Tail Of Probability Distribution Function Research Articles

Accurate analytic model for the weak lensing convergence one-point probability distribution function and its autocovariance

The one-point probability distribution function (PDF) is a powerful summary statistic for non-Gaussian cosmological fields, such as the weak lensing (WL) convergence reconstructed from galaxy shapes or cosmic microwave background (CMB) maps. Thus far, no analytic model has been developed that successfully describes the high-convergence tail of the WL convergence PDF for small smoothing scales from first principles. Here, we present a halo-model formalism to compute the WL convergence PDF, building upon our previous results for the thermal Sunyaev-Zel'dovich field. Furthermore, we extend our formalism to analytically compute the covariance matrix of the convergence PDF. Comparisons to numerical simulations generally confirm the validity of our formalism in the non-Gaussian, positive tail of the WL convergence PDF, but also reveal the convergence PDF's strong sensitivity to small-scale systematic effects in the simulations (e.g., due to finite resolution). Finally, we present a simple Fisher forecast for a Rubin-Observatory-like survey, based on our new analytic model. Considering the $\{A_s, \Omega_m, \Sigma m_\nu\}$ parameter space and assuming a Planck CMB prior on $A_s$ only, we forecast a marginalized constraint $\sigma(\Sigma m_\nu) \approx 0.08$ eV from the WL convergence PDF alone, even after marginalizing over parameters describing the halo concentration-mass relation. This error bar on the neutrino mass sum is comparable to the minimum value allowed in the normal hierarchy, illustrating the strong constraining power of the WL convergence PDF. We make our code publicly available at https://github.com/leanderthiele/hmpdf.

The Probability Distribution of Density Fluctuations in Supersonic Turbulence

Abstract A theoretical formulation is developed for the probability distribution function (pdf) of gas density in supersonic turbulence at steady state, connecting it to the conditional statistics of the velocity divergence. Two sets of numerical simulations are carried out, using either a Riemann solver to evolve the Euler equations or a finite-difference method to evolve the Navier–Stokes (N-S) equations. After confirming the validity of our theoretical formulation with the N-S simulations, we examine the effects of dynamical processes on the pdf, showing that the nonlinear term in the divergence equation amplifies the right pdf tail and reduces the left one, the pressure term reduces both the right and left tails, and the viscosity term, counterintuitively, broadens the right tail of the pdf. Despite the inaccuracy of the velocity divergence from the Riemann runs, we show that the density pdf from the Riemann runs is consistent with that from the N-S runs. Taking advantage of their higher effective resolution, we use Riemann runs with resolution up to 20483 to study the dependence of the pdf on the Mach number, , up to . The pdf width, σ s , follows the relation , with b ≈ 0.38. However, the pdf exhibits a negative skewness that increases with increasing , as the growth of the right tail with increasing tends to saturate. Thus, the usual prescription that combines a lognormal shape with a variance–Mach number relation greatly overestimates the right pdf tail at large , with important consequences for star formation models.

Review and comparison of two meta-model-based uncertainty propagation analysis methods in groundwater applications: polynomial chaos expansion and Gaussian process emulation

Gaussian process emulation (GPE) and polynomial chaos expansion (PCE) are tools for meta-model-based uncertainty propagation analysis (UPA) that have gained increasing attention in recent groundwater literature. Previous studies have shown that these two meta-models can provide satisfactorily accurate estimations of the model response in a wide range of groundwater UPA problems. However, PCE and GPE are based on very different mathematical concepts, and a question that arises is which one of these is more suitable for groundwater UPA. The current paper aims to provide an answer to this question by first presenting a theoretical comparison of the two meta-models, then reviewing previous comparisons of the two in other fields of engineering, and subsequently presenting an empirical comparison based on groundwater case studies. For this purpose, both meta-models are applied to two hypothetical test cases corresponding to seawater intrusion in coastal aquifers. The results show that: (1) GPE outperforms PCE in the estimation of the input–output relationships, by providing smaller root mean square errors. (2) In most cases assessed, PCE provides better accuracy in the estimation of mean, standard deviation and the entire shape and the tail of probability distribution functions. (3) Replicates of PCE show less statistical dispersion in the estimation of mean and standard deviations. (4) A trend of increase in the predominance of PCE over GPE can be identified as the probability distributions that are meant to be estimated become noisier and more multi-modal.

Mutually avoiding paths in random media and largest eigenvalues of random matrices.

