Multivariate Analysis - A Great Tool for a Multivariate World

In this short report the author tries to introduce in simple terms some Multivariate Analysis techniques and show their usefulness in the analysis of econometric data, namely in studies which aim at studying and comparing different regions inside a given country.publishersversionpublishe

The advantage of decomposing elaborate hypotheses on covariance matrices into conditionally independent hypotheses in building near-exact distributions for the test statistics

AbstractThe aim of this paper is to show how the decomposition of elaborate hypotheses on the structure of covariance matrices into conditionally independent simpler hypotheses, by inducing the factorization of the overall test statistic into a product of several independent simpler test statistics, may be used to obtain near-exact distributions for the overall test statistics, even in situations where asymptotic distributions are not available in the literature and adequately fit ones are not easy to obtain

Near-Exact Distributions – Needing Them and Building Them

Near-exact distributions are asymptotic distributions built using a different concept and technique in what concerns the approximation of the distribution of given statistics whose exact distribution has a complex structure and expression. In the present paper the author introduces, in simple terms and through two examples, the near-exact distributions as an alternative to common asymptotic distributions. First are introduced the characteristics and the general building process of these distributions. Then, through a couple of examples, a very simple first one and a more elaborate second one, we try to illustrate how in practice these distributions may be developed and to show their very good performance.As distribuições quase-exactas são distribuições assimptóticas construídas sobre uma abordagem diferente no que diz respeito ao princípio e à técnica da aproximação da distribuição de estatísticas cuja distribuição exacta tem uma estrutura e expressão complexas. No presente artigo o autor apresenta, em termos simples e através de dois exemplos, as distribuições quase-exactas como alternativa vantajosa às usuais distribuições assimptóticas. Em primeiro lugar são apresentadas as características e forma de construção destas distribuições e daí intuídas as suas vantagens em relação às usuais distribuições assimptóticas. Depois, através de dois exemplos, um primeiro muito simples e um segundo mais elaborado, tenta-se ilustrar como se constroem na prática estas distribuições e mostrar o seu excelente desempenho

Low order channel estimation for CDMA systems

New approaches and algorithms are developed for the identification and estimation of low order models that represent multipath channel effects in Code Division Multiple Access (CDMA) communication systems. Based on these parsimonious channel models, low complexity receivers such as RAKE receivers are considered to exploit these propagation effects and enhance the system performance. We consider the scenario where multipath is frequency selective slowly fading and where the channel components including delays and attenuation coefficients are assumed to be constant over one or few signalling intervals. We model the channel as a long FIR-like filter (or a tapped delay line filter) with the number of taps related to the ratio between the channel delay-spread and the chip duration. Due to the high data rate of new CDMA systems, the channel length in terms of the chip duration will be very large. With classical channel estimation techniques this will result in poor estimates of many of the channel parameters where most of them are zero leading to a reduction in the system performance. Unlike classical techniques which estimate directly the channel response given the number of taps or given an estimate of the channel length, the proposed techniques in this work will firstly identify the significant multipath parameters using model selection techniques, then estimate these identified parameters. Statistical tests are proposed to determine whether or not each individual parameter is significant. A low complexity RAKE receiver is then considered based on estimates of these identified parameters only. The level of significance with which we will make this assertion will be controlled based on statistical tests such as multiple hypothesis tests. Frequency and time domain based approaches and model selection techniques are proposed to achieve the above proposed objectives.The frequency domain approach for parsimonious channel estimation results in an efficient implementation of RAKE receivers in DS-CDMA systems. In this approach, we consider a training based strategy and estimate the channel delays and attenuation using the averaged periodogram and modified time delay estimation techniques. We then use model selection techniques such as the sphericity test and multiple hypotheses tests based on F-Statistics to identify the model order and select the significant channel paths. Simulations show that for a pre-defined level of significance, the proposed technique correctly identifies the significant channel parameters and the parsimonious RAKE receiver shows improved statistical as well as computational performance over classical methods. The time domain approach is based on the Bootstrap which is appropriate for the case when the distribution of the test statistics required by the multiple hypothesis tests is unknown. In this approach we also use short training data and model the channel response as an FIR filter with unknown length. Model parameters are then estimated using low complexity algorithms in the time domain. Based on these estimates, bootstrap based multiple hypotheses tests are applied to identify the non-zero coefficients of the FIR filter. Simulation results demonstrate the power of this technique for RAKE receivers in unknown noise environments. Finally we propose adaptive blind channel estimation algorithms for CDMA systems. Using only the spreading code of the user of interest and the received data sequence, four different adaptive blind estimation algorithms are proposed to estimate the impulse response of frequency selective and frequency non-selective fading channels. Also the idea is based on minimum variance receiver techniques. Tracking of a frequency selective varying fading channel is also considered.A blind based hierarchical MDL model selection method is also proposed to select non-zero parameters of the channel response. Simulation results show that the proposed algorithms perform better than previously proposed algorithms. They have lower complexity and have a faster convergence rate. The proposed algorithms can also be applied to the design of adaptive blind channel estimation based RAKE receivers

