Higher-order Spatial Correlations Research Articles

Measuring nonlocal three-body spatial correlations with Rydberg trimers in ultracold quantum gases

We measure nonlocal third-order spatial correlations in nondegenerate ultracold gases of bosonic ($^{84}\mathrm{Sr}$) and spin-polarized fermionic ($^{87}\mathrm{Sr}$) strontium through studies of the formation rates for ultralong-range trimer Rydberg molecules. The trimer production rate is observed to be very sensitive to the effects of quantum statistics with a strong enhancement of up to a factor of 6 (3!), in the case of bosonic $^{84}\mathrm{Sr}$ due to bunching and a marked reduction for spin-polarized fermionic $^{87}\mathrm{Sr}$ due to antibunching. The experimental results are compared to theoretical predictions and good agreement is observed. The present approach opens the way to in situ studies of higher-order nonlocal spatial correlations in a wide array of ultracold atomic-gas systems.

Quantifying microstructures of earth materials using higher-order spatial correlations and deep generative adversarial networks

The key to most subsurface processes is to determine how structural and topological features at small length scales, i.e., the microstructure, control the effective and macroscopic properties of earth materials. Recent progress in imaging technology has enabled us to visualise and characterise microstructures at different length scales and dimensions. However, one limitation of these technologies is the trade-off between resolution and sample size (or representativeness). A promising approach to this problem is image reconstruction which aims to generate statistically equivalent microstructures but at a larger scale and/or additional dimension. In this work, a stochastic method and three generative adversarial networks (GANs), namely deep convolutional GAN (DCGAN), Wasserstein GAN with gradient penalty (WGAN-GP), and StyleGAN2 with adaptive discriminator augmentation (ADA), are used to reconstruct two-dimensional images of two hydrothermally rocks with varying degrees of complexity. For the first time, we evaluate and compare the performance of these methods using multi-point spatial correlation functions—known as statistical microstructural descriptors (SMDs)—ultimately used as external tools to the loss functions. Our findings suggest that a well-trained GAN can reconstruct higher-order, spatially-correlated patterns of complex earth materials, capturing underlying structural and morphological properties. Comparing our results with a stochastic reconstruction method based on a two-point correlation function, we show the importance of coupling training/assessment of GANs with higher-order SMDs, especially in the case of complex microstructures. More importantly, by quantifying original and reconstructed microstructures via different GANs, we highlight the interpretability of these SMDs and show how they can provide valuable insights into the spatial patterns in the synthetic images, allowing us to detect common artefacts and failure cases in training GANs.

Mitigating ocean random medium effects in array processing by using a cross-correlator beamformer instrumentation

Ocean random medium effects such as internal waves can cause distortions to signal wave fronts that are highly detrimental to detecting and localizing signals with acoustic arrays. Motivated by the lucky imaging techniques used in astronomy [1], we have previously found evidence in real array data that when the data are looked at a much finer time scales, there can be brief moments when the wave front has little distortion [2]. However, the challenge has been developing metrics for ranking the quality of the data frames in order to exploit these lucky scintillations. In this paper, we propose a simple frame quality metric based on a cross-correlator instrumentation of the delay-sum beamformer [3] evaluated by arithmetically averaging the higher-order spatial correlation lags along the time delay trajectory corresponding to the beamformer focal location and then utilizing only the highest ranked frames in the localization. We connect this metric to beamformer focus sharpness by showing that the main lobe curvature increases monotonically with the number of lags used and demonstrate the approach on real underwater acoustic array data. [1] https://www.ast.cam.ac. uk/research/lucky [2] I.P. Kirsteins, JASA 143, 1976 (2018). [3] E. Ruigrok et al., “Cross-correlation beamforming,” J. Seismol. 21, 495–508 (2017).

