Estimation of Circular Statistics in the Presence of Measurement Bias

Circular statistics and Rayleigh tests are important tools for analyzing cyclic events. However, current methods are not robust to significant measurement bias, especially incomplete or otherwise non-uniform sampling. One example is studying 24-cyclicity but having data not recorded uniformly over the full 24-hour cycle. Our objective is to present a robust method to estimate circular statistics and their statistical significance in the presence of incomplete or otherwise non-uniform sampling. Our method is to solve the underlying Fredholm Integral Equation for the more general problem, estimating probability distributions in the context of imperfect measurements, with our circular statistics in the presence of incomplete/non-uniform sampling being one special case. The method is based on linear parameterizations of the underlying distributions. We simulated the estimation error of our approach for several toy examples as well as for a real-world example: analyzing the 24-hour cyclicity of an electrographic biomarker of epileptic tissue controlled for states of vigilance. We also evaluated the accuracy of the Rayleigh test statistic versus the direct simulation of statistical significance. Our method shows a very low estimation error. In the real-world example, the corrected moments had a root mean square error of [Formula: see text]. In contrast, the Rayleigh test statistic overestimated the statistical significance and was thus not reliable. The presented methods thus provide a robust solution to computing circular moments even with incomplete or otherwise non-uniform sampling. Since Rayleigh test statistics cannot be used in this circumstance, direct estimation of significance is the preferable option for estimating statistical significance.

Revisiting aliasing noise to build more robust sparsity in nonuniform sampling 2D-NMR

A continuing priority is to better understand and resolve the barriers to using nonuniform sampling (NUS) in challenging small molecule 2D NMR with subsampling of the Nyquist grid (a.k.a. coverage) below 50%. Possible causes for artifacts, often termed sampling noise, in 1D-NUS of 2D-NMR are revisited here, where weak aliasing artifacts are a growing concern as NUS becomes sparser. As NUS schedules become sparser, repeat sequences are shown to occur in the dense sampling regions early in the sampling schedule, causing aliasing artifacts in resulting spectra. An intuitive screening approach that detects patterns in sampling schedules based on a convolutional filter was implemented. Sampling schedules that have low proportions of repeat sequences show significantly reduced artifacts. Another route to remediate early repeat sequences is a short period of uniform sampling at the beginning of the schedule, which also leads to a significant suppression of unwanted sampling noise. Combining the repeat sequence filter with a survey of HSQC and LR-HSQMBC experiments, it is shown that very short initial uniform regions of about 2%-4% of the sampling space can ameliorate repeat sequences in sparser NUS and lead to robust spectral reconstructions by iterative soft thresholding (IST), even when the point spread function is unchanged. Using the principles developed here, a suite of 'one-click' schedules was developed for broader use.

An approach to fast 2D nuclear magnetic resonance at low concentration based on p-H2 -induced polarization and nonuniform sampling

Recent developments in para-hydrogen-induced polarization (PHIP) methods allow the nuclear magnetic resonance (NMR) detection of specific classes of compounds, down to sub-micromolar concentration in solution. However, when dealing with complex mixtures, signal resolution requires the acquisition of 2D PHIP-NMR spectra, which often results in long experimental times. This strongly limits the applicability of these 2D PHIP-NMR techniques in areas in which high-throughput analysis is required. Here, we present a combination of fast acquisition and nonuniform sampling that can afford a 10-fold reduction in measuring time without compromising the spectral quality. This approach was tested on a mixture of substrates at micromolar concentration, for which a resolved 2D PHIP spectrum was acquired in less than 3 min.

Benefits of time-resolved nonuniform sampling in reaction monitoring: The case of aza-Michael addition of benzylamine and acrylamide

Monitoring of chemical reactions is best carried out using methods that sample the test object at a rate greater than the time scale of the processes taking place. The recently proposed time-resolved nonuniform sampling (TR-NUS) method allows the use of two-dimensional (2D) nuclear magnetic resonance (NMR) spectra for this purpose and provides a time resolution equivalent to that achievable using one-dimensional spectra. Herein, we show that TR-NUS acquired data eliminates 2D spectral line disturbances and enables more accurate signal integration and kinetics conclusions. The considerations are exemplified with a seemingly simple aza-Michael reaction of benzylamine and acrylamide. Surprisingly, the product identification is possible only using 2D spectra, although credible monitoring requires TR-NUS.

