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.

table 7.1

At the page 218 there is a mathematical equation:

enter image description here

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.

  • 1
    $\begingroup$ $\exp$ is the exponential function: $\exp(x)=e^x$. $\endgroup$ May 24 '20 at 5:20
  • $\begingroup$ thank you very much. $\endgroup$
    – tahasozgen
    May 24 '20 at 5:26
  • 3
    $\begingroup$ $\exp(\text{stuff})$ is used for $e^\text{stuff}$ when "$\text{stuff}$" makes for an unwieldy exponent and/or would be illegible in a superscript font. The use of square brackets vs ordinary parentheses is likely just the editor's preference, again for legibility: nested parentheses can be hard to parse, so different types of brackets can help distinguish layers. Modern typesetting, which supports increasingly-large parentheses, tends to make this unnecessary, but it can be a stylistic choice. (That said, sometimes square brackets have a specific purpose besides grouping; context is key.) $\endgroup$
    – Blue
    May 24 '20 at 5:29
  • $\begingroup$ As for "What is the purpose of the equation?" ... You have the book. Doesn't it say? :) $\endgroup$
    – Blue
    May 24 '20 at 5:31
  • $\begingroup$ At the page 217, it is stated that "regression vector". Is it Vector autoregression? (en.wikipedia.org/wiki/Vector_autoregression) $\endgroup$
    – tahasozgen
    May 24 '20 at 5:33

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}

  • $\begingroup$ I would upvote this answer if it mentioned that this function is a proportionality constant away from the normal probability density function! That seems like an important thing to recognise. $\endgroup$ May 24 '20 at 7:02

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