Saturday, March 25, 2017

Reality vs nonsensical products (part 688 of Aleph-null)

Via Thunderf00t, I found this Waterseer-wannabe, which is about as feasible as the original Waterseer, that is not at all.



Obviously it's very important that the product is 3D-printed, rather than CNC-machined or heat-molded. 3D-printers, like the Internet Of Things, are magical incantations that can get around the laws of Physics. Or so one would think, given how credulous people become at the sound of these incantations.

Alas, as is usual with engineering, ugly numbers murder beautiful illusions:



Since the battery voltage is 12V, a 12kW Peltier effect cooler will require a 1000A current, which is likely to make Li-ion battery a bit... well, just watch what happens:



Engineering rule: when an electronic device starts outgassing, that's generally not a good thing.

Wednesday, March 22, 2017

The power of "equations"

If a picture is worth a thousand words, an equation is worth a thousand pages of text.

This was inspired by a livestream about free trade based on criticism of "original texts." (Basically Ricardo and Schumpeter.) The quotes aren't a diss on the texts themselves, but rather a way to emphasize that this is a type of scholarly pursuit in itself, though not the type used in modern economics, STEM, or pragmatic professional fields like business analytics or medicine.

What's the problem with the argumentation from these original texts? Simply put, the texts are long and convoluted, with many unnecessary diversions and some logical problems in the presentation. The valid arguments in these texts can be condensed in about one page of stated assumptions and two results about specialization.

It's not just that math's an efficient way to communicate, math has precise meaning and an inference process. It brings discipline and clarity to the texts and the inference process isn't open to debate. (Checks and corrections, yes; debate, no.)

Unfortunately, without math, the speaker's argument was essentially a sequence of variations on "Schumpeter points out that this assumption of Ricardo doesn't hold true," without the extra step of determining whether those assumptions are important to the final result or not. (We'll come back to this problem.)

Word-thinking about quantitative fields is generally to be avoided.

That was the inspiration, and this post isn't about free trade or the particular mode of thought of that speaker, but rather about the power of mathematical modeling, which I'm calling "equations" in the title.

Here's a reasonably robust statement: when the price of a commodity goes up, people buy less of that commodity. (Sometimes this is put as "demand goes down," which is incorrect, it's the demand quantity that goes down. Changes in demand are movements of an entire function.)

So, quantity is a decreasing function of price (and first-time readers of economics textbooks get confused because the charts have quantity in the $x$ axis and price in the $y$ axis). This has been known for a long time; what's the problem with that formulation, simplified to "when price rises, quantity falls"?

The problem, of course, is that there are many different types of decreasing function. Here are a few, for example (click for bigger):


Functions 1 to 4 represent four common behaviors of decreasing functions: the linear function has similar changes leading to similar effects; the convex function has decreasing effect of similar change (like most natural decay processes); the concave function has increasing effect of similar change (like the accelerating effect of a bank run on bank reserves); and the s-shaped function shows up in many diffusion processes (and is a commonly used price response function in marketing).

Functions 5 to 8 are variations on the convex function, showing increasing curvature. (Function 2 would fit between 5 and 6.) They're here to make the point that even knowing the general shape isn't enough: one must know the parameters of that shape.

That figure does have 2000 data points, since each function has 250 points plotted. (When talking about math, some people use drawing tools to make their "functions," I prefer to plot them from the mathematical formula; it's a habit of mine, not lying to the audience.) To describe them in text would take a long time (unless the text is a description of mathematical formulation), while they can be written simply as formulas; for example, the convex functions are all exponentials:

$\qquad y = 100 \, \exp(-\kappa \, x) $

with different values of $\kappa$. They are the type of exponential decay found in many processes, for example, where $x$ is time and $y(x) = \alpha \, y(x-1)$ with $y(0)>0$ models a process of decay with discrete-time rate $0 < \alpha < 1$. In case it's not obvious, $\kappa = -\log_{e}(\alpha)$.*

So, what does this have to do with reasoning?

Here we go back to the problem with arguments like "Schumpeter showed that Ricardo's assumption X was wrong." When a model is written out in equations, we have a sequence of steps leading to the result, each step tagged with either a know result, rules of math inference (say "$a \times b = a \times c$ simplifies to $b = c$ unless $a = 0$"), or an assumption of the model. This allows a reader to quickly see where a failed assumption will lead to problems and determine whether the assumption can be replaced with something true (or, as is the case with many of the assumptions made by Ricardo, is unnecessary for the result).

