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A New Method of Generating Multivariate Weibull Distributed Data Justin G Metcalf∗, K. James Sangstony, Muralidhar Rangaswamy∗, Shannon D Bluntz, Braham Himed∗ ∗Sensors Directorate, Air Force Research Laboratory ySensors and Electromagnetic Applications Laboratory, Georgia Tech Research Institute zRadar Systems Lab (RSL), University of Kansas Abstract—In order to fully test detector frameworks, it is be formed as a Gaussian random vector (real or complex) important to have representative simulated clutter data read- multiplied by a positive random variable [6], [7], [17]. In other ily available. While measured clutter data has often been fit words, a length L, zero mean, white SIRV X is formed as to the Weibull distribution, generation of simulated complex multivariate Weibull data with prescribed covariance structure X = V Z; (1) has been a challenging problem. As the multivariate Weibull distribution is admissible as a spherically invariant random where Z ∼ CN(0; I) is a length L vector and I is the vector for a specific range of shape parameter values, it can be L×L identity matrix. Therefore, clutter modeled as a SIRV is decomposed as the product of a modulating random variable locally Gaussian (i.e. a single draw from V ), but globally non- and a complex Gaussian random vector. Here we use this representation to compare the traditional method of generating Gaussian. As such, the modulating random variable provides multivariate Weibull data using the Rejection Method to a new a power modulation over the scene, resulting in an increased approximation of the modulating random variable that lends number of outlier samples as compared to the Gaussian itself to efficient computer generation. assumption. In addition to being experimentally validated, the SIRV I. INTRODUCTION model possesses several convenient properties. Perhaps the Classical detector theory relies on the assumption of a null most useful property of SIRVs is the property of closure under hypothesis distributed as independent, identically distributed linear transforms [6], [7]. Specifically, linearly transforming a Gaussian data [1]. However, sensing applications, radar in par- SIRV results in a SIRV with the same modulating random ticular, often encounter heavy-tailed distributions with a higher variable V , but with a different covariance matrix and mean frequency of outliers than is described by the Gaussian distri- vector. Therefore, if the modulating random variable V is bution (e.g. [2]–[5]). In the particular case of radar detection, both known and can be generated (e.g. via Matlab), then it is the Gaussian assumption is commonly violated by measured straightforward to generate efficiently the desired data with an radar clutter. Further complicating matters, measured clutter arbitrary mean and covariance matrix of any dimensionality. can often be fit to multiple possible statistical models equally Such simulation capability allows for the quick comparison of well [5]. In addition, measured data is necessarily limited detector performance with multiple models. in scope and new experimental data is expensive to obtain. The Weibull distribution has been suggested as a good fit Hence, it is important to have a library of clutter models with for heavy tailed, non-Gaussian clutter since at least 1969 [19]. which to test potential robust radar detectors before they are While Waloddi Weibull noted that the Weibull distribution validated with experimental data [6]–[9]. A library of clutter had no physical justification in general [20], the admissibility models is also useful to validate the emerging field of cognitive of the Weibull distribution as a SIRV provides a physical radar detection, which continues the legacy of knowledge- justification for the Weibull distribution to model radar clutter aided detector research [8], [10]. Specifically, data under test returns [6], [7], [15]–[17]. may be compared to the library of clutter models to provide a For SIRVs such as the K and Pareto distribution, the null hypothesis suggestion [8], [9], [11]. This null hypothesis modulating random variables are known and can be formed suggestion may be used to inform either a distribution specific from transformed Gamma distributed random variables [6], detector or to estimate directly the detection threshold [9], [7], [21]. Unfortunately, the general closed form solution to [11]–[14]. the modulating variable for the Weibull distribution is only The family of spherically invariant random processes known in the form of two equivalent infinite summations (SIRPs) has been shown to encompass the majority of the [17]. Previous work has used the norm of the Weibull SIRV models for non-Gaussian radar clutter, including the K, to generate Weibull data via the Rejection Method [6], [7]. Pareto/Student-t, and Weibull distributions [6], [7], [15]–[17]. However, this