Transcript Move Over, Big Data! How Small, Simple Models Can - SDM

Move Over, Big Data!
How Small, Simple Models Can Yield Big Insights
Richard C. Larson, Ph.D.,
[email protected]
Mitsui Professor of Engineering Systems and Director
of the Center for Engineering Systems Fundamentals,
MIT
September 8, 2014
© Richard C. Larson 2014
1
© Richard C. Larson 2014
2
© Richard C. Larson 2014
3
• Fishing in the Ocean….
• Random location? No strategy?
• Or, location and strategy based on prior analysis?
© Richard C. Larson 2014
4
In Trying to Make Sense of a
Sea of Data, We Need
Small Simple Models to
Guide our Search
© Richard C. Larson 2014
5
© Richard C. Larson 2014
6
From an MIT SDM alum:
• I work on big data “stuff” in my
day job and I think simple models
are too often discounted, often
due to bedazzlement by big data
trends, tools, and the quest for
the holy grail.
© Richard C. Larson 2014
7
A
D =C
N
F(y) = B(y) – [F(y-1) + F(y-2) + F(y-3)]
L = lW = r /(1- r )
implies W = (1/ l) r /(1- r) = (1/ m) /(1- r)
© Richard C. Larson 2014
8
• What we are not saying about Big Data
and Data Analytics…..
Small
• What we are saying about
small
Models
models…..
Big Data
• Ideally, in many applications, these two
approaches are complementary, going
hand in hand.
© Richard C. Larson 2014
9
•
•
•
•
•
Outline.
Flaws of Averages
Square root laws
Nonlinearities in Queueing
Case Study: Marrying Small Models
and Big Data Analysis
© Richard C. Larson 2014
10
If we are about to deal with lots of
data, averages will be important.
• An average is one of the simplest operations
on any dataset.
• We need to be savvy customers of averages!
© Richard C. Larson 2014
11
Flaws of Averages
• Simple model:
The average of N quantities,
X1, X2, …, XN.
• Average = (X1 +X2 +… + XN)/N.
• Simple, right?
© Richard C. Larson 2014
12
Flaws of Averages
© Richard C. Larson 2014
13
Flaws of Averages
• We tend to think in averages, often
to the point of believing that the
average is a constant describing all!
• Warning: Average River Depth is 4
feet!
• Mutual Fund: Average total annual
returns --- 7%.
© Richard C. Larson 2014
14
Flaws of Averages
• We’ve all heard the joke: When Bill Gates
walks into a crowded establishment, ON
AVERAGE everyone becomes a millionaire!
• The mean salary of a tech worker in San
Mateo County is $291,497.
• $81,000 of this is due to Mark Zuckerberg!
• Medians anyone?
© Richard C. Larson 2014
15
© Richard C. Larson 2014
16
Flaws of Averages
• Garrison Keeler: Lake Wobegon,
where all the women are strong, all
the men are good looking and all the
children are above average.
• Possible? Impossible?
© Richard C. Larson 2014
17
Flaws of Averages
• Movie Theaters: Estimate the fraction
of offered seats that are sold.
• Movie Theater Management: What
do they see?
• Typically – 5%
• Selection bias – occurs everywhere.
© Richard C. Larson 2014
18
Flaws of Averages
• Selection bias – occurs everywhere.
• Think of waking up and being a
chocolate chip in a chocolate chip
cookie!
–Your perceived distribution of chips in a
cookie
–Management’s experience..
© Richard C. Larson 2014
19
Flaws of Averages
• Selection bias – Extends to friends on
FaceBook.
• Yes, it is true that on average my
friends on FaceBook have more
friends than I do!
• How does this type of selection bias
extend into your business?
© Richard C. Larson 2014
20
Flaws of Averages
• Viral growth, R0.
• R0 initially from Germany, population
growth
• In epidemics, R0 is the average number
of new infections created by a newly
infected person when almost everyone is
susceptible to the disease.
© Richard C. Larson 2014
21
Flaws of Averages
• Suppose R0 = 2.0. Consider two very
different possibilities…
• 1: Every infection generates 2 more.
