Probability Distribution Function Research Articles

Inertial active harmonic particle with memory induced spreading by viscoelastic suspension.

We investigate the self-propulsion of an inertial active particle confined in a two-dimensional harmonic trap. The particle is suspended in a non-Newtonian or viscoelastic suspension with a friction kernel that decays exponentially with a time constant characterizing the memory timescale or transient elasticity of the medium. By solving the associated non-Markovian dynamics, we identify two regimes in parameter space distinguishing the oscillatory and non-oscillatory behavior of the particle motion. By simulating the particle trajectories and exactly calculating the steady-state probability distribution functions and mean square displacement; interestingly, we observe that with an increase in the memory time scale, the effective temperature of the environment increases. As a consequence, the particle becomes energetic and spread away from the center, covering larger space inside the confinement. On the other hand, with an increase in the duration of the activity, the particle becomes trapped by the harmonic confinement.

Assessing uncertainty in building material emissions using scenario-aware Monte Carlo simulation

Global greenhouse gas emissions from the built environment remain high, driving innovative approaches to develop and adopt building materials that can mitigate some of those emissions. However, life-cycle assessment (LCA) practices still lack standardized quantitative uncertainty assessment frameworks, which are urgently needed to robustly assess mitigation efforts. Previous works emphasize the importance of accounting for the three types of uncertainties that may exist within any quantitative assessment: parameter, scenario, and model uncertainty. Herein, we develop a quantitative uncertainty assessment framework that distinguishes between different types of uncertainties and suggest how these uncertainties could be handled systematically through a scenario-aware Monte Carlo simulation (MCS). We demonstrate the framework’s decision-informing power through a case study of two multilevel ordinary Portland cement (OPC) manufacturing scenarios. The MCS utilizes a first-principles-based OPC life-cycle inventory, which mitigates some of the model uncertainty that may exist in other empirical-based cement models. Remaining uncertainties are handled by scenario specification or sampling from developed probability distribution functions. We also suggest a standardized method for fitting distributions to parameter data by enumerating through and implementing distributions based on the Kolmogorov–Smirnov test. The level of detail brought by the high-resolution parameter breakdown of the model allows for developing emission distributions for each process of OPC manufacturing. This approach highlights how specific parameters, along with scenario framing, can impact overall OPC emissions. Another key takeaway includes relating the uncertainty of each process to its contributions to total OPC emissions, which can guide LCA modelers in allocating data collection and refinement efforts to processes with the highest contribution to cumulative uncertainty. Ultimately, the aim of this work is to provide a standardized framework that can provide robust estimates of building material emissions and be readily integrated within any uncertainty assessment.

Molecular cloud matching in CO and dust in M33. I. High-resolution hydrogen column density maps from Herschel

This study is aimed to contribute to a more comprehensive understanding of the molecular hydrogen distribution in the galaxy M33 by introducing novel methods for generating high angular resolution ($18.2''$, equivalent to $75\ pc $ for a distance of $847\ kpc $) column density maps of molecular hydrogen ($N_ H_2 $). M33 is a local group galaxy that has been observed with Herschel in the far-infrared (FIR) wavelength range from $70$ to $500\ Previous studies have presented total hydrogen column density maps ($N_ H $), using these FIR data (partly combined with mid-IR maps), employing various methods. We first performed a spectral energy distribution (SED) fit to the $160$, $250$, $350,$ and $500\ continuum data obtain $N_ H $, using a technique similar to one previously reported in the literature. We also use a second method which involves translating only the $250\ map into a $N_ H $ map at the same angular resolution of $18.2''$. An $N_ H_2 $ map via each method is then obtained by subtracting the component. Distinguishing our study from previous ones, we adopt a more versatile approach by considering a variable emissivity index, $ and dust absorption coefficient, $ This choice enables us to construct a $ map, thereby enhancing the depth and accuracy of our investigation of the hydrogen column density. We address the inherent biases and challenges within both methods (which give similar results) and compare them with existing maps available in the literature. Moreover, we calculate a map of the carbon monoxide $ CO(1-0) $-to-molecular hydrogen (htwo ) conversion factor ( factor), which shows a strong dispersion around an average value of $1.8 throughout the disk. We obtain column density probability distribution functions ($N$-PDFs) from the $N_ H $, $N_ H_2 $, and $N_ hi $ maps and discuss their shape, consisting of several log-normal and power-law tail components.