Recently, it was shown that the probability distribution function (PDF) of the free energy of a single continuum directed polymer (DP) in a random potential, equivalently to the height of a growing interface described by the Kardar-Parisi-Zhang (KPZ) equation, converges at large scale to the Tracy-Widom distribution. The latter describes the fluctuations of the largest eigenvalue of a random matrix, drawn from the Gaussian unitary ensemble (GUE), and the result holds for a DP with fixed end points, i.e., for the KPZ equation with droplet initial conditions. A more general conjecture can be put forward, relating the free energies of N>1 noncrossing continuum DP in a random potential, to the sum of the Nth largest eigenvalues of the GUE. Here, using replica methods, we provide an important test of this conjecture by calculating exactly the right tails of both PDFs and showing that they coincide for arbitrary N.

TURBULENCE-GENERATED PROTON-SCALE STRUCTURES IN THE TERRESTRIAL MAGNETOSHEATH

ABSTRACT Recent results of numerical magnetohydrodynamic simulations suggest that in collisionless space plasmas, turbulence can spontaneously generate thin current sheets. These coherent structures can partially explain the intermittency and the non-homogenous distribution of localized plasma heating in turbulence. In this Letter, Cluster multi-point observations are used to investigate the distribution of magnetic field discontinuities and the associated small-scale current sheets in the terrestrial magnetosheath downstream of a quasi-parallel bow shock. It is shown experimentally, for the first time, that the strongest turbulence-generated current sheets occupy the long tails of probability distribution functions associated with extremal values of magnetic field partial derivatives. During the analyzed one-hour time interval, about a hundred strong discontinuities, possibly proton-scale current sheets, were observed.

Statistics of extreme events in Chinese stock markets

We investigate the impact of financial factors on daily volume recurrent time intervals in the developing Chinese stock markets. The tails of probability distribution functions (PDFs) of volume recurrent intervals behave as a power-law, and the scaling exponent decreases with the increase of stock lifetime, which are similar to those in the US stock markets, and they are typical representatives of developed markets. The difference is that the power-law exponent values remain almost the same with the changes of market capitalization, mean volume, and mean trading value, respectively. These findings enrich the results for event statistics for financial markets.

TURBULENCE-INDUCED RELATIVE VELOCITY OF DUST PARTICLES. I. IDENTICAL PARTICLES

We study the relative velocity of inertial particles suspended in turbulent flows and discuss implications for dust particle collisions in protoplanetary disks. We simulate a weakly compressible turbulent flow, evolving 14 particle species with friction timescale, tau_p, covering the entire range of scales in the flow. The particle Stokes numbers, St, measuring the ratio of tau_p to the Kolmogorov timescale, are in the range from ~0.1 to ~800. Using simulation results, we show that the model by Pan & Padoan (PP10) gives satisfactory predictions for the rms relative velocity between identical particles. The probability distribution function (PDF) of the relative velocity is found to be highly non-Gaussian. The PDF tails are well described by a 4/3 stretched exponential function for particles with tau_p ~ 1-2 T_L, where T_L is the Lagrangian correlation timescale, consistent with a prediction based on PP10. The PDF approaches Gaussian only for very large particles with tau_p >~ 54 T_L. We split particle pairs at given distances into two types with low and high relative speeds, referred to as continuous and caustic types, respectively, and compute their contributions to the collision kernel. Although amplified by the effect of clustering, the continuous contribution vanishes in the limit of infinitesimal particle distance, where the caustic contribution dominates. The caustic kernel per unit cross section rises rapidly as St increases toward ~1, reaches a maximum at tau_p ~ 2 T_L, and decreases as tau_p^{-1/2} for tau_p >> T_L.

Tail-constraining stochastic linear–quadratic control: a large deviation and statistical physics approach

The standard definition of the stochastic risk-sensitive linear–quadratic (RS-LQ) control depends on the risk parameter, which is normally left to be set exogenously. We reconsider the classical approach and suggest two alternatives, resolving the spurious freedom naturally. One approach consists in seeking for the minimum of the tail of the probability distribution function (PDF) of the cost functional at some large fixed value. Another option suggests minimizing the expectation value of the cost functional under a constraint on the value of the PDF tail. Under the assumption of resulting control stability, both problems are reduced to static optimizations over a stationary control matrix. The solutions are illustrated using the examples of scalar and 1D chain (string) systems. The large deviation self-similar asymptotic of the cost functional PDF is analyzed.

The non-linear redshift space probability distribution function in models with local primordial non-Gaussianity

We use the ellipsoidal collapse approximation to investigate the nonlinear redshift space evolution of the density field with primordial non-Gaussianity of the local f_{nl}-type. We utilize the joint distribution of eigenvalues of the initial non-Gaussian shear field and evaluate the evolved redshift space probability distribution function (PDF). It is shown that, similar to the real space analysis, the underdense tail of the nonlinear redshift space PDF differs significantly from that for Gaussian initial conditions. We also derive the lowest order correction of the Kaiser's formulain the presence of a non-zero f_{nl}.

