INVERSE STEREOGRAPHIC HYPERBOLIC SECANT DISTRIBUTION: A NEW SYMMETRIC CIRCULAR MODEL BY ROTATED BILINEAR TRANSFORMATIONS

The inverse stereographic projection (ISP), or equivalently, bilinear transformation, is a method to produce a circular distribution based on an existing linear model. By the genesis of the ISP method, many important circular models have been provided by many researchers. In this study, we propose a new symmetric unimodal/bimodal circular distribution by the rotated ISP method considering the hyperbolic secant distribution as a baseline distribution. Rotation means that fixing the origin and rotating all other points the same amount counterclockwise. Considering the effect of rotation on the circular distribution to be obtained with the bilinear transformation, it is seen that it actually induces a location parameter in the obtained circular probability distribution. We analyze some of the stochastic properties of the proposed distribution. The methods for the parameter estimation of the new circular model and the simulation-based compare results of these estimators are extensively provided by the paper. Furthermore, we compare the fitting performance of the new model according to its well-known symmetric alternatives, such as Von-Misses, and wrapped Cauchy distributions, on a real data set. From the information obtained by the analysis on the real data, we say that the fitting performance of the new distribution is better than its alternatives according to the criteria frequently used in the literature.


INTRODUCTION
Circular or directional data are observed in various fields of science. Data on angular observations can often be associated with compass measurements. Additionally, daily, weekly, or hourly observations obtained in the specific time period 872 A. YILMAZ may be circular. Although it may seem attractive in some ways, processing and evaluating such data linearly can lead to false results. In directional data, the start and endpoints are neighbors despite having the furthest distance according to linear metric. As a simple example, the arithmetic mean of two angles 1 and 359 degrees is 180 degrees, although the circular average to be 0 degrees. Therefore, it requires a special class of distributions known as circular probability distributions to analyze such data.
Circular probability distributions are usually obtained by circularizing a known linear probability distribution. The two most common methods for circularization are wrapping and inverse stereographic projection (ISP). ISP method is based on bilinear transformations. Minh and Farnum [8] used bilinear transformations to map points on the unit circle in the complex plane into points on the real line. Thus, they used the stereographic projection as a transformation, to produces probability distributions on the real line by circular models. It was clear that by the inverse of this transformation (ISP), circular probability distributions could be obtained from probability distributions on the real line. Many studies on circular distributions obtained using the ISP method have been added to the literature. Yedlapalli, et al [11] used to transformation on double Weibull distribution to obtain a symmetric circular distribution. Kato and Jones [6] proposed a family of four-parameter distributions on the circle that contains the Von Mises and wrapped Cauchy distributions as special cases. Girija, et al [5] introduced stereographic double exponential distribution obtained by using double exponential (Laplace) distribution. The same authors introduced the stereographic logistic model [2] in a later study. Yedlapalli, et al [12] obtained semicircular (axial) model induced by using modified inverse stereographic projection on Quasi Lindley distribution. The projection method used in all these studies is based on the result obtained by Minh and Farnum [8] in a study in which they introduced the induction of linear models with Möbius transformations. Möbius transformation (bilinear, fractional linear or linear fractional transformation) provides very convenient methods of finding a one-to-one mapping of one domain into another. In a general form, Möbius transformation can be written as where a,b,c and d are complex or real valued coefficients and bc − ad = 0. This transformation was proposed by Minh and Farnum [8] as a new method of generating probability distributions, which maps every point on a real line onto the point on a unit circle. Their construction proceeds as follows. In order for T (z) to map the unit circle on the real line, the constraints Im (c) = 0, ad = cb, and a = 0 must be provided. Dividing all coefficients in Eq.(1) by a and imposing the requirement T (−1) = ∞ yields the transformation of the form where t ∈ (0, 1) , F − (t) = inf {x ∈ R : F (x) ≥ t} and F (.) is the cdf of random variable X.
Proposition 6. Let X be a symmetric random variable around E(X) = u. The random variable Θ defined as Θ = T −1 α (X) has a symmetrical distribution around −α.
In the following section, we show an application of the T −1 α transformation to hyperbolic secant distribution. We introduce the methods for estimating the location parameter induced by T −1 α in the relevant subsections. Also, that section includes the basic properties of the obtained circular distribution and an application to a real-life data set.

