搜索结果：Ratio

Ratio statistics and distributions are fundamental in various disciplines, including linear regression, metrology, nuclear physics, operations research, econometrics, biostatistics, genetics, and engineering. In this work, we introduce two novel computational approaches for evaluating ratio distributions using open data science tools and modern numerical quadratures. The first approach employs 1D double exponential quadrature of the Mellin convolution with/without barycentric interpolation, which is a very fast and efficient quadrature technique. The second approach utilizes 2D vectorized Broda-Khan numerical inversion of characteristic functions. It offers broader applicability by not requiring knowledge of PDFs or the independence of ratio constituents. The pilot numerical study, conducted in the context of Hake's ratio - a widely used measure of effect size and educational effectiveness in physics education - demonstrates the proposed methods' speed, accuracy, and reliability. The analytical and numerical explorations also provide more clarifying insight into the theoretical and empirical properties of Hake's ratio distribution. The proposed methods appear promising in a robust

The concept of the Quantum Ratio was born out of the efforts to find a simple but universal criterion if the center of mass (CM) of an isolated (microscopic or macroscopic) body behaves quantum mechanically or classically, and under which conditions. It is defined as the ratio between the quantum fluctuation range, which is the spatial extension of the pure-state CM wave function, and the linear size of the body (the space support of the internal, bound-state wave function). The two cases where the ratio is smaller than unity or much larger than unity, roughly correspond to the body's CM behaving classically or quantum mechanically, respectively. An important notion following from the introduction of quantum ratio is that the elementary particles (thus the electron and the photon) are quantum mechanical. This is so even when the environment-induced decoherence turns them into a mixed state. Decoherence (mixed state) and classical state should not be identified. This simple observation is further elaborated, by analyzing some atomic or molecular processes. It may have far-reaching implications on the way quantum mechanics works, e.g., in biological systems.

We propose an $O(\log n)$-approximation algorithm for the bipartiteness ratio of undirected graphs introduced by Trevisan (SIAM Journal on Computing, vol. 41, no. 6, 2012), where $n$ is the number of vertices. Our approach extends the cut-matching game framework for sparsest cut to the bipartiteness ratio, and requires only $\mathop{\mathrm{polylog}} n$ many single-commodity undirected maximum flow computations. Therefore, with the current fastest undirected max-flow algorithms, it runs in almost linear time. Along the way, we introduce the concept of well-linkedness for skew-symmetric graphs and prove a novel characterization of bipartiteness ratio in terms of well-linkedness in an auxiliary skew-symmetric graph, which may be of independent interest. As an application, we devise an $\tilde{O}(mn)$-time algorithm for the minimum uncut problem: given a graph whose optimal cut leaves an $η$ fraction of edges uncut, we find a cut that leaves only an $O(\log n \log(1/η)) \cdot η$ fraction of edges uncut, where $m$ is the number of edges. Finally, we propose a directed analogue of the bipartiteness ratio, and we give a polynomial-time algorithm that achieves an $O(\log n)$ approximation

The determination of an isotope ratio by secondary ion mass spectrometry (SIMS) traditionally involves averaging a number of ratios collected over the course of a measurement. We show that this method leads to an additive positive bias in the expectation value of the estimated ratio that is approximately equal to the true ratio divided by the counts of the denominator isotope of an individual ratio. This bias does not decrease as the number of ratios used in the average increases. By summing all counts in the numerator isotope, then dividing by the sum of counts in the denominator isotope, the estimated ratio is less biased: the bias is approximately equal to the ratio divided by the summed counts of the denominator isotope over the entire measurement. We propose a third ratio estimator (Beale's estimator) that can be used when the bias from the summed counts is unacceptably large for the hypothesis being tested. We derive expressions for the variance of these ratio estimators as well as the conditions under which they are normally distributed. Finally, we investigate a SIMS dataset showing the effects of ratio bias, and discuss proper ratio estimation for SIMS analysis.

This paper studies a boundary between local self-application and global self-certification. Irrational quantities are treated operationally, as procedures whose approximations are refined by effective update rules. The golden ratio $Φ$ is used as a model of stable local recurrence: the reciprocal update $R(x)=1+1/x$ has a unique positive fixed point and admits finite witnessed approximations. By contrast, global reflection asks a system to certify its own correctness uniformly. The proof-theoretic claim is therefore contrastive: primitive-recursive proof checking and local soundness preserve correctness through bounded checks and bounded witnesses, but they do not yield internal global reflection. No complexity advantage, decision procedure, or new reflection principle is claimed.

