Uniformly Most Powerful Invariant Research Articles

Conditionally optimal classification based on CFAR and invariance property for blind receivers

This paper proposes a new approach for finding the conditionally optimal solution (the classifier with minimum error probability) for the classification problem where the observations are from the multivariate normal distribution. The optimal Bayes classifier does not exist when the covariance matrix is unknown for this problem. However, this paper proposes a classifier based on the constant false alarm rate (CFAR) and invariance property. The proposed classifier is optimal conditionally as it has the minimum error probability in a subset of solutions. This approach has an analogy to hypothesis testing problems where uniformly most powerful invariant (UMPI) and uniformly most powerful unbiased (UMPU) detectors are used instead of the non-existing optimal UMP detector. Furthermore, this paper investigates using the proposed classifier for modulation classification as an application in signal processing.

Influence of the simplified stochastic model of TLS measurements on geometry-based deformation analysis

Abstract Terrestrial laser scanners (TLS) are powerful instruments that can be employed for deformation monitoring due to their high precision and spatial resolution in capturing 3D point clouds. Deformation detections from scatter point clouds can be based on different comparison methods, among which the geometry-based method is one of the most popular. Compared with approximating surfaces with predetermined geometric primitives, such as plane or sphere, the B-splines surface approximation offers a great flexibility and can be used to fit nearly every object scanned with TLS. However, a variance-covariance matrix (VCM) of the observations involved in approximating the scattered points to B-spline surfaces impact the results of a congruency test, which is the uniformly most powerful invariant (UMPI) test for discriminating between the null hypothesis of zero deformation and its alternative hypotheses. Consequently, simplified stochastic models may weaken the UMPI property. Based on Monte Carlo simulations, the impact of the heteroscedasticity and mathematical correlations often neglected in B-splines approximation are investigated. These correlations are specific in approximating TLS measurements when the raw measurements are transformed into Cartesian coordinates. The rates of rejecting the null hypothesis in a congruency test is employed to reflect the impact of unspecified VCMs on the power of the congruency test. The rejection rates are not sensitive to the simplification of the stochastic models, in the larger deformation area with higher point accuracy, while they are obviously influenced in the smaller deformation area with unfavourable geometries, i. e. larger uncertainties. A threshold ratio of estimated differences to the relative standard deviation highlights whereas the results of congruency test are reliable when using simplified VCMs. It is concluded that the simplification of the stochastic model has a significant impact on the power of the congruency test, especially in the smaller deformation area with larger uncertainties.

On the statistical power of Baarda’s outlier test and some alternative

Abstract Baarda’s outlier test is one of the best established theories in geodetic practice. The optimal test statistic of the local model test for a single outlier is known as the normalized residual. Also other model disturbances can be detected and identified with this test. It enjoys the property of being a uniformly most powerful invariant (UMPI) test, but is not a uniformly most powerful (UMP) test. In this contribution we will prove that in the class of test statistics following a common central or non-central χ2 distribution, Baarda’s solution is also uniformly most powerful, i.e. UMPχ2 for short. It turns out that UMPχ2 is identical to UMPI, such that this proof can be seen as another proof of the UMPI property of the test. We demonstrate by an example that by means of the Monte Carlo method it is even possible to construct test statistics, which are regionally more powerful than Baarda’s solution. They follow a so-called generalized χ2 distribution. Due to high computational costs we do not yet propose this as a ”new outlier detection method”, but only as a proof that it is in principle possible to outperform the statistical power of Baarda’s test.

Integration gain limitations in passive coherent location radars for manoeuvring target detection

The lack of transmission antenna gain in passive coherent location (PCL) systems can be circumvented somewhat by the fact that most broadcast waveforms are continuous waveforms and that the lengthening of the integration time can help to increase signal energy. However, with the increase of integration time, a PCL system encounters some limitations in detecting manoeuvring targets. To investigate this, the uniformly most powerful invariant (UMPI) detector presented by Zaimbashi et al. is considered. The detection performance of the UMPI detector in a closed-form expression for a manoeuvring target is obtained. The theoretical findings accompanied by simulation results demonstrate that a manoeuvring target can seriously degrade the detection performance of the UMPI detector when using long integration times.

