Optimal Number Of Measurements Research Articles

Sub-Nyquist sampling of sparse and correlated signals in array processing

This paper considers efficient sampling of simultaneously sparse and correlated (S&C) signals for automotive radar application. We propose an implementable sampling architecture for the acquisition of S&C at a sub-Nyquist rate. We prove a sampling theorem showing exact and stable reconstruction of the acquired signals even when the sampling rate is smaller than the Nyquist rate by orders of magnitude. Quantitatively, our results state that an ensemble M signals, composed of a-priori unknown latent R signals, each bandlimited to W/2 but only S-sparse in the Fourier domain, can be reconstructed exactly from compressive sampling only at a rate RSlogα⁡W samples per second. When R≪M and S≪W, this amounts to a significant reduction in sampling rate compared to the Nyquist rate of MW samples per second. This is the first result that presents an implementable sampling architecture and a sampling theorem for the compressive acquisition of S&C signals. We resort to a two-step algorithm to recover sparse and low-rank (S&L) matrix from a near optimal number of measurements. This result then translates into a signal reconstruction algorithm from a sub-Nyquist sampling rate.

A polynomial multiple variance method for impulse response measurement

This manuscript describes a methodology for measuring the first-order Volterra kernel of a discrete-time nonlinear system, that is, the impulse response for small signal amplitudes of the nonlinear system. In the proposed approach, multiple linear identifications are performed using the same excitation signal multiplied by different gains, and the first-order Volterra kernel is obtained from a polynomial interpolation of the measured values. Any linear identification method proposed in the literature can be used with this approach. The proposed approach is a multiple variance (MV) method that, in contrast to all other MV methods, aims to estimate one of the kernels of the Volterra model, the first-order kernel, with high precision. The manuscript discusses the proposed methodology, determines the mean square deviation (MSD) due to noise of the measured coefficients, and the value of the optimal gains that minimize the MSD. Remarkably, it is shown in the manuscript that the optimal gains assume only a reduced set of values that depends on the order of nonlinearity. The optimal number of measurements is also determined. The conditions under which the proposed methodology is more convenient than the classical linear methods are discussed. The experimental results demonstrate the proposed methodology and its strengths.

Morphology, accumulation and survival of Desmanthus under different planting densities and harvest heights

AbstractThe use of legumes is an important strategy for animal feeding, especially during the dry season. The aim was to evaluate the effect of planting densities (40,000; 15,625 and 10,000 plants ha−1) and harvest heights (20 and 40 cm) on the morphology, accumulation and survival of Desmanthus (Desmanthus pernambucanus [L.] Thellung), cultivated in subhumid tropical region, as well as to estimate the repeatability of the evaluated characteristics and the optimal number of measurements. The treatments were randomized in blocks, with subdivided plots and four replicates. Planting densities were evaluated in the plots and the harvest heights in the subplots. Eight harvests were carried out with an interval of 84 days. Morphological and productive characteristics and survival were evaluated. Cultivation under density of 40,000 plants ha−1 produced taller plants, with higher leaf area index (LAI = 0.98), light interception (LI = 49%), individual accumulations (18.8 g DM plant−1) and by area (576.5 kg DM ha−1 harvest−1), although resulting in reduced plant stand (66%). Harvest heights do not affect accumulation and survival. It is possible to reduce the number of measurements for stem diameter, number of leaflets per leaf (R2 = 95%), plant height, canopy diameter, LAI, LI, leaf length and width, number of leaves per branch, branch diameter and LBR (R2 = 90%), optimizing resources for future research. Desmanthus has potential for use in protein banks, being harvested in the rainy season, conserved and supplied in addition to animals, but plant growth is minimal during the dry season under rainfed conditions.

Low rank matrix recovery with adversarial sparse noise* *This work was supported in part by National Natural Science Foundation of China under Grant numbers 12071426, 11971427, 11901518, the NSAF of China under grant number U21A20426, and the Fundamental Research Funds for the Central Universities under Grant number 2020XZZX002-03.

Many problems in data science can be treated as recovering a low-rank matrix from a small number of random linear measurements, possibly corrupted with adversarial noise and dense noise. Recently, a bunch of theories on variants of models have been developed for different noises, but with fewer theories on the adversarial noise. In this paper, we study low-rank matrix recovery problem from linear measurements perturbed by ℓ 1-bounded noise and sparse noise that can arbitrarily change an adversarially chosen ω-fraction of the measurement vector. For Gaussian measurements with nearly optimal number of measurements, we show that the nuclear-norm constrained least absolute deviation (LAD) can successfully estimate the ground-truth matrix for any ω < 0.239. Similar robust recovery results are also established for an iterative hard thresholding algorithm applied to the rank-constrained LAD considering geometrically decaying step-sizes, and the unconstrained LAD based on matrix factorization as well as its subgradient descent solver.

