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Estimation of Critical Collapse Solutions to Black Holes with Nonlinear Statistical Models

The self-similar gravitational collapse solutions to the Einstein-axion-dilaton system have already been found out. Those solutions become invariants after combining the spacetime dilation with the transformations of internal SL(2, R). We apply nonlinear statistical models to estimate the functions that appear in the physics of Black Holes of the axion-dilaton system in four dimensions. These statistical models include parametric polynomial regression, nonparametric kernel regression and semi-parametric local polynomial regression models. Through various numerical studies, we reached accurate numerical and closed-form continuously differentiable estimates for the functions appearing in the metric and equations of motion.


Introduction
As the end state of gravitational collapse, black holes are defined by their mass, angular momentum as well as their charge. M. Choptuik [1] explored the so called critical phenomena in gravitational collapse as well as Choptuik scaling. He made a breakthrough in the subject of numerical relativity. Indeed, Choptuik scaling ( [1] and [2]) is a property that occurs in various systems which experience gravitational collapse. He discovered that there might be a fourth universal quantity that establishes the critical collapse. Choptuik followed the study of the spherically symmetric collapse of scalar field and explored a critical behaviour which demonstrates the discrete spacetime self-similarity. Through taking the amplitude of the scalar field p, he derived a critical value p crit where black hole forms as p exceeds p crit .
Also, as p goes beyond the threshold, the mass of the black hole M bh illustrates the scaling where the Choptuik exponent was found to be γ 0.37 [1] in four dimension and for a real scalar field. Various numerical computations with different matter content have also been discovered [3,4,5,6,7].
Motivated by String Theory, the axion-dilaton system can also experience the same gravitational collapse process. The study of Choptuik phenomenon in the axion-dilaton system was initiated in [8,9,10]. The AdS/CFT correspondence [11] is viewed as the first motivation to investigate critical collapse solutions especially for the axion-dilaton system.
The AdS/CFT correspondence correlates the critical exponent and the imaginary part of quasi normal modes as well as the dual conformal field theory [12]. The second motivation relies on the holographic description of black hole formation [13], particularly in the physics of black holes and their implications [14]. From the IIB string theory point of view, we look for the gravitational collapse for the special spaces that could asymptotically approach AdS 5 × S 5 . The matter field in the IIB string theory arises from the self-dual 5-form field strength and the axion-dilaton configuration. In a recent research [15,16], the whole families of Continuous Self-Similar (CSS) solutions of the Einstein-axion-dilaton system were explored for all the three conjugacy classes of SL (2,R). Some remarks about critical exponents and higher dimensional solutions have been made in [17] and [18]. For more details about the other systems experiencing gravitational collapse, readers are referred to [19,20,21,22,23].
To our best knowledge, there is no research article in the literature investigated the prop-erties of nonlinear statistical models to estimate the critical collapse functions in Einsteinaxion-dilaton. In this paper, for the first time, we utilize parametric polynomial regression, non-parametric kernel regression and semi-parametric local polynomial regression models to develop and a closed form and continuously differentiable functional forms of the critical collapse functions.
This article is organized as follows. We decribe the axion-dilaton system and its different Continuous Self-Similar ansatzë in Section 2. The initial conditions and Properties of the critical solutions for all three conjugacy classes are discussed in Sections 3 and 4, respectively. The nonlinear statistical estimation methods are then discussed in Section 5. The performance of proposed statistical models are finally investigated in Section 6.
Concluding remarks are presented in Section 7.