Invariant subspace detector in distributed multiple‐input multiple output radar: geometry gain helps improving moving target detection

To detect moving targets in a distributed multiple-input multiple output (MIMO) radars, it is preferable to null the clutter rather than whiten the clutter. This is mainly because clutter nulling approaches do not require range training data to form a sample covariance estimate of clutter in each transmit-receive pair, especially in the distributed MIMO radars with non-homogeneous clutter. Moreover, geometry diversity of the distributed MIMO helps improve moving target detection since for a given target velocity, different transmit-receive pairs produce different Doppler frequencies that are less likely to be all small and reside in the clutter nulling region. Based on these facts, the authors consider a subspace model for clutter and derive a uniformly most powerful invariant (UMPI) detector as an optimum invariant test. In this case, analytical expressions for calculating the false alarm and detection probabilities for Swerling 0 and Swerling 1 target models are derived in the closed-forms. Moreover, theoretical and numerical analyses of the proposed UMPI test are represented for several scenarios. More importantly, the simulation results show that the proposed subspace-based UMPI detector can attain the predetermined false alarm probability and superior detection performance in the presence of correlated clutter. Finally, authors consider a situation in which perfect waveform separations at the local receivers are no longer valid, and see the detection performance degradation of the proposed optimal detector designed for ideal waveform separations.

Is the locally best invariant test uniformly most powerful for a wider class of invariant tests?

In the context of a general regression model in which some regression coefficients are of interest and others are purely nuisance parameters, Bhowmik and King (5) constructed the locally best invariant (LBI) test against one-sided alternatives. This paper investigates whether this LBI test is uniformly most powerful invariant (UMPI) or not through simulation results. A test that is locally best invariant against one-sided alternative hypotheses is found to be uniformly most powerful invariant (UMPI) in a wider class of tests than the invariant tests for the standard F test. The results of a simulation study conducted to prove that the LBI test is UMPI are presented. Through simulation results this study proves that the LBI test is UMPI.

Point optimal testing: A survey of the post 1987 literature

In the absence of uniformly most powerful (UMP) tests or uniformly most powerful invariant (UMPI) tests, King [80] suggested the use of Point Optimal (PO) tests, which are most powerful at a chosen point under the alternative hypothesis. This paper surveys the literature and major developments on point optimal testing since 1987 and suggests some areas for future research. Topics include tests for which all nuisance parameters have been eliminated and dealing with nuisance parameters via (i) a weighted average of p values, (ii) approximate point optimal tests, (iii) plugging in estimated parameter values, (iv) using asymptotics and (v) integration. Progress on using point-optimal testing principles for two-sided testing and multi-dimensional alternatives is also reviewed. The paper concludes with thoughts on how best to deal with nuisance parameters under both the null and alternative hypotheses, as well as the development of a new class of point optimal tests for multi-dimensional testing.

Optimal invariant detection of a monochromatic plane wave with unknown amplitude, frequency, phase, direction of arrival and noise variance

This study deals with the problem of detecting a monochromatic plane wave with unknown amplitude, phase, temporal frequency and direction of arrival in complex white Gaussian noise with unknown variance. Depending on the natural invariance in scale, temporal modulation and spatial modulation of the problem, the uniformly most powerful invariant (UMPI) test is derived by using the statistical invariance principles. However, the UMPI test does not apply unless the signal-to-noise ratio (SNR) is known. However, it provides us with performance bound to evaluate any invariant test's performance when the SNR is unknown. Typically, the generalised likelihood ratio test (GLRT) and locally most powerful invariant (LMPI) test are derived as realisable suboptimal invariant tests with their performance comparison in different SNR through theoretical analysis. Computer simulation examples corroborate the authors analysis and indicate that the GLRT is close to the UMPI bound especially in the low probability-of-false-alarm (P FA) region of the receiver operating characteristic curve while the performance of the LMPI test is close to that of the UMPI test in the low SNR region.

Invariant Target Detection in Multiband FM-Based Passive Bistatic Radar

FM-based passive bistatic radars (PBRs) exploiting a single broadcasted channel suffer from time-varying detection performance due to the time-varying program content of the transmitted signal as well as the propagation condition of that channel. To circumvent this matter, it is a viable idea to exploit multiple broadcasted channels from a single transmitter for detection. In this case, we formulate the problem of detecting a target in the presence of interference signals such as receiver noise, direct signal, multipath/clutter echoes and interfering targets as a composite hypothesis test. To address this, we derive a multiband uniformly most powerful invariant (UMPI) test as an optimal invariant detector. Closed-form expressions for calculating false alarm probability and detection probability for the target models of Swerling 0 and Swerling 1 are derived. A thorough performance assessment is also given, and the results show the robustness of the proposed multiband detector against the time-varying program content of FM radio channels. In addition to the reliable detection performance, a multiband PBR system offers advantages in terms of combined diversity gain and frequency diversity gain as compared with a single band PBR system. The former happens due to the joint exploitation of multiple broadcasted channels for detection, while the latter occurs when independent returns are received from one target over different frequency channels.