The main power, however, is that mathematical notation forces the speaker to be precise, and inferences from mathematical models can be checked independently of subject matter expertise. A mathematician may not understand any of the economics involved, but will merrily check that a decay process of the kind $y(n)= \alpha \, y(n-1)$ can be described by an equation $y(n) = y(0) \, \exp(-\kappa \, n)$ and determine the relationship between $\kappa$ and $\alpha$.

From those precise models, one can make inferences that take into account details hidden by language. Consider the "price rises, quantity falls" text and compare it with the different decreasing functions in the figure above. The shape of the function, its slope and its curvature have different implications for how price changes affect a market, differences that are lost in the "price rises, quantity falls" formulation.

It bears repeating the first mentioned advantage: that hundreds of pages can be condensed in one page of equations. Once one's mind is used to processing equations, this is a very efficient way to learn new things. Stories about Port wineries in Portugal and textile factories in England may be entertaining, but they aren't necessary to understand specialization (which is what comparative advantage really is).

Math. It's a superpower mostly anyone can acquire. Sadly, most opt not to.


- - - - - Addendum - - - - -

No self-respecting economist would use the Ricardo comparative advantage argument for international trade now, particularly because it's so simple it can be understood by anyone. Most likely they'd use some variation of the magic factory example:

"Let's say a new technology that converts corn into cars is discovered and a factory is built in Iowa that can take ~ $\$20,000$ of corn and convert it into a car that costs $\$30,000$ to make in Michigan. Can we agree that this technology makes the US richer?

Now, move the factory to Long Beach, CA. Maybe there's a little more cost in moving the corn there, but we're still making the US richer, right?

Now, someone goes into the magic factory and discovers that it's really a depot: stores grain until it's sent to China on bulk carriers and receives cars made in China from RoRos during the night. The effect is the same as the magic factory, so it makes the US richer, right?"

There are many cons to this example, but it does make one issue clear: trade is in many respects just like a different technology.


- - - - - Footnote - - - - -

* It's obvious to me, because after decades of playing around with mathematical models, I grok most of these simple things. There are some people who mistake this well-developed and highly available knowledge (from practice) for ultra-high intelligence (rather than regular very high intelligence), a mistake I elaborate upon in this post. 😎

Tuesday, March 7, 2017

Deep understanding and problem solving

There's value in deep understanding.

Nope, I don't mean the difference between word thinkers and quantitative thinkers. Been there, done that. Nor the difference between different levels of expertise on technical matters; again, been there, done that.

No, we're talking the crème de la crème, experts that can adapt to changing situations or comprehend complexity across different fields, by being deep understanders.

Because any opportunity to mock those who purport to educate the masses by passing along material they don't understand, let us talk about Igon Values... ahem, eigenvalues and eigenvectors.

Taught in AP math classes or freshman linear algebra, the eigenvectors $\mathbf{x}_{i}$ and associated eigenvalues $\lambda_{i}$ of a square matrix $\mathbf{A}$ are defined as the solutions to $\mathbf{A} \, \mathbf{x}_{i} = \lambda_{i} \, \mathbf{x}_{i}$.

Undergrads learn that these represent something about the structure of the matrix, learn that the matrix can be diagonalized using them, how they appear in other places (principal components analysis and network centrality, for example).

But those who get to use these and other math concepts on a day-to-day basis, who get to really understand them, develop a deeper understanding of the meaning of the concepts. There's something important about how these objects relate to each other.

After a while, one realizes that there are structures and meta-structures that repeat across different problems, even across different fields. Someone said that after a lot of experience in one engineering (say, electrical), adapting to another (say, mechanical) revealed that while the nouns changed, the verbs were very similar.

This is what deep understanding affords: a quasi-intuitive grokking of a field, based on the regularities of knowledge across different fields.

For example: while many who have taken a linear algebra in college may vaguely recall what an eigenvalue is, those who understand the meaning of eigenvalues and eigenvectors for matrices will have a much easier time understanding the eigenfunctions of linear operators:


The structure [something that operates] [something operated upon] = [constant] [something operated upon] is common, and what it means is that the [something operated upon] is in some sense invariant with the [something that operates], other than the proportionality constant. That suggests that there's a hidden meaning or structure to the [something that operates] that can be elicited by studying the [something operated upon].

And this structure, mathematical as it might be, has a lot of applications outside of mathematics (and not just as a mathematical tool for formalizing technical problems). It's a basic principle of undestanding: what is invariant to a transformation tells us something deep about that transformation. (Again, invariant in "direction," so to speak, possibly a change of size or even sign.)

And this is itself a meta-principle: that the study of what changes and what's invariant in a particular set of problems gives some indications about latent structure to that set of problems. That latent structure may be a good point to start when trying to solve problems from this set.

Yep, really dumbing down this blog, pandering to the public...