Repeatability and the optimal number of measurements for screening of soybean cultivars under water deficit

Repeatability and the optimal number of measurements for screening of soybean cultivars under water deficit

Ultimate resolution limits of speckle-based compressive imaging.

Compressive imaging using sparsity constraints is a very promising field of microscopy that provides a dramatic enhancement of the spatial resolution beyond the Abbe diffraction limit. Moreover, it simultaneously overcomes the Nyquist limit by reconstructing an N-pixel image from less than N single-point measurements. Here we present fundamental resolution limits of noiseless compressive imaging via sparsity constraints, speckle illumination and single-pixel detection. We addressed the experimental setup that uses randomly generated speckle patterns (in a scattering media or a multimode fiber). The optimal number of measurements, the ultimate spatial resolution limit and the surprisingly important role of discretization are demonstrated by the theoretical analysis and numerical simulations. We show that, in contrast to conventional microscopy, oversampling may decrease the resolution and reconstruction quality of compressive imaging.

Signal separation under coherent dictionaries and [formula omitted]-bounded noise

In this paper, we discuss the compressed data separation problem. In order to reconstruct the distinct subcomponents, which are sparse in morphologically different dictionaries D1∈Rn×d1 and D2∈Rn×d2, we present a general class of convex optimization decoder. It can deal with signal separation under the corruption of different kinds of noises, including Gaussian noise (p=2), Laplacian noise (p=1), and uniformly bounded noise (p=∞).Although the restricted isometry property adapted to frames is a commonly used tool, the measurement number is suboptimal when p>2. Furthermore, the ℓp robust nullspace property adapted to Ψ, which is constructed by D1 and D2, may fail to work on data separation problem. Here we introduce the modified ℓp robust nullspace property adapted to Ψ (abbreviated as the modified (ℓp, Ψ)-RNSP). First of all, we show the robust recovery of signals based on the modified (ℓp, Ψ)-RNSP and the mutual coherence between D1 and D2. Besides, we find that Gaussian measurements meet the modified (ℓp, Ψ)-RNSP for any 1≤p≤∞, provided with the optimal number of measurements O(slog(d∕s)), where s is the sparsity level and d=d1+d2.Furthermore, we introduce another properly constrained ℓ1-analysis optimization model, called the Split Dantzig Selector. It can recover signals which are approximately sparse in terms of different frame representations, when the measurement matrix satisfies the modified (ℓp, Ψ)-RNSP. As a special case, when considering Gaussian white noise, the recovery error by the Split Dantzig Selector is Oslogdm. It outperforms the ℓ2-constrained model, whose recovery error is O(logm), if the sparsity level is small.

Analyzing cross-validation in compressed sensing with Poisson noise

In many compressive sensing reconstruction algorithms, a good choice of important parameters such as the optimal number of measurements (which is dependent upon the unknown signal sparsity) or the regularization parameter, is critical for successful signal recovery. Cross-validation provides a principled method of doing so, by dividing the measurements into a ‘reconstruction set’ to recover the signal given each candidate parameter value, and a ‘cross-validation set’ to determine in a purely data-driven manner as to which candidate parameter value is optimal. In previous work, such a technique has been theoretically analyzed for the case of noiseless compressive measurements or for the case of additive i.i.d. Gaussian noise in the measurements. This paper presents the first theoretical analysis of this technique for compressed sensing when the measurements are corrupted by Poisson noise, the dominant noise distribution in optical systems. Using two different methods, we present different types of theoretical bounds in the form of confidence intervals on the actual (unobservable) recovery error in terms of the (observable) cross-validation error. We show in particular that these bounds become tighter with increase in the underlying signal intensity or number of cross-validation measurements (if other factors are held constant). We validate our theoretical bounds with simulations and experimental results for Poisson compressed sensing with the widely used Lasso estimator.