Information propagation of focus wave mode localized waves in anisotropic turbulent seawater

The evolution of the information transfer capability of an optical system for underwater focused wave mode localized wave (FWMLW) in anisotropic weakly turbulent absorbing seawater is studied. By developing the probability distribution function as well as the detection probability of the vortex modes carried by the FWMLW and the average bit error rate of the FWMLW underwater system, the information capacity of the FWMLW system with a pointing error is modeled. Through a numerical analysis of the effects of turbulent seawater and optical system parameters on the built light intensity, the detection probability, and the information capacity models, we find that the FWMLW system has an optimal delay time determined by the spectrum bandwidth when the spectrum bandwidth is greater than 1. The information capacity of the FWMLW system is higher than that of the X localized wave system under the same turbulent seawater channel condition, and FWMLW is a better optical signal source for vortex mode division multiplexing underwater systems than a Bessel–Gaussian beam.

New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method.

Observed and future shifts in climate zone of Borneo based on CMIP6 models

Climate change has significantly altered the characteristics of climate zones, posing considerable challenges to ecosystems and biodiversity, particularly in Borneo, known for its high species density per unit area. This study aimed to classify the region into homogeneous climate groups based on long-term average behavior. The most effective parameters from the high-resolution daily gridded Princeton climate datasets spanning 65 years (1950–2014) were utilized, including rainfall, relative humidity (RH), temperatures (Tavg, Tmin, Tmax, and diurnal temperature range (DTR)), along with elevation data at 0.25° resolution. The FCM clustering method outperformed K-Mean and two Ward's hierarchical methods (WardD and WardD2) in classifying Borneo's climate zones based on multi-criteria assessment, exhibiting the lowest average distance (2.172–2.180) and the highest compromise programming index (CPI)-based correlation ranking among cluster averages across all climate parameters. Borneo's climate zones were categorized into four: ‘Wet and cold’ (WC) and ‘Wet’ (W) representing wetter zones, and ‘Wet and hot’ (WH) and ‘Dry and hot’ (DH) representing hotter zones, each with clearly defined boundaries. For future projection, EC-Earth3-Veg ranked first for all climate parameters across 961 grid points, emerging as the top-performing model. The linear scaling (LS) bias-corrected EC-Earth3-Veg model, as shown in the Taylor diagram, closely replicated the observed datasets, facilitating future climate zone reclassification. Improved performance across parameters was evident based on MAE (35.8–94.6%), MSE (57.0–99.5%), NRMSE (42.7–92.1%), PBIAS (100–108%), MD (23.0–85.3%), KGE (21.1–78.1%), and VE (5.1–9.1%), with closer replication of empirical probability distribution function (PDF) curves during the validation period. In the future, Borneo's climate zones will shift notably, with WC elongating southward along the mountainous spine, W forming an enclave over the north-central mountains, WH shifting northward and shrinking inland, and DH expanding northward along the western coast. Under SSP5-8.5, WC is expected to expand by 39% and 11% for the mid- and far-future periods, respectively, while W is set to shrink by 46%. WH is projected to expand by 2% and 8% for the mid- and far-future periods, respectively. Conversely, DH is expected to expand by 43% for the far-future period but shrink by 42% for the mid-future period. This study fills a gap by redefining Borneo's climate zones based on an increased number of effective parameters and projecting future shifts, utilizing advanced clustering methods (FCM) under CMIP6 scenarios. Importantly, it contributes by ranking GCMs using RIMs and CPI across multiple climate parameters, addressing a previous gap in GCM assessment. The study's findings can facilitate cross-border collaboration by providing a shared understanding of climate dynamics and informing joint environmental management and disaster response efforts.

