Abstract

This vignette discusses the implementation of the “Head/tail breaks” style (Jiang (2013)) in the classIntervals function. A step-by-step example is presented in order to clarify the method. A case study using spData::afcon is also included, as well as a test suite checking the performance and validation of the implementation.

Introduction

The Head/tail breaks, sometimes referred as ht-index (Jiang and Yin (2013)), is a classification scheme introduced by Jiang (2013) in order to find groupings or hierarchy for data with a heavy-tailed distribution.

Heavy-tailed distributions are heavily right skewed, with a minority of large values in the head and a majority of small values in the tail. This imbalance between the head and tail, or between many small values and a few large values, can be expressed as “far more small things than large things”.

Heavy tailed distributions are commonly characterized by a power law, a lognormal or an exponential function. Nature, society, finance (Vasicek (2002)) and our daily lives are full of rare and extreme events, which are termed “black swan events” (Taleb (2008)). This line of thinking provides a good reason to reverse our thinking by focusing on low-frequency events.

library(classInt)

#1. Characterization of heavy-tail distributions----
set.seed(1234)
#Pareto distribution a=1 b=1.161 n=1000
sample_par <- 1 / (1 - runif(1000)) ^ (1 / 1.161)
opar <- par(no.readonly = TRUE)
par(mar = c(2, 4, 3, 1), cex = 0.8)
plot(
  sort(sample_par, decreasing = TRUE),
  type = "l",
  ylab = "F(x)",
  xlab = "",
  main = "80/20 principle"
)
abline(h = quantile(sample_par, .8) ,
       lty = 2,
       col = "red3")
abline(v = 0.2*length(sample_par) ,
       lty = 2,
       col = "darkblue")
legend(
  "topleft",
  legend = c("F(x): p80", "x: Top 20%"),
  col = c("red3", "darkblue"),
  lty = 2,
  cex = 0.8
)

hist(
  sample_par,
  n = 100,
  xlab = "",
  main = "Histogram",
  col = "grey50",
  border = NA, 
  probability = TRUE
)
par(opar)

Breaking method

The method itself consists on a four-step process performed recursively until a stopping condition is satisfied. Given a vector of values var the process can be described as follows:

Compute mu = mean(var).
Break var into the tail (as var < mu) and the head (as var > mu).
Assess if the proportion of head over var is lower or equal than a given threshold (i.e. length(head)/length(var) <= thr)
If 3 is TRUE, repeat 1 to 3 until the condition is FALSE or no more partitions are possible (i.e. head has less than two elements expressed as length(head) < 2).

It is important to note that, at the beginning of a new iteration, var is replaced by head. The underlying hypothesis is to create partitions until the head and the tail are balanced in terms of distribution.So the stopping criteria is satisfied when the last head and the last tail are evenly balanced.

In terms of threshold, Jiang, Liu, and Jia (2013) set 40% as a good approximation, meaning that if the head contains more than 40% of the observations the distribution is not considered heavy-tailed.

The final breaks are the vector of consecutive mu.

Step by step example

We reproduce here the pseudo-code¹ as per Jiang (2019):

Recursive function Head/tail Breaks:
 Rank the input data from the largest to the smallest
 Break the data into the head and the tail around the mean;
 // the head for those above the mean
 // the tail for those below the mean
 While (head <= 40%):
 Head/tail Breaks (head);
End Function

A step-by-step example in R (for illustrative purposes) has been developed:


opar <- par(no.readonly = TRUE)
par(mar = c(2, 2, 3, 1), cex = 0.8)
var <- sample_par
thr <- .4
brks <- c(min(var), max(var))  #Initialise with min and max

sum_table <- data.frame(
  iter = 0,
  mu = NA,
  prop = NA,
  n_var = NA,
  n_head = NA
)
#Pars for chart
limchart <- brks
#Iteration
for (i in 1:10) {
  mu <- mean(var)
  brks <- sort(c(brks, mu))
  head <- var[var > mu]
  prop <- length(head) / length(var)
  stopit <- prop < thr & length(head) > 1
  sum_table = rbind(sum_table,
                    c(i, mu, prop, length(var), length(head)))
  hist(
    var,
    main = paste0("Iter ", i),
    breaks = 50,
    col = "grey50",
    border = NA,
    xlab = "",
    xlim = limchart
  )
  abline(v = mu, col = "red3", lty = 2)
  ylabel <- max(hist(var, breaks = 50, plot = FALSE)$counts)
  labelplot <- paste0("PropHead: ", round(prop * 100, 2), "%")
  text(
    x = mu,
    y = ylabel,
    labels = labelplot,
    cex = 0.8,
    pos = 4
  )
  legend(
    "right",
    legend = paste0("mu", i),
    col = c("red3"),
    lty = 2,
    cex = 0.8
  )
  if (isFALSE(stopit))
    break
  var <- head
}
par(opar)

