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统计代写|时间序列分析代写Time-Series Analysis代考|FUZZY SET THEORY

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统计代写|时间序列分析代写Time-Series Analysis代考|FUZZY SET THEORY

Zadeh (1965) created fuzzy set theory. Since its creation, it has achieved success in both theory and application. The impulse behind fuzzy set theory is primarily to supply a proper, stronger and quantitative structure to deal with ambiguity in individual understanding by expressing it in normal language (Dubois \& Prade, 1991). We encounter many situations in real life where inclusion and exclusion from a set are not clearly defined; for instance, the classes of tall boundaries of such sets are indistinguishable, and the transition from one member non-member appears slow rather than quick. A fuzzy set is defined as any set that permits its members to have dissimilar grades of membership (membership function) in the interval $[0,1]$. The mathematical description is shown below:

Let $\mathrm{X}$ be the collection of objects, then $\mathrm{A}$ is a fuzzy set of ordered pairs contained in $\mathrm{X}$ such that
$$A=\left{\left(x, \quad \mu_A(x)\right): \quad x \in X, \quad \mu_A(x): \quad X \rightarrow[0,1]\right}$$
where $\mu_{\mathrm{A}}$ (.) is the membership function of $\mathrm{A}$ which is defined as a function from $\mathrm{X}$ into $[0,1]$. The membership function provides a grade of membership of the element to the set. Fuzzy set has a graphical depiction which states how the transition from one to another takes place. Such type of graphical depiction is also known as a membership function.

A fuzzy system (FS) is defined as any fuzzy-based system which utilizes fuzzy logic as the source for information representation by using different forms of knowledge. We can model variables, interactions, systems and intermodal relationships by using many ways like membership function, shape analysis of membership function, etc. Membership functions are often used as the mathematical means of representing values for inference mechanism.

统计代写|时间序列分析代写Time-Series Analysis代考|MEMBERSHIP FUNCTION

A membership function is defined as the function which assigns values to the elements contained in a universal set that fall inside a specific range and point to the membership grade of these elements in the set. The larger the values, the larger the degrees of set membership. Such type of set defined by membership functions is known as a fuzzy set. The most generally used range of values for membership function is the unit interval i.e. $[0,1]$.
Note: For fuzzy set $A$, its membership function is represented by $\mu_A$
$$\mu_A: X \rightarrow[0,1]$$
Writing it in another way, if the function is represented by $A$, then it has the same form
$$A: X \rightarrow[0,1]$$

The features of a membership function are described below. The graphical representation is shown in Figure 11.1.

Core: It is defined as the region identified by full membership in set A, i.e., $\mu(x)=1$.
Boundary: It is defined as the region identified by partial membership in set A, i.e., $0<\mu(\mathrm{x})<1$. Support: It is defined as the region identified by non-zero membership in set A, i.e., $\mu(x)>0$.

Fuzzy logic is defined as a multi-valued logic that permits intermediary values to be labeled between traditional evaluations like high/low, yes/no, true/false, etc. FS translates these rules into their arithmetic equivalents. In this way, the work of the system designer and the computer is simplified and results in much more precise representation of the manner the system performs in the real world. Generally, fuzzy logic provides a straightforward way to reach a particular conclusion based upon ambiguous, imprecise, vague, noisy or missing input information.

Fuzzy logic could be understood as a superset of conventional (Boolean) logic that has been broadened to figure out the theory of partial truth that is truth values between “completely true” or “completely false”.

统计代写|时间序列分析代写Time-Series Analysis代考|FUZZY SET THEORY

Zadeh(1965)创立了模糊集合理论。自创立以来，它在理论和应用上都取得了成功。模糊集理论背后的推动力主要是提供一种适当的、更强的、定量的结构，通过用正常语言表达来处理个人理解中的模糊性(Dubois ＆ Prade, 1991)。在现实生活中，我们遇到了许多情况，其中一组的包容和排斥没有明确的定义;例如，这些集合的高边界的类是不可区分的，并且从一个成员非成员的过渡看起来很慢而不是很快。模糊集被定义为允许其成员在$[0,1]$区间内具有不同等级的隶属度(隶属度函数)的任何集合。数学描述如下:

$$A=\left{\left(x, \quad \mu_A(x)\right): \quad x \in X, \quad \mu_A(x): \quad X \rightarrow[0,1]\right}$$

统计代写|时间序列分析代写Time-Series Analysis代考|MEMBERSHIP FUNCTION

$$\mu_A: X \rightarrow[0,1]$$

$$A: X \rightarrow[0,1]$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。