Frequency curves of high and low flows in intermittent river basins for hydrological analysis and hydraulic design

AbstractUpper and lower percentiles of Flow Duration Curves (FDCs) of daily streamflow data were investigated to develop frequency curves. Upper percentiles with exceedance probability of 1, 5 and 10% (Q1, Q5, Q10) were used for high flows, and lower percentiles with non-exceedance probability of 10, 5 and 1% (Q90, Q95, Q99) for low flows. Median value (Q50) was covered to represent the average conditions of streamflow. A mixed frequency analysis based on the total probability theorem taking zero values into account was applied for the lower percentiles of FDC. Case studies were performed for three intermittent Streamflow Gauging Stations (SGSs) from Kucuk Menderes River Basin in western Turkey. An overall assessment of results shows that the best-fit probability distribution function does not change from one SGS to another considerably for low flows while each SGS has its own probability distribution function for high flows. Upper and lower percentiles, and median value were calculated at various return periods by using the identified probability distribution functions. The calculated values were plotted in the form of frequency curves of high flow percentiles and low flow percentiles. The frequency curves have a practically significant potential use in hydrological analysis, water resources management and hydraulic design under high and low flow conditions. They are yet open to further development for regionalization and their applicability can be extended to ungauged sites in river basins.

Estimating major merger rates and spin parameters ab initio via the clustering of critical events

Abstract We build a model to predict from first principles the properties of major mergers. We predict these from the coalescence of peaks and saddle points in the vicinity of a given larger peak, as one increases the smoothing scale in the initial linear density field as a proxy for cosmic time. To refine our results, we also ensure, using a suite of ∼400 power-law Gaussian random fields smoothed at ∼30 different scales, that the relevant peaks and saddles are topologically connected: they should belong to a persistent pair before coalescence. Our model allows us to (a) compute the probability distribution function of the satellite-merger separation in Lagrangian space: they peak at three times the smoothing scale; (b) predict the distribution of the number of mergers as a function of peak rarity: haloes typically undergo two major mergers (>1:10) per decade of mass growth; (c) recover that the typical spin brought by mergers: it is of the order of a few tens of percent.

A multiparametric study for size and stability of hybrid Fe 2 O 3 -NiO nanoparticles and their statistical transformation

A multiparametric study for size and stability of hybrid Fe ₂ O ₃ -NiO nanoparticles and their statistical transformation

Lateral Pelvis and Lumbar Motion in Seated and Standing Office Work and Their Association With Transient Low Back Pain.

To assess frontal plane motion of the pelvis and lumbar spine during 2 h of seated and standing office work and evaluate associations with transient low back pain. Although bending and twisting motions are cited as risk factors for low back injuries in occupational tasks, few studies have assessed frontal plane motion during sedentary exposures. Twenty-one participants completed 2 h of seated and standing office work while pelvic obliquity, lumbar lateral bending angles, and ratings of perceived low back pain were recorded. Mean absolute angles were compared across 15-min blocks, amplitude probability distribution functions were calculated, and associations between lateral postures and low back pain were evaluated. Mean pelvic obliquity (sit = 4.0 ± 2.8°, stand = 3.5 ± 1.7°) and lumbar lateral bending (sit = 4.5 ± 2.5°, stand = 4.1 ± 1.6°) were consistently asymmetrical. Pelvic obliquity range of motion was 4.7° larger in standing (13.6 ± 7.5°) than sitting (8.9 ± 8.7°). In sitting, 52% (pelvis) and 71% (lumbar) of participants, and in standing, 71% (pelvis and lumbar) of participants, were considered asymmetric for >90% of the protocol. Lateral postures displayed weak to low correlations with peak low back pain (R ≤ 0.388). The majority of participants displayed lateral asymmetries for the pelvis and lumbar spine within 5° of their upright standing posture. In short-term sedentary exposures, associations between lateral postures and pain indicated that as the range in lateral postures increases there may be an increased possibility of pain.

Effect of nonlinearity of discrete Langevin model on behavior of extremes in generated time series

In this work we analyze the influence of nonlinearity on the behavior of extremal values of time series generated by two discrete Langevin models: fixing the diffusion function in the first (M1), the probability distribution function in the second (M2). The extremes were generated by applying the run theory. A mathematical relationship was found between nonlinearity of models and means and distributions of run lengths and inter-extreme times as well as with the clustering of extremes. Furthermore, the Allan factor curves of the extremes suggest that the sequences of extremes are fractal for timescales up to the mean inter-extreme time. Our main findings are that the variation of the nonlinearity parameter in model M1 (leading to the increase of the distribution tail length) can cause a significant variation of the extreme characteristics and an increase of the clustering while the variation of the nonlinearity parameter in model M2 (with fixed distribution) has a little effect on extremes.

