Posts tagged with local linear approximation

A jet can be thought of as the infinitesimal germ of a section of some bundle or of a map between spaces.

Jets are a coordinate-free version of … Taylor series.

Michael BächtoldDavid CorfieldUrs Schreiber

 

Pictorial glossary

Bundle:

Vector Bundle Construction
image
image

Sections:
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imagepicture that illustrates the research done

Mapping between spaces:




memeengine replied to your post: Or can the influence of any ancestor ever fade down to zero? (Or, well… to arbitrarily small size?)

An AR[1] process can go arbitrarily ↓0 as time↑∞. But in real life the sins of the fathers set their childrens’ teeth on edge in a stiffer, more heriting, architectural way. (I was alluding to that with the new-parents post scriptum.)

I don’t know of any “grande classes of models” wherein you stay stuck from where the past put you, but for example you could say

  • if ($income >= 10 guineas) { $opportunities = 1000; } else { $opportunities = 3; }.
 

I’m neither an expert on history nor do I find the econometrics super compelling, but ∃ some theories of the past carving out a channel for the future.

  • In central Africa — the form of governance from 1000, 1500, 1800 AD still has an influence on GDP per capita today. Even once you statistically control for other “obvious” determinants of production power. (Here I learned the interesting term “ethnofract”—a measure of ethnic disparity.)
  • In the southeastern US — a county’s history of slave-holding has statistical relevance to. (Nathan Nunn is a co-author.)
  • In socialist Sweden — the class mobility from generation to generation is greater than in free-market United States.
  • According to Arvind Thornton, north-western European social norms of family size, structure, and intra-/inter-relationships set the stage for “industrial revolution” type anti-Malthusian family structures that inform major subcultures all over the OECD today. I.e., a small nuclear family wherefrom kids form their own families in a separate home; high value placed upon individualism; etc. 
  • In British-colonial Jamaica and French-colonial Haiti, an oligarchic political form in the 1700’s passed on poverty to its ‘fterbears. Sugar plantations financed European vacations, fine liberal educations, and leisure for the elites, but sugar is not an investment in the future. Indeed, growth markets might have overthrown the political structure by empowering the hands, so the positive-sum games were forestalled by the landed interests. (Story can be found on Daron Acemoglu’s website.)
  • (A similar argument has been applied to Europe in the Needham question: why did Europeans dominate the globe rather than Asians, when the Asians were ahead earlier on in the race? Perhaps because of the over-powerful Chinese government.)
  • In Louisiana, USA — juridical forms differ from the other 49 States. Since Louisiana’s French colonial history bequeathed it a civil-law rather than a common-law system of justice, not just its laws but the underlying reasoning for how they’re executed, differs orthogonally to other interstate legal variations.
  • Come to think of it: any common-law system, by design, to carve a river that the future will follow.

In all of the statistical examples we’ve got to ask if it’s possible to statistically control for parameter changes. To which the correct answer is: No. Well, maybe. Um, in a local sense any parameter change can be estimated as linear. If the underlying function is ≥once-differentiable. So, err. I’ll have to get back to you on that.

(I’ll look up paper links later…if you as a reader know the papers or related ones you could also do me the favour and post links in the Reply or Disqus Comments. =) )




"It’s easy to learn calculus and then forget what the point was.”
—Gilbert Strang

hi-res




The chief triumph of differential calculus is this:

Any nonlinear function can be approximated by a linear function.

(OK…pretty much any nonlinear function.) That approximation is the differential, aka the tangent line, aka the best affine approximation.  It is valid only around a small area but that’s good enough. Because small areas can be put together to make big areas. And short lines can make nonlinear* curves.

In other words, zoom in on a function enough and it looks like a simple line. Even when the zoomed-out picture is shaky, wiggly, jumpy, scrawly, volatile, or intermittently-volatile-and-not-volatile:

Fed funds rate history since 1990 -- back to 1949 available at www.economagic.com

Moreover, calculus says how far off those linear approximations are. So you know how tiny the straight, flat puzzle pieces should be to look like a curve when put together. That kind of advice is good enough to engineer with.

 

It’s surprising that you can break things down like that, because nonlinear functions can get really, really intricate. The world is, like, complicated.

So it’s reassuring to know that ideas that are built up from counting & grouping rocks on the ground, and drawing lines & circles in the sand, are in principle capable of describing ocean currents, architecture, finance, computers, mechanics, earthquakes, electronics, physics.

image

(OK, there are other reasons to be less optimistic.)


 

 

* What’s so terrible about nonlinear functions anyway? They’re not terrible, they’re terribly interesting. It’s just nearly impossible to generally, completely and totally solve nonlinear problems.

But lines are doable. You can project lines outward. You can solve systems of linear equations with the tap of a computer.  So if it’s possible to decompose nonlinear things into linear pieces, you’re money.

 

Two more findings from calculus.

  1. One can get closer to the nonlinear truth even faster by using polynomials. Put another way, the simple operations of + and ×, taught in elementary school, are good enough to do pretty much anything, so long as you do + and × enough times. 

  2. One can also get arbitrarily truthy using trig functions. You may not remember sin & cos but they are dead simple. More later on the sexy things you can do with them (Fourier decomposition).