Rao Test Research Articles

U-Statistics based on spacings

In this paper, we investigate the asymptotic theory for U-statistics based on sample spacings, i.e. the gaps between successive observations. The usual asymptotic theory for U-statistics does not apply here because spacings are dependent variables. However, under the null hypothesis, the uniform spacings can be expressed as conditionally independent Exponential random variables. We exploit this idea to derive the relevant asymptotic theory both under the null hypothesis and under a sequence of close alternatives. The generalized Gini mean difference of the sample spacings is a prime example of a U-statistic of this type. We show that such a Gini spacings test is analogous to Rao's spacings test. We find the asymptotically locally most powerful test in this class, and it has the same efficacy as the Greenwood statistic.

Parametric Rao Tests for Multichannel Adaptive Detection in Partially Homogeneous Environment

This paper considers the problem of detecting a multichannel signal in partially homogeneous environments, where the disturbances in both test signal and training signals share the same covariance matrix up to an unknown power scaling factor. Two different parametric Rao tests, referred to as the normalized parametric Rao (NPRao) test and the scale-invariant parametric Rao (SI-PRao) test, respectively, are developed by modeling the disturbance as a multichannel autoregressive (AR) process. The NPRao and SI-PRao tests entail reduced training requirements and computational efficiency, compared with conventional fully adaptive, covariance matrix based solutions. The SI-PRao test attains asymptotically a constant false alarm rate (CFAR) that is independent of the covariance matrix and power scaling factor of the disturbance. Comparisons with the covariance matrix based, scale-invariant generalized likelihood ratio test (GLRT), also known as the adaptive coherence estimator (ACE), are included. Numerical results show that the parametric Rao detectors, in particular the SI-PRao test, attain considerably better detection performance and use significantly less training than the ACE detector.

Generalized Matched Subspace Filter for Nonindependent Noise Based on ICA

We propose a generalization of the matched subspace filter for the detection of unknown signals in a background of non-Gaussian and nonindependent noise. The generalization is based on a modification of the Rao test by including a linear transformation derived from independent component analysis (ICA). Receiver operating characteristic (ROC) curves computed for simulated examples illustrate the significant improvement achieved with the generalized solution.

Performance analysis of a two-stage Rao detector

This paper deals with the problem of adaptive signal detection in the presence of Gaussian disturbance with unknown covariance matrix. A new two-stage Rao test detector is proposed, which is obtained by cascading a GLRT-based subspace detector (SD) and the Rao test. The statistical characterization for the proposed two-stage test statistic is provided, under both noise-only and signal-plus-noise hypotheses. The associated probability of false alarm ( P fa ) and probability of detection ( P d ) are derived in closed form. Performance of the proposed detector is demonstrated by simulation studies, wherein recently proposed detectors are involved for performance comparison. The results show that our detector can achieve better robustness with respect to (w.r.t.) the existing two-stage Rao test detector (AMF–RAO), and has better selectivity than the improved adaptive sidelobe blankers (WAS-ASB and KWAS-ASB) for small values of system parameters.

UMPI Test in SIRV Distribution for the Multi-Rank Signal Model

This letter mainly deals with the multi-rank signal detecting problem against Spherically Invariant Random Vector (SIRV) background with Invariance theory. It is proved that generalized likelihood ratio test (GLRT), Rao test and Wald test are all the Uniformly Most Powerful Invariant (UMPI) detectors in SIRV distributions under a mild technical condition.

Adaptive subspace detector for multi-input multi-output radar in the presence of steering vector mismatch

This study studies the problem of adaptive target detection by multi-input multi-output (MIMO) radar with multiple antennas at each of the receiver sites. MIMO radar has been postulated to have a number of advantages over its monostatic counterpart. However, many previous publications have not necessarily made valid assumptions in assessing performance potential. Here, the authors consider the issue of the overall MIMO radar system steering vector which should be aligned with the desired pointing direction. However, misalignment can occur because of calibration errors, receiver channel imperfections and beam pointing errors. To account for this misalignment, the steering vector is assumed to belong to a known linear subspace. At the design stage, the authors resort to a generalised likelihood ratio principle and Rao test criterion. Subsequently, MIMO versions of the subspace generalised likelihood ratio test (MIMO-SGLRT), the subspace adaptive beam-former orthogonal rejection test (MIMO-SABORT) and the subspace Rao (MIMO-SRao) detector are developed to improve the robustness of MIMO radar detection performance for the case of steering vector mismatch. The constant false alarm rate properties of the three detectors are demonstrated. Finally, the performance of the MIMO-SGLRT, MIMO-SABORT and MIMO-SRao detectors in both the matched and mismatched steering vector cases are numerically evaluated. The results show the robustness of the proposed detectors to steering vector mismatch.

