Wikipedia:Reference desk/Archives/Mathematics/2022 December 19

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

December 19

The page Conversion between quaternions and Euler angles says it deals with JPL quaternions, and I think it's not quite right

At the beginning of the section Definition, it's written "For the rest of this article, the "passive" JPL quaternion convention shall be used." There are two problems with this:

1) I don't think this is right, both the rotation matrix and the quaternion product formulas seem to be consistent with Hamilton matrices instead. Can someone else help confirming for me if this is in fact classical Hamilton quaternions or really the JPL convention?

2) The page Quaternions and spatial rotation makes very valid points on why Hamilton quaternions should be favored, so using only Hamilton quaternions here should be preferred.

PS.: I already made a comment in the talk section of this page. Ebernardes (talk) 11:30, 19 December 2022 (UTC)

I had not heard of this JPL convention before, and the article does not do a decent job of explaining it. (I guess this ought to be done either in the main Quaternion article or in a separate article. Assuming that the algebras are isomorphic, the isomorphism should be spelled out explicitly.) The Conversion article is IMO also lacking in another respect. There are two, dual approaches to thinking about rotations. In one you take a vector in the "before" coordinate system and you transform it into its representation in the "after" coordinate system. In a sense, you are rotating the coordinate systems; the vector stays put. In the other, there is only one coordinate system, and the vector rotates. The article is not explicit about the viewpoint. Also, unless the isomorphism between JPL and Hamilton ia trivial, I think it would be reasonable to describe the transformation under both conventions.  --Lambiam 21:13, 19 December 2022 (UTC)

I think it should just use Hamilton's system rather than confusing matters. There can be a small paragraph referring to JPL but using it generally is not in conformance with Wikipedia following the main sources. I agree with the business of making certain it is very clear what is being described and doing both but I would have thought that the change in the vector would be the main use. I've had enough problems with covariant and contravariant to know one has to be clear about things like this! NadVolum (talk) 13:36, 20 December 2022 (UTC)

Either viewpoint has its legitimate uses. When dealing with a jet propelled about in space the equations must handle the pitch, roll and yaw of the jet's axes, but for deriving the equations for the Coriolis force the change of coordinate system is pivotal.  --Lambiam 17:08, 20 December 2022 (UTC)

To be pedantic, roll–pitch–yaw are Tait–Bryan angles rather than Euler angles. But everyone calls them Euler angles. There are so many equally valid conventions and concomitant ambiguities in this area (say, the order in which you apply RPY), what's one more?

Anyway, the JPL thing is another matter — it's actually a different convention for quaternion arithmetic, not just their interpretation in terms of rotations. It's based on the observation that you can get essentially the same structure by taking ijk = 1 instead of ijk = −1, and someone thought that that way of presenting them would be "better" somehow. Predictably, it never caught on outside JPL itself, and if I had to guess I would bet JPL people first of all wish they'd just left it alone; they're probably extremely tired of figuring out whether a given usage is internal, using their convention, external using the normal convention, or external but someone has helpfully translated it for them. Fair warning to the tauists, though the fact that they've at least chosen a different letter is an enormous step to the good. --Trovatore (talk) 17:21, 20 December 2022 (UTC)

Or one can just have kji = -1 for the JPL version, ;-) Yes unless one has a very very good reason it's silly to have two different but similar conventions. The two conventions for spherical coordinates have given enough trouble and at least there it is just the order of the coordinates. NadVolum (talk) 20:46, 20 December 2022 (UTC)

when is it not “purely coincidental” anymore?

Hi! How do the insurance mathematicians find out, if a series of car accidents is criminal intent? What about this example:

• since he saw the last episode with Raul in Emergency Room (series) somebody (let's call him X) wants to catch train drivers, that are dangerous... X wants to use data warehousing for that task, but everybody tells him, that he is insane and a security hazard... X is fascinated by trains like Sheldon Cooper...
• they even bring X in front of a judge via a video conference... X's jump suit says: "SOLO" at his left chest pocket and all the other detainees have blank jump suits...
• at X's 42nd b-day a train crashes... that train should have been decommissioned at the 21st d-day of X's father some weeks after the crash... the train number was 4U9525 and the 19th letter in the alphabet is “S” and the 15th is “O” and the 12th is “L”... the train driver was allegedly insane and left lots of funny data in the databases of the train company long before the crash (he missed the duedate for his mandatory medical exam, he complained about weird colors inside the loco after each shift, he reported several other funny technical damages, he configured the loco to various kinds of self destruct, ...)...

