What is a model?

* representation/general "rule-of-thumb" of real-life(?) process
* representation usually a simplified version
* abstraction of reality but based on subject-matter understanding
[PURPOSEFUL REPRESENTATION OF REALITY]
[CARICATURE(sp?) OF REALITY]
[GE Box:  All models are wrong but some are useful ...]

* some predictive value
* often some mathematical/functional form
* + random elements incorporated via statistical ideas
* provides formalization of description => replication by
  others / communication
* limited in some ways: 1) assumptions  2) real data

* evaluate model performance => improve it with testing with data
* may be more than one model that explains the data =>
  comparing performance and selecting between competing
  models?  synthesizing estimates over models

* models vary between disciplines (both in origin and use)

* complexity, cost effectiveness, time constraints ...

* good models add to general understanding - generalizable

* model development is an iterative process

What do we learn from models?

* models have implications - predictions that implications for
  management, policy, social, physical, monetary

* understand relationships between variables


Example:  Suppose I asked you what is the probability of 2 boys
          and 2 girls in a family of 4 kids.  How would you
          answer this question?


Possible outcomes?
  BBBB
  GBBB
  BGBB
  ...

Assumptions?
  independence of outcome in family
  P(B)=P(G) = 1/2  *****
  single birth (no twins, triplets, quadruplets)
  
Strategies?
  1) enumerate outcomes - 
     use TREE diagram ... recognize probability of each branch
         (used idea of equal prob + independence)
     accumulate probs for outcomes of interest

6/16 satisfy the condition of interest

  2) consider some random variable -
     Y= number of girls in family = {0,1,2,3,4}

     Find P(Y=2)

     Is there a probaility model for this?  maybe ... binomial
     P(Y=2) = 4!/(2! 2!) * (1/2)^2 * (1- 1/2)^2

R ...
dbinom(2,4,prob=1/2)
[1] 0.375

   3)  simulation based ...

  i)  simulate family
 ii)  do this lots of times
iii)  determine proportion of simulated families with 2G,2B



14/36
3/8
pi.hat <- 14/36
B <- 2*sqrt(pi.hat*(1-pi.hat)/36)

pi.hat
B

# form CI
pi.hat + c(-1,1)*B

# simulated families to be within 5% or .5% of true proportion
B05 <- 4*(pi.hat)*(1-pi.hat)/(.05^2)
B005 <- 4*(pi.hat)*(1-pi.hat)/(.005^2)
B05
B005

B05
380.2469
B005
38024.69

#
# MC  problem HW 2 hints
#

BB <- runif(10)
BB
1-BB

max(BB, 1-BB)
pmax(BB, 1-BB)
pmin(BB, 1-BB)
?pmax

ratio <- pmax(BB, 1-BB)/pmin(BB, 1-BB)
ratio

hist(ratio)
BB <- runif(4000)
ratio <- pmax(BB, 1-BB)/pmin(BB, 1-BB)
hist(ratio)
BB <- runif(400)
ratio <- pmax(BB, 1-BB)/pmin(BB, 1-BB)
hist(ratio)
quantile(ratio)
boxplot(ratio)
boxplot(log10(ratio))

# only break between .2 and .8

BB <- runif(4000, min=0.2, max=0.8) 
ratio <- pmax(BB, 1-BB)/pmin(BB, 1-BB)
hist(ratio)
history()


# COMMENTS

# BONUS

t <- 50     # tagged in 1st sample
N <- 200    # population size

n <- 40     # size of the second sample

mypop <- rep(  c(0, 1), c(N-t, t))

mypop
  [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 [38] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 [75] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[112] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[149] 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[186] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

?sample

sample(mypop, size=n)
 [1] 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0
[39] 1 0

s.tagged <- sum(sample(mypop, size=n))
> s.tagged
[1] 10

N.hat <- t*n/s.tagged
N.hat

samp1 <- sample(mypop, size=n)
s.tagged <- sum(sample(mypop, size=n))
N.hat <- t*n/s.tagged
N.hat
samp2 <- sample(mypop, size=n)
rbind(samp1, samp2)

#  results matrix set up
res.mat<- matrix(nrow=2000, ncol=n, 0)
res.mat
dim(res.mat)

# filling the results matrix with
#   each row corresponding to a different sample (1)
for (irow in 1:2000) res.mat[irow,] <- sample(mypop, size=n)
res.mat[1:5,]

# figuring out number tagged in each sample (2)
?apply
s.tagged <- apply(res.mat, 1, sum)
length(s.tagged)
summary(s.tagged)

# estimating the population size (3)
Nhat <- t*n/s.tagged
hist(Nhat)
mean(Nhat)
summary(Nhat)
sd(Nhat)
B <- 2*sd(Nhat)/2000
B



# what about inverse sampling - s=10 fixed
#   and n varies

res.mat<- matrix(nrow=2000, ncol=N, 0)
for (irow in 1:2000) res.mat[irow,] <- sample(mypop, N)

cumsum(permute.pop)
csum <- cumsum(permute.pop)
1:200[csum==10]
(1:200)[csum==10]
min((1:200)[csum==10])







#
# even more homework comments
#


1/sqrt(2*pi)*exp(-x*x/2)

1/sqrt(2*pi)*exp(-x^2/2)

my.norm<- function(x)  (1/sqrt(2*pi))*exp(-x*x/2)

?log
pi
arcos(-1)
my.norm<- function(x)  (1/sqrt(2*pi))*exp(-x*x/2)
my.norm(0)
my.norm(1)
my.norm(-1)
dnorm(0)
dnorm(1)
?rnorm
e
xx <- runif(10, min=-2, max=2)
yy <- runif(10, min=0, max=0.4)
xx
yy
iunder <- as.numeric(yy <= 1/sqrt(2*pi)*exp(-xx^2))
iunder
iunder <- as.numeric(yy <= 1/sqrt(2*pi)*exp(-(xx^2)/2))