############################################################# # R code to accompany notes for lecture 8 (bootstrapping) # # 3 February 2011, 36-402 # ############################################################# ##### Code Example 1 ##### # Sketches of functions for performing basic bootstrap inference tasks # These are not very flexible and should probably be thought of as pseudo-code # or sketches for more complicated industrial-strength functions # Perform multiple independent simulations and calculate the same statistic # on each # Inputs: Number of bootstrap replicates (B) # Function to take a data set and return an estimate (statistic) # Function to produce a simulated data set (simulator) # Optional extra arguments for simulator, passed along by ... # Presumes: Output of simulator is formatted to be input for statistic # statistic needs no extra arguments # Output: Array of bootstrapped statistic values rboot <- function(B, statistic, simulator, ...) { tboots <- replicate(B, statistic(simulator(...))) return(tboots) } # Calculate bootstrap standard errors in a statistic # Inputs: Function to produce a simulated data set (simulator) # Function to calculate the statistic (statistic) # Number of bootstrap replicates (B) # Optional extra arguments to simulator() # Calls: rboot # Presumes: Output of simulator is formatted to be input for statistic # statistic needs no extra arguments # statistic() returns a numeric scalar # Output: the standard error bootstrap.se <- function(simulator, statistic, B, ...) { tboots <- rboot(B, statistic, simulator, ...) se <- sd(tboots) return(se) } ##### Code Example 2 ##### # See comments at the head of Code Example 1 # Bootstrap bias calculation # Inputs: function to make a simulated data set (simulator) # Function to calculate the statistic (statistic) # Number of bootstrap replicates (B) # Empirical value of statistic (t.hat) # Optional extra arguments for simulator (...) # Calls: rboot # Presumes: Output of simulator is formatted to be input for statistic # statistic needs no extra arguments # The mean function works on statistic's return values # Output: The bias bootstrap.bias <- function(simulator, statistic, B, t.hat, ...) { tboots <- rboot(B, statistic, simulator, ...) bias <- mean(tboots) - t.hat return(bias) } ##### Code Example 3 ###### # See comments at the head of Code Example 1 # Bootstrap confidence intervals # Applies the basic or "pivot" method, see notes # Inputs: function to make a simulated data set (simulator) # Function to calculate the statistic (statistic) # Number of bootstrap replicates (B) # Empirical value of statistic (t.hat) # One minus the desired confidence level (alpha) # Optional extra arguments for simulator (...) # Calls: rboot # Presumes: Output of simulator is formatted to be input for statistic # statistic needs no extra arguments # The quantile function works on statistic's return values # alpha is in [0,1] # Output: List with the lower and upper confidence limits (ci.lower, ci.upper) bootstrap.ci.basic <- function(simulator, statistic, B, t.hat, alpha, ...) { tboots <- rboot(B,statistic, simulator, ...) ci.lower <- 2*t.hat - quantile(tboots,1-alpha/2) ci.upper <- 2*t.hat - quantile(tboots,alpha/2) return(list(ci.lower=ci.lower,ci.upper=ci.upper)) } ##### Code Example 4 ##### # See comments at the head of Code Example 1 # Calculate p-value by bootstrapping # Inputs: function to calculate a test statistic (test) # Function to produce a surrogate data set (simulator) # Number of bootstrap replicates (B) # Observed value of test statistic (testhat) # Optional extra arguments to simulator (...) # Calls: rboot # Presumes: Output of simulator is formatted to be input for test # test needs no extra arguments # test returns a value for which the comparison >= is defined # Output: The p-value boot.pvalue <- function(test,simulator,B,testhat, ...) { testboot <- rboot(B=B, statistic=test, simulator=simulator, ...) p <- (sum(test >= testhat)+1)/(B+1) return(p) } ##### Code Example 5 ##### # See comments at the head of Code Example 1 # Calculate p-values by the "double bootstrap", correcting for the effects of # parameter estimation # Inputs: function to calculate a test statistic (test) # Function to produce a surrogate data set (simulator) # Number of first-level bootstrap replicates (B1) # Number of second-level bootstrap replicates (B2) # Function to estimate parameter from data (estimator) # Empirical estimate of parameter (thetahat) # Observed value of test statistic (testhat) # Optional extra arguments to simulator (...) # Calls: rboot, boot.pvalue # Presumes: Output of simulator is formatted to be input for test # test needs no extra arguments # test returns a value for which the comparison >= is defined # simulator takes an argument named theta # The output of estimator has the right format to be the theta argument # Output: The p-value doubleboot.pvalue <- function(test,simulator,B1,B2, estimator, thetahat, testhat, ...) { for (i in 1:B1) { xboot <- simulator(theta=thetahat, ...) thetaboot <- estimator(xboot) testboot[i] <- test(xboot) pboot[i] <- boot.pvalue(test,simulator,B2,testhat=testboot[i], theta=thetaboot, ...) } p <- (sum(testboot >= testhat)+1)/(B1+1) p.adj <- (sum(pboot <= p)+1)/(B1+1) } ###### Set up the Pareto law of wealth example # Data file and pareto.R are both available through the class website # Load the Pareto-related functions source("pareto.R") # Load the data wealth <- scan("wealth.dat") # Fit the distribution wealth.pareto <- pareto.fit(wealth,threshold=9e8) # pareto.fit() will go through a somewhat complicated procedure to estimate # the scaling threshold, if run with threshold="find" (the default value). # (See Clauset et al. 2009.) That's where this threshold came from. ##### Figure 2 ##### # See caption in notes # plot.survival.loglog is from pareto.R plot.survival.loglog(wealth,xlab="Net worth (dollars)", ylab="Fraction of individuals at or above that net worth") rug(wealth,side=1,col="grey") curve((302/400)*ppareto(x,threshold=9e8,exponent=2.34,lower.tail=FALSE), add=TRUE,lty=2,from=9e8,to=2*max(wealth)) How much uncertainty is there in this estimate of the exponent? Naturally, we'll bootstrap. We saw last time how to generate Pareto-distributed random variables using the quantile method; this, along with some related functions, is part of the file \texttt{pareto.R} on the course website. With that tool, parametric bootstrapping proceeds as in Code Example \ref{code:pareto.se}. ##### Code Example 6 ##### # Parametric bootstrap standard error and bias for the Pareto # Generate multiple data sets from the Pareto and re-estimate the exponent on # each one # Inputs: Number of bootstrap replicates (B) # Scaling exponent (exponent) # Threshold (x0) # Size of each data set (n) # Calls: rpareto and pareto.fit from pareto.R # Output: Vector of B estimates of the exponent rboot.pareto <- function(B,exponent,x0,n) { replicate(B,pareto.fit(rpareto(n,x0,exponent),x0)$exponent) } # Parametric boostrap standard error for the Pareto exponent # Inputs: Number of bootstrap replicates (B) # Scaling exponent (exponent) # Threshold (x0) # Size of each data set (n) # Calls: rboot.pareto # Output: The standard error pareto.se <- function(B,exponent,x0,n) { return(sd(rboot.pareto(B,exponent,x0,n))) } # Parametric boostrap bias for the Pareto exponent # Inputs: Number of bootstrap replicates (B) # Scaling exponent (exponent) # Threshold (x0) # Size of each data set (n) # Calls: rboot.pareto # Output: The bias pareto.bias <- function(B,exponent,x0,n) { return(mean(rboot.pareto(B,exponent,x0,n)) - exponent) } ##### Code Example 7 ##### # Parametric bootstrap confidence interval for the Pareto exponent # Uses the basic pivotal method # Inputs: Number of bootstrap replicates (B) # Scaling exponent (exponent) # Threshold (x0) # Size of each data set (n) # Calls: rboot.pareto # Output: list