FEATURES OF CALIBRATION OF FREQUENCY COMPARATORS

Modern precision measurements of time and frequency are the most accurate measurements among measurements of other physical quantities. They are impossible without the use of highly stable reference frequency sources (frequency standards) and frequency comparators.The paper presents the main results of research in the calibration of precision frequency comparators using high-precision frequency standards. The structural scheme and model of measurements, and also features of definition of the budget of the measurement uncertainty at calibration are described.The influence of the most significant influence values on the accuracy of measurement results is analyzed. The content of quantitative and qualitative indicators of corrections that must be taken into account during calibration to achieve the highest measurement accuracy is revealed. Practical results of researches of instability of the frequency entered by the comparator at zero difference of frequencies of input sinusoidal signals at frequency of 5 MHz from the cesium generator in a frequency bandwidth of 3 Hz on time intervals of 1 s, 10 s, 1 hour, and 1 day are resulted.Practical results include numerical values and plotted on the main indicators of frequency instability, such as: increase in phase difference between signals, relative frequency difference between signals, root mean square relative frequency deviation, root mean square relative two-sample frequency deviation (Allan deviation), and power spectrum of relative deviations of signal frequency.Uncertainty budgets were compiled and the results of calibration of the frequency comparator Ч7-308А/1 at the time intervals of measurements of 1 s, 10 s, 1 hour, and 1 day are given. The main advantages of calibration of comparators with the minimum difference of input frequencies and the optimal number of measurements at each time interval of measurement are shown.It is expedient to use the described method by each metrological laboratory which carries out calibration of frequency comparators.

Real-time temperature prediction in a cold supply chain based on Newton's law of cooling

Many goods, including pharmaceuticals, require close temperature monitoring. This is important not only for complying with regulations but also for guaranteeing safety of use. A particular challenge in controlling a product's temperature arises during transportation. In cold supply chains (SCs), temperature is maintained by refrigerated containers. However, many situations, e.g. cooling system failure, lead to ambient temperature changes, and this needs to be detected as early as possible to prevent product damage. Existing approaches to temperature prediction are confined to long-term forecasts with relatively stable ambient temperatures and/or rely on multiple sensors in the known fixed positions. Since interventions in a SC are required immediately, there is a need for methods that provide real-time predictions regarding regular ambient temperature instability, i.e. when the ambient temperature changes unexpectedly in the short term. We propose a novel method that extends the applicability of Newton's law of cooling (NLC) to changeable ambient temperatures based on a set of temperature stability conditions and a sensor measurement error. In the method, an optimal number of measurements that characterize stable ambient temperatures and improve prediction reliability are selected. We compare the adapted NLC with artificial neural networks and autoregressive moving average models with respect to deviation prediction, prediction error, and execution time. Our evaluation based on real-world data shows that the adapted NLC outperforms existing baseline methods. In contrast to existing solutions, our method does not require any knowledge about the positioning of products within the container, further increasing its practical value.

Phase retrieval of low-rank matrices by anchored regression

Abstract We study the low-rank phase retrieval problem, where our goal is to recover a $d_1\times d_2$ low-rank matrix from a series of phaseless linear measurements. This is a fourth-order inverse problem, as we are trying to recover factors of a matrix that have been observed, indirectly, through some quadratic measurements. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope and enforce the rank constraint through nuclear norm regularization. The result is a convex program in the space of $d_1 \times d_2$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank-$1$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that anchored regression returns an accurate estimate from a near-optimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also show how to create such an anchor in the phaseless blind deconvolution problem from an optimal number of measurements and present a partial result in this direction for the general rank problem.

Determining the dynamic loading on an open-top wagon with a two-pipe girder beam

To ensure the structural strength of open-top wagons, it has been proposed to introduce the concept of a traction device that could be implemented in open-top wagons with bearing elements made of round pipes. Feature of the concept is that the console parts of the girder beam are filled with a viscous substance with damping and anticorrosive properties. To convert the shock kinetic energy into the dissipation energy, the concept design includes a piston with two throttle valves (inlet and outlet).In order to determine the dynamic load on the bearing structure of an open-top wagon equipped with a concept design of the traction device, mathematical modeling was performed. A mathematical model of the open-top wagon dynamic load during shunting collision has been constructed. It was considered that the frame of an open-top wagon is exposed to a longitudinal load of 3.5 MN. Differential equations were solved in line with a Runge-Kutta method in the programming environment Mathcad. It was established that the maximum magnitude of acceleration that acts on an open-top wagon, taking the improvement into consideration, is about 30 m/s2. The proposed technical solutions make it possible to reduce the magnitude of dynamic load on a open-top wagon’s bearing structure at shunting collision by 25 %.The software CosmosWorks was used to perform computer simulation of the dynamic load on an open-top wagon. The finite element method was applied as a calculation technique. In this case, maximum accelerations amounted to about 37 m/s2 and were concentrated at the console parts of a girder beam.Adequacy of the developed models of dynamic loading on an open-top wagon’s bearing structure was tested against the Fisher criterion (F-criterion). The optimal number of measurements was defined based on the Student-Gorset criterion. The results from calculation have demonstrated that the hypothesis of adequacy is not rejected.This study will contribute to a decrease in the dynamic load on the bearing structures of open-top wagons in operation, as well as bring down the cost of unscheduled repairs. The current research enables the compilation of guidelines on designing innovative rolling stock with improved technical and economic indicators.