As it can be seen, in each iteration the resulting head gradually loses the high-tail property, until the stopping condition is met.

iter	mu	prop	n_var	n_head
1	5.6755	14.5%	1000	145
2	27.2369	21.38%	145	31
3	85.1766	19.35%	31	6
4	264.7126	50%	6	3

The resulting breaks are then defined as breaks = c(min(var), mu(iter=1), ..., mu(iter), max(var)).

Implementation in `classInt` package

The implementation in the classIntervals function should replicate the results:

ht_sample_par <- classIntervals(sample_par, style = "headtails")
brks == ht_sample_par$brks
#> [1] TRUE TRUE TRUE TRUE TRUE TRUE
print(ht_sample_par)
#> style: headtails
#> [1.000295,5.675463) [5.675463,27.23693) [27.23693,85.17664) [85.17664,264.7126) 
#>                 855                 114                  25                   3 
#> [264.7126,523.6254] 
#>                   3

As stated in Jiang (2013), the number of breaks is naturally determined, however the thr parameter could help to adjust the final number. A lower value on thr would provide less breaks while a larger thr would increase the number, if the underlying distribution follows the “far more small things than large things” principle.

opar <- par(no.readonly = TRUE)
par(mar = c(2, 2, 2, 1), cex = 0.8)

pal1 <- c("wheat1", "wheat2", "red3")

# Minimum: single break
print(classIntervals(sample_par, style = "headtails", thr = 0))
#> style: headtails
#> [1.000295,5.675463) [5.675463,523.6254] 
#>                 855                 145
plot(
  classIntervals(sample_par, style = "headtails", thr = 0),
  pal = pal1,
  main = "thr = 0"
)

# Two breaks
print(classIntervals(sample_par, style = "headtails", thr = 0.2))
#> style: headtails
#> [1.000295,5.675463) [5.675463,27.23693) [27.23693,523.6254] 
#>                 855                 114                  31
plot(
  classIntervals(sample_par, style = "headtails", thr = 0.2),
  pal = pal1,
  main = "thr = 0.2"
)

# Default breaks: 0.4
print(classIntervals(sample_par, style = "headtails"))
#> style: headtails
#> [1.000295,5.675463) [5.675463,27.23693) [27.23693,85.17664) [85.17664,264.7126) 
#>                 855                 114                  25                   3 
#> [264.7126,523.6254] 
#>                   3
plot(classIntervals(sample_par, style = "headtails"),
     pal = pal1,
     main = "thr = Default")

# Maximum breaks
print(classIntervals(sample_par, style = "headtails", thr = 1))
#> style: headtails
#> [1.000295,5.675463) [5.675463,27.23693) [27.23693,85.17664) [85.17664,264.7126) 
#>                 855                 114                  25                   3 
#>  [264.7126,391.279)  [391.279,523.6254] 
#>                   2                   1
plot(
  classIntervals(sample_par, style = "headtails", thr = 1),
  pal = pal1,
  main = "thr = 1"
)
par(opar)

The method always returns at least one break, corresponding to mean(var).

Case study

Jiang (2013) states that “the new classification scheme is more natural than the natural breaks in finding the groupings or hierarchy for data with a heavy-tailed distribution.” (p. 482), referring to Jenks’ natural breaks method. In this case study we would compare “headtails” vs. “fisher”, that is the alias for the Fisher-Jenks algorithm and it is always preferred to the “jenks” style (see ?classIntervals). For this example we will use the afcon dataset from spData package.

library(spData)
data(afcon, package = "spData")

Let’s have a look to the Top 10 values and the distribution of the variable totcon (index of total conflict 1966-78):


