Kernel Density Estimation

Question

In the Kernel density estimation formula below (from Wikipedia), what do the values of $x$ and $x_i$ represent?

$$ \hat f_h(x) = \frac{1}{n}\sum_{i=1}^n K_h(x-x_i) = \frac{1}{nh}\sum_{i=1}^n K\bigg(\frac{x-x_i}{h}\bigg) $$

I am implementing this formula but I am not sure about what they really represent. I know what values use for $x_i$, but I don't know what I'm passing in for $x$.

$x_i$ is a data value and the function is being evaluated at $x$. — Glen_b, Sep 19 '15 at 00:33
The formula is in this link: https://en.wikipedia.org/wiki/Kernel_density_estimation. I know what to use for xi, but i don't know what passing in x — Catia Cannata, Sep 19 '15 at 07:38
Typically, $x$ would be one of a grid of points on a range that at least covers that of your data set. — user603, Sep 19 '15 at 12:03
But if, for example, my xi are: 2, 4, 6, 8... etc what is my correspondinf x value? — Catia Cannata, Sep 19 '15 at 20:52
You should be building a function, using the $x_i$'s, so that you can pass in any $x$ and get a value. So, you create your function using only the $x_i$'s and if (say) your boss says "evaluate the function at $x=\pi$" you should be able to pass it to your function and get a value. — guy, Oct 06 '15 at 14:22
This tutorial explains it in details for noobie level enter link description here — , Nov 05 '15 at 09:56

score 3 · Answer 1 · answered May 30 '16 at 00:17

Since this doesn't have an answer yet, I'll expand my comment a little:

$x$ is the argument you're evaluating (calculating) the function (the density estimate) at.

$x_i$ the value of the $i$-th data point.

To draw the density, you'd normally evaluate $x$ across some reasonably fine grid of values.

But if, for example, my $x_i$ are: 2, 4, 6, 8... etc what is my corresponding x value?

$x$ will be whatever value you want to know $\hat{f}(x)$ at. This is like any other function -- If I said "here's a parabola, $g(x) = 3x^2-8x+5$" you're basically asking "but what value is $x$?". The answer is whatever value(s) you want to know $g$ at.

Kernel Density Estimation

1 Answers1