The influence patterns of carbon flux in different climatic zones in China —Based on the complex network approach

Research on ecosystem carbon flux can provide important methodological and strategic support for climate change mitigation. The existing studies focus on the calculation of carbon flux, ignoring the intertwined effects between regions. The quantification and analysis of the interaction patterns of carbon flux is crucial for understanding the global carbon cycle process, forecasting and coping with climate change. In this study, carbon flux network model sequences are established based on complex network theory using carbon flux data spanning from December 1, 2005, to November 30, 2020. The time delay effect is introduced to accurately quantify the influence patterns of carbon flux within climate zones across China. The findings indicate that the probability distribution function of the link weights during the various seasons of each year exhibits a bimodal distribution with distinct positive and negative components. The delay probability distribution function reveals the significant impact of delay effects, which are especially pronounced and mostly significant long-term lag effects in nodes with negative weights. Further, the results of the interactions of carbon flux among climate zones demonstrate that changes in carbon flux in the plateau and southern temperate regions have either positive or negative impacts on other climate zones. Therefore, controlling carbon flux changes in these climatic zones can effectively optimize the distribution of carbon flux. The modeling framework and results presented in this paper provide useful insights for the regulation and distribution optimization of carbon flux in China.

Caterpillar movement mediates spatially local interactions and determines the relationship between population density and contact.

While interactions in nature are inherently local, ecological models often assume homogeneity across space, allowing for generalization across systems and greater mathematical tractability. Density-dependent disease models are a prominent example of models that assume homogeneous interactions, leading to the prediction that disease transmission will scale linearly with population density. In this study, we examined how the scale of larval butterfly movement interacts with the resource landscape to influence the relationship between larval contact and population density in the Baltimore checkerspot (Euphydryas phaeton). Our study was inspired by the recent discovery of a viral pathogen that is transmitted horizontally among Baltimore checkerspot larvae. We used multi-year larvae location data across six Baltimore checkerspot populations in the eastern U.S. to test whether larval nests are spatially clustered. We then integrated these spatial data with larval movement data in different resource contexts to investigate whether heterogeneity in spatially local interactions alters the assumed linear relationship between larval nest density and contact. We used Correlated Random Walk (CRW) models and field observations of larval movement behavior to construct Probability Distribution Functions (PDFs) of larval dispersal, and calculated the overlap in these PDFs to estimate conspecific contact within each population. We found that all populations exhibited significant spatial clustering in their habitat use. Subsequent larval movement rates were influenced by encounters with host plants and larval age, and under many movement scenarios, the scale of predicted larval movement was not sufficient to allow for the "homogeneous mixing" assumed in density dependent disease models. Therefore, relationships between population density and larval contact were typically non-linear. We also found that observed use of available habitat patches led to significantly greater contact than would occur if habitat use were spatially random. These findings strongly suggest that incorporating larval movement and spatial variation in larval interactions is critical to modeling disease outcomes in E. phaeton. Epidemiological models that assume a linear relationship between population density and larval contact have the potential to underestimate transmission rates, especially in small populations that are already vulnerable to extinction.

On the Bivariate Extension of the Extended Standard U-quadratic Distribution

This paper derives the bivariate version of the extended standard U-quadratic (eSU) distribution using the method of compounding or pseudo family of distributions. The joint probability and cumulative distribution functions of the derived distribution are obtained and it is observed that the said distribution can generate bivariate shape with the following properties: $(i)$ $X$ and $Y$ have bathtub shapes; $(ii)$ $X$ has constant and $Y$ has bathtub shapes; and $(iii)$ $X$ has inverted bathtub and $Y$ has bathtub shapes. Moreover, some properties of the derived distribution such as the marginal distribution, conditional distributions, conditional moments, product and ratio moments are derived. Further, the maximum likelihood estimation is performed to estimate the parameters of the derived distribution. Also, a simulation study is carried out to evaluate the behavior of the estimates of the parameters. Moreover, we derive a new bivariate Kumaraswamy distribution and use it to simulate bivariate data with $X$ and $Y$ having bathtub shapes. Furthermore, A new bivariate version of the Cubic Transmuted Uniform (CTU) distribution is also derived. Finally, the proposed Bivariate eSU distribution is applied to simulated data and compared with the said Bivarite Cubic Transmuted Uniform distribution. The result shows that the proposed Bivariate eSU distribution provides a better fit for the said simulated dataset as compared with the Bivariate CTU distribution.