Induce Inverse Stereographic Hyperbolic Secant Model with Rotated Bilinear Transformations
Suppose X follows hyperbolic secant distribution, then cdf and pdf of X are respectively. This distribution is also called the inverse-cosh distribution because of the hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine function. Note that the pdf given by Eq.(12) is symmetrical around E (X) = 0. By considering the Eq.(8) and Eq.(9) with Eq.(12) and Eq.(11), we obtain the cdf and pdf of the inverse stereographic hyperbolic secant distributed random variable respectively, where v > 0 is the scale parameter and α ∈ [−π, π) is the location parameter. In the rest of this paper, a random variable Θ having cdf as in Eq.(13) and pdf as in Eq.(14) will be denoted as Θ ∼ ISHS (α, v) .   The inverse cdf of hyperbolic secant distribution is F −1 (t) = − 1 π log cot 2 1 2 πt . Thus, the quantile function of ISHS (α, v) can be easily obtained from Eq.(10) as where t ∈ (0, 1) .

Location, Dispersion and Median.
For a circular random variable, the pth cosine moment is defined as c p = E(cos pΘ), and the pth sine moment is defined as s p = E(sin pΘ) [7]. Thus, the mean direction is calculated as µ = atan (s 1 , c 1 ), where atan (., .) is quadrant inverse tangent function. The explicit analytical forms of c p and s p values can not be obtained for random variable Θ ∼ ISHS (α, v). However, according to proposition 6 and corollary 7, it is clear that The first trigonometric moments of Inverse Stereographic Hyperbolic Secant distribution are calculated numerically and presented in Figure 3. The following propositions give useful results for the location parameter of the ISHS distribution.
The length of mean direction vector is a measure of angular concentration around the mean and it is calculated as ρ = c 2 1 + s 2 1 . By using the value of ρ, the circular variance is calculated as V = 1 − ρ and the circular standard deviation calculated as σ = √ −2 ln ρ. These three characteristics are illustrated in Figure 4 for different values of v.
As a measure of asymmetry, the skewness coefficient for the circular distribution is calculated as According to the following proposition, the skewness coefficient of the ISHS (α, v) distribution are zero for every v > 0.
Proposition 10. All central sine moments of ISHS (α, v) distribution is zero.
Proof. Since g (θ; α, v) is periodic with period 2π and µ = −α, we have According to proposition 9 g (θ; α, v) = g (θ + α; 0, v) , and according to proposition 6 g (θ; 0, v) is an even function. Thus, we can write s p as The kurtosis coefficient of a circular distribution is calculated as  The median direction (M) and the interquartile range (Iqr) of ISHS (α, v) distribution are easily obtained from Eq.(15), as follows, respectively:

2.2.
Entropy. The entropy is a measure of variation or uncertainty of a random variable. Following the formal definition of the entropy, the entropy of the random variable Θ ∼ ISHS (α, v) is where Γ is any interval of length 2π. Since Θ is 2π periodic We could not get an explicit analytical form of the integral in Eq.(17). Therefore, we numerically calculated the H Θ with respect to v and illustrated in Figure 4. Note that the entropy of the circular uniform distribution is ln (2π) and this is the maximum entropy any circular distribution may have. Figure 4 shows that the maximum value of the H Θ is below this value. The entropy of the ISHS distribution attains its maximum value when the circular variance is maximized or equivalently angular concentration minimized. Thus one can write Since the minimum value of the first cosine moment is zero, the value of v * is obtained by solving the equation π 0 cos (θ) g (θ; 0, v) dθ = 0 with respect to v. Using the bisection method, we observed that v * 0.521567.  2.5. Inference. In this section, we consider estimating the unknown parameters of ISHS (α, v) distribution. We will use tree methods commonly used in the literature, such as, maximum likelihood (ml), weighted least-squares (ls) and moments estimation (me) methods. Finally, a Monte-Carlo simulation study will be given to show and compare the performance of ml, me and ls estimators.
2.5.1. Maximum Likelihood Estimation. Let Θ 1 , Θ 2 , ..., Θ n be a random sample from ISHS (α, v) distribution. By considering the random variables Θ i , i = 1, 2, ..., n, The logarithmic likelihood function of α and v can be written as If the first derivatives of this log-likelihood function with respect to parameters α and v are taken and equalized them to zero, then we have the following normal equations with respect to α and v. Where j n+1 is the expectation of the empirical distribution function of the ordered data, see Swain et al. [10]. Numerical methods can be used to minimize Eq.(21).

Method of Moment
Estimation. Let us start by expressing the sample trigonometric moments for circular data [7]. The pth order sample cosine moment is defined as and sample sine moment is defined as sin (pθ i ) .

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Now consider the random sample θ 1 , θ 2 , . . . , θ n from the ISHS (α, v) distribution. Moment estimates of α and v ( α M E and v M E ) are obtained from the collective solution of equations and S 1 − s 1 = 0 (23) by using numerical methods, where c 1 = E(cos Θ) and s 1 = E(sin Θ).

