Rao Test Research Articles

Looking back: Selected contributions by C. R. Rao to multivariate analysis

AbstractStatistician C. R. Rao made many contributions to multivariate analysis over the span of his career. Some of his earliest contributions continue to be used and built upon almost 80 years later, while his more recent contributions spur new avenues of research. The present article discusses these contributions, how they helped shape multivariate analysis as we see it today, and what we may learn from reviewing his works. Topics include his extension of linear discriminant analysis, Rao's perimeter test, Rao's U statistic, his asymptotic expansion of Wilks' statistic, canonical factor analysis, functional principal component analysis, redundancy analysis, canonical coordinates, and correspondence analysis. The examination of his works shows that interdisciplinary collaboration and the utilization of real datasets were crucial in almost all of Rao's impactful contributions.

Bayesian detection for distributed target with limited training data

The issue of subspace-based distributed target detection with limited training data is addressed in this study. We use the Bayesian method to tackle the issue by assuming that the covariance matrix follows an inverse Wishart distribution. According to the generalized likelihood ratio test, Rao test, and Wald test, three Bayesian detectors are designed. Real data and simulations both attest to the usefulness of the proposed detectors.

Moving-Target Detection for FDA-MIMO Radar in Partially Homogeneous Environments

This paper delves into the problem of moving-target detection in partially homogeneous environments (PHE) with unknown Gaussian disturbance using a frequency diverse array multiple-input multiple-output (FDA-MIMO) radar. Using training data, we have derived expressions for four adaptive detectors, including the one-step and two-step generalized likelihood ratio test (GLRT), two-step Rao (TRao) test, and two-step Wald (TWald) test criteria, respectively. All the proposed detectors are characterized by the constant false-alarm rate (CFAR). The theoretical analysis and simulation results validate the effectiveness of the proposed detectors.

Complex Parameter Rao and Wald Tests for Assessing the Bandedness of a Complex-Valued Covariance Matrix

Banding the inverse of a covariance matrix has become a popular technique for estimating a covariance matrix from a limited number of samples. It is of interest to provide criteria to determine if a matrix is bandable, as well as to test the bandedness of a matrix. In this paper, we pose the bandedness testing problem as a hypothesis testing task in statistical signal processing. We then derive two detectors, namely the complex Rao test and the complex Wald test, to test the bandedness of a Cholesky-factor matrix of a covariance matrix’s inverse. Furthermore, in many signal processing fields, such as radar and communications, the covariance matrix and its parameters are often complex-valued; thus, it is of interest to focus on complex-valued cases. The first detector is based on the complex parameter Rao test theorem. It does not require the maximum likelihood estimates of unknown parameters under the alternative hypothesis. We also develop the complex parameter Wald test theorem for general cases and derive the complex Wald test statistic for the bandedness testing problem. Numerical examples and computer simulations are given to evaluate and compare the two detectors’ performance. In addition, we show that the two detectors and the generalized likelihood ratio test are equivalent for the important complex Gaussian linear models and provide an analysis of the root cause of the equivalence.

Adaptive detection with training data in partially homogeneous environments for colocated MIMO radar

The issue of adaptive detection is considered for the colocated multi-input multi-output (MIMO) radar in this paper. Meanwhile, the background is partially homogeneous environments (PHE), where the power mismatch is present between the training data and test data. The training data in PHE are utilized to derive effective detectors on the basis of generalized likelihood ratio test, Rao, Wald, Gradient, and Durbin tests. Since the Durbin test-based detector coincides with the Rao test-based detector, the two-step design approach for the Durbin test is used for deriving the new detector. The outcomes of simulation experiments illustrate that the proposed detectors achieve superior effectiveness to existing approaches. Furthermore, the results also show that when the signal mismatch exists, the Wald and Durbin tests maintain robust characteristics. Meanwhile, the Rao test ensures selective property.

Rao test of polarimetric detection for targets with energy spillover in non-Gaussian clutter

In this paper, we address the problem of polarimetric detection for targets with energy spillover in non-Gaussian clutter. The polarization diversity and spillover of target energy to consecutive range bins are jointly exploited to enhance the performance of target detection. Two adaptive detectors are designed according to the Rao test, as well as its two-step implementation, i.e., 2S-Rao test. It is shown that the 2S-Rao test detector coincides with the existing generalized likelihood ratio test (GLRT) detector, whereas the Rao test detector is new, and outperforms the existing detectors in probability of detection. Experiments based on both simulated and real data illustrate the superiority of the proposed detector.

