model_thinking.3

Model Thinking
==============

https://www.coursera.org/modelthinking/class/


Section 3 - Aggregation
=======================

Aggregation can produce unexpected results
	saw this in Schelling, where macro behavior was very different from micro motives
	"More is different"
	Emergent properties (macro phenomena)
		wetness, from individual water molecules
		consciousness, from neurons, axons and dendrites

complexity of adding things up

We will analyze aggregation of 
	Actions - central limit theorem
		allows us to predict points, and understand data
	Single Rule - game of life
		allows us to understand patterns
	Family of Rules - cellular automata
		allows us to understand class of outcome
			equilibria, patterns, complexity, complete randomness/chaos
	Preferences - making collective choices
		work through logic


Central Limit Theorem
---------------------

independent actions of individual actors

probability distribution
	list of 
		all of the possible outcomes
		the probabilities of each of those outcomes

	e.g. # of people in a family that go to the beach

	#  P
	---------
	0  .10
	1  .15
	2  .40
	3  .15
	4  .20

	sum(P) always equals 1.0

CLT tells us that
	a whole bunch of individual, independent events will aggregate to a normal distribution (bell curve)
	the most likely outcome is the one in the middle 

	CLT allows us to predict a lot of things

binomial distribution
	a discrete probability distribution with just two possible outcomes
		pass/fail, heads/tails, live/die
	P - probability of outcome (heads, pass,live)
	N - number of trials

mean of a curve
	tells us the most likely outcome

	mean = pN
		15% probability of going to beach on a given day
		(0.15) * 1000

		if P = 0.5 - (outcomes are equally likely e.g. flipping a coin)
			mean reduces to N/2
				1000/2

Standard deviation of a curve
	σ (sigma) - tells us how wide/spread out the curve is

	it will always be the case that (just for a normal dist?)
		68% of the population will be within 1 σ
		95% of the population will be within 2 σ
		99% of the population will be within 3 σ

	σ  +/-%
	-------
	1  34.1
	2  13.6
	3   2.1
	4   0.1

	if you know the mean and the standard deviation, you can determine the range of probable outcomes
		e.g. mean = 100 persons show up at a meeting
		σ = 2
			68% probability between 98-102
			95% probability between 96-104
		σ = 15
			68% probability between 85-115
			95% probability between 70-130

	σ = √P(1-P)N
		if P = 0.5 - (outcomes are equally likely e.g. flipping a coin)
			σ reduces to √N/2

Example Boeing 747
	380 seats
	400 tickets sold
	90% ticketholders show up

	mean = (0.9)400 = 360
	σ = √(0.9)(0.1)400 = √36 = 6

	68% probability between 354-366
	95% probability between 348-372
	99% probability between 342-378

CLT says that
	when you add random variables
		that are independent
		with finite (bounded) variance
	they sum to normal distribution

Bimodal distribution

Not all things follow a normal distribution
	stock prices
		not independent - people buy/sell when others buy/sell


Six Sigma
---------

3.4/1,000,000

Example - Banana sales
	mean 500 lb/day
	σ 10 lb/day

	how many lbs needed to handle a 6σ surge?
		500 + 6(10) = 560

Example - sheet metal
	500-560mm required thickness
	530mm avg

	what must σ be to ensure that 6σ operation lies within the required thickness?
		560-530 = 30
		30/6 = 5


Game of Life
------------

one of a class of model called cellular automata

shows how things aggregate
	practice our thinking
	learn subtleties of aggregation
leads to some surprising conclusions

shows how hard it is to infer at the macro level from what's going on at the micro level

John Conway - Cambridge mathematician

center of tic-tac-toe grid 
	eight neighbors
	iterations based on neighbors
		if the cell is off
			turn on if exactly three neighbors are on
		if the cell is on
			if <2 or >3 neighbors on, turn off
			otherwise, stay on

patterns
	beacon - two 2x2 squares connected at LR/TL corners
	figure 8 - two 3x3 squares connected at LR/TL corners
	f-pimento - XX
                   XX
                    X
		produces gliders
		almost seems like it's alive
		compare to human brain 
			has neurons that follow very simple rules
			that are connected in such a way 
			that they produce really novel patterns
				memory, thought, cognition, personality, 

simple things, following simple rules, can create incredibly elaborate patterns

emergent patterns

Understand class of outcome
	1) fixed points (steady state)
	2) alternation (blinking)
	3) randomness
	4) complexity (repeating patterns)