Software for reconstruction of nonuniformly sampled NMR data

Nonuniform sampling (NUS) of multidimensional NMR experiments is a powerful tool to obtain high-resolution spectra with less instrument time. With NUS, only a subset of the points needed for conventional Fourier transformation is recorded, and sophisticated algorithms are needed to reconstruct the missing data points. During the last decade, several software packages implementing the reconstruction algorithms have emerged and been refined and now result in spectra of almost similar quality as spectra from conventionally recorded and processed data. However, from the number of literature references to the reconstruction methods, many more multidimensional NMR spectra could presumably be recorded with NUS. To help researchers considering to start using NUS for their NMR experiments, we here review 13 different reconstruction methods found in five software packages (CambridgeCS, hmsIST, MddNMR, NESTA-NMR, and SMILE). We have compared how the methods run with the provided example scripts for reconstructing a nonuniform sampled three-dimensional ¹⁵ N-NOESY-HSQC at sampling densities from 5% to 50%. Overall, the spectra are all of similar quality above 20% sampling density. Thus, without any particular knowledge about the details of the reconstruction algorithms, significant reduction in the experiment time can be achieved. Below 20% sampling density, the intensities of particular weak peaks start being affected. MddNMR's IST with virtual echo and the SMILE algorithms still reproduced the spectra with the highest accuracy of peak intensities.

Nonuniformly sampled exclusively-13 C/15 N 4D solid-state NMR experiments: Assignment and characterization of IKe phage capsid

An important step in the process of protein research by NMR is the assignment of chemical shifts. In the coat protein of IKe bacteriophage, there are 53 residues making up a long helix resulting in relatively high spectral ambiguity. Assignment thus requires the collection of a set of three-dimensional (3D) experiments and the preparation of sparsely labeled samples. Increasing the dimensionality can facilitate fast and reliable assignment of IKe and of larger proteins. Recent progress in nonuniform sampling techniques made the application of multidimensional NMR solid-state experiments beyond 3D more practical. 4D ¹ H-detected experiments have been demonstrated in high-fields and at spinning speeds of 60 kHz and higher but are not practical at spinning speeds of 10-20 kHz for fully protonated proteins. Here, we demonstrate the applicability of a nonuniformly sampled 4D ¹³ C/¹⁵ N-only correlation experiment performed at a moderate field of 14.1 T, which can incorporate sufficiently long acquisition periods in all dimensions. We show how a single CANCOCX experiment, supported by several 2D carbon-based correlation experiments, is utilized for the assignment of heteronuclei in the coat protein of the IKe bacteriophage. One sparsely labeled sample was used to validate sidechain assignment of several hydrophobic-residue sidechains. A comparison to solution NMR studies of isolated IKe coat proteins embedded in micelles points to key residues involved in structural rearrangement of the capsid upon assembly of the virus. The benefits of 4D to a quicker assignment are discussed, and the method may prove useful for studying proteins at relatively low fields.

CG-SENSE revisited: Results from the first ISMRM reproducibility challenge

PURPOSE

The aim of this work is to shed light on the issue of reproducibility in MR image reconstruction in the context of a challenge. Participants had to recreate the results of "Advances in sensitivity encoding with arbitrary k-space trajectories" by Pruessmann et al. METHODS: The task of the challenge was to reconstruct radially acquired multicoil k-space data (brain/heart) following the method in the original paper, reproducing its key figures. Results were compared to consolidated reference implementations created after the challenge, accounting for the two most common programming languages used in the submissions (Matlab/Python).

RESULTS

Visually, differences between submissions were small. Pixel-wise differences originated from image orientation, assumed field-of-view, or resolution. The reference implementations were in good agreement, both visually and in terms of image similarity metrics.

DISCUSSION AND CONCLUSION

While the description level of the published algorithm enabled participants to reproduce CG-SENSE in general, details of the implementation varied, for example, density compensation or Tikhonov regularization. Implicit assumptions about the data lead to further differences, emphasizing the importance of sufficient metadata accompanying open datasets. Defining reproducibility quantitatively turned out to be nontrivial for this image reconstruction challenge, in the absence of ground-truth results. Typical similarity measures like NMSE of SSIM were misled by image intensity scaling and outlier pixels. Thus, to facilitate reproducibility, researchers are encouraged to publish code and data alongside the original paper. Future methodological papers on MR image reconstruction might benefit from the consolidated reference implementations of CG-SENSE presented here, as a benchmark for methods comparison.

