Cumulative Entropy

مقاله حاضر به بررسی تأثیر الگوریتم‌های یادگیری ماشین در مدیریت ریسک و بازدهی پورتفلیوهای سرمایه‌گذاری پرداخته است. در دنیای مدرن سرمایه‌گذاری، اهمیت مدیریت ریسک و حداکثر کردن بازدهی پورتفلیوها به منظور دستیابی به اهداف سرمایه‌گذاری به طرز چشم‌گیری افزایش یافته است. در این زمینه، الگوریتم‌های یادگیری ماشین به عنوان ابزارهای کلیدی برای بهبود تصمیم‌گیری‌های سرمایه‌گذاری مورد توجه قرار گرفته‌اند.در ابتدا، معرفی و تبیین نقش مهم پورتفلیوهای سرمایه‌گذاری به عنوان ابزار اصلی مدیریت سرمایه و تحلیل ریسک و بازدهی آنها صورت گرفته است. سپس، نقش الگوریتم‌های یادگیری ماشین در بهبود مدیریت ریسک از طریق پیش‌بینی ریسک بازار و تطبیق سریع با شرایط متغیر مورد بررسی قرار گرفته است.در بخش دیگری از مقاله، نحوه استفاده از الگوریتم‌های یادگیری ماشین در بهبود بازدهی پورتفلیوها مورد بحث و بررسی قرار گرفته است. نتایج نشان می‌دهد که استفاده از الگوریتم‌های یادگیری ماشین در مدیریت ریسک و بهبود بازدهی پورتفلیوهای سرمایه‌گذاری، منجر به افزایش دقت تصمیم‌گیری و کاهش تأثیر عوامل غیرقابل پیش‌بینی در بازار می‌شود. این رویکرد نوین باعث می‌شود که سرمایه‌گذاران توانایی بهتری در مدیریت ریسک و بهره‌وری در بازدهی پورتفلیوهای خود پیدا کنند، که به عنوان یک ابزار اساسی در موفقیت در عرصه سرمایه‌گذاری محسوب می‌شود.

In the present communication we introduce a dynamic measure of inaccuracy between two past lifetime distributions over the interval (0, t). Based on proportional reversed hazard rate model (PRHRM), a characterization problem for this dynamic inaccuracy measure has been studied. An upper bound to the dynamic measure of inaccuracy H * (f, g; t) has also been derived. Keywords Kerridge inaccuracy • Past entropy • Measure of discrimination • Reversed hazard rate function • Proportional reversed hazards model 1 Introduction Let X and Y be two non-negative random variables representing time to failure of two systems with p.d.f., respectively, f (x) and g(x). Let F(x) = P(X ≤ x) and G(y) = P(Y ≤ y) be failure distributions, μ F (x) = f (x) F(x) and μ G (x) = g(x) G(x) be reversed hazard rates, andF(x) = 1 − F(x) andḠ(x) = 1 − G(x) be survival functions of X and Y , respectively. Shannon's (1948) measure of uncertainty associated with the random variable X , Kullback's (1959) measure of discrimination of X about Y , and Kerridge (1961) measure of inaccuracy are given by

This article introduces a new method to estimate the least absolutes linear regression model's parameters, which considers optimization problems based on the weighted aggregation operators of ordered least absolute deviations. In the optimization problem, weighted aggregation of orderd fitted least absolute deviations provides data analysis to identify the outliers while considering different fitting functions simultaneously in the modeling problem. Accordingly, this approach is not affected by outlier observations and in any problem proportional to the number of potential outliers selects the best model estimator with the optimal breakdown point among a set of other candidate estimators. The performance and the goodness-of-fit of the proposed approach are investigated, analyzed and compared in modeling analytical dataset and a real value dataset in hydrology engineering at the presence of outliers. Based on the results of the sensitivity analysis, the properties of unbiasedness and efficiency of the estimators are obtained.