Knowledge-Aided Wideband Radar Target Detection in the Heterogeneous Clutter

In this paper, the problem of wideband radar target detection in the heterogeneous environment is addressed. The wideband radar return of target with range migration is characterized as a subband model, and the heterogeneous clutter is described with a hierarchical Bayesian model. Both the prior knowledge of the clutter power and the covariance matrix and the dependence of the primary data and the secondary data are characterized with the inverse gamma and the inverse complex Wishart distribution, respectively. Based on the target and the clutter models, knowledge-aided maximum posterior ratio test (KAMAPRT), knowledge-aided Rao (KARao) test and knowledge-aided Wald (KAWald) test for wideband radar target detection in heterogeneous clutter are proposed. Finally, the performance of the proposed detectors is evaluated by simulations with both the simulated clutter generated by the probability model and the synthesized clutter from a real synthetic aperture radar (SAR) complex image. The results show that the proposed knowledge-aided detectors are effective for wideband radar target detection in the heterogeneous clutter.

Bayesian Detectors for Wideband Radar Target With Range Walking in Compound Gaussian Clutter

In this paper, the problem of the wideband radar target detection with the range walking in compound Gaussian clutter without the secondary data is addressed. The wideband radar target return with the range walking is represented as a canonical linear model in the multi-rank subspace, and the clutter is modeled as a two-dimensional random process with separable complex inverse Wishart distributed random covariance matrices in the pulse and range-frequency domains. The generalized likelihood ratio test (GLRT), the Wald test and the Rao test based on the above target and clutter models are designed under the Bayesian framework. The Laplace approximation is adopted to compute the second-kind confluent hypergeometric function of matrix argument (SKCHFMA) in the proposed detectors. Finally, the performance of the detectors is validated by the simulations.

Adaptive detection based on gradient test and Durbin test in spectrally symmetric interference

In this paper, we consider the problem of detecting a signal in ground clutter dominated environments with a symmetrically structured power spectral density. We take advantage of interference’s spectrum symmetry to develop decision schemes. The gradient test, Durbin test and their two-step (2S) variants are deduced. The 2S-gradient test and the 2S-Durbin test coincide with the 2S generalized likelihood ratio test, and the Durbin test coincide with the Rao test for the same detection problem, while the gradient test is a new detector. Numerical results, based on simulated and real data, show that the gradient test can perform better than the existing ones in sample-starved scenarios, and the detectors exploiting the interference spectrum symmetry outperform traditional detectors without exploiting the interference spectral symmetry.

Improved Model-Based Rao and Wald Test for Adaptive Range-Spread Target Detection

This article addresses the problem of the detection of range-spread targets in the presence of Gaussian disturbance which are in possession of unidentified covariance matrices. The detectors have been derived by resorting to a design composed of two steps. Based on the Rao test and Wald test, the corresponding strategies of detection were respectively derived, assuming the expression of disturbance covariance matrix has been obtained. Afterwards, the unknown parameters in the detectors were estimated on the basis of both the primary and the training data, utilizing the autoregressive property of the disturbance. A remarkable characteristic of the Rao and Wald detectors is they both asymptotically attain constant false-alarm rate (CFAR) in respect of the disturbance covariance matrix. Finally, we completed a performance assessment by utilizing the simulated data, and the result demonstrated the effectiveness of the existing proposals compared with the detectors previously proposed.

Distributed Target Detection in Unknown Interference.

Interference can degrade the detection performance of a radar system. To overcome the difficulty of target detection in unknown interference, in this paper we model the interference belonging to a subspace orthogonal to the signal subspace. We design three effective detectors for distributed target detection in unknown interference by adopting the criteria of the generalized likelihood ratio test (GLRT), the Rao test, and the Wald test. At the stage of performance evaluation, we illustrate the detection performance of the proposed detectors in the presence of completely unknown interference (not constrained to lie in the above subspace). Numerical examples indicate that the proposed GLRT and Wald test can provide better detection performance than the existing detectors.

Target detection with persymmetric subspace models for steering vector mismatches in MIMO radars

In colocated multiple-input multiple-output radar, the problem of detecting a target embedded in Gaussian noise with unknown covariance matrix is examined for the mismatched case where the true transmit and receive steering vectors are not aligned with the nominal ones, respectively. We adopt subspace models to take into account the steering vector mismatches. With exploitation of persymmetry, adaptive detectors without requiring training data are proposed according to the criteria of generalized likelihood ratio test (GLRT), Wald test, and Rao test. The proposed GLRT and Wald test exhibit constant false alarm rate properties against the noise covariance matrix. Interestingly, we find out that the Rao test does not exist for the detection problem considered. Simulation results show that the proposed detectors bear stronger robustness to the steering vector mismatches than their counterparts.

