Fuzzy logic for the management of uncertainty pdf

They were there just to for representing and manipulating fuzzy terms was called put him under constant scrutiny but Zadeh never said die, he fuzzy logic, and Zadeh became the Master of fuzzy logic.

However, The well-known American Professor Bart Kosko highlighted over time began to gain enough supporters, which led to these the differences between Eastern and Western philosophies theories, being extended again and again, settling firmly regarding the concept of truth, summing up in opposition among the most innovative scientists, and especially among against Aristotle Buddha.

In fact, Kosko said that Western the best professionals, more than anywhere else, initially in philosophy, Aristotle's successor, has accepted uncritically Japan and then South Korea, China and India. Europe and the the bivalent as the system that is useful, but overly simplifying States have been incorporated into this new math, but more complex reality.

Put simply: what has won in simplicity is lost slowly. Always have accepted the strict unity of As a matter of picturesque, if you will, but true, we can tell opposites, of what they call as we know the yin and yang. Zadeh, in his time met with executives from II. Of course, it will be considered a very clear model of According to this theory, thus fuzzy logic, we have a transfer intelligence and vision. Zadeh's intention was to create a formalism which is [0, 1].

The study led to the methodology of fuzzy logic and its applications, as of various thinkers from many different disciplines, who, like known today.

The paradox of all sets that do not contain themselves, that is Interestingly, in fuzzy logic, statements are not absolutely true very famous and was proposed by Bertrand Russell or absolutely false.

And the variables or categories by Werner Heisenberg. This is attributed a value to vagueness at an appropriate level of detail and suggests that each of these features diffuse, and that value is accentuated or 69 www. An engineer might define the depend on these functions. The Fuzzy Rules define a set of with gradual effects and variation. The feature of working overlapping patches that relate a full range of inputs to a full with border regions is not fully defined itself in this area of range of outputs.

In that sense, the fuzzy system approximates mathematics. Bart Kosko In , with the basic theory of Zadeh fuzzy controllers and proved this uniform convergence theorem by showing that other researchers began applying fuzzy logic to various enough small fuzzy patches can sufficiently cover the graph of mechanical and industrial processes, improving existing any function or input or output relation.

The theorem also hitherto. Several research groups were established at shows that we can pick in advance the maximum error of the universities fuzzy Japanese. That is, Professors Terano and approximation and be sure there exist a finite number of fuzzy Shibata in Tokyo, with Professors Tanaka and Asai in Osaka, rules that achieve it.

And recent advances in Neural Networks, made vast contributions to both the development of fuzzy in terms of programs that learn from experience , and logic theory and its applications. Professor Ebrahim Mandani Genetic Algorithms as programs evolve over time are in the United Kingdom, designed the first fuzzy controller for certainly a fitting complement to fuzzy logic. This Another key reason for increased research in this field would actually emerged in Wanted relations between the two techniques, Further, Hitachi used a fuzzy controller for the Sendai train thereby obtaining Neuro-Fuzzy Systems, which use learning control, which used an innovative system created by man.

Then, as we say, appeared genetic algorithms, efficiently. It was right there in values in response to inputs from surrounding neurons and the country Nippon and South Korea, where more height synapses. That is all about Neural Networks NN.

The neural Fuzzy Logic has been, creating close partnerships between network acts like a computer, because it maps inputs to government, universities and industries. Parallel to the study outputs. The neurons and synapses may be silicon of the applications of fuzzy logic, Professors Takagi and components, or equations in software, that simulate their Sugeno developed the first approach to construct Fuzzy behavior. Supervised learning, reached through supervised Rules, from training data.

The user provides the first set of rules, which the The applications of fuzzy logic in everyday life since then neural net refines by running through hundreds of thousands grow rapidly. In fact, already part of it. For example, some of inputs, slightly varying the fuzzy sets each time to see how brands of washing machine using fuzzy logic are Electrolux, well the system performs.

The technique also is performance and to ignore others. Manufacturers do not want the members of which resemble one another. There may be no to give much publicity to the fuzziness implicit in these given right or wrong response or way to organize the data.

It is developments for an obvious reason. To say that their cars' in advance. Fuzzy Modeling is many times used to modify the brake controlled by fuzzy logic does not belong to the class of knowledge of an expert into a mathematical model. The messages that can sell more cars. Also as a tool that can assist B.

In many fields of 70 www. However, to study these phenomena, there is a need to construct a suitable mathematical model, a process that To plan is to forecast and to forecast is to dig into the future. These are linked with multifaceted activities thereby inviting Fuzzy modeling is a many more direct and natural approach fuzzy logic to making it superb. The notion of some people is for transforming the linguistic description into such model.