In Reliability Theory, uncertainty is measured by the Shannon entropy. Recently, in order to analyze the variability of such measure, varentropy has been introduced and studied. In this paper we define a new concept of varentropy for past lifetime distributions as the variability of uncertainty in past lifetime. We analyze the relation of this varentropy with the corresponding past entropy and other concepts of interest in reliability and we study the effect of linear and monotonic transformations on it. Finally, we evaluate past varentropy for random variables following a proportional reversed hazard rate model.

In this paper, we consider the information content of maximum ranked set sampling procedure with unequal samples (MRSSU) in terms of Tsallis entropy which is a nonadditive generalization of Shannon entropy. We obtain several results of Tsallis entropy including bounds, monotonic properties, stochastic orders, and sharp bounds under some assumptions. We also compare the uncertainty and information content of MRSSU with its counterpart in the simple random sampling (SRS) data. Finally, we develop some characterization results in terms of cumulative Tsallis entropy and residual Tsallis entropy of MRSSU and SRS data.

Recently, a new concept of entropy called generalized cumulative entropy of order n was introduced and studied in the literature. It is related to the lower record values of a sequence of independent and identically distributed random variables and with the concept of reversed relevation transform. In this paper, we provide some further results for the generalized cumulative entropy such as stochastic orders, bounds and characterization results. Moreover, some characterization results are derived for the dynamic generalized cumulative entropy. Finally, it is shown that the empirical generalized cumulative entropy of an exponential distribution converges to normal distribution.

Objective: One of the most important issues for all investors, including individual and institutional investors in the stock market, is finding the optimal portfolio. Identifying the optimal portfolio in the stock market can be considered a two-objective optimization problem. This problem maximizes and minimizes the return and risk of the portfolio, respectively. Like other multi-objective optimization problems, the portfolio optimization problem can be solved by multi-objective evolutionary algorithms (MOEAs). Accordingly, the Non-dominated Sorting Genetic Algorithm () and Strength Pareto Evolutionary Algorithm (2), as the two advanced algorithms of multiobjective evolution algorithms, can be used to solve this problem. These algorithms identify the optimal solution by ranking and archiving solutions located on the Pareto frontier. The purpose of this research is to compare the performance of and 2 in mean-variance and mean-semi-variance approaches to identify the optimal stock portfolio. Methods: This research investigated 241 stocks enlisted on the Tehran Stock Exchange (TSE). It was conducted within 174 months from September 2006 to March 2019. The researchers first identified the optimal portfolio using NSGAII and SPEA2 algorithms through two approaches including mean-variance and mean-semi-variance. Then, by conducting a statistical hypothesis test on the average Sharp ratio of extracted portfolios, the performance of NSGAII and SPEA2 algorithms were compared. To confirm the research findings, a robustness test was done by comparing the performance of the SPEA2 algorithm with the traditional Markowitz model. Also, to ensure the stability of research findings, the performance of two algorithms in the mean-variance and meansemi variance approaches were compared with quarterly data ending March 2022. Results: According to the obtained results, the SPEA2 algorithm has better performance than the NSGAII algorithm in both approaches. Backtesting the real data for the quarter ending inMarch 2022 confirmed the findings of the present study. Also by doing robustness tests, the researchers found the SPEA2 algorithm as the superior algorithm in this research with better performance than Markowitz's basic model. Conclusion: The results indicated that the 2 algorithm has better performance in selecting the optimal portfolio than the NSGAII algorithm in both the mean-variance and mean-semi variance approaches. Regardless of how the stock returns are distributed, this study recommends that individual and institutional investors use the SPEA2 algorithm to determine the optimal portfolio arrangement.

Biradar and Santosha (2014) proposed maximum ranked set sampling procedure with unequal samples (MRSSU) to estimate the mean of the exponential distribution. In this paper, we consider information measures of MRSSU in terms of Shannon entropy, Rényi entropy and Kullback-Leibler (KL) information. We also compare the uncertainty and information content of MRSSU with simple random sampling and ranked set sampling data. Finally, we develop some characterization results in terms of cumulative entropy and failure entropy of MRSSU.

