Monte Carlo Resampling (a.k.a Bootstrapping)

Bootstrap: “to pull oneself up by one's bootstraps”; to seemingly do the impossible

• Much of early statistics was about what to do when we didn't have enough data

• Now we are more likely to have too much data

• In early statistics we needed computational shortcuts

• Now we have immense computational power

• Bootstrap resampling allows us harness computational power to get an infinite amount of data

We load some data

# load the data
colClasses=c('POSIXct','numeric','numeric',
             'numeric','numeric','numeric',
             'numeric')
# load the sales data
path<-"C:/Users/dgn2/Documents/R/IS606/"
salesFile<-"sales.csv"
sales <- read.csv(paste0(path,salesFile),
                  header=TRUE,
                  stringsAsFactors=FALSE,
                  colClasses=colClasses)
# load the price and cost details
detailsFile<-"details.csv"
details <- read.csv(paste0(path,detailsFile),
                    header=TRUE,
                    stringsAsFactors=FALSE)

# extract the price and cost of each product
turkeyPrice<-details$price[2]
turkeyCost<-details$cost[2]

# define the function to compute revenue, expense, and P&L
pnlUnderScenario<- function(demand,supply,pricePerUnit,
                            costPerUnit){
  expense<-supply*costPerUnit
  unitsSold<-demand
  flag<-supply-demand<0
  unitsSold[flag]<-supply[flag]
  revenue<-unitsSold*pricePerUnit
  pnl<-sum(revenue-expense)
  }

Let's take a look at the P&L

# turkey revenue, expense, and P&L
turkeyDemand<-sales[,3]
turkeySupply<-sales[,6]
turkeyExpense<-turkeySupply*turkeyCost
turkeyUnitsSold<-turkeyDemand
turkeyFlag<-turkeySupply-turkeyDemand<0
turkeyUnitsSold[turkeyFlag]<-turkeySupply[
  turkeyFlag]
turkeyRevenue<-turkeyUnitsSold*turkeyPrice
turkeyPnl<-turkeyRevenue-turkeyExpense

We sample from the data with replacement

# set the parameters for resampling
dimension<-dim(sales)
nRows<-dimension[1]
nCols<-dimension[2]
nPaths<-1000
# create the data resampling index
resampleIndex<-sample(1:nRows,
                      nRows*nPaths,
                      replace=TRUE,
                      prob=NULL)

# create the resampled data for each column
resampledData<-sales[resampleIndex,]

Re-organize the data so that we have nRows by nPath matrix

# reshape the demand data
turkeyDemandPaths<-data.frame(matrix(
  resampledData[,3],nrow=nRows,ncol=nPaths))
# reshape the supply data
turkeySupplyPaths<-data.frame(matrix(
  resampledData[,6],nrow=nRows,ncol=nPaths))

Each row represents a point in time and each column represents an alternate reality consistent with the variability of observed observations

Say we care about the expected value of supply versus demand.

turkeyCumulativeDemand<-cumsum(
  turkeyDemandPaths)
turkeyCumulativeSupply<-cumsum(
  turkeySupplyPaths)

Resampling gives us a number of alternate realities consistent with the variation in the data

From these paths / alternate realities we can determine the distribution of the statistic of interest (in this case, the expected value)

turkeyExpectedDemand<-mean(as.numeric(
  turkeyCumulativeDemand[nRows,]/nRows))
turkeyExpectedSupply<-mean(as.numeric(
  turkeyCumulativeSupply[nRows,]/nRows))

The expected demand for turkey significantly exceeded supply (22.0643615 versus 17.2389231).

We find the distribution of P&L by resampling the original supply and demand.

# find the distribution of P&L using the original supply 
# and demand
turkeyPnLUnderScenarios<-0

for (pathIndex in 1:nPaths){
  turkeyPnLUnderScenarios[pathIndex]<-
    pnlUnderScenario(
    turkeyDemandPaths[,pathIndex],
    turkeySupplyPaths[,pathIndex],turkeyPrice,
    turkeyCost)
  }

# maximize turkey P&L for each price scenario
turkeySupplyRange<-min(turkeyDemand):
  max(turkeyDemand)
nScenarios<-length(turkeySupplyRange)
turkeyPnLUnderSupplyScenarios<-matrix(rep(0,
  nPaths*nScenarios),nrow=nPaths,ncol=nScenarios)
for (scenarioIndex in 1:nScenarios){
  turkeySupplyScenario<-rep(turkeySupplyRange[
    scenarioIndex],nRows,nrow=nRows,ncol=1)
  for (pathIndex in 1:nPaths){
    turkeyPnLUnderSupplyScenarios[pathIndex,
      scenarioIndex]<-pnlUnderScenario(
    turkeyDemandPaths[,pathIndex],
      turkeySupplyScenario,
      turkeyPrice,turkeyCost)
  }
}

We can determine the distribution of P&L for a range of supply scenarios, then find the supply that maximizes expected P&L for each product.

# compute the expected P&L under each scenario
turkeyExpectedPnLUnderScenario<-colMeans(
  turkeyPnLUnderSupplyScenarios)
# compute the percentile bounds
lowerBound<-apply(turkeyPnLUnderSupplyScenarios, 2, 
                  quantile, probs = c(0.05))
upperBound<-apply(turkeyPnLUnderSupplyScenarios, 2, 
                  quantile, probs = c(0.95))
# find the max P&L
turkeyMaxExpectedPnLUnderScenario<-max(
  turkeyExpectedPnLUnderScenario)
# find the max P&L index
turkeyMaxIndex<-turkeyExpectedPnLUnderScenario==
  turkeyMaxExpectedPnLUnderScenario
# find the optimal supply
turkeyOptimalSupply<-turkeySupplyRange[
  turkeyMaxIndex]

• Visualize variation
• Determine the distribution of any statistic/parameter; helps determine whether a result could be explained just by variation (i.e., significance)
• Solve path-dependent problems

Most Important Points:

• Anyone can use this to get the right answer
• Anyone can understand this
• Computational power is cheaper than the skills needed to get the right answer the traditional way!
• Welcome to the era of computation!

Efron, Bradley; Tibshirani, Robert J. (1993). An introduction to the bootstrap, New York: Chapman & Hall.

Simon, J. L. The Philosophy and Practice of Resampling Statistics http://www.juliansimon.com/writings/Resampling_Philosophy/

Simon, J. L. (1997): Resampling: The New Statistics http://www.resample.com/intro-text-online/