Narrow Results

In this paper we describe RooFitUnfold, an extension of the RooFit statistical software package to treat unfolding problems, and which includes most of the unfolding methods that commonly used in particle physics. The package provides a common interface to these algorithms as well as common uniform methods to evaluate their performance in terms of bias, variance and coverage. In this paper we exploit this common interface of RooFitUnfold to compare the performance of unfolding with the Richardson–Lucy, Iterative Dynamically Stabilized, Tikhonov, Gaussian Process, bin-by-bin and inversion methods on several example problems.

I review the excursion set theory with particular attention toward applications to cold dark matter halo formation and growth, halo abundance, and halo clustering. After a brief introduction to notation and conventions, I begin by recounting the heuristic argument leading to the mass function of bound objects given by Press and Schechter. I then review the more formal derivation of the Press–Schechter halo mass function that makes use of excursion sets of the density field. The excursion set formalism is powerful and can be applied to numerous other problems. I review the excursion set formalism for describing both halo clustering and bias and the properties of void regions. As one of the most enduring legacies of the excursion set approach and one of its most common applications, I spend considerable time reviewing the excursion set theory of halo growth. This section of the review culminates with the description of two Monte Carlo methods for generating ensembles of halo mass accretion histories. In the last section, I emphasize that the standard excursion set approach is the result of several simplifying assumptions. Dropping these assumptions can lead to more faithful predictions and open excursion set theory to new applications. One such assumption is that the height of the barriers that define collapsed objects is a constant function of scale. I illustrate the implementation of the excursion set approach for barriers of arbitrary shape. One such application is the now well-known improvement of the excursion set mass function derived from the "moving" barrier for ellipsoidal collapse. I also emphasize that the statement that halo accretion histories are independent of halo environment in the excursion set approach is not a general prediction of the theory. It is a simplifying assumption. I review the method for constructing correlated random walks of the density field in the more general case. I construct a simple toy model to illustrate that excursion set theory (with a constant barrier height) makes a simple and general prediction for the relation between halo accretion histories and the large-scale environments of halos: regions of high density preferentially contain late-forming halos and conversely for regions of low density. I conclude with a brief discussion of the importance of this prediction relative to recent numerical studies of the environmental dependence of halo properties.

In this paper, we summarize some of the main observational challenges for the standard Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological model and describe how results recently presented in the parallel session “Large-scale Structure and Statistics” (DE3) at the “Fourteenth Marcel Grossman Meeting on General Relativity” are related to these challenges.

Interpreting research is an important skill to ensure one can maintain their practise with current evidence. The technicalities of statistics can be daunting and thus, this article aims to provide a clear overview of key statistical tests that a surgeon will encounter. It highlights the various study designs, summary statistics and comparative tests that are used in clinical research. Furthermore, it provides a guide to determine which statistical method is most appropriate for various study designs. Overall, it aims to act as an introductory text to supplement further reading into the more advanced statistical methodologies.

Level of Evidence: Level V

In this study, a new regression method called Kappa regression is introduced to model conditional probabilities. The regression function is based on Dombi’s Kappa function, which is well known in fuzzy theory. Here, we discuss how the Kappa function relates to the Logistic function as well as how it can be used to approximate the Logistic function. We introduce the so-called Generalized Kappa Differential Equation and show that both the Kappa and the Logistic functions can be derived from it. Kappa regression, like binary Logistic regression, models the conditional probability of the event that a dichotomous random variable takes a particular value at a given value of an explanatory variable. This new regression method may be viewed as an alternative to binary Logistic regression, but while in binary Logistic regression the explanatory variable is defined over the entire Euclidean space, in the Kappa regression model the predictor variable is defined over a bounded subset of the Euclidean space. We will also show that asymptotic Kappa regression is Logistic regression. The advantages of this novel method are demonstrated by means of an example, and afterwards some implications are discussed.

