Resulting Probability Distribution Research Articles

A tool for simulating collision probabilities of animals with marine renewable energy devices.

The mathematical problem of establishing a collision probability distribution is often not trivial. The shape and motion of the animal as well as of the the device must be evaluated in a four-dimensional space (3D motion over time). Earlier work on wind and tidal turbines was limited to a simplified two-dimensional representation, which cannot be applied to many new structures. We present a numerical algorithm to obtain such probability distributions using transient, three-dimensional numerical simulations. The method is demonstrated using a sub-surface tidal kite as an example. Necessary pre- and post-processing of the data created by the model is explained, numerical details and potential issues and limitations in the application of resulting probability distributions are highlighted.

Optimal and Robust Quantum Metrology Using Interaction-Based Readouts.

Useful quantum metrology requires nonclassical states with a high particle number and (close to) the optimal exploitation of the state's quantum correlations. Unfortunately, the single-particle detection resolution demanded by conventional protocols, such as spin squeezing via one-axis twisting, places severe limits on the particle number. Additionally, the challenge of finding optimal measurements (that saturate the quantum Cramér-Rao bound) for an arbitrary nonclassical state limits most metrological protocols to only moderate levels of quantum enhancement. "Interaction-based readout" protocols have been shown to allow optimal interferometry or to provide robustness against detection noise at the expense of optimality. In this Letter, we prove that one has great flexibility in constructing an optimal protocol, thereby allowing it to also be robust to detection noise. This requires the full probability distribution of outcomes in an optimal measurement basis, which is typically easily accessible and can be determined from specific criteria we provide. Additionally, we quantify the robustness of several classes of interaction-based readouts under realistic experimental constraints. We determine that optimal and robust quantum metrology is achievable in current spin-squeezing experiments.

Quasi-probability distributions for observables in dynamic systems

We develop a general framework to investigate fluctuations of non-commuting observables. To this end, we consider the Keldysh quasi-probability distribution (KQPD). This distribution provides a measurement-independent description of the observables of interest and their time-evolution. Nevertheless, positive probability distributions for measurement outcomes can be obtained from the KQPD by taking into account the effect of measurement back-action and imprecision. Negativity in the KQPD can be linked to an interference effect and acts as an indicator for non-classical behavior. Notable examples of the KQPD are the Wigner function and the full counting statistics, both of which have been used extensively to describe systems in the absence as well as in the presence of a measurement apparatus. Here we discuss the KQPD and its moments in detail and connect it to various time-dependent problems including weak values, fluctuating work, and Leggett-Garg inequalities. Our results are illustrated using the simple example of two subsequent, non-commuting spin measurements.

Skeletal Shape Correspondence Through Entropy.

We present a novel approach for improving the shape statistics of medical image objects by generating correspondence of skeletal points. Each object's interior is modeled by an s-rep, i.e., by a sampled, folded, two-sided skeletal sheet with spoke vectors proceeding from the skeletal sheet to the boundary. The skeleton is divided into three parts: the up side, the down side, and the fold curve. The spokes on each part are treated separately and, using spoke interpolation, are shifted along that skeleton in each training sample so as to tighten the probability distribution on those spokes' geometric properties while sampling the object interior regularly. As with the surface/boundary-based correspondence method of Cates et al., entropy is used to measure both the probability distribution tightness and the sampling regularity, here of the spokes' geometric properties. Evaluation on synthetic and real world lateral ventricle and hippocampus data sets demonstrate improvement in the performance of statistics using the resulting probability distributions. This improvement is greater than that achieved by an entropy-based correspondence method on the boundary points.

A geometrical optimization method applied to a heaving point absorber wave energy converter

A methodology for the geometrical optimization of wave energy converters (WEC) based on statistical analysis methods and the hydrodynamics of the system in the frequency domain is presented. The optimization process has been applied on a one-body heaving point absorber for a nearshore region of the Rio de Janeiro coast. The sea characteristics have been described using a five-year wave hindcast and are based on a third generation wind wave model WAVEWATCH III. The optimization procedure is performed based on the resultant wave spectrum and joint probability distribution. The optimization process aims at maximizing both WEC absorbed power and absorption bandwidth when providing a natural period close to the predominant wave periods of the sea site. The optimized geometry of the WEC is determined by running a few simulations in the frequency domain and using the design of experiment (DOE) method. The software ANSYS-AQWA is used for the hydrodynamic diffraction analysis, and the DOE method is applied through the Minitab software to determine the optimized geometry. The two primary advantages of this optimization method are the reduced computational time and the possibility of performing parametric analyses for the WEC geometry.