The cumulative entropy is an information measure which is alternative to the differential entropy and is connected with a notion of reliability theory. Indeed, the cumulative entropy of a random lifetime X can be expressed as the expectation of its mean inactivity time evaluated at X. After a brief rieview of its main properties, in this paper we relate the cumulative entropy to the cumulative inaccuracy, and provide some inequalities based on suitable stochastic orderings. We also show a characterization property of the dynamic version of the cumulative entropy. In conclusion, a stochastic comparison between the empirical cumulative entropy and the empirical cumulative inaccuracy is investigated.

The cumulative Kullback-Leibler information has been proposed recently as a suitable extension of Kullback-Leibler information to the cumulative distribution function. In this paper, we obtain various results on such a measure, with reference to its relation with other information measures and notions of reliability theory. We also provide some lower and upper bounds. A dynamic version of the cumulative Kullback-Leibler information is then proposed for past lifetimes. Furthermore, we investigate its monotonicity property, which is related to some new concepts of relative aging. Moreover, we propose an application to the failure of nanocomponents. Finally, in order to provide an application in image analysis, we introduce the empirical cumulative Kullback-Leibler information and prove an asymptotic result.

Biradar and Santosha (2014) proposed maximum ranked set sampling procedure with unequal samples (MRSSU) to estimate the mean of the exponential distribution. In this paper, we consider information measures of MRSSU in terms of Shannon entropy, Rényi entropy and Kullback-Leibler (KL) information. We also compare the uncertainty and information content of MRSSU with simple random sampling and ranked set sampling data. Finally, we develop some characterization results in terms of cumulative entropy and failure entropy of MRSSU.

The concept of mean inactivity time plays a crucial role in reliability, risk theory and life testing. In this regard, we introduce a weighted mean inactivity time function by considering a non-negative weight function. Based on this function, we provide expressions for the variance of transformed random variable and the weighted generalized cumulative entropy. The latter concept is an important measure of uncertainty which is shift-dependent and is of interest in certain applied contexts, such as reliability or mathematical neurobiology. Moreover, based on the comparison of mean inactivity times of a certain function of two lifetime random variables, we introduce and study a new stochastic order in terms of the weighted mean inactivity time function. Several characterizations and preservation properties of the new order under shock models, random maxima and renewal theory are discussed.

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

Objective: The main object of this study is to use a new measure of risk called Cumulative entropy in the Markowitz portfolio optimization model and solve this model using Particle swarm optimization (PSO) to optimize the portfolio of Petrochemical companies by Applying the data consist of monthly returns of the Fifteen petrochemical companies in Tehran Stock Exchange from 2013 to 2019. The Markowitz model uses the variance as a risk measure by default. in this study, a new measure of risk called Cumulative Entropy is introduced. This measure can be used in many issues without considering the limitations of variance (standard deviation). Methods: The Markowitz model is one of the most important models for solving portfolio optimization problems, but this model has many disadvantages. The Markowitz optimization problem can be solved by simple mathematical programming models when the number of assets to be invested and the market constraints are small, but when the real-world conditions and constraints are taken into account, the problem becomes complex and difficult. One of the methods that have solved human ambiguities in recent years in solving many optimization problems and has been successful in responding to complex problems is the so-called intelligent methods and algorithms. Intelligent methods that were introduced to eliminate the shortcomings of classical (traditional) optimization methods with a comprehensive and random search, largely guarantees the possibility of achieving better results. Due to the mentioned problems in this research, a new criterion by the name Cumulative entropy is introduced which can be used as an alternative to variance in the Markowitz meanvariance optimization model as a risk criterion. Also, due to the mentioned problems for the Markowitz model, in this research, the meta-innovative particle cumulative motion (PSO) algorithm will be used to optimize the stock portfolio for Petrochemical companies stocks. Results: As can be seen in the PSO algorithm, the average value of the stock return function is less than the average value of the stock return function in the Markowitz model, while the