A Higher-Order Motif-Based Spatiotemporal Graph Imputation Approach for Transportation Networks

Due to the incomplete coverage and failure of traffic data collectors during the collection, traffic data usually suffers from information missing. Achieving accurate imputation is critical to the operation of transportation networks. Existing approaches usually focus on the characteristic analysis of temporal variation and adjacent spatial representation, and the consideration of higher-order spatial correlations and continuous data missing attracts more attentions from the academia and industry. In this paper, by leveraging motif-based graph aggregation, we propose a spatiotemporal imputation approach to address the issue of traffic data missing. First, through motif discovery, the higher-order graph aggregation model was presented in traffic networks. It utilized graph convolution network (GCN) to polymerize the correlated segment attributes of the missing data segments. Then, the multitime dimension imputation model based on bidirectional long short-term memory (Bi-LSTM) incorporated the recent, daily-periodic, and weekly-periodic dependencies of the historical data. Finally, the spatial aggregated values and the temporal fusion values were integrated to obtain the results. We conducted comprehensive experiments based on the real-world dataset and discussed the case of random and continuous data missing by different time intervals, and the results showed that the proposed approach was feasible and accurate.

Exact closed-form analytic wave functions in two dimensions: Contact-interacting fermionic spinful ultracold atoms in a rapidly rotating trap

Exact two-dimensional analytic wave functions for an arbitrary number $N$ of contact-interacting lowest-Landau-level (LLL) spinful fermions are derived with the use of combined numerical and symbolic computational approaches via analysis of exact Hamiltonian numerical diagonalization data. Closed-form analytic expressions are presented for two families of zero-interaction-energy states at given total angular momentum and total spin $0 \leq S \leq N/2$ in the neighborhood of the $\nu=1$ filling, covering the range from the maximum density droplet to the first quasihole. Our theoretical predictions for higher-order spatial and momentum correlations reveal intrinsic polygonal, multi-ring crystalline-type structures, which can be tested with ultracold-atom experiments in rapidly rotating traps, simulating quantum Hall physics (including quantum LLL skyrmions).

Learning complex texture discrimination.

Higher-order spatial correlations contribute strongly to visual structure and salience, and are common in the natural environment. One method for studying this structure has been through the use of highly controlled texture patterns whose obvious structure is defined entirely by third- and higher-order correlations. Here we examine the effects that longer-term training has on discrimination of 17 such texture types. Training took place in 14 sessions over 42 days. Discrimination performance increased at different rates for different textures. The time required to complete a visit reduced by 25.4% (p=0.0004). Factor analysis was applied to data from the learning and experienced phases of the experiment. This indicated that the gain in speed was accompanied by an increase in the number of mechanisms contributing to discrimination. Learning was not affected by sleep quality but was affected by extreme tiredness (p<0.01). The improved discrimination and speed were retained for 2.5 months. Overall, the effects were consistent with perceptual learning. The observed learning is likely related to the adaptation of innate mechanisms that underlie our ability to identify nonredundant, visually salient structure in natural images. It may involve cortical V2 and appears to involve increased strength, speed, and breadth of connections within our internal representation of this complex perceptual space.

Nonlinear Imaging with All-Dielectric Metasurfaces.

Nonlinear metasurfaces incorporate many of the functionalities of their linear counterparts such as wavefront shaping, but simultaneously they perform nonlinear optical transformations. This dual functionality leads to a rather unintuitive physical behavior which is still widely unexplored for many photonic applications. The nonlinear processes render some basic principles governing the functionality of linear metasurfaces. Exemplarily, the superposition principle and the geometric optics approximation become not directly applicable to nonlinear metasurfaces. On the other hand, nonlinear metasurfaces facilitate new phenomena that are not possible in the linear regime. Here, we study the imaging of objects through a dielectric nonlinear metalens. We illuminate objects by infrared light and record their generated images at the visible third-harmonic wavelengths. We revisit the classical lens theory and suggest a generalized Gaussian lens equation for nonlinear imaging, verified both experimentally and analytically. We also demonstrate experimentally higher-order spatial correlations facilitated by the nonlinear metalens, resulting in additional image features.

Higher-order spatial correlation coefficients of ultrasonic backscattering signals using partial cross-correlation analysis.

