Talk:Multimodal distribution

It might be nice to split up the section on bimodale indices, as there are a few by now and it's not obvious on a first glance, where one ends and the next begins. (I would do it myself, but I barely know anything about the wikipedia mark-up language.) — Preceding unsigned comment added by 193.174.18.4 (talk) 11:09, 29 June 2015 (UTC)

I added nl:Bimodale distributie for the Dutch version to the article, but it doesn't show up? That's strange...

It's there. It worked. Nederlands is listed in the sidebar. Deco 19:49, 20 June 2006 (UTC)

What happens when we have two local maxima, but these maxima are different? For example, the experiment "roll a d6, if d6 = 1, then sample from a N(100,1), else sample from a N(0,1)". I would call this bimodal, even though the peak near x = 100 is 1/5 higher than the peak near x = 0. Albmont 10:09, 2 January 2007 (UTC)

Yes, I think it is misleading that Figure 1 shows the two peaks as being equal. I made it clear in the text at least that we are only talking about local maxima. I also linked to mode (statistics) and edited that page to note that local as well as global maxima of a pdf are commonly referred to as modes. Eclecticos 03:15, 21 January 2007 (UTC)

The article says (more or less) that $X=\alpha Y+(1-\alpha )Z$ is usually a bimodal distribution. This is not precise: the correct way is that the variable whose density is $f_{X}=\alpha f_{Y}+(1-\alpha )f_{Z}$ is usually bimodal. Albmont 14:28, 1 February 2007 (UTC)

Also does a bimodal distribution have to be continuous, it can also be discrete, right?

Sex and height probably not a good example

The article currently states: "A good example [of a bimodal distribution] is the height of a person. The heights of males form a roughly normal distribution, as do those of females. Each of these distributions is unimodal. However, if we plot a single histogram of the entire population, we see two peaks—one for males and one for females." I suspect the statement about there existing two peaks in the overall histogram is empirically false in this case of the heights of women and men. The effect size article demonstrates the use of Cohen's d and Hedges's ĝ by stating the effect size of the sex difference in height as calculated from data from a UK sample, giving values of d=1.72 and ĝ=1.76. However, if you plot the sum of two normal distributions with equal variances and means separated by 1.76 standard deviations (try it), there is in fact only one local maximum; you don't really see bimodality in the overall population distribution unless the means of the sexes are separated by two or more standard deviations. I suggest that a better example be chosen to illustrate this article, unless the data in the effect size article is wrong (in which case that article should be edited), or I am somehow mistaken. Z. M. Davis 17 December 2007 (UTC)

I agree. Note that the sum of two shifted Gaussian distributions with the same standard deviation will be unimodal unless the means differ by more than the standard deviation since mean +/-σ are the inflection points of the Gaussian distribution. —Ben FrantzDale (talk) 15:37, 30 January 2008 (UTC)

I found a published article showing you're both entirely correct, so I've just edited the article accordingly, added the ref and given some other examples instead. Qwfp (talk) 16:30, 17 September 2008 (UTC)

Sum of Normals

The terminology "Sum of Normals" is used here on the talk page, and ALSO in the article, when Mixture of Normals is meant. The sum of two normals is always a normal distribution which is always UNIMODAL, whereas a Mixture of normals is usually bimodal. Can someone double check and correct? — Preceding unsigned comment added by Asaduzaman (talk • contribs) 07:31, 5 July 2015 (UTC)

References

I tried to edit the References page but could not. The article:

Ray S, Lindsay BG (2005) The topography of multivariate normal mixtures. Ann Stat 33 (5) 2042–2065

Is available from following link: https://projecteuclid.org/euclid.aos/1132936556 but I could not insert this link because when I click on edit the page comes up nearly blank Asaduzaman (talk) 08:04, 5 July 2015 (UTC)

Measure?

Is there a quantitative measure of bimodalness or multimodalness? Kurtosis would tell us something, I suppose... —Ben FrantzDale (talk) 15:37, 30 January 2008 (UTC)

Links to this page?

