Statistical Interpretation of a Data Adaptive Clutter Subspace Estimation Algorithm

Being equipped with a millimeter-wave radar allows a low-flying helicopter to sense the surroundings in real time, which significantly increases its safety. However, nonhomogeneous clutter environments, such as a multiple target situation and a clutter edge environment, can dramatically affect the radar signal detection performance. In order to improve the radar signal detection performance in nonhomogeneous clutter environments, this paper proposes a new automatic censored cell averaging CFAR detector. The proposed CFAR detector does not require any prior information about the background environment and uses the hypothesis test of the first-order difference (FOD) result of ordered data to reject the unwanted samples in the reference window. After censoring the unwanted ranked cells, the remaining samples are combined to form an estimate of the background power level, thus getting better radar signal detection performance. The simulation results show that the FOD-CFAR detector provides low loss CFAR performance in a homogeneous environment and also performs robustly in nonhomogeneous environments. Furthermore, the measured results of a low-flying helicopter validate the basic performance of the proposed method.

Space-time adaptive processing (STAP) holds tremendous potential for the new generation airborne surveillance radar, in which the phased array antennas and pulse Doppler processing mode are adopted. A new STAP approach using the multiple-beam and multiple Doppler channels is presented here for airborne phased array radar. The approach with space-time multiple-beam (STMB) architecture is robust to array errors and has very low system degrees of freedom (DOFs). Hence, it has low sample support requirement and it is very suitable for the practical planar phased array radar under nonhomogeneous clutter environments. Meanwhile, a new nonhomogeneous detector (NHD) based on the correlation dimension (CD) is also proposed here, which is used as an effective method to screen tracing data prior to detection processing. It can further improve the performance of the STAP approach in the severely nonhomogeneous clutter environments. Therefore, a scheme that incorporates the correlation dimension nonhomogeneity detector (CD-NHD) with the STMB is recommended, which we term CD-NHD-STMB. The experimental simulation results indicate that: 1) the STMB processor is robust to array element error and has high performance under nonhomogeneous clutter environments; 2) the CD-NHD is also effective on the nonhomogeneous clutter. As a result, the CD-NHD-STMB scheme is robust to array element error and nonhomogeneous clutter, and therefore available for airborne phased array radar applications.

The performance of space–time adaptive processing (STAP) may degrade significantly when some of the training samples are contaminated by the signal-like components (outliers) in nonhomogeneous clutter environments. To remove the training samples contaminated by outliers in nonhomogeneous clutter environments, a robust nonhomogeneous training samples detection method using the sparse-recovery (SR) with knowledge-aided (KA) is proposed. First, the reduced-dimension (RD) overcomplete spatial–temporal steering dictionary is designed with the prior knowledge of system parameters and the possible target region. Then, the clutter covariance matrix (CCM) of cell under test is efficiently estimated using a modified focal underdetermined system solver (FOCUSS) algorithm, where a RD overcomplete spatial–temporal steering dictionary is applied. Third, the proposed statistics are formed by combining the estimated CCM with the generalized inner products (GIP) method, and the contaminated training samples can be detected and removed. Finally, several simulation results validate the effectiveness of the proposed KA-SR-GIP method.

The EM algorithm is a widely used technique for finding maximum likelihood (ML) estimates when the data are not fully observed. Despite its popularity for computing ML estimates in unrestricted problems and the need for simplified computations for problems with equality and inequality restrictions, there have been few applications of the algorithm to restricted ML estimation. The EM algorithm is presented for restricted ML estimation and provides its applications to the probit model under equality and inequality restrictions using two small data sets.

Finite mixtures of multivariate skew t (MST) distributions have proven to be useful in modelling heterogeneous data with asymmetric and heavy tail behaviour. Recently, they have been exploited as an effective tool for modelling flow cytometric data. A number of algorithms for the computation of the maximum likelihood (ML) estimates for the model parameters of mixtures of MST distributions have been put forward in recent years. These implementations use various characterizations of the MST distribution, which are similar but not identical. While exact implementation of the expectation-maximization (EM) algorithm can be achieved for `restricted' characterizations of the component skew t-distributions, Monte Carlo (MC) methods have been used to fit the `unrestricted' models. In this paper, we review several recent fitting algorithms for finite mixtures of multivariate skew t-distributions, at the same time clarifying some of the connections between the various existing proposals. In particular, recent results have shown that the EM algorithm can be implemented exactly for faster computation of ML estimates for mixtures with unrestricted MST components. The gain in computational time is effected by noting that the semi-infinite integrals on the E-step of the EM algorithm can be put in the form of moments of the truncated multivariate non-central t-distribution, similar to the restricted case, which subsequently can be expressed in terms of the non-truncated form of the central t-distribution function for which fast algorithms are available. We present comparisons to illustrate the relative performance of the restricted and unrestricted models, and demonstrate the usefulness of the recently proposed methodology for the unrestricted MST mixture, by some applications to three real datasets.