Optimal Number of Measurements in a Linear System With Quadratically Decreasing SNR

We consider the design of a linear sensing system with a fixed energy budget assuming that the sampling noise is the dominant noise source. The energy constraint implies that the signal energy per measurement decreases linearly with the number of measurements. When the maximum sampling rate of the sampling circuit is chosen to match the designed sampling rate, the assumption on the noise implies that its variance increases approximately linearly with the sampling rate (number of measurements). Therefore, the overall SNR per measurement decreases quadratically in the number of measurements. Our analysis shows that, in this setting there is an optimal number of measurements. This is in contrast to the standard case, where noise variance remains unchanged with sampling rate, in which case more measurements imply better performance. Our results are based on a state evolution technique of the well-known approximate message passing algorithm. We consider both the sparse (e.g. Bernoulli-Gaussian and least-favorable distributions) and the non-sparse (e.g. Gaussian distribution) settings in both the real and complex domains. Numerical results corroborate our analysis.

How to reliably diagnose arterial hypertension: lessons from 24 h blood pressure monitoring

Background: Hypertension is a common condition in modern society. As blood pressure fluctuates with time, a single blood pressure measurement is useless to diagnose hypertension. Nevertheless, no well-defined number of measurements is often used for this purpose. Diagnosis and therapeutic control of hypertension are therefore suboptimal.Objective: To determine the number and timing of measurements needed to give a trustworthy approximation of an individual’s average blood pressure.Methods: In this observational study 306 clinically indicated 24h ABPM datasets were analysed. Hypertension was defined as a daytime blood pressure mean exceeding 135/85 mm Hg. Kappa coefficients determined the best time of day for measuring blood pressure. The optimal number of measurements was estimated using canonical correlation.Results: 162 (53%) patients were diagnosed with hypertension. Kappa statistics indicated that measuring during the afternoon gave the best agreement with the 24h blood pressure mean (κ = 0.78). According to canonical correlation, about 8–10 blood pressure readings give enough information for hypertension diagnosis.Conclusions: Eight to ten blood pressure measurements between 01:00 and 05:00 p.m. are sufficient to give a clinically useful approximation of the daytime mean blood pressure and therefore for diagnosing hypertension accurately. Future research should determine the ideal dispersion of measurements and include patient characteristics which could influence the required number and timing of measurements. These results may increase the future importance of telemonitoring in diagnosing hypertension.

Technical Performance of Two-Dimensional Shear Wave Elastography for Measuring Liver Stiffness: A Systematic Review and Meta-Analysis.

ObjectiveTo assess the technical performance of two-dimensional shear wave elastography (2D-SWE) for measuring liver stiffness.Materials and MethodsThe Ovid-MEDLINE and EMBASE databases were searched for studies reporting the technical performance of 2D-SWE, including concerns with technical failures, unreliable measurements, interobserver reliability, and/or intraobserver reliability, published until June 30, 2018. The pooled proportion of technical failure and unreliable measurements was calculated using meta-analytic pooling via the random-effects model and inverse variance method for calculating weights. Subgroup analyses were performed to explore potential causes of heterogeneity. The pooled intraclass correlation coefficients (ICCs) for interobserver and intraobserver reliability were calculated using the Hedges-Olkin method with Fisher's Z transformation of the correlation coefficient.ResultsThe search yielded 34 articles. From 20 2D-SWE studies including 6196 patients, the pooled proportion of technical failure was 2.3% (95% confidence interval [CI], 1.3–3.9%). The pooled proportion of unreliable measurements from 20 studies including 6961 patients was 7.5% (95% CI, 4.7–11.7%). In the subgroup analyses, studies conducting more than three measurements showed fewer unreliable measurements than did those with three measurements or less, but no intergroup difference was found in technical failure. The pooled ICCs for interobserver reliability (from 10 studies including 517 patients) and intraobserver reliability (from 7 studies including 679 patients) were 0.87 (95% CI, 0.82–0.90) and 0.93 (95% CI, 0.89–0.95), respectively, suggesting good to excellent reliability.Conclusion2D-SWE shows good technical performance for assessing liver stiffness, with high technical success and reliability. Future studies should establish the quality criteria and optimal number of measurements.