with lower and upper confidence limits (ci.lower,ci.upper) pareto.ci <- function(B,exponent,x0,n,alpha) { tboot <- rboot.pareto(B,exponent,x0,n) ci.lower <- 2*exponent - quantile(tboot,1-alpha/2) ci.upper <- 2*exponent - quantile(tboot,alpha/2) return(list(ci.lower=ci.lower, ci.upper=ci.upper)) } # Code Example 8 # Bootstrapped Kolmogorov-Smirnov test for the Pareto distribution # Use the bootstrap to find a valid p-value for the KS test, correcting # for estimating the exponent # Calculate the KS test statistic on the tail of the data # Input: Data set (data) # Estimated scaling exponent (exponet) # Threshold (x0) # Calls: ppareto from pareto.R # Presumes: data is a vector of numerical values # exponent is a single number (>= 1) # x0 is a positive number # At least one value in data >= x0 # Output: the KS statistic ks.stat.pareto <- function(data, exponent, x0) { ks.test(data[data>=x0], ppareto, exponent=exponent, threshold=x0)$statistic } # Calculate p-values in the bootstrapped KS test # Inputs: Number of bootstrapped replicates (B) # Data set (data) # Estimated scaling exponent (exponet) # Threshold (x0) # Calls: ks.stat.pareto from above, rpareto and pareto.fit from pareto.R # Presumes: B is a positive integer # data is a vector of numerical values # exponent is a single number (>= 1) # x0 is a positive number # At least one value in data >= x0 # Output: The p-value ks.pvalue.pareto <- function(B, data, exponent, x0) { testhat <- ks.stat.pareto(data, exponent, x0) testboot <- vector(length=B) for (i in 1:B) { xboot <- rpareto(length(data),exponent=exponent,threshold=x0) exp.boot <- pareto.fit(xboot,threshold=x0)$exponent testboot[i] <- ks.stat.pareto(xboot,exp.boot,x0) } p <- (sum(testboot >= testhat)+1)/(B+1) return(p) } ##### Code Example 9 ##### # Nonparametric bootstrap confidence intervals for the Pareto exponent # Utility function for resampling from a vector # Resamples its argument with replacement # Inputs: Vector to resample from (x) # Presumes: x is a vector which sample() can handle # Output: A vector of the same length as x resample <- function(x) { sample(x,size=length(x),replace=TRUE) } # Resample data and fit a Pareto distribution to it multiple times # Inputs: Number of bootstrap replicates (B) # Data vector to resample (data) # Threshold for the Pareto distribution (x0) # Calls: pareto.fit in pareto.R, resample from above # Presumes: data is a vector of positive numbers # x0 is a positive scalar # At least some values in data are >= x0 # B is a positive integer # Output: Vector of estimated scaling exponents resamp.pareto <- function(B,data,x0) { replicate(B,pareto.fit(resample(data),threshold=x0)$exponent) } # Pareto exponent confidence intervals from resampling # Basic pivotal method # Inputs: Number of bootstrap replicates (B) # Data vector to resample (data) # One minus confidence level (alpha) # Threshold for the Pareto distribution (x0) # Calls: resamp.pareto above # Presumes: data is a vector of positive numbers # x0 is a positive scalar # At least some values in data are >= x0 # B is a positive integer # alpha is in [0,1] # Output: List with lower and upper confidence limits (ci.lower,ci.upper) resamp.pareto.CI <- function(B,data,alpha,x0) { thetahat <- pareto.fit(data,threshold=x0)$exponent thetaboot <- resamp.pareto(B,data,x0) ci.lower <- thetahat - (quantile(thetaboot,1-alpha/2) - thetahat) ci.upper <- thetahat - (quantile(thetaboot,alpha/2) - thetahat) return(list(ci.lower=ci.lower,ci.upper=ci.upper)) } #### Set up example for section 4.1 #### library(MASS) data(geyser) geyser.lm <- lm(waiting~duration,data=geyser) # Resample data points from the geyser data # Notice that it's sufficient to resample row numbers from the data frame, # then take those rows # Inputs: None # Calls: resample (above) # Presumes: geyser exists and has rows and columns # Output: Resampled data frame resample.geyser <- function() { sample.rows <- resample(1:nrow(geyser)) return(geyser[sample.rows,]) } # Linearly regress waiting on duration and return the coefficient vector # Inputs: Data frame on which to estimate (data) # Presumes: data is a data frame # data has columns named waiting and duration, containing numerical values # Output: Vector of estimated intercept and slope est.waiting.on.duration <- function(data) { fit <- lm(waiting ~ duration, data=data) return(coefficients(fit)) } #### Code Example 10 #### # Nonparametric bootstrapped confidence intervals for the linear model of # Old Faithful, by resampling data points # Basic pivot method # Inputs: Number of bootstrap replicates (B) # One minus the desired confidence level (alpha) # Calls: resample.geyser, est.waiting.on.duration # Presumes: geyser.lm exists, has suitable coefficients # B is a positive integer # alpha is in [0,1] # Output: 2x2 array of upper and lower limits for intercept and slope geyser.lm.cis <- function(B,alpha) { tboot <- replicate(B,est.waiting.on.duration(resample.geyser())) low.quantiles <- apply(tboot,1,quantile,probs=alpha/2) high.quantiles <- apply(tboot,1,quantile,probs=1-alpha/2) low.cis <- 2*coefficients(geyser.lm) - high.quantiles high.cis <- 2*coefficients(geyser.lm) - low.quantiles cis <- rbind(low.cis,high.cis) return(cis) } #### Set up example for section 4.2 #### library(np) # Kernel regression of waiting time on duration # Slow, 0.6 seconds on my laptop. Could probably speed up by telling # npreg not to put so much effort into optimizing the bandwidth, but haven't # experimented with this. # Inputs: Data frame (data) # Calls: npreg from np # Presumes: data is a data frame # data has columns of numerical values called waiting and duration # Output: The fitted npregression object npr.waiting.on.duration <- function(data) { bw <- npregbw(waiting ~ duration, data=data) fit <- npreg(bw) # The natural thing to do would be to say ### fit <- npreg(waiting~duration,data=data) # but for obscure reasons this does not work when we pass in data as an # argument return(fit) } ##### Code Example 11 ##### # Calculate pointwise confidence bands for kernel regression of Old Faithful # by resampling data points # Because we resample data points, the training set will be different for # each bootstrap replicate. Thus fitted() values will not be comparable # across replicates. Instead, define a grid of points, evenly spaced along # the duration axis, and evaluate each kernel regression model on these # points. # Extends just slightly beyond the data evaluation.points <- seq(from=0.8,to=5.5,length.out=200) # Wrapper for evaluating a kernel regression on the grid # Input: regression model object (npr) # Presumes: npr has a predct() method # evaluation.points exists # npr's predict() method takes an exdat argument # Returns: The predictions eval.npr <- function(npr) { return(predict(npr,exdat=evaluation.points)) } # Get the predictions on the grid-points for our first model main.curve <- eval.npr(geyser.npr) # Pointwise kernel regression confidence bands by resampling data-points # Applies the basic pivotal method to the predictions at each point on the # evaluation grid # Inputs: Number of bootstrap replicates (B) # One minus confidence level (alpha) # Calls: eval.npr, npr.waiting.on.duration, resample.geyser # Output: list containing two arrays # cis has the lower and upper confidence limits for each evaluation point # tboot has all B curves at the evaluation points npr.cis <- function(B,alpha) { tboot <- replicate(B,eval.npr(npr.waiting.on.duration(resample.geyser()))) low.quantiles <- apply(tboot,1,quantile,probs=alpha/2) high.quantiles <- apply(tboot,1,quantile,probs=1-alpha/2) low.cis <- 2*main.curve - high.quantiles high.cis <- 2*main.curve - low.quantiles cis <- rbind(low.cis,high.cis) # Currently the evaluation points correspond to columns of cis but to the # rows of tboot; it'd be nicer to have them oriented the same way, so # transpose one of them return(list(cis=cis,tboot=t(tboot))) } ##### Figure 