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Whenever Clouds Become Clocks
Preston S. Fan
School of Management, New York Institute of Technology
October 2013
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 1 / 29
Motivation
x
y
1
-1
θ
sinθ
π6
π3
2π3
5π6
7π6
4π3
5π3
11π6
π2
3π2
π 2π
30 60 90 120 150 180 210 240 270 300 330 360
−1
−0.5
0.5
1
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 2 / 29
An Identity
What have you seen through this identity?
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 3 / 29
An Identity
What have you seen through this identity?
The Identity
cos2θ + sin2θ = 1
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 4 / 29
The Manifesto of Clocks
cos2θ + sin2θ = 1
This trigonometric identity seems
Pythagoras’ Theorem, in a right triangle;
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 5 / 29
The Manifesto of Clocks
cos2θ + sin2θ = 1
This trigonometric identity seems
Pythagoras’ Theorem, in a right triangle;
An equation for a circle: a unit circle...
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 6 / 29
A Pendulum12mv2
1 + mgh1 = 12mv2
2 + mgh2 ⇒ cos2θ + sin2θ = 1
θ
mg
v
l
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 8 / 29
The Manifesto of Clocks
cos2θ + sin2θ = 1
This identity also seems
Conservation of energy, or the first law of thermodynamics;
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 9 / 29
The Manifesto of Clocks
cos2θ + sin2θ = 1
This identity also seems
Conservation of energy, or the first law of thermodynamics;
In a projection
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 10 / 29
The Manifesto of Clocks
cos2θ + sin2θ = 1
This identity also seems
Conservation of energy, or the first law of thermodynamics;
In a projection θ is the angel between regressand y onto columnspace of regressors Xβ
cosθ might be treated as the coefficientIn analysis of variances, cos2θ seems R2
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 11 / 29
Questions
Abuse of Popper’s Metaphor,
What have we done to Clock in to the Clouds for past centuries?
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 13 / 29
How Clouds become Clocks
Let’s make a slight variation
What about thiscosθ + sinθ =?
cosθ + sinθ = sin(θ + π2 ) + sinθ
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 14 / 29
How Clouds become Clocks
How would you interpret f (x) = a0sinθ0 + a1sinθ1 + a2sinθ2 + · · ·
or f (x) = a0sin(π2 x
)+ a1sin(3π
2 x) + a2sin(5π2 ) + · · ·
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 15 / 29
How Clouds become Clocks
How would you interpret f (x) = a0sinθ0 + a1sinθ1 + a2sinθ2 + · · ·
or f (x) = a0sin(π2 x
)+ a1sin(3π
2 x) + a2sin(5π2 ) + · · ·
Recall the sauces of a cuisine you prefer2 spoons of sugar, 1 spoon of salt, 0.25 spoon of red chili, · · ·
Make sure you have a list (spectrum) of all sauces
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 16 / 29
How Clouds become Clocks
How would you interpret f (x) = a0sinθ0 + a1sinθ1 + a2sinθ2 + · · ·
or f (x) = a0sin(π2 x
)+ a1sin(3π
2 x) + a2sin(5π2 ) + · · ·
A full list of potential saucesa0=2 spoons of sugar (sinθ0);a1=1 spoon of salt (sinθ1);a2=0.25 spoon of red chili (sinθ2);· · ·
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 17 / 29
How Clouds become Clocks
How would you interpret f (x) = a0sinθ0 + a1sinθ1 + a2sinθ2 + · · ·
or f (x) = a0sin(π2 x
)+ a1sin(3π
2 x) + a2sin(5π2 ) + · · ·
This is how we make musicThe sounds of cello, piano, saxophone differ in their ingredients of”sines” and proportions
This has been how we make clouds clocks
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 18 / 29
How Clouds become Clocks
It requires a complete list of potential ingredients...
In the language of Nicolas Bourbaki, it calls for Arrow-Debrewprices for all time periods and for all states of the world.
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 19 / 29
How Clouds become Clocks
Let’s make another slight variation
Another Identity
cosθ + isinθ = eiθ
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 20 / 29
How Clouds become Clocks
Let’s make another slight variation
Another Identity
cosθ + isinθ = eiθ
eiθ proxies rotation θ counter-clockwisely∣∣eiθ∣∣ = ∣∣∣∣cosθ −sinθ
sinθ cosθ
∣∣∣∣ = cos2θ + sin2θ = 1
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 21 / 29
How Clouds become Clocks
Without loss of generality,
Another Identity
cos(nθ) + isin(nθ) = einθ
Without loss of intuition,What is cos(nθ) + isin(nθ) = einθ?
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 22 / 29
How Clouds become Clocks
Without loss of generality,
Another Identity
cos(nθ) + isin(nθ) = einθ
Without loss of intuition,Multiplication n times proxy rotation by nθ[
cosθ −sinθsinθ cosθ
]n
=
[cos(nθ) −sin(nθ)sin(nθ) cos(nθ)
]= einθ
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 23 / 29
Equal Temperament
Each round has been divided in 12 sections with angle of 2π12
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 25 / 29
Equal Temperament
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Each round has a cycle of 12 notes C ]C D ]D E F ]F G ]G A ]A B
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 26 / 29
Equal Temperament
Gˇ 4ˇ ˇ 4̌ ˇ ˇ 4̌ ˇ 4ˇ ˇ 4̌ ˇ
”12 equal temperament”(J. S. Bach, The Well-Tempered Clavier )
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 27 / 29
Homunculus
In the world of clocks, no heterogeneity...
Statistical significance tests
”Happy families are all alike; every unhappy family is unhappy in itsown way.” (L. Tolstoy, Anna Karenina)
Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 28 / 29
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