Do Dogs Really Know Calculus

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Do Dogs Really Know Calculus?
Presented by Eric Reyes Under the direction of
Mark H. Inlow Rose-Hulman Institute of Technology
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Overview

• The story of a professor and his dog
• A standard approach
• The hurdle
• Measurement error
• Learning to jump as we run
• Two approaches
• Application

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Background
(courtesy of Tim Pennings)
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Data Collection
Lake Michigan
Ball
Starting Point
Elvis Enters Water
Tape Measure
Beach
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Lake Michigan
Ball
B
Where r running speed s swimming speed
Elviss starting point
A
C
D
Beach
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Derivation of Optimal Path

• Collected 5 observations of Elviss running and
  swimming speeds
• Used an average of 3 best times to compute
  running speed (r) and
• swimming speed (s)
• Found derivative of Time function, and solved
  for y using values of r and s
• calculated above

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Derivation of Optimal Path
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Derivation of Optimal Path
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Derivation of Optimal Path
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What Does it Mean to Know Calculus?

• On Average, Elvis performance jitters about
  his optimal path

(Simulation)
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Simple Linear Model
Model
Deterministic
Stochastic
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Does Elvis Really Know Calculus?
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Measurement Error
Model
observed x value
true x value
measurement error
The measurement error will bias the slope toward
zero.
(Simulation)
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First Approach
Because of the data collection methods, we test
for the presence of measurement error using a
different estimator for the slope.
Therefore, this estimate is consistent,
regardless of measurement error.
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Assumptions

• Expected value of error in xs equal to 0
• Y-intercept of zero
• Independent observations

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Does Elvis Really Know Calculus?
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Second Approach

• Do not put your faith in what statistics say
  until you have carefully considered what they do
  not say.
• - William Watt

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Second Approach
Since we desire to remove the assumption that the
intercept is zero, we use a top-third,
bottom-third slope estimator.
Therefore, this estimate is also consistent,
regardless of measurement error.
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Assumptions

• Expected value of error in xs equal to 0
• Independent observations

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Question

• How do we test if intercept is zero?

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Results

• The value of a problem is not so much coming up
  with the answer as in the ideas and attempted
  ideas it forces on the would be solver.
• - I.N. Herstein

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Results

• After top-third, bottom-third created, we end up
  with a small sample size

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Does Elvis Really Know Calculus?
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Applications

• Correlation between breast-cancer incidence and
  saturated-fat intake
• The method for determining proper dieting is a
  self-report form. This has a lot of
  measurement error (Rosner).
• Any observational data that depends on
  self-reported results.

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All this over a game of fetch?
Questions?
(Courtesy of Tim Pennings)