Trigonometric Functions Above and Below of the Double-Slit Barrier

The Double-Slit Experiment with individual electrons represents the central mystery of quantum mechanics and was voted by readers of Physics World as the most beautiful experiment in physics. In our trigonometric model we have divided the Young-Feynman double-slit interferometer into two zones: Aristotelian mixing chamber (the space between the entrance slit S0 and above the double-slit barrier with slits S1 and S2 where particles have been reflected (Euclid) and/or randomly deviated – parenklisis of Epicurus) and Plato’s projection chamber (the space below the double-slit barrier and above the projection screen). Aristotelian mixing chamber represents the classical hidden variable: individual particles have been reflected and/or randomly deviated in this zone and later pass through the double-slit barrier in such a way that they create probability distributions of single particles described by the very well known trigonometric relations. The double-slit barrier serves as a “self-organizing filter” for reflected and/or deviated individual particles and creates an order in the population densities among individual particles. For the experimental evaluation of this trigonometric model we propose to insert an internal and external “sfumato mirrors” for the quantitative manipulation of probability distributions of individual particles. The concept “sfumato” was inspired by the Old Master Leonardo da Vinci (see his picture Mona Lisa) (“sfumato” means “smoke”). Sfumato mirrors represent the classical observable variable and their influence on the resulting probability densities of individual particles can be trigonometrically described. Sfumato mirrors must not absorb any particles. In another experimental modification we can insert below one slit of the double-slit barrier an “absorption chiaroscuro mask” to fully remove those penetrating particles from the system and thus manipulate with the probability distribution observed with one opened slit (“chiaroscuro” means “light-dark”). It could be of interest to observe the resulting probability distribution of individual particles registered on the projection screen from one opened slit S1 or S2. The absorption chiaroscuro mask must not reflect particles back to the Aristotelian mixing chamber or to the Plato´s projection chamber. The results of these experiments will decide if we are just going up in a blind alley”.

Robust MIMO Beamforming for Cellular and Radar Coexistence

In this letter, we consider the coexistence and spectrum sharing between downlink multi-user multiple-input-multiple-output (MU-MIMO) communication and an MIMO radar. For a given performance requirement of the downlink communication system, we design the transmit beamforming such that the detection probability of the radar is maximized. While the original optimization problem is non-convex, we exploit the monotonically increasing relationship of the detection probability with the non-centrality parameter of the resulting probability distribution to obtain a convex lower-bound optimization. The proposed beamformer is designed to be robust to imperfect channel state information (CSI). Simulation results verify that the proposed approach facilitates the coexistence between radar and communication links, and illustrates a scalable tradeoff between the two systems' performance.

On the inequivalence of the CH and CHSH inequalities due to finite statistics

Different variants of a Bell inequality, such as CHSH and CH, are known to be equivalent when evaluated on nonsignaling outcome probability distributions. However, in experimental setups, the outcome probability distributions are estimated using a finite number of samples. Therefore the nonsignaling conditions are only approximately satisfied and the robustness of the violation depends on the chosen inequality variant. We explain that phenomenon using the decomposition of the space of outcome probability distributions under the action of the symmetry group of the scenario, and propose a method to optimize the statistical robustness of a Bell inequality. In the process, we describe the finite group composed of relabeling of parties, measurement settings and outcomes, and identify correspondences between the irreducible representations of this group and properties of outcome probability distributions such as normalization, signaling or having uniform marginals.