This paper presents methods for performing steganography and steganalysis using a statistical model of the cover medium. The methodology is general, and can be applied to virtually any type of media. It provides answers for some fundamental questions that have not been fully addressed by previous steganographic methods, such as how large a message can be hidden without risking detection by certain statistical methods, and how to achieve this maximum capacity. Current steganographic methods have been shown to be insecure against simple statistical attacks. Using the model-based methodology, an example steganography method is proposed for JPEG images that achieves a higher embedding efficiency and message capacity than previous methods while remaining secure against first order statistical attacks. A method is also described for defending against "blockiness" steganalysis attacks. Finally, a model-based steganalysis method is presented for estimating the length of messages hidden with Jsteg in JPEG images.

The ability to calculate precise likelihood ratios is fundamental to science, from Quantum Information Theory through to Quantum State Estimation. However, there is no assumption-free statistical methodology to achieve this. For instance, in the absence of data relating to covariate overlap, the widely used Bayes’ theorem either defaults to the marginal probability driven “naive Bayes’ classifier”, or requires the use of compensatory expectation-maximization techniques. This paper takes an information-theoretic approach in developing a new statistical formula for the calculation of likelihood ratios based on the principles of quantum entanglement, and demonstrates that Bayes’ theorem is a special case of a more general quantum mechanical expression.

The surface electromyography (SEMG) is a complicated biomedical signal, generated during voluntary or involuntary muscle activities and these muscle activities are always controlled by the nervous system. In this paper, the processing and analysis of SEMG signals at multiple muscle points for different operations were carried out. Myoelectric signals were detected using designed acquisition setup which consists of an instrumentation amplifier, filter circuit, an amplifier with gain adjustment. Further, Labview${}^{Ⓡ}$-based data programming code was used to record SEMG signals for independent activities. The whole system consists of bipolar noninvasive electrodes, signal acquisition protocols and signal conditioning at different levels. This work uses recorded SEMG signals generated by biceps and triceps muscles for four different arm activities. Feature extraction was done on the recorded signal for investigating the voluntary muscular contraction relationship for exercising statistic measured index method to evaluate distance between two independent groups by directly addressing the quality of signal in separability class for different arm movements. Thereafter repeated factorial analysis of variance technique was implemented to evaluate the effectiveness of processed signal. From these results, it demonstrates that the proposed method can be used as SEMG feature evaluation index.

In dermatology, the optical coherence tomography (OCT) is used to visualize the skin over few millimeters depth. These images are affected by speckle, which can alter their interpretation, but which also carries information that characterizes locally the visualized tissue. In this paper, we propose to differentiate the skin layers by modeling locally the speckle in OCT images. The performances of four probability density functions (Rayleigh, Lognormal, Nakagami and Generalized Gamma) to model the distribution of speckle in each skin layer are analyzed. From this study, we propose to classify the pixels of OCT images using the estimated parameters of the most appropriate distribution. Quantitative results with 30 images are compared to the manual delineations of five experts. The results confirm the potential of the method to generate useful data for robust segmentation.

Within the general setting of algebraic quantum field theory, a new approach to the analysis of the physical state space of a theory is presented; it covers theories with long range forces, such as quantum electrodynamics. Making use of the notion of charge class, which generalizes the concept of superselection sector, infrared problems are avoided. In fact, on this basis one can determine and classify in a systematic manner the proper charge content of a theory, the statistics of the corresponding states and their spectral properties. A key ingredient in this approach is the fact that in real experiments the arrow of time gives rise to a Lorentz invariant infrared cutoff of a purely geometric nature.