# Top10
knitr::kable(head(afcon[order(afcon$totcon, decreasing = TRUE),c("name","totcon")],10))

	name	totcon
EG	EGYPT	5246
SU	SUDAN	4751
UG	UGANDA	3134
CG	ZAIRE	3087
TZ	TANZANIA	2881
LY	LIBYA	2355
KE	KENYA	2273
SO	SOMALIA	2122
ET	ETHIOPIA	1878
SF	SOUTH AFRICA	1875


opar <- par(no.readonly = TRUE)
par(mar = c(4, 4, 3, 1), cex = 0.8)
hist(afcon$totcon,
     n = 20,
     main = "Histogram",
     xlab = "totcon",
     col = "grey50",
     border = NA, )
plot(
  density(afcon$totcon),
  main = "Distribution",
  xlab = "totcon",
)
par(opar)

The data shows that EG and SU data present a clear hierarchy over the rest of values. As per the histogram, we can confirm a heavy-tailed distribution and therefore the “far more small things than large things” principle.

As a testing proof, on top of “headtails” and “fisher” we would use also “quantile” to have a broader view on the different breaking styles. As “quantile” is a position-based metric, it doesn’t account for the magnitude of F(x) (hierarchy), so the breaks are solely defined by the position of x on the distribution.

Applying the three aforementioned methods to break the data:

brks_ht <- classIntervals(afcon$totcon, style = "headtails")
print(brks_ht)
#> style: headtails
#>   one of 91,390 possible partitions of this variable into 5 classes
#>    [147,1350.619) [1350.619,2488.6)   [2488.6,3819.8)   [3819.8,4998.5) 
#>                27                10                 3                 1 
#>     [4998.5,5246] 
#>                 1
#Same number of classes for "fisher"
nclass <- length(brks_ht$brks) - 1
brks_fisher <-  classIntervals(afcon$totcon, style = "fisher",
                               n = nclass)
print(brks_fisher)
#> style: fisher
#>   one of 91,390 possible partitions of this variable into 5 classes
#>    [147,693.5) [693.5,1474.5)  [1474.5,2618)  [2618,3942.5)  [3942.5,5246] 
#>             12             17              8              3              2

brks_quantile <- classIntervals(afcon$totcon, style = "quantile",
                                n = nclass)
print(brks_quantile)
#> style: quantile
#>   one of 91,390 possible partitions of this variable into 5 classes
#>       [147,604)     [604,833.6)  [833.6,1137.2) [1137.2,1877.4)   [1877.4,5246] 
#>               8               9               8               8               9

pal1 <- c("wheat1", "wheat2", "red3")
opar <- par(no.readonly = TRUE)
par(mar = c(2, 2, 2, 1), cex = 0.8)
plot(brks_ht, pal = pal1, main = "headtails")
plot(brks_fisher, pal = pal1, main = "fisher")
plot(brks_quantile, pal = pal1, main = "quantile")
par(opar)

It is observed that the top three classes of “headtails” enclose 5 observations, whereas “fisher” includes 13 observations. In terms of classification, “headtails” breaks focuses more on extreme values.

The next plot compares a continuous distribution of totcon re-escalated to a range of [1,nclass] versus the distribution across breaks for each style. The continuous distribution has been offset by -0.5 in order to align the continuous and the discrete distributions.

#Helper function to rescale values
help_reescale <- function(x, min = 1, max = 10) {
  r <- (x - min(x)) / (max(x) - min(x))
  r <- r * (max - min) + min
  return(r)
}
afcon$ecdf_class <- help_reescale(afcon$totcon,
                                  min = 1 - 0.5,
                                  max = nclass - 0.5)
afcon$ht_breaks <-  cut(afcon$totcon,
                        brks_ht$brks,
                        labels = FALSE,
                        include.lowest = TRUE)

afcon$fisher_breaks <-  cut(afcon$totcon,
                            brks_fisher$brks,
                            labels = FALSE,
                            include.lowest = TRUE)

afcon$quantile_break <-  cut(afcon$totcon,
                             brks_quantile$brks,
                             labels = FALSE,
                             include.lowest = TRUE)

opar <- par(no.readonly = TRUE)
par(mar = c(4, 4, 1, 1), cex = 0.8)
plot(
  density(afcon$ecdf_class),
  ylim = c(0, 0.8),
  lwd = 2,
  main = "",
  xlab = "class"
)
lines(density(afcon$ht_breaks), col = "darkblue", lty = 2)
lines(density(afcon$fisher_breaks), col = "limegreen", lty = 2)
lines(density(afcon$quantile_break),
      col = "red3",
      lty = 2)
legend("topright",
       legend = c("Continuous", "headtails",
                  "fisher", "quantile"),
  col = c("black", "darkblue", "limegreen", "red3"),
  lwd = c(2, 1, 1, 1),
  lty = c(1, 2, 2, 2),
  cex = 0.8
)
par(opar)

