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)

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.

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
<- 1 / (1 - runif(1000)) ^ (1 / 1.161)
sample_par <- par(no.readonly = TRUE)
opar 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)
```

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`

.

We reproduce here the pseudo-code^{1} 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:

```
<- par(no.readonly = TRUE)
opar par(mar = c(2, 2, 3, 1), cex = 0.8)
<- sample_par
var <- .4
thr <- c(min(var), max(var)) #Initialise with min and max
brks
<- data.frame(
sum_table iter = 0,
mu = NA,
prop = NA,
n_var = NA,
n_head = NA
)#Pars for chart
<- brks
limchart #Iteration
for (i in 1:10) {
<- mean(var)
mu <- sort(c(brks, mu))
brks <- var[var > mu]
head <- length(head) / length(var)
prop <- prop < thr & length(head) > 1
stopit = rbind(sum_table,
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)
<- max(hist(var, breaks = 50, plot = FALSE)$counts)
ylabel <- paste0("PropHead: ", round(prop * 100, 2), "%")
labelplot 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
<- head
var
}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))`

.

`classInt`

packageThe implementation in the `classIntervals`

function should
replicate the results:

```
<- classIntervals(sample_par, style = "headtails")
ht_sample_par == ht_sample_par$brks
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.

```
<- par(no.readonly = TRUE)
opar par(mar = c(2, 2, 2, 1), cex = 0.8)
<- c("wheat1", "wheat2", "red3")
pal1
# 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)`

.

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

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 |

```
<- par(no.readonly = TRUE)
opar 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:

```
<- classIntervals(afcon$totcon, style = "headtails")
brks_ht print(brks_ht)
#> style: headtails
#> one of 91,390 possible partitions of this variable into 5 classes
#> [147,1350.619) [1350.619,2488.6) [2488.6,3819.8) [3819.8,4998.5)
#> 27 10 3 1
#> [4998.5,5246]
#> 1
#Same number of classes for "fisher"
<- length(brks_ht$brks) - 1
nclass <- classIntervals(afcon$totcon, style = "fisher",
brks_fisher n = nclass)
print(brks_fisher)
#> style: fisher
#> one of 91,390 possible partitions of this variable into 5 classes
#> [147,693.5) [693.5,1474.5) [1474.5,2618) [2618,3942.5) [3942.5,5246]
#> 12 17 8 3 2
<- classIntervals(afcon$totcon, style = "quantile",
brks_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
<- c("wheat1", "wheat2", "red3")
pal1 <- par(no.readonly = TRUE)
opar 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
<- function(x, min = 1, max = 10) {
help_reescale <- (x - min(x)) / (max(x) - min(x))
r <- r * (max - min) + min
r return(r)
}$ecdf_class <- help_reescale(afcon$totcon,
afconmin = 1 - 0.5,
max = nclass - 0.5)
$ht_breaks <- cut(afcon$totcon,
afcon$brks,
brks_htlabels = FALSE,
include.lowest = TRUE)
$fisher_breaks <- cut(afcon$totcon,
afcon$brks,
brks_fisherlabels = FALSE,
include.lowest = TRUE)
$quantile_break <- cut(afcon$totcon,
afcon$brks,
brks_quantilelabels = FALSE,
include.lowest = TRUE)
<- par(no.readonly = TRUE)
opar par(mar = c(4, 4, 1, 1), cex = 0.8)
plot(
density(afcon$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.

```
<- c("#FE9F6D99",
custompal "#DE496899",
"#8C298199",
"#3B0F7099",
"#00000499")
$cex_points <- help_reescale(afcon$totcon,
afconmin = 1,
max = 5)
<- par(no.readonly = TRUE)
opar 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$cex_points,
pch = 20,
col = "grey50",
main = "Continuous",
)
<- (max(afcon$totcon) - min(afcon$totcon)) / 4
mcont <- 1:5 * mcont - (mcont - min(afcon$totcon))
legcont
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.

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
<- 7 / (1 - runif(5000000)) ^ (1 / 14)
paretodist #Exponential dist
<- rexp(5000000)
expdist #Lognorm
<- rlnorm(5000000)
lognormdist #Weibull
<- rweibull(5000000, 1, scale = 5)
weibulldist #LogCauchy "super-heavy tail"
<- exp(rcauchy(5000000, 2, 4))
logcauchdist #Remove Inf
<- logcauchdist[logcauchdist < Inf]
logcauchdist
#Normal dist
<- rnorm(5000000)
normdist #Left-tailed distr
<-
leftnorm sample(rep(normdist[normdist < mean(normdist)], 3), size = 5000000)
#Uniform distribution
<- runif(5000000) unifdist
```