4 ##### # Plot confidence bands for the kernel regression model of Old Faithful # The next line takes 14 minutes to run on my laptop; it's needed for the # rest of the plot, but don't just run it automatically! ##### geyser.npr.cis <- npr.cis(B=800,alpha=0.05) # Set up plotting window, but don't plot anything. plot(0,type="n",xlim=c(0.8,5.5),ylim=c(0,100), xlab="Duration (min)", ylab="Waiting (min)") # Add thin grey lines for the resampled curves for (i in 1:800) { lines(evaluation.points,geyser.npr.cis$tboot[i,],lwd=0.1,col="grey") } # Lower confidence limit lines(evaluation.points,geyser.npr.cis$cis[1,]) # Upper confidence limit lines(evaluation.points,geyser.npr.cis$cis[2,]) # Initial estimate on full data lines(evaluation.points,main.curve) # Tick marks on x axis for where the data were rug(geyser$duration,side=1) # Scatterplot of actual values points(geyser$duration,geyser$waiting) ##### Set up example for section 4.3 ##### oecd.lm <- lm(growth ~ initgdp + popgro + inv, data=oecdpanel) ##### Code Example 12 ##### # Confidence intervals for multiple linear regression by resampling residuals # Resample residuals from the linear model for the OECD data # Inputs: none # Calls: resample # Presumes: oecdpanel exists, is a data frame # oecd.lm exists, has fitted() and residuals() method # Output: Data frame with old input variables and new growth values resample.residuals.oecd <- function() { # Resampling residuals leaves the independent variables alone, so copy them new.frame <- oecdpanel # Take the old fitted values, and perturb them by resampling the residuals new.growths <- fitted(oecd.lm) + resample(residuals(oecd.lm)) # Make these the new values of the response new.frame$growth <- new.growths # We're done return(new.frame) } # Wrapper for estimating the OECD linear model on a data frame # Inputs: Data frame (data) # Presumes: data has appropraite columns # Returns: Vector of linear regression coefficients oecd.estimator <- function(data) { fit <- lm(growth~initgdp + popgro + inv, data=data) return(coefficients(fit)) } # Confidence intervals by resampling residuals # Basic pivotal method # Input: Number of bootstrap replicates (B) # One minus desired confidence level(alpha) # Calls: oecd.estimator, resample.residuals.oecd # Presumes: B is a positive integer # alpha is in [0,1] # Output: array of upper and lower confidence limits for each coefficient oecd.lm.cis <- function(B,alpha) { tboot <- replicate(B,oecd.estimator(resample.residuals.oecd())) low.quantiles <- apply(tboot,1,quantile,probs=alpha/2) high.quantiles <- apply(tboot,1,quantile,probs=1-alpha/2) low.cis <- 2*coefficients(oecd.lm) - high.quantiles high.cis <- 2*coefficients(oecd.lm) - low.quantiles cis <- rbind(low.cis,high.cis) return(cis) } ##### Example for section 6 ##### # Nonparametric bootstrapping does badly on things where changing a single # data point can drastically change the result, like extremes of distributions # Here we show that using resampling to get confidence intervals for the # maximum of a uniform distribution is an EPIC FAIL # Calculate actual coverage probability of what looks like a 95% CI # Presume we know X~Unif(0,theta), and are trying to estimate theta # In reality, theta is fixed at 1 # The MLE is max(x) # Draw 1000 bootstrap replicates by resampling x, and take the max on each # Find the quantiles of these re-estimates and correspond 95% CI # Check if the CI covers 1 (the true value of theta) # Inputs: None # Calls: resample # Outputs: TRUE if the CI covers 1, FALSE otherwise is.covered <- function() { x <- runif(100) max.boot <- replicate(1e3,max(resample(x))) # all() takes a vector of Boolean quantities and returns TRUE if all are TRUE # The any() function similarly returns TRUE if any of its arguments are TRUE all(1 >= 2*max(x) - quantile(max.boot,0.975), 1 <= 2*max(x) - quantile(max.boot,0.025)) } sum(replicate(1000,is.covered())) # I got 19 rather than about 950