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The true covariance matrix is a 120 × 120 identity matrix with Kronecker delta structure. We estimate this covariance matrix for N = 10 , 50 , and 100 independent sets of random samples by using 15 × 8 slicing.
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Nonsingular estimation of high dimensional covariance matrices is an
important step in many statistical procedures like classification, clustering,
variable selection an future extraction. After a review of the essential
background material, this paper introduces a technique we call slicing for
obtaining a nonsingular covariance matrix of high dime...
Context in source publication
Context 1
... this example, we will use the heatmap of the true and es- timated covariance matrices under different scenarios to see that slicing gives a reasonable description of the variable variances and covariances. In Figure 3 the true covariance matrix is a 120 × 120 identity matrix, we estimate this covariance matrix for N = 10, 50, and 100 independent sets of random samples by using 15 × 8 slicing. In Figure 4 the true covariance matrix is a 120 × 120 block diagonal matrix with Kronecker delta structure. ...
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Citations
... In practice, the GLRT is used based on the assumption that the sample size N is large while the sample dimensions {K, L} are small. However, when the sample support is limited (in particular, when N ≤ KL) , the GLRT degenerates due to the ill-conditioning of the estimated covariance matrix [22]. A way to reduce these limitations will be presented next. ...
... The GLRT-based detection scheme in Equation (7) assumes no structure for the covariance matrix, except that it is symmetric. In order to improve the conditioning for small sample support, in this section we exploit an a-priori suitable structure for the covariance matrix based on the underlying physical topology [22]. Taking this into effect, we propose two detection techniques in Sections 4.1 and 4.2 that exploit the embedded correlation structure based on the SPKP and MPKP, respectively. ...
... Keeping the above facts and discussion in [22,24] into considerations, the overall (inter-antenna plus inter-receiver) spatial covariance Σ 1 can be represented with the help of the SPKP model as: ...
This paper considers a novel fusion rule for spectrum sensing scheme for a cognitive radio network with multi-antenna receivers. The proposed scheme exploits the fact that when any primary signal is present, measurements are spatially correlated due to presence of inter-antenna and inter-receiver spatial correlation. In order to exploit this spatial structure, the generalized likelihood ratio test (GLRT) operates with the determinant of the sample covariance matrix. Therefore, it depends on the sample size N and the dimensionality of the received data (i.e., the number of receivers K and antennas L). However, when the dimensionality fK; Lg is on the order, or larger than the sample size N, the GLRT degenerates due to the ill-conditioning of the sample covariance matrix. In order to circumvent this issue, we propose two techniques that exploit the inner spatial structure of the received observations by using single pair and multi-pairs Kronecker products. The performance of the proposed detectors is evaluated by means of numerical simulations, showing important advantages with respect to the traditional (i.e., unstructured) GLRT approach.
... By using this prior information, we can reduce the demand for large sample support. Having said this, convenient structures that can be assumed for the spatio-temporal modeling of multiantenna measurements are those given by persymmetric and Kronecker product structures [79,80,81]. In [34], the authors exploited the Toeplitz structure of the covariance matrix by assuming wide-sense stationarity. ...
... However, when the sample support is limited (in particular, when N KL) , the GLRT degenerates due to the ill-conditioning of the estimated covariance matrix [79]. A way to reduce these limitations will be presented next. ...
... The GLRT based detection scheme in (6.52) assumes no structure for the covariance matrix, except that the covariance matrix is symmetric. However, in order to circumvent the issue of ill-conditioning problems, one may assume an a-priori suitable structure on the covariance matrix [79]. Hence, we propose two detection techniques in Sections 6.2.3.1 and 6.2.3.2 that exploit the embedded correlation structure based on the single-pair Kronecker product (SPKP) and multi-pairs Kronecker product (MPKP), respectively. ...
This thesis is focused on multi-sensor event detection schemes in Wireless Sensor Network (WSN). In the WSN, sensors individually take their observations, do some initial local processing and then send the result to the fusion center. Apart from this information, the fusion center may have prior information such as the known positions of sensors, the topology of the network or structures, features and patterns present in the received observations. In order to achieve better detection performance, the fusion center must exploit all this prior information in the best possible way. Keeping this in mind, in this thesis we focus on the exploitation of available a-priori information with the aim of developing enhanced and robust detection mechanisms. For instance, we exploit the fact that in most applications, the signal emitted from the event becomes a local physical phenomenon that only a"ects a small subset of sensors. Moreover, the a"ected sensors will be located close to the event as well as close to each other in the form of a spatial cluster. Hence, novel detection schemes are presented with a two-fold motivation: !rst, the exploitation of the relevant set of sensors, which helps in rejecting the noise; second, to take advantage of the signal correlation by using a-priori information about the positions of sensors. Based on these results, we move one step further and concentrate on the exploitation of both spatial and temporal correlation in the received observations. In this case, we propose detection schemes that explore di"erent matrix structures embedded in the observed covariance matrix, which naturally arise due to the underlying topologies of the multiple sensors. Numerical results are presented for all of the proposed schemes that show important advantages compared to traditional schemes
... In practice, the GLRT is used based on the assumption that the sample size is large while the sample dimension is small. In many applications, where the sample support available for estimating the covariance matrix is limited, the GLRT may degenerate due to a singular and ill-conditioned sample covariance matrix [5,6]. ...
... In order to circumvent these ill-conditioning problems, one may assume an a-priori structure on the covariance matrices involved herein, based on the underlying layout of the sensor field. For instance, a convenient structure that can be assumed is the Kronecker product structure [5], which can model the lattice-type spatial structure that appears in networks whose sensors are grouped in clusters. By grouping in clusters and using Kronecker product structure, a nonsingular estimate of the required covariance matrix can be more easily obtained [5], and this leads to more robust and stable detection tests. ...
... For instance, a convenient structure that can be assumed is the Kronecker product structure [5], which can model the lattice-type spatial structure that appears in networks whose sensors are grouped in clusters. By grouping in clusters and using Kronecker product structure, a nonsingular estimate of the required covariance matrix can be more easily obtained [5], and this leads to more robust and stable detection tests. Recently, the concept of exploiting the Kronecker product structure of the covariance matrix has received a lot of interest in statistics [7,8,6] . ...
In collaborative spectrum sensing, spatial correlation in the measurements obtained by sensors can be exploited by adopting Generalized Likelihood Ratio Test (GLRT). In this process the GLRT provides a test statistics that is normally based on the sample covariance matrix of the received signal samples. Unfortunately, problems arise when the dimensions of this matrix become excessively large, as it happens in the so-called large-scale wireless sensor networks. In these circumstances, a huge amount of samples are needed in order to avoid the ill-conditioning of the GLRT, which degenerates when the dimensionality of data is equal to the sample size or larger. To circumvent this problem, we modify the traditional GLRT detector by decomposing the large spatial covariance matrix into small covariance matrices by using properties of the Kronecker Product. The proposed detection scheme is robust in the case of high dimensionality and small sample size. Numerical results are drawn, which show that the proposed detection schemes indeed outperform the traditional approaches when the dimension of data is larger than the sample size.