The spatial correlation properties of ultrasonic backscattering signals in random media have important implications. For example, they can be used for microstructural characterization and flaw detection in engineering materials. However, the traditional spatial correlation coefficient (SCC) is only a leading order quantity that does not capture the true spatial correlations of random media. This is caused by neglecting confounding variables such as non-zero means or other non-zero odd-order moments. Here, the SCC is generalized from zeroth- to general-order through partial cross-correlation analysis. A series of indicators are defined to quantify the SCC curve at zero time lag, and the maximum time shift curve, which are both functions of lateral separation between two sensor positions. A stainless-steel specimen and a focused ultrasonic transducer are used to verify the method. Scattering measurements show that the higher-order SCC can consistently capture spatial correlations whereas the zeroth-order SCC is inadequate. The zeroth-order SCC is shown to predict a step size that can be more than six times too large. Thus, the present method can provide better understanding of statistical correlations and conditions to measure uncorrelated backscattering signals.

ANCA: Anharmonic Conformational Analysis of Biomolecular Simulations

Anharmonicity in time-dependent conformational fluctuations is noted to be a key feature of functional dynamics of biomolecules. Although anharmonic events are rare, long-timescale (μs–ms and beyond) simulations facilitate probing of such events. We have previously developed quasi-anharmonic analysis to resolve higher-order spatial correlations and characterize anharmonicity in biomolecular simulations. In this article, we have extended this toolbox to resolve higher-order temporal correlations and built a scalable Python package called anharmonic conformational analysis (ANCA). ANCA has modules to: 1) measure anharmonicity in the form of higher-order statistics and its variation as a function of time, 2) output a storyboard representation of the simulations to identify key anharmonic conformational events, and 3) identify putative anharmonic conformational substates and visualization of transitions between these substates.

Material structure-property linkages using three-dimensional convolutional neural networks

The core materials knowledge needed in the accelerated design, development, and deployment of new and improved materials is most accessible when cast in the form of computationally low cost (reduced-order) and reliable process-structure-property (PSP) linkages. Quantification of the material structure (also referred as microstructure) is the core challenge in this task. Conventionally, microstructure quantification has been addressed using highly simplified measures suggested by the governing physics, with the list of measures often suitably augmented by the intuition of the materials expert. In this paper, we develop an objective (data-driven) approach to efficiently and accurately link a three-dimensional (3-D) microstructure to its effective (homogenized) properties. Our method employs a 3-D convolutional neural network (CNN) to learn the salient features of the material microstructures that lead to good predictive performance for the effective property of interest. We then utilize 3-D CNN learned features as estimators of higher-order spatial correlations, and formulate an integrated framework combining 3-D CNN features with 2-point spatial correlations. In this work, we created an extremely large microstructure-property benchmark dataset of 5900 microstructures, and demonstrated that our CNN based approach not only learns interpretable microstructure features, but also leads to improved accuracy in property predictions for new microstructures, while achieving a dramatic reduction in the computation time.

Recurrent Network Dynamics; a Link between Form and Motion.

To discriminate visual features such as corners and contours, the brain must be sensitive to spatial correlations between multiple points in an image. Consistent with this, macaque V2 neurons respond selectively to patterns with well-defined multipoint correlations. Here, we show that a standard feedforward model (a cascade of linear–non-linear filters) does not capture this multipoint selectivity. As an alternative, we developed an artificial neural network model with two hierarchical stages of processing and locally recurrent connectivity. This model faithfully reproduced neurons’ selectivity for multipoint correlations. By probing the model, we gained novel insights into early form processing. First, the diverse selectivity for multipoint correlations and complex response dynamics of the hidden units in the model were surprisingly similar to those observed in V1 and V2. This suggests that both transient and sustained response dynamics may be a vital part of form computations. Second, the model self-organized units with speed and direction selectivity that was correlated with selectivity for multipoint correlations. In other words, the model units that detected multipoint spatial correlations also detected space-time correlations. This leads to the novel hypothesis that higher-order spatial correlations could be computed by the rapid, sequential assessment and comparison of multiple low-order correlations within the receptive field. This computation links spatial and temporal processing and leads to the testable prediction that the analysis of complex form and motion are closely intertwined in early visual cortex.

