Error Pdf Research Articles

A non‐parametric way to estimate observation errors based on ensemble innovations

AbstractPrevious studies that inferred the observation error statistics from the innovation statistics can only provide the second moment of the error probability density function (pdf). However, the observation errors are sometimes non‐Gaussian, for example, for observation operators with unknown representation errors, or for bounded observations. In this study, we propose a new method, the Deconvolution‐based Observation Error Estimation (DOEE), to infer the full observation error pdf. DOEE does not rely on linear assumptions on the observation operator, the optimality of the data assimilation algorithm, or implicit Gaussian assumptions on the error pdf. The main assumption of DOEE is the availability of an ensemble of background forecasts following the independent and identically distributed (i.i.d.) assumption. We conduct idealized experiments to demonstrate the ability of the DOEE to accurately retrieve a non‐Gaussian (bimodal, skewed, or bounded) observation error pdf. We then apply the DOEE to construct a state‐dependent observation error model for satellite radiances by stratifying the observation errors based on cloud amount. In general, we find that the observation error pdfs in many categories are skewed. By adding a new predictor, total column water vapor (TCWV), into the state‐dependent model, we find that for cloudy pixels, when TCWV is small, the observation error pdfs are quite similar and Gaussian‐like, whereas when TCWV is large, the observation error pdfs differ for different cloud amount, while all of them are positively biased. This result suggests that exploring other predictors, like cloud type, might improve the stratification of the observation error model. We also discuss ways to include a non‐parametric observation error pdf into modern data assimilation schemes, including a consideration of the strong‐constraint 4D‐Var perspective, as well as the implications for other observation types including observations with bounded range.

Impact of the LHCb 8 [formula omitted], [formula omitted], and asymmetry data on the CT14HERA2 PDFs

First, we calculate the total cross-sections and differential cross-sections of the W and Z productions at next-to-leading order QCD using ResBos and CT14HERA2 PDFs, and compare with the measurements from the LHCb detector at a center-of-mass energy of 8 TeV and find good agreement between the experimental data and theoretical predictions. Then, we analyze the impact of LHCb 8 TeV W, Z, and asymmetry data on CT14HERA2 PDFs( g(x,Q), u(x,Q), d(x,Q), ū(x,Q), d̄(x,Q), d(x,Q)/u(x,Q) ) using the Error PDF Updating Method Package. Through this analysis, we find that the LHCb 8 TeV W and Z data prefer softer gloun and d(x,Q) at large x region, but have no significant impacts on u(x,Q) PDF.

Determining the helicity structure of the nucleon at the Electron Ion Collider in China

Understanding how sea quarks behave inside a nucleon is one of the most important physics goals of the proposed Electron-Ion Collider in China (EicC), which is designed to have a 3.5 GeV polarized electron beam (80% polarization) colliding with a 20 GeV polarized proton beam (70% polarization) at instantaneous luminosity of 2 × 1033cm−2s−1. A specific topic at EicC is to understand the polarization of individual quarks inside a longitudinally polarized nucleon. The potential of various future EicC data, including the inclusive and semi-inclusive deep inelastic scattering data from both doubly polarized electron-proton and electron-3He collisions, to reduce the uncertainties of parton helicity distributions is explored at the next-to-leading order in QCD, using the Error PDF Updating Method Package (ePump) which is based on the Hessian profiling method. We show that the semi-inclusive data are well able to provide good separation between flavour distributions, and to constrain their uncertainties in the x > 0.005 region, especially when electron-3He collisions, acting as effective electron-neutron collisions, are taken into account. To enable this study, we have generated a Hessian representation of the DSSV14 set of PDF replicas, named DSSV14H PDFs.