• 2: A new infection has a 50% chance of
generating 4 new infections and a 50%
chance of generating none.
• Can you picture the temporal dynamics of
each case?
© Richard C. Larson 2014
22
Ebola Summer 2014
© Richard C. Larson 2014
23
Flaws of Averages
• “Outliers”: What to do with them?
• Many say, clip them off, they distort the
analysis, they mislead intuition.
• But, “outliers” have determined the course
of human history.
– Meteors hitting Planet Earth
– Richter 9 and above earthquakes
– Financial collapses.
© Richard C. Larson 2014
24
Earthquakes
• Richter Scale: logarithmic. Each whole
number step in the magnitude scale
corresponds to the release of about 31
times more energy than the amount
associated with the preceding whole
number value.
© Richard C. Larson 2014
25
Seismic Energy Yield
5.6 kg (12.4 lb)
32 kg (70 lb)
178 kg (392 lb)
1 metric ton
5.6 metric tons
32 metric tons
178 metric tons
1 kiloton
5.6 kilotons
32 kiloton
178 kilotons
1 megaton
5.6 megatons
50 megatons
178 megatons
1 gigaton
5.6 gigatons
32 gigatons
Example
Hand grenade
Construction site blast
WWII conventional
bombs
late WWII conventional
bombs
WWII blockbuster bomb
Massive Ordnance Air
Blast bomb
Chernobyl nuclear
disaster, 1986
Small atomic bomb
Average tornado (total
energy)
Nagasaki atomic bomb
Little Skull Mtn., NV
Quake, 1992
Double Spring Flat, NV
Quake, 1994
Northridge quake, 1994
Tsar Bomba, largest
thermonuclear weapon
ever tested
Landers, CA Quake, 1992
San Francisco, CA
Quake, 1906
Anchorage, AK Quake,
1964
2004 Indian Ocean
earthquake
© Richard C. Larson 2014
Richter Magnitude
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
10.0
26
Flaws of Averages: Summary Points
• Averages can be deceiving.
• Treating a distribution as its average value
usually results in incorrect inferences.
• Averages as experienced by one
population may be very different from
those experienced by another.
• Ignore “outliers” at your peril.
© Richard C. Larson 2014
27
And we haven’t even
considered…
• Regression to the Mean
• Variance
• Exponential smoothing
–Example: Baseball batting averages
• And much more…
© Richard C. Larson 2014
28
One More Average:
Based on Dimensionality Arguments
• Mean travel distance in a city, N police cars,
area A.
A
D =C
N
• This is a Square Root Law.
• In our analysis of Big Data, we can look for this
type of behavior.
© Richard C. Larson 2014
29
Let’s Now Switch:
From Averages to a
Simple Operational Model
© Richard C. Larson 2014
30
What Kinds of Queues Occur in
Systems of Interest to ESD?
Queues,
Queues
Everywhere!
© Richard C. Larson 2014
31
Queueing System
Arriving
Customers
Queue of
Waiting
Customers
SERVICE
FACILITY
Departing Customers
© Richard C. Larson 2014
32
Queues, Queues Everywhere!
• Queueing Theory: 100 Years Old!
• Most queues are complicated, and folks
want to simulate almost all of the detail.
• And today there are numerous files of
Big Data drawn from queues.
• But let’s look at simple models first! –
to Guide Us
© Richard C. Larson 2014
33
It May be Little, But It’s The Law!
L = lW
• L= Time average number of customers in the
system, both in queue and in service
• l = Average rate of arrivals of customers into
the system
• W = Mean time spent by a customer in the
system, both in queue and in service
© Richard C. Larson 2014
34
L = lW
• Formula applies in all sorts of places, including
those not normally thought of as queues.
• Example: Annual rate of new hires of
assistant professors in a university.
• MIT: L = 1,000 tenure-track faculty members
W= mean duration of a faculty career
• If W moves upwards from 20 to 22 years, l
moves down accordingly, since L = 1,000
remains constant.
© Richard C. Larson 2014
35
The M/M/k Queue
Queue notation: Input/Service/Servers
• First M: “Memoryless” input
process, meaning Poisson process
• Second M: “Memoryless” service
time, meaning exponential
probability density function
• k = number of servers.
