Fuzzy Mathematics In Pattern Classification Pdf

Fuzzy Mathematics

Fuzzy logic is an extension or a superset of the Boolean logic – aimed at maintaining the concept of the "partial truth," i.e. expression values ranging from "completely truthful" to "completely untruthful" (from 0 to 1).

From: Minerals Engineering , 2015

Online Diagnosis of PEM Fuel Cell by Fuzzy C-Means Clustering

Damien Chanal , ... Marie-Cécile Péra , in Reference Module in Earth Systems and Environmental Sciences, 2021

Fuzzy Logic (FL)

Fuzzy logic can be very powerful to describe complex systems, mainly with ambiguities and non-linearities. FL tends to represent the human reflection by combining: IF–THEN rules and logical operators AND–OR, with the objective to convert numeric value to fuzzy set (Fuzzification process). Thanks to "if–then" rules, a membership is assigned to each fuzzy set then converted to numerical value (Defuzzification process). Hissel et al. (2004) proposed a diagnosis model of PEMFC using fuzzy logic to detect water management problems.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128197233000998

Advanced Modelling Techniques Studying Global Changes in Environmental Sciences

Wout Van Echelpoel , ... Peter L.M. Goethals , in Developments in Environmental Modelling, 2015

6.3.4.3 Additional remarks

Fuzzy logic models have been shown to perform similarly when compared with random forests (a specific type of decision tree), although when considering transparency, fuzzy logic models scored better because of their ability to combine ecological relevance with reasonable interpretability (Mouton et al., 2011). Drawbacks of fuzzy logic are the increase in complexity with an increasing number of predictors (Ahmadi-Nedushan et al., 2006), the loss of information due to data discretisation, and the possibility that the implementation of expert knowledge rules is time intensive (Kompare et al., 1994).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444635365000089

Data Mining and Knowledge Discovery

Sally I. McClean , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.B.4 Fuzzy Logic

Fuzzy logic maintains that all things are a matter of degree and challenges traditional two-valued logic which holds that a proposition is either true or it is not. Fuzzy Logic is defined via a membership function that measures the degree to which a particular element is a member of a set. The membership function can take any value between 0 and 1 inclusive.

In common with a number of other Artificial Intelligence methods, fuzzy methods aim to simulate human decision making in uncertain and imprecise environments. We may thus use Fuzzy Logic to express expert opinions that are best described in such an imprecise manner. Fuzzy systems may therefore be specified using natural language which allows the expert to use vague and imprecise terminology. Fuzzy Logic has also seen a wide application to control theory in the last two decades.

An important use of fuzzy methods for Data Mining is for classification. Associations between inputs and outputs are known in fuzzy systems as fuzzy associative memories or FAMs. A FAM system encodes a collection of compound rules that associate multiple input statements with multiple output statements We combine such multiple statements using logical operators such as conjunction, disjunction and negation.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B0122274105008450

SECONDARY BATTERIES – LEAD– ACID SYSTEMS | Modeling

M. Cugnet , ... B.Y. Liaw , in Encyclopedia of Electrochemical Power Sources, 2009

Fuzzy Logic-Based Models

Introduction to Fuzzy Logic

In order to understand FL, it is important to realize that data may be characterized in two different ways: crisp or fuzzy. A crisp data is expressed in a certain, defined manner, e.g., a battery open-circuit voltage (OCV) of 12.5 V. In contrast, a fuzzy data is depicted in an indefinite, often descriptive way, e.g., the battery is 'deeply discharged'. This linguistic description can cover a range of OCVs, and the degree to which a crisp data point falls into the fuzzy set of 'deeply discharged' is indicated by a fit value (fuzzy unit) between zero and one. The fit value is sometimes called the degree of membership. Figure 2 shows an example of various fuzzy subsets or membership functions of OCV. Depicted is the degree of membership of various OCVs to the fuzzy subsets 'deeply discharged', 'normally discharged', and 'shallowly discharged'. The process of assigning membership functions to sets of data is referred to as 'fuzzification'.

