#### Functions for continuous power law or Pareto distributions # Revision history at end of file ### Standard R-type functions for distributions: # dpareto Probability density # ppareto Probability distribution (CDF) # qpareto Quantile function # rpareto Random variable generation ### Functions for fitting: # pareto.fit Fit Pareto to data # .pareto.fit.threshold Determine scaling threshold and then fit # --- not for direct use, call pareto.fit instead # .pareto.fit.ml Fit Pareto to data by maximum likelihood # --- not for direct use, call pareto.fit instead # pareto.loglike Calculate log-likelihood under Pareto # .pareto.fit.regression.cdf Fit Pareto data by linear regression on # log-log CDF (disrecommended) # --- not for direct use, call pareto.fit instead # loglogslope Fit Pareto via regression, extract scaling # exponent # loglogrsq Fit Pareto via regression, extract R^2 ### Functions for testing: # ### Functions for visualization: # plot.eucdf.loglog Log-log plot of the empirical upper cumulative # distribution function, AKA survival function # plot.survival.loglog Alias for plot.eucdf.loglog ### Back-stage functions, not intended for users: # .ks.dist.for.pareto Find Kolmogorov-Smirnov distance between fitted # and empirical distribution; called by # .pareto.fit.threshold # .ks.dist.fixed.pareto Find K-S distance between given Pareto and # empirical distribution # Probability density of Pareto distributions # Gives NA on values below the threshold # Input: Data vector, lower threshold, scaling exponent, "log" flag # Output: Vector of (log) probability densities dpareto <- function(x, threshold = 1, exponent, log=FALSE) { # Avoid doing limited-precision arithmetic followed by logs if we want # the log! if (!log) { prefactor <- (exponent-1)/threshold f <- function(x) {prefactor*(x/threshold)^(-exponent)} } else { prefactor.log <- log(exponent-1) - log(threshold) f <- function(x) {prefactor.log -exponent*(log(x) - log(threshold))} } d <- ifelse(x=threshold] d <- suppressWarnings(ks.test(data,ppareto,threshold=threshold,exponent=exponent)) # ks.test complains about p-values when there are ties, we don't care return(as.vector(d\$statistic)) } # Estimate scaling exponent of Pareto distribution by maximum likelihood # Input: Data vector, lower threshold # Output: List giving distribution type ("pareto"), parameters, log-likelihood .pareto.fit.ml <- function (data, threshold) { data <- data[data>=threshold] n <- length(data) x <- data/threshold alpha <- 1 + n/sum(log(x)) loglike = pareto.loglike(data,threshold,alpha) ks.dist <- .ks.dist.fixed.pareto(data,threshold=threshold,exponent=alpha) fit <- list(type="pareto", exponent=alpha, xmin=threshold, loglike = loglike, ks.dist = ks.dist, samples.over.threshold=n) return(fit) } # Calculate log-likelihood under a Pareto distribution # Input: Data vector, lower threshold, scaling exponent # Output: Real-valued log-likelihood pareto.loglike <- function(x, threshold, exponent) { L <- sum(dpareto(x, threshold = threshold, exponent = exponent, log = TRUE)) return(L) } # Log-log plot of the survival function (empirical upper CDF) of a data set # Input: Data vector, lower limit, upper limit, graphics parameters # Output: None (returns NULL invisibly) plot.survival.loglog <- function(x,from=min(x),to=max(x),...) { plot.eucdf.loglog(x,from,to,...) } plot.eucdf.loglog <- function(x,from=min(x),to=max(x),type="l",...) { # Use the "eucdf" function (below) x <- sort(x) x.eucdf <- eucdf(x) # This is nice if the number of points is small... plot(x,x.eucdf(x),xlim=c(from,to),log="xy",type=type,...) # Should check how many points and switch over to a curve-type plot when # it gets too big invisible(NULL) } # Calculate the upper empirical cumulative distribution function of a # one-dimensional data vector # Uses the standard function ecdf # Should, but does not yet, also produce a function of class "stepfun" # (like ecdf) # Input: data vector # Output: a function eucdf <- function(x) { # Exploit built-in R function to get ordinary (lower) ECDF, Pr(X<=x) x.ecdf <- ecdf(x) # Now we want Pr(X>=x) = (1-Pr(X<=x)) + Pr(X==x) # If x is one of the "knots" of the step function, i.e., a point with # positive probability mass, should add that in to get Pr(X>=x) # rather than Pr(X>x) away.from.knot <- function(y) { 1 - x.ecdf(y) } at.knot.prob.jump <- function(y) { x.knots = knots(x.ecdf) # Either get the knot number, or give zero if this was called # away from a knot k <- match(y,x.knots,nomatch=0) if ((k==0) || (k==1)) { # Handle special cases if (k==0) { prob.jump = 0 # Not really a knot } else { prob.jump = x.ecdf(y) # Special handling of first knot } } else { prob.jump = x.ecdf(y) - x.ecdf(x.knots[(k-1)]) # General case } return(prob.jump) } # Use one function or the other x.eucdf <- function(y) { baseline = away.from.knot(y) jumps = sapply(y,at.knot.prob.jump) ifelse (y %in% knots(x.ecdf), baseline+jumps, baseline) } return(x.eucdf) } # Calculate valid p-value for the goodness of fit of a power-law # tail to a data set, via simulation # Input: data vector (x), number of replications (m) # Output: p-value pareto.tail.ks.test <- function(x,m) { x.pt <- pareto.fit(x,threshold="find") x0 <- x.pt\$xmin # extract parameters of fitted dist. alpha <- x.pt\$exponent ntail <- sum(x>=x0) # How many samples in the tail? n <- length(x) ptail <- ntail/n # Total prob. of the tail # Carve out the non-tail data points body <- x[x < x0] # Observed value of KS distance: d.ks <- x.pt\$ks.dist # KS statistics of resamples: r.ks <- replicate(m,.ks.resimulate.pareto.tail(n,ptail,x0,alpha,body)) p.value <- sum(r.ks >= d.ks)/m return(p.value) } # Resimulate from a data set with a Pareto tail, estimate on # the simulation and report the KS distance # Inputs: Size of sample (n), probability of being in the tail (tail.p), # threshold for tail (threshold), power law exponent (exponent), # vector giving values in body (data.body) # Output: KS distance .ks.resimulate.pareto.tail <- function(n,tail.p,threshold,exponent,data.body) { # Samples come from the tail with probability ptail, or else from the body # decide randomly how many samples come from the tail tail.samples <- rbinom(1,n,tail.p) # Draw the samples from the tail rtail <- rpareto(tail.samples,threshold,exponent) # Draw the samples from the body (with replacement!) rbody <- sample(data.body,n-tail.samples,replace=TRUE) b <- c(rtail,rbody) b.ks <- pareto.fit(b,threshold="find")\$ks.dist return(b.ks) } ### The crappy linear regression way to fit a power law # The common procedure is to fit to the binned density function, which is even # crappier than to fit to the complementary distribution function; this # currently only implements the latter # First, produce the empirical complementary distribution function, as # a pair of lists, {x}, {C(x)} # Then regress log(C) ~ log(x) # and report the slope and the R^2 # Input: Data vector, threshold # Output: List with distributional parameters and information about the # fit .pareto.fit.regression.cdf <- function(x,threshold=1) { # Discard data under threshold x <- x[x>=threshold] n <- length(x) # We need the different observed values of x, in order distinct_x <- sort(unique(x)) x.eucdf <- eucdf(x) upper_probs <- x.eucdf(distinct_x) loglogfit <- lm(log(upper_probs) ~ log(distinct_x)) intercept <- as.vector(coef(loglogfit)[1]) # primarily useful for plotting slope <- as.vector(-coef(loglogfit)[2]) # Remember sign of parameterization # But that's the exponent of the CDF, that of the pdf is one larger # and is what we're parameterizing by slope <- slope+1 r2 <- summary(loglogfit)\$r.squared loglike <- pareto.loglike(x, threshold, slope) ks.dist <- .ks.dist.fixed.pareto(x,threshold=threshold,exponent=slope) result <- list(type="pareto", exponent = slope, rsquare = r2, log_x = log(distinct_x), log_p = log(upper_probs), intercept = intercept, loglike = loglike, xmin=threshold, ks.dist = ks.dist, samples.over.threshold=n) return(result) } # Wrapper function to just get the exponent estimate loglogslope <- function(x,threshold=1) { llf <- .pareto.fit.regression.cdf(x,threshold) exponent <- llf\$exponent return(exponent) } # Wrapper function to just get the R^2 values loglogrsq <- function(x,threshold=1) { llf <- .pareto.fit.regression.cdf(x,threshold) r2 <- llf\$rsquare return(r2) } # Revision history: # no release 2003 First draft # v 0.0 2007-06-04 First release # v 0.0.1 2007-06-29 Fixed "not" for "knot" typo, thanks to # Nicholas A. Povak for bug report # v 0.0.2 2007-07-22 Fixed bugs in plot.survival.loglog, thanks to # Stefan Wehrli for report # v 0.0.3 2008-03-02 Realized R has a "unique" function; added # estimating xmin via method in minimal KS dist. # v 0.0.4 2008-04-24 Made names of non-end-user functions start # with period, hiding them in workspace # v 0.0.5 2011-02-03 Suppressed the warning ks.test produces about # not being able to calculate p-values in the # presence of ties