2023-02-28

There are far more ordinary people (say, 80 percent) than extraordinary people (say, 20 percent); this is often characterized by the 80/20 principle, based on the observation made by the Italian economist Vilfredo Pareto in 1906 that 80% of land in Italy was owned by 20% of the population. A histogram of the data values for these phenomena would reveal a right-skewed or heavy-tailed distribution. How to map the data with the heavy-tailed distribution?
Jiang (2013)

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)
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:

1. Compute mu = mean(var).
2. Break var into the tail (as var < mu) and the head (as var > mu).
3. Assess if the proportion of head over var is lower or equal than a given threshold (i.e. length(head)/length(var) <= thr)
4. 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
End Function

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


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,
)
#Pars for chart
limchart <- brks
#Iteration
for (i in 1:10) {
mu <- mean(var)
brks <- sort(c(brks, mu))
stopit <- prop < thr & length(head) > 1
sum_table = rbind(sum_table,
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)
#> [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))
#> [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))
#> [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
#> [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
pal = pal1,
main = "thr = Default")

# Maximum breaks
print(classIntervals(sample_par, style = "headtails", thr = 1))
#> [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)
#>   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(afcontotcon, 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) } afconecdf_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)

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",
"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)
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],
)
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)
}
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",
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 0.552
Pareto headtails 5,000,000 0.00 2 0.5444
Pareto headtails 5,000,000 0.75 15 0.5545
Pareto sample fisher 2,000 0.40 12 0.0215
Pareto sample headtails 2,000 0.40 8 3e-04
Exponential headtails 5,000,000 0.40 16 0.4702
Exponential headtails 5,000,000 0.00 2 0.4205
Exponential headtails 5,000,000 1.00 17 0.4244
Exponential quantile 5,000,000 0.40 24 0.7006
Weibull headtails 5,000,000 0.40 16 0.4134
Weibull headtails 5,000,000 0.00 2 0.4511
Weibull headtails 5,000,000 1.00 17 0.48
LogCauchy headtails 4,991,187 0.40 6 0.4903
LogCauchy headtails 4,991,187 0.00 2 0.4117
LogCauchy headtails 4,991,187 1.00 6 0.3666
Normal headtails 5,000,000 0.40 2 0.3752
Normal headtails 5,000,000 0.00 2 0.4164
Normal headtails 5,000,000 1.00 17 0.6555
Left Normal headtails 5,000,000 0.40 2 0.4753
Left Normal headtails 5,000,000 -100.00 2 0.5718
Left Normal headtails 5,000,000 500.00 21 0.658
Uniform headtails 5,000,000 0.70 22 0.4538
Uniform headtails 5,000,000 0.00 2 0.4203
Uniform headtails 5,000,000 1.00 22 0.4706

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.

1. The method implemented in classInt corresponds to head/tails 1.0 as named in this article.↩︎