A piecewise linear function in two dimensions (top) and the convex polytopes on which it is linear (bottom).
(Source: Wikipedia)
hi-res
Posts tagged with linear
hi-res
SETUP (CAN BE SKIPPED)
We start with data (how was it collected?) and the hope that we can compare them. We also start with a question which is of the form:
- how much tax increase is associated with how much tax avoidance/tax evasion/country fleeing by the top 1%?
- how much traffic does our website lose (gain) if we slow down (speed up) the load time?
- how many of their soldiers do we kill for every soldier we lose?
- how much do gun deaths [suicide | gang violence | rampaging multihomicide] decrease with 10,000 guns taken out of the population?
- how much more fuel do you need to fly your commercial jet 1,000 metres higher in the sky?
- how much famine [to whom] results when the price of low-protein wheat rises by $1?
- how much vegetarian eating results when the price of beef rises by $5? (and again distributionally, does it change preferentially by people with a certain culture or personal history, such as they’ve learned vegetarian meals before or they grew up not affording meat?) How much does the price of beef rise when the price of feed-corn rises by $1?
- how much extra effort at work will result in how much higher bonus?
- how many more hours of training will result in how much faster marathon time (or in how much better heart health)?
- how much does society lose when a scientist moves to the financial sector?
- how much does having a modern financial system raise GDP growth? (here ∵ the
X~ branchy and multidimensional, we won’t be able to interpolate in Tufte’s preferred sense) - how many petatonnes of carbon per year does it take to raise the global temperature by how much?
- how much does $1000 million spent funding basic science research yield us in 30 years?
- how much will this MBA raise my annual income?
- how much more money does a comparable White make than a comparable Black? (or a comparable Man than a comparable Woman?)
- how much does a reduction in child mortality decrease fecundity? (if it actually does)
- how much can I influence your behaviour by priming you prior to this psychological experiment?
- how much higher/lower do Boys score than Girls on some assessment? (the answer is usually “low
|β|, with lowp” — in other words “not very different but due to the high volume of data whatever we find is with high statistical strength”)
bearing in mind that this response-magnitude may differ under varying circumstances. (Raising morning-beauty-prep time from 1 minute to 10 minutes will do more than raising 110 minutes to 120 minutes of prep. Also there may be interaction terms like you need both a petroleum engineering degree and to live in one of {Naija, Indonesia, Alaska, Kazakhstan, Saudi Arabia, Oman, Qatar} in order to see the income bump. Also many of these questions have a time-factor, like the MBA and the climate ones.)
As Trygve Haavelmo put it: using reason alone we can probably figure out which direction each of these responses will go. But knowing just that raising the tax rate will drive away some number of rich doesn’t push the debate very far—if all you lose is a handful of symbolic Eduardo Saverins who were already on the cusp of fleeing the country, then bringing up the Laffer curve is chaff. But if the number turns out to be large then it’s really worth discussing.
In less polite terms: until we quantify what we’re debating about, you can spit bollocks all day long. Once the debate is quantified then the discussion should become way more intelligent, less derailing to irrelevant theoretically-possible-issues-which-are-not-really-worth-wasting-time-on.
So we change one variable over which we have control and measure how the interesting thing responds. Once we measure both we come to the regression stage where we try to make a statement of the form “A 30% increase in effort will result in a 10% increase in wage” or “5 extra minutes getting ready in the morning will make me look 5% better”. (You should agree from those examples that the same number won’t necessarily hold throughout the whole range. Like if I spend three hours getting ready the returns will have diminished from the returns on the first five minutes.)
Avoiding causal language, we say that a 10% increase in (your salary) is associated with a 30% increase in (your effort).
MAIN PART (SKIP TO HERE IF SKIMMING)
The two numbers that jump out of any regression table output (e.g., lm in R) are p and β.
βis the estimated size of the linear effectpis how sure we are that the estimated size is exactlyβ. (As in golf, a lowpis better: more confident, more sure. Lowpcan also be stated as a hight.)
