The Use of Conditional Probability Functions in Range Data Analysis and Simulation

A random variable Y 1 is said to be smaller than Y 2 in the increasing concave stochastic order if for all increasing concave functions for which the expected values exist, and smaller than Y 2 in the increasing convex order if for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions of a response variable Y given a covariate X, solely under the assumption that the conditional distributions are increasing in x in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the K-sample case and for continuously distributed X.

Determine animal performance and eco-nomic net returns for 6 bermudagrass-based pasture graz-ing systems in the US Southern Great Plains. Animal performance data from a 4-yr split-plot grazing study conducted in south-central Oklahoma were used to evaluate 6 bermudagrass-based stocker cattle grazing systems: bermudagrass forage with 112 kg N/ha (100 lb/acre) and a feed supplement under continuous stocking (BNSC), bermudagrass forage with 112 kg N/ha and a feed supplement grazed in a 28-d rest rotation (BNSR), bermudagrass forage without N fer-tilizer with a feed supplement under continuous stocking (BSC), bermudagrass forage without N fertilizer with a feed supplement grazed in a 28-d rest rotation (BSR), ber-mudagrass forage interseeded with grazing-tolerant alfalfa under continuous stocking (BAC), and bermudagrass for-age interseeded with grazing-tolerant alfalfa grazed in a 28-d rest rotation (BAR). Enterprise budgets and mixed effect models were used to estimate means for measures of animal performance and net returns. Total gain was greater in the BNSC system (324 kg/ha) than the BNSR and BAR sys-tems with gains of 307 and 299 kg/ha. Interseeded alfalfa with rotational grazing (BAR) had the greatest net re-turn of $44/ha ($18/acre). The next-best system was BAC with a net return of −$56/ha. The conventional systems (BNSC and BNSR) had low net returns due to fertilizer and supplemental feed costs, and rotational systems (BSR and BNSR) had low net returns due to greater water, fenc-ing, and labor costs. The bermudagrass systems with interseeded alfalfa performed better eco-nomically than the conventional bermudagrass systems. The alfalfa rotational system was more economical than conventional continuously stocked systems because giving alfalfa time to recover from preferential grazing, extended its stand life.

There is growing interest in the field of digital soil mapping (DSM) to quantify the underlying uncertainty of point predictions through different probabilistic prediction models. Uncertainty in DSM is often described in the format of a prediction interval (PI). Yet, PIs or uncertainty estimates in general are only of value for end users if they have good quality, i.e. when they are reliable. Reliability refers to the consistency between predicted conditional probabilities and observed frequencies of independent validation data. Ideally, PIs are also sharp, which refers to the concentration of probabilistic information, i.e. the narrowness of conditional probability distributions. The prediction interval coverage probability (PICP) is currently used in DSM to assess the reliability of PIs but it is ignorant to a potential one-sided bias of its bounding quantiles. Therefore, we propose to complement the current validation procedure with new metrics suggested in the broader probabilistic literature. This includes metrics that do not only evaluate uncertainty estimates in PI format but also quantiles or full conditional probability distributions. The newly proposed metrics are the quantile coverage probability (QCP), the probability integral transform (PIT) and so-called proper scoring rules for relative comparisons. Examples of scoring rules are the continuous ranked probability score (CRPS), which can be decomposed into a reliability part (RELI) and the interval score (IS). Sharpness can be evaluated through the prediction interval width (PIW). We illustrated the use of the various metrics in a case-study using soil pH data from The Land Use and Coverage Area Frame Survey (LUCAS). Thereby, uncertainty estimates of five different models were compared: Kriging with external drift (KED), quantile regression forest (QRF), quantile regression post-processing of a random forest (QRPP RF), quantile regression neural network (QRNN) and a reference null-model (NM).  KED, NM and QRPP RF showed very good reliability according to QCP, PICP, PIT and RELI. QRF was slightly pessimistic in the centre and QRNN very overoptimistic at the edges of the conditional probability distributions. Despite this, QRF performed best according to mean CRPS and mean IS because it produced fewer outliers at the edges. As expected, NM had the lowest sharpness, i.e. the largest PIW values. Sharpness of the other models was overall similar but QRNN had sharper predictions at the edges and QRF was less sharp in the centre of the conditional probability distributions. Lastly, we also generated PIW maps to indicate the spatial uncertainty of the five prediction models. The spatial variability of PIW was larger for QRF, QRNN and QRPP RF in comparison to KED. Whereas with NM, PIW was completely uniform. 