Most of this book concerns the interpretation of regression models, with the understanding that they can be fit to data fairly automatically using R and Bugs. However, it can be useful to understand some of the theory behind the model fitting, partly to connect to the usual presentation of these models in statistics and econometrics. This chapter outlines some of the basic ideas of likelihood and Bayesian inference and computation, focusing on their application to multilevel regression. One point of this material is to connect multilevel modeling to classical regression; another is to give enough insight into the computation to allow you to understand some of the practical computational tips presented in the next chapter. Least squares and maximum likelihood estimation We first present the algebra for classical regression inference, which is then generalized when moving to multilevel modeling. We present the formulas here without derivation; see the references listed at the end of the chapter for more. Least squares The classical linear regression model is y i = X i β + ∊ i , where y and ∊ are (column) vectors of length n , X is a n × k matrix, and β is a vector of length k . The vector β of coefficients is estimated so as to minimize the errors ∊ i .

Most of the current QTL mapping methods are developed for the backcross and F2 populations. The inter-crossed F3 population is also popular and has been well devised in genetic study. As the F3 population can provide more recombinants than the backcross and F2 populations, it can facilitate the detection of closely linked QTL. In this paper, we propose a statistical model to estimate the effects and positions of QTL for the F3 population by considering the differences in the genome structure between the F3 and F2 populations. As the genotypes of QTL are usually unknown, the statistical model is a finite mixture model. By treating the mixture model as an incomplete data problem, the EM algorithm is implemented to obtain the maximum likelihood estimates of the parameters. Simulations were used to illustrate the performance of the proposed method and evaluate the relative efficiency in the F3 population (as comparing to that of the F2 population) in QTL mapping under several factors, such as sample size, genetic distance between QTL (and markers), the proportion of trait variance contributed by QTL (i.e. heritability). It is found that the QTL mapping by using the F3 population is more powerful and precise in separating closely linked QTL and in estimating the parameters as compare to the use of the F2 population, especially for small sample size, close genetic distance and low heritability. The paper can help outline a strategy of detecting tightly linked QTL for the advanced intercross populations to improve the resolution of genetic architecture of quantitative traits.

Many diseases show dichotomous phenotypic variation but do not follow a simple Mendelian pattern of inheritance. Variances of these binary diseases are presumably controlled by multiple loci and environmental variants. A least-squares method has been developed for mapping such complex disease loci by treating the binary phenotypes (0 and 1) as if they were continuous. However, the least-squares method is not recommended because of its ad hoc nature. Maximum Likelihood (ML) and Bayesian methods have also been developed for binary disease mapping by incorporating the discrete nature of the phenotypic distribution. In the ML analysis, the likelihood function is usually maximized using some complicated maximization algorithms (e.g. the Newton-Raphson or the simplex algorithm). Under the threshold model of binary disease, we develop an Expectation Maximization (EM) algorithm to solve for the maximum likelihood estimates (MLEs). The new EM algorithm is developed by treating both the unobserved genotype and the disease liability as missing values. As a result, the EM iteration equations have the same form as the normal equation system in linear regression. The EM algorithm is further modified to take into account sexual dimorphism in the linkage maps. Applying the EM-implemented ML method to a four-way-cross mouse family, we detected two regions on the fourth chromosome that have evidence of QTLs controlling the segregation of fibrosarcoma, a form of connective tissue cancer. The two QTLs explain 50-60% of the variance in the disease liability. We also applied a Bayesian method previously developed (modified to take into account sex-specific maps) to this data set and detected one additional QTL on chromosome 13 that explains another 26% of the variance of the disease liability. All the QTLs detected primarily show dominance effects.

We investigate an alternative model for signals encountered in acoustic environments to the traditional Gaussian process. The sound signals in this case are assumed to be sub-Gaussian of impulsive nature. The noise encountered in these environments predominantly stems from reverberation or multipath effects, which makes it significantly dependent on the source. Hence, the noise is also modeled as jointly sub-Gaussian. The Levy alpha-stable distribution, of characteristic exponent 0.5 and index of symmetry 1, is used together with a multivariate Gaussian density to derive the sub-Gaussian process. Based on this density, the stochastic Maximum Likelihood (ML) estimator is formulated. A separable solution of the estimator is given. Subsequently, simulations demonstrating the performance gains relative to the Gaussian-based ML estimator are provided, as well as a comparison of the two methods on localization of real sounds gathered with a 20- and 41-microphone arrays.

