Talk:Kernel (statistics)

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class Mid Priority  Field: Probability and statistics
Please update this rating as the article progresses, or if the rating is inaccurate. Please also add comments to suggest improvements to the article.

Contents

[edit] A question from 167.191.250.81

triweight--is that the same as tricube? Which I understood to be:

K(u)=\left(1-\left|u^3\right|\right)^3

[edit] Why is this approach "non-parametric"

I have to admit I'm not as familiar with this subject as I ought to be. The stuff I remember about non-parametric statistics includes things like the rank-sum test, which really does not make any assumptions about the distribution of the two samples beyond the null hypothesis (both samples drawn from identical populations?).

After reading this article and a couple of related articles, it appears that the "kernel" is a sort of prior distribution. So why are these methods called non-parametric? And why isn't that question addressed somewhere in Wikipedia? Or have I just been reading the wrong articles? DavidCBryant 16:43, 11 August 2007 (UTC)

Actually, this question is addressed in Wikipedia. See Non-parametric statistics. Solarapex 23:14, 10 October 2007 (UTC)

[edit] Munaf Kernel is incorrect?

Something is wrong with this kernel. I could not find any references to this kernel. The current formula states:

K(u)=\frac{45}{64}(1-u^2)^4\ 1_{(|u|\leq1)}

But it does not satisfy the requirement:

\int K(u) du = 1

The one that satisfies this requirement is:

K(u)=\frac{315}{256}(1-u^2)^4\ 1_{(|u|\leq1)}

Otherwise, the Munaf kernel should be (to follow the ratio):

K(u)=\frac{45}{64}(1-u^4)^2\ 1_{(|u|\leq1)}

[edit] Generalization

In general, Epanechnikov, Quartic, Triweight, and Munaf can be generalized as:

K(u)=B (1-u^{2m})^n\ 1_{(|u|\leq1)} ,

where

B = \frac{1}{2 \times \sum_{j=0}^{n} C_j^n \frac{(-1)^j}{2mj + 1} } ,

C_j^n - the number of combinations.

I haven't seen this in books. So I have to warn you, this may be original research. Solarapex 23:14, 10 October 2007 (UTC)

Assuming this was incorrectly copied from a correct source, my money is on the formula (45/65)(1-u^4)^2. In light of the lack of sources and importance ("Munaf kernel" only gets a Google hit on this page), I've removed it from the article.  --Lambiam 11:07, 29 October 2007 (UTC)

[edit] Proposed merge of Kernel (statistics) and Kernel smoother

Isn't a merge with Kernel density estimation more appropriate?  --Lambiam 20:11, 16 March 2008 (UTC)

I think they are different things, although perhaps the redundancy between them could be reduced somehow. --Zvika (talk) 05:32, 17 March 2008 (UTC)
The procedure of kernel density estimation clearly is a form of kernel smoothing. Could you give a hint what you see as an essential difference?  --Lambiam 19:30, 17 March 2008 (UTC)
It seems to me that both the setting and the applications are different. In one case we have want to estimate a function from noisy measurements, and in the other we want to estimate a pdf from iid realizations of its random variable. The technique itself is similar. However, I am not an expert on this, so barring any further resistance, go ahead and do what you think is right. --Zvika (talk) 07:03, 18 March 2008 (UTC)

In kernel density estimation you are actually trying to estimate the number of measurements inside each window, when with kernel smoother you are estimating the values of data points (and not the number of data points). In some sense, both techniques are similar (density estimation is a particular case of kernel smoother, when all the measurement points has the value 1). Anyway, I think that kernel density estimation is an important topic, and it should have its own article --Anry11 (talk) 17:13, 18 March 2008 (UTC)

There is also the page density estimation to consider for overlap. Melcombe (talk) 11:56, 9 April 2008 (UTC)

[edit] Missing inverse in the definition

The last line of the definition states that "If K is a kernel, then so is the function K* defined by K*(u) = λ−1K(λu), where λ > 0." Isn't the λ "inside" the function supposed to be inverted as well? I'm pretty sure that this is the case but since I haven't written here before I don't dare to change it myself. —Preceding unsigned comment added by 217.10.116.172 (talk) 21:57, 15 April 2008 (UTC)

No, I think the definition is correct as it is: this is the way a scaling factor is used. However, please don't hesitate in the future to make changes yourself (even if you are not sure of them); in the worst case, someone will come along and correct you. --Zvika (talk) 04:05, 16 April 2008 (UTC)
This was indeed an error. I've fixed it.  --Lambiam 05:06, 24 April 2008 (UTC)
You're right. I must have been confused that day :) --Zvika (talk) 10:46, 24 April 2008 (UTC)