It can be observed that the distribution of “headtails” breaks is also heavy-tailed, and closer to the original distribution. On the other extreme, “quantile” provides a quasi-uniform distribution, ignoring the totcon hierarchy

In terms of data visualization, we compare here the final map using the techniques mentioned above. On this plotting exercise:

cex of points are always between 1 and 5.
For the continuous approach, no classes are provided. This plot will be used as the reference.
For all the rest of styles, col and cex on each point is defined as per the class of that point.

custompal <- c("#FE9F6D99",
               "#DE496899",
               "#8C298199",
               "#3B0F7099",
               "#00000499")

afcon$cex_points <- help_reescale(afcon$totcon,
                                  min = 1,
                                  max = 5)
opar <- par(no.readonly = TRUE)
par(mar = c(1.5, 1.5, 2, 1.5), cex = 0.8)
# Plot continuous
plot(
  x = afcon$x,
  y = afcon$y,
  axes = FALSE,
  cex = afcon$cex_points,
  pch = 20,
  col = "grey50",
  main = "Continuous",
)

mcont <- (max(afcon$totcon) - min(afcon$totcon)) / 4
legcont <- 1:5 * mcont - (mcont - min(afcon$totcon))

legend("bottomleft",
       xjust = 1,
       bty = "n",
       legend = paste0("   ",
                  round(legcont, 0)
                  ),
       col = "grey50",
  pt.cex = seq(1, 5),
  pch = 20,
  title = "totcon"
)
box()

plot(
  x = afcon$x,
  y = afcon$y,
  axes = FALSE,
  cex = afcon$ht_breaks,
  pch = 20,
  col = custompal[afcon$ht_breaks],
  main = "headtails"
)
legend(
  "bottomleft",
  xjust = 1,
  bty = "n",
  legend = paste0("   ",
                  round(brks_ht$brks[2:6],0)
                  ),
  col = custompal,
  pt.cex = seq(1, 5),
  pch = 20,
  title = "totcon"
)
box()

plot(
  x = afcon$x,
  y = afcon$y,
  axes = FALSE,
  cex = afcon$fisher_breaks,
  pch = 20,
  col = custompal[afcon$fisher_breaks],
  main = "fisher"
)
legend(
  "bottomleft",
  xjust = 1,
  bty = "n",
  legend = paste0("   ",
                  round(brks_fisher$brks[2:6],0)
                  ),
  col = custompal,
  pt.cex = seq(1, 5),
  pch = 20,
  title = "totcon"
)
box()

plot(
  x = afcon$x,
  y = afcon$y,
  axes = FALSE,
  cex = afcon$quantile_break,
  pch = 20,
  col = custompal[afcon$quantile_break],
  main = "quantile"
)
legend(
  "bottomleft",
  xjust = 1,
  bty = "n",
  legend = paste0("   ",
                  round(brks_quantile$brks[2:6],0)
                  ),
  col = custompal,
  pt.cex = seq(1, 5),
  pch = 20,
  title = "totcon"
)
box()

par(opar)

As per the results, “headtails” seems to provide a better understanding of the most extreme values when the result is compared against the continuous plot. The “quantile” style, as expected, just provides a clustering without taking into account the real hierarchy. The “fisher” plot is in-between of these two interpretations.

It is also important to note that “headtails” and “fisher” reveal different information that can be useful depending of the context. While “headtails” highlights the outliers, it fails on providing a good clustering on the tail, while “fisher” seems to reflect better these patterns. This can be observed on the values of Western Africa and the Niger River Basin, where “headtails” doesn’t highlight any special cluster of conflicts, “fisher” suggests a potential cluster. This can be confirmed on the histogram generated previously, where a concentration of totcon around 1,000 is visible.

Testing and benchmark

On this section the performance of the “headtails” implementation is tested, in terms of speed and handling of corner cases. A small benchmark with another styles is also presented.

