _1\). If the Hausdorff distance involves \(\rho _2\), (Cholaquidis et al. 2020) prove that \(\lim _{n\rightarrow \infty }d_H(\partial G_f(t),\partial \hat{G}_f^{'}(t))=0\) and \(\lim _{n\rightarrow \infty }d_H(G_f(t),\hat{G}_f^{'}(t))=0, \text{ a.s }\). The existing monotone relationship between chordal and geodesic distances guarantees the consistency of the plug-in estimator in Cholaquidis et al. (2020) also when the Hausdorff distance depends on \(\rho _1\) instead of \(\rho _2\). Therefore, if the target is the reconstruction of a set, the Hausdorff metric (defined from the chordal distance) can be seen as a suitable error criteria in the directional setting.

The first step to reconstruct the HDR in Definition 1 for a given \(\tau \in (0,1)\) is to estimate the threshold \(f_\tau \). As in the Euclidean case, numerical integration methods could be also used in the directional setting in order to approximate its value. However, when the dimension increases, the computational cost becomes a major issue due to the complexity of the numerical integration algorithms considered on high dimensional spaces. An alternative approach for estimating \(f_{\tau }\) with a feasible computational cost is described next.

As before, let X be a random vector with directional density f and let \(Y = f(X)\) be the random vector obtained by transforming X by its own density function. Since \(\mathbb {P}(f(X)\ge f_\tau )=1-\tau \), \(f_{\tau }\) is exactly the \(\tau -\) quantile of Y, following (Hyndman 1996), \(f_{\tau }\) can be estimated as a sample quantile from a set of independent and identically distributed random vectors with the same distribution as Y.

In particular, if \(\mathcal X_n=\{X_1,\ldots ,X_{n}\}\) denotes a set of independent observations in \(S^{d-1}\) from a density f, \(\{f(X_1), \ldots , f(X_n)\}\) is a set of independent observations from the distribution of Y. Let \(f_{(j)}\) be the \(j-\)th largest value of \(\{f(X_i)\}_{i=1}^n\) so that \(f_{(j)}\) is the (j/n) sample quantile of Y. We shall use \(f_{(j)}\) as an estimate of \(f_\tau \). Specifically, we choose \(\hat{f}_\tau = f_{(j)}\) where \(j = \lfloor \tau n\rfloor \). Cadre et al. (2009) study the convergence of \(\hat{f}_\tau \) to \(f_\tau \) in the linear setting.

Obviously, if f is a known function, the observations can be pseudorandomly generated and the estimation of \(f_\tau \) could be made arbitrarily accurate by increasing n. In practice, f is often unknown and we have as only information a random sample of points \(\mathcal {X}_n\) from an unknown density f. From this sample, we propose first to determine the kernel estimator \(f_n\) in (4). If n is large enough, then calculate the set \(\{f_n(X_1),\ldots ,f_n(X_n)\}\) in order to estimate f empirically. If n is moderate, it may be preferable to generate observations \(\mathcal X_n=\{X_l,\ldots ,X_N\}\) of large size N from \(f_n\). For small values of n it may not be possible to get a reasonable density estimate. Besides, with few observations and no prior knowledge of the underlying density, there seems little point in attempting to summarize the sample space (see Wand and Jones 1995 for some discussion on the number of observations needed for a reasonable linear density estimate). Note that the problem here is not with the density quantile algorithm (that give results to an arbitrary degree of accuracy given a density), but with estimating the density from insufficient data.

Once the threshold \(f_\tau \) is estimated, plug-in methods reconstruct the \(100(1 - \tau )\)% HDR namely \(L(f_\tau )\) in (7) asFigure 7 shows the circular kernel estimator \(f_n\) (grey color) calculated from a sample \(\mathcal {X}_{250}\) generated from the second model (black color) in Fig. 4 and different empirically approximated 50% circular regions (grey color, top). The boxplot of the transformed values denoted by \(\{f_n(X_1),\ldots ,f_n(X_{250})\}\) is also shown (bottom). The dotted lines represent the quantiles that determine the corresponding 50% (probability coverage) circular region. Note that only the estimated HDR (left), \(\hat{L}(\hat{f}_\tau )\), is able to show the existence of the five existing modes.

Apart from the consistency of \(\hat{f}_\tau \), Cadre et al. (2009) establish the exact convergerce rate (considering the distance in measure \(d_\mu \) as error criteria) for Euclidean HDRs. The extension of these results to the directional setting does not seem straightforward. However, if \(\hat{f}_{\tau }-\)consistency remains true, we could prove that \(\hat{L}(\hat{f}_\tau )\) is also a \(d_H-\)consistent estimator of \(L(f_\tau )\) in \(S^2\) under the assumptions of Corollary 1 and condition (T) in Cuevas et al. (2006). To complete the proof is only necessary to apply a triangle inequality on \(d_H(L(f_\tau ),\hat{L}(\hat{f}_\tau ))\).