Investigating and improving student understanding of the expectation values of observables in quantum mechanics

The expectation value of an observable is an important concept in quantum mechanics since measurement outcomes are, in general, probabilistic and we only have information about the probability distribution of measurement outcomes in a given quantum state of a system. However, we find that upper-level undergraduate and PhD students in physics have both conceptual and procedural difficulties when determining the expectation value of a physical observable in a given quantum state in terms of the eigenstates and eigenvalues of the corresponding operator, especially when using Dirac notation. Here we first describe the difficulties that these students have with determining the expectation value of an observable in Dirac notation. We then discuss how the difficulties found via student responses to written surveys and individual interviews were used as a guide in the development of a quantum interactive learning tutorial (QuILT) to help students develop a good grasp of the expectation value. The QuILT strives to help students integrate conceptual understanding and procedural skills to develop a coherent understanding of the expectation value. We discuss the effectiveness of the QuILT in helping students learn this concept from in-class evaluations.

Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting

Gene expression is inherently stochastic. Advanced single-cell microscopy techniques together with mathematical models for single gene expression led to important insights in elucidating the sources of intrinsic noise in prokaryotic and eukaryotic cells. In addition to the finite size effects due to low copy numbers, translational bursting is a dominant source of stochasticity in cell scenarios involving few short lived mRNA transcripts with high translational efficiency (as is typically the case for prokaryotes), causing protein synthesis to occur in random bursts.In the context of gene regulation cascades, the Chemical Master Equation (CME) governing gene expression has in general no closed form solution, and the accurate stochastic simulation of the dynamics of complex gene regulatory networks is a major computational challenge. The CME associated to a single gene self regulatory motif has been previously approximated by a one dimensional time dependent partial integral differential equation (PIDE). However, to the best of our knowledge, multidimensional versions for such PIDE have not been developed yet. Here we propose a multidimensional PIDE model for regulatory networks involving multiple genes with self and cross regulations (in which genes can be regulated by different transcription factors) derived as the continuous counterpart of a CME with jump process. The model offers a reliable description of systems with translational bursting. In order to provide an efficient numerical solution, we develop a semilagrangian method to discretize the differential part of the PIDE, combined with a composed trapezoidal quadrature formula to approximate the integral term.We apply the model and numerical method to study sustained stochastic oscillations and the development of competence, a particular case of transient differentiation attained by certain bacterial cells under stress conditions. We found that the resulting probability distributions are distinguishable from those characteristic of other transient differentiation processes. In this way, they can be employed as markers or signatures that identify such phenomena from bacterial population experimental data, for instance. The computational efficiency of the semilagrangian method makes it suitable for purposes like model identification and parameter estimation from experimental data or, in combination with optimization routines, the design of gene regulatory networks under molecular noise.

Probability distributions of continuous measurement results for conditioned quantum evolution

We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the post-selection in the end of the evolution. The statistics may drastically differ from the non-conditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of non-conditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, evaluate and discuss the distributions in experimentally relevant setups. We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of non-trivial probability distributions. We reveal peaks (or dips) at half-quantized values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and sudden jump in time-integrated output. We present and discuss the numerical evaluation of the probability distribution aiming at extend- ing the analytic results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence.

Investigating and improving student understanding of the probability distributions for measuring physical observables in quantum mechanics

A solid grasp of the probability distributions for measuring physical observables is central to connecting the quantum formalism to measurements. However, students often struggle with the probability distributions of measurement outcomes for an observable and have difficulty expressing this concept in different representations. Here we first describe the difficulties that upper-level undergraduate and PhD students have with the probability distributions for measuring physical observables in quantum mechanics. We then discuss how student difficulties found in written surveys and individual interviews were used as a guide in the development of a quantum interactive learning tutorial (QuILT) to help students develop a good grasp of the probability distributions of measurement outcomes for physical observables. The QuILT strives to help students become proficient in expressing the probability distributions for the measurement of physical observables in Dirac notation and in the position representation and be able to convert from Dirac notation to position representation and vice versa. We describe the development and evaluation of the QuILT and findings about the effectiveness of the QuILT from in-class evaluations.

