<s>
Head/tail	B-Application
breaks	I-Application
is	O
a	O
clustering	B-Algorithm
algorithm	I-Algorithm
for	O
data	O
with	O
a	O
heavy-tailed	O
distribution	O
such	O
as	O
power	O
laws	O
and	O
lognormal	O
distributions	O
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
is	O
not	O
just	O
for	O
classification	O
,	O
but	O
also	O
for	O
visualization	O
of	O
big	O
data	O
by	O
keeping	O
the	O
head	O
,	O
since	O
the	O
head	O
is	O
self-similar	O
to	O
the	O
whole	O
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
can	O
be	O
applied	O
not	O
only	O
to	O
vector	O
data	O
such	O
as	O
points	O
,	O
lines	O
and	O
polygons	O
,	O
but	O
also	O
to	O
raster	O
data	O
like	O
digital	O
elevation	O
model	O
(	O
DEM	O
)	O
.	O
</s>
<s>
The	O
head/tail	B-Application
breaks	I-Application
is	O
motivated	O
by	O
inability	O
of	O
conventional	O
classification	O
methods	O
such	O
as	O
equal	O
intervals	O
,	O
quantiles	O
,	O
geometric	O
progressions	O
,	O
standard	O
deviation	O
,	O
and	O
natural	O
breaks	O
-	O
commonly	O
known	O
as	O
Jenks	B-Algorithm
natural	I-Algorithm
breaks	I-Algorithm
optimization	I-Algorithm
or	O
k-means	B-Algorithm
clustering	I-Algorithm
to	O
reveal	O
the	O
underlying	O
scaling	O
or	O
living	O
structure	O
with	O
the	O
inherent	O
hierarchy	O
(	O
or	O
heterogeneity	O
)	O
characterized	O
by	O
the	O
recurring	O
notion	O
of	O
far	O
more	O
small	O
things	O
than	O
large	O
ones	O
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
uses	O
the	O
mean	O
or	O
average	O
to	O
dichotomize	O
a	O
dataset	O
into	O
small	O
and	O
large	O
values	O
,	O
rather	O
than	O
to	O
characterize	O
classes	O
by	O
average	O
values	O
,	O
which	O
is	O
unlike	O
k-means	B-Algorithm
clustering	I-Algorithm
or	O
natural	O
breaks	O
.	O
</s>
<s>
Through	O
the	O
head/tail	B-Application
breaks	I-Application
,	O
a	O
dataset	O
is	O
seen	O
as	O
a	O
living	O
structure	O
with	O
an	O
inherent	O
hierarchy	O
with	O
far	O
more	O
smalls	O
than	O
larges	O
,	O
or	O
recursively	O
perceived	O
as	O
the	O
head	O
of	O
the	O
head	O
of	O
the	O
head	O
and	O
so	O
on	O
.	O
</s>
<s>
Recursive	O
function	O
Head/tail	B-Application
Breaks	I-Application
:	O
</s>
<s>
The	O
resulting	O
number	O
of	O
classes	O
is	O
referred	O
to	O
as	O
ht-index	O
,	O
an	O
alternative	O
index	O
to	O
fractal	B-Application
dimension	O
for	O
characterizing	O
complexity	O
of	O
fractals	B-Application
or	O
geographic	O
features	O
:	O
the	O
higher	O
the	O
ht-index	O
,	O
the	O
more	O
complex	O
the	O
fractals	B-Application
.	O
</s>
<s>
The	O
criterion	O
to	O
stop	O
the	O
iterative	O
classification	O
process	O
using	O
the	O
head/tail	B-Application
breaks	I-Application
method	O
is	O
that	O
the	O
remaining	O
data	O
(	O
i.e.	O
,	O
the	O
head	O
part	O
)	O
are	O
not	O
heavy-tailed	O
,	O
or	O
simply	O
,	O
the	O
head	O
part	O
is	O
no	O
longer	O
a	O
minority	O
(	O
i.e.	O
,	O
the	O
proportion	O
of	O
the	O
head	O
part	O
is	O
no	O
longer	O
less	O
than	O
a	O
threshold	O
such	O
as	O
40%	O
)	O
.	O