Reconstruction of Periodic Band Limited Signals from Non-Uniform Samples with Sub-Nyquist Sampling Rate

Important state parameters, such as torque and angle obtained from the servo control and drive system, can reflect the operating condition of the equipment. However, there are two problems with the information obtained through the network from the control and drive system: the low sampling rate, which does not meet the sampling theorem and the nonuniformity of the sampling points. By combing equivalent sampling and nonuniform signal reconstruction theory, this paper proposes a reconstruction method for signal obtained from servo system in periodic reciprocating motion. Equivalent sampling combines the low rate and nonuniform samples from multiple periods into one single period, so that the equivalent sampling rate is far increased. Then, the nonuniform samples with high density are further resampled to meet the reconstruction conditions. This step can avoid the amplitude error in the reconstructed signal and increase the possibility of successful reconstruction. Finally, the reconstruction formula derived from basis theory is applied to recover the signal. This method has been successfully verified by the simulation signal of the robot swing process and the actual current signal collected on the robot arm testbed.

Developing nonuniform sampling strategies to improve sensitivity and resolution in 1,1-ADEQUATE experiments

Nonuniform sampling (NUS) strategies are developed for acquiring highly resolved 1,1-ADEQUATE spectra, in both conventional and homodecoupled (HD) variants with improved sensitivity. Specifically, the quantile-directed and Poisson gap methods were critically compared for distributing the samples nonuniformly, and the quantile schedules were further optimized for weighting. Both maximum entropy and iterative soft thresholding spectral estimation algorithms were evaluated. All NUS approaches were robust when the degree of data reduction is moderate, on the order of a 50% reduction of sampling points. Further sampling reduction by NUS is facilitated by using weighted schedules designed by the quantile method, which also suppresses sampling noise well. Seed independence and the ability to specify the sample weighting in quantile scheduling are important in optimizing NUS for 1,1-ADEQUATE data acquisition. Using NUS yields an improvement in sensitivity, while also making longer evolution times accessible that would be difficult or impractical to attain by uniform sampling. Theoretical predictions for the sensitivity enhancements in these experiments are in the range of 5-20%; NUS is shown to disambiguate weak signals, reveal some ⁿ J_CC correlations obscured by noise, and improve signal strength relative to uniform sampling in the same experimental time. This work presents sample schedule development for applying NUS to challenging experiments. The schedules developed here are made available for general use and should facilitate the broader utilization of ADEQUATE experiments (including 1,1-, 1,n-, and HD- variants) for challenging structure elucidation problems.

EventDTW: An Improved Dynamic Time Warping Algorithm for Aligning Biomedical Signals of Nonuniform Sampling Frequencies

The dynamic time warping (DTW) algorithm is widely used in pattern matching and sequence alignment tasks, including speech recognition and time series clustering. However, DTW algorithms perform poorly when aligning sequences of uneven sampling frequencies. This makes it difficult to apply DTW to practical problems, such as aligning signals that are recorded simultaneously by sensors with different, uneven, and dynamic sampling frequencies. As multi-modal sensing technologies become increasingly popular, it is necessary to develop methods for high quality alignment of such signals. Here we propose a DTW algorithm called EventDTW which uses information propagated from defined events as basis for path matching and hence sequence alignment. We have developed two metrics, the error rate (ER) and the singularity score (SS), to define and evaluate alignment quality and to enable comparison of performance across DTW algorithms. We demonstrate the utility of these metrics on 84 publicly-available signals in addition to our own multi-modal biomedical signals. EventDTW outperformed existing DTW algorithms for optimal alignment of signals with different sampling frequencies in 37% of artificial signal alignment tasks and 76% of real-world signal alignment tasks.

The Automatic Proportionator Estimator Is Highly Efficient for Estimation of Total Number of Sparse Cell Populations

Estimation of total number of a population of cells that are sparsely distributed in an organ or anatomically-defined region of interest represents a challenge for conventional stereological methods. In these situations, classic fractionator approaches that rely on systematic uniform random sampling are highly inefficient and, in many cases, impractical due to the intense sampling of the organ and tissue sections that is required to obtain sufficient counts for an acceptable level of precision. The proportionator, an estimator based on non-uniform sampling theory, marries automated image analysis with stereological principles and is the only estimator that provides a highly efficient and precise method to address these challenging quantification problems. In this paper, the practical considerations of the proportionator estimator and its implementation with Proportionator™ software and digital slide imaging are reviewed. The power of the proportionator as a stereological tool is illustrated in its application to the estimation of the total number of a very rare (~50/vertebrae) and sparsely distributed population of osteoprogenitor cells in mouse vertebral body. The proportionator offers a solution to neuroscientists interested in quantifying total cell number of sparse cell populations in the central and peripheral nervous system where systematic uniform random sampling-based stereological estimators are impractical.