Fuel for debating ancient economies. Calculating wood consumption at urban scale in Roman Imperial times

Estimating wood extraction rates from forests based on archaeological and historical evidence is an important step in evaluating the sustainability of past social-ecological systems. In this paper, we present a calculation tool to estimate human wood resource use for a selected location during a defined period in the past. We illustrate the method by its application to the ancient town of Sagalassos (South-west Anatolia, Turkey) during the Roman Imperial period, with a focus on pottery production and the Roman Baths. Based on archaeological data, thermodynamic formulas and calorific values, an estimation is provided of the amount of wood used within a time step of one year. Because quantitative information on ancient technology and lifestyle is rather scarce and uncertain, input values consist of ranges. In order to take this uncertainty into account, a Monte Carlo procedure is included, offering a probability distribution of possible outcomes. Our results indicate that wood consumption in 2nd century Sagalassos was quite high, with a lifestyle including frequent hot bathing, export driven pottery production and a climate that required heating during winter months. Based on the available woodland area, we conclude that the community of Sagalassos was intensively using the surrounding forests.

A Bayesian Examines the Lady Tasting Tea (Abridged)

This technical note accompanies the case/class on “The Lady Tasting Tea” (LTT). It describes how a Bayesian would update his or her prior probability for the probability the LTT can correctly distinguish cups of tea based on whether the milk was added first or last. The long-form note (UVA-QA-0769) describes both a discrete model (LTT has p of 0.5 or 0.8) and a more complicated continuous model (p is beta distributed). This abridged note describes only the discrete model. The case/class itself provides a very useful analogy with which students can explore the elements of statistical inference. Excerpt UVA-QA-0768 Jun. 13, 2011 A Bayesian Examines the Lady Tasting Tea (Abridged) Before I describe what Bayesian statistics can add to the discussion of the lady tasting tea (LTT), let me remind you where the classical statisticians left us. They used the binomial distribution to calculate the probability distribution of the number the LTT gets correct (in 10 trials) under two scenarios. The first scenario, called the null hypothesis, is that she is guessing. Guessing would mean P, the probability that she correctly identifies which of two cups has had milk poured into it first and which had tea in it first, is 0.5. The second scenario, called the alternative hypothesis, is that she is skilled. To keep things from getting too complicated, we assume that skilled means P = 0.8 for each trial. The resulting probability distributions are given in Table 1. With these probabilities, the classical statistician can make probability statements about what will happen undereither scenario. Table 1. Probability distributions for number correct in 10 trials. . . .

A Bayesian Examines the Lady Tasting Tea

This technical note accompanies the case/class on “The Lady Tasting Tea” (LTT). It describes how a Bayesian would update his or her prior probability for the probability the LTT can correctly distinguish cups of tea based on whether the milk was added first or last. This long-form note describes both a discrete model (LTT has p of 0.5 or 0.8) and a more complicated continuous model (p is beta distributed). The abridged note (UVA-QA-0768) describes only the discrete model. The case/class itself provides a very useful analogy with which students can explore the elements of statistical inference. Excerpt UVA-QA-0769 Jun. 13, 2011 A Bayesian Examines the Lady Tasting Tea Before I describe what Bayesian statistics can add to the discussion of the lady tasting tea (LTT), let me remind you where the classical statisticians left us. They used the binomial distribution to calculate the probability distribution of the number the LTT gets correct (in 10 trials) under two scenarios. The first scenario, called the null hypothesis, is that she is guessing. Guessing would mean P, the probability that she correctly identifies which of two cups has had milk poured into it first and which had tea in it first, is 0.5. The second scenario, called the alternative hypothesis, is that she is skilled. To keep things from getting too complicated, we assume that skilled means P = 0.8 for each trial. The resulting probability distributions are given in Table 1. With these probabilities, the classical statistician can make probability statements about what will happen undereither scenario. Table 1. Probability distributions for number correct in 10 trials. . . .

Instrumental Divergence and the Value of Control.

A critical aspect of flexible choice is that alternative actions yield distinct consequences: Only when available action alternatives produce distinct outcome states does discrimination and selection between actions allow an agent to flexibly obtain the currently most desired outcome. Here, we use instrumental divergence – the degree to which alternative actions differ with respect to their outcome probability distributions – as an index of flexible instrumental control, and assess the influence of this novel decision variable on choice preference. In Experiment 1, when other decision variables, such as expected value and outcome entropy, were held constant, we found a significant preference for high instrumental divergence. In Experiment 2, we used an “auto- vs. self-play” manipulation to eliminate outcome diversity as a source of behavioral preferences, and to contrast flexible instrumental control with the complete absence of voluntary choice. Our results suggest that flexible instrumental control over decision outcomes may have intrinsic value.