A Deformation Analysis Method for Sluice Structure Based on Panel Data

Deformation, as the most intuitive index, can reflect the operation status of hydraulic structures comprehensively, and reasonable analysis of deformation behavior has important guiding significance for structural long-term service. Currently, the health evaluation of dam deformation behavior has attracted widespread attention and extensive research from scholars due to its great importance. However, given that the sluice is a low-head hydraulic structure, the consequences of its failure are easily overlooked without sufficient attention. While the influencing factors of the sluice’s deformation are almost identical to those of a concrete dam, nonuniform deformation is the key issue in the sluice’s case because of the uneven property of the external load and soil foundation, and referencing the traditional deformation statistical model of a concrete dam cannot directly represent the nonuniform deformation behavior of a sluice. In this paper, we assume that the deformation at various positions of the sluice consist of both overall and individual effects, where overall effect values describe the deformation response trend of the sluice structure under external loads, and individual effect values represent the degree to which the deformation of a single point deviates from the overall deformation. Then, the random coefficient model of panel data is introduced into the analysis of sluice deformation to handle the unobservable overall and individual effects. Furthermore, the maximum entropy principle is applied, both to approximate the probability distribution function of individual effect extreme values and to determine the early warning indicators, completing the assessment and analysis of the nonuniform deformation state. Finally, taking a project as an example, we show that the method proposed can effectively identify the overall deformation trend of the sluice and the deviation degree of each measuring point from the overall deformation, which provides a novel approach for sluice deformation behavior research.

Ensemble variational Bayesian approximation for the inversion and uncertainty quantification of Darcy flows in heterogeneous porous media with random parameters

The inversion and uncertainty quantification of physical parameters are important for many science and engineering problems. For nonlinear parameter inversion problems of flows in hydrocarbon reservoirs, estimating the true posterior probability distribution function for the random parameter is difficult when the dimension of the parameter is relatively high. Inversion methods such as gradient-based or heuristic optimization methods searching for the maximum a posterior are generally point-estimate, while uncertainty quantification using MCMC is computationally expensive. In the current study, a new ensemble variational Bayesian approximation method is developed to estimate the posterior probability density using an ensemble of realizations regarded as a trial sample distribution. The objective is to optimize the trial sample distribution such that the evidence lower bound is maximized indicating that the distance between the trial and true posterior distribution is minimized. A subspace is built using principle component analysis based on finite samples to obtain a diagonal positive-definite covariance matrix, and ensemble-based data assimilation is used to optimize the trial distribution efficiently. Heterogeneous 2D and 3D test cases of transient single- and two-phase Darcy flows are presented for validation.

Evaluation of multiple machine learning models for ASR expansion of concrete

Machine learning algorithms (MLA) show promise in predicting alkali-silica reaction (ASR) expansion. Nevertheless, the predictive performance of different machine learning algorithms (MLAs) varies across various engineering scenarios and datasets. Given that, this study was devoted to constructing an appropriate predictive model for ASR expansion of concrete. Multiple MLAs including evolutionary-based, support vector-based, assembly-based, and neural network-based algorithms, were employed to establish predictive models. An optimization technique, the particle swarm optimization algorithm, was also utilized and integrated with the aforementioned MLA models to effectively search for optimal hyperparameters and topological structures. The predictive performance was evaluated using statistical methods, which showed that the extreme gradient boosting (XGBoost) model had the best fitting goodness among the standard algorithms, with an R2 value of 0.9695. Moreover, the Particle Swarm Optimization (PSO) model significantly improved the predictive performance of all Machine Learning Algorithm (MLA) models. The error distribution analysis was then performed. There are fewer large errors in the predicted values of the PSO-XGBoost model. Monte Carlo simulation and probability distribution functions were used to thoroughly analyze the stability of each model after multiple independent runs. The results show that all models have excellent stability. The PSO-XGBoost model performed the best and was subsequently employed to conduct feature analysis to determine the significance of each input parameter and capture the underlying evidence of ASR expansion.