Fuzzy set theory explicitly depicts a method to categorize measured data using linguistic variables such as 'deeply discharged', 'normally discharged', and 'shallowly discharged', as exemplified above. It accounts for the uncertainty, or fuzziness, inherent in such a linguistic description by using multivalued sets. This representation offers a qualitative rather than a numerical description of a system. This intuitive representation allows a relatively easy logical development of a model with a fuzzy algorithm compared to crisp numerical computations. This ease should not however undermine its powerful capability in solving complex control and modeling problems.

Fuzzy logic usually uses 'if–then' rules to perform the inference procedure describing the relationship between input and output variables. For example, in hybrid electric vehicle (HEV), a simple battery management strategy might look like this:

•: IF battery is shallowly discharged, THEN use only the electric motor;
•: IF battery is normally discharged, THEN use the engine, electric motor, and regenerative braking;
•: IF battery is deeply discharged, THEN use only the engine and regenerative braking.

The membership functions and rule set may be prescribed by an expert or created by an ANN algorithm.

Applications of the Fuzzy Logic Approach

In the last decade, some FL methods have been used to estimate the state-of-health (SoH) and SoC of batteries; therefore, FL can offer simple and powerful ways to model battery characteristics. Impedance measurements combined with FL data analysis have been applied by P. Singh and D. E. Reisner to lead–acid defibrillator batteries to estimate SoC. This approach has also been used to estimate the SoH of LABs in uninterruptible power supply (UPS). Another example of FL is the estimate of the valve-regulated lead–acid (VRLA) battery capacity by assessing the discharge profile in the coup de fouet region, as proposed by P. E. Pascoe and A. H. Anbuky. The latter corresponds to a transient behavior of LAB in the beginning of a discharge or charge regime, characterized by the trough and the plateau feature in the voltage profiles.

Impedance measurements and voltage response measurements can be combined with FL data analysis to estimate the battery state with good accuracy. This approach has already been used by P. Singh and D. E. Reisner to estimate the SoC of spirally wound 2 V lead–acid cells and implemented efficiently by S. Malkhandi on onboard microprocessors in the vehicles, where measurements and SoC/SoH calculations can both be performed in real time. FL algorithms for battery monitoring systems can be deployed rapidly with relatively sparse data sets first and then improved as more data are acquired, either by ANN-based algorithms or evolutionary computation methodologies. In these methods, the battery monitoring system essentially 'learns' about the performance of the battery system and adapts the estimation of SoC or SoH based on the evaluation of the battery parameters upon, e.g., aging.

Figure 3 displays the errors in real-time SoC estimation using voltage, current, and temperature data of an 80 Ah 12 V VRLA battery. In this experiment a fully charged battery is discharged following a set current profile and then fully recharged. This profile is made up of C/6 and C/8 rates with different time intervals, selected randomly. The results indicate that the estimation error is well within 5% from the fifth discharge cycle and on.

The FL approach offers robustness, fast development time, and high efficiency. It is also generic and applicable not only to the LAB but also to a wide variety of other battery chemistries.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444527455001519

Geophysics for Petroleum Engineers

Fred Aminzadeh , Shivaji N. Dasgupta , in Developments in Petroleum Science, 2013

5.5 Fuzzy Logic

FL is a computational tool that deals with the linguistic and qualitative nature of information to a computer. It is generalization of the classical or Aristotelian logic of "A thing either is or is not." FL, goes beyond the rigid boundaries of "black" or white and allows the gray area which is the more realistic situation in many cases. The binary language of Boolean algebra, used by nearly all types of modern digital computers, is based directly on the true and false logical variables of conventional logic. Using this logic, computers are able to manipulate precise facts that have been reduced to strings of zeros and ones. The multivalued nature of FL has been employed to allow computers to deal with the "real world" vagueness associated with linguistic and qualitative information.