Distributed estimation is examined for energy-harvesting wireless sensor networks, where the energy available at the sensors are converted entirely from ambient sources. Here, each sensor takes a local measurement of the common parameter of interest and, by adopting an analog-forwarding transmission strategy, forwards a scaled version of it to the fusion center. A sensor transmits in a given time slot only if its available energy is sufficient to support the transmission of its scaled local observation. The scaling factor is properly chosen to exploit the tradeoff between the probability that a sensor transmits and the quality of the received signal at the fusion center. However, even when a sensor does not transmit, the fact that it remains silent still contains information regarding the magnitude of the observation. The maximum-likelihood estimator (MLE) is adopted at the fusion center to exploit this knowledge and is derived based on the statistics of the energy arrivals. Two cases are considered: the case with no batteries at the sensors and the case with finite-capacity batteries at the sensors. Since the sensors' available energy is unknown at the fusion center, the computation of the MLE will require averaging over the energy arrival at all sensors and in all time slots, which can be intractable in the case with batteries. A particle-filtering based expectation-maximization (EM) algorithm is proposed to address this issue. The effectiveness of our proposed schemes is demonstrated through Monte Carlo simulations.

Computation: Expectation-Maximization Algorithm

In this paper, a computationally efficient algorithm is presented for joint maximum likelihood (ML) timing and carrier phase estimation of OFDM systems employing M-PSK modulation scheme with additive Gaussian noise, based on the expectation-maximization (EM) algorithm. A nondata-aided (NDA) scheme is considered for the joint timing and phase synchronizer which maximizes the low SNR limit of the likelihood function averaged over the M-PSK signal constellation. For this, an EM algorithm is derived which estimates the timing offset and the phase rotations of each subcarrier iteratively and which converges to the true ML estimation of the unknown timing and phase. It is shown that the algorithm becomes independent of the signal-to-noise ratio for both low and high SNR cases. The algorithm is applied to the QPSK modulated OFDM systems and it is concluded that for SNR values greater than 10 dB the convergence is achieved in first iteration and for SNR values less than 10 dB, at most in three iterations. It is also concluded that the convergence is independent of the initial starting points.

In this paper, we consider moving target detection using a distributed multiple-input multiple-output (MIMO) radar on stationary platforms in nonhomogeneous clutter environments. Our study is motivated by the fact that the multistatic transmit-receive configuration in a distributed MIMO radar causes nonstationary clutter. Specifically, the clutter power for the same test cell may vary significantly from one transmit-receive pair to another, due to azimuth-selective backscattering of the clutter. To account for these issues, a new nonhomogeneous clutter model, where the clutter resides in a low-rank subspace with different subspace coefficients (and hence different clutter power) for different transmit-receive pair, is introduced and the relation to a general clutter model is discussed. Following the proposed clutter model, we develop a generalized-likelihood ratio test (GLRT) for moving target detection in distributed MIMO radar. The GLRT is shown to be a constant false alarm rate (CFAR) detector, and the test statistic is a central and noncentral Beta variable under the null and alternative hypotheses, respectively. Simulations are provided to demonstrate the performance of the proposed GLRT in comparison with several existing techniques.

In order to deal with the problem space-time adaptive processing (STAP) performance degradation of an airborne phased array system caused by the serious shortage of independent and identical distributed (IID) training samples in the nonhomogeneous clutter environment, an improved direct data domain method based on sparse Bayesian learning is proposed in this paper, which only uses a single snapshot data of a cell under test (CUT) to suppress the clutter and has fast computational speed. Firstly, three hyper-parameters required to obtain the sparse solution are derived. Secondly, the comparative analysis of their iterative formulas is made, and the piecewise iteration of hyper-parameter that has an obvious influence on the computational complexity of obtaining sparse solution is presented. Lastly, with the approximate prior information of the target, the clutter sparse solution is given and its covariance matrix is effectively estimated to calculate the adaptive filter weight and realize the clutter suppression. Simulation results verify that the proposal can dramatically decrease the computational burden while keeping the superior heterogeneous clutter suppression performance.

This study describes the basic framework of EM algorithm and gives how to apply EM algorithm to solve the problem of maximum-likelihood parameters estimation combining with the models of HMM and PCFG. In the process of statistics natural language, one kind of problem is often encountered that is how to solve the parameter's maximum-likelihood estimation when observation data is incomplete. EM algorithm is the classical method to solve this problem. Finally, the advantages and disadvantages of EM algorithm are discussed.