Beyond radial profiles: Using log-normal distributions to model the multiphase circumgalactic medium

Abstract Recent observations and simulations reveal that the circumgalactic medium (CGM) surrounding galaxies is multiphase, with the gas temperatures spanning a wide range at most radii, ∼104 K to the virial temperature (∼106 K for Milky Way). Traditional CGM models using simple density profiles are inadequate at reproducing observations that indicate a broad temperature range. Alternatively, a model based on probability distribution functions (PDFs) with parameters motivated by simulations can better match multi-wavelength observations. In this work, we use log-normal distributions, commonly seen in the simulations of the multiphase interstellar and circumgalactic media, to model the multiphase CGM. We generalize the isothermal background model by Faerman et al. 2017 to include more general CGM profiles. We extend the existing probabilistic models from 1D-PDFs in temperature to 2D-PDFs in density-temperature phase space and constrain its parameters using a Milky Way-like Illustris TNG50-1 halo. We generate various synthetic observables such as column densities of different ions, UV/X-ray spectra, and dispersion and emission measures. X-ray and radio (Fast Radio Burst) observations mainly constrain the hot gas properties. However, interpreting cold/warm phase diagnostics is not straightforward since these phases are patchy, with inherent variability in intercepting these clouds along arbitrary lines of sight. We provide a tabulated comparison of model predictions with observations and plan to expand this into a comprehensive compilation of models and data. Our modeling provides a simple analytic framework that is useful for describing important aspects of the multiphase CGM.

Parametric description of intermittent probability distribution functions in solar wind and magnetohydrodynamic turbulence

ABSTRACT In this work, we find empirical evidence that the scale-dependent statistical properties of solar wind and magnetohydrodynamic (MHD) turbulence can be described in terms of a family of parametric probability distribution functions (PDFs) known as Normal Inverse Gaussian (NIG). Understanding these PDFs is one of the most important goals in turbulence theory, as they are inherently connected to the intermittent properties of solar wind turbulence. We investigate the properties of PDFs of Elsasser increments based on a large statistical sample from solar wind observations and high-resolution numerical simulations of MHD turbulence. In order to measure the PDFs and their corresponding properties, three experiments are presented: fast and slow solar wind for experimental data and a simulation of reduced MHD (RMHD) turbulence. Conditional statistics on a 23-yr-long sample of WIND data near 1 au and high-resolution pseudo-spectral simulation of steadily driven RMHD turbulence on a $2048^3$ mesh are used to construct scale-dependent PDFs. The empirical PDFs are fitted to NIG distributions, which depend on four free parameters. Our analysis shows that NIG distributions accurately capture the evolution of the PDFs, with scale-dependent parameters, from large scales characterized by a Gaussian distribution, turning to exponential tails within the inertial range and stretched exponentials at dissipative scales. We also show that empirically-measured NIG parameters exhibit well-defined scaling properties that are similar across the three empirical data sets, which may be indicative of universal behaviour.

Improving and extending non-Poissonian distributions for satellite galaxies sampling in HOD: applications to eBOSS ELGs

ABSTRACT Halo occupation distribution (HOD) models help us to connect observations and theory, by assigning galaxies to dark matter haloes. In this work, we study one of the components of HOD models: the probability distribution function (PDF), which is used to assign a discrete number of galaxies to a halo, given a mean number of galaxies. For satellite galaxies, the most commonly used PDF is a Poisson Distribution. PDFs with super-Poisson variances have also been studied, allowing for continuous values of variances. This has not been the case for sub-Poisson variances, for which only the nearest integer distribution, with a single variance, has been used in the past. In this work, we propose a distribution based on the binomial one, which provides continuous sub-Poisson variances. We have generated mock galaxy catalogues from two dark-matter only simulations, unit and outerrim, with HOD models assuming different PDFs. We show that the variance of the PDF for satellite galaxies affects the one-halo term of the projected correlation function, and the count-in-cells (CIC) one point statistics. We fit the clustering of eBOSS emission line galaxies, finding results consistent with the literature, when we only vary the parameter controlling the PDF variance and the fraction of satellites. Using a mock catalogue as a reference, we have also included both the clustering and CIC to constrain the parameters of the HOD model. CIC can provide strong constraints to the PDF variance of satellite galaxies.