Wary that regression tables spit out many, many numbers (like Durbin-Watson statistic, F statistic, Akaike Information, and more) specifically to measure potential problems with interpreting β and p naïvely, here are pictures of the textbook situations where p and β can be interpreted in the straightforward way:
First, the standard cases where the regression analysis works as it should and how to read it is fairly obvious:
(NB: These are continuous variables rather than on/off switches or ordered categories. So instead of “Followed the weight-loss regimen” or “Didn’t follow the weight-loss regimen” it’s someone quantified how much it was followed. Again, actual measurements (how they were coded) getting in the way of our gleeful playing with numbers.)
Second, the case I want to draw attention to: a small statistical significance doesn’t necessarily mean nothing’s going on there.
The code I used to generate these fake-data and plots.
If the regression measures a high β but low confidence (high p), that is still worth taking a look at. If regression picks up wide dispersion in male-versus-female wages—let’s say double—but we’re not so confident (high p) that it’s exactly double because it’s sometimes 95%, sometimes 180%, sometimes 310%, we’ve still picked up a significant effect.
The exact value of β would not be statistically significant or confidently precise due to a high p but actually this would be a very significant finding. (Try it the same with any of my other examples, or another quantitative-comparison scenario you think up. It’s either a serious opportunity, or a serious problem, that you’ve uncovered. Just needs further looking to see where the variation around double comes from.)
You can read elsewhere about how awful it is that p<.05 is the password for publishable science, for many reasons that require some statistical vocabulary. But I think the most intuitive problem is the one I just stated. If your geiger counter flips out to ten times the deadly level of radiation, it doesn’t matter if it sometimes reads 8, sometimes 5, and sometimes 15—the point is, you need to be worried and get the h*** out of there. (Unless the machine is wacked—but you’d still be spooked, wouldn’t you?)
FOLLOW-UP (CAN BE SKIPPED)
The scale of β is the all-important thing that we are after. Small differences in βs of variables that are important to your life can make a huge difference.
- Think about getting a 3% raise (1.03) versus a 1% wage cut (.99).
- Think about twelve in every 1000 births kill the mother versus four in every 1000.
- Think about being 5 minutes late for the meeting versus 5 minutes early.
Order-of-magnitude differences (like 20 versus 2) is the difference between fly and dog; between life in the USA and near-famine; between oil tanker and gas pump; between Tibet’s altitude and Illinois’; between driving and walking; even the Black Death was only a tenth of an order of magnitude of reduction in human population.
Keeping in mind that calculus tells us that nonlinear functions can be approximated in a local region by linear functions (unless the nonlinear function jumps), β is an acceptable measure of “Around the current levels of webspeed” or “Around the current levels of taxation” how does the interesting thing respond.
Linear response magnitudes can also be used to estimate global responses in a nonlinear function, but you will be quantifying something other than the local linear approximation.
![going the long way
What does it mean when mathematicians talk about a bijection or homomorphism?
Imagine you want to get from X to X′ but you don’t know how. Then you find a “different way of looking at the same thing” using ƒ. (Map the stuff with ƒ to another space Y, then do something else over in image ƒ, then take a journey over there, and then return back with ƒ ⁻¹.)
The fact that a bijection can show you something in a new way that suddenly makes the answer to the question so obvious, is the basis of the jokes on www.theproofistrivial.com.
In a given category the homomorphisms Hom ∋ ƒ preserve all the interesting properties. Linear maps, for example (except when det=0) barely change anything—like if your government suddenly added another zero to the end of all currency denominations, just a rescaling—so they preserve most interesting properties and therefore any linear mapping to another domain could be inverted back so anything you discover over in the new domain (image of ƒ) can be used on the original problem.
All of these fancy-sounding maps are linear:
Fourier transform
Laplace transform
taking the derivative
Box-Müller
They sound fancy because whilst they leave things technically equivalent in an objective sense, the result looks very different to people. So then we get to use intuition or insight that only works in say the spectral domain, and still technically be working on the same original problem.
Pipe the problem somewhere else, look at it from another angle, solve it there, unpipe your answer back to the original viewpoint/space.