Testing has been performed over the following distributions:

Heavy-tailed distributions

Pareto
Exponential
Log-normal
Weibull
Log-Cauchy, also known as super-heavy tail distribution (Falk, Huesler, and Reiss (2011), p. 80, Fraga Alves, Haan, and Neves (2009))

Non heavy-tailed distributions

Normal (non heavy-tailed)
Truncated Normal (left-tailed)
Uniform distribution

#Init samples
set.seed(2389)

#Pareto distributions a=7 b=14
paretodist <- 7 / (1 - runif(5000000)) ^ (1 / 14)
#Exponential dist
expdist <- rexp(5000000)
#Lognorm
lognormdist <- rlnorm(5000000)
#Weibull
weibulldist <- rweibull(5000000, 1, scale = 5)
#LogCauchy "super-heavy tail"
logcauchdist <- exp(rcauchy(5000000, 2, 4))
#Remove Inf 
logcauchdist <- logcauchdist[logcauchdist < Inf]

#Normal dist
normdist <- rnorm(5000000)
#Left-tailed distr
leftnorm <-
  sample(rep(normdist[normdist < mean(normdist)], 3), size = 5000000)

#Uniform distribution
unifdist <- runif(5000000)

Let’s define a helper function and proceed to run the whole test suite:

testresults <- data.frame(
  Title = NA,
  style = NA,
  nsample  = NA,
  thresold = NA,
  nbreaks = NA,
  time_secs = NA
)

benchmarkdist <-
  function(dist,
           style = "headtails",
           thr = 0.4,
           title = "",
           plot = FALSE) {
    init <- Sys.time()
    br <- classIntervals(dist, style = style, thr = thr)
    a <- Sys.time() - init
    test <- data.frame(
      Title = title,
      style  = style,
      nsample  = format(length(br$var), 
                        scientific = FALSE, big.mark = ","),
      thresold = thr,
      nbreaks = length(br$brks) - 1,
      time_secs = as.character(round(a,4))
    )
    testresults <- unique(rbind(testresults, test))
    
    if (plot) {
      plot(
        density(br$var,
                from = quantile(dist,.0005),
                to = quantile(dist,.9995)
                ),
        col = "black",
        cex.main = .9,
        main = paste0(
          title,
          " ",
          style,
          ", thr =",
          thr,
          ", nbreaks = ",
          length(br$brks) - 1
        ),
        ylab = "",
        xlab = ""
      )
      abline(v = br$brks,
             col = "red3",
             lty = 2)
    }
    return(testresults)
  }
opar <- par(no.readonly = TRUE)
par(mar = c(2, 2, 2, 2), cex = 0.8)

# Pareto----
testresults <- benchmarkdist(paretodist, title = "Pareto", plot = TRUE)
testresults <- benchmarkdist(paretodist, title = "Pareto", thr = 0)
testresults <- benchmarkdist(paretodist, title = "Pareto", thr = .75, plot = TRUE)

#Sample 2,000 obs
set.seed(1234)
Paretosamp <- sample(paretodist, 2000)
testresults <- benchmarkdist(Paretosamp,
                             title = "Pareto sample",
                             style = "fisher",
                             plot = TRUE)
testresults <- benchmarkdist(Paretosamp,
                             title = "Pareto sample",
                             style = "headtails",
                             plot = TRUE)


#Exponential----
testresults <- benchmarkdist(expdist, title = "Exponential", plot = TRUE)
testresults <- benchmarkdist(expdist, title = "Exponential", thr = 0)
testresults <- benchmarkdist(expdist, title = "Exponential", thr = 1)
testresults <- benchmarkdist(expdist, title = "Exponential",
                             style = "quantile", plot = TRUE)

#Weibull-----
testresults <- benchmarkdist(weibulldist, title = "Weibull", plot = TRUE)
testresults <- benchmarkdist(weibulldist, title = "Weibull", thr = 0)
testresults <- benchmarkdist(weibulldist, title = "Weibull", thr = 1)

#Logcauchy
testresults <- benchmarkdist(logcauchdist, title = "LogCauchy", plot = TRUE)
testresults <- benchmarkdist(logcauchdist, title = "LogCauchy", thr = 0)
testresults <- benchmarkdist(logcauchdist, title = "LogCauchy", thr = 1)

#Normal----
testresults <- benchmarkdist(normdist, title = "Normal", plot = TRUE)
testresults <- benchmarkdist(normdist, title = "Normal", thr = 0)
testresults <- benchmarkdist(normdist, title = "Normal", thr = 1, plot = TRUE)

#Truncated Left-tail Normal----
testresults <- benchmarkdist(leftnorm, title = "Left Normal", plot = TRUE)
testresults <- benchmarkdist(leftnorm, title = "Left Normal", thr = -100)
testresults <- benchmarkdist(leftnorm, title = "Left Normal", plot = TRUE, thr = 500)

