Nearest-neighbor Distribution Research Articles

Spatial relationships in the urothelial and head and neck tumor microenvironment predict response to combination immune checkpoint inhibitors

Immune checkpoint inhibitors (ICI) can achieve remarkable responses in urothelial cancer (UC), which may depend on tumor microenvironment (TME) characteristics. However, the relationship between the TME, usually characterized by immune cell density, and response to ICI is unclear. Here, we quantify the TME immune cell densities and spatial relationships (SRs) of 24 baseline UC samples, obtained before pre-operative combination ICI treatment, using multiplex immunofluorescence. We describe SRs by approximating the first nearest-neighbor distance distribution with a Weibull distribution and evaluate the association between TME metrics and ipilimumab+nivolumab response. Immune cell density does not discriminate between response groups. However, the Weibull SR metrics of CD8+ T cells or macrophages to their closest cancer cell positively associate with response. CD8+ T cells close to B cells are characteristic of non-response. We validate our SR response associations in a combination ICI cohort of head and neck tumors. Our data confirm that SRs, in contrast to density metrics, are strong biomarkers of response to pre-operative combination ICIs.

An optimization in big data time series prediction method by Parzen estimation with Spark

With the development and change of big data related technologies, more and more large amounts of data need to be analyzed. Now there are companies like Google, Yahoo, etc. Frameworks such as MapReduce, Hadoop, Spark, etc. are developed for processing large amounts of data. In this paper, relevant discussions and researches are carried out on time series forecasting under the new era of big data. Now there are time series forecasting methods based on map reduce, Hadoop, spark data processing framework, including nearest neighbor distribution method, neural network method, etc., which have made quite good achievements in big data time series forecasting. By reading the relevant research literature, it is universally acknowledged that the Sparks framework has good application prospects and potential in predicting big data time series. As a result, this paper is mainly aimed at the optimization and improvement of the big data time series forecasting method on the basis of the spark framework. The author noticed that most of the default configurations of spark clusters are generated by default or automatically, rather than the optimal solution obtained after algorithm optimization, so there is still room for improvement in this regard. In this regard, this paper proposes a kernel method for visual data processing of related configurations and parameters, and then optimizes the default data configuration as much as possible to improve the accuracy and feasibility of the big data time series prediction method on the basis of the spark framework. In this paper, the optimized scheme is used to forecast the domestic electricity consumption in the past five years, and the results show that the optimized scheme has a good improvement performance on the basis of the original method.

Missing levels in intermediate spectra

We derive an expression for the nearest-neighbor spacing distribution P(s) of the energy levels of quantum systems with intermediate dynamics between regularity and chaos and missing levels due to random experimental errors. The expression is based on the Brody distribution, the most widely used for fitting mixed spectra as a function of one parameter. By using Monte Carlo simulations of intermediate spectra based on the β-Hermite ensemble of random matrix theory (RMT), we evaluate the quality of the formula and its suitability for fitting purposes. Estimations of the Brody parameter and the fraction of missing levels can be obtained by a least-square two-parameter fitting of the experimental P(s). The results should be important to distinguish the origins of deviations from RMT in experimental spectra.

Observation of Weibull, Lognormal, and Gamma Distributions in Electrodeposited Cu and Cu-Ag Particles.

In this work, the nearest-neighbor distances and Voronoi cell features of Cu-Ag deposits were analyzed and fitted with Lognormal, Weibull, and Gamma distributions. The nearest-neighbor distance distributions of the samples were compared with those of complete spatially random points, showing spatial inhomogeneity due to the nucleation exclusion effect. The radial distribution function was calculated, showing both influences from the grain size and the nucleation exclusion effect. Voronoi cells were generated based on the shape of the grains. The size, occupancy, and coordination of the Voronoi cells were examined and fitted. The results show that although the Cu-Ag deposits seemed to be governed by the instantaneous nucleation mode, the spatial distribution of the nuclei was more impacted by the nucleation exclusion effect than the Cu-only samples. This behavior is also justified by the grain size distribution generated with Voronoi cell size and occupancy distributions.