Correlated spectroscopic imaging of calf muscle in three spatial dimensions using group sparse reconstruction of undersampled single and multichannel data

PURPOSE

To implement a 5D (three spatial + two spectral) correlated spectroscopic imaging sequence for application to human calf.

THEORY AND METHODS

Nonuniform sampling was applied across the two phase encoded dimensions and the indirect spectral dimension of an echo planar-correlated spectroscopic imaging sequence. Reconstruction was applied that minimized the group sparse mixed ℓ2,1-norm of the data. Multichannel data were compressed using a sensitivity map-based approach with a spatially dependent transform matrix and utilized the self-sparsity of the individual coil images to simplify the reconstruction.

RESULTS

Single channel data with 8× and 16× undersampling are shown in the calf of a diabetic patient. A 15-channel scan with 12× undersampling of a healthy volunteer was reconstructed using 5 virtual channels and compared to a fully sampled single slice scan. Group sparse reconstruction faithfully reconstructs the lipid cross peaks much better than ℓ1 minimization.

CONCLUSION

COSY spectra can be acquired over a 3D spatial volume with scan time under 15 min using echo planar readout with highly undersampled data and group sparse reconstruction.

Accelerated five-dimensional echo planar J-resolved spectroscopic imaging: Implementation and pilot validation in human brain

PURPOSE

To implement an accelerated five-dimensional (5D) echo-planar J-resolved spectroscopic imaging sequence combining 3 spatial and 2 spectral encoding dimensions and to apply the sequence in human brain.

METHODS

An echo planar readout was used to acquire a single spatial and a single spectral dimension during one readout. Nonuniform sampling was applied to the two phase-encoded spatial directions and the indirect spectral dimension. Nonlinear reconstruction was used to minimize the ℓ1-norm or the total variation and included a spectral mask to enhance sparsity. Retrospective reconstructions at multiple undersamplings were performed in phantom. Ten healthy volunteers were scanned with 8× undersampling and compared to a fully sampled single slice scan.

RESULTS

Retrospective reconstruction of fully sampled phantom data showed excellent quality at 4×, 8×, 12×, and 16× undersampling using either reconstruction method. Reconstruction of prospectively acquired in vivo scans with 8× undersampling showed excellent quality in the occipito-parietal lobes and good quality in the frontal lobe, consistent with the fully sampled single slice scan.

CONCLUSION

By utilizing nonuniform sampling with nonlinear reconstruction, 2D J-resolved spectra can be acquired over a 3D spatial volume with a total scan time of 20 min, which is reasonable for in vivo studies.

Sensitivity gains, linearity, and spectral reproducibility in nonuniformly sampled multidimensional MAS NMR spectra of high dynamic range

Recently, we have demonstrated that considerable inherent sensitivity gains are attained in MAS NMR spectra acquired by nonuniform sampling (NUS) and introduced maximum entropy interpolation (MINT) processing that assures the linearity of transformation between the time and frequency domains. In this report, we examine the utility of the NUS/MINT approach in multidimensional datasets possessing high dynamic range, such as homonuclear (13)C-(13)C correlation spectra. We demonstrate on model compounds and on 1-73-(U-(13)C,(15)N)/74-108-(U-(15)N) E. coli thioredoxin reassembly, that with appropriately constructed 50% NUS schedules inherent sensitivity gains of 1.7-2.1-fold are readily reached in such datasets. We show that both linearity and line width are retained under these experimental conditions throughout the entire dynamic range of the signals. Furthermore, we demonstrate that the reproducibility of the peak intensities is excellent in the NUS/MINT approach when experiments are repeated multiple times and identical experimental and processing conditions are employed. Finally, we discuss the principles for design and implementation of random exponentially biased NUS sampling schedules for homonuclear (13)C-(13)C MAS correlation experiments that yield high-quality artifact-free datasets.

Impact of hydrostatic pressure on an intrinsically disordered protein: a high-pressure NMR study of α-synuclein

