A Bayesian nonparametric model for bounded directional data on the positive orthant of the unit sphere

Abstract

Directional data appears in several branches of research. In some cases, those directional variables are only defined in subsets of the K-dimensional unit sphere. For example, in some applications, angles as measured responses are limited on the positive orthant. Analysis on subsets of the K-dimensional unit sphere is challenging and nowadays there are not many proposals that discuss this topic. Thus, from a methodological point of view, it is important to have probability distributions defined on bounded subsets of the K-dimensional unit sphere. Specifically, in this paper, we introduce a nonparametric Bayesian model to describe directional variables restricted to the first orthant. This model is based on a Dirichlet process mixture model with multivariate projected Gamma densities as kernel distributions. We show how to carry out inference for the proposed model based on a slice sampling scheme. The proposed methodology is illustrated using simulated data sets as well as a real data set.

1. Introduction

In recent years, interest in analyzing data representing directions has increased. These type of measurements are known as directional data and appear in several areas of knowledge such as biology, geology, meteorology, ecology and environmental sciences. Directional data are related to unit vectors in the $(K+1)$-dimensional space ${\mathbb{R}}^{K+1}$. Thus, these kinds of data can be represented by K angles and their natural sample space is the K-dimensional unit sphere, ${\mathbb{S}}^K$. In the cases where K = 1 and K = 2, directional data are named circular data and spherical data, respectively. Since the unit sphere is topologically different from the Euclidean space, a proper analysis of directional data requires appropriate statistical methods that consider inherent properties from the corresponding sample spaces. Contributions range from graphical methods to the development of new statistical models to describe directional observations. For a survey, the reader is referred to [3,6,10,13,16,18,19,21,27,28,30].

Although there has been an increasing development in directional data modeling, most recent proposals have focused on defining new parametric models on the entire unit sphere and the development of probability distributions restricted to subsets of the unit sphere has been overlooked. In many instances, there are applications relating angle measurements on subsets of the unit sphere. An example of the previous situation is axial data, where the corresponding sample space turns out to be the interval $(0,\pi ]$. In the same context, there are phenomena where the interest lies in angles defined only on the positive orthant of the unit sphere. Some applications of the latter can be found in the analysis of incidence and refraction angles of the archerfish [33]. In sports science, particularly in baseball, the study of angles related to achieving home runs is relevant [32]; in kinesiology, the relation of the extension angle of the human knee with the possibility of suffering injuries is a topic of general interest [29]; in environmental sciences, the tilt angle inclination of photovoltaic cells is analyzed to optimize power generation. In phenology, the analysis of the behavior of solar-to-sensor angles of backscatter light is important in the study of its correlation with some phenological metrics [24]. On the other hand, directional data present characteristics such as multimodality, which could not be well fitted by standard parametric models. There are undoubtedly multimodal parametric models (see, for example, [2], [15] and [22]), which can be fitted for describing multimodal behaviors of directional data. However, the fitting of the above models may require particular procedures, even more when sequential processes are applied. In general instances, where the dataset has high skewness or kurtosis and several modes, it may be preferable to consider semiparametric or nonparametric models as an alternative (see, for example, [21] and [26]).

From a theoretical point of view, there are several basic approaches to generate circular distributions. One way is defining a distribution directly on the unit sphere, as with the von Mises–Fisher or Kent or Fisher–Bingham distributions. Another method is based on projecting a multivariate distribution initially defined on the $(K+1)$-dimensional space onto the corresponding unit sphere ${\mathbb{S}}^K$. Let $\mathit{Y}$ be a random $(K+1)$-dimensional vector such that P $r(\mathit{Y}=\mathbf{0})=0$. Then $\mathit{U}=\mathit{Y}||\mathit{Y}|{|}^{-1}$ is a random point on the $K-$dimensional unit sphere, ${\mathbb{S}}^K$. Two important instances are those in which $\mathit{Y}$ has a $(K+1)$-variate Normal distribution or a Gamma distribution. In the first case, $\mathit{U}$ is said to have a multivariate projected Normal distribution and, in the second case, $\mathit{U}$ is said to have a multivariate projected Gamma distribution. While the projected Normal distribution has received a lot of attention, the projected Gamma distribution has only been studied by Núñez-Antonio and Geneyro [22], who mentioned that the family of projected Gamma distributions is appropriate for analyzing particular directional data defined only on the positive orthant.

In this paper, we propose a new nonparametric model to describe directional data restricted to the first orthant of the K-dimensional unit sphere; that means, directional data whose corresponding angles belong to the interval $(0,\pi /2]$. The proposed model is a Dirichlet process (DP) mixture model based on multivariate projected Gamma distributions. As discussed below, this model allows greater flexibility to perform analyses for directional random variables defined in the first orthant with behaviors such as multimodality or high skewness and kurtosis.

The rest of this paper proceeds as follows. In Section 2, the multivariate projected Gamma model is briefly reviewed. In Section 3, the proposed model based on the Dirichlet process mixture of multivariate projected Gamma densities is presented, as well as the way to perform Bayesian inferences. Section 4 illustrates our proposal with several simulated examples and a real dataset on solar-to-sensor angles of backscatter light. Finally, Section 5 provides some concluding remarks.

2. The projected gamma distribution

As mentioned above, one technique used to define directional distributions is radially projecting on the unit sphere ${\mathbb{S}}^K$ a multivariate distribution initially defined on ${\mathbb{R}}^{K+1}$. Since that technique does not establish any additional restrictions on the original multivariate distribution, radial projection can be used to define distributions on specific subsets of ${\mathbb{S}}^K$. Following this idea, Núñez-Antonio and Geneyro [22] have developed the multivariate projected Gamma distribution which is a model only defined on the positive orthant. They show how to carry out inference based on a Gibbs sampling scheme after the introduction of suitably chosen latent variables. The details of that Gibbs sampler are given in Núñez-Antonio and Geneyro [22], but we briefly describe the basis of the algorithm here. The starting point is the joint density

$$\begin{array}{rl}{f}_{\mathit{Y}}(\mathit{y}|\mathit{\alpha },\mathit{\beta })& =\prod _{k=1}^{K+1}\mathrm{Ga}(y_k|{\alpha }_k,{\beta }_k),\\ & \equiv G{a}_{K+1}(\mathit{y}|\mathit{\alpha },\mathit{\beta })\end{array}$$ (1)