The NUTs and Bolts of squashed holography

We evaluate the partition function of the free O(N) model on a two-parameter family of squashed three spheres. We also find new solutions of general relativity with negative cosmological constant and the same double squashed boundary geometry and analyse their thermodynamic properties. Remarkably, both systems exhibit a qualitatively similar behaviour over the entire configuration space of boundary geometries. Recent formulations of dS/CFT enable one to interpret the field theory partition function as a function of the two squashing parameters as the Hartle-Hawking wave function in a minisuperspace model of anisotropic deformations of de Sitter space. The resulting probability distribution is normalisable and globally peaked at the round three sphere, with a low amplitude of boundary geometries with negative scalar curvature.

A SELECTION OF GIANT RADIO SOURCES FROM NVSS

ABSTRACT Results of the application of pattern-recognition techniques to the problem of identifying giant radio sources (GRSs) from the data in the NVSS catalog are presented, and issues affecting the process are explored. Decision-tree pattern-recognition software was applied to training-set source pairs developed from known NVSS large-angular-size radio galaxies. The full training set consisted of 51,195 source pairs, 48 of which were known GRSs for which each lobe was primarily represented by a single catalog component. The source pairs had a maximum separation of and a minimum component area of 1.87 square arcmin at the 1.4 mJy level. The importance of comparing the resulting probability distributions of the training and application sets for cases of unknown class ratio is demonstrated. The probability of correctly ranking a randomly selected (GRS, non-GRS) pair from the best of the tested classifiers was determined to be 97.8 ± 1.5%. The best classifiers were applied to the over 870,000 candidate pairs from the entire catalog. Images of higher-ranked sources were visually screened, and a table of over 1600 candidates, including morphological annotation, is presented. These systems include doubles and triples, wide-angle tail and narrow-angle tail, S- or Z-shaped systems, and core-jets and resolved cores. While some resolved-lobe systems are recovered with this technique, generally it is expected that such systems would require a different approach.

SU‐F‐T‐450: The Investigation of Radiotherapy Quality Assurance and Automatic Treatment Planning Based On the Kernel Density Estimation Method

Purpose:To develop a fast automatic algorithm based on the two dimensional kernel density estimation (2D KDE) to predict the dose‐volume histogram (DVH) which can be employed for the investigation of radiotherapy quality assurance and automatic treatment planning.Methods:We propose a machine learning method that uses previous treatment plans to predict the DVH. The key to the approach is the framing of DVH in a probabilistic setting. The training consists of estimating, from the patients in the training set, the joint probability distribution of the dose and the predictive features. The joint distribution provides an estimation of the conditional probability of the dose given the values of the predictive features. For the new patient, the prediction consists of estimating the distribution of the predictive features and marginalizing the conditional probability from the training over this. Integrating the resulting probability distribution for the dose yields an estimation of the DVH. The 2D KDE is implemented to predict the joint probability distribution of the training set and the distribution of the predictive features for the new patient. Two variables, including the signed minimal distance from each OAR (organs at risk) voxel to the target boundary and its opening angle with respect to the origin of voxel coordinate, are considered as the predictive features to represent the OAR‐target spatial relationship. The feasibility of our method has been demonstrated with the rectum, breast and head‐and‐neck cancer cases by comparing the predicted DVHs with the planned ones.Results:The consistent result has been found between these two DVHs for each cancer and the average of relative point‐wise differences is about 5% within the clinical acceptable extent.Conclusion:According to the result of this study, our method can be used to predict the clinical acceptable DVH and has ability to evaluate the quality and consistency of the treatment planning.

Experimental Test of Heisenberg's Measurement Uncertainty Relation Based on Statistical Distances.

Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenberg's original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [P. Busch et al., Phys. Rev. Lett. 111, 160405 (2013); P. Busch et al., Phys. Rev. A 89, 012129 (2014)]. Here we reformulate their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin system. The relation reveals that the worst-case inaccuracy is tightly bounded from below by the incompatibility of target observables, and is verified by the experiment employing joint measurement in which two compatible observables designed to approximate two incompatible observables on one qubit are measured simultaneously.