Non-perturbative statistical theory of intermittency in ITG drift wave turbulence with zonal flows

The probability distribution functions (PDFs) of momentum flux and zonal flow formation in ion-temperature-gradient (ITG) turbulence are investigated in two different models. The first is a general five-field model (ni, ϕ, Ti, Te, vi∥) where a reductive perturbation method is used to derive dynamical equations for drift waves and a zonal flow. The second is a reduced two-field model (ϕ, Ti) that has an exact non-linear solution (bipolar vortex soliton). In both models the exponential tails of the zonal flow PDFs are found with the same scaling (), but with different coefficients cZF. The PDFs of momentum flux is, however, found to be qualitatively different with the scaling (PDF ∼ exp{−cMRs}), where s = 2 and s = 3/2 in the five and two-field models, respectively.

Probability distribution function for self-organization of shear flows

The first prediction of the probability distribution function (PDF) of self-organized shear flows is presented in a nonlinear diffusion model where shear flows are generated by a stochastic forcing while diffused by a nonlinear eddy diffusivity. A novel nonperturbative method based on a coherent structure is utilized for the prediction of the strongly intermittent exponential PDF tails of the gradient of shear flows. Numerical simulations using Gaussian forcing not only confirm these predictions but also reveal the significant contribution from the PDF tails with a large population of supercritical gradients. The validity of the nonlinear diffusion model is then examined using a threshold model where eddy diffusivity is given by discontinuous values, elucidating an important role of relative time scales of relaxation and disturbance in the determination of the PDFs.

The non-linear probability distribution function in models with local primordial non-Gaussianity

We use the spherical evolution approximation to investigate non-linear evolution from the non-Gaussian initial conditions characteristic of the local fnl model. We provide an analytic formula for the non-linearly evolved probability distribution function (PDF) of the dark matter which shows that the underdense tail of the non-linear PDF in the fnl model should differ significantly from that for Gaussian initial conditions. Measurements of the underdense tail in numerical simulations may be affected by discreteness effects, and we use a Poisson counting model to describe this effect. Once this has been accounted, our model is in good quantitative agreement with the simulations. In principle, our calculation is an important first step in programs which seek to reconstruct the shape of the initial PDF from observations of large-scale structures in the Lyα forest and the galaxy distribution at later times.

Decadal‐scale changes in the tails of probability distribution functions of climate variables in Switzerland

AbstractAn analysis of several Swiss climatological sites reveals that a substantial change in the behaviour of pressure, minimum and maximum temperature extremes has occurred in the past two decades. Extreme cold tails defined by the 10% quantiles of temperature drop by a factor of 2 or 3, while the upper tails (beyond the 90% quantile) exhibit a four‐ or five‐fold increase in all seasons. Pressure shows contrasting behaviour, with increases in wintertime highs and summertime lows, while precipitation shows little change. On the basis of the observed datasets, temperature biases related to extremes of pressure or precipitation have been computed, as well as for joint combinations of precipitation and pressure extremes. The most dominant bias is associated with periods without rainfall, during which temperatures are at least 1 °C warmer than otherwise. Changes in the behaviour of joint combinations of extreme pressure and precipitation regimes also have a discernible influence on temperatures. Copyright © 2008 Royal Meteorological Society

Nonperturbative models of intermittency in edge turbulence

A theory of the probability distribution function (PDF) tails of the blob density in plasma edge turbulence is provided. A simplified model of the fast convective radial transport is used. The theoretically predicted PDF tails corroborate earlier measurements of edge transport, further confirming the strongly non-Gaussian feature of edge transport. It is found that increasing the cross-sectional spatial scale length (Lx and Ly) of the blob results in larger transport, whereas increasing the toroidal scale length (Lz) decreases the PDF. The results imply that the PDF decreases for larger blob speed vb.

Analytical theory of the probability distribution function of structure formation

The probability distribution function (PDF) tails of the zonal flow structure formation and the PDF tails of momentum flux are computed by incorporating the effect of a shear flow in ion-temperature-gradient (ITG) turbulence. The bipolar vortex soliton (modon) is assumed to be the coherent structure responsible for bursty and intermittent events driving the PDF tails. It is found that stronger zonal flows are generated in ITG turbulence than Hasegawa–Mima turbulence, as well as further from marginal stability. This suggests that although ITG turbulence has a higher level of heat flux, it also more likely generates stronger zonal flows, leading to a self-regulating system. It is also shown that shear flows can significantly reduce the PDF tails of Reynolds stress and structure formation.