Rao Test of Distributed Targets in K-distributed Clutter

该文针对高分辨雷达体制下，点目标分裂成分布式目标所带来的检测问题，提出了基于Rao检测的分布式目标自适应检测算法。将分布式目标建模为子空间信号，目标不仅在距离维上扩展同时也在Doppler频率维上扩展。Rao检测算法只需对H0假设条件下的未知参数进行最大似然估计，在构造检测器的过程中运用两步法检测策略，有效地减少了计算量和复杂度。最后，用Monte Carlo仿真了该算法的检测性能，并与以前的检测器相比较验证了新提出的检测器对分布式目标在K分布杂波中的有效性。

Parametric Adaptive Radar Detector with Enhanced Mismatched Signals Rejection Capabilities

We consider the problem of adaptive signal detection in the presence of Gaussian noise with unknown covariance matrix. We propose a parametric radar detector by introducing a design parameter to trade off the target sensitivity with sidelobes energy rejection. The resulting detector merges the statistics of Kelly's GLRT and of the Rao test and so covers Kelly's GLRT and the Rao test as special cases. Both invariance properties and constant false alarm rate (CFAR) behavior for this detector are studied. At the analysis stage, the performance of the new receiver is assessed and compared with several traditional adaptive detectors. The results highlight better rejection capabilities of this proposed detector for mismatched signals. Further, we develop two two-stage detectors, one of which consists of an adaptive matched filter (AMF) followed by the aforementioned detector, and the other is obtained by cascading a GLRT-based Subspace Detector (SD) and the proposed adaptive detector. We show that the former two-stage detector outperforms traditional two-stage detectors in terms of selectivity, and the latter yields more robustness.

A Rao Test With Enhanced Selectivity Properties in Homogeneous Scenarios

This correspondence focuses on the design of a selective receiver for homogeneous scenarios. To this end, at the design stage it is assumed that the cell under test contains a noise-like interferer in addition to thermal noise, clutter, and to the possible signal of interest; a set of secondary data, free of signal components, is available: such data share a common covariance matrix that is equal to that of thermal noise plus clutter in the cell under test. Under the above assumptions, an adaptive detector implementing the RAO test is designed. A preliminary performance assessment, conducted assuming that the cell under test and the secondary data share one and the same covariance matrix of the overall disturbance (including possible noise-like interferers), shows that the newly proposed detector can provide enhanced rejection capabilities with respect to its natural competitor (the W-ABORT). It can also outperform the W-ABORT for matched signals.

Full Positional Accuracy Analysis of Spatial Data by Means of Circular Statistics

AbstractThe positional error in spatial data is defined as a vector by comparing the coordinates between the true position and the measured position. The standard tests to assess the positional accuracy use only the magnitude of the vector and omit the azimuth. This article suggests that the use of both values allows a much more complete analysis of the positional error. A set of tests is proposed that are relevant for this purpose and demonstrate that some important features are not identified by the common procedures. The test samples come from two datasets. The first is obtained from the comparison of 100 homologous points in two conventional maps, and the second one comes from the geometric calibration of a photogrammetric scanner. The results are analyzed and discussed, showing that important issues such as error anisotropy are detected only by means of the circular statistics tests and density maps of distribution. Therefore, tests that assess the goodness of fit for uniform distribution in azimuths, such as Rayleigh and Rao tests, give low probabilities (P = 0 and P > 0.01). Moreover, density maps working with both magnitude and angle can locate the outlier candidate and offer more information about the spatial distribution of error.

Subspace detection for range and Doppler distributed targets with Rao and Wald tests

For the detection problem caused by the point target broken up into the distributed target in the high-range resolution radar system, adaptive detection algorithms based on the Rao and the Wald tests are proposed. The distributed target, which is modeled as a subspace random signal, may be distributed both in range and also in Doppler frequency axes. The clutter is modeled as a spherically invariant random process (SIRP). The unknown parameters are estimated by maximum likelihood estimation only under hypothesis H 0 in Rao test, while only under hypothesis H 1 in Wald test. Interestingly the proposed non-adaptive detectors from different deriving progresses have the same final form, which perform an incoherent processing over all the cells under test and ensure CFAR property with respect to the unknown statistics of the clutter texture component. In the end, performances of the proposed adaptive detectors are assessed through Monte Carlo simulations and are shown to have better detection performance compared with existing similar detectors.