Through the introduction of the "membership function" concept μ(x) the degree of belonging of the variable x to a given set allows tremendous flexibility in representing imprecise data and linguistic rules. The characteristics, or shape, of membership functions can be chosen based on mathematical convenience or how accurately they describe a linguistic or physical phenomenon. Figure 5.10 shows three typical membership functions, triangular, trapezoidal, and Gaussian. We also show how these functions can describe physical properties with linguistic qualifiers like porosity of "about" 2%, porosity "approximately between" 4% and 6%, and porosity of "roughly" 13%.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444506627000056

Remote Sensing and Fuzzy Logic Approach for Artificial Recharge Studies in Hard Rock Terrain of South India

C.K. Muthumaniraja , ... M. Chinnamuthu , in GIS and Geostatistical Techniques for Groundwater Science, 2019

8.3.1 Fuzzy Logic and Membership Functions

Fuzzy logic is commonly used in spatial planning in order to allow the spatial objects on a map to be treated as members of a set. In a classic case, which is sometimes called "crisp," an object either belongs to a set or not. However, in fuzzy set theory a candidate object can take on membership values between 0 and 1, which reflects a degree of membership (Zadeh, 1965; Bonham-Carter, 1996). The benefit of fuzzy logic is that a new analysis, or a change in the rules or the criteria is not required, which saves time and effort. In fuzzy systems, values are indicated by a number (called a truth value) in the range from 0 to 1, where 0.0 represents absolute falseness and 1.0 represents absolute truth. While this range evokes the idea of probability, fuzzy logic and fuzzy sets operate quite differently from probability.

Using the integration of two or more parameters with fuzzy membership functions for the same set, the different operators can be employed to combine the membership values together. It was found that five operators were useful for combining exploration datasets, namely the fuzzy AND, fuzzy OR, fuzzy algebraic product, fuzzy algebraic sum, and fuzzy gamma operator (Tangestani, 2001). The fuzzy operators are as follows:

Fuzzy AND: This is equivalent to a Boolean AND (logical intersection) operation on classical set values:

(8.1) $μ combination = MIN (μ_{A}, μ_{B}, \dots \dots \dots μ_{N})$

Fuzzy OR: This is equivalent to a Boolean OR (logical union) on classical set values:

(8.2) $μ combination = MAX (μ_{A}, μ_{B}, \dots \dots \dots \dots . . μ_{N})$

Fuzzy algebraic product: The combined membership function is defined as:

(8.3) $μ combination = \prod_{i = 1}^{n} μi$

where μ _i is the fuzzy membership function for the Ith map, i = 1, 2, 3…., n maps are to be combined. The combined fuzzy membership values tend to be very small with this operator, due to the effect of multiplying several numbers < 1. Nevertheless, all the contributing membership values have an effect on the result, unlike the fuzzy AND or fuzzy OR operators.

Fuzzy algebraic sum: This operator is complementary to the fuzzy product, being defined as:

(8.4) $μ combination = 1 - \prod_{i = 1}^{n} (1 - μi)$

The result is always larger (or equal to) the largest contributing fuzzy membership value. The effect is therefore "increasive." The "increasive" effect of combining several favorable pieces of evidence is automatically limited by the maximum value of 1.0. Fuzzy algebraic product is an algebraic product, but fuzzy algebraic sum is not an algebraic summation.

Gamma operation: This is defined in terms of the fuzzy algebraic product and the fuzzy algebraic sum by the representation:

(8.5) $μ combination = {(FAS)}^{γ} * {(FAP)}^{1 - γ}$

where γ is a parameter chosen in the range (0, 1). When γ is 1 the combination is the same as the fuzzy algebraic sum, and when γ is 0 the combination is equal to the fuzzy algebraic product. Judicious choice of the γ produces output values that ensure a flexible compromise between the "increasing" tendencies of the fuzzy algebraic sum and the "decreasing" effects of the fuzzy algebraic product where γ is a parameter chosen in the range (0, 1).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128154137000080