The impact of pressure on the backbone (15) N, (1) H and (13) C chemical shifts in N-terminally acetylated α-synuclein has been evaluated over a pressure range 1-2500 bar. Even while the chemical shifts fall very close to random coil values, as expected for an intrinsically disordered protein, substantial deviations in the pressure dependence of the chemical shifts are seen relative to those in short model peptides. In particular, the nonlinear pressure response of the (1) H(N) chemical shifts, which commonly is associated with the presence of low-lying "excited states", is much larger in α-synuclein than in model peptides. The linear pressure response of (1) H(N) chemical shift, commonly linked to H-bond length change, correlates well with those in short model peptides, and is found to be anticorrelated with its temperature dependence. The pressure dependence of (13) C chemical shifts shows remarkably large variations, even when accounting for residue type, and do not point to a clear shift in population between different regions of the Ramachandran map. However, a nearly universal decrease in (3) JHN-Hα by 0.22 ± 0.05 Hz suggests a slight increase in population of the polyproline II region at 2500 bar. The first six residues of N-terminally acetylated synuclein show a transient of approximately 15% population of α-helix, which slightly diminishes at 2500 bar. The backbone dynamics of the protein is not visibly affected beyond the effect of slight increase in water viscosity at 2500 bar.

A SARS Method for Reliable Spectrum Sensing in Multiband Communication Systems

This paper introduces a novel method of spectrum sensing in communication systems that utilizes nonuniform sampling in conjunction with a suitable spectral analysis tool. It is referred to here as spectral analysis for randomized sampling (SARS). Owing to the deployment of nonuniform sampling, the proposed technique can accomplish the sensing task by using sampling rates well below the ones demanded by uniform-sampling-based digital signal processing (DSP). The effect of the cyclostationary nature of the incoming digital communication signal on the adequacy of the adopted periodogram-type estimator for the spectrum sensing operation is addressed. The statistical characteristics of the estimator are presented. General reliability conditions on the length of the required signal observation window, i.e., sensing time, for a chosen sampling rate or vice versa are provided amid a sought system performance. The impact of the presence of noise and processing transmissions with various power levels on the derived dependability recommendations is given. The analytical results are illustrated by numerical examples. This paper establishes a new framework for efficient spectrum sensing where considerable savings on the sampling rate and number of processed samples can be attained.

Sensitivity enhancement for maximally resolved two-dimensional NMR by nonuniform sampling

Resolving NMR signals which are separated in frequency on the order of their line widths requires obtaining the time domain free induction decay for a maximum time tmax = πT2 , where T2 is the transverse relaxation time of the given signals. Unfortunately, samples acquired beyond ∼1.26T2 contribute more noise than signal to the data; and samples in the range of about (0.75-1.26)× T2 have a negligible effect on the signal-to-noise ratio (SNR). Therefore, one must sacrifice SNR to reach evolution times of πT2 . One can preserve resolution in a shorter total experimental time by selecting a reduced set of samples from the Nyquist grid according to an exponential probability density which is on the order of the T2 of the signals. This practice is widely termed nonuniform sampling (NUS). We derive analytic theory for the enhancement of the intrinsic SNR of NUS time domain data compared with uniformly sampled data when the total experimental times are equivalent. This theory is general for any tmax and exponential weighting and is further carefully validated with simulations. Enhancements of SNR in the time domain on the order of twofold are routinely available when tmax ∼ πT2 and are reflected in the subsequent maximum entropy reconstructed spectra. SNR enhancement by NUS is demonstrated to be helpful in enabling the acquisition of HMQC spectra of dilute bile salts in which high resolution in the indirect carbon dimension is required.

Knowledge-based nonuniform sampling in multidimensional NMR

The full resolution afforded by high-field magnets is rarely realized in the indirect dimensions of multidimensional NMR experiments because of the time cost of uniformly sampling to long evolution times. Emerging methods utilizing nonuniform sampling (NUS) enable high resolution along indirect dimensions by sampling long evolution times without sampling at every multiple of the Nyquist sampling interval. While the earliest NUS approaches matched the decay of sampling density to the decay of the signal envelope, recent approaches based on coupled evolution times attempt to optimize sampling by choosing projection angles that increase the likelihood of resolving closely-spaced resonances. These approaches employ knowledge about chemical shifts to predict optimal projection angles, whereas prior applications of tailored sampling employed only knowledge of the decay rate. In this work we adapt the matched filter approach as a general strategy for knowledge-based nonuniform sampling that can exploit prior knowledge about chemical shifts and is not restricted to sampling projections. Based on several measures of performance, we find that exponentially weighted random sampling (envelope matched sampling) performs better than shift-based sampling (beat matched sampling). While shift-based sampling can yield small advantages in sensitivity, the gains are generally outweighed by diminished robustness. Our observation that more robust sampling schemes are only slightly less sensitive than schemes highly optimized using prior knowledge about chemical shifts has broad implications for any multidimensional NMR study employing NUS. The results derived from simulated data are demonstrated with a sample application to PfPMT, the phosphoethanolamine methyltransferase of the human malaria parasite Plasmodium falciparum.