A that, policy is only applicable to vast companies or firms but it fuzzy model represents the real system in a form that does not hold true. It is in fact the concern of individuals as corresponds closely to the way humans perceive it. Thus, the well. It could be a nuclear family system where a policy going model is easily understandable, and each parameter has a to church only at the end of each month becomes a formalized readily perceivable meaning.

The model can easily be altered procedure. That is their policy and binding. So it is imperative to incorporate new phenomena, and if its behavior is different in the public sector. Fuzzy logic provides an alternative way to represent linguistic and subjective attributes of the real world in computing in so Furthermore, the mathematical procedures used in fuzzy far as variables are concerned.

It is able to be applied to modeling have been tried and tested many times, and their control systems and other applications in order to improve the techniques are relatively well documented. What is fuzzy logic is not in itself, which logic and leads to faster and simpler program development of has a precise mathematical definition, but the world over system controllers. Fuzzy logic provides decision support system tool for managers in their various places of work or offices.

The general notion about decision making is that, it is coupled III. Thus; fuzzy decision typology. In all levels of organizations, concept of fuzziness as a result of which the truth or falsity are firms, enterprises including civil service institutions, policy is only extreme cases.

By fuzziness understand the fact that a applicable. This is due to the fact that, it is the fundamental proposition may be partially true and partially false part of any mid-term or long term planning. A person is not just high or low, but partially may participate in both features, so that only above and below Fuzzy Policy making is a multidisciplinary and subject area, certain heights necessarily called it higher or lower bound, not a discipline; it borrows from other natural and basic and while in the intermediate zone of both heights exist as a social science disciplines in order to develop study in the area.

It seems intuitively The contributory disciplines include mathematics, social clear that the concept of fuzziness is rooted in most of our work, statistics, economics, political science, management, ways of thinking and speaking. Another separate issue is the valuation that each individual Unlike other decision methods, the fuzzy can adaptively find granted such a fuzzy character the glass half full or half a suitable policy for a country or the place where policy wants empty , which depend on subjective psychological issues and to be implemented Fathi et al, The fuzzy principle states that everything is a matter of degree.

Policy is the study of services and the welfare of individuals All propositions acquire a truth value between one true and involved. In general terms, it looks at the idea of social zero false , inclusive.

The allocation of these extreme values welfare, country's issues and its relationship to politics and will only be given in the case of logical truths or falsehoods or society. For instance, in Iran some pioneer universities have a specific university courses The arguments for introducing the concept of fuzziness in and studies on policy and policy making universities like logic have already been naked, but it will be indispensable to 71 www.

The practical and philosophical outcomes stemming from such inaugurations. Slack-Based Measurement that is useful for representing the implications The situation of leptokurtic and imprecise risk factors modelled as fuzzy random of risk variables, generalizing [7], is analyzed in Reference [8] where linear portfolio VaR and expected shortfall ES are computed.

In [9] VaR is estimated by using probabilistic fuzzy systems PFSs which are semi-parametric methods that combines a linguistic description of the system behavior with statistical properties of the data; the approach to designing probabilistic fuzzy VaR models is compared with GARCH model and statistical back testing always accepts PFS models after tuning, whereas GARCH models may be rejected in some situations.

A possibilistic portfolio model is proposed in [10] as an expansion of the possibilistic mean-variance model by with VaR constraint and risk-free investment are computed taking the assumption that the expected rate of returns is a fuzzy number. In [11] the authors suggest an evolving possibilistic fuzzy modeling ePFM approach to estimate VaR; data from the main global equity market indexes are used to estimate VaR using ePFM and the performance of ePFM is compared with traditional VaR benchmarks producing encouraging results.

A growing interest for researches and practitioners is directed to VaR estimation in the case of operational risk, in [12] the intrinsic properties of the data as fuzzy sets are related to the linguistic variables of the observed data external , allowing an organization to supervise operational VaR over time. The notion of credibilistic average VaR is detailed in [13] where simulation algorithms support its use in many real problems of risk analysis.

An alternative nonparametric approach based on maxitive kernel estimation of VaR is studied in [14] where the obtained interval-valued VaR estimates are the key factors to lead to accurate decisions involving uncertainty.

Axioms , 9, 98 3 of 15 Nevertheless, we believe that the modeling of uncertainty through fuzzy logic in decision making and risk management deserves in-depth analysis; just to mention some of our contributions, we developed the rigorous use of the extension principle for fuzzy-valued functions in [15] where we show that fuzzy financial option prices can capture the unavoidable uncertainty of several stylized facts in real markets and the subjective believes of the investor.

