Wikipedia:Reference desk/Archives/Mathematics/2015 August 30

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August 30

ROC curves

I would appreciate your feedback about whether my understanding of ROC-curves is correct. I also have a couple of questions at the end. As this post clearly shows, my mathematical capabilities are limited, so please be gentle, and cautious about introducing terminology that I might have trouble understanding. If necessary, please translate my statements into more conventional terminology.

In my understanding, a ROC curve is a plot of true positive rates (TPR, Y-axis) vs false positive rates (FPR, X-axis), when the cutoff (CO) between what is considered a positive and a negative observation is varied such that it covers all reasonable values. The tangent at a given point (CO), can be estimated as

ΔTPR(CO)/ΔFPR(CO),

hence TPR(CO)/FPR(CO) is the derived function of the ROC curve, and the ROC-curve the antiderivative of the function TPR(CO)/FPR(CO).

In the following, I'll use the ROC curve in a medical context, and let 'm' represent the observed value of a diagnostic test, for which we have a ROC curve availabe. Then

TPR(CO)/FPR(CO) = p(m = CO ± eps|Disease)/p(m = CO ± eps|No disease) = the likelihood ratio function.

Q1: Am I right in thinking that the probability ratio in the previous line should be the probablity of 'm' being close to CO (in my notation ± eps), and not greater than CO?

Q2: I've read a couple of places (such as here: Choi BC (1998) Am J Epidemiol 148:1127–32. PMID 9850136) that it is valid to draw lines between several points on the ROC curve, say corresponding to "negative", "weak positive" and "strong positive", and that the slope of each line is a valid estimate of the likelihood ratio for test results that fall within the corresponding interval. Sounds reasonable, but exactly why is that so?

Q3: I've read many places (including in our article) that the area under the curve corresponds to the probability of a randomly chosen diseased individual getting a higher test result than a randomly chosen individual without the disease. Again, this sounds reasonable, but exactly why is it so?

Thanks, NorwegianBlue ^talk 14:32, 30 August 2015 (UTC)

help me coin a term

I dabble in programming. I stink at coming up with names for my variables and functions, because my thinking about the concepts is mostly nonverbal.

One of my projects involves fitting an arc to the tangent angles and curvatures of given endpoints. Curvature, as you know, is the first derivative of tangent angle with respect to arc length.

I've now found a good reason to adapt the algo to allow for a varying number of constraints, either adding derivatives beyond curvature or omitting curvature (and sometimes angle) at either end or both.

Accordingly, instead of separate fields named theta (angle) and kappa (curvature), I need a name for a list (of varying length) that begins [θ, κ ...]. What would you call this list?

The first coiner of the winning name (unless it's thetalist) will be permanently remembered in a comment in the code. —Tamfang (talk) 23:19, 30 August 2015 (UTC)

• While we're here, does the first derivative of curvature (with respect to arc length) have a common name? —Tamfang (talk) 23:54, 30 August 2015 (UTC)

I remember a college prof calling change of acceleration, "jerk", like pushing an accelerator pedal quickly. If we consider motion in a circle as constant acceleration, then changing the curvature could be a "jerk" function, like turning the steering wheel quickly? Tom Ruen (talk) 00:06, 31 August 2015 (UTC)

OH! It's real! Jerk_(physics) and next comes Jounce?! Tom Ruen (talk) 00:07, 31 August 2015 (UTC)

Knot theory has Writhe, which is of course not exactly what your list is conveying, but you might find the term evocative. SemanticMantis (talk) 01:44, 31 August 2015 (UTC)

• Maybe trend, or some nautical metaphor to go along with others in the code (bow, port, stern, starboard). —Tamfang (talk) 05:42, 31 August 2015 (UTC)

• I'd call the list a jet. Sławomir
  Biały 10:15, 31 August 2015 (UTC)

• I'd rather use a word from another semantic domain entirely than an almost-accurate word from the same domain. You get partial credit for starting the mental chain that led to rudder which is the front-runner for now. —Tamfang (talk) 21:25, 31 August 2015 (UTC)

• Btw, you can define a curve by tangential angle or curvature as a function of arc length, but more often than not there are simpler ways to do it. There are notable exceptions such as the circle, the cycloid, epi- and hypocycloids, and the catenary, but even for an ellipse finding curvature as a function of arc length is difficult. On the other hand, curves defined by a simple relationship between curvature and arc length can be complex and hard to compute, for example the Euler spiral (aka Cornu spiral). What you're really talking about is expressing the Whewell equation as a power series. The Whewell equation and other types of so-called intrinsic equations have applications in the solution of certain physical problems, but for computer graphics it's apparently more practical to use compound polynomial approximations as in Bézier splines. --RDBury (talk) 19:55, 31 August 2015 (UTC)

• I'm getting satisfactory results with polynomial spirals (of which the Euler spiral is the quadratic case): $z(t)=z_{0}+\textstyle \int _{0}^{t}e^{if(s)}\,ds$, implemented as Taylor series — and then, for graphic output, approximated with cubics – or, put another way, the fancy stuff is there to help me design the cubics. Fortunately it's not a realtime application. (It uses complex time!) —Tamfang (talk) 21:25, 31 August 2015 (UTC)

I would collectively call them all constraints. StuRat (talk) 05:21, 1 September 2015 (UTC)

• Ah, but these are not the only constraints in the project. —Tamfang (talk) 08:15, 1 September 2015 (UTC)

• Well, if the location, angle, curvature, etc., are all specified for specific points, then those can be called "point constraints". If the angle, curvature, etc. are specified, but without a specific location, those can be called "floating constraints". Where the constraints are at the end of a curve, they can also be called "end constraints". Hopefully any other constraints on the project can be given names that contrast with these StuRat (talk) 17:03, 4 September 2015 (UTC)

• What would you call the constraint(s) that the arc must pass within a certain distance of some points and not within the same distance of other points? (These constraints are more permanent than the curvature &c.) —Tamfang (talk) 09:11, 5 September 2015 (UTC)