The impact of ATLAS and CMS single differential top-quark pair measurements at TeV on CTEQ-TEA PDFs * *The work of S. Dulat was supported by the National Natural Science Foundation of China (11965020, 11847160)

By applying the Error PDF Updating Method, we analyze the impact of the absolute and normalized single differential cross-sections for top-quark pair production data from the ATLAS and CMS experiments at the Large Hadron Collider, at a center-of-mass energy of TeV, on the CT14HERA2 PDFs. We find that the top quark pair single differential distributions provide minor constraints on the CT14HERA2 gluon PDF when the nominal CT14HERA2 inclusive jet production data are included in the fit. Larger constraints on the gluon distribution are present when the jet data are removed (CT14HERA2mJ) and/or when increased weights are given to the top data in the CT14HERA2 fits. The weighted data provide significant constraints on the CT14HERA2mJ gluon PDF, which are comparable to those obtained from inclusive jet production data. Furthermore, we examine the top quark mass sensitivity of the top-quark pair single differential distributions.

The Impact of Single Top Data on CT14nnlo PDFs

In this paper we study the impact of ATLAS 8 TeV t-channel single top quark production data on CT14nnlo PDFs. First, we calculate the total cross section, cross section ratio, differential cross section, and normalized differential cross section as a function of transverse momentum pT and rapidity y, at next-to-leading order QCD by MadGraph using CT14nnlo PDFs. And then we update CT14nnlo PDFs and observables by applying Error PDF Updating method. We find that down quark PDF increased roughly about 3% in 10− 4 < x < 4 × 10− 3 range at Q = 1.3 GeV, when weight factor equals to six. Furthermore, for data and theory comparison, we get slightly better agreement after updating observables.

Design of modeling error PDF based fuzzy neural network for effluent ammonia nitrogen prediction

To predict the effluent ammonia nitrogen (NH4-N) of wastewater treatment process (WWTP), the soft computing methods are widely used, in which the mean square error (MSE) is usually adopted as the performance criterion. However, the MSE based methods cannot fully utilize the statistic information of data and are vulnerable to the nonzero-mean noise. To address these issues, the modeling-error probability density function based fuzzy neural network (PDF-FNN) is proposed in this paper. Firstly, the modeling error PDF criterion is generated to minimize the spatial deviation between the modeling error distribution and the predefined target. Then, a gradient descent method with adaptive learning rate is presented to update the parameters of PDF-FNN. Furthermore, the convergence of PDF-FNN is analyzed from a mathematical point of view. Finally, a nonlinear system modeling and the effluent NH4-N prediction in WWTP are applied to prove the effectiveness of the proposed PDF-FNN. The results indicate that the PDF-FNN has better prediction accuracy and model stability than other methods, especially in the noisy environment.

QCD analysis of CMS W + charm measurements at LHC with and implications for strange PDF * * Supported by the National Natural Science Foundation of China (11965020)

We calculate cross-sections and cross-section ratios of a charm quark production in association with a W gauge boson at next-to-leading order QCD using MadGraph and CT10NNLO, CT14NNLO, and MSTW2008NNLO PDFs. We compare the results with measurements from the CMS detector at the LHC at a center-of-mass energy of 7 TeV. Moreover, we calculate absolute and normalized differential cross-sections as well as differential cross-section ratios as a function of the lepton pseudorapidity from the W boson decay. The correlation between the CT14NNLO PDFs and predictions for charm data are studied as well. Furthermore, by employing the error PDF updating method proposed by the CTEQ-TEA group, we update CT14NNLO PDFs, and analyze the impact of CMS 7 TeV charm production data to the original CT14NNLO PDFs. By comparison of the , , , , , and PDFs at GeV and GeV for the CT14NNLO and CT14NNLO+Wc, we see that the error band of the PDF is reduced in the region , and the error band of PDF is also slightly reduced at region .

The quaternion minimum error entropy algorithm with fiducial point for nonlinear adaptive systems