Can that insurance mathematics be used for that example, too? Thx. Bye. Homer Landskirty (talk) 20:39, 19 December 2022 (UTC)

Under certain assumptions, statisticians can compute the likelihood that a striking pattern will arise purely coincidentally. See Null hypothesis – encompassing the hypothesis that no foul play is involved – and Statistical hypothesis testing. The validity of this approach depends on the quality of the use of mathematical modelling for the relevant aspects of a usually confusing and murky jumble of supposedly known facts of reality. A statistician cannot (in their capacity as a mathematician) judge whether a model is adequate (no unwarranted assumptions of statistical independence!) and adequately applied (no cherry picking! beware of selection bias!); all they can hopefully do, combining their professional expertise with common sense, is point out that a model or its application appears to be inadequate. If the model is not appropriate, applying it for hypothesis testing can have tragic consequences and destroy lives – see the case of Lucia de Berk. Also, in no way has a statistician shown "criminal intent" if the null hypothesis is rejected (which in a criminal case requires a confidence level that is far higher than commonly used in statistical hypothesis testing). They have shown that an observed pattern was unlikely to arise by pure chance, but patterns also arise from non-random processes without underlying criminal intent. The task of constructing a plausible mathematical model for the events of the given example, let alone an adequate one, appears beyond hope to me.  --Lambiam 21:47, 19 December 2022 (UTC)

Yes, all statistics can do is raise a suspicion that there may be malicious activity and without further investigation as to causes all you can say is that you have an "outlier". On top of this, the vast majority of people don't know how to correctly apply statistical reasoning, or even what constitutes valid statistical reasoning. Because of this, what may seem to be compelling circumstantial evidence against a suspect may in fact be a simple probabilistic fallacy. For example, someone is murdered and evidence shows that the murderer has AB- blood type. An acquaintance of the victim has AB- blood type and is arrected and put on trial. The prosecutor reasons something like this, the chances of someone having AB- blood type is 1%, so the chance of two people, the suspect and the murderer, having this blood type is .01%. But the reasoning is false; since the murderer is known to have the blood type, that probability is 100%, and so the probability that a random acquaintance having the blood type is still 1%. Further, since victim may have had many acquaintances, say 30, that brings the probability that one of them has the blood type up to 30%, which makes the so-called coincidence more of a commonplace occurrence. Even when an outlier is not a coincidence there may be an innocent explanation. For example by examining prescription data it may be found that a certain doctor prescribes opiates at twice the average rate of other doctors in the same area. But if the doctor specializes in providing palliative care to the terminally ill, the explanation is clear. There is also just plain statistical variation which, even if unlikely, will occur with high probability if there are a larger number test cases. For example if you want to find people with ESP you might administer a test and select those people where their success rate at guessing pictures on cards has a .01% chance of occurring by chance. This may seem like a valid test, but if you give the test to 20,000 people then the chances are that at least one of will pass and appear to have ESP. This kind of spurious statistical reasoning is common in pseudoscience and conspiracy theories, and one reason they become popular is that many people aren't able to distinguish between a valid statistical argument and a fallacious one. A classic example of this is the supposed connection between Vaccines and autism. Finally, statistical inference is easily thrown off by so-called black swan events. In other words an outlier can occur because of an unusual circumstance that isn't accounted for in any preexisting model. For example you might look at drowning statistics with the assumption that they are unrelated individual events. But the Titanic, SS Eastland, General Slocum and similar maritime disasters show that this is not a valid assumption. --RDBury (talk) 01:30, 20 December 2022 (UTC)

Bayesian inference may be relevant. catslash (talk) 02:15, 20 December 2022 (UTC)

Apart from the difficulty of establishing a prior, this road just as much requires an adequate model.  --Lambiam 09:40, 20 December 2022 (UTC)

do u mean something like this? In germany they tried to find out, if nuclear power plants cause tumors by using historical wind and pollution and vacation data in order to find out, what the individual radiation dose might have been... if they only used a circle with the plant in the middle, the tumor rate was (almost) normal... Homer Landskirty (talk) 18:51, 20 December 2022 (UTC)

It sounds like one would expect a fairly typical application of classical frequentist inference in the style of Fisher. Select a number of areas that have similar and typical characteristics, except that they can be divided into two classes: those that are downwind from a nuclear power plant, and those that are not. On a priori grounds one expects in either case the incidence of tumors to follow a Poisson distribution. One can then use any of several tests (see the answers here) to test whether the rate (adjusted for the size of the population in the areas) in the first class is significantly higher than that in the second class.  --Lambiam 20:01, 20 December 2022 (UTC)

See Insurance fraud #Detecting insurance fraud for relevant references: "The first step is to identify suspicious claims that have a higher possibility of being fraudulent. ... Regardless of the source, the next step is to refer these claims to investigators for further analysis. ...many companies use computers and statistical analysis to identify suspicious claims for further investigation. ...suspicious claims are identified by comparing data about the claim to expected values. "

"Once is happenstance. Twice is coincidence. Three times is enemy action."