Wald- and Rao-Based Detection for Maritime Radar Targets in Sea Clutter With Lognormal Texture

Non-Gaussian sea clutter causes conventional detectors designed in Gaussian clutter to suffer detection performance degradation, and some nuisance parameters cause the uniformly most powerful test to be unavailable. To improve the detection performance of maritime radar targets, we investigate the design of adaptive detectors in correlated non-Gaussian sea clutter via using suboptimal tests. The non-Gaussian sea clutter is modelled as a product of lognormal-distributed texture and complex Gaussian speckle. Two adaptive radar target detectors are developed by using the suboptimal two-step Wald and Rao tests. Specifically, a non-adaptive detector is derived by the Wald or Rao test when the clutter texture and speckle covariance matrix are assumed to be known in the first step; then the clutter parameters known in the first step are estimated, and the true parameters of the detector obtained in the first step are replaced with the estimated values. Theoretical proof and experimental verification indicate that the two proposed detectors have the constant false alarm property with regard to the clutter speckle covariance matrix and the clutter average power. Numerical results on simulated and measured radar data show that the proposed Rao-based detector outperforms its competitors, and has the stronger robustness to the signal mismatch compared to the proposed Wald-based detector.

Persymmetric Detection of Radar Targets in Nonhomogeneous and Non-Gaussian Sea Clutter

This article addresses the detection problem of radar targets embedded in nonhomogeneous and non-Gaussian sea clutter. Nonhomogeneity leads to insufficiency of secondary data for estimating the clutter speckle covariance matrix, and non-Gaussianity causes sea clutter to become spiky. In this article, the persymmetry of the clutter covariance matrix is adopted to alleviate the requirement of secondary data, and the prior distribution of clutter texture is exploited to tackle the clutter non-Gaussianity. Based on such clutter knowledge, three adaptive detectors are proposed according to the principles of the generalized likelihood ratio test, the Wald test, and the Rao test. It is proven that three detectors ensure constant false alarm rate (CFAR) properties with respect to both the clutter speckle covariance matrix and the clutter power mean. Simulation experiments show that three detectors outperform their competitors.

Persymmetric Range-Spread Targets Detection in Compound Gaussian Sea Clutter With Inverse Gaussian Texture

This letter addresses the persymmetric adaptive detection problem of range-spread targets in compound Gaussian sea clutter. Based on the generalized likelihood ratio test (GLRT), Rao test, and Wald test, three novel compound Gaussian detectors are developed. Specifically, the sea clutter is modeled as compound Gaussian with inverse Gaussian distribution, and the detectors are derived by the two-step maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posterior</i> (MAP) procedures. In the first step, we assume the clutter covariance matrix (CCM) and the inverse Gaussian texture are known and derive the proposed detectors’ test statistics. In the second step, we use the persymmetric property and MAP criterion to estimate the CCM and inverse Gaussian texture parameters. Then the proposed detectors are proved to be constant false alarm rate (CFAR) detectors with respect to the real CCM. The simulation experiments are conducted by comparing the proposed detectors with their counterparts on both synthetic data and real sea clutter data. The numerical results indicate that the novel detectors exhibit better detection performance than their competitors.

Polarimetric Target Detection in Compound Gaussian Sea Clutter With Inverse Gaussian Texture

This letter addresses the polarimetric target detection problem in compound Gaussian sea clutter. In view of the heavy-tailed characteristic of sea clutter, we model the sea clutter as compound Gaussian distribution with inverse Gaussian texture (IGCG). We propose three polarimetric compound Gaussian detectors based on the two-step Rao test, the Wald test, and the generalized likelihood ratio test (GLRT). Specifically, we assume that the polarimetric clutter covariance matrix (PCCM) and the inverse Gaussian textures are known in the first step, and the test statistics of the proposed polarimetric detectors are derived. In the second step, we use the training data to estimate PCCM and maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> (MAP) to estimate the inverse Gaussian textures, and we obtain the fully adaptive detectors. Then, we give proof of the constant false alarm rate (CFAR) properties of the proposed detectors. We validate the performance of the proposed polarimetric detectors by conducting experiments based on simulated and real sea clutter data. Finally, the numerical results indicate that the proposed detectors exhibit better detection performance than their competitors and are robust when the mismatched signal occurs.

Complex Parameter Rao, Wald, Gradient, and Durbin Tests for Multichannel Signal Detection

In the problem of multichannel signal detection, when it comes to the detector design criteria apart from the generalized likelihood ratio test, the traditional method is to cascade the real and imaginary parts of the parameters, and then substitute them into the real parameter statistics. This method is not succinct, and sometimes may be cumbersome and difficult to handle. Recently, a complex parameter Rao test was introduced by Kay and Zhu without the need of cascading the real and imaginary parts of the complex parameters when there is no nuisance parameter. Inspired by this work, we move a further step toward the complex parameter statistics of the Rao, Wald, gradient, and Durbin tests both with and without nuisance parameters, and derive the relationships between their real and complex parameter statistics. Moreover, for a special Fisher information matrix which often holds in practice, we derive a series of simple forms of the complex parameter statistics for the above four criteria, and discuss their application conditions in linear multivariate complex circular Gaussian distribution. Finally, several application examples are given to confirm the proposed schemes.