A lower bound on the number of mechanisms for discriminating fourth and higher order spatial correlations

The human visual system must employ mechanisms to minimize informational redundancy whilst maximizing dynamic range and maintaining that which is behaviorally relevant [1,2]. Previous research has concentrated on two-point correlation properties, as captured by spatial frequency and orientation tuning. There has been less research into higher-order correlations although they may inform us about cortical functioning [3]. Isotrigon textures can be used to probe the sensitivity of the human visual system. The obvious structure in isotrigons is exclusively due to 4th and higher-order spatial correlations [4]. Thus, in order to discriminate isotrigons from noise, it is necessary to identify higher-order structure. Although artificially generated, the same structural features that give isotrigons salience also create salience in natural images [2]. Factor analysis can be used to infer the number of underlying independent neurological mechanisms which govern isotrigon discrimination. In this study, mean performance functions were calculated for two subjects using ten new isotrigons (VnL2) (Figure ​(Figure1A).1A). Two forms of factor analysis identified 3 principal factors (Figure ​(Figure1B)1B) [5]. Previous studies support that the number of mechanisms is less than 10 [6], and more likely 2-4 [7,8]. Such mechanisms may represent some combination of recursive or rectifying processes. Simple models of cortical processing, based on recursion, can discriminate isotrigons [9]. The formation of recursively applied products is physiologically plausible and can occur via dendritic back-propagation or dendritic spiking [10]. Figure 1 A: Mean performance functions for all subjects by texture type, presented separately according to subject and texture size. For All16 (16x16 textures) and All32 (32x32) error bars are SE for n = 6 subjects. Glider shapes are shown in the bottom panel. ...

Developing and validating an isotrigon texture discrimination task using Amazon Mechanical Turk

The human visual system must employ mechanisms to minimize informational redundancy whilst maintaining that which is behaviorally relevant [1,2]. Previous research has concentrated on two-point correlations via spatial frequency and orientation tuning. Higher-order correlations are less studied, but they may inform us about cortical functioning [3]. Isotrigon textures can be used to probe the sensitivity of the human visual system as their structure is exclusively due to 4th and higher-order spatial correlations [4]. Although artificially generated, the same features that give isotrigons salience also create salience in natural images [2]. We implemented an isotrigon discrimination task using the crowdsourcing platform Amazon Mechanical Turk (mTurk) [5]. An important secondary aim was to evaluate the suitability of mTurk for visual psychometric studies as very few exist [6]. 960 HITs were uploaded to mTurk and 121 naive subjects participated. Based on data quality, 91% of HITs were retained at a cost of $0.132 AUD per HIT. The mTurk data was compared to two supervised lab datasets. Lab and mTurk performance functions were very similar (Figure ​(Figure1A)1A) and highly correlated (Figure ​(Figure1B).1B). Bland-Altman plots were examined and the mean lab/mTurk coefficient of repeatability was 15.5%. Factor analysis was performed on the combined data and 2 principal factors were identified. Previous studies support that the number of mechanisms is less than 10 [7] and more likely 2-4 [8,9]. The congruence between the lab and mTurk data is striking considering the unsupervised mode of delivery. In conclusion, mTurk is an underutilized platform for visual psychometric research which can produce data of comparable quality to lab samples at reduced cost and increased scale. Figure 1 1A: Color map of median texture discrimination performance versus isotrigon texture type. Lab datasets DC (84 HITs) and M1-6 (270 HITs). mTurk datasets Live1 (480 HITs) and Live2 (480 HITs). 1B: Pearson's correlation coefficients between datasets (abbreviations ...

