Specific Probability Density Function Research Articles

Gravitational lensing of the cosmic microwave background by non-linear structures

Weak gravitational lensing changes the angular power spectra of the cosmic microwave background (CMB) temperature and polarization in a characteristic way containing valuable information for cosmological parameter estimation and weak lensing reconstructions. So far, analytical expressions for the lensed CMB power spectra assume the probability density function (PDF) of the lensing excursion angle to be Gaussian. However, coherent light deflection by nonlinear structures at low redshifts causes deviations from a pure Gaussian PDF. Working in the flat-sky limit we develop a method for computing the lensed CMB power spectra which takes these non-Gaussian features into account. Our method does not assume any specific PDF but uses instead an expansion of the characteristic function of the lensing excursion angle into its moments. Measuring these in the CMB lensing deflection field obtained from the Millennium Simulation we show that the change in the lensed power spectra is only at the 0.1% - 0.4% level on very small scales (below 4 arcmin) and demonstrate that the assumption of a Gaussian lensing excursion angle PDF is well applicable.

Gamma random field simulation by a covariance matrix transformation method

In studies involving environmental risk assessment, Gaussian random field generators are often used to yield realizations of a Gaussian random field, and then realizations of the non-Gaussian target random field are obtained by an inverse-normal transformation. Such simulation process requires a set of observed data for estimation of the empirical cumulative distribution function (ECDF) and covariance function of the random field under investigation. However, if realizations of a non-Gaussian random field with specific probability density and covariance function are needed, such observed-data-based simulation process will not work when no observed data are available. In this paper we present details of a gamma random field simulation approach which does not require a set of observed data. A key element of the approach lies on the theoretical relationship between the covariance functions of a gamma random field and its corresponding standard normal random field. Through a set of devised simulation scenarios, the proposed technique is shown to be capable of generating realizations of the given gamma random fields.

WE‐C‐204B‐06: Tumor Specific Reconstruction of Motion Probability Density Function from 4DCT

Purpose: The actual delivered dose to a moving tumor can deviate from prescribed dose not only during each fraction, but during each IMRT field. This deviation can be anticipated and incorporated into a treatment plan if the tumor specific motion probability density function (PDF) is accurately identified. In this study, a novel technique in determining the actual, tumor specific PDF of a moving gross tumor volume (GTV) is described and confirmed through phantom experiments using a quantitative approach of 4DCT imaging. Methods and Materials: We hypothesize that a PDF of the GTV can be obtained through weighted sums of phase‐sorted 4DCT images. Experimental validation is performed using a lung phantom attached to a programmable motion platform. The ground truth motion PDF of the targets was calculated through convolution of the motion pattern and the GTV dimension along the direction of motion. These PDF convolutions are compared to the normalized, reconstructed CT numbers obtained from averaging ten phase sorted 4DCT images after background subtraction. Results: The PDF reconstructed from averaged, weighted sets of 4DCT images was almost identical to the ground truth PDF in all comparisons made. Through sums of phase sorted 4DCT images, the resulting CT number represents a relative probability of finding some portion of the GTV in each geometric location (or voxel). In physical phantom reconstructions, the Pearson correlation coefficient between CT‐based PDF and the ground truth PDF was greater than .97 in all cases considered. An actual patient PDF was also evaluated in frontal and sagittal planes. Conclusions: Combining phase sorted CT images is an effective method in obtaining the actual tumor PDF as seen by an IMRT field. With the patient specific PDF, treatment planning to ensure proper target coverage can be achieved through intra‐field modulation, dose rate, and monitoring initial beam‐on phase.

Adaptive Polynomial Chaos for Gas Turbine Compression Systems Performance Analysis

The design of a gas turbine, or one of its constituentmodules, is generally approachedwith some specific operating condition in mind (its design point). Unfortunately, engine components seldom exactly meet their specifications and do not operate at just one condition, but over a range of power settings. This simplification can then lead to a product that exhibits performance worse than nominal in real-world conditions. The integration of some consideration of robustness as an active part of the design process can allow products less sensitive to the presence of the noise factors commonly found in real-world environments to be obtained. To become routinely used as a design tool, minimization of the time required for robustness analysis is paramount. In this study, a nonintrusive polynomial chaos formulation is used to evaluate the variability in the performance of a genericmodular-core compression system for a three-spool modern gas turbine engine subject to uncertain operating conditions with a defined probability density function. The standard orthogonal polynomials from the Askey scheme are replaced by a set of orthonormal polynomials calculated relative to the specific probability density function, improving the convergence of the method.