Spectrum Sensing Under Illegal Spectrum Access Behaviors in Multiple Authorized Users Scenario

In this paper, we detect illegal spectrum access behaviors in the dynamic spectrum sharing of the cognitive wireless networks, in which multiple authorized users (AUs) work in the same channel. First, we detect whether there are users occupy the channel, and if the channel is occupied, then recognize whether there is the illegal user (IU) and infer the number of AUs roughly. In the light of the thorny challenge that legal spectrum utilization behaviors and illegal spectrum access behaviors coexist probabilistically, the spectrum sensing problem is formulated as a mathematical model of ternary hypothesis test. Moreover, when multiple AUs are working at the same time, it is extremely difficult to detect IU. To tackle the problem, taking into account the various potential combinations of AUs and IU, we exploit a two-step detector and derive related detection schemes based on the generalized likelihood ratio test (GLRT), the Rao test, and the Wald test following the generalized multi-hypothesis Neyman-Pearson (GMNP) criterion respectively. Additionally, we design a cooperative spectrum sensing (CSS) scheme on the basis of the global GMNP criteria, in which the distributed detection framework composed of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${K}$ </tex-math></inline-formula> sensing nodes and a fusion center (FC). Finally, to verify the performance of the proposed detection scheme in a variety of parameter configurations, we provide comprehensive simulations and find that there will be at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$({{2^{N}}-1})$ </tex-math></inline-formula> detection valleys when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> AUs work at the same time. Compared with single-sensing node detection, the CSS scheme can significantly improve the detection performance of the proposed detection schemes, especially in the upper bound and valley of the detection performance.

Robust density power divergence based tests in multivariate analysis: A comparative overview of different approaches

Hypothesis testing is one of the fundamental paradigms of statistical inference. The three canonical hypothesis testing procedures available in the statistical literature are the likelihood ratio (LR) test, the Wald test and the Rao (score) test. All of them have good optimality properties and past research has not identified any of these three procedures to be a clear winner over the other two. However, the classical versions of these tests are based on the maximum likelihood estimator (MLE), which, although the most optimal estimator asymptotically, is known for its lack of robustness under outliers and model misspecification. In the present paper we provide an overview of the analogues of these tests based on the minimum density power divergence estimator (MDPDE), which presents us with an alternative option that is strongly robust and highly efficient. Since these tests have, so far, been mostly studied for univariate responses, here we primarily focus on their performances for several important hypothesis testing problems in the multivariate context under the multivariate normal model family.

Age and growth of early-life-stage Atlantic tarpon (Megalops atlanticus) from the northcentral Gulf of Mexico.

Age and growth of early-life-stage Atlantic tarpon Megalops atlanticus collected from Mississippi coastal waters in the northcentral Gulf of Mexico (GOM) are described using otolith microstructure analysis. Tarpon leptocephali (n= 95, 16.0-27.8 mm standard length, LS ) collected from June throughOctober 2013-2018, ranged in age from 22 to 43 days (mean=30.9± 0.5 days). Leptocephalus somatic growth rates ranged 0.46-1.24 mm day-1 (mean=0.76 ± 0.02 mm day-1 ), and leptocephalus otolith growth rates ranged 1.78-3.97 μm day-1 (mean=2.58 ± 0.04 μm day-1 ). Growth rates were inversely correlated to leptocephalus age, indicating the shrinkage phase associated with leptocephalus metamorphosis. Juvenile tarpon (n= 358, 50-359 mm fork length, LF ) were collected from August through December 2007-2018. Juveniles exhibited a positive allometric relationship (adjusted R2 = 0.99, P< 0.001) between length and mass. The age of 100 juveniles (71-277 mm LF ) ranged from 76 to 174 days. Juvenile growth rate was estimated as 1.56 ± 0.11 mm day-1 . Significant (P< 0.001) linear relationships were found between juvenile age and otolith metrics, including otolith mass (R2 = 0.81) and radius (R2 = 0.68). Evaluation of the backcalculated hatch dates suggests that specimens in the collection hatched from late May through mid-September with slight peaks during July and August. A Rao's Spacing Test of Uniformity indicates the presence of significant lunar periodicity in leptocephalus hatch dates (n= 95, U= 250.1, P< 0.05), with 50% of the leptocephali hatched within 5 days (before or after) of the full moon. This study fills critical gaps in the scientific knowledge of tarpon and provides estimates of early-life-history metrics for an iconic game fish at the northernmost extent of its GOM range.