Likelihood of the sky

Catalogs with celestial positions and uncertainties are among the primary data products of observational astronomy. Consequently, the (cross-)matching problem, namely, the question of how many (and which) pairs of positions pertain to the same celestial source, remains a central part of this scientific exploitation. We address the matching problem for two catalogs from a purely geometric point of view by adopting the concept of point processes and nearest-neighbor distributions. Suitable expressions for the nearest-neighbor distributions were derived and used to model the distribution of spatial offsets in the sky. Thus, we are able to estimate a total number of matching pairs, along with individual matching probabilities between sources in any two catalogs of positions and uncertainties. We demonstrate the workings of the model using mock data and apply it to cross-matching the X-ray sources of the eROSITA Final Equatorial-Depth survey with opticalGaiacounterparts.

FACAM: A Fast and Accurate Clustering Analysis Method for Protein Complex Quantification in Single Molecule Localization Microscopy

Single molecule localization microscopy (SMLM) enables the analysis and quantification of protein complexes at the nanoscale. Using clustering analysis methods, quantitative information about protein complexes (for example, the size, density, number, and the distribution of nearest neighbors) can be extracted from coordinate-based SMLM data. However, since a final super-resolution image in SMLM is usually reconstructed from point clouds that contain millions of localizations, current popular clustering methods are not fast enough to enable daily quantification on such a big dataset. Here, we provide a fast and accurate clustering analysis method called FACAM, which is modified from the Alpha Shape method (a point dataset analysis method used in many fields). By taking advantage of parallel computation, FACAM is able to process millions of localizations in less than an hour, which is at least 10 times faster than the popular DBSCAN method. Furthermore, FACAM adaptively determines the segmentation threshold, and thus overcomes the problem of user-defined parameters. Using simulation and experimental datasets, we verified the advantages of FACAM over other reported clustering methods (including Ripley’s H, DBSCAN, and ClusterViSu).

Investigation of the Properties of First Nearest Neighbors’ Graphs

In this study we present a benchmark of statistical distributions of the first nearest neighbors in random graphs. We consider distribution of such graphs by the number of disconnected fragments, fragments by the number of involved nodes, and nodes by their degrees. The statements about the asymptotic properties of these distributions for graphs of large dimension are proved. The problem under investigation is to estimate the probability of realization of a certain structure of the first nearest neighbors graph depending on the distribution function of distances between the elements of the studied set. It is shown that, up to isomorphism, the graph of the first nearest neighbors does not depend on the distance distribution. This fact makes it possible to conduct numerical experiments on the construction of basic statistics based on a uniform distribution of distances and obtain tabulated data as a result of numerical modeling. We also discuss the approximation of the distribution of graph vertices by degrees, which allows us to estimate the proportion of randomness for a particular structure resulting from clustering elements of a certain set by the nearest neighbor method. The asymptotic analysis of the fragment distribution is discussed.

Mean nearest-neighbor distance in random 2D binary mixtures

The two-dimensional, mean nearest-neighbor distances, in random sequential addition hard disc computer-generated patterns with mixtures of two circle diameters, have been studied. The mean nearest-neighbor distance at a given area fraction is a quadratic polynomial function of the circle diameter standard deviation. Deviations from the uniform case may be either positive or negative, with large deviations (10–25%) at large circle diameter standard deviations. Nearest-neighbor distance distributions for binary mixture patterns may be well fit by skewed normal distributions.

Disordered Heterogeneous Universe: Galaxy Distribution and Clustering across Length Scales

