Hasil untuk "Probabilities. Mathematical statistics"

H. Touchette

Abstract The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein’s theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.

J. Rice

D. Horvitz, D. Thompson

M. Sugeno

G. Baecher

Alberto Leon-Garcia

K. Marvel, B. Cook, E. Weng et al.

<p>An understanding of Earth's past climate can help put current and future changes into historical context. Widely used tree ring-based drought atlases generally target the Palmer Drought Severity Index or other metrics of soil moisture and/or drought risk. These indices reflect contemporaneous meteorological conditions, and it is possible to extract information about temperature and precipitation given the existing reconstructions. Here, we present a fully Bayesian inverse method that infers a joint posterior for monthly mean temperature and precipitation given tree ring-based PDSI reconstructions from the North American Drought Atlas. The method is skillful at reconstructing early twentieth century conditions when compared to instrumental measurements from the CRU TS dataset. Moreover, the reconstructions can capture the complex temporal and multivariate covariance structure between monthly regional temperatures and precipitation. By reconstructing regional temperature and precipitation for the last millennium, we identify the driest and wettest years and decades in each region. Our results highlight the unique nature of the 1930s Dust Bowl drought in central Kansas and the late twentieth century pluvial in the North American southwest.</p>

Vanessa Anggriawan, Ferry Jaya Permana, Benny Yong

Disasters that occur in Indonesia lead to financial loss. One approach to mitigating the financial impact is through the utilization of natural disaster insurance. Although natural disasters occur with a relatively small frequency, the associated losses are substantial. Insurance companies need to carefully consider the characteristics of natural disaster data, as these events can lead to significant claims and potentially result in the bankruptcy of insurance companies. Insurance companies can reduce the risk of bankruptcy by transferring some risk to reinsurance companies. In this paper, the disaster reinsurance premium is determined by considering both the mortality and economic risks using the peaks over threshold (POT) model under the standard deviation principle. The Poisson, generalized Pareto, and lognormal distributions are used to determine the premium, with parameters estimated using the maximum likelihood method. A simulation analysis is conducted using synthetic data generated with RStudio software, which includes the frequency of floods per year over 20 years, as well as the number of deaths and the number of houses damaged in each flood event. The threshold is determined using the percentage method, where 10% of the data is considered extreme values. The POT model is applied to various retention cases. The simulation results show that the risk of the number of damaged houses has a greater impact on the premium amount that the insurance company must pay to the reinsurance company than the risk of the number of deaths. Additionally, cases with retention values below the threshold result in the highest reinsurance premiums, while cases with retention values above the threshold result in the lowest reinsurance premiums. This paper also shows that the reinsurance premium changes almost linearly with the increase in the extreme value percentage. This study is among the first to apply the peaks over threshold model in combination with multiple distributions for reinsurance premium estimation in the Indonesian context. The findings provide new insights into the sensitivity of reinsurance premiums to damage thresholds and retention levels, offering a practical tool for insurers in disaster-prone regions.

Noor Hafizah Khairul Anuar, Mohd Amri Md Yunus, Muhammad Ariff Baharudin et al.

Environmental factors like temperature, solar irradiance, and rain may influence the health and productivity of stingless bees. This paper aims to investigate the best approaches applied in meliponiculture to predict beehive health and products based on environmental variables and bee activity data. The data on temperature, humidity, rain, beehive weight, and bee activity traffic utilized in this project were monitored in real-time and saved on the Google Spreadsheet platform. The dataset extracted from the 6th of January 2024 to the 5th of February 2024, at a 15-minute time interval comprising a total of 2577 data points was analyzed using various deep learning approaches for best RMSE performance. A single-layer LSTM model with 50 units produced the best RMSE performance of 0.039, representing that the beehive weight was accurately predicted. This predictive capability can help farmers determine the optimum harvesting time based on weight forecasts, ensuring maximum yield and quality. Additionally, by providing early warnings of unwanted conditions such as swarming or potential attacks, this method significantly enhances the ability of beekeepers to take proactive measures to protect their colonies, safeguarding both bee populations and the livelihoods of farmers.

J I Gisbtarani, M N Huda, H Setiyaningsih et al.