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

```
<- data.frame(
testresults Title = NA,
style = NA,
nsample = NA,
thresold = NA,
nbreaks = NA,
time_secs = NA
)
<-
benchmarkdist function(dist,
style = "headtails",
thr = 0.4,
title = "",
plot = FALSE) {
<- Sys.time()
init <- classIntervals(dist, style = style, thr = thr)
br <- Sys.time() - init
a <- data.frame(
test 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))
)<- unique(rbind(testresults, test))
testresults
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(no.readonly = TRUE)
opar par(mar = c(2, 2, 2, 2), cex = 0.8)
# Pareto----
<- benchmarkdist(paretodist, title = "Pareto", plot = TRUE)
testresults <- benchmarkdist(paretodist, title = "Pareto", thr = 0)
testresults <- benchmarkdist(paretodist, title = "Pareto", thr = .75, plot = TRUE)
testresults
#Sample 2,000 obs
set.seed(1234)
<- sample(paretodist, 2000)
Paretosamp <- benchmarkdist(Paretosamp,
testresults title = "Pareto sample",
style = "fisher",
plot = TRUE)
<- benchmarkdist(Paretosamp,
testresults title = "Pareto sample",
style = "headtails",
plot = TRUE)
#Exponential----
<- 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",
testresults style = "quantile", plot = TRUE)
#Weibull-----
<- benchmarkdist(weibulldist, title = "Weibull", plot = TRUE)
testresults <- benchmarkdist(weibulldist, title = "Weibull", thr = 0)
testresults <- benchmarkdist(weibulldist, title = "Weibull", thr = 1)
testresults
#Logcauchy
<- benchmarkdist(logcauchdist, title = "LogCauchy", plot = TRUE)
testresults <- benchmarkdist(logcauchdist, title = "LogCauchy", thr = 0)
testresults <- benchmarkdist(logcauchdist, title = "LogCauchy", thr = 1)
testresults
#Normal----
<- benchmarkdist(normdist, title = "Normal", plot = TRUE)
testresults <- benchmarkdist(normdist, title = "Normal", thr = 0)
testresults <- benchmarkdist(normdist, title = "Normal", thr = 1, plot = TRUE)
testresults
#Truncated Left-tail Normal----
<- 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)
testresults
#Uniform----
<- 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)
testresults par(opar)
# Results
::kable(testresults[-1, ], row.names = FALSE) knitr
```

Title | style | nsample | thresold | nbreaks | time_secs |
---|---|---|---|---|---|

Pareto | headtails | 5,000,000 | 0.40 | 15 | 0.5586 |

Pareto | headtails | 5,000,000 | 0.00 | 2 | 0.5295 |

Pareto | headtails | 5,000,000 | 0.75 | 15 | 0.5592 |

Pareto sample | fisher | 2,000 | 0.40 | 12 | 0.0196 |

Pareto sample | headtails | 2,000 | 0.40 | 8 | 3e-04 |

Exponential | headtails | 5,000,000 | 0.40 | 16 | 0.4547 |

Exponential | headtails | 5,000,000 | 0.00 | 2 | 0.3724 |

Exponential | headtails | 5,000,000 | 1.00 | 17 | 0.4065 |

Exponential | quantile | 5,000,000 | 0.40 | 24 | 0.7121 |

Weibull | headtails | 5,000,000 | 0.40 | 16 | 0.4277 |

Weibull | headtails | 5,000,000 | 0.00 | 2 | 0.4585 |

Weibull | headtails | 5,000,000 | 1.00 | 17 | 0.4912 |

LogCauchy | headtails | 4,991,187 | 0.40 | 6 | 0.5134 |

LogCauchy | headtails | 4,991,187 | 0.00 | 2 | 0.4214 |

LogCauchy | headtails | 4,991,187 | 1.00 | 6 | 0.3842 |

Normal | headtails | 5,000,000 | 0.40 | 2 | 0.3906 |

Normal | headtails | 5,000,000 | 0.00 | 2 | 0.4259 |

Normal | headtails | 5,000,000 | 1.00 | 17 | 0.6012 |

Left Normal | headtails | 5,000,000 | 0.40 | 2 | 0.5093 |

Left Normal | headtails | 5,000,000 | -100.00 | 2 | 0.5749 |

Left Normal | headtails | 5,000,000 | 500.00 | 21 | 0.626 |

Uniform | headtails | 5,000,000 | 0.70 | 22 | 0.4728 |

Uniform | headtails | 5,000,000 | 0.00 | 2 | 0.4215 |

Uniform | headtails | 5,000,000 | 1.00 | 22 | 0.4693 |

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`

.

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.
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The method implemented in

`classInt`

corresponds to head/tails 1.0 as named in this article.↩︎