where $\mathit{Y}=(y_1,y_2,\dots ,y_{K+1}{)}^t$ is a vector of dimension $(K+1)$, $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_{K+1}{)}^t$, $\mathit{\beta }=({\beta }_1,\dots ,{\beta }_{K+1}{)}^t$ and $\mathrm{Ga}(y_k|{\alpha }_k,{\beta }_k)$ is a Gamma density given by $\mathrm{Ga}(y_k|{\alpha }_k,{\beta }_k)=\frac{{\beta }_k^{{\alpha }_k}}{\mathrm{\Gamma }({\alpha }_k)}{y}_k^{{\alpha }_k-1}\exp (-{\beta }_k{y}_k)$. By applying a spherical coordinates transformation $\mathit{Y}\to (R,\mathbf{\Theta })$, and integrating out R, the variable $\mathbf{\Theta }=({\theta }_1,\dots ,{\theta }_K{)}^t$ has a multivariate projected Gamma distribution with density defined as

$$\begin{array}{rl}{\mathit{PG}}_K(\mathit{\theta }|\mathit{\alpha },\mathit{\beta })& =\frac{\mathrm{\Gamma }(A){\beta }_{K+1}^{{\alpha }_{K+1}}}{(B{)}^A\mathrm{\Gamma }({\alpha }_{K+1})}(\prod _{j=1}^K\frac{{\beta }_j^{{\alpha }_j}}{\mathrm{\Gamma }({\alpha }_j)}(\cos {\theta }_j{)}^{{\alpha }_j-1}(\sin {\theta }_j{)}^{(\sum _{h=j+1}^{K+1}{\alpha }_h)-1})\\ & {\mathit{I}}_{(0,\pi /2{]}^K}(\mathit{\theta })\end{array}$$ (2)

where

$$A=\sum _{j=1}^{K+1}{\alpha }_j\mathrm{and}B={\beta }_1\cos {\theta }_1+\sum _{m=2}^K({\beta }_m\cos {\theta }_m\prod _{j=1}^{m-1}\sin {\theta }_j)+{\beta }_{K+1}\prod _{j=1}^K\sin {\theta }_j.$$

The vector parameters $(\mathit{\alpha },\mathit{\beta }{)}^t$ are not identifiable, and Núñez-Antonio and Geneyro [22] propose to consider an additional constraint given by $\mathit{\beta }=(1,{\beta }_2,{\beta }_3,\dots ,{\beta }_{K+1}{)}^t$. Despite that restriction, the model maintains enough flexibility to describe several behaviors on the positive orthant.

Given a sample of directional vectors $\{{\mathit{\theta }}_1,{\mathit{\theta }}_2,\dots ,{\mathit{\theta }}_n\}$, where ${\mathit{\theta }}_i=({\theta }_{i1},{\theta }_{i2},\dots ,{\theta }_{iK})$, to make inferences on parameters $(\mathit{\alpha },\mathit{\beta })$, Núñez-Antonio and Geneyro [22] propose to consider $R_i$, $i=1,\dots ,n$, as suitable latent variables so that each vector ${\mathit{y}}_i=(r_i,{\mathit{\theta }}_i)$ follows a distribution as (1). It can be shown that the conditional posterior distribution of $R_i$ is

$$f(r_i|{\mathit{\theta }}_i,\mathit{\alpha },\mathit{\beta })=\mathit{Ga}(\sum _{j=1}^{K+1}{\alpha }_j,\cos {\theta }_{i1}+\sum _{m=2}^K({\beta }_m\cos {\theta }_{\mathrm{im}}\prod _{j=1}^{m-1}\sin {\theta }_{\mathrm{ij}})+{\beta }_{K+1}\prod _{j=1}^K\sin {\theta }_{\mathrm{ij}}).$$

Then considering the complete data ${\mathit{D}}_{\mathit{n}}=\{({\mathit{\theta }}_i,r_i){\}}_{i=1}^n$, posterior conditional distributions for each component of the vectors $\mathit{\alpha }$ and $\mathit{\beta }$ are obtained and a Gibbs sampler can be carried out.

3. Directional DP mixture model

In this section, we introduce a new nonparametric model for directional variables defined only in the first orthant. The model is based on the DP mixture model of multivariate projected Gamma distributions. Firstly, we provide a brief introduction to DP mixture models.

3.1. DP mixture model

From a Bayesian framework, a nonparametric model is a model defined on an infinite-dimensional parameter space. The application of the nonparametric Bayesian methodology was limited by its computational complexity. It was not until the last decade of the 20th century, with computational development and advances in Markov chain Monte Carlo methods (MCMC), that this methodology attracted the interest of researchers because of its application in several areas. In order to consider infinite-dimensional initial distributions, it is necessary to define random probability measures (RPM). One of the most important RPM is the Dirichlet process (DP), which was proposed by Ferguson in 1973 [8]. The DP is denoted as $\mathrm{DP}(M,G_0)$, where M>0 is a precision parameter and $G_0$ is a base measure. An important result related to a DP is the stick-breaking construction proposed by Sethuraman [31], where any $G\sim \mathrm{DP}(M,G_0)$ can be represented as

$$\begin{array}{rl}G(\cdot )& =\sum _{h=1}^{\mathrm{\infty }}{w}_h{\delta }_{m_h}(\cdot ),m_h\mathrm{i}.\mathrm{i}.\mathrm{d}{G}_0,\end{array}$$ (3)

$$\begin{array}{rl}{w}_1=v_1,w_h& =v_h\prod _{1\le l<h}(1-v_l),v_h\mathrm{i}.\mathrm{i}.\mathrm{d}\mathrm{Beta}(1,M).\end{array}$$ (4)

It is well known that random distribution functions chosen from a DP are almost surely discrete [4]. In order to extend these kind of priors to the case of absolutely continuous random distribution functions, DP mixture models (DPM) were first considered by Lo [17]. Thus, absolutely continuous random density functions can be considered as

$$f_G(\mathit{x})=\int f(\mathit{x}|\mathit{\gamma })\mathrm{dG}(\mathit{\gamma })$$

where $G\sim \mathrm{DP}(M,G_0)$, and $f(\mathit{x}|\mathit{\gamma })$ is a density belonging to a family of continuous densities indexed by $\mathit{\gamma }$. From a hierarchical approach, DPM models can be defined as

$$\begin{array}{rl}\mathit{x}|{\mathit{\gamma }}_j& \sim f(\cdot |{\mathit{\gamma }}_j)\\ {\mathit{\gamma }}_j|G& \sim G\\ G& \sim \mathrm{DP}(M,G_0).\end{array}$$