For example: the Gaussian (normal) cumulative distribution function is monotone, hence injective (one-to-one), hence invertible.
By contrast the Gaussian probability distribution function (the “default” way of looking at a “normal Bell Curve”) fails the horizontal line test, hence is many-to-one, hence cannot be totally inverted.
So in this case, integrating once ∫[pdf] = cdf made the function “mathematically nicer” without changing its interesting qualities or altering its inherent nature.
“Going the long way” can be easier than trying to solve a problem directly.](http://25.media.tumblr.com/tumblr_m2pk08lcbr1qc38e9o1_400.gif)
going the long way
What does it mean when mathematicians talk about a bijection or homomorphism?
Imagine you want to get from X to X′ but you don’t know how. Then you find a “different way of looking at the same thing” using ƒ. (Map the stuff with ƒ to another space Y, then do something else over in image ƒ, then take a journey over there, and then return back with ƒ ⁻¹.)
The fact that a bijection can show you something in a new way that suddenly makes the answer to the question so obvious, is the basis of the jokes on www.theproofistrivial.com.
In a given category the homomorphisms Hom ∋ ƒ preserve all the interesting properties. Linear maps, for example (except when det=0) barely change anything—like if your government suddenly added another zero to the end of all currency denominations, just a rescaling—so they preserve most interesting properties and therefore any linear mapping to another domain could be inverted back so anything you discover over in the new domain (image of ƒ) can be used on the original problem.
All of these fancy-sounding maps are linear:
- Fourier transform
- Laplace transform
- taking the derivative
- Box-Müller
They sound fancy because whilst they leave things technically equivalent in an objective sense, the result looks very different to people. So then we get to use intuition or insight that only works in say the spectral domain, and still technically be working on the same original problem.
Pipe the problem somewhere else, look at it from another angle, solve it there, unpipe your answer back to the original viewpoint/space.
For example: the Gaussian (normal) cumulative distribution function is monotone, hence injective (one-to-one), hence invertible.
By contrast the Gaussian probability distribution function (the “default” way of looking at a “normal Bell Curve”) fails the horizontal line test, hence is many-to-one, hence cannot be totally inverted.
So in this case, integrating once ∫[pdf] = cdf made the function “mathematically nicer” without changing its interesting qualities or altering its inherent nature.
“Going the long way” can be easier than trying to solve a problem directly.
Just playing with z² / z² + 2z + 2
on WolframAlpha. That’s Wikipedia’s example of a function with two poles (= two singularities = two infinities). Notice how “boring” line-only pictures are compared to the the 3-D ℂ→>ℝ picture of the mapping (the one with the poles=holes). That’s why mathematicians say ℂ uncovers more of “what’s really going on”.
As opposed to normal differentiability, ℂ-differentiability of a function implies:
- infinite descent into derivatives is possible (no chain of
C¹ ⊂ C² ⊂ C³ ... Cωlike usual)
nice Green’s-theorem type shortcuts make many, many ways of doing something equivalent. (So you can take a complicated real-world situation and validly do easy computations to understand it, because a squibbledy path computes the same as a straight path.)
Pretty interesting to just change things around and see how the parts work.
- The roots of the denominator are
1+iand1−i(of course the conjugate of a root is always a root sinceiand−iare indistinguishable) - you can see how the denominator twists
- a fraction in ℂ space maps lines to circles, because lines and circles are turned inside out (they are just flips of each other: see also projective geometry)
- if you change the
z^2/to az/or a1/you can see that. - then the Wikipedia picture shows the poles (infinities)
Complex ℂ→ℂ maps can be split into four parts: the input “real”⊎”imaginary”, and the output “real“⊎”imaginary”. Of course splitting them up like that hides the holistic truth of what’s going on, which comes from the perspective of a “twisted” plane where the elements z are mod z • exp(i • arg z).
- Apparently this is related to fluid mechanics and also QM somehow? And of course, tachyons!
- Angles. All it took was angles (angle-preserving maps = conformal maps = holomorphic = ℂ-differentiable). Compare: diffeomorphism.
ℂ→ℂ mappings mess with my head…and I like it.
