James Simons, Renaissance Technologies - Gib Bassett

The Goat in the CityGib Bassett

November 2021

The Goat in the City• The Goat on the Farm

• Goat Moves to City• City Distance and L1 (not L2)

• Manhattan vs Chicago

The Ladies Diary: Marvelous!

(Backstory: Least Squares and Least Absolute Deviations--

Gauss, Boscovich, Simpson.

The Goat in the City (cont.)

Discovering Connections

Solving the City Problem

Variations of the problem with alternative street grids

The Goat on the Farm on the Hill. The Goat in the City on a Hill.

Farm and City in n-dimensions

The Goat on the Farm

James SimonsRenaissance Technologies


Quanta Magazine, December 2020


What rope length allows the goat to access half the area of the farm?

Origins of the Goat Problem

• Goes Way back• 1894

• 1748

The Ladies Diary 1748

• The Ladies Diary: Containing many Delightful and Entertaining Particulars Peculiarly adapted for the Use and Diversion of the FAIR-SEX

•Published in England for 136 years, 1704-1840

More below……


Farm Distance: Euclidean, L2

𝑑 𝑿𝑨, 𝑿𝑩 = 𝑥1𝐴 − 𝑥1

𝐵 2 + 𝑥2𝐴 − 𝑥2

𝐵 2.

What happens when the Goat Moves to the City (Chicago)?

City Travel: Restricted to

Streets

City Distance is L1

Trip from (0,0) to 𝑿𝐴= [𝑥1

𝐴, 𝑥2

𝐴].

Unless 𝑿𝐴

is on the same street (directly E/W, or N/S) of the origin, a straight-line trip is not feasible.

The trip requires two legs with the first ending at an intermediate location 𝑿𝐵

that must be:

(i) straight-line accessible away from the origin, and

(ii) straight-line accessible towards 𝑿𝐴

.

The first leg is from [0,0] to 𝑿𝐵= [𝑥1

𝐴, 0], and the second leg from 𝑿

𝐵to 𝑿

𝐴. So,

City Distance L1

𝑑𝐶𝑖𝑡𝑦1 0, 𝑿𝐴 = 𝑑 0, 𝑿𝐵 + 𝑑 𝑿𝐵 , 𝑿𝑨 =

𝑥1𝐴 2 + 𝑥2

𝐴 2

= 𝑥1𝐴 + 𝑥2

𝐴

L2 vs L1(Goats/Pigs can’t fly)

Name

• Manhattan

• Taxi Cab

• Uber

• I prefer Chicago

Chicago Metric because:

(i) its streets are also a grid

(ii) it is outside my window

(iii) goats and Chicago go together

A Chicago goat was the cause of the curse on the Cubs that made them losers for a hundred years.

Tavern owner Sam Siannis hexed the team after being denied admission to the ballpark (accompanied by his pet goat).

Sam later was the inspiration for the famous John Belushi and Dan Ackroyd SNL sketch, “Cheeseburgers, Cheeseburgers, Cheeseburgers” which were sold at his Bill Goat Tavern, not to be confused with the Girl and the Goat restaurant and its famous chef, Stefani Izard, Top Chef Champion, and a candidate for GOAT, like MJ.

Sam Siannis at The Billy Goat

William “Billy Goat” Siannis with his Goat at Wrigley Field

Belushi and Ackroyd on SNL

Why L1?

Might lead to different, interesting answers to the goat problem?

Illustrate how different distance metrics can lead to different results

Least Squares and Least Absolute Values

Estimation: Does X matter?

X

YL1 Yes

L2 No

Does X matter?

X

YL1 Yes Positive

L2 Yes Negative

Does X matter?

X

YL2 Yes

L1 No

• Least squares invented in 1800• Gauss, LeGendre

• L1 invented in 1750s

• In 1750s Roger Boscovich proposes estimation using minimization of absolute values.

• Figuring out the distance to the sun based on noisy observations from various Earth observatories.

Roger Boscovich and Thomas Simpson• Thomas Simpson: English Mathematician

(1710-1761), teaching at the Royal Military Academy

• Corresponds with Boscovich regarding the L1 minimization problem.