The studies of disordered heterogeneous media and galaxy cosmology share a common goal: analyzing the disordered distribution of particles and/or building blocks at microscales to predict physical properties of the medium at macroscales, whether it be a liquid, colloidal suspension, composite material, galaxy cluster, or entire Universe. The theory of disordered heterogeneous media provides an array of theoretical and computational techniques to characterize a wide class of complex material microstructures. In this work, we apply them to describe the disordered distributions of galaxies obtained from recent suites of dark matter simulations. We focus on the determination of lower-order correlation functions, void and particle nearest-neighbor functions, certain cluster statistics, pair-connectedness functions, percolation properties, and a scalar order metric to quantify the degree of order. Compared to analogous homogeneous Poisson and typical disordered systems, the cosmological simulations exhibit enhanced large-scale clustering and longer tails in the void and particle nearest-neighbor functions, due to the presence of quasi-long-range correlations imprinted by early Universe physics, with a minimum particle separation far below the mean nearest-neighbor distance. On large scales, the system appears hyperuniform, as a result of primordial density fluctuations, while on the smallest scales, the system becomes almost antihyperuniform, as evidenced by its number variance. Additionally, via a finite-scaling analysis, we compute the percolation threshold of the galaxy catalogs, finding this to be significantly lower than for Poisson realizations (at reduced density ηc=0.25 in our fiducial analysis compared to ηc=0.34), with strong dependence on the mean density; this is consistent with the observation that the galaxy distribution contains voids of up to 50% larger radius. However, the two sets of simulations appear to share the same fractal dimension on scales much larger than the average intergalaxy separation, implying that they lie in the same universality class. We also show that the distribution of galaxies is a highly correlated disordered system (relative to the uncorrelated Poisson distribution), as measured by the τ order metric. Finally, we consider the ability of large-scale clustering statistics to constrain cosmological parameters, such as the Universe’s expansion rate, using simulation-based inference. Both the nearest-neighbor distribution and pair-connectedness function (which includes contributions from correlation functions of all order) are found to considerably tighten bounds on the amplitude of quantum-mechanical fluctuations from inflation at a level equivalent to observing 25 times more galaxies. The pair-connectedness function in particular provides a useful alternative to the standard three-particle correlation, since it contains similar large-scale information to the three-point function, can be computed highly efficiently, and can be straightforwardly extended to small scales (though likely requires simulation-based modeling). This work provides the first application of such techniques to cosmology, providing both a novel system to test heterogeneous media descriptors and a tranche of new tools for cosmological analyses. A range of extensions are possible, including implementation on observational data; this will require further study on various observational effects, necessitating high-resolution simulations.6 MoreReceived 16 July 2022Revised 2 January 2023Accepted 10 January 2023DOI:https://doi.org/10.1103/PhysRevX.13.011038Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasLarge scale structure of the UniverseGravitation, Cosmology & AstrophysicsInterdisciplinary Physics

Dimensional crossover in the nearest-neighbor statistics of random points in a quasi-low-dimensional system

In this work, we study the effects of geometric confinement on the point statistics in a quasi-low-dimensional system. Specifically, we focus on the nearest-neighbor statistics. Accordingly, we have performed comprehensive numerical simulations of binomial point process on quasi-one-dimensional rectangle strips for different values of the confinement ratio defined as the ratio of the strip width to the mean nearest-neighbor distance. We found that the nearest-neighbor distance distributions (NNDDs) conform to an extreme value Weibull distribution with the shape parameter depending on the confinement ratio, while the process intensity remains constant. This finding reveals the reduction of effective spatial degrees of freedom in a quasi-low-dimensional system under the geometric confinement. The scale dependence of the number of effective spatial degrees of freedom is found to obey the crossover ansatz. We stress that the functional form of the crossover ansatz is determined by the nature of the studied point process. Accordingly, different physical processes in the quasi-low-dimensional system obey different crossover ansatzes. The relevance of these results for quasi-low-dimensional systems is briefly highlighted.

Nearest neighbor spatial statistics as a framework for analyzing spatial patterns from image data.