Monte-Carlo Simulation Study.
We perform some Monte-Carlo experiments to compare the performance of ml, ls, and me estimators in different sample sizes. We consider n = 50, 100, 500, and 1000 sample sizes and the repitation of the simulation is set as 100 times in each sample size. The algorithm below has been run for different parameter sets and the results are shown in Table 1.
Step 1. Select n and set values of the parameters α and v.
Step 3. Calculate Q (u n×1 ) → θ n×1 , where Q (.) as Eq.(15). According to the results in Table 1, it is seen that the Bias and MSE values decrease to zero as the sample size increases for the estimation of parameters α and v by all three methods. This shows that the estimates are precise and accurate, hence, we say that it is consistent and unbiased. It is known that ml estimators are asymptotically unbiased estimators. So, the results in Table 1 agree with expectations for ml estimators. In addition, simulation results show that the other estimators have the same characteristics. 2.6. Real Data Example. In this section, we study the modeling behavior of the ISHS distribution on a real-life dataset. We consider the termite mounds data in Appendix B.13 (set 7) of Fisher [3]. The data consist of n = 66 termite mounds orientations of Amitermes laurensis in the Cape York Peninsula, North Queensland. We obtained the parameter estimates by using Matlab's 'fmincon' and 'fsolve' subroutines. In these subroutines, the parameter ranges were chosen as wide as possible to avoid local maxima. The initial values were set to −m 1 = − atan S 1 , C 1 for α and 1 for v. In order to make comparisons, we chosed the Von-Mises (VM) and Wrapped Cauchy (WC) distributions as well-known alternatives from the location family for modeling symmetrical circular data. Table 2 shows the parameter estimates for each models, and Figure 5 illustrates the fitted pdfs and cdfs. The ISHS parameters were estimated with three methods; ml, me and ls. Table 2 also includes the mean direction and resultant length estimates for each models, and the values of these characteristics obtained from the sample.  Table 3 contains Log-likelihood (LL), the Akaike and Bayesian information criteria (AIC and BIC), Watson's U 2 (W 2 ) statistics values, Kolmogorov-Smirnov (KS) and Chi square tests statistics with p-values. Here, it is seen that the data fit all the distributions selected (p > 0.05). However, it can be said that the proposed ISHS model is the model that best fits the data since it has the smallest values in all model selection criteria. Plots of the fitted densities are shown in Figure 5. Left panel of this figure represents the circular data plot, fitted pdfs of the ISHS distribution with ml, me and ls estimates, fitted vm and wc models. The arrow points out the sample mean resultant vector m 1 = atan S 1 , C 1   Figure 5. Plots for termite mounds data. Circular data plot, fitted circular pdfs (left) linear histogram and fitted pdfs (center), empirical cdf and fitted cdfs (rigth).

Cumulative Distibution Function
All models estimated the average orientation of the mounds to be almost south. The ISHS model with ME estimates gave the mean orientation and resultant length the same as in the sample. This is an expected result for moment estimators. Same thing valid for the VM model. However, when we compare the modeling performances with the values in Table 3, we see that the ISHS model is better than both the VM and the WC model.

Conclusion
After Minh and Farnum [8] introduced the ISP method, a number of researchers have introduced many circular distributions by employing the ISP method. In some of these (for example; [1], [4] and [6]), the authors added a location parameter to the circular distributions in their studies. In fact, the location parameter to be added to the circular probability distributions obtained by the ISP method corresponds to the rotation property of bilinear transforms. Here, the rotation means fixing the origin and rotating all other points by the same amount and counterclockwise. In this study, we considered rotation in bilinear transformations and used the rotated inverse stereographic projection (T −1 α ) to obtain a new circular model. Thus, we showed that the circular model to be obtained by the T −1 α (X) transformation will naturally belong to the location family of the distributions. Before the section including the application of the method, we gave some propositions and theorems that are useful when the transformation is applied to especially symmetric distributions. In the study, we applied T −1 α to the hyperbolic secant distribution. Thus, we obtained a symmetrical circular distribution with two parameters. One of these parameters is the location parameter and induced by rotated inverse stereographic projection T −1 α . To estimate the unknown parameters of the introduced distribution, the maximum likelihood, the weighted least squares, and the moment estimators are obtained. By a conducted Monte Carlo simulation study, we show that, as the sample size increases, both Bias and MSE values decrease for all estimation methods. Finally, we used the introduced distribution on a real dataset. To compare the fitting performance, we considered the Von-Mises distribution (also known as the circular normal distribution) and Wrapped Cauchy distribution as well-known symmetric alternatives. We observed that the fitting performance of the obtained distribution according to the measures frequently used in the literature is better than both Von-Mises and Wrapped Cauchy distribution.

Declaration of Competing Interests
The author declares that he has no known competing financial interests or personal relationships that could affect the work reported in this article.