Adaptive Detection in Gaussian Interference With Unknown Covariance After Reduction by Invariance

In this paper, we address the problem of adaptive detection in homogeneous Gaussian interference, with unknown covariance matrix, after reduction by invariance. Starting from a maximal invariant statistic, which contains all the information for the synthesis of an invariant detector, we devise the Rao test, generalized likelihood ratio test (GLRT), and Durbin test. Moreover, we compare their decision statistics with those of the receivers designed according to the same criteria from the raw data (i.e., before reduction by invariance). We prove that the GLRT in the original data space is statistically equivalent to the GLRT designed after reduction by invariance (under a very mild assumption) and coincide with the conditional uniformly most powerful invariant (C-UMPI) test, obtained conditioning on an ancillary part of the maximal invariant statistic. As to the Rao and Durbin criteria, when they are applied after reduction by invariance, lead to detectors different from the counterparts devised in the raw data domain. At the analysis stage, the performance of the receivers devised in the invariant domain is analyzed in comparison with that of the counterparts synthesized from the original observations.

The Effect of Exertion Level on Activation Patterns and Variability of Trunk Muscles During Multidirectional Isometric Activities in Upright Posture

An experimental design to investigate activation patterns of trunk muscles during multidirectional exertions. To evaluate trunk muscle activation patterns in varying directions and moment magnitudes during an isometric task, and to investigate the effects of angle and level of isometric exertion on the electromyography (EMG) variability of trunk muscles in upright posture. Few studies have investigated trunk muscle activation patterns in multidirectional exertions with different moment magnitudes. A total of 12 asymptomatic male subjects were participated in the study. The EMG activity of 10 selected trunk muscles was collected in the 3 seconds end point matching tasks in 8 angles and 3 magnitudes of exertion. Trunk muscle activation patterns were examined using EMG tuning curves and measuring preferred direction (mean vector direction) and the index of spatial focus. The effect of exertion level on these measures was investigated by Rao test. The effects of angle and level of exertion on the EMG variability of trunk muscles were tested by analysis of variance with repeated measures design. No significant difference in EMG tuning curves, preferred direction, and the index of spatial focus was found for each muscle studied across exertion levels (P > 0.05). The index of spatial focus of most muscles studied was not changed with increasing moment magnitude. EMG variability of trunk muscles was significantly affected by angle and level of exertion and their interaction effect (P < 0.001). Consistent activation patterns of trunk muscles were found within and among subjects in different moment magnitudes. The index of spatial focus indicated that probably no shift to a higher co-contraction strategy has been adapted with increasing moment magnitude. The results suggested that increased EMG variability of trunk muscles in asymmetric exertions may be associated with lower trunk controllability during combined exertions.

On the Invariance, Coincidence, and Statistical Equivalence of the GLRT, Rao Test, and Wald Test

Three common techniques to discriminate between alternatives in a binary hypothesis testing problem are: the generalized likelihood ratio test (GLRT), the Rao test, and the Wald test. In this paper, we investigate some characteristics of the corresponding decision statistics and provide their expressions for some problems of particular interest in statistical signal processing. First of all, we focus on the invariance of the Rao and Wald tests with respect to transformations leaving the testing problem unaltered. Then, we introduce necessary and sufficient conditions in order for their decision statistics to coincide with twice the logarithm of the GLRT statistic. Finally, we present some detection problems, usually encountered in practical signal processing applications, where the decision variables of the three quoted tests are equivalent, namely related by strictly monotonic transformations.

Detection of Single Scatterers in Multidimensional SAR Imaging

Multidimensional synthetic aperture radar (SAR) imaging is a technique based on coherent SAR data combination for space (full 3-D) and space deformation-velocity (4-D) analysis. It is an extension of the concepts of SAR interferometry and differential interferometry SAR and offers new options for the analysis and monitoring of ground scenes. In this paper, we consider the problem of detecting single scatterers for localization and monitoring issues. To this end, we resort to a constant false alarm rate (CFAR) detection scheme which can be synthesized according to three different design criteria: generalized likelihood ratio test, Rao test, and Wald test. At the analysis stage, the performance of the aforementioned detector is compared to that of a previously proposed CFAR scheme, based on the multi-interferogram complex coherence and widely used in persistent scatterer interferometry. The analysis is conducted both on simulated and on real SAR data, acquired by ERS-1/2 satellites. Finally, Cramer-Rao lower bounds for the estimation of the scatterer elevation and velocity are provided.

