I am reading book "Fuzzy Logic With Engineering Applications, Wiley" written by Timothy J. Ross. I am reading chapter 7 and in this chapter, "Batch Least Squares Algoritm" has been defined. It illustrates the development of a nonlinear fuzzy model for the data in Table 7.1 using the Batch Least Squares algorithm.
At the page 218 there is a mathematical equation:
I have two questions: 1- As you can see, there is a "exp" phrase (in the red rectangle). What is this? Is it the exponential function? (https://en.wikipedia.org/wiki/Exponential_function)
At the link, "What is the meaning of $\exp(\,\cdot\,)$?" it was stated at the link that it is an exponential function, but I noticed that ordinary paranthesis has been used. In my equation, square brackets is used.
2- What is the purpose of the equation?
Thanks in advance.
The exponential function $\exp: \Bbb{R} \to \Bbb{R}$ is the function $\exp(x) = e^x$. There is no difference between $(\cdot)$ and $[\cdot]$ here. It is just a way to make things look nicer, and attempt to clarify the order in which the brackets should be read.
For example, that red thing you circled can also be written as: \begin{align} \exp\left(-\dfrac{1}{2} \left( \dfrac{x_1 - c_1^1}{\sigma_1^1}\right)^2 \right) \quad \text{or} \quad e^{-\frac{1}{2} \left( \frac{x_1 - c_1^1}{\sigma_1^1}\right)^2} \end{align} These are all correct, but which one "looks the nicest"? Well, to the author, it seems his/her favourite is \begin{align} \exp\left[-\dfrac{1}{2} \left( \dfrac{x_1 - c_1^1}{\sigma_1^1}\right)^2 \right]. \end{align}