Hence, if the stick-breaking representation of G is considered, then the DPM model can be represented as

$$f(\mathit{x})=\int f(\mathit{x}|\mathit{\gamma })\mathrm{dG}(\mathit{\gamma })=\sum _{j=1}^{\mathrm{\infty }}{w}_{j}f(\mathit{x}|{\mathit{\gamma }}_j).$$ (5)

3.2. The projected gamma DP mixture model

Consider a multivariate Gamma DP mixture model given by

$$\begin{array}{rl}\mathit{Y}|\mathit{\alpha },\mathit{\beta }& \sim G{a}_{K+1}(\mathit{y}|\mathit{\alpha },\mathit{\beta })\\ \mathit{\alpha },\mathit{\beta }|G& \sim G\\ G& \sim \mathrm{DP}(M,G_0),\end{array}$$

where $G{a}_{K+1}(\mathit{y}|\mathit{\alpha },\mathit{\beta })$ is a multivariate Gamma distribution as defined in (1) and $G_0$ is specified by

$$G_0(\mathit{\alpha },\mathit{\beta })=\prod _{k=1}^{K+1}\mathit{Ga}({\alpha }_k|c_k,d_k)\prod _{l=2}^{K+1}\mathit{Ga}({\beta }_l|a_l,b_l).$$

Here, we assume that $a_l$, $b_l$, $c_k$ and $d_k$ for all ls and ks are known and we set a prior distribution $M\sim \mathrm{Ga}(a_0,b_0)$ for the concentration parameter of the DP. Therefore, from (5) the density of $\mathit{Y}$ can be expressed as

$$f(\mathit{y})=\sum _{j=1}^{\mathrm{\infty }}{w}_{j}G{a}_{K+1}(\mathit{y}|{\mathit{\alpha }}_j,{\mathit{\beta }}_j).$$

Now, if we define a directional variable $\mathbf{\Theta }$ by projecting $\mathit{Y}$ onto the unit sphere as outlined in Section 2, the directional variable $\mathbf{\Theta }$ follows an infinite mixture of multivariate projected Gamma distributions,

$$f(\mathit{\theta }|\mathit{\alpha },\mathit{\beta })=\sum _{j=1}^{\mathrm{\infty }}{w}_{j}P{G}_K(\mathit{\theta }|{\mathit{\alpha }}_j,{\mathit{\beta }}_j).$$

We call that model the projected Gamma DP mixture model.

3.3. Bayesian inference

Suppose we are given a sample of data ${\mathit{\theta }}_i=({\theta }_{i1},{\theta }_{i2},\dots ,{\theta }_{iK})$ for $i=1,2,\dots ,n$ as previously defined. Following the idea described in Section 2 we can consider complete data $\{{\mathit{y}}_1,\dots ,{\mathit{y}}_n\}$ for carrying out inferences, where ${\mathit{y}}_i=(y_{i1},y_{i2},\dots ,y_{i(K+1)})$. That means, the completed data has a DPM of multivariate Gamma distributions as previously defined and we can use nonparametric inference techniques for this type of models. In order to do that, we use a slice sampling algorithm developed by Kalli et al. [14]. The basic idea of this approach is to introduce further latent variables so that, conditional on these, each ${\mathit{y}}_i$ can be represented as being generated from a single multivariate Gamma distribution. Details of the original slice sampler algorithm can be found in Walker [34], but we describe the basic ideas of that algorithm for our model.

Given the stick-breaking representation for $G\sim \mathrm{DP}(M,G_0)$, we can write

$$f(\mathit{y}|\mathit{w},\mathit{\alpha },\mathit{\beta })=\sum _{j=1}^{\mathrm{\infty }}{w}_{j}G{a}_{K+1}(\mathit{y}|{\mathit{\alpha }}_j,{\mathit{\beta }}_j).$$

Hence, firstly, if latent variables $u_i$ are introduced such that the joint density of $({\mathit{y}}_i,u_i)$ given $(\mathit{w},\mathit{\alpha },\mathit{\beta })$ is given by

$$f(\mathit{y},\mathit{u}|\mathit{w},\mathit{\alpha },\mathit{\beta })=\sum _{j=1}^{\mathrm{\infty }}\mathbb{I}(u<w_j)G{a}_{K+1}(\mathit{y}|{\mathit{\alpha }}_j,{\mathit{\beta }}_j),$$

then, integrating over $\mathit{u}$ we obtain our density $f(\mathit{y}|\mathit{w},\mathit{\alpha },\mathit{\beta })$. In addition, given u, the number of components is finite with the indexes being $A_u=\{j:w_j>u\}$. Thus,

$$f(\mathit{y}|\mathit{u},\mathit{w},\mathit{\alpha },\mathit{\beta })=\sum _{j=1}^{\mathrm{\infty }}{N}_u^{-1}G{a}_{K+1}(\mathit{y}|{\mathit{\alpha }}_j,{\mathit{\beta }}_j),$$

where $N_u$ is the size of $A_u$, which is a finite set. Secondly, as it is usually done in mixture settings, for each $i=1,\dots ,n$, indicator variables $d_i$ are introduced which indicates which of these finite number of components provide the i-th observation. Then, given $d_i$, the variables ${\mathit{y}}_i$ follow a multivariate Gamma distribution $G{a}_{K+1}(\cdot |{\mathit{\alpha }}_{d_i},{\mathit{\beta }}_{d_i})$, where $w_{d_i}$ is the corresponding mixture weight of the $d_i$-th component. Thus, the completed likelihood function is proportional to

$$f(\{{\mathit{y}}_i,u_i,d_i{\}}_{i=1}^n|\mathit{w},\mathit{\alpha },\mathit{\beta })=\prod _{i=1}^n\mathbb{I}(u_i<w_{d_i})G{a}_{K+1}({\mathit{y}}_i|{\mathit{\alpha }}_{d_i},{\mathit{\beta }}_{d_i}),$$ (6)

and this allows a simple Gibbs sampling scheme for inferences. However, as Kalli et al. [14] pointed out, the Walker's algorithm [34] has some problems and proposed to use a general class of slice sampler, which in our case is defined by the following completed likelihood

$$f(\{{\mathit{y}}_i,u_i,d_i{\}}_{i=1}^n|\mathit{w},\mathit{\alpha },\mathit{\beta })=\prod _{i=1}^n{\xi }_{d_i}^{-1}\mathbb{I}(u_i<{\xi }_{d_i})w_{d_i}G{a}_{K+1}({\mathit{y}}_i|{\mathit{\alpha }}_{d_i},{\mathit{\beta }}_{d_i}),$$

where ${\xi }_1,{\xi }_2,\dots$ is any positive decreasing sequence, which plays a role between balance of efficiency and computational time.