© Richard C. Larson 2014
36
M/M/1 Queue
1
Wµ
1- r
Mean Wait vs. Rho
25
Queue Explodes!
20
W
15
Series1
Note the Elbow!
10
5
0
0
0.2
0.4
0.6
0.8
1
Rho
Rho = r = fraction of time that server is
busy serving customers
© Richard C. Larson 2014
37
Elbow as we Increase the Number
of Servers (k = 1,2,3,9,16)
© Richard C. Larson 2014
38
Do You See Why Large Call Centers
are More Productive?
© Richard C. Larson 2014
39
What Do You See as a Role for
Big Data Analysis Here?
© Richard C. Larson 2014
40
D = Deterministic
© Richard C. Larson 2014
41
Averages in Queues
• Performance degrades as arrival rate
increases and/or mean service time
increases.
• Performance degrades as Variance of
time between arrivals increases and/or
variance of the service time increases.
• Can you think of examples?
© Richard C. Larson 2014
42
Now for Final Switch:
From Queueing Overview
to Case Study
© Richard C. Larson 2014
43
Queue Inference Engine:
A Personal Big Data–Small Models Experience
• It started with Reams of Old-fashioned Paperbased Computer Printouts
© Richard C. Larson 2014
44
Queue Inference Engine:
• Big Data:
– Time ATM card inserted;
– Time ATM Transaction completed.
© Richard C. Larson 2014
45
Queue Inference Engine:
• Knowing the probability properties of the
Arrival Process, a “Poisson Process,” we were
able to derive a mathematically valid
algorithm to determine many statistics of
customers’ queue delays.
• It’s called an O(N3) algorithm, since the
number of computations grows as the 3rd
power of the number of customers in a busy
period.
© Richard C. Larson 2014
46
Queue Inference Engine:
• Imagine receiving your monthly bank statement
and with it is a statement of the times you
spent waiting in bank queues. The queues
could include both those involving human
tellers and automatic teller machines (ATMs).
• With the technology of the Queue Inference
Engine (QIE) such an innovation is now well
within the realm of possibility.
© Richard C. Larson 2014
47
Queue Inference Engine:
• With our first results published in 1990, Dr. David
Simchi-Levi and others call this one of the first
applications of Big Data analysis to modern-day
problems: “This is just a beautiful example of how
data drive new research…” (Simchi-Levi, 2014)
• But this “QIE” Big Data algorithm could not have
been derived without marrying Small Models (with
their behavior) with Big Data recursive thinking.
© Richard C. Larson 2014
48
Big Data and Small Models
© Richard C. Larson 2014
49
© Richard C. Larson 2014
50
References
• Larson, R.C., "The Queue Inference Engine: Deducing Queue Statistics
From Transactional Data." Management Science 36(5):586-601, May
1990.
• Larson, Richard C., QUEUE INFERENCE ENGINE, chapter in Encyclopedia of
Operations Research and Management Science, Centennial Edition, Saul I.
Gass and Carl M. Harris (eds.), Kluwer, Boston, 2001, pp.674-679.
• Jones, Lee K. and Richard C. Larson, "Efficient Computation of Probabilities
of Events Described by Order Statistics and Applications to Queue
Inference." ORSA Journal on Computation., vol. 7, no. 1, Winter 1995, pp.
89-100.
• Gross, Donald and Richard C. Larson, “Queuing Systems,” in International
Encyclopedia of Business and Management (IEBM), 2nd edition, 8-volume
set, Malcolm Warner, ed., Thomson Learning, London, U.K., 2001, pp.
5502-5513.
• Larson, Richard C. and Mauricio Gomez Diaz, “Nonfixed Retirement Age
for University Professors: Modeling Its Effects on New Faculty Hires,”
Service Science, V. 4, No. 1, March 2012, p. 69-78.
• Simchi-Levi, David. “OM Research: From Problem-Driven to Data-Driven
Research,” M&SOM, 16 (1) 2014 pp. 2-10.
© Richard C. Larson 2014
51