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Mathematical Models & Movies: A Sneak Preview. Ron Buckmire [email protected] Occidental College Los Angeles, CA. Outline. Introduction to Cinematic Box-Office Dynamics Important variables and concepts Graphs of typical movie data Presentation of Edwards-Buckmire Model (EBM) Drawbacks of EBM - PowerPoint PPT Presentation
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Mathematical Models & Movies: A Sneak Preview
Occidental CollegeLos Angeles, CA
Outline• Introduction to Cinematic Box-Office Dynamics
– Important variables and concepts– Graphs of typical movie data
• Presentation of Edwards-Buckmire Model (EBM)
• Drawbacks of EBM• Derivation of Modified EBM• Numerical Results Using Modified EBM• Future (and Past) Work
– The Holy Grail: A Priori Prognostication– Sequels: Parent-Child Relationship
• Conclusions• References and Acknowledgements
Introduction: Cinematic Box-Office DynamicsImportant variables
• G(t) : cumulative gross receipts of the movie
• S(t) : number of screens movie is exhibited• A(t) : normalized weekly revenue ($ per
screen average)• t : time in number of weeks
Important concepts• A and S have quasi-exponential
profiles•
Actual Movie Data: The Expendables (2010)
Actual Movie Data: Taken (2009)
Actual Movie Data: The Love Guru (2008)
Actual Movie Data: Spider-Man 3 (2007)
Actual Movie Data: Open Season (2006)
The original Edwards-Buckmire model SA
dt
dG
max* A
ASS
dt
dSS
AGHPMS
S
Ddt
dA A
2
%)1(
0)0( AA
0)0( SS
0)0( G
EBM Parameters
Dimensionless EBM SA
dt
dG AS
dt
dS
AGS
S
dt
dA
~
~~
where
*
~S
M )1(
1~
DS
A
SP
ASH
max*
2%~
Typical solution curves
Drawbacks of EBM• H% varies with time
• Parameter ( ) estimates are difficult to make and somewhat arbitrary
• Most movies have a contract period in which screens is constant, i.e.S’=0
• S and A actual data more erratic than first thought; G is relatively smooth
Modifying the EBM (J. Ortega-Gingrich)• Uses an Economics-inspired
“demand” model• Incorporates fixed contract
periods when screens are constant
• Greatly modifies the A equation• Both versions of EBM have 3
unknown parameters
Deriving the new A equation
Consider a Demand function D(t)=S(t)Ap(t) which satisfies
Where Ap is the revenue per screen if everyone whowanted to see the film, saw it, i.e. “A potential”
Recall that G’=SA and assume that G could satisfy the IVP
Which leads to
and
The selected form of μ(S) used is given below (a=1/T),
T is total number of movie theaters in North America (~4,000)
The function μ(S) should satisfy the following conditions
We apply the product rule to A and Ap
Derivation: Doing The Math
Modified EBM
Comparing Original EBM to Modified EBM
SAdt
dG
ASdt
dS
AGS
S
dt
dA
~
~~
Numerical Calculations• Analyzed119 movies from 2005-2010
(minimum final gross $50m)• All dollars adjusted for inflation to 2005• Used Mathematica to generate
numerical solutions to the modified EBM
• Attempted to find “global” values of parameters that would minimize std. dev. in difference between computed G∞ and actual G∞ while minimizing error
Numerical Results: (N=119)
Distribution of G Computed/G Actual as Histogram
mean=1.0389, std. dev.=0.158
Numerical Results: (N=119)
Numerical Results: Using Global ParametersThe Expendables (2010)
Numerical Results: Using Global ParametersTaken (2009)
Numerical Results: Using Global ParametersThe Love Guru (2008)
Numerical Results: Using Global ParametersSpider-Man 3 (2007)
Numerical Results: Using Global ParametersOpen Season (2006)
Numerical Results: Using Chosen ParametersThe Expendables (2010)
Numerical Results: Using Chosen ParametersOpen Season (2006)
Future Work“The Holy Grail”: Predict the
opening weekend gross before the movie is released
The sequel problem: predict the gross of a sequel based on the parent’s characteristics
The Sequel Problem• Considered a subset of the a priori
prediction problem with (possibly) more known information
• Main assumption is opening weekend revenue, A0, must depend on awareness of film (which probably depends on marketing, M)
Conclusions• Predicting the final accumulated
gross of any given movie before it is released is a hard problem
• The original EBM should probably be modified to be less movie-specific and the modified EBM changed to be more movie-specific
Acknowledgements• Joint work with Occidental
College students Jacob Ortega-Gingrich ’13 and Rohan Shah ’07
• Many thanks to David Edwards and the staff and faculty of University of Delaware