The plug-in reconstruction of the directional HDRs in (9) involves the calculation of the kernel density estimator in (4) that is known to be heavily dependent on the selection of h. The existing methods for selecting an optimal value for h aim for minimizing some error criterion on the target density f, but they are not specifically designed for the estimation of HDRs. The goal of this section is to propose the first selector of h specifically designed for HDRs reconstruction.

A bootstrap bandwidth selector focused on the problem of reconstructing HDRs is introduced in what follows. The idea is to use an error criterion that quantifies the differences between the theoretical region and its plug-in reconstruction. In the real-valued setting, Samworth and Wand (2010) propose one of the first bandwidth selectors for HDRs estimation studying an relatively uncommon distance (depending on both \(\mu \) and g) between these sets. In this work, we consider the classical Hausdorff distance (introduced in Sect. 2.2) between the boundaries of the HDR and the corresponding estimator.

In the directional case, the closed expression of \(d_H(\partial L(f_\tau ),\partial \hat{L}(\hat{f}_\tau ))\) is not known. However, it could be estimated through a bootstrap procedure. Therefore, a new bandwidth selector can be established aswhere \(\mathbb {E}_B\) denotes the bootstrap expectation with respect to random samples \(\mathcal X_n^*=\{X_1^*,\ldots ,X_n^*\}\) generated from the directional kernel \(f_n\) that, of course, is dependent on a pilot bandwidth and also on the choice of the distance \(\rho \) in Eq. (6).

Figure 12 shows the theoretical HDR for model S3 (see Sect. 4.2) when \(\tau =0.5\) (first and second columns). Moreover, the plug-in estimator \(\hat{L}(\hat{f}_\tau )\) obtained from a sample of size \(n=1000\) and considering the bandwidth proposed in García-Portugués (2013) when \(\tau =0.5\) is also represented (third column). Note that, for this sample size, only the largest mode is detected. In this particular case, the Hausdorff error is smaller if the HDR is reconstructed from a cross-validation bandwidth designed for density estimation (fourth and fifth columns). A relevant issue appears when \(h_1\) is estimated from imprecise HDR estimators. Remember that the minimization procedure considered for determining \(h_1\) involves the boundary of the set \(\hat{L}(\hat{f}_\tau )\). If this set is poorly approximated the resulting bandwidth surely will not provide competitive results. Therefore, largest sample sizes will be considered in this section for avoiding this problem.

Another point that is worth to mention is that diverse bandwidths selectors emerge from the consideration of the different choices of \(\rho \) in the definition of the Hausdorff distance. In fact, other bandwidths could be defined if, for example, \(d_H\) in (10) is replaced by a completely different error criteria such as the distance in measure \(d_\mu \) that, unlike Hausdorff distance, does not take in account the connected components of a set only composed by a isolated point. Therefore, we could propose as many bandwidths as existing distances between sets attending to the specific properties and characteristics of each distance.

A collection of 9 circular densities (models C1 to C9) have been considered in this simulation study. These models are mixture of different circular distributions and they correspond to densities 5, 6, 7, 8, 10, 11, 16, 19 and 20 fully described in Oliveira et al. (2014). Figure 9 shows these densities and the thresholds \(f_\tau \) for \(\tau =0.2\), \(\tau =0.5\) and \(\tau =0.8\) through dotted circles.

A total of 250 random samples of sizes \(n=500\) and \(n=1000\) were generated for each of these models. From each sample, circular HDRs are reconstructed for \(\tau =0.2\), \(\tau =0.5\) and \(\tau =0.8\). Results for \(\tau =0.2\) and \(\tau =0.8\) are exposed in Appendix C.1. The behavior of plug-in methods that emerge from the consideration of different bandwidth parameters will be checked. Apart from \(h_1\), we will consider the circular rule-of-thumb by Taylor (2008) (\(h_2\)); its improved version by Oliveira et al. (2012) (namely \(h_3\)); cross-validation methods (likelihood \(h_4\) and least squares \(h_5\)) introduced by Hall et al. (1987) and the bootstrap bandwidth (\(h_6\)) presented by Di Marzio et al. (2011). Note that for computing \(h_1\), a pilot bandwidth is required. In this study, \(h_3\) has been taken as a pilot, and \(B=200\) resamples are considered for obtaining \(h_1\).

For each method and each sample, the estimation error is measured by computing the Hausdorff distance (\(d_H\)) between the boundaries of estimated HDR and the frontier of theoretical set. As a reference, note that the maximum value of this criteria in \(S^1\) is 2 (the length of the diameter of the circle).