Quantitatively describing spatial patterns in biological substrates is a difficult task that requires careful consideration of labeling methods, imaging modality, and the motivating hypothesis. This problem is commonly addressed using intensity correlation paradigms which correlate the brightness between fluorescence images of co-labeled biomolecules as a measure of their colocalization. These approaches ultimately characterize the superposition of fluorescence signals and thus are limited to binary assessment of spatial relationships at the scale of the image resolution. We’ve previously demonstrated an image analysis framework, which leverages nearest neighbor (NN) distribution functions (F, G, and L) for richer, more informative quantification of the spatial distribution of immunofluorescently imaged biomolecules on all spatial scales captured within the image. Briefly, biomolecule signals and relevant biological structures are segmented, biomolecules are localized relative to landmarks, which are either structures or other biomolecules, by measuring NN distances between biomolecule and landmark voxels, and finally their global spatial association is assessed by comparing the resulting distance distribution to a null hypothesis distribution, in our case a homogenous Poisson distribution which represents complete spatial randomness. The sign of the difference between the distributions indicates repulsion or attraction of the biomolecule to the landmark and the magnitude indicates the relationship's strength. Ongoing efforts aim to further validate this framework by characterizing its sensitivity to image segmentation parameters and signal concentration, and the impact on measured spatial relationships of our methods for generating the null distribution and measuring distances within an image of anisotropic resolution. Preliminary results indicate our framework is robust over a wide range of input parameters and data quality, and our custom methods introduce errors that are smaller than the image's precision. Thus, this approach has wide applicability for scientific image analysis.

Tracer-field cross-correlations with k-nearest neighbour distributions

ABSTRACT In astronomy and cosmology significant effort is devoted to characterizing and understanding spatial cross-correlations between points – e.g galaxy positions, high energy neutrino arrival directions, X-ray and AGN sources, and continuous fields – e.g. weak lensing meiand Cosmic Microwave Background maps. Recently, we introduced the k-nearest neighbour (kNN) formalism to better characterize the clustering of discrete (point) data sets. Here, we extend it to the point – field cross-correlations analysis. It combines kNN measurements of the point data set with measurements of the field smoothed at many scales. The resulting statistics are sensitive to all orders in the joint clustering of the points and the field. We demonstrate that this approach, unlike the 2-pt cross-correlation, can measure the statistical dependence of two data sets even when there are no linear (Gaussian) correlations between them. We further demonstrate that this framework is far more effective than the two point function in detecting cross-correlations when the continuous field is contaminated by high levels of noise. For a particularly high level of noise, the cross-correlation between haloes and the underlying matter field in a cosmological simulation, between 10 h−1 Mpc and 30 h−1 Mpc, is detected at >5σ significance using the technique presented here, when the two-point cross-correlation significance is ∼1σ. Finally, we show that kNN cross-correlations of haloes and the matter field can be well modelled on quasi-linear scales using the Hybrid Effective Field Theory (HEFT) framework, with the same set of bias parameters as are used for 2-pt cross-correlations. The substantial improvement in the statistical power of detecting cross-correlations using this method makes it a promising tool for various cosmological applications.

Comparison of Solute Nanoclusters in Fe–Cr–N Tempered Martensite Observed by Different Atom Probe Instruments

AbstractThe solute clustering behaviors in Fe-1.0mass%Cr-0.3mass%N tempered martensite were comparatively studied by using three atom probe instruments, i.e. LEAP 4000 HR, 5000 XR, and 5000 XS. The microstructural features of nanoclusters were measured under the same conditions in voltage pulse mode, while statistical analysis including nearest neighbor distribution, radial distribution, frequency distribution, and cluster size distribution of Cr atoms were performed on the reconstructed datasets. The results revealed the superior capability of the LEAP 5000 XR to detect Cr nanoclusters in tempered martensite compared with the LEAP 4000 HR as a result of the improvement in detection efficiency and hit detection performance, whereas the high level of background in the mass-to-charge spectrum measured by the LEAP 5000 XS in voltage pulse mode limits its application in the detection of solute nanoclusters.

