10 simulation

Hadley Wickham

Stat405Simulation

Thursday, 23 September 2010

1. Homework comments

2. Mathematical approach

3. More randomness

4. Random number generators


Just graded your organisation and code, and focused my comments there.

Biggest overall tip: use floating figures (with \figure{...}) with captions. Use \ref{} to refer to the figure in the text.

Captions should start with brief description of plot (including bin width if applicable) and finish with brief description of what the plot reveals.

Will grade captions more aggressively in the future.

Homework


Code

Gives explicit technical details.

Your comments should remind you why you did what you did.

Most readers will not look at it, but it’s very important to include it, because it means that others can check your work.


Mathematical approach

Why are we doing this simulation? Could work out the expected value and variance mathematically. So let’s do it!

Simplifying assumption: slots are iid.


calculate_prize <- function(windows) { payoffs <- c("DD" = 800, "7" = 80, "BBB" = 40, "BB" = 25, "B" = 10, "C" = 10, "0" = 0)

same <- length(unique(windows)) == 1 allbars <- all(windows %in% c("B", "BB", "BBB"))

if (same) { prize <- payoffs[windows[1]] } else if (allbars) { prize <- 5 } else { cherries <- sum(windows == "C") diamonds <- sum(windows == "DD")

prize <- c(0, 2, 5)[cherries + 1] * c(1, 2, 4)[diamonds + 1] } prize}


slots <- read.csv("slots.csv", stringsAsFactors = F)

# Calculate empirical distributiondist <- table(c(slots$w1, slots$w2, slots$w3))dist <- dist / sum(dist)

slots <- names(dist)


poss <- expand.grid( w1 = slots, w2 = slots, w3 = slots, stringsAsFactors = FALSE)

poss$prize <- NAfor(i in seq_len(nrow(poss))) { window <- as.character(poss[i, 1:3]) poss$prize[i] <- calculate_prize(window)}


Your turn

How can you calculate the probability of each combination?

(Hint: think about subsetting. Another hint: think about the table and character subsetting. Final hint: you can do this in one line of code)

Then work out the expected value (the payoff).


poss$prob <- with(poss, dist[w1] * dist[w2] * dist[w3])

(poss_mean <- with(poss, sum(prob * prize)))

# How do we determine the variance of this# estimator?


More randomness


Sample

Very useful for selecting from a discrete set (vector) of possibilities.

Four arguments: x, size, replace, prob


How can you?

Choose 1 from vector

Choose n from vector, with replacement

Choose n from vector, without replacement

Perform a weighted sample

Put a vector in random order

Put a data frame in random order


# Choose 1 from vectorsample(letters, 1)

# Choose n from vector, without replacementsample(letters, 10)sample(letters, 40)

# Choose n from vector, with replacementsample(letters, 40, replace = T)

# Perform a weighted samplesample(names(dist), prob = dist)


# Put a vector in random ordersample(letters)

# Put a data frame in random orderslots[sample(1:nrow(slots)), ]


Your turn

Source of randomness in random_prize is sample. Other options are:

runif, rbinom, rnbinom, rpois, rnorm, rt, rcauchy

What sort of random variables do they generate and what are their parameters? Practice generating numbers from them.


Function Distribution Parameters

runif Uniform min, max

rbinom Binomial size, prob

rnbinom Negative binomial size, prob

rpois Poisson lambda

rnorm Normal mean, sd

rt t df

rcauchy Cauchy location, scale


Distributions

Other functions

• r to generate random numbers

• d to compute density f(x)

• p to compute distribution F(x)

• q to compute inverse distribution F-1(x)


# Easy to combine random variables

n <- rpois(10000, lambda = 10)x <- rbinom(10000, size = n, prob = 0.3)qplot(x, binwidth = 1)

p <- runif(10000)x <- rbinom(10000, size = 10, prob = p)qplot(x, binwidth = 0.1)

# cf.qplot(runif(10000), binwidth = 0.1)


# Simulation is a powerful tool for exploring # distributions. Easy to do computationally; hard # to do analytically

qplot(1 / rpois(10000, lambda = 20))qplot(1 / runif(10000, min = 0.5, max = 2))

qplot(rnorm(10000) ^ 2)qplot(rnorm(10000) / rnorm(10000))

# http://www.johndcook.com/distribution_chart.html


http://www.johndcook.com/distribution_chart.html

http://www.johndcook.com/distribution_chart.html

Your turn


RNGComputers are deterministic, so how

do they produce randomness?



How do computers generate random numbers?

They don’t! Actually produce pseudo-random sequences.

Common approach: Xn+1 = (aXn + c) mod m

(http://en.wikipedia.org/wiki/Linear_congruential_generator)


http://en.wikipedia.org/wiki/Linear_congruential_generator




next_val <- function(x, a, c, m) { (a * x + c) %% m}

x <- 1001(x <- next_val(x, 1664525, 1013904223, 2^32))

# http://en.wikipedia.org/wiki/List_of_pseudorandom_number_generators

# R uses# http://en.wikipedia.org/wiki/Mersenne_twister


http://en.wikipedia.org/wiki/List_of_pseudorandom_number_generators




# Random numbers are reproducible!

set.seed(1)runif(10)

set.seed(1)runif(10)

# Very useful when required to make a reproducible# example that involves randomness


Atmospheric radio noise: http://www.random.org. Use from R with random package.

Not really important unless you’re running a lottery. (Otherwise by observing a long enough sequence you can predict the next value)

True randomness


http://www.random.org




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10 simulation