Lift-off velocity of saltating sand particles in wind-blown sand located at 1.0 mm above the sand bed surface was measured using a phase Doppler particle analyzer in a wind tunnel. The results show that the probability distribution of lift-off velocity can be expressed as a lognormal function, while that of lift-off angle follows an exponential function. The probability distribution of lift-off angle conditioned for each lift-off velocity also follows an exponential function, with a slope that becomes steeper with increasing lift-off velocity. This implies that the probability distribution of lift-off velocity is strongly dependent on the lift-off angle. However, these lift-off parameters are generally treated as an independent joint probability distribution in the literature. Numerical simulations were carried out to investigate the effects of conditional versus independent joint probability distributions on the vertical sand mass flux distribution. The simulation results derived from the conditional joint probability distribution agree much better with experimental data than those from the independent ones. Thus, it is better to describe the lift-off velocity of saltating sand particles using the conditional joint probability distribution. These results improve our understanding of saltation processes in wind-blown sand.

A priori structural information is incorporated into determinantal phasing techniques to improve phase prediction accuracy in resolution ranges where atomic or isotropic group-scatterer assumptions are not valid. For this purpose a conditional joint probability distribution to triplet order for any set of normalized structure factors of space groups P1 and P\bar 1 is derived. The covariance of two normalized structure factors from the original set is calculated. A more general conditional joint probability distribution, involving covariance matrices of any order, is further derived. Numerical tests are performed employing ideal models consisting of several atomic groups of known stereochemistry but with random positions and orientations. The results indicate that the inclusion of stereochemical information improves the accuracy of phase prediction. The relative merit of this strategy in either one of or both normalization and covariance calculations for different resolutions is discussed.

This paper explores how to exploit shape information to perform object class recognition. We use a sparse part-based model to describe object categories defined by shape. The sparseness allows the relative spatial relationship between parts to be described simply. It is possible, with this model, to highlight potential locations of the object and its parts in novel images. Subsequently these areas are examined by a more flexible shape model that measures if the image data provides evidence of the existence of boundary/connecting curves between connected hypothesized parts. From these measurements it is possible to construct a very simple cost function which indicates the presence or absence of the object class. The part-based model is designed to decouple variations due to affine warps and other forms of shape deformations. The latter are modeled probabilistically using conditional probability distributions which describe the linear dependencies between the location of a part and a subset of the other parts. These conditional distributions can then be exploited to search efficiently for the instances of the part model in novel images. Results are reported on experiments performed on the ETHZ shape classes database that features heavily cluttered images and large variations in scale.

Conditional probability distributions and state spaces of a special class of distributed parameter systems subject to a special class of distributive random inputs are denned. Fokker—Planck equation technique is employed to obtain explicit solutions of conditional probability density functions.

It is well known that for finite-sized networks, onestep retrieval in the autoassociative Willshaw net is a suboptimal way to extract the information stored in the synapses. Iterative retrieval strategies are much better, but have hitherto only had heuristic justification. We show how they emerge naturally from considerations of probabilistic inference under conditions of noisy and partial input and a corrupted weight matrix. We start from the conditional probability distribution over possible patterns for retrieval. This contains all possible information that is available to an observer of the network and the initial input. Since this distribution is over exponentially many patterns, we use it to develop two approximate, but tractable, iterative retrieval methods. One performs maximum likelihood inference to find the single most likely pattern, using the (negative log of the) conditional probability as a Lyapunov function for retrieval. In physics terms, if storage errors are present, then the modified iterative update equations contain an additional antiferromagnetic interaction term and site dependent threshold values. The second method makes a mean field assumption to optimize a tractable estimate of the full conditional probability distribution. This leads to iterative mean field equations which can be interpreted in terms of a network of neurons with sigmoidal responses but with the same interactions and thresholds as in the maximum likelihood update equations. In the absence of storage errors, both models become very similiar to the Willshaw model, where standard retrieval is iterated using a particular form of linear threshold strategy.