In this paper, we develop a kernel adaptive filter for quaternion domain data, based on information theoretic learning cost function which could be useful for quaternion based kernel applications of nonlinear filtering. The new algorithm is based on error entropy function with Fiducial point and is referred to as the quaternion minimum error entropy with fiducial point (QKMEEF) algorithm. In our previous work we developed quaternion kernel adaptive filter based on minimum error entropy referred to as the quaternion KMEE (QKMEE) algorithm [1]. Since entropy does not change with the mean of the distribution, the algorithm may converge to a set of optimal weights without having zero mean error. Traditionally, to make the zero mean output error, the output during testing session was biased with the mean of errors of training session. However, for non-symmetric or heavy tails error PDF the estimation of error mean is problematic. The minimum error entropy criterion, minimizes Renyi’s quadratic entropy of the error between the filter output and desired response or indirectly maximizing the error information potential. Here, the approach is applied to quaternions. Adaptive filtering in quaterion domain intrinsically incorporates component-wise real valued cross-correlation or the coupling within the dimensions of the quaternion input. We apply generalized Hamilton-real (GHR) calculus that is applicable to Hilbert space for evaluating the cost function gradient to develop the Quaternion Minimum Error Entropy Algorithm with Fiducial points. Simulation results are used to show the behavior of the new algorithm (QKMEEF) when signal is non-Gaussian in presence of unimodal noise versus bi-modal noise distributions. Simulation results also show that the new algorithm Quat-KMEEF can track and predict the 4-Dimensional non-stationary process signals where there are correlations between components better than quadruple real-valued KMEEF and Quat-KLMS algorithms.

Stochastic-Based Analyses of Noncoherent Code Tracking Loops Over Wireless Fading Circumstances

For band-limited pseudonoise signals, a noncoherent digital delay-lock loop (DDLL) equipped with a discrete automatic gain control algorithm under a RAKE scheme over frequency-selective fading channels is studied. Its associated statistical characteristics are analyzed based on the Nakagami- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> stochastic process. The transition timing error probability density function (pdf) of the Chapman-Kolmogorov equation is evaluated to drive an analytical expression for the steady-state timing error pdf. The mean time to lose lock and the mean square tracking error with its corresponding Cramér-Rao lower bound (CRLB) under frequency-selective Nakagami- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> fading environments are scrutinized, whereas their numerical results are confirmed by time-domain computer simulations. Moreover, the tracking performance of the DDLL in the presence of Doppler shifts is also examined.

Entropy and Correntropy Against Minimum Square Error in Offline and Online Three-Day Ahead Wind Power Forecasting

This paper reports new results in adopting entropy concepts to the training of neural networks to perform wind power prediction as a function of wind characteristics (speed and direction) in wind parks connected to a power grid. Renyi's entropy is combined with a Parzen windows estimation of the error pdf to form the basis of two criteria (minimum entropy and maximum correntropy) under which neural networks are trained. The results are favorably compared in online and offline training with the traditional minimum square error (MSE) criterion. Real case examples for two distinct wind parks are presented.

Statistical Analysis of Wind Power Forecast Error

Wind power forecast error usually has been assumed to have a near Gaussian distribution. With a simple statistical analysis, it can be shown that this is not valid. To obtain a more appropriate probability density function (pdf) of the wind power forecast error, an indirect algorithm based on the Beta pdf is proposed. Measured one-year time series from two different wind farms are used to generate the forecast data. Three different forecast scenarios are simulated based on the persistence approach. This makes the results comparable to other forecast methods. It is found that the forecast error pdf has a variable kurtosis ranging from 3 (like the Gaussian) to over 10, and therefore it can be categorized as fat-tailed. A new approximation function for the parameters of the Beta pdf is proposed because results from former publications could not be confirmed. Besides, a linear approximation is developed to describe the relationship between the persistence forecast and the related mean measured power. An energy storage system (ESS), which reduces the forecast error and smooths the wind power output, is considered. Results for this case show the usefulness of the proposed forecast error pdf for finding the optimum rated ESS power.

Is the diaphragm motion probability density function normally distributed?