</s>
<s>
This	O
process	O
is	O
called	O
head/tail	B-Application
breaks	I-Application
1.0	O
.	O
</s>
<s>
In	O
head/tail	B-Application
breaks	I-Application
2.0	O
the	O
threshold	O
only	O
applies	O
to	O
the	O
overall	O
heads	O
 '	O
percentage	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
there	O
is	O
data	O
distributed	O
in	O
such	O
a	O
way	O
that	O
it	O
has	O
a	O
clearly	O
defined	O
head	O
and	O
tail	O
during	O
the	O
first	O
and	O
second	O
iteration	O
(length(head )	O
/	O
(length(data )	O
<	O
20%	O
)	O
but	O
a	O
much	O
less	O
well	O
defined	O
long	O
tailed	O
distribution	O
for	O
the	O
third	O
iteration	O
(	O
60%	O
in	O
the	O
head	O
)	O
,	O
head/tail	B-Application
breaks	I-Application
2.0	O
allows	O
the	O
iteration	O
to	O
continue	O
into	O
the	O
fourth	O
iteration	O
which	O
can	O
be	O
distributed	O
30%	O
head	O
-	O
70%	O
tail	O
again	O
and	O
so	O
on	O
.	O
</s>
<s>
As	O
long	O
as	O
the	O
overall	O
threshold	O
is	O
not	O
surpassed	O
the	O
head/tail	B-Application
breaks	I-Application
classification	O
holds	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
ht-index	O
,	O
the	O
following	O
indices	O
are	O
also	O
derived	O
with	O
the	O
head/tail	B-Application
breaks	I-Application
.	O
</s>
<s>
Two	O
unified	O
metrics	O
(	O
UM1	O
and	O
UM2	O
)	O
were	O
proposed	O
in	O
an	O
AAAG	O
paper	O
for	O
characterizing	O
the	O
fractal	B-Application
nature	O
of	O
geographic	O
features	O
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
can	O
be	O
used	O
to	O
characterize	O
the	O
scale-free	O
phenomena	O
,	O
which	O
are	O
in	O
the	O
majority	O
.	O
</s>
<s>
As	O
the	O
head/tail	B-Application
breaks	I-Application
method	O
can	O
be	O
used	O
iteratively	O
to	O
obtain	O
head	O
parts	O
of	O
a	O
data	O
set	O
,	O
this	O
method	O
actually	O
captures	O
the	O
underlying	O
hierarchy	O
of	O
the	O
data	O
set	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
we	O
divide	O
the	O
array	O
(	O
19	O
,	O
8	O
,	O
7	O
,	O
6	O
,	O
2	O
,	O
1	O
,	O
1	O
,	O
1	O
,	O
0	O
)	O
with	O
the	O
head/tail	B-Application
breaks	I-Application
method	O
,	O
we	O
can	O
get	O
two	O
head	O
parts	O
,	O
i.e.	O
,	O
the	O
first	O
head	O
part	O
(	O
19	O
,	O
8	O
,	O
7	O
,	O
6	O
)	O
and	O
the	O
second	O
head	O
part	O
(	O
19	O
)	O
.	O
</s>
<s>
The	O
term	O
‘	O
natural	O
cities’	O
refers	O
to	O
the	O
human	O
settlements	O
or	O
human	O
activities	O
in	O
general	O
on	O
Earth	O
's	O
surface	O
that	O
are	O
naturally	O
or	O
objectively	O
defined	O
and	O
delineated	O
from	O
massive	O
geographic	O
information	O
based	O
on	O
head/tail	O
division	O
rule	O
,	O
a	O
non-recursive	O
form	O
of	O
head/tail	B-Application
breaks	I-Application
.	O
</s>
<s>
Also	O
note	O
that	O
the	O
structure	O
of	O
natural	O
cities	O
can	O
be	O
considered	O
to	O
be	O
fractal	B-Application
in	O
nature	O
.	O
</s>
<s>
It	O
is	O
important	O
when	O
head/tail	B-Application
breaks	I-Application
are	O
being	O
used	O
to	O
generate	O
natural	O
cities	O
,	O
that	O
the	O
data	O