Axion-dilaton Configuration
One can combine the axion a and dilaton φ field into a single complex field τ ≡ a + ie −φ , and its coupling to four dimensional gravity is given by where R is the scalar curvature. The above action describes the effective action of type II string theory [24,25]. This action respects SL(2,R) symmetry, which means that if we consider the following the action remains invariant where a, b, c, d are real parameters satisfying ad − bc = 1. The equations of motion can be read as follows We have looked for critical solutions by dealing with spherical symmetry and continuous self-similarity. Following [8,9,10], one can choose the metric as We might consider a scale invariant variable as z ≡ −r/t and hence the Continuous Self-Similarity of the metric actually means that all functions u(t, r), b(t, r) can be expressed just in terms of z, that is, b(t, r) = b(z), u(t, r) = u(z).
This continuous self-similarity condition for τ was described in detail in [26]. The axion-dilaton system does have a global SL(2, R)-symmetry which is broken to a SL(2, Z) by taking into account the non-perturbative phenomena in type II string theory. If we take the quantum effects, SL(2,R) symmetry reduces to SL(2,Z) for which it is believed to be non-perturbative symmetry of String Theory [27,28,29]. Therefore, one might compensate the action by means of an SL(2, R)-transformation, that is τ (t, z) must respect the following with α 0,1,2 real numbers. The above equation has two roots that are related to compensating the scaling transformation. Having set that, we find three different ansatzë, which are related to the fact that the chosen SL(2, R)-transformation is either an elliptic, hyperbolic or parabolic transformation. The elliptic ansatz is defined as where ω is a real constant that will be known by the regularity conditions for the critical solution. On the other hand, for the hyperbolic case, τ (t, r) is given by Eventually the parabolic ansatz is illustrated by τ (t, r) = f (z) + ω log(−t). Note that, the function f (z) needs to satisfy |f (z)| < 1 for the elliptic case, whereas Imf (z) > 0 for hyperbolic and parabolic cases.

Equations of motion and initial conditions
In this section, we first study the equations of motion and then explain the properties of solutions. Replacing CSS into the equations of motion, we derive a system of differential equations just for b(z), f (z). Using Einstein equations for angular variables, one can express u(z), b(z) just in terms of f (z), which means that Hence u(z) and its derivatives can be eliminated from equations of motion. The other equations of motion involve b(z), f (z). Hence we are left with various ordinary differential Since in this paper, we are looking for estimation of the function of b(z) and real and Imaginary part of f (z) for elliptic and hyperbolic cases in 4 dimension, we just generate those equations as follows. Indeed the equations of motion are derived in [26]. The equations for the elliptic case are Using time scaling one can set b(t, 0) = 1. In the elliptic case, by writing f (z) = f m (z)e ifa(z) , the regularity conditions imply: The above equations are invariant under a global phase of f (z) so we can choose For hyperbolic case the equations are determined by They are invariant under a constant scaling f → λf and applying regularity at the origin we find that f (z = 0) should vanish. Thus the initial conditions for the hyperbolic case are: Finally in the parabolic ansatz the equations of motion are invariant under arbitrary shifts of f (z).

Properties of the critical solutions
The properties of the solutions, the physical and geometrical behaviours of the solutions for elliptic case within details were explained in [9,10]. Naturally, for the hyperbolic case the same properties are being held. In all equations we have five singularities where z = ±0 corresponds to origin and we have dealt with them by making the regularity conditions. On the other hand, the point z = ∞ related to t = 0. By change of variables and redefinition of the fields f (z), b(z), one can show that [30] the equations remain regular there as well.
The singularities b(z ± ) = ±z ± are the locations where the homothetic Killing vector is null, as explained in [26]. For b(z + ) = z + the solution must be smooth within this surface and we need to have the continuity of f, b in this region. b(z + ) = z + is related to the homothetic horizon and it is indeed a mere coordinate singularity [31,26] so τ must be finite across it which gets interpreted as the finiteness of f (z) once z → z + . Another constraint comes from the fact that the vanishing of the divergent part of f (z) generates one complex valued constraint at z + that can be defined by G(b(z + ), f (z + ), f (z + )) = 0 where the definitions of G are given in Equations (49)-(51) of [15]. If we use regularity at z = 0 as well as the residual symmetries, then we find out the initial conditions b(0) = 1, f (0) = 0 and the value of f (0) is shown by where x 0 is a real parameter. Thus, we have two constraints as the real and imaginary parts of G must be vanished and two parameters (ω, x 0 ) to be known.
The entire solutions for hyperbolic case in four and five dimensions have been derived in [16]. These solutions are obtained by making use of numerical integration from the equations of motion. For instance, for four dimensional elliptic case, just one solution is determined [10,26] and it is given by Using this new search methodology, we are able to explore the entire families of solutions for hyperbolic case in 4 dimension which are three cases called α, β, γ solutions. The α solution is given by The β solution is determined by Finally γ solution is explored to be

Statistical Estimation Methods
Throughout this section, we use the following notations to present the statistical estimation methods. Let (X, y) denote a multivariate random variable from a random sample of size n.
Suppose y = (y 1 . . . , y n ) and X = (x 1 , . . . , x p ) represent, respectively, the vector of response variable of size n and (n × p) dimensional design matrix with p explanatory variables where rank(X) = p < n.