- Auric Goldfinger, in "Goldfinger" by Ian L. Fleming (1908-1964) Modocc (talk) 17:39, 20 December 2022 (UTC)

so the references to (1) b-day, (2) d-day and (3) SOLO prove hostility? or are the links too weak and indirect...?

what about the remainder of the modus operandi? (ignoring all the hints since weeks or even months... who knows if they even gave that train driver a worrisome phone call before his last tour...) Homer Landskirty (talk) 10:04, 21 December 2022 (UTC)

I haven't followed your characters and games. They remind me of role-playing though in which some randomness is included (with dice). Modocc (talk) 17:49, 21 December 2022 (UTC)

so... it might be, that a lot of people had 3 or more properties, that seem to be related to that accident... e. g. about 400.000 people had their 42nd b-day at that day (where “42” has a special meaning via The Hitchhiker′s Guide to the Galaxy... maybe many other numbers from 1 to 99 have a special cultural meaning...)... and maybe even more people have a property that can be easily converted into “9525” or that is related to 2015-04-23 or to some other prominent property of that train... and maybe many people would like to be a data-firefighter or secret agent... Homer Landskirty (talk) 18:51, 20 December 2022 (UTC)

By the creative application of numerology, the dedicated practioner can conjure up a fantastic buffet of the most incredible coincidences out of thin air.  --Lambiam 20:11, 20 December 2022 (UTC)

ok... is there a method to distinguish “numerology” from “stochastics”? Homer Landskirty (talk) 09:50, 21 December 2022 (UTC)

One is incoherent hogwash based on irrational beliefs, the other a mathematical discipline. Is the question how to distinguish incoherent hogwash from a mathematical discipline?  --Lambiam 13:01, 21 December 2022 (UTC)

yup... *nods head* 😋 Homer Landskirty (talk) 17:29, 21 December 2022 (UTC)

How can we prove that Joseph Biden is a Great Satan? Well, using A = 1, B = 2, ..., Z = 26, we calculate:

Joseph Biden = 10+15+19+5+16+8+2+9+4+5+14 = 107

a Great Satan = 1+7+18+5+1+20+19+1+20+1+14 = 107

But why does this calculation use the name in the form "Joseph Biden"? Why not "Joe Biden" or "Joseph Robinette Biden Jr."? And why "a Great Satan"? Why not "the Great Satan" or just "Great Satan"? The problem is that there are no rules; the numerologist can keep trying until a hit is found. By Littlewood's law, miraculous events are actually commonplace if one does not fix the rules in advance. For numerologists and conspiracy theorists alike, the rules are fluid; they are made up on the spot when convenient (which is when they signal something remarkable), and they are just as easily ignored when less convenient (the remarkable item was not remarkable after all). Humans tend to perceive meaningful connections between unrelated things; some only when smoking weed, some others all the time; see Apophenia. If the rules are not clear and can be changed at will, all bets are off, and any conclusions are worthless. In contrast, in a mathematical discipline, the rules are clearly defined and must be applied consistently.  --Lambiam 18:32, 21 December 2022 (UTC)

numerology = ambiguously unconvincing fuzziness (each = 145)

mathematics = precisely vindicating actuality (each = 112)

Certes (talk) 20:49, 21 December 2022 (UTC)

Data dredging can be applied to generate both hypotheses which you consider absurd, and hypotheses in which you confidently believe. You don't know that Joseph Biden isn't Beelzebub, or that his name isn't a sign to the cognoscenti. If you were presented with this claim, and had no prior opinion, how would you apply statistical testing to dismiss it? catslash (talk) 00:08, 22 December 2022 (UTC)

so it is not possible to prove a difference between a real accident (a train driver that suffers the blow-out of one of his hidden aneurysms, that nobody could knew of in advance...) and a mass murder (a train driver that is manipulated via secret phone calls from the bad guys, so that he blows-out exactly at the b-day of a «persona non grata», after the bad guys set the train's end-of-service-date to the d-day of the father of that «persona non grata» (which they presented in a “SOLO” saying jump suit to a federal judge), and the train's number to something like “FOR YOU SOLO”...)...? Homer Landskirty (talk) 06:17, 22 December 2022 (UTC)

That is a non sequitur. There are other means of proving a difference than based on wacky notions of "too much of a coincidence". Proving that a theory proposed by the defence holds no water is bread-and-butter for the prosecution in criminal cases. The detectives on a case may prove the existence of a mastermind by retrieving phone records, and so on and so forth. What one may consider a proof is based on evidence. It is impossible to prove that Hitler was not manipulated by telepathic antisemitic Ganymedians circling the Earth in UFOs. At some point one has to discard theories that are not supported by any evidence. This is not an issue of the (ab)use of statistics, so I consider the discussion closed.  --Lambiam 10:03, 22 December 2022 (UTC)