Tables 1 and 2 show the means and the standard deviations of the 250 estimation errors obtained when \(\tau =0.5\) from samples of sizes \(n=500\) and \(n=1000\), respectively. Bold numbers correspond to the lowest mean errors obtained for each density. Taking into account the variety of models considered, exhibiting different features, it is not surprising that all of the bandwidth selectors are the best ones for some model, showing \(h_1\) a competitive behavior in all cases. In fact, it is the best one for models C3, C5, C6 and C8 (with \(n=1000\)).

Figure 10 shows the violin plots of Hausdorff errors obtained for some of the simulation models when \(\tau =0.5\) (\(n=1000\)). It shows that \(h_2\) is the selector that presents a worst behavior for models C3 and C6. Furthermore, its variance is again specially large for model C3.

Finally, it is worth to mention that results in Appendix C.1 shows that the competitive behavior of \(h_1\) improves considerably when high values of \(\tau \) are selected. This is not a minor question when the goal is to estimate the biggest modes of a distribution.

For the spherical scenario, 9 density models have been considered. These models, namely S1 to S9, are mixtures of von Mises-Fisher densities on the sphere, allowing to represent complex structures showing multimodality and/or asymetry. Parameters of mixtures are fully established in Table 8 in Appendix B for reproducibility. Moreover, these densities are also implemented in the R package HDiR. Figure 11 shows them and the corresponding HDRs for \(\tau =0.2\), \(\tau =0.5\) and \(\tau =0.8\).

For sample sizes \(n=500\), \(n=1500\) and \(n=2500\), 200 random samples were generated from models S1 to S9. From each sample, HDRs are reconstructed for \(\tau =0.2\), \(\tau =0.5\) and \(\tau =0.8\). As before, results for \(\tau =0.2\) and \(\tau =0.8\) are contained in Appendix C.2. The performance of different plug-in methods that emerge from the consideration of different bandwidth parameters discussed in this work is checked. Apart from \(h_1\), cross-validation bandwidth selectors for data on a sphere \(S^{d-1}\) (\(h_5\)) and the plug-in bandwidth selector introduced by García-Portugués (2013) (\(h_7\)) are taken into account. In this case, a total of \(B=50\) resamples are established for estimating the proposed bootstrap bandwidth \(h_1\), taking \(h_5\) as a pilot bandwidth.

For each method and each sample, the estimation error is again measured calculating the Hausdorff distance between the boundaries of estimated HDR and the frontier of theoretical set. As reference, note that the maximum value of both criteria \(S^2\) is also 2. In this case, this upper bound coincides exactly with the length of the diameter of the sphere.

Tables 3, 4 and 5 contains the results for \(\tau =0.5\) when \(n=500\), \(n=1500\) and \(n=2500\), respectively. Bold numbers correspond to the lowest mean errors obtained for each density. The proposed selector \(h_1\) is the best or second best in all cases. In fact, \(h_1\) and \(h_5\) usually behave similarly and \(h_7\) is the worst selector for S3.

Figure 13 contains the violin plots of Hausdorff errors for some of the considered models when \(\tau =0.5\) and \(n=1500\). Remark that \(h_1\) and \(h_5\) usually present similar results, see densities S5 and S9. However, \(h_1\) is clearly more competitive for models S1 and S8.

To conclude, simulations in Appendix C.2 allows to confirm the good performance of the selector \(h_1\) when small or big values of \(\tau \) are considered for spherical data.

Adaptation to changing beach environments for the real example on sandhoppers introduced in Sect. 1.1 is analyzed from HDRs estimation perspective. HDRs are estimated for \(\tau =0.8\) disaggregating the sandhoppers data taking into account the categories of variables specie, sex, time of day and month of year. As consequence, a total of 24 set estimators are determined, numbered E1 to E24. Variables combinations yielding this group classification are presented in Table 7 in Appendix A.

Note that the estimated HDRs correspond to the largest modes of the orientation distributions. Hausdorff distances between the boundaries of these 24 sets are able to establish the degree of dissimilarity of HDRs. In general, large distances between the boundaries of two sets indicate the existence of modes in different directions. If the categories of all variables with the exception of one are fixed, it is possible to check if the different values of the non-fixing variable has some influence in sandhoppers orientation through the comparison of the estimated HDRs. The upper triangular matrix in Table 6 contains the Hausdorff distances (defined from \(\rho _1\)) between boundaries. The largest distances are represented in blue color. Grey color is used in order to depict the next largest values. Furthermore, Table 6 (top) contains some of the estimated HDRs that present the largest distances.