Charged-particle thermonuclear reaction rates: I. Monte Carlo method and statistical distributions

A method based on Monte Carlo techniques is presented for evaluating thermonuclear reaction rates. We begin by reviewing commonly applied procedures and point out that reaction rates that have been reported up to now in the literature have no rigorous statistical meaning. Subsequently, we associate each nuclear physics quantity entering in the calculation of reaction rates with a specific probability density function, including Gaussian, lognormal and chi-squared distributions. Based on these probability density functions the total reaction rate is randomly sampled many times until the required statistical precision is achieved. This procedure results in a median (Monte Carlo) rate which agrees under certain conditions with the commonly reported recommended “classical” rate. In addition, we present at each temperature a low rate and a high rate, corresponding to the 0.16 and 0.84 quantiles of the cumulative reaction rate distribution. These quantities are in general different from the statistically meaningless “minimum” (or “lower limit”) and “maximum” (or “upper limit”) reaction rates which are commonly reported. Furthermore, we approximate the output reaction rate probability density function by a lognormal distribution and present, at each temperature, the lognormal parameters μ and σ. The values of these quantities will be crucial for future Monte Carlo nucleosynthesis studies. Our new reaction rates, appropriate for bare nuclei in the laboratory, are tabulated in the second paper of this issue (Paper II). The nuclear physics input used to derive our reaction rates is presented in the third paper of this issue (Paper III). In the fourth paper of this issue (Paper IV) we compare our new reaction rates to previous results.

SU-FF-T-620: Assessment of Dose Reconstruction Methods for Prostate Image Guided Radiotherapy Protocols

Purpose: Many dosimetric evaluations of prostate intrafraction motion disregard effects of treatment duration and translations of the surrounding body per fraction. However the probability of motion has been shown to increase with time and image guidance strategies transfer interfractional uncertainties from the prostate to the surrounding body. We evaluate the dosimetric effect of these two factors for an online image guidance protocol. Method and Materials: Using intrafractional measurements from 22 IGRT patients, we calculated a probability density function (PDF) for each beam and for all beams combined excluding residual setup error. The effect of treatment duration was evaluated by computing cumulative dose using 2 convolution methods: population PDF and beam specific PDF. The calculations were performed on both a 30 cm diameter phantom and a patient. We evaluated the dose discrepancy introduced when PDFs were created using various motion magnitudes. The effect of relative motion from surrounding body was evaluated on the PTV dose discrepancy when the surrounding body was translated anterior and posterior by 0‐2cm, and after 20 fractions with the surrounding body displaced randomly about the mean using a multivariate normal distribution with various σ. Results: Differences in D99 between dose convolved with a beam specific PDF vs. a single PDF were ∼0.3% of mean PTV dose. The discrepancy was ∼5 times smaller than the dose discrepancy between static and convolved dose. Translation of the surrounding body increased dose to target, however the effect was small for clinically relevant translations (∼0.3% change in D99 for ±5mm AP translation). Random translations of surrounding body over multiple fractions caused differences in D99 of ∼0.3%, but with no apparent trend. Conclusions: Time dependency and surrounding body motion have modest dosimetric effects (0.3%) to the target, and should be accounted for when high accuracy is desired for dosimetric evaluation of intrafraction motion.

WE-E-AUD A-07: Should a Gaussian Probability Density Function Be Used to Approximate Respiration Induced Dosimetric Effects for Proton Radiotherapy?

