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I don't understand the main function of the Parzen Window

Let $u=[u_1, u_2,..., u_d]$ and define a window function

$φ(u)=\left\{ \begin{array}{l l} 1 & \quad \text{$|u_j|<\frac{1}{2}$ , $j=1,2,...,d$}\\ 0 & \quad \text{otherwise} \end{array} \right.$

What exactly it means?

I found lots of presentation on the internet, so please do not direct me to one.

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  • $\begingroup$ When you edit a question such that parts of existing answers no longer make sense, please mark the edit as such. $\endgroup$
    – joriki
    Dec 12, 2012 at 18:14
  • $\begingroup$ Also, editing other people's posts while ignoring their comments is not exactly the best style. $\endgroup$
    – joriki
    Dec 13, 2012 at 6:05
  • $\begingroup$ My edit of your your answer is the sign that I didn't ignored your comment. I even wrote why I edited it. Take it easy man. You got your points, I got my answer. (I tried to answer my question an hour after I posted it but I couldn't because I'm new here, even though I accepted your answer and not my) $\endgroup$
    – Bush
    Dec 13, 2012 at 6:39
  • $\begingroup$ I am taking it easy :-) Relative to how annoying I find it that so many people on this site just edit around in posts (their own and others) without ever replying to comments, my reaction was quite mild :-) My annoyance was also perhaps furthered by the fact that you asked a question about what exactly something means but didn't bother to make sure that you quoted it exactly; that sort of thing often wastes a lot of time, and it would have been so easy for you to avoid. Feel free to accept your own answer if you find it more helpful; points are not so important. $\endgroup$
    – joriki
    Dec 13, 2012 at 8:44
  • $\begingroup$ I agree. Need to recheck anything I write, especially quote, so people will not waste their time on stupid mistakes. I learnt from this case, thank's :) $\endgroup$
    – Bush
    Dec 13, 2012 at 9:14

2 Answers 2

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It means that $\varphi(u)$ is $1$ in a unit hypercube centred on the origin and $0$ outside (and on the boundary).

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$u$ is a $d$ dimensional sample, we choose a "window" which is: a line with length h when $d=1$, a square with edge $h=2$ ($h^2$ space), an hypercube with edge $h=2$. The "window" is centered at the point $u$ and we normalize is so $h=1$.

Now our function assign 1 to each other sample that is close to $u$ and 0 to samples that are far.

Close means that the sample $x$ is in the range of $u-\frac{1}{2}h\leq{x}\leq{u+\frac{1}{2}h}$

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