Ecological Model Types

T. Prato , in Developments in Environmental Modelling, 2016

Abstract

A fuzzy logic-based, adaptive management (FLAM) model is developed that allows managers of coupled natural and human systems to determine preferred management actions over time when they are uncertain about the extent of future climate change and system responses to climate change and management actions. The FLAM model uses (1) data from adaptive management experiments and expert judgment, surveys, and/or simulation models to estimate multiple attributes of system responses to climate change and management actions, (2) fuzzy TOPSIS to determine the preferred management action for each climate change scenario within time periods, and (3) the minimax regret criterion to determine the preferred management actions across climate change scenarios within time periods. Application of the model is demonstrated for a hypothetical national park whose managers want to determine the best adaptive management strategy for increasing the number of backcountry campsites over time.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444636232000104

ADVANCED LOG INTERPRETATION TECHNIQUES

Toby Darling , in Well Logging and Formation Evaluation, 2005

5.5 FUZZY LOGIC

"Fuzzy logic" is a technique that assists in facies discrimination, and that may have particular application in tying together petrophysical and seismic data. In this chapter, the basic technique will be explained, together with a worked example to illustrate the principle. Consider a situation in which one is using a GR (gamma ray) log to discriminate sand and shale. With the conventional approach, one would determine a cutoff value below which the lithology should be set to sand and above which it should be set to shale. To use fuzzy logic, one would do the following:

1: In some section of the well where sand and shale can be identified with complete confidence, one would generate a "learning set," that is, create a new log in which the values are set to 0 or 1 depending on whether the formation is sand or shale.
2: Over the interval defined by the learning set, one would separate all the bits of GR log corresponding to sand and shale, respectively.
3: For the sand facies, a histogram would be made of all the individual GR readings. To this distribution would be fitted a mathematical function (most commonly a normal distribution) that would capture the mean and spread of the data points. This is often called a "membership function."
4: The same would be done for all the shale values, generating a new membership function with its own mean and spread.
5: Both membership functions would be normalized so that the area underneath them is unity.

The resulting distributions would look like Figure 5.5.1.

Now, supposing one were trying to determine whether a new interval of formation, having a GR reading of x, belonged to the class "sand" or "shale." Using the functions shown in Figure 5.5.1, one would simply enter the graph at the GR axis at the value x and read off the relative probabilities of the interval belonging to either class. The interval would be assigned to the one having the greatest probability. Moreover, one can assign a confidence level based on the relative probabilities.

Having understood the principle of fuzzy logic with one variable, it is easy to see how it might be extended to more than two classes (e.g., sand, silt, and shale) and with more than one input variable (e.g., GR, density, neutron). Since it is not practical to plot more than two variables on a graph, the actual allocation is performed in a computer program in the N dimensional space corresponding to the N variables. Obviously for the method to work well, it is necessary that the membership function not overlap much in the N dimensional space they occupy. Also, the method does not work well with parameters that vary gradually with depth.

The advantage of the method over other approaches such as neural networks is that one is able to see, through plotting the membership functions with respect to a certain variable, whether or not it is applicable to include a certain variable or not. Also, the method can generate a confidence level for the output classification, as well as a "second choice." The use of fuzzy logic has been mainly in acoustic and elastic impedance modeling, where one can investigate whether or not, for instance, there is any acoustic impedance contrast between oil- and gas-filled sandstones. If there is, the membership functions may be used as input to a seismic cube for allocating facies types to parts of the seismic volume, thereby showing up potential hydrocarbon zones.

Fuzzy logic may also be useful to allocate certain facies types to the logs, as for instance a basis of applying a different poroperm model. In my experience with using fuzzy logic, I have often found that one starts out with too many facies, which then are found to overlap each other. Also, the effect of adding more log types as variables, which may be only loosely related to the properties one is interested in, is generally detrimental. In many respects, fuzzy logic is similar to the statistical analyses packages described earlier. In common with these, it has the advantage over deterministic techniques in that it can handle a lot of variables impartially and simultaneously. However, also in common with those packages, it can easily generate rubbish unless great care is taken with the input.