Purpose: To compare the dose distributions generated by convolving a static dose distribution using a patient specific respiratory probability density function (R-PDF) with those generated using a generic Gaussian PDF (G-PDF) for proton therapy of lung cancer. Method and Materials: The R-PDFs were obtained by identifying the centroid motion of the targets from the 4D-CT scans of a phantom (CIRS Model 008 Dynamic Thorax) and a representative lung cancer patient. The CMS XiO® Treatment Planning System commissioned with 208 MeV nominal proton beam data from a passive scattering beam line at a proton therapy center was used for the static dose calculation. The dose convolution results from four different G-PDFs with standard deviations (SD) of 0.2, 0.3, 0.4, and 0.5 multiplying by the peak-to-peak motion amplitude (letter “A”, 1.60cm in the phantom and 1.75cm in the patient) were compared to the R-PDF convolved dose distributions using a commercial dosimetry analysis package (OmniPro I'mRT). Results: Respiration-induced dose error was 29% and 16% of the prescribed dose (PD) compared to the static doses in the phantom and patient, respectively. The G-PDF with SD of 0.4A most closely approximates the R-PDF whilst the maximum dose disagreements (MDDs) between the convolved doses using the two methods were 6% and 4% of the PD in phantom and patient, respectively. When G-PDFs with SD of 0.2A and 0.5A were used to approximate the R-PDF, the resulting MDDs were 19% and 12% in the phantom respectively, and 12% and 10% in the patient, respectively. When the G-PDF with SD of 0.3A was used to approximate the R-PDF, the resulting MDDs were 10% in the phantom and 8% in the patient. Conclusion: A Gaussian function should not be used to approximate a patient specific respiratory PDF since it can lead to clinically significant dose errors.

An approximation and its applications in wireless networks performance analysis

AbstractIn the literature, there are two common assumptions for the tele‐traffic parameter in analyzing the wireless network performance, that is, the tele‐parameter follows a specific probability density function (pdf) and additionally the pdf exists closed‐form Laplace Transform (LT). However, taking into account the cell irregular shape, the specific pdf may be unavailable while only the measured statistical moments are available. Moreover, the pdf function may not exist a closed‐form LT, for example, lognormal distribution function. In this paper, based on the Central Limit Theorem and hyper‐Erlang universal approximation property, we propose an approximation method applicable in the situations when only the statistical moments are available or LT of pdf does not exist. We then employ the technique in diverse applications, including the performance analysis of wireless network and the cost evaluation of mobility management. Extensive numerical examples demonstrate the good approximation capability to the exact formula and the simulation results. Copyright © 2006 John Wiley & Sons, Ltd.

Maximum and Minimum Likelihood Hebbian Learning for Exploratory Projection Pursuit

In this paper, we review an extension of the learning rules in a Principal Component Analysis network which has been derived to be optimal for a specific probability density function. We note that this probability density function is one of a family of pdfs and investigate the learning rules formed in order to be optimal for several members of this family. We show that, whereas we have previously (Lai et al., 2000s Fyfe and MacDonald, 2002) viewed the single member of the family as an extension of PCA, it is more appropriate to view the whole family of learning rules as methods of performing Exploratory Projection Pursuit. We illustrate this on both artificial and real data sets.

Magnetic Fields in the Quiet Sun: Observational Discrepancies and Unresolved Structure

Observations of magnetically sensitive lines in the visible and the infrared yield apparently contradictory values for the intrinsic field strength in the internetwork quiet Sun. It is shown that this discrepancy can be understood if one assumes that the magnetic field is not homogeneous over the resolution element. The difference between visible and infrared measurements may be used to set constrains on the subpixel distribution of field strengths. We suggest a specific probability density function that seems to satisfy the existing observational constraints.

Morphological Hopfield Networks

This paper reports on the investigation of the effects of neuronal shape, at both individual cell and network level, on the behavior of neuronal systems. More specifically, two-dimensional biologically realistic neuronal networks are obtained that take explicity into account the position and morphology of neuronal cells, with the respective behavior for associative recall being simulated through a diluted version of Hopfield's model. While a specific probability density function is used for the placement of the cell bodies, images of real neuronal cells (namely alpha and beta ganglion cells from the cat retina) are used to obtain biologically realistic models. Several morphological measures – including fractal dimension, the excluded volume, and integral geometry functionals–are estimated for the considered cells, and their values are correlated with the potential of the network for associative recall, which is quantified in terms of the overlap between distorted version of the trained patterns and their original version. Such an approach allows the quantitative and objective characterization of the relationship between neuronal shape and function, an important issue in neuroscience. The obtained results substantiate interesting relationships between the neural morphology and function as determined by the performance of the network.