Since the construction of the weights $\mathit{w}$ (defined as in (4)) depends directly on sampling of the variables $\mathit{v}$, the latter will be used in the description of sampling process. Thus, to generate samples of the joint posterior distribution using a Gibbs sampler algorithm, the variables that need to be sampled at each step are ${\mathit{\alpha }}_j,{\mathit{\beta }}_j,v_j$ and $d_i,u_i$, for $j=1,\dots$ and $i=1,\dots ,n$, respectively. In this paper, we take ${\xi }_j=e^{-j}$, then ξ and v are conditionally independent and our Gibbs sampler for completed data model turns out to be

• (0)Set initial values for $d_1,d_2,\dots ,d_n$ and $r_1,r_2,\dots ,r_n$.
• (1)Applying the standard k-dimensional spherical coordinate transformation $\mathit{y}\to (\mathit{r},\mathit{\theta })$, construct ${\mathit{y}}_i=(r_i,{\mathit{\theta }}_i)$ ∀ $i=1,\dots ,n$. See, for example, Blumenson [5].
• (2)Simulate $({\mathit{\alpha }}_j,{\mathit{\beta }}_j)$ from $\pi ({\mathit{\alpha }}_j,{\mathit{\beta }}_j|\cdots )\propto G_0(\mathit{\alpha },\mathit{\beta })\prod _{d_i=j}G{a}_{K+1}({\mathit{y}}_i|{\mathit{\alpha }}_{d_i},{\mathit{\beta }}_{d_i})$
• (3)Update $v_j$ from $\pi (v_j)\propto \mathrm{Be}(1+\sum _{i=1}^n\mathbb{I}(d_i=j),M+\sum _{i=1}^n\mathbb{I}(d_i>j))$
• (4)Simulate $u_i\sim \mathrm{Unif}(0,{\xi }_{d_i})$
• (5)Simulate $d_i$ from $P(d_i=q|\cdots )\propto {\xi }_q^{-1}\mathbb{I}(q:{\xi }_q>u_i)w_{q}G{a}_{K+1}({\mathit{y}}_i|{\mathit{\alpha }}_q,{\mathit{\beta }}_q)$
• (6)Update $M|\mathit{d}$.
• (7)Update $r_i$ from $f(r_i|{\mathit{\theta }}_i,{\mathit{\alpha }}_{d_i},{\mathit{\beta }}_{d_i})$. ∀ $i=1,\dots ,n.$
• (8)Repeat from (1) until convergence.

Here $\mathbb{I}(H)$ is the indicator function on the set H and $\mathrm{Be}(\cdot ,\cdot )$ represents a Beta distribution.

It should be noticed that the set of all ${\mathit{\alpha }}_j,{\mathit{\beta }}_j,v_j$ variables to be sampled is infinite. However, as Kalli et al. [14] show, it is only necessary to obtain samples up to an appropriate number N, at each step of the algorithm. Consider the set of $\{d_i{\}}_{i=1}^n$, such that $d_i$ is the largest integer that meets $u_i<{\xi }_{d_i}=e^{-d_i}$. In this case, since the sequence of ${\xi }_j$ is positive decreasing, it is possible to obtain all the necessary $d_i$. Thus, the set of integers q for the simulation of $P(d_i=q|\cdot )$ in step 5 can be gotten. Therefore $N=\max \{d_i\}$ can be defined, where $d_i=d_i+\lfloor -\ln (\mathrm{Unif}(0,1))\rfloor$ and $\lfloor \cdot \rfloor$ represents the floor function.

Step 6 can be carried out following a scheme introduced in Escobar and West [7]. Firstly, we can sample a latent variable $\eta |M\sim B(M+1,n)$ and then

$$M|\eta ,\mathit{d}\sim \psi \mathrm{Ga}(a_0+N,b_0-\log \eta )+(1-\psi )\mathrm{Ga}(a_0+N-1,b_0-\log \eta )$$

where the weight is such that $\psi /(1-\psi )=(a_0+N-1)/(n(b_0-\log \eta ))$.

Once convergence is reached, an approximation to the posterior predictive distribution for the directional variable $\mathit{\theta }$ can be gotten. As will be shown in the examples, we have observed that a good approximation can be obtained by estimating the predictive density using

$$\hat{f}(\mathit{\theta }|{\mathit{\theta }}_1,\dots ,{\mathit{\theta }}_n)=\frac{1}{T}\sum _{r=1}^{T}P{G}_K(\mathit{\theta }|{\mathit{\alpha }}^{(r)},{\mathit{\beta }}^{(r)})$$ (7)

where T is the number of iterations in the algorithm and $({\mathit{\alpha }}^{(r)},{\mathit{\beta }}^{(r)})$ are the projected Gamma parameters of the related mixture components which are sampled at each step of the algorithm.

4. Illustrations

In this section, we use five simulated examples and one real data set to illustrate the performance of the proposed methodology. In all of these examples, we impose non-informative prior assumptions by setting for all models $a_0=2$, $b_0=8$, and for the projected Gamma parameters, we set $c_m=a_m=1$ and $d_m=b_m=0.05$, for all necessary subscripts m. We use the R language and environment ([1]) to simulate the corresponding data sets and to carry out all of the analyses in this section.

Example 4.1

In this example, we examine data simulated from a finite mixture of three univariate projected Gamma densities $\mathit{P}{\mathit{G}}_{\mathit{1}}(\theta |\mathit{\alpha },\mathit{\beta })$. Specifically, we consider a data sample of angles θ of size n = 500 from the following model:

$$f(\theta )=0.2\mathit{P}{\mathit{G}}_{\mathit{1}}(\theta |{\mathit{\alpha }}_1,{\mathit{\beta }}_1)+0.5\mathit{P}{\mathit{G}}_{\mathit{1}}(\theta |{\mathit{\alpha }}_2,{\mathit{\beta }}_2)+0.3\mathit{P}{\mathit{G}}_{\mathit{1}}(\theta |{\mathit{\alpha }}_3,{\mathit{\beta }}_3),$$

where ${\mathit{\alpha }}_1=(5,8)$, ${\mathit{\alpha }}_2=(20,10)$, ${\mathit{\alpha }}_3=(8,12)$, ${\mathit{\beta }}_1=(1,6)$, ${\mathit{\beta }}_2=(1,0.5)$ and ${\mathit{\beta }}_3=(1,0.5)$. That model produces a trimodal data set. Figure 1 shows a linear histogram of that data set and the true density.

