Wikipedia:Reference desk/Archives/Mathematics/2021 May 5

= May 5 =

Orientation of confusion matrix
asked me on Talk:Confusion_matrix about whether confusion matrices should have actual (ground truth) values as rows, and predicted values as columns, counter to the ground-truth-as-columns convention here on Wikipedia.

Danilosilva128 cited several references using the ground-truth-as-rows convention. Is anyone here aware of a reputable style guide that mandates one or the other?

Thanks, cm&#610;&#671;ee&#9094;&#964;a&#671;&#954; 15:14, 5 May 2021 (UTC)

P.S. Pinging


 * I looked at the first eight research papers I could find that used the term "confusion matrix" for which I had access to the full text. Of these, two followed the WP convention, the other six its transpose. Clearly, there is no strict standard. Aside: I do not like the term "predicted". It is not prediction, more guessing, so "guessed class" would be clearer. Or, since this is the class produced by a classification system as output, in response to a stimulus provided as input, perhaps "output class"? --Lambiam 17:03, 5 May 2021 (UTC)
 * Not a topic I'm familiar enough with to have an informed opinion, sorry. —David Eppstein (talk) 17:18, 5 May 2021 (UTC)
 * Textbooks too do not follow one standard. I see no clear reason to prefer one variant over the other, but it may be good to point out in the article that both variants are found in the literature. --Lambiam 19:55, 5 May 2021 (UTC)
 * Thanks, Lambiam and David Eppstein. I'll discuss separately with Danilosilva128 how to proceed. Good point about mentioning that both are found in literature. Cheers, cm&#610;&#671;ee&#9094;&#964;a&#671;&#954; 23:43, 6 May 2021 (UTC)

Is there a "name" for this triangle?
Consider a right triangle with side lengths 1/2 and 1, respectively. The hypotenuse is thus $$\sqrt{(1/2)^2 + 1^2} = 1.1180339887[...]$$. Now add 1/2 to that and you get the golden ratio! In other words, $$\sqrt{(1/2)^2 + 1^2} + 1/2 = \sqrt{5/4} + 1/2 = 1.6180339887[...]$$ What I'd like to know is if there a name for this particular triangle and moreover if any interesting properties are known of it? Earl of Arundel (talk) 21:52, 5 May 2021 (UTC)

The square of 1.118 is 1.249924. Pythagorean triples must be made of 3 integers, and in this case, multiply 1/2, 1, and 1.118 all by 1000 and you'll get 500, 1000, 1118; these can be halved to get 250, 500, 559. This clearly isn't a Pythagorean triple because it has 2 even legs and an odd hypotenuse. Left side is 312500; right side is 312481. Georgia guy (talk) 23:51, 5 May 2021 (UTC)


 * The 1.118[...] part (the square being 5/4) is actually irrational. So by definition it couldn't possibly be scaled to ANY Pythagorean triple to begin with.
 * Earl of Arundel (talk) 00:27, 6 May 2021 (UTC)
 * The golden ratio is (1+√5)/2, so yes, if you add 1/2 to the length of the hypotenuse (=√5/2) then you get the golden ratio. If memory serves then Euclid used such a triangle to construct the golden ratio and from that to construct a regular pentagon, but I don't know of any specific name given to the traiangle; if Euclid ever gave it a name it was probably lost to history. It's not hard to construct an infinite number of right triangles whose sides are integer combinations of 1 and φ, Pythagophinarean triples if you will. One such is (1, 2, 2φ-1) which is your triangle scaled by a factor of 2. Others are (2φ-1, 2, 3), (4, 3φ-4, 5φ-4) and (4φ+1, 4φ-2, 6φ-1). --RDBury (talk) 02:21, 6 May 2021 (UTC)
 * Interesting!
 * Earl of Arundel (talk) 06:39, 6 May 2021 (UTC)
 * Two more with only one irrational side and with single-digit integers only: (2, 6φ−3, 7) and (1, 8φ−4, 9). --Lambiam 08:47, 6 May 2021 (UTC)
 * No triangles were harmed in Euclid's construction. The construction from a a right triangle with side lengths 1/2 and 1 is shown in the "Golden Ratio in Geometry" article on Cut-the-knot, in the diagram below the text "". (The triangle's hypotenuse is the dotted line.) --Lambiam 06:14, 6 May 2021 (UTC)
 * Aha, so it was hiding in plain sight! Also found in the "Dividing a line segment by exterior division" subsection of Golden_ratio. Thanks again, Lambiam. :)
 * Earl of Arundel (talk) 06:39, 6 May 2021 (UTC)