The Effects of Gridding Algorithms on the Statistical Moments and Their Trends of Daily Surface Air Temperature*

Abstract This paper explores the effects from averaging weather station data onto a grid on the first four statistical moments of daily minimum and maximum surface air temperature (SAT) anomalies over the entire globe. The Global Historical Climatology Network–Daily (GHCND) and the Met Office Hadley Centre GHCND (HadGHCND) datasets from 1950 to 2010 are examined. The GHCND station data exhibit large spatial patterns for each moment and statistically significant moment trends from 1950 to 2010, indicating that SAT probability density functions are non-Gaussian and have undergone characteristic changes in shape due to decadal variability and/or climate change. Comparisons with station data show that gridded averages always underestimate observed variability, particularly in the extremes, and have altered moment trends that are in some cases opposite in sign over large geographic areas. A statistical closure approach based on the quasi-normal approximation is taken to explore SAT’s higher-order moments and point correlation structure. This study focuses specifically on relating variability calculated from station data to that from gridded data through the moment equations for weighted sums of random variables. The higher-order and nonlinear spatial correlations up to the fourth order demonstrate that higher-order moments at grid scale can be determined approximately by functions of station pair correlations that tend to follow the usual Kolmogorov scaling relation. These results can aid in the development of constraints to reduce uncertainties in climate models and have implications for studies of atmospheric variability, extremes, and climate change using gridded observations.

Angular power spectra with finite counts

Angular anisotropy techniques for cosmic diffuse radiation maps are powerful probes, even for quite small data sets. A popular observable is the angular power spectrum; we present a detailed study applicable to any unbinned source skymap S(n) from which N random, independent events are observed. Its exact variance, which is due to the finite statistics, depends only on S(n) and N; we also derive an unbiased estimator of the variance from the data. First-order effects agree with previous analytic estimates. Importantly, heretofore unidentified higher-order effects are found to contribute to the variance and may cause the uncertainty to be significantly larger than previous analytic estimates---potentially orders of magnitude larger. Neglect of these higher-order terms, when significant, may result in a spurious detection of the power spectrum. On the other hand, this would indicate the presence of higher-order spatial correlations, such as a large bispectrum, providing new clues about the sources. Numerical simulations are shown to support these conclusions. Applying the formalism to an ensemble of Gaussian-distributed skymaps, the noise-dominated part of the power spectrum uncertainty is significantly increased at high multipoles by the new, higher-order effects. This work is important for harmonic analyses of the distributions of diffuse high-energy gamma-rays, neutrinos, and charged cosmic rays, as well as for populations of sparse point sources such as active galactic nuclei.

A lower bound on the number of mechanisms for discriminating fourth and higher order spatial correlations

Research on single striate cortical neurons has often concentrated on their responses to stimuli defined by two-point correlations. Texture discrimination studies using a relatively small palette of isotrigon textures have indicated that we are sensitive to third and higher-order spatial correlations.To further evaluate the underlying mechanisms of texture discrimination subjects discriminated random binary noise patterns from ten new isotrigon texture types.Factor analysis revealed that as few as three mechanisms may govern the detection of fourth and higher order image structure. This supports the findings of previous studies using different isotrigon textures.The computation of higher-order correlations by the brain is neurophysiologically plausible. The mechanisms identified in this study may represent some short range nonlinear combination of recursive and/or rectifying processes.

Fourth-Order Spatial Correlation of Thermal Light

We investigate the fourth-order spatial correlation properties of pseudo-thermal light in the photon counting regime, and apply the Klyshko advanced-wave picture to describe the process of four-photon coincidence counting measurement. We deduce the theory of a proof-of-principle four-photon coincidence counting configuration, and find that if the four randomly radiated photons come from the same radiation area and are indistinguishable in principle, the fourth-order correlation of them is 24 times larger than that when four photons come from different radiation areas. In addition, we also show that the higher-order spatial correlation function can be decomposed into multiple lower-order correlation functions, and the contrast and visibility of low-order correlation peaks are less than those of higher orders, while the resolutions all are identical. This study may be useful for better understanding the four-photon interference and multi-channel correlation imaging.