The ratio of Bayesian evidences is a popular tool in cosmology to compare different models. There are however several issues with this method: Bayes' ratio depends on the prior even in the limit of non-informative priors, and Jeffrey's scale, used to assess the test, is arbitrary. Moreover, the standard use of Bayes' ratio is often criticized for being unable to reject models. In this paper, we address these shortcoming by promoting evidences and evidence ratios to frequentist statistics and deriving their sampling distributions. By comparing the evidence ratios to their sampling distributions, poor fitting models can now be rejected. Our method additionally does not depend on the prior in the limit of very weak priors, thereby safeguarding the experimenter against premature rejection of a theory with a uninformative prior, and replaces the arbitrary Jeffrey's scale by probability thresholds for rejection. We provide analytical solutions for some simplified cases (Gaussian data, linear parameters, and nested models), and we apply the method to cosmological supernovae Ia data. We dub our method the FB method, for Frequentist-Bayesian.

Proximal Policy Optimization (PPO) has become the predominant algorithm for on-policy reinforcement learning due to its scalability and empirical robustness across domains. However, there is a significant disconnect between the underlying foundations of trust region methods and the heuristic clipped objective used in PPO. In this paper, we bridge this gap by introducing the Bounded Ratio Reinforcement Learning (BRRL) framework. We formulate a novel regularized and constrained policy optimization problem and derive its analytical optimal solution. We prove that this solution ensures monotonic performance improvement. To handle parameterized policy classes, we develop a policy optimization algorithm called Bounded Policy Optimization (BPO) that minimizes an advantage-weighted divergence between the policy and the analytic optimal solution from BRRL. We further establish a lower bound on the expected performance of the resulting policy in terms of the BPO loss function. Notably, our framework also provides a new theoretical lens to interpret the success of the PPO loss, and connects trust region policy optimization and the Cross-Entropy Method (CEM). We additionally extend BPO to Grou

The risk ratio is a popular tool for summarizing the relationship between a binary covariate and outcome, even when outcomes may be dependent. Investigations of infectious disease outcomes in cohort studies of individuals embedded within clusters -- households, villages, or small groups -- often report risk ratios. Epidemiologists have warned that risk ratios may be misleading when outcomes are contagious, but the nature and severity of this error is not well understood. In this study, we assess the epidemiologic meaning of the risk ratio when outcomes are contagious. We first give a structural definition of infectious disease transmission within clusters, based on the canonical susceptible-infective epidemic model. From this standard characterization, we define the individual-level ratio of instantaneous risks (hazard ratio) as the inferential target, and evaluate the properties of the risk ratio as an estimate of this quantity. We exhibit analytically and by simulation the circumstances under which the risk ratio implies an effect whose direction is opposite that of the true individual-level hazard ratio. In particular, the risk ratio can be greater than one even when the covaria

The photometric mass ratios of eclipsing binaries are usually estimated by light-curve modeling with an iterative method. We propose a new method for estimating the photometric mass ratio of an overcontact binary using the derivatives of a light curve, which provides a reasonable uncertainty value. The method mainly requires only the time interval value between two local extrema found in the third derivative of a light curve, with no need of an iterative procedure. We applied the method to a sample of real overcontact binary data and compared the estimated mass ratios with their spectroscopically determined values. The comparison showed that our estimated mass ratios for $\sim $67% of the samples agreed with their spectroscopic mass ratios within the estimated uncertainties, and the errors for 95% of them are within $\pm 0.1$. Our method should be useful for estimating mass ratios for numerous overcontact eclipsing binaries found with existing and future surveys, as well as for the light-curve analysis of each system.

Sunspot engravings made in the Maunder minimum are used to evaluate the fine structure of sunspots. Based on 78 images of the full solar disk and 77 images of individual sunspots, we have evaluated the ratio of penumbral-to-umbral area (P/U) to be 4.2 $\pm$ 2.5 and 3.8 $\pm$ 2.3, respectively. These results are consistent with previous estimates before, during, and after the Maunder minimum, as well as with the P/U ratio observed in the largest sunspot groups in solar cycle 24. This suggests that the near-surface convection mode has most likely remained unchanged since the early seventeenth century. We also found that schematic sunspot drawings tend to underestimate the P/U ratio.