It is a matter of interest the use of VaR as a factor fixing decision-making believes for risk-averse investors, for example in [16], the process of recovering investment opportunities with projects that have been rejected when applying the criterion of the Value-at-Risk method, is studied. In [17] we took three financial time series and we modeled them with two non-parametric smoothing techniques defined in terms of expectile and quantile fuzzy-valued Fuzzy-transform, respectively obtained by minimizing a least squares L2 -norm operator and a L1 -norm operator.

The relevance of expectiles in risk management is also focused in [18] where it is shown that several limits of VaR and expected shortfall can be overcome by expectiles because they are the only elicitable law-invariant coherent risk measures. The goal of the present paper is to highlight potentialities of fuzzy numbers in VaR estimation which is a fundamental topic in risk management because the amount of capital to be allocated in case of future losses has to be identified carefully—if it is too small then it does not cover from adverse events and if it is too large then the allocated capital can not be used for other crucial activities.

To reach the goal we apply two results which provide instruments for detecting time series properties and for modelling their uncertainty. The first VaR estimation is produced by the mentioned smoothing techniques, based on Fuzzy-transform and evaluated by performing a rolling window analysis along twelve years of daily returns, during which we count the number of violations and we compare it to the traditional historical simulation.

The second approach addresses non parametric methods and, as in [19], we propose a method to estimate quantiles through a nonparametric estimates of the cumulative distribution function deduced by using the Average Cumulative Function ACF which plays the role of the double kernel smoothing in the mentioned paper. The choice of ACF, introduced in [20] as an alternative representation of fuzzy numbers, is justified in terms of an interesting link which can be established between ACF-representation and quantile functions, without requiring distributional assumptions.

In [21] we extend the ACF analysis and in particular we clarify the crucial role of ACF in determining the membership function from experimental data. Again we compare the number of violations when VaR is defined in terms of ACF, with the same number obtained through the historical simulation of VaR.

The paper is organized into seven sections. The fourth section gives the basic knowledge of the Average Cumulative Function which allows the interpretation of the fuzzy quantile function in terms of VaR; in Section 5 experiments are given with the same time series as in Section 3, in relation to which the comparison is detailed in Section 6.

Highlights on possible future researches are investigated in Section 7. Fuzzy-Transform Smoothing In [17] two non-parametric smoothing methodologies are introduced; the expectile fuzzy-valued F-transform is based on the classical F-transform obtained by minimizing a least squares L2 -norm operator, while the quantile fuzzy-valued F-transform is based on the L1 -type F-transform, obtained by minimizing an L1 -norm operator.

In addition, the robustness is confirmed within time series of various types. We detail some useful preliminaries explaining the theoretical steps in the discrete case which fits the time series experiments in Section 3. Axioms , 9, 98 6 of 15 Figure 2. Ak x if! Its main advantages can be phrased as follows— 1 generally the implementation is very easy, 2 it does not depend on parametric assumptions on returns distribution implying that it can accommodate wide tails, skewness and any other non-normal features in financial observations.

On the other hand, the strongest weakness is its completely dependence on the data set. Axioms , 9, 98 7 of 15 Many contributions find that historical simulation underestimates risk in an unusually quiet period and it is sometimes slow to reflect market turbulence.

The non parametric smoothing methods based on F-transform can weaken these distortions as we show in some simple experiments. Figure 3. Figure 4. In Figure 7 the left and right minimizers in the smoothing procedure based on L1 norm minimization are represented. We consider rolling windows, each one made up of observations two …nancial years.

Graphical representation of the minimizers in Equation 2 when developing Figure expectile 5: Graphical representation of the minimizers in equation 2 when de- smoothing.

Figure 6 shows that the bandwidth of the quantile smoothing is slightly smaller than the expectile one, especially for the downside returns which mostly affect the Value at Risk; more evident differences could be appreciated in case of higher volatile time series. Violations percentages A back-testing procedure is adopted to compare the three VaRs in terms of violations which are the unpredicted losses with the actual losses realized on each day. The back-testing analysis takes into account rolling windows for a total number of VaR values.

The ACF-representation and its properties are studied in [20] where the rela- 10 tionship between ACF and quantile functions, which we apply to Value at Risk hereafter, is proved and resumed in what follows. Minimizers of Equation 8 in quantile smoothing. Figure 7: Minimizers of equation 8 in quantile smoothing. A back-testing procedure is adopted to compare the three VaRs in terms of violations which are and with the actual losses realized on each day.