Multifractal scaling of flux penetration in the iron-based superconductor Ba(Fe0.93Co0.07)2As2

The penetration of magnetic flux fronts in the optimally doped iron-based superconductor Ba(Fe${}_{1\ensuremath{-}x}$Co${}_{x}$)${}_{2}$As${}_{2}$ ($x=0.07\ifmmode\pm\else\textpm\fi{}0.005$) is studied by means of magneto-optical imaging and Bitter decoration. The higher-order analysis of roughening and growth of the magnetic flux front reveals anomalous scaling properties, indicative of non-Gaussian correlations of the disorder potential. While higher-order spatial correlation functions reveal multifractal behavior for the roughening, the usual Kardar-Parisi-Zhang growth exponent is found. Both exponents are found to be independent of temperature. The scaling behavior is manifestly different from that found for other modes of flux penetration, such as that mediated by avalanches, suggesting that multiscaling is a powerful tool for the characterization of roughened interfaces. We propose a scenario for vortex penetration based on two-dimensional percolation and cluster aggregation for an inhomogeneously disordered superconductor.

A multiscale maximum entropy moment closure for locally regulated space–time point process models of population dynamics

The prevalence of structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based only on mean densities (local or global). Individual-based models (IBMs) were introduced during the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite useful, the capability to follow each individual usually comes at the expense of analytical tract ability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space-time point processes with local regulation seem to cover the middle ground between analytical tract ability and a higher degree of biological realism. This approach has shown that simplified representations of fecundity, local dispersal and density-dependent mortality weighted by the local competitive environment are sufficient to generate spatial patterns that mimic field observations. Continuum approximations of these stochastic processes try to distill their fundamental properties, and they keep track of not only mean densities, but also higher order spatial correlations. However, due to the non-linearities involved they result in infinite hierarchies of moment equations. This leads to the problem of finding a 'moment closure'; that is, an appropriate order of (lower order) truncation, together with a method of expressing the highest order density not explicitly modelled in the truncated hierarchy in terms of the lower order densities. We use the principle of constrained maximum entropy to derive a closure relationship for truncation at second order using normalisation and the product densities of first and second orders as constraints, and apply it to one such hierarchy. The resulting 'maxent' closure is similar to the Kirkwood superposition approximation, or 'power-3' closure, but it is complemented with previously unknown correction terms that depend mainly on the avoidance function of an associated Poisson point process over the region for which third order correlations are irreducible. This domain of irreducible triplet correlations is found from an integral equation associated with the normalisation constraint. This also serves the purpose of a validation check, since a single, non-trivial domain can only be found if the assumptions of the closure are consistent with the predictions of the hierarchy. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the truncated hierarchy to predict equilibrium values for mildly aggregated spatial patterns. However, the maxent closure performs comparatively poorly in segregated ones. Although the closure is applied in the context of point processes, the method does not require fixed locations to be valid, and can in principle be applied to problems where the particles move, provided that their correlation functions are stationary in space and time.

Large fluctuations of the macroscopic current in diffusive systems: A numerical test of the additivity principle

Most systems, when pushed out of equilibrium, respond by building up currents of locally conserved observables. Understanding how microscopic dynamics determines the averages and fluctuations of these currents is one of the main open problems in nonequilibrium statistical physics. The additivity principle is a theoretical proposal that allows to compute the current distribution in many one-dimensional nonequilibrium systems. Using simulations, we validate this conjecture in a simple and general model of energy transport, both in the presence of a temperature gradient and in canonical equilibrium. In particular, we show that the current distribution displays a Gaussian regime for small current fluctuations, as prescribed by the central limit theorem, and non-Gaussian (exponential) tails for large current deviations, obeying in all cases the Gallavotti-Cohen fluctuation theorem. In order to facilitate a given current fluctuation, the system adopts a well-defined temperature profile different from that of the steady state and in accordance with the additivity hypothesis predictions. System statistics during a large current fluctuation is independent of the sign of the current, which implies that the optimal profile (as well as higher-order profiles and spatial correlations) are invariant upon current inversion. We also demonstrate that finite-time joint fluctuations of the current and the profile are well described by the additivity functional. These results suggest the additivity hypothesis as a general and powerful tool to compute current distributions in many nonequilibrium systems.