I am in the process of trying to implement Kernel Density Estimation for a project but I am struggling to understand some notation.

I have the following function

$$f_h(x) = \frac{1}{hN}\sum_{i=1}^NK(\frac{x - x_i}{h})$$

How is do I use this function/How do I read it?

My assumption is that the result of the sum is substituted in the place of $N$ but I wanted to clarify before I proceeded.

I am using this on a histogram.

I have attempted to find an answer through Google but couldn't find anything.

Any help would be greatly appreciated.

  • $\begingroup$ Are you asking how to interpret the the expression? $\endgroup$ – user113529 Mar 16 '17 at 14:17
  • $\begingroup$ @user43687 Yes, sorry if I wasn't to clear on that. $\endgroup$ – Mykrosoft Mar 16 '17 at 14:22

The function $f_h$ appears to depend on more than just $h$. You must be given $N$-points $x_1$, $x_2$ ,..., $x_N$ (I assume real numbers, but you have not specified the domain) and another function $K(x)$

Given these and a nonzero real number $h\neq0$, $f_h$ is the function which assigns each real number $x$ to the sum $$K\left(\frac{x-x_1}{h}\right)+K\left(\frac{x-x_2}{h}\right)+...+K\left(\frac{x-x_N}{h}\right)$$ and divides the result by $Nh$.

  • $\begingroup$ Having this explained actually makes me feel a little silly. This is extremely obvious thinking about it, however I have never seen a the sum notation used in this way before. Thank you. $\endgroup$ – Mykrosoft Mar 16 '17 at 14:37
  • $\begingroup$ @Mykrosoft To paraphrase Mark Twain: you should never feel silly about not understanding something. You should feel silly when you are certain about something which is wrong. $\endgroup$ – user113529 Mar 17 '17 at 4:46

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