Generalized CFAR Property and UMP Invariance for Adaptive Signal Detection

In this paper we consider adaptive detection of a signal embedded in additive disturbance whose multivariate distribution belongs to a very general class, including many statistical models commonly adopted for radar disturbance. We introduce the concept of generalized Constant False Alarm Rate (CFAR) and show that a class of receivers sharing some invariances complies with the quoted property. Then, we devise the Generalized Likelihood Ratio Test (GLRT) and prove that, under some mild technical conditions, it coincides with that obtained under the Gaussian assumption for the observations. We also deal with the existence of the Uniformly Most Powerful Invariant (UMPI) detector either using the Wijsman theorem or directly computing the maximal invariant Likelihood Ratio (LR). At the analysis stage, we focus on a compound matrix variate model for the disturbance component, which is a natural generalization of the Spherically Invariant Random Vector (SIRV). In this context, we assess the performance of some well known invariant decision rules also in comparison with the Most Powerful Invariant (MPI) detector. The results highlight that some among the analyzed receivers exhibit a performance level very close to the MPI test.

Constructing Locally Best Invariant Tests of the Linear Regression Model Using the Density Function of a Maximal Invariant

In the context of the linear regression model in which som e regression coefficients are of interest and others are purely nuisance parameters, we derive the density function of a maximal invariant statistic after eliminating the nuisance parameters by the principle of invariance argu ment. This allows the construction of a range of optimal test statistics including the locally best invariant (LBI) test wh ich is equivalent to the well-known one-sided t-test. The resultant LBI test is also found to be uniformly most powerfu l invariant (UMPI).

Invariant detection of orthogonal frequency division multiplexing signals with unknown parameters for cognitive radio applications

The authors propose a computationally efficient spectrum sensing solution for an orthogonal frequency division multiplexing (OFDM) signal in a frequency-selective fading channel with additive white Gaussian noise (AWGN). The authors’ assumption is that the data symbols, channel coefficients and the noise variance are all unknown. The nature of the problem leads them to find an invariant detector. The optimum one is uniformly most powerful invariant (UMPI); their effort shows that this test does not exist, as the final decision statistic depends on some unknown parameters; although, they derive the most powerful invariant (MPI) detector, implanting these parameters, to provide an upper bound for invariant detectors. Instead, the authors develop the generalised likelihood ratio test (GLRT), substituting the unknown parameters by their maximum likelihood (ML) estimates in the Neyman–Pearson likelihood ratio. Furthermore, they propose a computationally efficient implementation of the resulting detector. Simulation results show a slight decrease in efficiency against gaining so much computational complexity (CC) improvement. For more comparisons they also derive the conventional energy detector, namely, generalised energy detector (GED) for a binary phase shift keying (BPSK) signal, to show the adequacy of the GLR detector performance which would be demonstrated by simulation results.

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.

Uniformly most powerful invariant detection in spherically invariant random vector distributed clutter

This study deals with the problem of detecting a radar target with unknown complex amplitude in the presence of clutter modelled as a spherically invariant random vector (SIRV). It is proved that, under a quite mild technical condition, the normalised matched filter (NMF) is the uniformly most powerful invariant (UMPI) detector in SIRV disturbance. The developed proof is based on the theory of total positive functions of order two.

Invariant detection for short-code QPSK DS-SS signals

This paper focuses on the optimal detection of quadrature phase-shift keying (QPSK) direct sequence spread spectrum (DS-SS) signals in additive white Gaussian noise (AWGN) with unknown parameters. We consider the invariant detection problem using the complex Gaussian mixture and the modulo-shift signal model, and derive constant-false-alarm-rate (CFAR) invariant detectors such as uniformly most-powerful invariant (UMPI) test and other sub-optimal invariant tests such as the generalized likelihood ratio test (GLRT). An approximation of the UMPI test under low signal-noise-ratio scenarios yields an asymptotic locally most-powerful invariant (ALMPI) test, which has similar computational complexity as the cycle feature detectors. The ALMPI test is computationally efficient compared with the UMPI and GLRT tests. Moreover, further approximation to the ALMPI test leads to the incoherent weighted multi-cycle detectors, which serve as the performance upper bound of all the spreading sequence period detectors based on second-order cyclostationary statistics. Simulation results demonstrate that the proposed ALMPI test exhibits better detection performance than cycle feature detectors with finite observation samples.