An inverse problem analysis for the determination of probalistic parameters of concrete behavior modeled by a statistical approach

When the Monte Carlo method is used to simulate the heterogeneous behavior of concrete in the framework of a finite element probabilistic analysis, N samples of the vector of random variables (tensile strength, Young's modulus, etc.) are generated from a specific probability density function. If the uncertainties of these material parameters are assumed to vary spatially following a normal distribution, the N samples corresponding to a simulation are function of the mean and the standard deviation that define the Gauss density function. The problem is that these statistical moments are not known,a priori, for the characteristic volume of the finite elements for which the problem has been discretized. In this paper an algorithm is proposed to evaluate the parameters characterizing the statistical distribution (e. g. for a normal distribution: the mean and the standard deviation) for a given response of the structure (for instance, a mean load-displacement curve) following an inverse analysis procedure. A very simple mechanical system is used to verify the feasibility of the procedure. By way of example, it is shown that this kind of inverse problem for the identification of statistical parameters is suitable for concrete.

Stochastic gradient optimization of importance sampling for the efficient simulation of digital communication systems

Importance sampling (IS) techniques offer the potential for large speed-up factors for bit error rate (BER) estimation using Monte Carlo (MC) simulation. To obtain these speed-up factors, the IS parameters specifying the simulation probability density function (PDF) must be carefully chosen. With the increased complexity in communication systems, analytical optimization of the IS parameters can be virtually impossible. We present a new IS optimization algorithm based on stochastic gradient techniques. The formulation of the stochastic gradient descent (SGD) algorithm is more general and system-independent than other existing IS methodologies, and its applicability is not restricted to a specific PDF or biasing scheme. The effectiveness of the SGD algorithm is demonstrated by two examples of communication systems where the IS techniques have not been applied before. The first example is a communication system with diversity combining, slow nonselective Rayleigh fading channel, and noncoherent envelope detection. The second example is a binary baseband communication system with a static linear channel and a recursive least square (RLS) linear equalizer in the presence of additive white Gaussian noise (AWGN).

Locally optimum rank detection of correlated random signals in additive noise

Nonparametric detection of a zero-mean random signal in additive noise is considered. The locally optimum detector based on signs and ranks of observations is derived, for good weak-signal detection performance under any specified noise probability density function. This detector is shown to have interesting similarities to the locally optimum detector for random signals. It may also be viewed as a generalization of the locally optimum rank detector for known signals. Examples of the test statistic of the detector are given for some specific noise probability density functions. Asymptotic and finite sample-size performance of the locally optimum rank detector is also considered. >

Channel Modeling and Threshold Signal Processing in Underwater Acoustics: An Analytical Overview

An overview of underwater acoustic channel modeling and threshold signal processing is presented, which emphasizes the inhomogeneous, random, and non-Ganssian nature of the generalized channel, combined with appropriate weak-signal detection and estimation. Principal attention is given to the formal structuring of the scattered and ambient acoustic noise fields, as well as that of the desired signal, including both fading and Doppler "smear" phenomena. The role of general receiving arrays is noted, as well as their impact on spatial and temporal signal processing and beam forming, as indicated by various performance measures in detection and estimation. The emphasis here is on limiting optimum threshold systems, with some attention to suboptimum cases. Specific first-order probability density functions (pdf's) for the non-Ganssian components of typical underwater acoustic noise environments are included along with their field covariances. Several examples incorporating these pdf's are given, to illustrate the applications and general methods involved. The fundamental role of the detector structure in determining the associated optimum estimators is noted: the estimators arc specific linear or nonlinear functionals of the original optimum detector algorithm, depending on the criterion (i.e., minimization of the chosen error or cost function) selected. Results for both coherent and incoherent modes of reception are presented, reflecting the fact that frequently signal epoch is not known initially at the receiver. To supplement the general discussion, a selected list of references is included, to provide direct access to specific detailed problems, techniques, and results, for which the present paper is only a guide.