Let ${ℳ}\subset {ℝ}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${ℳ}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${ℳ}$ is greater than $\tau$. Moreover, let $\mu$ be a probability measure on ${ℳ}$ whose density on ${ℳ}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {ℳ}$, $j=1,2,\dots ,N$ be $N$ independent random samples from distribution $\mu$. Also, let ${\xi }_j$, $j=1,2,\dots ,N$ be independent random samples from a Gaussian random variable in ${ℝ}^n$ having covariance ${\sigma }^{2}I$, where $\sigma$ is less than a certain specified function of $d,V$ and $\tau$. We assume that we are given the data points $y_j=x_j+{\xi }_j,$$j=1,2,\dots ,N$, modeling random points of ${ℳ}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${{ℳ}}_o\subset {ℝ}^n$ whose Hausdorff distance to ${ℳ}$ is less than $\mathrm{\Delta }$ for $\mathrm{\Delta }>Cd{\sigma }^2/\tau$ and whose reach is greater than $c\tau /d^6$ with universal constants $C,c>0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on ${\mathrm{\Delta }}^{-1}$ and exponentially on $d$.

The ubiquitous presence of complexity in nature makes it necessary to seek new mathematical tools which can probe physical systems beyond linear or perturbative approximations. The random matrix theory is one such tool in which the statistical behavior of a system is modeled by an ensemble of its replicas. This paper is an attempt to review the basic aspects of the theory in a simplified language, aimed at students from diverse areas of physics.

The goal of this work is the further development of neoclassical analysis, which extends the scope and results of the classical mathematical analysis by applying fuzzy logic to conventional mathematical objects, such as functions, sequences, and series. This allows us to reflect and model vagueness and uncertainty of our knowledge, which results from imprecision of measurement and inaccuracy of computation. Basing on the theory of fuzzy limits, we develop the structure of statistical fuzzy convergence and study its properties. Relations between statistical fuzzy convergence and fuzzy convergence are considered in the First Subsequence Theorem and the First Reduction Theorem. Algebraic structures of statistical fuzzy limits are described in the Linearity Theorem. Topological structures of statistical fuzzy limits are described in the Limit Set Theorem and Limit Fuzzy Set theorems. Relations between statistical convergence, statistical fuzzy convergence, ergodic systems, fuzzy convergence and convergence of statistical characteristics, such as the mean (average), and standard deviation, are studied in Secs. 2 and 4. Introduced constructions and obtained results open new directions for further research that are considered in the Conclusion.

An exact solution is presented to a model that mimics the crowding effect in financial markets which arises when groups of agents share information. We show that the size distribution of groups of agents has a power law tail with an exponential cut-off. As the size of these groups determines the supply and demand balance, this implies heavy tails in the distribution of price variation. The moments of the distribution are calculated, as well as the kurtosis. We find that the kurtosis is large for all model parameter values and that the model is not self-organizing.

Functional Class Scoring (FCS) is a network-based approach previously demonstrated to be powerful in missing protein prediction (MPP). We update its performance evaluation using data derived from new proteomics technology (SWATH) and also checked for reproducibility using two independent datasets profiling kidney tissue proteome. We also evaluated the objectivity of the FCS p-value, and followed up on the value of MPP from predicted complexes. Our results suggest that (1) FCS $p$-values are non-objective, and are confounded strongly by complex size, (2) best recovery performance do not necessarily lie at standard $p$-value cutoffs, (3) while predicted complexes may be used for augmenting MPP, they are inferior to real complexes, and are further confounded by issues relating to network coverage and quality and (4) moderate sized complexes of size 5 to 10 still exhibit considerable instability, we find that FCS works best with big complexes. While FCS is a powerful approach, blind reliance on its non-objective $p$-value is ill-advised.

This chapter covers the statistical techniques using linear regression for quantitative outcomes, logistic regression for qualitative outcomes and Cox regression for time-to-event outcomes. Examples of result interpretation and presentation by means of tables for univariate and multivariate analyses were shown.

We examine the evolution of the spatial counts-in-cells distribution of galaxies and show that the form of the galaxy distribution function does not change significantly as galaxies merge and evolve. In particular, bound merging pairs follow a similar distribution to that of individual galaxies. From the adiabatic expansion of the universe we show how clustering, expansion and galaxy mergers affect the clustering parameter b. We also predict the evolution of b with respect to redshift.