The momentum flux probability distribution function for ion-temperature-gradient turbulence

There has been overwhelming evidence that coherent structures play a critical role in determining the overall transport in a variety of systems. We compute the probability distribution function (PDF) tails of momentum flux and heat flux in ion-temperature-gradient turbulence, by taking into account the interaction among modons, which are assumed to be coherent structures responsible for bursty and intermittent events, contributing to the PDF tails. The tail of PDF of momentum flux R=⟨vxvy⟩ is shown to be exponential with the form exp{−ξR3∕2}, which is broader than a Gaussian, similar to what was found in the previous local studies. An analogous expression with the same functional dependence is found for the PDF tails of heat flux. Furthermore, we present a detailed numerical study of the dependence of the PDF tail on the temperature and density scale lengths and other physical parameters through the coefficient ξ.

Probability distribution function of a forced passive tracer in the lower stratosphere

The probability distribution function (PDF) of a passive tracer, forced by a “mean gradient”, is studied. First, we take two theoretical approaches, the Lagrangian and the conditional closure formalisms, to study the PDFs of such an externally forced passive tracer. Then, we carry out numerical simulations for an idealized random flow on a sphere and for European Center for Medium-Range Weather Forecasts (ECMWF) stratospheric winds to test whether the mean-gradient model can be applied to studying stratospheric tracer mixing in midlatitude surf zones, in which a weak and poleward zonal-mean gradient is maintained by tracer leakage through polar and tropical mixing barriers, and whether the PDFs of tracer fluctuations in midlatitudes are consistent with the theoretical predictions. The numerical simulations show that when diffusive dissipation is balanced by the mean-gradient forcing, the PDF in the random flow and the Southern-Hemisphere PDFs in ECMWF winds show time-invariant exponential tails, consistent with theoretical predictions. In the Northern Hemisphere, the PDFs exhibit non-Gaussian tails. However, the PDF tails are not consistent with theoretical expectations. The long-term behavior of the PDF tails of the forced tracer is compared to that of a decaying tracer. It is found that the PDF tails of the decaying tracer are time-dependent, and evolve toward flatter than exponential.

Single-Point Velocity Statistics of Forced and Decaying Two-Dimensional Turbulence

The single-point (SP) velocity statistics are investigated in forced and decaying two-dimensional turbulence in a flowing soap film. It is shown that the probability distribution functions (PDF) in both cases deviate from a Gaussian distribution, which is normally anticipated in turbulent fluid flows. In the forced turbulence case, the tail of the SP velocity PDF decays faster than Gaussian (termed the sub-Gaussian) and can be correlated with the forcing statistics on small scales. In the decaying-turbulence case, the SP velocity PDF evolves from a sub-Gaussian to a super-Gaussian behavior as a function of time. However, in all times, the locally averaged vorticity remains normally distributed. While our forced turbulence data may be explained by a recent theory proposed by Falkovich et al., the decaying-turbulence data remain unexplained.

Shifts of means are not a proxy for changes in extreme winter temperatures in climate projections

Changes in the severity of extreme weather events under the influence of the enhanced greenhouse effect could have disproportionally large effects compared to changes in the mean climate. Here, we explored the meteorological circumstances of extremes and changes therein using two 49-member climate model ensembles for reference (1961–1990) and scenario (2051–2080) greenhouse-gas concentrations. We have focused on daily-mean surface-air temperatures over the Northern Hemisphere in January. Over large parts of the continents, changes in the one-in-10-year temperature events are influenced at least as much by changes in the shape of the probability distribution functions (PDFs) as by shifts in the mean. In coastal areas, this is largely attributable to changes in the large-scale circulation, for those types of extremes linked to infrequent wind directions. In other areas, the inhomogeneous mean warming, increasing inland and polewards, affects the tails of the local temperature PDFs. Temperature extremes in widely different regions were found to be linked by a large-scale circulation anomaly pattern, which resembles the Arctic Oscillation. In the scenario ensemble, this anomaly pattern favors its positive phase, leading to enhanced probabilities of westerly winds in a belt around the Northern Hemisphere.

Search of exoplanetary radio signals in the presence of strong interference: enhancing sensitivity by data accumulation

We develop a statistical approach aimed at the detection of weak sporadic pulses on the noise background. The results are applied to modeling the observational time series where pulsed radio emissions have to be recognized against the sky background fluctuations. The proposed methodology demonstrates the efficiency of using the statistics of peak values (integrated tail of probability distribution function of the intensity) for the purpose of signal detection. It is established that the highest sensitivity is reached with this method at low values of the filling factor (duty cycle) of the pulsed signals. If in addition the pulses have sufficiently high intensity, the discussed approach performs better than simple integration over the observational time. Then we discuss the possibility of detecting radio pulses from exoplanetary magnetospheres, especially from known “hot Jupiters” found by radial velocity measurements in the visible and we report our results from extensive observations of several candidate exoplanets with the world largest decameter telescope UTR-2. Although no detection of pulsed emission from exoplanets has been found to date, the analysis demonstrates the feasibility of detection with more stable receivers and longer observational time.