isomorphismes

Cylinder = line-segment × disc

C = | × ●

The “product rule” from calculus works as well with the boundary operator ∂ as with the differentiation operator ∂.

∂C  =   ∂| × ●   +   | × ∂●

>Oops. Typo. Sorry, I did this really late at night! cos and sin need to be swapped back.

Oops. Another typo. Wrong formula for circumference.

I was re-reading Michael Murray’s explanation of cointegration:

and marvelling at the calculus.

Calculus blows my mind sometimes. Like, hey guess how much we can do with subtraction.

— protëa(@isomorphisms) March 28, 2013

Of course it’s not any subtraction. It’s subtracting a function from a shifted version of itself. Still doesn’t sound like a universal revolution.

(But of course the observation that the lagged first-difference will be zero around an extremum (turning point), along with symbolic formulæ the (infinitesimal) first-differences of a function, made a decent splash.)

Jeff Ryan wrote some R functions that make it easy to first-difference financial time series.

Here’s how to do the first differences of Goldman Sachs’ share price:

require(quantmod)
getSymbols("GS")
gs <- Ad(GS)
plot(  gs - lag(gs)  )

Look how much more structured the result is! Now all of the numbers are within a fairly narrow band. With length(gs) I found 1570 observations. Here are 1570 random normals plot(rnorm(1570, sd=10), type="l") for comparison:

Not perfectly similar, but very close!

Looking at the first differences compared to a Gaussian brings out what’s different between public equity markets and a random walk. What sticks out to me is the vol leaping up aperiodically in the $GS time series.

I think I got even a little closer with drawing the stdev’s from a Poisson process plot(rnorm(1570, sd=rpois(1570, lambda=5)), type="l")

but I’ll end there with the graphical futzing.

What’s really amazing to me is how much difference a subtraction makes.

Saying derivative is “slope” is a nice pedant’s lie, like the Bohr atom

which misses out on a deeper and more interesting later viewpoint:

The “slope” viewpoint—and what underlies it: the “charts” or “plots” view of functions as ƒ(x)–vs–x—like training wheels, eventually need to come off. The “slope” metaphor fails

• for pushforwards,
• on surfaces,
   
• on curves γ that double back on themselves
   
• my vignettes about integrals,
• and, in my opinion, it’s harder to “see” derivatives or calculus in a statistical or business application, if you think of “derivative = slope”. Since you’re presented with reams of numbers rather than pictures of ƒ(x)–vs–x, where is the “slope” there?

“Really” it’s all about diff’s. Derivatives are differences (just zoomed in…this is what lim ∆x↓0 was for) and that viewpoint works, I think, everywhere.

I half-heartedly tried making these drawings in R with the barcode package but they came out ugly. Even uglier than my handwriting—so now enjoy the treat of my ugly handwriting.

Step back to Descartes’ definition of a function. It’s an association between two sets. And the language we use sounds “backwards” to that of English. If I say “associate a temperature number to every point over the USA”

then that should be written as a function ƒ: surface → temp.,

(or we could say ƒ: ℝ²→ℝ with ℝ²=(lat,long) )

The \to arrow and the “maps to” phrasing are backwards of the way we speak.

• “Assign a temperature to the surface” —versus— “Map each surface point to a temperature element from the set of possible temperatures”.

{elf, book, Kraken, 4^π^e} … no, I’m not sure where that came from either. But I think we can agree that such a set is unstructured.

Great. I drew above a set “without other structure” as the source (domain) and a branched, partially ordered weirdy thing as the target (codomain). Now it’s possible with some work to come up with a calculus like the infinitesimal one on ℝ→ℝ functions that’s taught to many 19-year-olds, but that takes more work. But for right now my point is to make that look ridiculous and impossible. Newton’s calculus is something we do only with a specific kind of Cartesian mapping: where both the from and the to have Euclidean concepts of straight-line-ness and distance has the usual meaning from maths class. In other words the Newtonian derivative applies only to smooth mappings from ℝ to ℝ.

Let’s stop there and think about examples of mappings.

(Not from the real world—I’ll do another post on examples of functions from the real world. For now just accept that numbers describe the world and let’s consider abstractly some mappings that associate, not arbitrarily but in a describable pattern, some numbers to other numbers.)

(I didn’t have a calculator at the time but the circle values for [1,2,3,4,5,6,7] are [57°,114°,172°,229°,286°,344°,401°=41°].)

I want to contrast the “map upwards” pictures to both the Cartesian pictures for structure-less sets

and to the normal graphical picture of a “chart” or “plot”.

Notice what’s obscured and what’s emphasised in each of the picture types. The plots certainly look better—but we lose the Cartesian sense that the “vertical” axis is no more vertical than is the horizontal. Both ℝ’s in ƒ: ℝ→ℝ are just the same as the other.

And if I want to compose mappings? As in the parabola picture above (first the square function, then an affine recentering). I can only show the end result of g∘ƒ rather than the intermediate result.

Whereas I could line up a long vertical of successive transformations (like one might do in Excel except that would be column-wise to the right) and see the results of each “input-output program”.