In any legal chess position, we define an attacked-square entropy $S$ for either Black or White pieces in terms of the square occupation probability $p_i=m_i∕M$, where $m$ is the number of all possible movements to square $i$ (free or occupied by an opponent’s piece) and $M$ is the total mobility defined as the sum of all possible movements. Thus, each attacked square contributes to the entropy according to its received “firepower” concentration. A simpler nonsquare-dependant equiprobable entropy $S_e$ in terms of equal probabilities $p_e=1∕M$ always yields $S_e-S\ge 0$. On average, the difference $S_e-S$ is very large in the Opening phase and $S$ decreases faster for lower ranked players after move 25. A major cause of the reduction of $S$ during a game is material loss, which is an irreversible process. By game outcome, gaps in average $〈S〉$ among winners, draws and losers are larger for Amateur players than for Elite players, both in the Middlegame and Endgame. Statistically, Elite players exhibit narrower dispersions in $S$. Also, the entropy rates of the Elite level fluctuate much less than the entropies of other levels. Density of attacks in the four-square central zone is very high in the Opening, specially for Elite players.

Applying the powerful thin film brick-wall model to the general Kerr–Newman black hole, we find out that the entropy calculation result can also satisfy the area theorem. Moreover, the area theorem is not only satisfied for the global black hole, but also for every area cell on its horizon, that means, every cell on the horizon contributes its own part of entropy if we choose a same temperature-related radial cutoff ε'. This new thin film brick-wall model can be used to calculate dynamic black hole which has different temperatures on the horizon. It tells us that the horizon is exactly the statistical origin of a black hole entropy, the total entropy of a black hole is just the sum of all the contributions from every area cell. For a Kerr–Newman black hole, there is also an important difference between the thin film brick-wall model and the original one, that is, we do not need any angular cutoff in the thin film model, and this makes the physical meaning clearer.

We continue the study of Confined Vortex Surfaces (CVS) that we introduced in the previous paper. We classify the solutions of the CVS equation and find the analytical formula for the velocity field for arbitrary background strain eigenvalues in the stable region. The vortex surface cross-section has the form of four symmetric hyperbolic sheets with a simple equation $\left|y\right||x{|}^{\mu }=\text{const}$ in each quadrant of the tube cross-section ($xy$ plane).

We use the dilute gas approximation for the vorticity structures in a turbulent flow, assuming their size is much smaller than the mean distance between them. We vindicate this assumption by the scaling laws for the surface shrinking to zero in the extreme turbulent limit. We introduce the Gaussian random background strain for each vortex surface as an accumulation of a large number of small random contributions coming from other surfaces far away. We compute this self-consistent background strain, relating the variance of the strain to the energy dissipation rate.

We find a universal asymmetric distribution for energy dissipation. A new phenomenon is a probability distribution of the shape of the profile of the vortex tube in the $xy$ plane. This phenomenon naturally leads to the “multifractal” scaling of the moments of velocity difference $v(r_1)-v(r_2)$. More precisely, these moments have a nontrivial dependence of $n$, $log\mathrm{\Delta }r$, approximating power laws with effective index $\zeta (n,log\mathrm{\Delta }r)$. We derive some general formulas for the moments containing multidimensional integrals. The rough estimate of resulting moments shows the log–log derivative $\zeta (n,log\mathrm{\Delta }r)$ which is approximately linear in $n$ and slowly depends on $log\mathrm{\Delta }r$. However, the value of effective index is wrong, which leads us to conclude that some other solution of the CVS equations must be found. We argue that the approximate phenomenological relations for these moments suggested in a recent paper by Sreenivasan and Yakhot are consistent with the CVS theory. We reinterpret their renormalization parameter $\alpha \approx 0.95$ in the Bernoulli law $p=-\frac{1}{2}\alpha v^2$ as a probability to find no vortex surface at a random point in space.