Applications of the Topological Data Analysis in Real Life

Statistical topology inference is a branch of algebraic topology that analyzes the geometric structure's global topological properties underlying a point cloud dataset. There is an increasing need to analyze massive data sets and screen large databases to address real-world problems. A central challenge to modern applied mathematics is the need to generate tools to simplify the data in high dimensional order to extract the important features or the relationships while performing the analysis. A growing field of study at the intersection of algebraic topology, computational geometry, and statistics is topological data analysis (TDA) inference. This study applies TDA tools to test hypothesis between two high-dimensional data sets. Hypothesis testing is one of the most important topics of statistical topology inference. A proposed test was created, which was built on the nearest-neighbor function. Three tests such as (Hypothesis testing based on persistent homology, hypothesis testing based on persistent landscapes, and hypothesis testing based on density estimation) based on TDA, are discussed. Moreover, a modification of these tests was proposed. Monte Carlo simulation was conducted to compare the power of the previous tests. We displayed the use of TDA tools in hypothesis testing. It was proposed that this test might be established based on the nearest neighbor distance function. Furthermore, a suggested modification for the present tests based on TDA was introduced. Finally, the tests specified in the vignette were enabled by two empirical applications within the biology field. We demonstrated the efficacy of the above tests on the heart disease dataset from Statlog and the Wisconsin breast cancer dataset.

Linear level repulsions near exceptional points of non-Hermitian systems

The nearest-neighbor level-spacing distributions are a fundamental quantity of disordered systems and are classified into different universality classes. They are the Wigner-Dyson and the Poisson functions for extended and localized states in Hermitian systems, respectively. The distributions follow the Ginibre functions for the non-Hermitian systems whose eigenvalues are complex and away from exceptional points (EPs). However, the level-spacing distributions of disordered non-Hermitian systems near EPs are still unknown, and a corresponding random matrix theory is absent. Here, we show another class of universal level-spacing distributions in the vicinity of EPs of non-Hermitian Hamiltonians. Two distribution functions, ${P}_{\text{SP}}(s)$ for the symmetry-preserved phase and ${P}_{\text{SB}}(s)$ for the symmetry-broken phase, are needed to describe the nearest-neighbor level-spacing distributions near EPs. Surprisingly, both ${P}_{\text{SP}}(s)$ and ${P}_{\text{SB}}(s)$ are proportional to $s$ for small $s$, or linear level repulsions, in contrast to cubic level repulsions of the Ginibre ensembles. For disordered non-Hermitian tight-binding Hamiltonians, ${P}_{\text{SP}}(s)$ and ${P}_{\text{SB}}(s)$ can be well described by a surmise ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{\text{ep}}(s)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{1}sexp[\ensuremath{-}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{2}{s}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}}]$ in the thermodynamic limit (infinite systems) with a constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}$ that depends on the localization nature of states at EPs rather than the dimensionality of non-Hermitian systems and the order of EPs.

Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at noninteger β.

A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature β. For 2×2 matrices with Gaussian distribution we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective β_{eff}(β) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of β. It reproduces the 2D Poisson distribution at β=0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values β=1.4 and β=2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.

Fluctuation properties of the eigenfrequencies and scattering matrix of closed and open unidirectional graphs with chaotic wave dynamics.

We present experimental and numerical results for the fluctuation properties in the eigenfrequency spectra and of the scattering matrix of closed and open unidirectional quantum graphs, respectively. Unidirectional quantum graphs, that are composed of bonds connected by reflectionless vertices, were introduced by Akila and Gutkin [Akila and Gutkin, J. Phys. A: Math. Theor. 48, 345101 (2015)1751-811310.1088/1751-8113/48/34/345101]. The nearest-neighbor spacing distribution of their eigenvalues was shown to comply with random-matrix theory predictions for typical chaotic systems with completely violated time-reversal invariance. The occurrence of short periodic orbits confined to a fraction of the system, that lead in conventional quantum graphs to deviations of the long-range spectral correlations from the behavior expected for typical chaotic systems, is suppressed in unidirectional ones. Therefore, we pose the question whether such graphs may serve as a more appropriate model for closed and open chaotic systems with violated time-reversal invariance than conventional ones. We compare the fluctuation properties of their eigenvalues and scattering matrix elements and observe especially in the long-range correlations larger deviations from random-matrix theory predictions for the unidirectional graphs. These are attributed to a loss of complexity of the underlying dynamic, induced by the unidirectionality.