Polynomial Regression Model
Linear regression models are among the most popular statistical methods for modelling data. One can address the relationship between response variable y and the explanatory variables x 1 , . . . , x p by the linear regression model where β is the unknown parameters (hence-after called coefficients) of the model, x i indicates the transpose of x i and that i iid ∼ N (0, 1); i = 1, . . . , n; that is, the error terms are independent and identically distributed from standard normal distribution [32].
It is easily seen that linear regression model (23), as a parametric method, translates the prediction problem of response function y = g(x) (as a function of explanatory variable x) to the estimation problem of the unknown parameters/coefficients of the model. Least square (LS) method [32] is one of the most common approaches to estimate the coefficients of model (23). Given the design matrix X and response vector y from n observations, the least squares estimate of β is given by where || · || 2 2 denotes the l 2 norm. It is easy to show that the solution to (24) is given by Once model (23) is trained, the response can be predicted at a new value x new by It is evident that the functional forms of |f 0 (z)|, arg(f 0 (z)) and b 2 0 (z) − z 2 , as our underlying statistical population to be estimated are clearly nonlinear functions of spacetime. Hence, the simple linear regression model (23) based on z is not flexible enough to estimate the nonlinearity of critical collapse functions. One can employ polynomial regression model to deal with the nonlinear critical collapse functions. Polynomial regression model enables us to incorporate the higher orders of explanatory variable x to approximate better the nonlinear response function y = g(x). Polynomial regression model of order l is given by where β = (β 0 , . . . , β l ) represent the unknown coefficients of the model. Note that, we only focus on the main effects of explanatory variables (and their higher orders) in estimating the critical functions. First, in the estimating of the critical functions, there is only a single explanatory variable -that is the space-time z. Hence, no interaction term is defined in the regression models. When there is a single explanatory variable in the regression, the higher orders of the explanatory variable can very well accommodate the nonlinearity of the population. Finally, the interaction terms typically contribute to the refinement of the estimates at the price of introducing more parameters in the model and reducing the degree of freedom in the estimation. For the above reasons, throughout these manuscript, we do not include the interaction terms in the statistical models.
Polynomial regression model, as a special case of model (23), can be written as a linear model again by where the columns of matrix Z are the copies of explanatory variable x i taken to various powers j = 0, . . . , l. Similarly, from least squares method (24), the polynomial regression at x new is produced by Polynomial regression model provides a flexible solution to estimate a nonlinear function at the price of higher orders of explanatory variable x in the model. Therefore, the estimation performance of the polynomial proposal depends on the order of the polynomial regression l. In Section 6, we perform a cross validation to select the best order of the polynomial estimators.

Kernel Regression Model
Linear regression and polynomial regression models translate the estimation problem of the response function y = g(x) to the estimation problem of parameters β. Non-parametric regression models can be considered as another approach to estimate the nonlinear critical however, in this manuscript, we focus only on Epanechnikov kernel [33] given by where h denotes the bandwidth parameter of the kernel function K and I is an indicator function such that I(u) = 1 if u=true otherwise I(u) = 0. It is at once apparent that the kernel function (29) tunes the width of neighbourhoods based on the bandwidth parameter h. When an observation falls out of the bandwidth, the kernel function assigns very small weight to the observation to reduce its impact on the function estimate. Given a training data set of size n with explanatory variable x and response y, the Nadaraya-Watson kernel regression estimate [34,35] the response function at new observation x new as where K h (·) is obtained from (29). Note that the kernel regression estimator (30) deals with the nonlinearity of response function at the price of selecting the bandwidth parameter. To this end, when small h is selected, the weights assigned by kernel regression estimator are more concentrated around the new observation (i.e., shorter neighbourhoods will be declared). In contrast, when large h is selected, the weights will be more spread out and consequently wider neighbourhoods are declared. Hence, for a given training data, we carry out a cross validation to find out the optimal bandwidth.