The population of Australian sheep blowflies, Lucilia cuprina, in Australia is of concern to many researchers because it causes several problems. These problems occur in the sheep industry where there is a term "flystrike" in the industry. Flystrike is a fly attack on sheep that causes myiasis on the sheep's skin, affecting the quality and quantity of wool. In the worst cases, the sheep may die if not treated. This issue has attracted researcher to conduct a population control study of fly growth to suppress flystrike in the Australian sheep industry. In this paper, fly growth will be approached using a difference equation to better represent the industry’s situation. This equation will be analyzed using its approximate solution that is obtained through linearization of perturbation method, Cardano’s formula, and Galois solution’s method. By studying fly growth, Australian sheep farmers may find it easier to handle and prevent fly infestations using the solution.

Amanda Fitria, Said Iskandar Al Idrus, Asiah Asiah et al.

Psychologists and child education experts always remind the importance of knowing interests and talents from an early age to provide a stimulus to children from an early age, because the provision of the stimulus affects the future of children. This study aims to (1) To make calculations in mathematical models to calculate and analyze the interests of children's talents using applications. (2) To create a web-based information system that can facilitate teachers and parents in determining the interests of children's talents at the Al-Ikhlas Taqwa Plus Elementary School using the Certainty Factor method. The method used in this research is the research and development (R&D) method using the certainty factor. The population in this study were all students of SD Plus Al Ikhlas Taqwa Medan T.P 2021/2022 starting from grades 3-6. Sampling was done by purposive sampling technique. Data collection was carried out by interviews, material expert test questionnaires, and media experts, and the results of the children's talent interest questionnaire were processed using the Certainty Factor. The results of this study are the results of interest and talent analysis based on 7 intelligence criteria and also the highest summary results from several criteria with one of the tests yielding a percentage of 93.58% in the field of linguistics.

R. Ross, H. P. Hudson

Prefatory .—It is somewhat surprising that so little mathematical work should have been done on the subject of epidemics, and, indeed, on the distribution of diseases in general. Not only is the theme of immediate importance to humanity, but it is one which is fundamentally connected with numbers, while vast masses of statistics have long been awaiting proper examination. But, more than this, many and indeed the principal problems of epidemiology on which preventive measures largely depend, such as the rate of infection, the frequency of outbreaks, and the loss of immunity, can scarcely ever be resolved by any other methods than those of analysis. For example, infections diseases may perbaps be classified in three groups: (1) diseases such as leprosy, tuberculosis, and (?) cancer, which fluctuate comparatively little from month to month, though they may slowly increase or decrease in the course of years; (2) diseases such as measles, scarlatina, malaria, and dysentery, which, though constantly present in many countries, flare up in epidemics at frequent intervals; and (3) diseases such as plague or cholera, which disappear entirely after periods of acute epidemicity. To what are these differences due? Why, indeed, should epidemics occur at all, and why sbould not all infections diseases belong to the first group and always remain at an almost flat rate? Behind these phenomena there must be causes which are of profound importance to mankind and which probably can be ascertained only by those principles of careful computation which have yielded such brilliant results in astronomy, physics, and mechanics. Are the epidemics in the second class of diseases due (1) to a sudden and simultaneous increase of infectivity in the causative agents living in affected persons; or (2) to changes of environment which favour their dissemination from person to person; or (3) merely to the increase of suscep­tible material in a locality due to the gradual loss of acquired immunity in the population there; or to similar or other causes? And why should diseases of tbs third class disappear, as they undoubtedly do, and diseases of the first class remain so persistently?—all questions which immediately and obviously present themselves for examination.

Khozin Mu'tamar, Janson Naiborhu, Roberd Saragih

In this article, we present a control design on a SIRD model with treatment in infected individuals. The SIRD model with treatment is obtained from literature study and the parameter model is obtained  from covid-19 daily case in the Riau province using the Particle Swarm Optimization method. The control design is carried out based on the backstepping method combined with feedback linearization based on input and output (IOFL). The SIRD model which is a nonlinear system will be transformed into a normal form using IOFL. Each variable is then stabilized Lyapunov using virtual control which at the same time generates a new state variable. This stage will be carried out iteratively until the last state variable is stabilized using a real control function. This control function is then applied to the SIRD model using the inverse of IOFL transformation. The simulation results compared with the Pontryagin Minimum Principle (PMP) method show that by selecting the appropriate control parameters, backstepping obtains better control performance which is a smaller number of infected populations.