New measurements of compressional and shear wave velocities to hydrostatic pressures of 1 GPa are summarized for 678 rocks. Emphasis was placed on obtaining high‐accuracy velocity measurements, which are shown to be critical in calculating Poisson's ratios from velocities. The rocks have been divided into 29 major groups for which velocities, velocity ratios, and Poisson's ratios are presented at several pressures. Observed Poisson's ratios for the monomineralic rocks compare favorably with theoretical Poisson's ratios calculated from single‐crystal elastic constants. Plagioclase feldspar composition is important in understanding rock Poisson's ratios, since Poisson's ratio of albite increases from 0.28 to a predicted value of 0.31 for anorthite. Fe substitution for Mg in pyroxene and olivine also increases Poisson's ratio. Plotting rock compressional wave velocities versus Poisson's ratios reveals a triangular distribution bounded by quartzite with low compressional wave velocity and low Poisson's ratio, dunite with high compressional wave velocity and intermediate Poisson's ratio, and serpentinite with low compressional wave velocity and high Poisson's ratio. For common plutonic igneous rocks, there is a clear trend relating Poisson's ratio to composition, in which Poisson's ratio for granitic rocks increases from 0.24 to 0.29 as composition changes to gabbro and then decreases with decreasing plagioclase and increasing olivine contents to 0.25 in dunite. Changes in Poisson's ratio with progressive metamorphism of mafic and pelitic rocks correlate reasonably well with mineral reactions. There is no simple correlation between Poisson's ratio and felsic and mafic rock compositions; however, a linear correlation of increasing Poisson's ratio with decreasing SiO 2 content is observed for rocks with 55 to 75 wt % SiO 2 . Average Poisson's ratios for continental and oceanic crusts are estimated to be 0.265 and 0.30, respectively.

This study addresses the multi-item multi-period order allocation problem under all-unit quantity discounts (AUQD) and blending ratios. A manufacturer makes a single product that requires mixing/assembling multiple ingredients/components with pre-determined blending ratios. We consider a single supplier offering quantity-based discounts which introduces non-linearities to the problem. The objective is to minimize procurement cost which includes purchasing, inventory, and ordering costs. We develop a solution procedure that systematically generates a finite dominating set (FDS) of order quantities guaranteed to include an optimal solution to the problem. A Mixed Integer Linear Programming (MILP) model based on the FDS. Our procedure guarantees optimality and eliminates the need for nonlinear discount modeling. Numerical experiments demonstrate that the proposed MILP achieves optimal solutions with significantly reduced computational effort, up to 99% faster for large-scale instances compared to conventional formulations. Sensitivity analyses reveal that the model dynamically adapts to changes in holding costs, shifting between bulk-purchasing and just-in-time strategies, and identif

This work studies the influence of an external magnetic field on hadron yields and fluctuations in a hadron resonance gas by performing calculations within an updated version of the open-source Thermal-FIST package. The presence of the magnetic field has a sizable influence on certain hadron yield ratios. Most notably, it leads to enhanced $p/π$ and suppressed $n/p$ ratios, which may serve as a magnetometer in heavy-ion collisions. By attributing the centrality dependence of the $p/π$ ratio in Pb-Pb collisions at 5.02~TeV measured by the ALICE Collaboration entirely to the magnetic field, its maximal strength at freeze-out is estimated to be $eB \simeq 0.12$~GeV$^2 \simeq 6.3 m_π^2$ in peripheral collisions. The magnetic field also enhances various conserved charge susceptibilities, which is qualitatively consistent with recent lattice QCD data and is driven in the HRG model by the increase of hadron densities. However, the variances of hadrons do not show any enhancement when normalized by the means, therefore, measurements of second-order fluctuations in heavy-ion collisions appear to offer limited additional information about the magnetic field relative to mean multiplicities.

We discuss - in what is intended to be a pedagogical fashion - generalized "mean-to-risk" ratios for portfolio optimization. The Sharpe ratio is only one example of such generalized "mean-to-risk" ratios. Another example is what we term the Fano ratio (which, unlike the Sharpe ratio, is independent of the time horizon). Thus, for long-only portfolios optimizing the Fano ratio generally results in a more diversified and less skewed portfolio (compared with optimizing the Sharpe ratio). We give an explicit algorithm for such optimization. We also discuss (Fano-ratio-inspired) long-short strategies that outperform those based on optimizing the Sharpe ratio in our backtests.