In particular, Hausdorff distance between regions 5 and 11 is equal to 1.91 (close to 2, the maximum value of Hausdorff distance). According to Table 7, the variable configuration 5 corresponds to the largest orientation modes for females of the specie Talitrus saltator when the orientation is measure in noon during October. Region 11 refers to same measurements taken in April. Therefore, the month can be seen as variable that has influence on the orientation for sandhoppers.

Hausdorff distance between regions 5 and 6 is equal to 1.93. According to Table 7, set 6 also corresponds to the HDR for females of the specie Talitrus saltator but, in this case, when the orientation is registered in morning during October. Then, the moment of the day also seems a factor with influence on the sandhoppers behavior.

Several cells in Table 6 are represented in pink color. All of them corresponds to considerable large values of distances (larger than 1.00) and they are used to analyze briefly the influence of each of the variables in the dataset. Under the same values of the rest of variables Talitrus saltator and Talorchestia brito present different behaviors. For instance, distances between sets 5 and 17 or 3 and 15 correspond to this situation. Sets 5 and 17 can be compared using their representations in Table 6 (top).

The importance of the sex variable for the specie Talitrus saltator can be also seen considering the Hausdorff distances of the sets 2 and 5, 3 and 6 or 18 and 15. According to images in Table 6, these sets present their largest modes in completely different directions. Note that the role of the variable month is clearly remarkable. The relatively high values of the distances between sets 1 and 7 and 6 and 12 or 14 and 20 for the species Talitrus saltator and Talorchestia brito also corresponds to the existence of modes in different directions. Finally, the importance of the moment of the day for the Talitrus saltator can be studied through the distances between sets 4 and 5 or 4 and 6. Remark that set 4 has two connected components while set 5 only presents one.

Finally, the analysis of Hausdorff distances for the two species of sandhoppers shows that the median of the Talitrus saltator in Hausdorff distance is 0.76, clearly bigger than the median of Talorchestia brito that is equal to 0.52. Therefore, Talitrus saltator presents more differentiated orientations, depending on the time of day, period of year and sex, with respect to Talorchestia brito. Therefore, conclusions in Scapini et al. (2002) are corroborated from this perspective.

According to the theory of plate tectonics, Earth is an active planet. Its surface is composed of about 15 individual plates that move and interact, constantly changing and reshaping Earth’s outer layer. These movements are usually the main cause of volcanoes and earthquakes. Seismologists have related these natural phenomena to the boundaries of tectonic plates because they tend to occur there, see Selley et al. (2004). In fact, the concentration of earthquake epicenters traces the filamentary network of fault lines and, consequently, they could be analyzed alternatively from the perspective of nonparametric filamentary structure estimation (see, for instance, Genovese et al. 2012). Moreover, tectonic hazards can provoque important damages (destroy buildings, infrastructures or even cause deaths). Therefore, it is important to detect which areas are specially risky. As an illustration, the recent world earthquakes distribution is analyzed next through HDRs estimation.

Figure 14 shows the margins of the tectonic plates (grey color) and the geographical coordinates (red points) of a total of 272 medium and strong earthquakes registered between 1th October 2004 and 9th April 2020 already introduced in Sect. 1.1. Note that most of events are exactly located on the plates boundaries.

Our main goal is to detect which areas are really problematic nowadays. In Sect. 1.1, we show that the largest mode is located on the Southeast Europe considering a value of \(\tau =0.8\). However, a more general view on earthquakes distribution could be obtained if more HDRs are reconstructed for a range of values of \(\tau \). Specifically, they were estimated choosing \(\tau _1=0.1\), \(\tau _2=0.3\), \(\tau _3=0.5\), \(\tau _4=0.7\) and \(\tau _5=0.9\). The bandwidth parameter used is the proposed in García-Portugués (2013). The corresponding contours are also represented in Fig. 14 using blue colors. An interactive representation of these HDRs can be seen in Appendix D.

The two smallest contours (dark blue colors) corresponds to density regions with probability at least \(1-\tau _5=0.1\) and \(1-\tau _4=0.3\), respectively. Therefore, they match with the greatest modes of earthquakes world distribution and they identify the more risky parts of the world. They are located on Europe. Concretely, on the boundaries intersection for the Eurasian and African Plates. Note that the second of these regions even includes the frontier of the Arabian Plate. Contours for \(\tau _2=0.3\) and \(\tau _3=0.5\) are related to Indo-Australian Plate and margins of Philippine Sea and Pacific Plates appears when \(\tau _1=0.1\).

As for America, the most problematic area is detected in Central America. Concretely, it is mainly located on the frontiers of Cocos, Nazca and Caribbean Plates. According to the contours shown, this region belongs to the zone of the world where the 70% (\(1-\tau _2\)%) of earthquakes are registered. If \(\tau _1=0.1\) is considered then Pacific, North and South American plates appears as risky areas.