We apply our Bayesian nonparametric approach to analyze that data. Using the simulation algorithm described in the previous section, convergence diagnostics led us to stop the process after 150, 000 iterations, discarding the first 120, 000 as burn-in. From the remaining 30, 000 iterations, we kept one observation every 30 iterations as part of the final sample. Figure 2 compares the true density with the estimated predictive density using (7). We can observe that the proposed model captures the multimodality of the data.

Figure 1.

True density (solid line) and data sample (histogram) for Example 4.1.

Figure 2.

True density (solid black line); estimated density (dashed blue line); 0.95 pointwise credible bands (dot-dashed red lines).

In addition, in order to have an idea of the variability of the process, at each s-th iteration in convergence state, a posterior realization of the random measure for θ as a finite mixture of projected Gamma distributions is obtained. Thus, once these realizations $f^{(s)}$ are available, it is possible to obtain 95% joint pointwise intervals all over the interval $(0,\pi /2]$. In this work, we will call these intervals as pointwise credible bands. For this example, the corresponding 0.95 pointwise credible bands for $f(\theta )$ are shown in Figure 2.

In general, the estimator for the predictive density and the pointwise credible bands capture the shape of the true density and describe this type of data qualitatively well.

As a measure of performance for our proposal, we also calculate the number of non-empty components derived from our approach. In each convergence iteration of the algorithm, we get the number of non-empty components, and the mode of these 1000 estimates was J = 3. That estimate (the mode) matches the actual number of components. Table 1 shows the corresponding estimated probability for the number of components obtained through our proposal.

Table 1.

Estimated probability of non-empty components for data from Example 4.1.

	Non-empty components
	$J=3$	$J=4$	$J=5$	$J=6$
Probability	0.581	0.342	0.070	0.007

In the framework of DPM models, the number of components used to estimate the predictive distribution does not necessarily coincide with the number of modes or groups (see, for example, [11] and [20]). In this example, both numbers are equal. This result was foreseen since the data modes are well defined by the data set sample. In addition, the data set was directly sampled from a mixture of projected Gamma densities.

It is worth pointing out that our proposed methodology does not require that observations follow a mixture of projected Gamma distributions. In the next two examples, we analyze the performance of our proposal to describe data sets defined in the first quadrant, but whose distributions are different from a mixture of univariate projected Gamma distributions.

Example 4.2

For this second example, we consider circular observations (univariate angles) from a mixture of truncated von Mises distributions (see, [9]). Specifically, we use the truncated von Mises density defined by

$$\mathit{tvM}\mathit{(}\theta \mathit{|}\mu \mathit{,}\kappa \mathit{,}\mathit{a}\mathit{,}\mathit{b}\mathit{)}\mathit{=}\frac{\mathit{exp}\{\kappa \cos \mathit{(}\theta \mathit{-}\mu \mathit{)}\}}{{\displaystyle {\int }_{\mathit{a}}^{\mathit{b}}}\mathit{exp}\{\kappa \cos \mathit{(}\theta \mathit{-}\mu \mathit{)}\}\mathrm{d}\theta }\mathit{.}$$

Using an acceptance-rejection algorithm, we simulate a sample of size n = 500 of circular data from the next model:

$$\begin{array}{rl}f(\theta )& =0.15\mathit{tvM}\mathit{(}\theta \mathit{|}\mu \mathit{=}\pi \mathit{/}\mathit{12}\mathit{,}\kappa \mathit{=}\mathit{130}\mathit{,}\mathit{0}\mathit{,}\pi \mathit{/}\mathit{2}\mathit{)}\mathit{+}\mathit{0.5}\mathit{tvM}\mathit{(}\theta \mathit{|}\mu \mathit{=}\pi \mathit{/}\mathit{4}\mathit{,}\kappa \mathit{=}\mathit{25}\mathit{,}\mathit{0}\mathit{,}\pi \mathit{/}\mathit{2}\mathit{)}\\ & +0.35\mathit{tvM}\mathit{(}\theta \mathit{|}\mu \mathit{=}\mathit{9}\pi \mathit{/}\mathit{24}\mathit{,}\kappa \mathit{=}\mathit{100}\mathit{,}\mathit{0}\mathit{,}\pi \mathit{/}\mathit{2}\mathit{)}\mathit{.}\end{array}$$

Figure 3 shows a histogram for these data and the true density. It can be noticed that the components for the model have a significant overlap between its corresponding densities, and particularly the mode associated with the second component is dimmed despite the weight assigned to this component is the largest. We carry out our Bayesian nonparametric proposal. The algorithm was implemented and stopped after 200,000 iterations, discarding the first 160, 000 as burn-in. From the remaining 40, 000, we kept one observation every 40 iterations.

The true density with the posterior predictive density $\hat{f}(\theta )$ and, the corresponding 0.95 pointwise credible bands are shown in Figure 4. It can be noticed our projected Gamma DP mixture model performs well, despite the fact that observations follow a mixture of truncated von Mises distributions.

Figure 3.

True density (solid line) and histogram for data set of Example 4.2.

Figure 4.

Example 4.2. True density of truncated von Mises mixture (solid black line); estimated predictive density (dashed blue line); 0.95 pointwise credible bands (dot-dashed red lines).