Detection of spatial clustering in the 1000 richest SDSS DR8 redMaPPer clusters with nearest neighbor distributions

ABSTRACT Distances to the k-nearest-neighbor (kNN) data points from volume-filling query points are a sensitive probe of spatial clustering. Here, we present the first application of kNN summary statistics to observational clustering measurement, using the 1000 richest redMaPPer clusters (0.1 ≤ z ≤ 0.3) from the SDSS DR8 catalog. A clustering signal is defined as a difference in the cumulative distribution functions (CDFs) of kNN distances from fixed query points to the observed clusters versus a set of unclustered random points. We find that the k = 1, 2-NN CDFs of redMaPPer deviate significantly from the randoms’ across scales of 35 to 155 Mpc, which is a robust signature of clustering. In addition to kNN, we also measure the two-point correlation function for the same set of redMaPPer clusters versus random points, which shows a noisier and less significant clustering signal within the same radial scales. Quantitatively, the χ2 distribution for both the kNN-CDFs and the two-point correlation function measured on the randoms peak at χ2 ∼ 50 (null hypothesis), whereas the kNN-CDFs (χ2 ∼ 300, p = 1.54 × 10−36) pick up a much more significant clustering signal than the two-point function (χ2 ∼ 100, p = 1.16 × 10−6) when measured on redMaPPer. Finally, the measured 3NN and 4NN CDFs deviate from the predicted k = 3, 4-NN CDFs assuming an ideal Gaussian field, indicating a non-Gaussian clustering signal for redMaPPer clusters, although its origin might not be cosmological due to observational systematics. Therefore, kNN serves as a more sensitive probe of clustering complementary to the two point correlation function, providing a novel approach for constraining cosmology and galaxy–halo connection.

Local Order in AgAuCuPdPt High-Entropy Alloy Surfaces

Synthesis of high-entropy alloys often involves heating precursors to above 1000 °C such that entropy stabilization takes effect. The resulting alloys are characterized by random element distribution at the bulk lattice positions. Surfaces may however be more ordered, even at high temperatures. We explore the local order at (111), (100), and (533) surfaces of the equimolar face-centered cubic (fcc) AgAuCuPdPt high-entropy alloy at high temperatures and thermal equilibrium. We find that the local order is significantly increased at the (100) surface and to a lesser extent at the (533) and (111) surfaces compared to bulk AgAuCuPdPt. The (100) surface both segregates with increased amounts of Au and Ag and less Pd and Pt and has a more ordered distribution of nearest-neighbor atom pairs. The (111) surface segregates with increased amounts of Au and Ag and less Cu, Pd, and Pt, but the nearest-neighbor distribution is mostly random. The (100)-type step edge of the (533) surface resembles the (100) surface. The degree of surface order seems linked to the structure ensemble’s energy distribution width, so we suggest that the width can be used to estimate the degree of order in high-entropy alloys with minimal computational efforts.

Structure of cosmic web in non-linear regime: the nearest neighbour and spherical contact distributions

ABSTRACT In non-linear scales, the matter density distribution is not Gaussian. Consequently, the widely used two-point correlation function is not adequate anymore to capture the matter density field’s entire behaviour. Among all statistics beyond correlation functions, the spherical contact (or equivalently void function), and nearest neighbour distribution function seem promising tools to probe matter distribution in non-linear regime. In this work, we use haloes from cosmological N-body simulations, galaxy groups from the volume-limited galaxy group and central galaxies from mock galaxy catalogues, to compare the spherical contact with the nearest neighbour distribution functions. We also calculate the J-function (or equivalently the first conditional correlation function), for different samples. Moreover, we consider the redshift evolution and mass-scale dependence of statistics in the simulations and dependence on the magnitude of volume-limited samples in group catalogues as well as the mock central galaxies. The shape of the spherical contact probability distribution function is nearly skew-normal, with skewness and kurtosis being approximately 0.5 and 3, respectively. On the other hand, the nearest neighbour probability distribution function is nearly lognormal, with logarithmic skewness and kurtosis being approximately 0.1 and 2.5, respectively. Accordingly, the spherical contact distribution function probes larger scales compared to the nearest neighbour distribution function, which is influenced by details of structures. We also find a linear relation between the mean and variance of the spherical contact probability distribution function in simulations and mock galaxies, which could be used as a distinguishing probe of cosmological models.