Local Regression Model
As another approach, we study the local regression model to earn the curvature of the response function y = g(x). Local regression, as a semi-parametric approach, combines the parametric advantages of the polynomial regression and non-parametric properties of the kernel regression in estimating the response function. Local regression can be intuitively explained by the Taylor expansion of g(x) around x new as follows: where kernel regression can be viewed as an estimator which only utilizes the constant term to approximate g(x). Local regression method exploits the high orders of x by polynomial regression and then estimates the coefficients (of the polynomial regression) by the kernel regression in each neighbourhood.
Given a training data set of size n with explanatory variable x and response variable y, the local regression estimator [33] of order l at x new is given by where the vector of coefficients estimate β(x new ) = β 0 (x new ), . . . , β l (x new ) is obtained as a solution to: where K h (·) obtained from (29).
The estimator (31) can be used to estimate the response function at any value of x new .
Accordingly, the estimator (31) enables us to approximate the response function y = g(x) throughout the domain. Note that the local regression smoother (31) requires two tuning parameters. These tuning parameters include the bandwidth parameter of kernel part h and the order of the polynomial part l.   MSE of local regression method with l = 1 and bandwidth h (presented in parenthesis) in estimating the critical collapse response function g(z) = |f 0 (z)| in elliptic and hyperbolic cases based on a training sample of size n = 100.
Using the optimization techniques of [10] a numerical search is carried out to find the critical solution on various intervals in the domain of forward singularity ( [0, z + ]). Accordingly, they showed that there was a unique critical solution in elliptic space. This results in the interval [0, 2.5] as the domain of the critical collapse functions in elliptic space in four dimension, where this unique solution was also confirmed in [15]. Similarly, R. Antonelli and E. Hatefi [15]  In the statistical analysis, we considered space-time z as the single explanatory variable (x) and the realization of the critical collapse functions b 2 0 −z 2 , |f 0 |, arg f 0 as the responses (observed from the corresponding critical function) in the regression models. We fitted one regression model for each critical function. We generated independently (i.e., with replacement) training samples of size n = 100 from each population. For the validation of the estimation, we generated (independent from the training data) test data (x test , y test ) of size n = 100 from the entire domains of the critical functions. As described in Section 5, to estimate the critical function by the polynomial regression, we first applied equation (25) to the training data and estimated the coefficients of the model. Using the estimated coefficients β and (26), we then predicted the response of the critical function ( y test ) at x test . For Kernel regression method, we applied equation (30) to the training data and predicted the value of the critical function ( y test ) at test point x test . According to the definition of the local regression, the coefficients of the model are treated as the functions of the test data. Hence, we used the training data and estimated the coefficients using (32). From (31) and the estimated the coefficients β(x test ), we then predicted the critical collapse function ( y test ) at x test . We finally implemented the above prediction procedures sequentially for all the points in test data to estimate all the critical collapse functions over their entire domains.
To assess the accuracy of the proposed estimators, we use the measure of square root of the mean squared errors √ MSE as follows Note that the trained model will be more accurate in estimating the critical collapse response

Conclusion
The black hole solutions of axion-dilaton the system were recently investigated in elliptic and hyperbolic cases in four and five dimensions [15]. It is crucial for researchers to estimate the functional form of the critical collapse functions. These estimates pave the path to make the critical solutions, critical exponents, the mass of Black Holes and universality of Choptuik exponents more tractable. To our best knowledge, no research article in the literature investigated the properties of nonlinear statistical models in estimating the critical collapse functions in Einstein-axion-dilaton.
In this paper, we employed parametric polynomial regression, non-parametric kernel regression and semi-parametric local polynomial regression for the first time to estimate the functional forms of the critical collapse functions. From numerical studies, we observe that the local regression estimators outperform the kernel and polynomial counterparts in estimating almost all critical collapse functions in elliptic and hyperbolic domains. While local and kernel methods estimate more accurately the critical collapse function, the polynomial regression method enables us to obtain the closed-form and continuously differentiable estimates for the critical functions. Given the closed forms of critical functions, a pressing question is if one can algebraically derive the critical exponents for the axion-dilaton system. Note that these methods are applied not only for Einstein-axion-dilaton system and similar solutions but also for other potential systems. These methods are generic and can be used to any matter content for any space-time dimensions. This is a path that we plan to follow in the near future.              MSE of kernel regression method evaluated at bandwidth h (presented in parenthesis) in estimating the critical collapse response function g(z) = arg f 0 (z) in elliptic and hyperbolic cases based on a training sample of size n = 100.