Example 4.3

In this example, we analyze circular observations (univariate angles) from a mixture of truncated projected normal distributions (tPN). Following the works of Fernandez-Gonzalez [9] and Núñez-Antonio and Gutiérrez-Peña [23], we defined a truncated projected normal distribution as:

$$\mathit{tPN}\mathit{(}\theta \mathit{|}\mathit{\mu }\mathit{,}\lambda \mathit{,}\mathit{a}\mathit{,}\mathit{b}\mathit{)}\propto \mathit{C}[\mathit{1}\mathit{+}\frac{\mathrm{d}\mathit{b}\mathbf{\Phi }\mathit{(}\mathrm{d}\mathit{b}\mathit{)}}{\mathit{\varphi }\mathit{(}\mathrm{d}\mathit{b}\mathit{)}}]{\mathit{I}}_{\mathit{(}\mathit{a}\mathit{,}\mathit{b}\mathit{)}}\mathit{(}\theta \mathit{)}\mathit{,}$$

where $\mathit{\mu }=({\mu }_1,{\mu }_2{)}^t$ is a vector defined on the truncated interval $(a,b)$ and a, $b\in (0,2\pi ]$, $\lambda \in {\mathbb{R}}^{+}$ and, $\mathit{\varphi }(\cdot )$ and $\mathbf{\Phi }(\cdot )$ are the standard normal density function and the standard normal cumulative distribution function, respectively. Here $d=\sqrt{{\cos }^2\theta +\lambda {\sin }^2\theta }$, and

$$\begin{array}{rl}b& =\frac{{\mu }_1\cos \theta +\lambda {\mu }_2\sin \theta }{{\cos }^2\theta +\lambda {\sin }^2\theta }\\ C& =\frac{{\lambda }^{1/2}\exp \{-\frac{1}{2}({\mu }_1^2+\lambda {\mu }_2^2)\}}{2\pi d^2}\end{array}$$

For this example, using an acceptance-rejection algorithm we simulated a sample of size n = 500 of angles ${\theta }_i\in (0,\pi /2]$ from the following model

$$\begin{array}{rl}f(\theta )& =0.2\mathit{tPN}\mathit{(}\theta \mathit{|}{\mathit{\mu }}_{\mathit{1}}\mathit{,}{\lambda }_{\mathit{1}}\mathit{,}\mathit{0}\mathit{,}\pi \mathit{/}\mathit{2}\mathit{)}\mathit{+}\mathit{0.5}\mathit{tPN}\mathit{(}\theta \mathit{|}{\mathit{\mu }}_{\mathit{2}}\mathit{,}{\lambda }_{\mathit{2}}\mathit{,}\mathit{0}\mathit{,}\pi \mathit{/}\mathit{2}\mathit{)}\\ & +0.3\mathit{tPN}\mathit{(}\theta \mathit{|}{\mathit{\mu }}_{\mathit{3}}\mathit{,}{\lambda }_{\mathit{3}}\mathit{,}\mathit{0}\mathit{,}\pi \mathit{/}\mathit{2}\mathit{)}\mathit{,}\end{array}$$

where ${\mathit{\mu }}_1=(13,4{)}^t$, ${\mathit{\mu }}_2=(3,5{)}^t$, ${\mathit{\mu }}_3=(5,10{)}^t$, ${\lambda }_1=2$, ${\lambda }_2=0.6$ and ${\lambda }_3=4$.

We ran our algorithm for 150, 000 iterations. We discarded the first 120, 000 as burn-in. From the remaining 30, 000 iterations, we kept one observation every 30 iterations. Figure 5 compares the true density and the posterior predictive estimator of $f(\theta )$. In addition, the corresponding 0.95 pointwise credible bands are shown. It can be noticed that our proposal based on the DPM of projected Gamma model describes that type of data quite well.

Figure 5.

True density (solid black line); estimated density (dashed blue line); 0.95 pointwise credible bands (dot-dashed red lines).

Results from Example 4.2 and Example 4.3 show our proposed model is able to describe data sets whose probability distribution is not necessary a mixture of projected Gamma densities.

Example 4.4

In this example, we examine data from a mixture of three bivariate projected Gamma densities, $P{G}_2(\mathit{\theta }|\mathit{\alpha },\mathit{\beta })$. From Section 2, the density $P{G}_2(\mathit{\theta }|\mathit{\alpha },\mathit{\beta })$ is given by

$$\begin{array}{rl}& {\mathit{PG}}_2(\mathit{\theta }|\mathit{\alpha },\mathit{\beta })\\ & =\frac{\mathrm{\Gamma }(\sum _{i=1}^3{\alpha }_i){\beta }_1^{{\alpha }_1}{\beta }_2^{{\alpha }_2}{\beta }_3^{{\alpha }_3}(\cos {\theta }_1{)}^{{\alpha }_1-1}(\cos {\theta }_2{)}^{{\alpha }_2-1}(\sin {\theta }_1{)}^{({\alpha }_2+{\alpha }_3)-1}(\sin {\theta }_2{)}^{{\alpha }_3-1}}{\mathrm{\Gamma }({\alpha }_1)\mathrm{\Gamma }({\alpha }_2)\mathrm{\Gamma }({\alpha }_3)({\beta }_1\cos {\theta }_1+{\beta }_2\sin {\theta }_1\cos {\theta }_2+{\beta }_3\sin {\theta }_1\sin {\theta }_2{)}^{({\alpha }_1+{\alpha }_2+{\alpha }_3)}}\\ & {\mathit{I}}_{(0,\pi /2{]}^2}(\mathit{\theta }),\end{array}$$

where $\mathit{\theta }=(\theta ,\varphi )$, $\mathit{\alpha }=({\alpha }_1,{\alpha }_2,{\alpha }_3)$ and $\mathit{\beta }=({\beta }_1=1,{\beta }_2,{\beta }_3)$. We simulated a sample of size n = 1000 of spherical data $\mathit{\theta }=(\theta ,\varphi )$ from the next model

$$f(\mathit{\theta })=0.2{\mathit{PG}}_2(\mathit{\theta }|{\mathit{\alpha }}_1,{\mathit{\beta }}_1)+0.5{\mathit{PG}}_2(\mathit{\theta }|{\mathit{\alpha }}_2,{\mathit{\beta }}_2)+0.3{\mathit{PG}}_2(\mathit{\theta }|{\mathit{\alpha }}_3,{\mathit{\beta }}_3),$$

where ${\mathit{\alpha }}_1=(6.9,11,19.2)$, ${\mathit{\beta }}_1=(1,5.3,9.2)$, ${\mathit{\alpha }}_2=(4.5,8.7,7.8)$, ${\mathit{\beta }}_2=(1,1.3,2.2)$ and ${\mathit{\alpha }}_3=(4.8,13,10)$, ${\mathit{\beta }}_3=(1,3.2,1.3)$.