Compressive Sampling Rate Optimization for Multiple Measurement Vector Problems

Joint sparse recovery is a recently proposed signal processing approach which has become conventional due to its capability of solving a wide range of optimization problems. However, despite the appropriate performance of existing joint sparse recovery algorithms, all of them suffer from the same drawback: They collect pre-defined number of compressed measurements, which may be too small to precisely reconstruct the row support, or may be unnecessarily large which leads to the wastage of sampling resources. To address this issue and unlike the previous studies, this paper proposes a novel compressed sensing-based method to adaptively adjust the optimal sampling rate of joint sparse recovery problem. A data-driven approach based on Monte Carlo simulation is introduced to obtain the minimum number of samples which is required for precise row support estimation using several well-known joint sparse recovery techniques. Then, a sequential joint sparse recovery framework is developed where the first step predicts the optimal number of measurements and the second step reconstructs signal vectors applying the determined sampling rate. Numerical simulations investigate the effectiveness of suggested method to reduce both the required number of measurements and average algorithm runtime, without losing the recovery performance.

On the Gap Between Restricted Isometry Properties and Sparse Recovery Conditions

We consider the problem of recovering sparse vectors from underdetermined linear measurements via $\ell _{p}$ -constrained basis pursuit. Previous analyses of this problem based on generalized restricted isometry properties have suggested that two phenomena occur if $p\neq 2$ . First, one may need substantially more than $s \log (en/s)$ measurements (optimal for $p=2$ ) for uniform recovery of all $s$ -sparse vectors. Second, the matrix that achieves recovery with the optimal number of measurements may not be Gaussian (as for $p=2$ ). We present a new, direct analysis, which shows that in fact neither of these phenomena occur. Via a suitable version of the null space property, we show that a standard Gaussian matrix provides $\ell _{q}/\ell _{1}$ -recovery guarantees for $\ell _{p}$ -constrained basis pursuit in the optimal measurement regime. Our result extends to several heavier-tailed measurement matrices. As an application, we show that one can obtain a consistent reconstruction from uniform scalar quantized measurements in the optimal measurement regime.

The Rate-Distortion Optimized Compressive Sensing for Image Coding

Compressive Sensing (CS) is an emerging technology which can encode a signal into a small number of incoherent linear measurements and reconstruct the entire signal from relatively few measurements. Different from former coding scheme which distortion mainly comes from quantizer, distortion and bit rate are related to quantization and compressive sampling in the compressive sensing based image coding schemes. Moreover, the total number of bits is often constrained in the practical application. Therefore under the given bit rate how to balance the number of measurements and quantization step size to minimization the distortion is a great challenge. In this paper, a fast Lagrange multiplier solving method is proposed for the compressive sensing based image coding scheme. Then using the solved Lagrange multiplier, the optimal number of measurements and quantization step size are determined based on the rate-distortion criteria. Experimental results show that the proposed algorithm improves objective and subjective performances substantially.

Economic design of the upper-sided synthetic chart with measurement errors

Considering a quadratic Taguchi loss function for poor quality products, as well as measurement errors in the process characteristic, this paper proposes an economic model for the upper-sided synthetic chart allowing multiple measurements for each sample item. The steady-state ARL of this chart is computed using a Markov chain method and the optimal number of measurements are obtained through the minimisation of the total cost model. For the proposed economic and economic statistical model, numerical examples are presented to evaluate the effect of measurement errors on the upper-sided synthetic chart. Finally, the sensitivity analyses of the optimal chart parameters and the cost of the upper-sided synthetic chart are also investigated using different input parameter values. All the above results are verified through a wide range of the input parameter values. Two illustrative examples are also provided.

Optimal Number of Contact Angle Measurements on Pyrite Surface

Contact angle measurement is a simple and quick method that expresses the surface wettability. Resulting values vary and depend on many factors that significantly affect the interpretation of results. In this paper a set of seventeen samples from different deposits with different genesis were subjected to a statistical evaluation, to determine the optimal number of measurements needed to obtain the desired standard deviation.