On the anti-causal path of the path probability method by Kikuchi

The effect of sawteeth on plasma performance and transport in the plasma of tokamak is an important problem in the fusion field. Sawtooth oscillations can trigger off heat and turbulence pulses that propagate into the edge plasma, and thus enhancing the edge shear flow and inducing a transition from low confinement mode to high confinement mode. The influences of turbulence spreading and symmetry breaking on edge shear flow with sawtooth crashes are observed in the J-TEXT tokamak. The edge plasma turbulence and shear flow are measured using a fast reciprocating electrostatic probe array. The experimental data are analyzed using some methods such as conditional average and probability distribution function. After sawtooth crashes, the heat and turbulence pulses in the core propagate to the edge, with the turbulence pulse being faster than the heat pulse. The attached figures (a)–(e) show the core electron temperature, and the edge electron temperature, turbulence intensity, turbulence drive and spreading rates, Reynolds stress and its gradient, and shearing rates, respectively. After sawtooth crashes, the edge electron temperature increases and the edge turbulence is enhanced, with turbulence preceding temperature. The enhanced edge turbulence is mainly composed of two parts: the turbulence driven by local gradient and the turbulence spreading from core to edge. The development of the estimated turbulence spreading rate is prior to that of the turbulence driving rate. The increase in the turbulence intensity can cause the turbulent Reynold stress and its gradient to increase, thereby enhancing shear flows and radial electric fields. Turbulence spreading leads the edge Reynolds stresses to develop and the shear flow to be faster than edge electron temperature. The Reynolds stress arises from the symmetry breaking of the turbulence wave number spectrum. After sawtooth collapses, the joint probability density function of radial wave number and poloidal wave number of turbulence intensity becomes highly skewed and anisotropic, exhibiting strong asymmetry, which can be seen in attached figures (f) and (g). The development of turbulence spreading flux at the edge is also prior to the particle flux driven by turbulence, indicating that turbulent energy transport is not simply accompanied by turbulent particle transport. These results show that the turbulence spreading and symmetry breaking can enhance turbulent Reynolds stress, thereby driving shear flows, after sawtooth has crashed.

Using the two-point Edgeworth series up to second order in the linear rms density fluctuation we construct the weakly non-linear conditional probability distribution function for the density field around an overdense region. This requires calculating the two-point analogues of the skewness parameter S3. We test the dependence of the two-point skewness on distance from the peak for scale-free power spectra and Gaussian smoothing. The statistical features of such a conditional distribution are given as the values obtained within linear theory corrected by the terms that arise as a result of weakly non-linear evolution. The expected density around the peak is found to be always below the linear prediction while its dispersion is always larger than in the linear case. For large enough overdensities the weakly non-linear corrections can be more significant than the peak constraint introduced by Bardeen et al. We apply these results to the spherical model of collapse as developed by Hoffman & Shaham and find that in general the effect of weakly non-linear interactions is to decrease the scale from which a peak gathers mass and therefore also the mass itself. In the case of an open universe this results in steepening of the final profile of the virialized proto-object.

[Departement_IRSTEA]Eaux [TR1_IRSTEA]ARCEAU

An objective of longitudinal analysis is to estimate the conditional distributions of an outcome variable through a regression model. The approaches based on modeling the conditional means are not appropriate for this task when the conditional distributions are skewed or cannot be approximated by a normal distribution through a known transformation. We study a class of time-varying transformation models and a two-step smoothing method for the estimation of the conditional distribution functions. Based on our models, we propose a rank-tracking probability and a rank-tracking probability ratio to measure the strength of tracking ability of an outcome variable at two different time points. Our models and estimation method can be applied to a wide range of scientific objectives that cannot be evaluated by the conditional mean-based models. We derive the asymptotic properties for the two-step local polynomial estimators of the conditional distribution functions. Finite sample properties of our procedures are investigated through a simulation study. Application of our models and estimation method is demonstrated through an epidemiological study of childhood growth and blood pressure. Supplementary materials for this article are available online.

Efficient seismic fragility analysis method utilizing ground motion clustering and probabilistic machine learning

Joint probability distribution of winds and waves from wave simulation of 20 years (1989-2008) in Bohai Bay