method is dependent on the dimensionality of However, it should be noted that the lognormal distribution, the desired vector, and can be computationally infeasible for while experimentally validated, is not admissable as a SIRP high dimensionalities or low values of the shape parameter. In [6], [15], [18]. A spherically invariant random vector (SIRV) contrast, here we bound a novel approximation of the modu- is a sample drawn from a SIRP. By definition, a SIRV may lating random variable using one of the infinite summations from [17]. This bound may then be used in conjunction with generated, the Method of Norms may be used to generate the the Rejection Method to generate previously computationally random variable. The Method of Norms requires the ability infeasible instantiations of Weibull distributed data. to generate random variables distributed according to the norm of the desired random variable and the generation of II. TRADITIONAL WEIBULL DATA GENERATION random vectors from a different SIRV distribution (typically A. The Weibull Distribution the Gaussian distribution). The method is summarized as [6], The univariate envelope of the Weibull distribution (i.e. the [9] amplitude of a complex Weibull random variable) is defined 1) Generate a white, zero mean complex Gaussian random as vector with identity covariance matrix, denotedp as z. b−1 b H fR(r) = abr exp(−ar ); r > 0; (2) 2) Compute the norm of z as rZ = jjzjj = z z. 3) Generate a random variable distributed according to the a > 0 p where is the scale parameter of the distribution, and the desired norm of the SIRV x, r = jjxjj = xH x. shape parameter is 0 < b ≤ 2. Notably, for b = 1 the Weibull X 4) Generate x as x = z rX . coincides with the exponential distribution, and for b = 2 the rZ Weibull coincides with the Rayleigh distribution. While the norm of the multivariate Weibull distribution is It can be shown that the envelope of the complex multivari- given in (3), the generation of random variables distributed ate Weibull distribution is [6], [7], [17] according to (3) is not a straightforward task. L L k 2 (−1) X a kb−1 b C. Generating Data from an Unknown Distribution via the fR(r) = Ck r exp −ar ; (3) Γ(L) k! Rejection Method k=1 where In practice, the most common methods of generating arbi- trary random variables are via the transformation of a random k mb X m k Γ 1 + C = (−1) 2 (4) variable that can be generated (e.g. transformation of Gaussian k m mb m=1 Γ 1 + 2 − L or Uniform distributed data) or by using the inverse of the cumulative distribution function (cdf) [23]. Let R be the and Γ(t) is the Gamma function. To simplify the examination random variable to be generated, and assume that the inverse of the two parameter Weibull distribution, consider a nor- cdf of R is unavailable. Further, let U be a random variable malized form of the distribution where the scale parameter 1 p that can be readily generated and whose scaled pdf bounds the is restricted to enforce E[r2] = L. This normalization can pdf of R. In other words, f (r) ≤ kf (r); 8r; k > 0. be accomplished by relating the scale parameter to the shape R U1 The Rejection Method [6], [9], [23] is then: parameter b as 1) Generate a sample u1. b=2 2 2) Generate u ∼ U(0; kf (u )). a = Γ + 1 : (5) 2 U1 1 b 3) If u2 ≤ fR(u1), then accept the point (i.e. r = u1) 4) Otherwise, reject u . Unless noted otherwise, hereafter the scale parameter is as- 1 sumed to be set according to (5). The overall process is illustrated in Figure 1. The Rejection For the univariate case, Weibull distributed data is readily Method thus performs a uniform sampling under kfU1 (u1), generated via a zero-memory non-linear (ZMNL) transform and rejects the points that fall between kfU1 (u1) and fR(r). [22]. However, the ZMNL transform does not lend itself well For efficient use of the Rejection Method, it is important to to generating correlated vector valued data. The difficulty in minimize the quantity kfU1 (r) − fR(r). generating vector valued data is due to the fact that the ZMNL does not afford independent control over the first order pdf and correlation structure. Therefore, the Method of Norms was used in [6], [7], [9] to generate Weibull distributed data. B. Method of Norms SIRVs are notable in that a length L SIRV may be gen- eralized into a set of spherical coordinates. These spherical coordinates are constructed from a set of L independent random variables, of which only one random variable in the set has a distribution dependent on the SIRV distribution [6]. As might be expected from the form of (1), there is thus one ”degree of freedom” that differentiates individual SIRV distributions. This random variable is by definition the norm of the distribution. Therefore, if the exact distribution Fig. 1: Rejection Method of the modulating random variable is not known or able to be D.

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