• Simpson paper on the L1 method in 1757.

Backstory Summary

•Roger Boscovich proposes L1 half a century prior to least squares

•Thomas Simpson is the other major player in the development of L1

Back to The Goat Problem

The Ladies Diary

Just Published2021

Enlightenment Books and Journals

•Astronomical Dialogues between a Gentleman and a Lady, 1719

•Newtonianism for Ladies, 1737

All the issues of Ladies Diary online……

• Browsing.

• Page 3 of the 1748 issue…..

All Persons who are pleased to be CONTRIBUTORS, by answering the ENIGMAS, QUESTIONS, ETC., in this Diary, or by sending new Enigmas, Questions, Paradoxes, or other Subjects fitting for this WORK, are desired to send their Solutions with them before the End of April, 1748, di-rected for the Author, at Mr. Simpson’s, at Stationers Hall, LONDON.

• Is this Mr. Simpson the Thomas Simpson who in a few years would correspond with Boscovich about the development of L1?

The Ladies Diary and Thomas Simpson

• Submitted problems as early as 1733

• Correspondent for Solutions for many years including the “Goat” year, 1748.

• He will become the editor of the Diary 1751 to 1760.

• More: In 1760 while visiting London Boscovich meets with Simpson and discusses the L1 minimization problem.

London, 1760

Simpson talking to Boscovich,

“Roger, enough about absolute values: got any good math problems for my Ladies Diary?”

Doubling Down

• Simpson is math correspondent of the Ladies Diary when the “Goat” problem first appears.

•Could Simpson be the author of the “Goat Problem”?

Upnorensis!?

Does not look like Simpson.

But Wait!!

Simpson Literature (not related to Statistics)

• Many contributors to the Ladies Diary used aliases

• Many contributors to the Diary used aliases

Including Simpson!

Simpson’s AKA

• Patrick O'Cavanaugh

• Kubernetes

• Anthony Shallow

• Hurlothrumbo

• Timothy Doodle Esq.

•Marmaduke

• Hodgson

Simpson’s AKA

• Patrick O'Cavanaugh

• Kubernetes

• Anthony Shallow

• Hurlothrumbo

• Timothy Doodle Esq.

•Marmaduke

• Hodgson

➢Upnorensis (not!)

History Recap: A new connection……

• Thomas Simpson of L1 fame is the math correspondent to the Ladies Diary in 1748.

Back to the Goat in the City

Goat tied to boundary with rope length equal city radius

Goat at the Southern Boundary¼ Coverage with 1.0 Uber(Intersection of Unit Balls)

Goat at the Southern Boundary

Goat at the Southeast Boundary!!

Goat at the Southeast Boundary½ the City can be accessed with 1.0 Uber

(The solution to the Goat Problem!!)

Farm vs City

• The Goat on the Farm needs a rope of about 1.16 length to visit half the sites on the Farm• The solution is the same for any location on the farm boundary.

• The Goat in the City with a rope of 1.0 can visit half the City sites!• The solution is dependent on the location of the goat on the city boundary.

• Needs to be at SE,SW,NW, or NE boundary

What happens with different street configurations?

A different grid:NE/SW

Solution is the same--but the distance is Chebyshev

L∞, Chebyshev distance

𝑑𝐶𝑖𝑡𝑦2 𝑿𝐴, 𝑿𝐵 = 2max{|(𝑥1𝐴 − 𝑥1

𝐵|, 𝑥2𝐴 − 𝑥2

𝐵 }

What happens if the Farm or City is on a hill? (Travel up the hill is more difficult

than down the hill)

Goats Love Hills!

City and Farm on a HillsideUnit Balls

The Goat in the City in Rn?

Summary

• The Goat in the City Illustrates how different distance metrics can lead to different solutions.

--The Goat in the City can visit half the city with a travel length equal to the city radius

• Thomas Simpson--a major player in the development of the city distance metric for estimation--was the math correspondent of The Ladies Diary in 1748 where and when The Goat Problem originated.

• https://gib.people.uic.edu/Goat.pdf

Thanks.

James Simons, Renaissance Technologies - Gib Bassett

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