The results are shown in Table 1 which offers the possibility of some observations: first implying that the -Average Cumulative function -ACF of u is de…ned of all, the number of violations is slightly reduced when a smoothing methodology is applied and for a …xed value of 2 [0; 1] as the following convex combination of FuL and the reduction is more evident when the quantile one is performed, confirming its results extensively analyzed in [17].

Table 1. Violations percentages. Average Cumulative Function The ACF-representation and its properties are studied in [20] where the relationship between ACF and quantile functions, which we apply to Value at Risk hereafter, is proved and resumed in what follows.

Value at Risk Based on ACF The ACF representation allows obtaining the guess quantiles directly from the time series because it has the same properties of a cumulative distribution function CDF of a real random variable X, defined on the same support domain. Consequently, VaR[ X; 0. The percentage of violations for VaR ACF as shown in Table 2 is considerably smaller in every year and the reduction becomes crucial in and when the high volatility in financial markets caused unsuitable capital allocations.

Table 2. Percentages of violations. Closing Considerations The adopted back-testing procedure studies the performance of the three values VaREXP , VaRQU A , VaR ACF in terms of the number of violations over values when they are compared to the value VaR Hist ; violations have a strong concrete meaning because they detect all the days in which the actual losses are not predicted by the VaR estimation implying a scarce amount of capital for facing adverse situations.

The limitation of a single asset can have a strong impact because an index is generally less volatile than a stock price, as it is computed as a weighted mean, and the smoothing effect could be theoretically mostly effective when the volatility is higher. In VaR estimation through the ACF, the investigation of the properties which relate the non parametric estimation and the uncertainty modeling deserves a further appropriate theoretical investigation devoted to the combination between probability and possibility settings.

However, preliminary results strongly encourage the adoption of models based on Fuzzy set theory for calculating VaR and possibly several more risk measures, due to their capability to contemplate uncertainty in a rigorous way. The paradigm of uncertainty is crucial not only in financial time series but in the whole field of data science which includes all real life data, from social media to medicine, from ecology to history.

Paths for Future Research The amount of promising research paths for uncertainty theories in risk management is huge as described in [22] , here we just give a flavour of the most appealing from our point of view.

Vagueness sources in real data can be viewed as a mixture of stochastic and fuzzy approaches as in [23] where a given interval describes the available information regarding the evolution of certain variables is vague.

Also the possibility of introducing random-fuzzy variables as in [24] in the evaluation of risk measures deserves more efforts in order to be combined with the stochastic nature of the traditional model. In order to extend the research from a unique stock to a portfolio, a possible research is to apply the introduced non parametric estimation in the case of a portfolio selection model with VaR constraint under different attitudes, as studied in [25].

A challenging future research path can be the extension of the proposed approaches within the field of neutrosophic logic derived from the seminal philosophical paper by Samarandache in [26] which is compared in [27] with well-known frameworks for reasoning with uncertainty and vagueness.

The concept of set highlights the main difference between fuzzy logic and neutrosophic logic; a fuzzy set is defined through association with a scale of grades of membership, its generalization is called intuitionistic fuzzy set and it assigns two values called membership degree and non-membership degree respectively, when an additional parameter for neutrality is introduced, then the neutrosophic fuzzy set is defined.

In particular, many contributions based on neutrosophic statistics broadly detailed in [28] are decisively successful when data have indeterminate observations. In social sciences as Economics and Management the scarce availability of certain data can be approached through neutrosophic statistics as in [33], where an efficient and flexible acceptance criterion is introduced for a two-stage process for multiple lines.

Also in medical science uncertainty is a common feature and neutrosophic statistics is a rigorous way to model it as it is shown in [34] where uncertain data from an healthcare department are analyzed and in [35] where an investigation is introduced to classify when patients have the diabetes or not.

In addition, widening the scenario of uncertainty modelling, the tail VaR metric on a tree network, as shown in [36], computes the probability of the loss and its severity in an uncertain environment. Author Contributions: The authors contributed equally in writing the manuscript.

All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The authors would like to thank the editors and the anonymous reviewers for their meaningful and constructive suggestions that have led to the present improved version of the paper.

Conflicts of Interest: The authors declare no conflict of interest. References 1. Jorion, P. Abad, P. A comprehensive review of Value at Risk methodologies. Shayya, R. Cherubini, U. Fuzzy value-at-risk: Accounting for market liquidity. Notes , 30, — Zmeskal, Z. Value at risk methodology under soft conditions approach fuzzy-stochastic approach.