Invariant activity detection of a constant magnitude signal with unknown parameters in white Gaussian noise

The authors propose invariant tests for the detection of a complex signal with unknown constant amplitude and unknown phase variation in additive white Gaussian noise (AWGN). The authors show that in this problem, the uniformly most powerful invariant (UMPI) detector does exist only if the number of samples N is two. For more than two samples N≥3, the authors derive the most powerful invariant (MPI) detector in known signal-to-noise ratio (SNR) and use its performance as the upper bound benchmark for any invariant test. In addition, the authors derive the generalised likelihood ratio (GLR) detector and evaluate its performance against the MPI performance bound. This detector is very simple and represents the ratio of the L1-norm to the L2-norm of the data. Simulation results illustrate the close performances of the two detectors even at low SNRs, whereas in contrast to the MPI test the SNR is not required in the proposed GLR test. In order to understand why the knowledge of SNR is not so important in this detection problem, the authors also derive the GLR test for the case of known SNR. Interestingly, the resulting GLR detector (derived for the case of known SNR) turns out to be equivalent to the one derived for unknown SNR, i.e. a knowledge of the SNR is not used in any of the GLR tests. This reveals why the knowledge of the SNR is not so useful in this detection problem.

Efficient two-sided nonsimilar invariant tests in IV regression with weak instruments

Abstract As Nelson and Startz [Nelson, C.R., Startz, R., 1990a. The distribution of the instrumental variable estimator and its t ratio when the instrument is a poor one. Journal of Business 63, S125–S140; Nelson, C.R., Startz, R., 1990b. Some further results on the exact small sample properties of the instrumental variables estimator. Econometrica 58, 967–976] dramatically demonstrated, standard hypothesis tests and confidence intervals in instrumental variables regression are invalid when instruments are weak. Recent work on hypothesis tests for the coefficient on a single included endogenous regressor when instruments may be weak has focused on similar tests. This paper extends that work to nonsimilar tests, of which similar tests are a subset. The power envelope for two-sided invariant (to rotations of the instruments) nonsimilar tests is characterized theoretically, then evaluated numerically for five IVs. The power envelopes for similar and nonsimilar tests differ theoretically, but are found to be very close numerically. The nonsimilar test power envelope is effectively achieved by the Moreira [Moreira, M.J., 2003. A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048] conditional likelihood ratio test, so that test is effectively uniformly most powerful invariant (UMPI). We also provide a new nonsimilar test, P ∗ , which has χ 1 2 critical values, is asymptotically efficient under strong instruments, involves only elementary functions, and is very nearly UMPI.

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.

A Fast Multiple-Source Detection and Localization Array Signal Processing Algorithm Using the Spatial Filtering and ML Approach

We propose a computationally efficient algorithm for detection of multiple signals that also gives a rough estimation of their direction of arrivals (DOAs). The narrowband received signals from a uniform linear array are first filtered by a set of orthogonal filters, e.g., by a fast Fourier transformation, in order to separate the sources into multiple spatial intervals. This transformation converts the complicated multihypothesis problem of source detection and localization into multiple binary hypothesis testing problems. For an additive white Gaussian noise (AWGN) environment, the maximum-likelihood (ML) solution of these interrelated tests requires substantially less computational complexity than that of the multihypothesis problem. For each spatial interval, a binary test detects the presence of a single source and thus gives a rough localization. We employ generalized-likelihood ratio (GLR) tests as the detection criterion, assuming that the number of sources, their power, and the noise variance are all unknown. We also show that the optimal uniformly most powerful invariant (UMPI) detector does not exist. However, we derive a UMPI detector that uses some extra information and as a result provides an upper bound performance for evaluation of any invariant detector. Simulations illustrate that the proposed noniterative GLR test performs efficiently for various number of observed data snapshots and signal-to-noise-ratios (SNR)s, and its performance is comparable to the upper bound performance. We used the proposed algorithm for the initialization of the iterative implementation of the standard ML localization. This combination is a high-resolution localization algorithm with a low computational complexity

Optimality in detecting targets with unknown location

An optimal test does not exist for the problem of detecting a known target with unknown location in additive Gaussian noise. A common solution uses a generalised likelihood ratio testing (GLRT) formalism, where a maximum likelihood estimate of the unknown location parameter is used in a likelihood ratio test. The performance of this test is commonly assessed by comparing it to the ideal matched filter, which assumes the target location known in advance. This comparison is of limited utility, however, since the fact that the location is unknown has a significant effect on the detectability of the target. We demonstrate that a uniformly most powerful invariant (UMPI) optimal test exists for a specific class of unknown target location problems, where observations are discrete and shifts are defined circularly. Since this approach explicitly models the location as unknown, an assessment of the suboptimality of competing tests becomes meaningful. It is shown that for certain examples in this class the GLRT performance is negligibly different from that of the optimal test.