T. Bradley

This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals recover classical marginal probabilities. In general, these reduced densities will have rank higher than one, and their eigenvalues and eigenvectors will contain extra information that encodes subsystem interactions governed by statistics. We decode this information, and show it is akin to conditional probability, and then investigate the extent to which the eigenvectors capture "concepts" inherent in the original joint distribution. The theory is then illustrated with an experiment that exploits these ideas. Turning to a more theoretical application, we also discuss a preliminary framework for modeling entailment and concept hierarchy in natural language, namely, by representing expressions in the language as densities. Finally, initial inspiration for this thesis comes from formal concept analysis, which finds many striking parallels with the linear algebra. The parallels are not coincidental, and a common blueprint is found in category theory. We close with an exposition on free (co)completions and how the free-forgetful adjunctions in which they arise strongly suggest that in certain categorical contexts, the "fixed points" of a morphism with its adjoint encode interesting information.

D. Astuti, L. Anggraeni, F. Setyawan

Probability is one of the mathematical concepts that are abstract for students. Probability concepts used in daily life. As a basis of statistics, students used the probability concept. This topic is one of the compulsory subjects learned in the Mathematics Education Study Program of Universitas Ahmad Dahlan. However, there are still many students who have difficulties in understanding the concept of probability. This study aims to describe students’ misconceptions in solving Probability problems. The research used descriptive method. The subjects in this study were the fourth-semester students of Mathematics Education Study Program Class C of Universitas Ahmad Dahlan. The results of this study found that there are four misconceptions made by students in solving Probability problems. First, students’ error in interpreting questions. Second, students’ errors in the procedure of proving the theorems of probabilities. Third, students’ misconceptions in the application of Bayes’ rules while they are solving problems. Fourth, students’ error in calculating the possibility of an event.

R. Larsen

Xingyu Yan, Xiaolong Pu, Yingchun Zhou et al.

The unified weighing scheme for the local-linear smoother in analysing functional data can deal with data that are dense, sparse or of neither type. In this paper, we focus on the convergence rate of functional principal component analysis using this method. Almost sure asymptotic consistency and rates of convergence for the estimators of eigenvalues and eigenfunctions have been established. We also provide the convergence rate of the variance estimation of the measurement error. Based on the results, the number of observations within each curve can be of any rate relative to the sample size, which is consistent with the earlier conclusions about the asymptotic properties of the mean and covariance estimators.

J.T. Ferreira , A. Bekker , M. Arashi

This paper paves the way when the assumption of normality is challenged within the wireless communications systems arena. Innovative results pertaining to the distributions of quadratic forms and their associated eigenvalue density functions for the complex elliptical family are derived, which includes an original Rayleigh-type representation of channels. The presented analytical framework provides computationally convenient forms of these distributions. The results are applied to evaluate an important information-theoretic measure, namely channel capacity. Superior performance in terms of higher capacity of the wireless channel is obtained when considering the underlying complex matrix variate t distribution compared to the usual complex matrix variate normal assumption.

Daniel Adrian, Diann Reischman, Kirk Anderson et al.

Maps are a primary method of displaying statistical data that comes from a geographical frame. Maps are esthetically appealing and make it easier to identify geographic patterns in a dataset. However, few introductory statistical texts and courses explicitly present maps as a way to display data. In this article, we will present examples of different types of statistical maps and illustrate how these maps can be used in the instruction of an introductory statistics course.

Yosra Yousif, Faiz Ahmed Mohamed Elfaki, Meftah Hrairi

In the studies that involve competing risks, somehow, masking issues might arise. That is, the cause of failure for some subjects is only known as a subset of possible causes. In this study, a Bayesian analysis is developed to assess the effect of risks factor on the Cumulative Incidence Function (CIF) by adopting the proportional subdistribution hazard model. Simulation is conducted to evaluate the performance of the proposed model and it shows that the model is feasible for the possible applications.