Compressed sensing, which involves reconstruction of sparse signals from an under-determined linear system, has been recently applied to problems in group testing to save on the number of tests administered during a pandemic or other resource-constrained scenarios. In practical group testing in time-constrained situations, the results of two different groups can sometimes be mistakenly exchanged by a technician. This is called ‘permutation noise’ and it presents challenges in determining the signal vector containing p health status values of the participating subjects from the results on n ≪ p pooled tests. In this paper, we present a method to determine the health status values in a manner that is robust to a small number of such permutations. The technique is based on a ‘debiased’ form of the robust LASSO estimator, with which we carefully design hypothesis tests in order (i) to identify the unhealthy subjects (based on non-zero values in the health status signal vector), and (ii) to identify the pooled measurements which were corrupted by permutation noise. Furthermore, we present an algorithm to correct the permutations in the pooled tests and subsequently reconstruct the signal vector from the corrected measurements. We further provide empirical results showing the efficacy of both the identification and correction of permutation errors, and show that it is superior to many intuitive baseline techniques.

Quasi (pseudo/approximate) periodic signals often occur in natural settings, particularly when a periodic signal is recorded with noise. In this work, we develop a new dynamical equation system to construct a family of Quasi-periodic Gaussian Processes (QPGP). We provide a computationally inexpensive algorithm based on a newly derived likelihood function for parameter estimation. Our approach also simplifies the signal forecasting. We illustrate via a simulation study that the proposed QPGP estimation strategy is faster as well and more accurate than existing constructions. Unlike these existing models, the proposed approach extends to multiple families of kernels, which we illustrate by modeling sunspot data with both periodic Matérn and MacKay’s covariance kernel. This shows the exclusive advantage of the new QPGP family proposed in this work.

A practical constraint that comes in the way of spectrum estimation of a continuous time stationary stochastic process is the minimum separation between successively observed samples of the process. When the underlying process is not band-limited, sampling at any uniform rate leads to aliasing, while certain stochastic sampling schemes, including Poisson process sampling, are rendered infeasible by the constraint of minimum separation. It is shown in this paper that, subject to this constraint, no point process sampling scheme is alias-free for the class of all spectra. It turns out that point process sampling under this constraint can be alias-free for band-limited spectra. However, the usual construction of a consistent spectrum estimator does not work in such a case. Simulations indicate that a commonly used estimator, which is consistent in the absence of this constraint, performs poorly when the constraint is present. These results should help practitioners in rationalizing their expectations from point process sampling as far as spectrum estimation is concerned, and motivate researchers to look for appropriate estimators of bandlimited spectra.

In the matter of selection of sample time points for the estimation of the power spectral density of a continuous time stationary stochastic process, irregular sampling schemes such as Poisson sampling are often preferred over regular (uniform) sampling. A major reason for this preference is the well-known problem of inconsistency of estimators based on regular sampling, when the underlying power spectral density is not bandlimited. It is argued in this paper that, in consideration of a large sample property like consistency, it is natural to allow the sampling rate to go to infinity as the sample size goes to infinity. Through appropriate asymptotic calculations under this scenario, it is shown that the smoothed periodogram based on regularly spaced data is a consistent estimator of the spectral density, even when the latter is not band-limited. It transpires that, under similar assumptions, the estimators based on uniformly sampled and Poisson-sampled data have about the same rate of convergence. Apart from providing this reassuring message, the paper also gives a guideline for practitioners regarding appropriate choice of the sampling rate. Theoretical calculations for large samples and Monte-Carlo simulations for small samples indicate that the smoothed periodogram based on uniformly sampled data have less variance and more bias than its counterpart based on Poisson sampled data.