What GoL teaches us
	self-organized patterns appear without a designer
		gliders, blinkers, glider guns...
	emergence - functionalities appear
		patterns that have, or seem to have functions
		gliders, glider guns, counters, computers
		consciousness and cognition are emergent phenomena in the brain
	help us get the logic right
		simple rules produce incredible phenomena



Cellular Automata
-----------------

one dimensional array along the X axis
	each cell only has two neighbors - left, right
	allows the Y axis to represent time
		following a cell downward let's you see its state at each iteration

8 possible states
	000
	001
	010
	011
	100
	101
	110
	111

	each of these states can result in one of two outcomes - on or off
	thus, there are 2**8 (256) possible rule sets, or universes
		exhaustively studied by Wolfram in ANKoS
	rule sets numbered 0-255 

4 classes of behavior
	1) fixed points (steady state)
	2) alternation (blinking)
	3) randomness
	4) complexity (repeating patterns)

Rule 30 produces perfect randomness
rule 110 produces complex patterns

"It from bit"
	anything and everything in be universe can be generated by a series of yes/no questions
		thus represented by a string of binary digits

Langton's λ
	tells us what these outcomes look like
	simpler approach than Wolfram
	number or percentage of things that are on (e.g. 3 or 3/8)
	0/8 - everything dies
	1/8 - alternation, blinking
	4/8 - (rule 30) chaotic, random
	5/8 - (rule 110) complex

	λ	sets	CIII	CIV
	0	1	0	0
	1	8	0	0
	2	28	2	0
	3	56	4	1
	4	70	20	4
	5	56	4	1
	6	28	2	0
	7	8	0	0
	8	1	0	0

	chaos (class 3) can only occur when λ is 2-6
	complexity (class 4) can only occur when λ is 3-5
	thus chaos and complexity may arise when there are intermediate levels of interdependence

	complexity in stock markets, due to interdependence of individual actors

Summary
	simple rules can combine to form anything
	"it from bit"
	complexity and randomness require interdependency


Preference Aggregation
----------------------

need a different mathematical structure than that used for aggregating actions/numbers

preferences
	revealed actions - give people a choice and askwhat they prefer
	preference ordering
		complete, ranked listing of someone's preferences within a category/class
	transitive, rational - cannot be a conflict/inconsistency in the ordering

preference aggregation methods
	tally method - tally votes at each position (1st, 2nd...)
	pairwise method - for each unique pair, tally preferences

Condorcet's paradox
	individual preferences are consistent, but aggregate preferences may not be
	severe implications
		when we think about voting, we could get a random outcome
		people might want to vote strategically
		creates incentives, opportunities, conditions to manipulate the system


========================================

Statistics
----------

population - The total set of observations that can be made
sample     - a set of observations drawn from a population
parameter  - a measurable characteristic of a population, such as a mean or standard deviation.
statistic  - a measurable characteristic of a sample, such as a mean or standard deviation.
sampling method - a procedure for selecting sample elements from a population.
random number - a number determined totally by chance, with no predictable relationship to any other number.
random number table - a list of numbers


Measures of Central Tendency - mean, median, mode
Proportions and Percentages - degree to which a population possesses a particular attribute

N - a population
n - a sample
_
X - the population mean
μ - the population mean
_
x - the sample mean
P - The proportion of elements in the population that has a particular attribute.
p - The proportion of elements in the sample that has a particular attribute.
Q - The proportion of elements in the population that does not have a specified attribute. Note that Q = 1 - P.
q - The proportion of elements in the sample that does not have a specified attribute. Note that q = 1 - p.


Measures of Variability - range, variance, standard deviation

σ² - The variance of the population.
σ  - The standard deviation of the population.
s² - The variance of the sample.
s  - The standard deviation of the sample.

Range = Maximum value - Minimum value

Variance of a random variable
	σ² = Σ ( Xi - μ )² / N
	              _
	s² = Σ ( xi - x )² / ( n - 1 )

Variance of proportions
	σ² = PQ / n
	s² = pq / ( n - 1 )

Standard deviation
	σ = √σ²
	s = √s²

Sampling Distribution of the Mean
	 _
	μx = μ
	 _
	σx = σ * sqrt( 1/n - 1/N )
	   = σ / sqrt(n) for large N