Exercise 5.2

Fuzzy Logic

1: Set up a fuzzy logic model to distinguish between net and non-net on the basis of GR using the data from the core as a learning set.
2: Apply the model to the lower half of the entire logged interval. Compare the average net/gross with that derived using the conventional analyses.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B978075067883450005X

Spatial Prediction of Groundwater Level Using Models Based on Fuzzy Logic and Geostatistical Methods

Ata Allah Nadiri , ... Keyvan Naderi , in GIS and Geostatistical Techniques for Groundwater Science, 2019

7.4.1 Performances of Fuzzy Models

To investigate the performance of SFL and MFL, these techniques were implemented using the following steps: (i) Gaussian membership functions for inputs were used and data were classified by clustering methods—SC is used in the SFL model and FCM in the MFL model to develop fuzzy rules; (ii) data clustering methods minimized NRMSE to identify the optimum cluster radius and the number of rules; (iii) the defuzzification method used centroid calculation to produce crisp output; and (iv) the groundwater level in each observation well was identified using the least squares optimization technique for both models. Fig. 7.4 illustrates schematically the implementation of the SFL model from input data to the generation of groundwater levels following the four rules.

SFL Results: Fig. 7.5 shows the results for SFL, in which the optimum cluster radius is 0.6 m and it generates four fuzzy if-then rules and establishes four Gaussian membership functions for the SFL model. The scale of NRMSE in Fig. 7.5 is logarithmic for clarity.

MFL Results: For the MFL model, the FCM clustering method rendered six clusters and six rules based on the minimum average NRMSE and its value at the optimum cluster radius is found to be 0.022 m. Notably, SFL and MFL were carried out using the same data sets with the identical input and output variables.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128154137000079

Industrial intelligent controllers

Peng Zhang , in Advanced Industrial Control Technology, 2010

7.3.3 Fuzzy industrial controllers

This subsection briefly introduces three important types of fuzzy logic controllers in industrial control applications.

(1) Embedded fuzzy controllers

The concept of fuzzy logic makes feasible the use of a fuzzy controller built on a microcontroller or microprocessor chipset or industrial computers. Manufacturers have recognized the power of fuzzy logic and have created fuzzy kernels and support tools. In some industrial applications, such as motion control, electrical drives, temperature and humidity stabilization, the fuzzy controller receives information from the controlled process via analog-digital converters and controls it through digital-analog converters, as in Figure 7.22.

(2) Fuzzy three-term (PID-like) controllers

The industrial three-term (PID-like) controller constitutes the backbone of industrial control, having been in use for over a century. They can take a number of forms, from the early mechanical to later hydraulic, pneumatic, analog and digital versions. The modern form takes the form of multitasking discrete-time three-term (PID) algorithms embedded in almost all industrial PLCs and RTUs.

The advent of fuzzy control motivated many researchers to reconsider this controller in the hope that "fuzzification" would improve its domain of effectiveness. This is a case of a technology retrofit, in which computational intelligence is applied to improve a well-known device. The result has been a new generation of intelligent three-term (PID-like) controllers that are more robust. Fuzzy logic can be applied in a number of ways. One obvious approach is to fuzzify the gains of the three-term (PID) controller by establishing rules whereby these gains are varied in accordance with the operating state of the closed system.

It is often more convenient to partition the three-term controller into two independent fuzzy sub-controllers that separately generate the signals uPD and uI that correspond to the proportional plus derivative term and the integral term respectively. The result is a fuzzy proportional plus derivative sub-controller FPD (fuzzy proportional and differentiate) in parallel with a fuzzy integral controller FPI (fuzzy proportional and integral) as shown in Figure 7.23. Both sub-controllers are fed with the error and its derivative. The second sub-controller requires an integrator (or accumulator) to generate the integral term.

(3) FPGA-based fuzzy logic controllers

A recent hardware design and implementation of fuzzy logic controllers builds all fuzzy control modules on a field-programmable gate array (FPGA) chipset. The general layout of the controller chip in a unity feedback control system is shown in Figure 7.24. Generally, the proposed controller accepts both the output of the plant (yp) and the desired output (yd), as digital signals, and delivers a digital control action signal as an output. The design also accepts four digital signals that represent the gain coefficients needed by the controller (proportional gain Kp, derivative gain Kd, integral gain Ki, and output gain Ko), and two one-bit signals to select the type of the controller. Figure 7.25 shows a view of its FPGA chipset.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9781437778076100075