During radiotherapy treatment planning, the margins given to the clinical target volume to form the planning target volume accounts for internal motion and set-up error. Most margin formulas assume that the underlying distributions are independent and normal. Clinical data suggests that the set-up error probability density function (pdf) can be considered to have an approximately normal distribution. However, there is evidence that internal motion does not have a normal distribution. Thus, in general, a convolution of the two pdfs should be performed to determine the total geometric error. The goals of this article were to (1) determine if the internal motion pdf due to respiration can be characterized using a normal distribution, and (2) if not, determine if the total geometric uncertainty for combining internal motion and set-up error can be characterized by a normal distribution. Sixty fluoroscopy diaphragm motion data sets were obtained using three breathing training types: free breathing, audio instruction, and visual feedback. Diaphragm motion was used as a surrogate for liver and lung cancer motion. The data were analyzed with normality tests in the following groups: (1) single motion measurements, (2) combined motion measurements for each patient, and (3) combined motion measurements for all patients. Following this analysis, the diaphragm motion pdfs were convolved with a set-up error pdf, and the standard deviation of the set-up error pdf at which the total geometric error pdf became normal was determined. At set-up error standard deviation values of at least 0.27 and 0.1 cm for free breathing, 0.57 and 0.42 cm for audio instruction, and 0.55 and 0 cm for visual feedback, for single motion measurements and combined motion measurements for each patient, respectively, total geometric error pdfs became approximately normal. When the motion measurements for all the patients were combined, diaphragm motion pdfs were approximately normal for all feedback types. Therefore, for treatment planning purposes in the absence of individual patient measurements, the diaphragm motion pdf can be considered an approximately normal distribution. However, care should be taken when determining a margin based on individual patients measurements as the total geometric error will, in general, not be normally distributed.

Numerical Method for Determining the Quantization Error PDF of Single-Bit&amp;lt;tex&amp;gt;$Sigma Delta$&amp;lt;/tex&amp;gt;Modulators

Existing models for the quantizer of /spl Sigma//spl Delta/ modulators make assumptions on the probability density function (pdf) of the quantization error, or some other convenient signal of the modulator. In this paper, a method for the determination of this pdf for single-bit /spl Sigma//spl Delta/ modulators is presented. First, a numerical method is proposed in order to solve the simplified equation for the quantization error pdf for first-order systems considering noiseless and noisy dc input signals. Then, it is shown how most practical high-order (>2)/spl Sigma//spl Delta/ modulators, resulting from well-established design methods, can be modeled as first-order systems plus an additive noise source at the input. Hence, their quantization error pdf is analyzed using the proposed method. Simulation results are shown to be in considerable agreement with those of the proposed method.

Space-time structure and dynamics of the forecast error in a coastal circulation model of the Gulf of Lions

The probability density function (pdf) of forecast errors due to several possible error sources is investigated in a coastal ocean model driven by the atmosphere and a larger-scale ocean solution using an Ensemble (Monte Carlo) technique. An original method to generate dynamically adjusted perturbation of the slope current is proposed. The model is a high-resolution 3D primitive equation model resolving topographic interactions, river runoff and wind forcing. The Monte Carlo approach deals with model and observation errors in a natural way. It is particularly well-adapted to coastal non-linear studies. Indeed higher-order moments are implicitly retained in the covariance equation. Statistical assumptions are made on the uncertainties related to the various forcings (wind stress, open boundary conditions, etc.), to the initial state and to other model parameters, and randomly perturbed forecasts are carried out in accordance with the a priori error pdf. The evolution of these errors is then traced in space and time and the a posteriori error pdf can be explored. Third- and fourth-order moments of the pdf are computed to evaluate the normal or Gaussian behaviour of the distribution. The calculation of Central Empirical Orthogonal Functions (Ceofs) of the forecast Ensemble covariances eventually leads to a physical description of the model forecast error subspace in model state space. The time evolution of the projection of the Reference forecast onto the first Ceofs clearly shows the existence of specific model regimes associated to particular forcing conditions. The Ceofs basis is also an interesting candidate to define the Reduced Control Subspace for assimilation and in particular to explore transitions in model state space. We applied the above methodology to study the penetration of the Liguro-Provençal Catalan Current over the shelf of the Gulf of Lions in north-western Mediterranean together with the discharge of the Rhône river. This region is indeed well-known for its intense topographic and atmospheric forcings.

Noise Study of an Asynchronous Delta Sigma Modulator

The steady-state probability density function of the error of an asynchronous delta sigma modulator in the presence of a dc input and additive Gaussian noise with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> power spectrum is obtained by solving the Fokker-Planck equation. The error pdf plots are presented for different system parameters.