Passive detection of correlated subspace signals in two MIMO channels

In this paper, we consider a two-channel multiple-input multiple-output passive detection problem, in which there is a surveillance array and a reference array. The reference array is known to carry a linear combination of broadband noise and a subspace signal of known dimension, but unknown basis. The question is whether the surveillance channel carries a linear combination of broadband noise and a subspace signal of the same dimension, but unknown basis, which is correlated with the subspace signal in the reference channel. We consider a second-order detection problem where these subspace signals are structured by an unknown, but common, p-dimensional random vector of symbols transmitted from sources of opportunity, and then received through unknown M × p matrices at each of the M-element arrays. The noises in each channel have spatial correlation models ranging from arbitrarily correlated to independent with identical variances. We provide a unified framework to derive the generalized likelihood ratio test for these different noise models. In the most general case of arbitrary noise covariance matrices, the test statistic is a monotone function of canonical correlations between the reference and surveillance channels.I. Santamaría and J. Vía have received funding from Ministerio de Economía y Competitividad (MINECO) of Spain, and AEI/FEDER funds of the E.U. under projects TEC2013-47141-C4-3-R (RACHEL), TEC2016-75067-C4-4-R (CARMEN) and TEC2016-81900-REDT (KERMES). The research of Haonan Wang was partially supported by NSF grant DMS-1521746

Analysis Techniques for Diffusion Tensor Imaging Data

A recent protocol innovation with magnetic resonance imaging (MRI) has resulted in diffusion tensor imaging (DTI). The approach holds tremendous promise for improving our understanding of neural pathways, especially in the brain. MRIs work by recording displacements at a molecular level. The DTI protocol highlights the distribution of water molecules (in three dimensions). In a medium with free water motion, the diffusion of water molecules is expected to be isotropic, the same in all directions. With water embedded innonhomogeneous tissue, motion is expected to be anisotropic, not the same in all directions, and might show preferred directions of mobility. DTI fully characterizes diffusion anisotropy locally in space, thus providing rich detail about tissue microstructure. However, little has been done to define metrics or describe credible statistical methods for analyzing DTI data. This dissertation will show that the Geisser-Greenhouse sphericity estimator can be approximated by a squared beta distribution. Noise will also be added to show these fits also work for simulated diffusion tensors. Diagnostics are extremely important prior to analyzing these data. There are various regions, especially in the brain, where the distribution of the fractional anisotropy values could be bimodal. This is most likely due to partial voluming affects in imaging, where a voxel (volume of space) may incorporate more than the region of interest. However, the bimodal distribution can also be the result of picking up both white and grey matter in the region. If checks are not done prior to the analysis, all the results may be incorrect, since the main assumption (approximate F) would not be valid. By using diagnostic approaches like QQ-envelop and SiZer, one can examine whether the approximations are reasonable. If appropriate, the methodology previously discussed can be used. However, if these approximations do not apply, new methods will be necessary to analyze the data. Different methods for analyzing the data will be considered, these methods will include: finding an approximate bimodal distribution and the DiProPerm (Direction PROjection PERMutation) test

Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets

Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components. For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper. We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles. Quaternions are isomorphic to an algebra of structured 4x4 real matrices. This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms. A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance. Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs. Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces

Spectrum Sensing Algorithms for Cognitive Radio Applications

Future wireless communications systems are expected to be extremely dynamic, smart and capable to interact with the surrounding radio environment. To implement such advanced devices, cognitive radio (CR) is a promising paradigm, focusing on strategies for acquiring information and learning. The first task of a cognitive systems is spectrum sensing, that has been mainly studied in the context of opportunistic spectrum access, in which cognitive nodes must implement signal detection techniques to identify unused bands for transmission. In the present work, we study different spectrum sensing algorithms, focusing on their statistical description and evaluation of the detection performance. Moving from traditional sensing approaches we consider the presence of practical impairments, and analyze algorithm design. Far from the ambition of cover the broad spectrum of spectrum sensing, we aim at providing contributions to the main classes of sensing techniques. In particular, in the context of energy detection we studied the practical design of the test, considering the case in which the noise power is estimated at the receiver. This analysis allows to deepen the phenomenon of the SNR wall, providing the conditions for its existence and showing that presence of the SNR wall is determined by the accuracy of the noise power estimation process. In the context of the eigenvalue based detectors, that can be adopted by multiple sensors systems, we studied the practical situation in presence of unbalances in the noise power at the receivers. Then, we shift the focus from single band detectors to wideband sensing, proposing a new approach based on information theoretic criteria. This technique is blind and, requiring no threshold setting, can be adopted even if the statistical distribution of the observed data in not known exactly. In the last part of the thesis we analyze some simple cooperative localization techniques based on weighted centroid strategies

Examining the performance of trend surface models for inference on Functional Magnetic Resonance Imaging (fMRI) data

The current predominant approach to neuroimaging data analysis is to use voxels as units of computation in a mass univariate approach which does not appropriately account for the existing spatial correlation and is plagued by problems of multiple comparisons. Therefore, there is a need to explore alternative approaches for inference on neuroimaging data that accurately model spatial autocorrelation, potentially providing better type I error control and more sensitive inference. In this project we examine the performance of a trend surface modeling (TSM) approach that is based on a biologically relevant parcellation of the brain. We present our results from applying the TSM to both task fMRI and resting-state fMRI and compare the latter to the results from the parametric software, FSL. We demonstrate that the TSM provides better Type I error control, as well as sensitive inference on task data.The current predominant approach to neuroimaging data analysis is to use voxels as units of computation in a mass univariate approach which does not appropriately account for the existing spatial correlation and is plagued by problems of multiple comparisons. Therefore, there is a need to explore alternative approaches for inference on neuroimaging data that accurately model spatial autocorrelation, potentially providing better type I error control and more sensitive inference. In this project we examine the performance of a trend surface modeling (TSM) approach that is based on a biologically relevant parcellation of the brain. We present our results from applying the TSM to both task fMRI and resting-state fMRI and compare the latter to the results from the parametric software, FSL. We demonstrate that the TSM provides better Type I error control, as well as sensitive inference on task data

One-bit target detection in collocated MIMO Radar and performance degradation analysis

Target detection is an important problem in multipleinput multiple-output (MIMO) radar. Many existing target detection algorithms were proposed without taking into consideration the quantization error caused by analog-to-digital converters (ADCs). This paper addresses the problem of target detection for MIMO radar with one-bit ADCs and derives a Rao's test-based detector. The proposed method has several appealing features: 1) it is a closed-form detector; 2) it allows us to handle sign measurements straightforwardly; 3) there are closed-form approximations of the detector's distributions, which allow us to theoretically evaluate its performance. Moreover, the closed-form distributions allow us to study the performance degradation due to the one-bit ADCs, yielding an approximate 2 dB loss in the low-signal-to-noise-ratio (SNR) regime compared to 34-bit ADCs. Simulation results are included to showcase the advantage of the proposed detector and validate the accuracy of the theoretical results.The work of David Ramírez was supported in part by the Ministerio de Ciencia e Innovación, jointly with the European Commission (ERDF) under Grant PID2021-123182OB-I00 (EPiCENTER) and in part by the Comunidad de Madrid under Grant Y2018/TCS-4705 (PRACTICO-CM). The work of Lei Huang was supported in part by the National Science Fund for Distinguished Young Scholars under Grant 61925108 and in part by the Joint fund of the National Natural Science Foundation of China and Robot Fundamental Research Center of Shenzhen Government under Grant U1913203