I tried to find this page (because I couldn't remember it's name) via other pages in statistics in wikipedia, and nothing came up. It's not even mentioned in Continuous probability distribution or the Statistics navbox. Maybe we could link this page from other pages in order to interconnect it to them? Rhetth (talk) 22:38, 4 August 2008 (UTC)

what is a bimodal distribution?

It says that it is a distribution with two modes. However, the multimodal figure gives a distribution with 4 peaks but 2 of them are well higher than the other two. So my question is this, does it really have to have many modes, or just many local maxima? Or perhaps something else? Isn't a mode the largest peak? MATThematical (talk) 00:17, 8 March 2010 (UTC)

Ants

I deleted the nice graphic as there seems to be a scale error with it - basically the decimal points have gone AWOL.

Figure 2. Histogram of body lengths of 300 weaver ant workers.

Nick Connolly (talk) 05:11, 15 April 2010 (UTC)

Well spotted. Now fixed. Qwfp (talk) 07:35, 15 April 2010 (UTC)

sooper-dooper.Nick Connolly (talk) 04:49, 16 April 2010 (UTC)

References

Pearson Test mis-stated?

The Pearson test is written as kurtosis -(skew)^2 > 1, or equivalently kurtosis > skew^2+1 . However, just above on the page, the Sarles coefficient , (skew^2+1)/kurtosis , is described as "The logic behind this coefficient is that a bimodal distribution will have very low kurtosis, an asymmetric character, or both - all of which increase this coefficient." . The Pearson test as written is exactly the opposite of this statement.

Can someone clarify the situation? Cellocgw (talk) 15:28, 23 August 2013 (UTC)

Self-contradictory wording on equal variances

Currently a section says:

===Mixture of two normal distributions with equal variances===

In the case of equal variance, the mixture is unimodal if and only if

$d\leq 1$

or

$\left\vert \log(1-p)-\log(p)\right\vert \geq 2\log(d-{\sqrt {d^{2}-1}})+2d{\sqrt {d^{2}-1}}$

where p is the mixing parameter and d is

$d={\frac {\left\vert \mu _{1}-\mu _{2}\right\vert }{2{\sqrt {\sigma _{1}\sigma _{2}}}}}$

where μ₁ and μ₂ are the means of the two normal distributions and σ₁ and σ₂ are their standard deviations.

This starts out by assuming equal variances and then gives a formula with potentially different variances. This should be made consistent one way or the other. Loraof (talk) 21:44, 23 June 2017 (UTC)

Should this be "jointly" normal?

The section Multimodal distribution#Mixture of two normal distributions says

It is not uncommon to encounter situations where an investigator believes that the data comes from a mixture of two normal distributions. Because of this, this mixture has been studied in some detail.[17]

Should this be

==Mixture of two marginal distributions from a joint normal distribution==

It is not uncommon to encounter situations where an investigator believes that the data comes from a mixture of two marginal distributions from a jointly normal distribution. Because of this, this mixture has been studied in some detail.[17]

Not all pairs of normally distributed variables share the nice properties of pairs of variables that are jointly normally distributed. Loraof (talk) 22:07, 23 June 2017 (UTC)

I think that it doesn't need to be jointly normally distributed, since we don't draw from both distributions at the same time. Is this right? Loraof (talk) 00:17, 24 June 2017 (UTC)

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Link to MAP-Test wrong

Under https://wiki.riteme.site/wiki/Multimodal_distribution#General_tests the MAP-Test links to a test of the same name but for testing the flushing of toilets. — Preceding unsigned comment added by 194.147.52.101 (talk) 09:44, 22 July 2020 (UTC)

Chaudhuri and Agrawal index is incorrectly attributed

Resolved

I don't see why this index is being attributed to Chaudhuri and Agrawal. As they note in the linked reference, the relevant portion is a standard application of Otsu's method, which consists of optimizing the indicated quantity. The index is not original to them, and they have not claimed that it is. AldenMB (talk) 01:51, 11 July 2022 (UTC)

I went ahead and changed the relevant section. AldenMB (talk) 18:10, 7 August 2022 (UTC)