terça-feira, 1 de fevereiro de 2011

Power Law Exponent

One main property of these fractals (or another way to express their main property, scalability) is that the ratio of two exceedances is going to be the ratio of the two numbers to the negative power of the power exponent.

Let us illustrate this. Say that you "think" that only 96 books a year will sell more than 250,000 copies (which is what happened last year), and that you "think" that the exponent is around 1.5. You can extrapolate to estimate that around 34 books will sell more than 500,000 copies -- simply 96 times (500,000/250,000)^(-1.5). We can continue, and note that around 8 books should sell more than a million copies, here 96 times (l,000,000/250,000)^(-1.5).


Table 2 illustrates the impact of the highly improbable. It shows the contributions of the top 1 percent and 20 percent to the total. The lower the exponent, the higher those contributions. But look how sensitive the process is: between 1.1 and 1.3 you go from 66 percent of the total to 34 percent. Just a 0.2 difference in the exponent changes the result dramatically -- and such a difference can come from a simple measurement error. This difference is not trivial: just consider that we have no precise idea what the exponent is because we cannot measure it directly. All we do is estimate from past data or rely on theories that allow for the building of some model that would give us some idea -- but these models may have hidden weaknesses that prevent us from blindly applying them to reality.

(The Black Swan, Nassim Nicholas Taleb)

Scale Invariance

Emily Dickinson

A word is dead

A word is dead
When it is said,
Some say.

I say it just
Begins to live
That day.

I'm Nobody! Who are you?

I'm Nobody! Who are you?
Are you -- Nobody -- Too?
Then there's a pair of us!
Don't tell! they'd advertise -- you know!

How dreary -- to be -- Somebody!
How public -- like a Frog --
To tell one's name -- the livelong June --
To an admiring Bog!

There is another sky

There is another sky,
Ever serene and fair,
And there is another sunshine,
Though it be darkness there;
Never mind faded forests, Austin,
Never mind silent fields -
Here is a little forest,
Whose leaf is ever green;
Here is a brighter garden,
Where not a frost has been;
In its unfading flowers
I hear the bright bee hum:
Prithee, my brother,
Into my garden come!

The Geometry of Nature

Triangles, squares, circles, and the other geometric concepts that made many of us yawn in the classroom may be beautiful and pure notions, but they seem more present in the minds of architects, design artists, modern art buildings, and schoolteachers than in nature itself. That's fine, except that most of us aren't aware of this. Mountains are not triangles or pyramids; trees are not circles; straight lines are almost never seen anywhere. Mother Nature did not attend high school geometry courses or read the books of Euclid of Alexandria. Her geometry is jagged, but with a logic of its own and one that is easy to understand.

(The Black Swan, Nassim Nicholas Taleb)