#Uniform----
testresults <- benchmarkdist(unifdist, title = "Uniform", plot = TRUE, thr = 0.7)
testresults <- benchmarkdist(unifdist, title = "Uniform", thr = 0)
testresults <- benchmarkdist(unifdist, title = "Uniform", plot = TRUE, thr = 1)
par(opar)

# Results
knitr::kable(testresults[-1, ], row.names = FALSE)

Title	style	nsample	thresold	nbreaks	time_secs
Pareto	headtails	5,000,000	0.40	15	0.6368
Pareto	headtails	5,000,000	0.00	2	0.5429
Pareto	headtails	5,000,000	0.75	15	0.6518
Pareto sample	fisher	2,000	0.40	12	0.0182
Pareto sample	headtails	2,000	0.40	8	3e-04
Exponential	headtails	5,000,000	0.40	16	0.5016
Exponential	headtails	5,000,000	0.00	2	0.4388
Exponential	headtails	5,000,000	1.00	17	0.5074
Exponential	quantile	5,000,000	0.40	24	0.7512
Weibull	headtails	5,000,000	0.40	16	0.5902
Weibull	headtails	5,000,000	0.00	2	0.4161
Weibull	headtails	5,000,000	1.00	17	0.4857
LogCauchy	headtails	4,991,187	0.40	6	0.527
LogCauchy	headtails	4,991,187	0.00	2	0.4371
LogCauchy	headtails	4,991,187	1.00	6	0.5043
Normal	headtails	5,000,000	0.40	2	0.5623
Normal	headtails	5,000,000	0.00	2	0.4973
Normal	headtails	5,000,000	1.00	17	0.5446
Left Normal	headtails	5,000,000	0.40	2	0.6323
Left Normal	headtails	5,000,000	-100.00	2	0.6091
Left Normal	headtails	5,000,000	500.00	21	0.6657
Uniform	headtails	5,000,000	0.70	22	0.5204
Uniform	headtails	5,000,000	0.00	2	0.3966
Uniform	headtails	5,000,000	1.00	22	0.4633

The implementation works as expected, with a good performance given the size of the sample, and also compares well with another current implementations in classIntervals.

References

Falk, Michael, Juerg Huesler, and Rolf-Dieter Reiss. 2011. Laws of Small Numbers: Extremes and Rare Events. Laws of Small Numbers: Extremes and Rare Events. https://doi.org/10.1007/978-3-0348-0009-9.

Fraga Alves, Maria, L. D. Haan, and Claudia Neves. 2009. “Statistical Inference for Heavy and Super-Heavy Tailed Distributions.” J. Stat. Plan. Inf. 139 (January): 213–27.

Jiang, Bin. 2013. “Head/Tail Breaks: A New Classification Scheme for Data with a Heavy-Tailed Distribution.” The Professional Geographer 65 (3): 482–94. https://doi.org/10.1080/00330124.2012.700499.

———. 2019. “A Recursive Definition of Goodness of Space for Bridging the Concepts of Space and Place for Sustainability.” Sustainability 11 (15): 4091. https://doi.org/10.3390/su11154091.

Jiang, Bin, Xintao Liu, and Tao Jia. 2013. “Scaling of Geographic Space as a Universal Rule for Map Generalization.” Annals of the Association of American Geographers 103 (4): 844–55. https://doi.org/10.1080/00045608.2013.765773.

Jiang, Bin, and Junjun Yin. 2013. “Ht-Index for Quantifying the Fractal or Scaling Structure of Geographic Features.” Annals of the Association of American Geographers 104 (3): 530–40. https://doi.org/10.1080/00045608.2013.834239.

Taleb, Nassim Nicholas. 2008. The Black Swan: The Impact of the Highly Improbable. 1st ed. London: Random House.

Vasicek, Oldrich. 2002. “Loan Portfolio Value.” Risk, December, 160–62.

Head/Tails breaks

Diego Hernangomez

2025-01-08

Abstract

Introduction

Breaking method

Step by step example

Implementation in `classInt` package

Case study

Testing and benchmark

References

Head/Tails breaks

Diego Hernangomez

2025-01-08

Abstract

Introduction

Breaking method

Step by step example

Implementation in classInt package

Case study

Testing and benchmark

References

Implementation in `classInt` package