Figure 6(a) shows the corresponding contour plots in the $(\theta ,\varphi )$-plane for the true density. It can be appreciated that variability components produce three modes. We run our Bayesian nonparametric proposal, convergence diagnostics led us to stop the algorithm after 300, 000 iterations, discarding the first 240, 000 as burn-in. From the remaining 60, 000, we kept one observation every 60 iterations as part of the final realizations. Figure 6(b) shows the contour plots on the $(\theta ,\varphi )$-plane for the corresponding posterior predictive density $\hat{f}(\theta ,\varphi )$. It can be noticed how closely $\hat{f}(\theta ,\varphi )$ and the true model resemble each other. Particularly, $\hat{f}(\theta ,\varphi )$ captures the three-modal behavior of the actual density. In addition, the contour plot of the $\hat{f}(\theta ,\varphi )$ together with the observed data are represented in Figure 7(a,b) on the $(\theta ,\varphi )$-plane and on the first orthant of the unit sphere ${\mathbb{S}}^2$, respectively.

Figure 6.

Contour plot of true density and estimated density in $(\theta ,\varphi )$-plane. Example 4.4. (a) True density contour plot. (b) Estimated density contour plot.

Figure 7.

Data set and contour plots for Example 4.4. (a) Data set (blue dots) and estimated density contour plot (solid black lines) in $(\theta ,\varphi )$ plane. (b) Data set (blue dots) and contour plot of estimated density over ${\mathbb{S}}^2$ (solid black lines).

Like the Example 4.1, we also estimate the true number of components. In each convergence iteration of the algorithm, we calculate the number of non-empty components, and the mode of these 1000 estimates was J = 3. In this case, that estimate exactly matches the correct number of components, too. Table 2 shows the corresponding estimated probability for the numbers of components obtained through our proposal.

Table 2.

Estimated probability of non-empty components for data from Example 4.4.

	Non-empty components
	$J=3$	$J=4$	$J=5$
Probability	0.509	0.458	0.033

Results of this section show that the proposed projected Gamma DP mixture model is appropriate for analyzing directional data only defined on the first orthant of ${\mathbb{S}}^k$.

Example 4.5

In the mixture model approach, a potential problem in density estimation is overfitting. That means, the estimation carried out with a model could collect noise, inducing non-existent modes in the estimation. In order to analyze that issue, we simulated a sample of size n = 500 from a projected Gamma distribution, $\mathit{P}{\mathit{G}}_{\mathit{1}}(\theta |\mathit{\alpha },\mathit{\beta })$, with $\mathit{\alpha }=(5,8)$ and $\mathit{\beta }=(1,6)$. Figure 8 shows the corresponding histogram of the simulated sample. Initially, from the histogram features, a multimodal model could be probable for the true model from which the sample was obtained. However, it is worth mentioning that the true model, $\mathit{P}{\mathit{G}}_{\mathit{1}}(\theta |\mathit{\alpha },\mathit{\beta })$, is unimodal.

Figure 8.

Simulated dataset sampled from an unimodal model (histogram) and 100 estimated predictive densities (red points) obtained from the proposed approach.

In order to analyze a possible overfitting for the predictive density estimated from our approach, we run the proposed algorithm 100 times. At each of these repetitions, we estimated the predictive density. In all cases, the estimated predictive distribution obtained was unimodal, adequately describing the data set. The previous analysis suggests our approach does not induce overfitting. Figure 8 shows the corresponding 100 estimated predictive densities.

4.1. Real data example

Phenology is the study of the life cycle events of plants and animals caused by environmental changes. These studies are very important for ecological monitoring and analysis of the impact of climate change on the environment. In recent years, technological advances have made it possible to obtain a large amount and variety of data. In particular, satellite-derived metrics allowed the phenological analysis to be expanded to a broader scale.

One of these metrics is the Normalized Difference Vegetation Index (NDVI), which is an indicator of the amount of vegetation and its health in a specific area. This indicator mathematically compares the amount of absorbed visible red light and reflected near-infrared light. In this context, backscatter light is considered when the satellite sensor is aligned with the incident illumination and is reflected at a phase angle lower than $\pi /2$, which is called backscatter solar-to-sensor angle. The analysis of behavior of this angle is important in the study of its correlation with some phenological metrics, such as NDVI peaks [24]. A database of 243, 422 backscatter solar-to-sensor angles in a pinyon-juniper ecosystem in Grand Canyon National Park was released for public access by the U.S. Geological Survey [25].

For this example, we apply our methodology for analyzing a sample of size n = 2000 of backscatter solar-to-sensor angles from the database cited above. The sample obtained has special characteristics such as multimodality. We fitted a DPM of projected Gamma model and produced the estimators for the predictive density and the 0.95 pointwise credible bands, which are shown in Figure 9 along with the histogram for this data set. It can be noticed that these estimators adequately describe the general behavior of this type of data. Remarkably, the variability described from the pointwise credible bands in the first half of the data could lead to a model with two, three or four modes, as a probable model to the predictive distribution. In general, measures of goodness-of-fit for describing real phenomena through inferences based on models are useful.

Figure 9.

Estimated density (dashed blue line) and 0.95 pointwise credible bands (dot-dashed red lines) for sample of backscatter solar-to-sensor angle.

As an objective measure of comparison, we compute the logarithm of the pseudo marginal likelihood (LPML) for some existing circular distributions, which is a goodness-of-fit measure originally suggested by Geisser and Eddy [12]. The considered models are shown in Table 3. Here $\mathrm{PN}(\theta |\mathit{\mu })$ denotes a projected normal model, $\mathrm{tPN}(\theta |\mathit{\mu },\lambda ,0,\pi /2)$ and $\mathrm{tvM}(\theta |\mu ,\kappa ,0,\pi /2)$, a projected normal model and a von Mises truncated on the interval $(0,\pi /2]$, respectively. The Model 1 and Model 2 are models defined all over (unbounded) the unit circle (see Núñez-Antonio et al. [21]). On the other hand, Model 1, Model 3, Model 5 and Model 6 are nonparametric models, which can describe multimodal and asymmetric behaviors. In addition, we must mention the fitting of Model 1, Model 5 and Model 6 become a model with only two probable modes (the last two modes from the real data histogram, see Figure 10) for the predictive distribution. Table 3 shows the corresponding value of the statistics LPML for each of these models. It can be seen the worst fitting (smaller LPML value) is obtained from Model 2, which is an unimodal and symmetric model. The next better model is the projected Gamma model (Model 4), which can describe some symmetric, asymmetric, and unimodal or bimodal patterns of behavior (see Núñez-Antonio and Geneyro [22]). Despite Model 1, Model 3, Model 5 and Model 6 being nonparametric models, the best fitting according to the LPML is obtained from Model 3. It should be mentioned that our proposed methodology is quite robust regarding the sensitivity of the LPML, under different non-informative prior specifications for the parameters $\mathit{\alpha }$ and $\mathit{\beta }$ of the projected Gamma DP mixture model (Model 3). That means, the corresponding obtained values of the statistics LPML result in a range from 371 to 377. Finally, it is worth mentioning that the fit of the projected Gamma DP mixture model (Model 3) is approximately 20% more computationally expensive than the truncated projected normal DP mixture model (Model 5) and comparable to the truncated von Mises DP mixture model (Model 6).