Likelihood-Based Hypothesis Tests for Brain Activation Detection From MRI Data Disturbed by Colored Noise: A Simulation Study

Functional magnetic resonance imaging (fMRI) data that are corrupted by temporally colored noise are generally preprocessed (i.e., prewhitened or precolored) prior to functional activation detection. In this paper, we propose likelihood-based hypothesis tests that account for colored noise directly within the framework of functional activation detection. Three likelihood-based tests are proposed: the generalized likelihood ratio (GLR) test, the Wald test, and the Rao test. The fMRI time series is modeled as a linear regression model, where one regressor describes the task-related hemodynamic response, one regressor accounts for a constant baseline and one regressor describes potential drift. The temporal correlation structure of the noise is modeled as an autoregressive (AR) model. The order of the AR model is determined from practical null data sets using Akaike's information criterion (with penalty factor 3) as order selection criterion. The tests proposed are based on exact expressions for the likelihood function of the data. Using Monte Carlo simulation experiments, the performance of the proposed tests is evaluated in terms of detection rate and false alarm rate properties and compared to the current general linear model (GLM) test, which estimates the coloring of the noise in a separate step. Results show that theoretical asymptotic distributions of the GLM, GLR, and Wald test statistics cannot be reliably used for computing thresholds for activation detection from finite length time series. Furthermore, it is shown that, for a fixed false alarm rate, the detection rate of the proposed GLR test statistic is slightly, but (statistically) significantly improved compared to that of the common GLM-based tests. Finally, simulations results reveal that all tests considered show seriously inferior performance if the order of the AR model is not chosen sufficiently high to give an adequate description of the correlation structure of the noise, whereas the effects of (slightly) overmodeling are observed to be less harmful.

Evaluation of neurite outgrowth anisotropy using a novel application of circular analysis

Precise axon growth is required for making proper connections in development and after injury. One method of studying axon guidance and growth is through in vitro outgrowth assays that present controlled microenvironments. In this study, we applied circular statistical methods to evaluate directional neurite response. Visualization of data on a circular scale allows more accurate representation of the data, as neurite angles are inherently expressed on a circle. Here, the direction of neurite outgrowth from dorsal root ganglia derived neurons on different substrate types was quantitatively measured. Further, simulations of datasets with known circular parameters reflecting expected neurite angle distributions from different substrate types were also generated. Circular statistical methods were utilized and compared to linear statistical models widely used in the neuroscience literature. For small samples, Rao's spacing test showed the smallest occurrence of Type I errors (false positives) when tested against simulated uniform distributions. V-test and Rayleigh's test showed highest statistical power when tested against a unimodal distribution with known and unknown mean direction, respectively. For bimodal samples, Watson's U 2-test showed the highest statistical power. Overall, circular statistical uniformity tests showed higher statistical power than linear non-parametric tests, particularly for small samples ( n = 5). Circular analysis methods represent a useful tool for evaluation of directionality of neurite outgrowth with applications including: (1) assessment of neurite outgrowth potential; (2) determination of isotropy of cellular responses to single and multiple cues and (3) determination of the relative strengths of cues present in a complex environment.

Determination of digital watermarking optimal detection threshold based on asymptotically optimal detector

The magnitude of detection thresh will influence the effect of watermarking detector.The asymptotically optimal detector is a blind watermarking detection adopting Rao test,but its detection thresh is determined by experience.In order to make asymptotically optimal detection thresh more objective and accurate,the minimum error probability criterion was analyzed.The detection thresh based on asymptotically optimal detector was theoretically deduced to gain an expression of optimal detection thresh which made error rate of watermarking detection system lowest.Simulation results indicate that this method is adaptive and objective and can make error rate of watermarking detection system theoretically minimum.

A New Nonstationarity Detector

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A new test to determine the stationarity length of a locally wide sense stationary Gaussian random process is proposed. Based on the modeling of the process as a time-varying autoregressive process, the time-varying model parameters are tested using a Rao test. The use of a Rao test avoids the necessity of obtaining the maximum likelihood estimator of the model parameters under the alternative hypothesis, which is intractable. Computer simulation results are given to demonstrate its effectiveness and to verify the asymptotic theoretical performance of the test. Applications are to spectral analysis, noise estimation, and time series modeling. </para>

Coincidence of the Rao Test, Wald Test, and GLRT in Partially Homogeneous Environment

This letter deals with the problem of detecting a signal known up to a scaling factor in the presence of partially homogeneous Gaussian disturbance with unknown covariance matrix. It is proved that the Rao test and the Wald test coincide with the generalized likelihood ratio test (GLRT) previously derived. Otherwise stated, the Rao test, the Wald test, and the GLRT are all equivalent to the uniformly most powerful invariant (UMPI) detector.