(Además, I have a languishing draft post called “How I Got to Gobbledegook” which shows how much simpler a sequence of transforms can be rather than “a forbidding formula from a textbook”.)

Another weakness of the “charts” approach is that whereas "Stay the same" command ought to be the simplest one (it’s a null command), it gets mapped to the 45˚ line:

Here’s the familiar parabola / x² plot “my way”: with the numbers written out so as to equalise the target space and the source space.

Now the “new” tool is in hand let’s go back to the calculus. Now I’m going to say “derivative=pulse” and that’s the main point of this essay.

Considering both the source ℝ→ and the target →ℝ on the same footing, I’ll call the length of the arrows the “mapping strength”. In a convex mapping like square the diffs are going to increase as you go to the right.

OK now in the middle of the piece, here is the main point I want to make about derivatives and calculus and how looking at numbers written on the paper rather than plots makes understanding a push forward possible. And, in my opinion, since in business the gigantic databases of numbers are commoner than charts making themselves, and in life we just experience stimuli rather than someone making a chart to explain it to us, this perspective is the more practical one.

I’m deliberately alliding the concepts of diff as

• difference
• R’s diff function
• differential (as in differential calculus or as in linear approximation)

Linear

I’m still not quite done with the “my style of pictures” because there’s another insight you can get from writing these mappings as a bar code rather than as a “chart”. Indeed, this is exactly what a rug plot does when it shows histograms.

Here are some strip plots = rug plots = carpet plots = barcode plots of nonlinear functions for comparison.

The main conclusion of calculus is that nonlinear functions can be approximated by linear functions. The approximation only works “locally” at small scales, but still if you’re engineering the screws holding a plane together, it’s nice to know that you can just use a multiple (linear function) rather than some complicated nonlineary thingie to estimate how much the screws are going to shake and come loose.

For me, at least, way too many years of solving y=mx+b obscured the fact that linear functions are just multiples. You take the space and stretch or shrink it by a constant multiple. Like converting a currency: take pesos, divide by 8, get dollars. The multiple doesn’t change if you have 10,000 pesos or 10,000,000 pesos, it’s still the same conversion rate.

So in a neighborhood or locality a linear approximation is enough. That means that a collection of linear functions can approximate a nonlinear one to arbitrary precision.

That means we can use computers!

Square

I can’t use the example of self times self so many times without exploring the concept a bit. Squares to me seem so limited and boring. No squizzles, no funky shapes, just boring chalkboard and rulers.

But that’s probably too judgmental.

After all there’s something self-referential and almost recursive about repeated applications of the square function. And it serves as the basis for Euclidean distance (and standard deviation formula) via the Pythagorean theorem.

How those two are connected is a mystery I still haven’t wrapped my head around. But a cool connection I have come to understand is that between:

• a variety of inverse square laws in Nature
• a curve that is equidistant from a point and a line
• and the area of a rectangle which has both sides equal.

I guess first of all one has to appreciate that “parabola” shouldn’t necessarily have anything to do with x•x. Hopefully that’s become more obvious if you read the sections above where I point out that the target ℝ isn’t any more “vertical” than is the source ℝ.

The inverse-square laws show up everywhere because our universe is 3-dimensional. The surface of a 3-dimensional ball (like an expanding wave of gravitons, or an expanding wave of photons, or an expanding wave of sound waves) is 2-dimensional, which means that whatever “force” or “energy” is “painted on” the surface, will drop off as the square rate (surface area) when the radius increases at a constant rate. Oh. Thanks, Universe, for being 3-dimensional.

What’s most amazing about the parabola—gravity connection is that it’s a metaphor that spans across both space and time. The curvature that looks like a-plane-figure-equidistant-to-a-line-and-a-point is curving in time.

Keeping tabs on storm surges at tidesonline.noaa.gov and not liking what I’m seeing. That air pressure graph….

— Charlie Loyd (@vruba) October 29, 2012

(I had to look up MLLW—it means mean lower low water. As in there’s a lower high tide, a higher low tide, a lower low tide, and a higher high tide.) Normal tide cycle from Wikipedia:

For comparison, a gauge not experiencing a major storm right now: http://tidesonline.noaa.gov/plotcomp.shtml?station_info=9449880+Friday+Harbor,+WA#

— Charlie Loyd (@vruba) October 29, 2012

The GIF near the end of en.wikipedia.org/wiki/Storm_sur… shows a modeled storm surge dramatically.

— Charlie Loyd (@vruba) October 29, 2012

The number of isobar lines on hpc.ncep.noaa.gov/sfc/lrgnamsfcw… should give some sense of how unusually low that is.

— Charlie Loyd (@vruba) October 29, 2012

It’s dated 1500Z = 11:00 a.m. EDT.

— Charlie Loyd (@vruba) October 29, 2012

Want to see what a hurricane looks like on a seismometer? You’re seeing the ocean wave energy traveling to PA. twitter.com/volcanoclast/s…

— Ian Saginor (@volcanoclast) October 30, 2012

(Source: basecase.org)