Table 3.

LPML goodness-of-fit measures for real data.

Model	$f(\theta )$	Type	LPML
Model 1:	$\sum _{j=1}^{\mathrm{\infty }}{w}_j\mathrm{PN}(\theta |{\mathit{\mu }}_j)$	unbounded (multimodal)	344.04
Model 2:	$\mathrm{PN}(\theta |\mathit{\mu })$	unbounded (unimodal)	231.56
Model 3:	$\sum _{j=1}^{\mathrm{\infty }}{w}_{j}P{G}_1(\theta |{\mathit{\alpha }}_j,{\mathit{\beta }}_j)$ (our model)	bounded (multimodal)	373.90
Model 4:	$P{G}_1(\theta |\mathit{\alpha },\mathit{\beta })$	bounded	286.50
Model 5:	$\sum _{j=1}^{\mathrm{\infty }}{w}_j\mathrm{tPN}(\theta |{\mathit{\mu }}_j,{\lambda }_j,0,\pi /2)$	bounded (multimodal)	348.87
Model 6:	$\sum _{j=1}^{\mathrm{\infty }}{w}_j\mathrm{tvM}(\theta |{\mu }_j,{\kappa }_j,0,\pi /2)$	bounded (multimodal)	335.49

Figure 10.

Estimated predictive densities from six proposed models for the backscatter solar-to-sensor angles. (a) Model 1 (b) Model 2 (c) Model 3 (d) Model 4 (e) Model 5 (f) Model 6.

Figure 10 shows the corresponding predictive distribution and pointwise credible bands for each of the proposed models for making inferences about the backscatter solar-to-sensor angles. In addition, Table 4 shows the corresponding estimated probabilities of non-empty components for all nonparametric models. As mentioned above, initially a model with two or three or four modes could be appropriated to describe the backscatter solar-to-sensor angles. However, an analysis of the results from Table 3, Table 4 and Figure 10 leads to select a four-mode model as the best model. In this case, the best fitting is obtained from the Model 3. Thus, the best model to describe the backscatter solar-to-sensor angles turns out to be the projected Gamma DP mixture model.

Table 4.

Estimated probability of non-empty components of DPM models for the real data example.

		Non-empty components
		$J=3$	$J=4$	$J=5$	$J=6$
Model 1:	$\sum _{j=1}^{\mathrm{\infty }}{w}_j\mathrm{PN}(\theta |{\mathit{\mu }}_j)$	0.170	0.681	0.124	0.025
Model 3:	$\sum _{j=1}^{\mathrm{\infty }}{w}_{j}P{G}_1(\theta |{\mathit{\alpha }}_j,{\mathit{\beta }}_j)$	–	0.988	0.012	–
Model 5:	$\sum _{j=1}^{\mathrm{\infty }}{w}_j\mathrm{tPN}(\theta |{\mathit{\mu }}_j,{\lambda }_j,0,\pi /2)$	0.856	0.135	0.009	–
Model 6:	$\sum _{j=1}^{\mathrm{\infty }}{w}_j\mathrm{tvM}(\theta |{\mu }_j,{\kappa }_j,0,\pi /2)$	0.282	0.435	0.220	0.063

Once a projected Gamma DP mixture model has been adjusted and selected to describe the backscatter solar-to-sensor angles, some inferences can be made. It is possible to conclude that there are four angle values that may be correlated with NDVI index value peaks and consequently indicate possible different vegetation health status in the pinyon-juniper ecosystem in Grand Canyon National Park. These clustering values are given by the modes of the final predictive distribution, which are determinated by 0.183, 0.341, 0.593 and 0.814 radians. Thus, once these clustering values have been defined, appropriate correlation studies in regions around these values can be carried out by ecological researchers.

On the other hand, as is pointed out in [24], a goal of phenological studies related with NDVI is to detect timing changes of life cycle events during a year or from year-to-year. From the specifical angular regions described previously, a range of possible NDVI index values associated with these regions can be calculated. Then, it is possible to decide about the existence of atypical data in future samples of NDVI index and analyze the possible causes of these atypical records.

The results obtained in this section suggest our proposed model can be suitable for analyzing real directional data defined on the first orthant of the K-dimensional unit sphere.

5. Concluding remarks

This work introduces a new Bayesian nonparametric approach for analyzing directional data defined only on the first orthant of the K-dimensional unit sphere, ${\mathbb{S}}^K$. The proposed model is based on the Dirichlet Process mixture where a multivariate projected Gamma density is considered as base distribution.

Our nonparametric proposal considers a particular multivariate projected Gamma distribution where an independent structure defines the corresponding unprojected multivariate distribution. Initially, this choice of unprojected multivariate distribution seems restrictive for describing data with complex behaviors. However, the proposed model is able to describe data sets generated even by mixtures from different distributions, such as truncated von Mises and truncated projected Normal distributions.

Some extensions of the proposed approach are possible. Firstly, from a theoretical frame, it would be interesting to develop an extension of the model without the independent structure originally contemplated. Secondly, due to the close relationship that exists between the first orthant of the K-dimensional unit sphere and the corresponding K-dimensional unit simplex, it would be interesting to develop a methodology that allows the analysis of compositional data through the approach proposed in this paper. Work is currently in progress for these problems.

Funding Statement

The work of the first author was supported by CONACYT, Mexico. The second author was partially supported from CONACYT, through Sistema Nacional de Investigadores, Mexico. The support received from the Department of Mathematics of the Metropolitan Autonomous University, Iztapalapa Unit is also gratefully acknowledged. Finally, the authors are grateful to the anonymous reviewers for their detailed and insightful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.