Omega ratio, defined as the probability-weighted ratio of gains over losses at a given level of expected return, has been advocated as a better performance indicator compared to Sharpe and Sortino ratio as it depends on the full return distribution and hence encapsulates all information about risk and return. We compute Omega ratio for the normal distribution and show that under some distribution symmetry assumptions, the Omega ratio is oversold as it does not provide any additional information compared to Sharpe ratio. Indeed, for returns that have elliptic distributions, we prove that the optimal portfolio according to Omega ratio is the same as the optimal portfolio according to Sharpe ratio. As elliptic distributions are a weak form of symmetric distributions that generalized Gaussian distributions and encompass many fat tail distributions, this reduces tremendously the potential interest for the Omega ratio.

Sharpe ratio (sometimes also referred to as information ratio) is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the (excess) net return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is likely to be error prone because of statistical estimation errors. In this paper, we provide various tests to measure the quality of the Sharpe ratios. By quality, we are aiming at measuring whether a manager was indeed lucky of skillful. The test assesses this through the statistical significance of the Sharpe ratio. We not only look at the traditional Sharpe ratio but also compute a modified Sharpe insensitive to used Capital. We provide various statistical tests that can be used to precisely quantify the fact that the Sharpe is statistically significant. We illustrate in particular the number of trades for a given Sharpe level that provides statistical significance as well as the impact of auto-correlation by providing reference tables that

Several sources of information suggest that human beings evolved on a diet with a ratio of omega-6 to omega-3 essential fatty acids (EFA) of approximately 1 whereas in Western diets the ratio is 15/1-16.7/1. Western diets are deficient in omega-3 fatty acids, and have excessive amounts of omega-6 fatty acids compared with the diet on which human beings evolved and their genetic patterns were established. Excessive amounts of omega-6 polyunsaturated fatty acids (PUFA) and a very high omega-6/omega-3 ratio, as is found in today's Western diets, promote the pathogenesis of many diseases, including cardiovascular disease, cancer, and inflammatory and autoimmune diseases, whereas increased levels of omega-3 PUFA (a lower omega-6/omega-3 ratio), exert suppressive effects. In the secondary prevention of cardiovascular disease, a ratio of 4/1 was associated with a 70% decrease in total mortality. A ratio of 2.5/1 reduced rectal cell proliferation in patients with colorectal cancer, whereas a ratio of 4/1 with the same amount of omega-3 PUFA had no effect. The lower omega-6/omega-3 ratio in women with breast cancer was associated with decreased risk. A ratio of 2-3/1 suppressed inflammation in patients with rheumatoid arthritis, and a ratio of 5/1 had a beneficial effect on patients with asthma, whereas a ratio of 10/1 had adverse consequences. These studies indicate that the optimal ratio may vary with the disease under consideration. This is consistent with the fact that chronic diseases are multigenic and multifactorial. Therefore, it is quite possible that the therapeutic dose of omega-3 fatty acids will depend on the degree of severity of disease resulting from the genetic predisposition. A lower ratio of omega-6/omega-3 fatty acids is more desirable in reducing the risk of many of the chronic diseases of high prevalence in Western societies, as well as in the developing countries.

We study boundaries arising from limits of ratios of transition probabilities for random walks on relatively hyperbolic groups. We extend, as well as determine significant limitations of, a strategy employed by Woess for computing ratio-limit boundaries for the class of hyperbolic groups. On the one hand we employ results of the second and third authors to adapt this strategy to spectrally non-degenerate random walks, and show that the closure of minimal points in $R$-Martin boundary is the unique smallest invariant subspace in ratio-limit boundary. On the other hand we show that the general strategy can fail when the random walk is spectrally degenerate and adapted on a free product. Using our results, we are able to extend a theorem of the first author beyond the hyperbolic case and establish the existence of a co-universal quotient for Toeplitz C*-algebras arising from random walks which are spectrally non-degenerate on relatively hyperbolic groups. Finally, we exhibit an example of a relatively hyperbolic group carrying two random walks such that the ratio limit boundaries are not equivariantly homeomorphic and no two equivariant quotients of their respective Toeplitz C*-algebr