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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

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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 Pendulum

θ

mg

v

l

Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 7 / 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

Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 12 / 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

ρ = erθ

Preston S. Fan (NYIT) Whenever Clouds Become Clocks October 2013 24 / 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