is	O
not	O
aggregated	O
afterwards	O
.	O
</s>
<s>
An	O
example	O
of	O
this	O
is	O
seen	O
in	O
a	O
research	O
where	O
head/tail	B-Application
breaks	I-Application
were	O
used	O
to	O
extract	O
natural	O
cities	O
,	O
but	O
they	O
were	O
aggregated	O
to	O
administrative	O
borders	O
,	O
which	O
following	O
that	O
concluded	O
that	O
the	O
cities	O
do	O
not	O
follow	O
Zipf	O
's	O
law	O
.	O
</s>
<s>
It	O
was	O
found	O
that	O
coloring	O
based	O
head/tail	B-Application
breaks	I-Application
is	O
more	O
favorable	O
than	O
those	O
by	O
other	O
classifications	O
.	O
</s>
<s>
Other	O
applications	O
of	O
Head/tail	B-Application
breaks	I-Application
:	O
</s>
<s>
To	O
identify	O
urban	O
function	O
graphs	O
,	O
note	O
that	O
this	O
paper	O
applies	O
head/tail	B-Application
breaks	I-Application
on	O
a	O
Gaussian	O
kernel	B-General_Concept
density	I-General_Concept
estimation	I-General_Concept
which	O
reduces	O
the	O
accuracy	O
of	O
the	O
head/tail	B-Application
breaks	I-Application
method	O
.	O
</s>
<s>
Essentially	O
a	O
natural	O
cities	O
approach	O
is	O
taken	O
but	O
the	O
initial	O
data	O
chosen	O
to	O
perform	O
head/tail	B-Application
breaks	I-Application
on	O
has	O
been	O
reduced	O
beforehand	O
.	O
</s>
<s>
For	O
a	O
better	O
representation	O
of	O
urban	O
function	O
graphs	O
head/tail	B-Application
breaks	I-Application
may	O
be	O
applied	O
as	O
the	O
first	O
step	O
in	O
delineating	O
these	O
areas	O
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
are	O
applied	O
to	O
separate	O
the	O
less	O
dense	O
stops	O
where	O
few	O
people	O
exit	O
,	O
from	O
the	O
major	O
stops	O
where	O
the	O
most	O
people	O
exit	O
.	O
</s>
<s>
Detecting	O
hierarchical	O
crowd	O
data	O
with	O
different	O
clustering	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
Reducing	O
carbon	O
emissions	O
by	O
dividing	O
urban	O
spaces	O
using	O
head/tail	B-Application
breaks	I-Application
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
can	O
serve	O
as	O
a	O
main	O
indicator	O
that	O
phenomena	O
are	O
distributed	O
long	O
tailed	O
and	O
that	O
Paretian	O
thinking	O
should	O
favor	O
Gaussian	O
thinking	O
in	O
geographic	O
spaces	O
.	O
</s>
<s>
For	O
example	O
within	O
biodiversity	O
and	O
pedodiversity	O
studies	O
where	O
there	O
seem	O
to	O
be	O
fractal	B-Application
relationships	O
such	O
as	O
taxa-area	O
relationships	O
.	O
</s>
<s>
This	O
can	O
be	O
identified	O
and	O
highlighted	O
by	O
using	O
head/tail	B-Application
breaks	I-Application
.	O
</s>
<s>
In	O
image	O
feature	O
and	O
texture	O
extraction	O
,	O
certain	O
algorithms	O
like	O
the	O
discrete	O
pulse	O
transform	O
,	O
where	O
LULU	O
smoothing	O
is	O
used	O
to	O
extract	O
the	O
features	O
,	O
can	O
be	O
sped	O
up	O
by	O
using	O
head/tail	B-Application
breaks	I-Application
in	O
the	O
algorithm	O
by	O
separating	O
large	O
features	O
and	O
smaller	O
features	O
more	O
effectively	O
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
are	O
an	O
aid	O
in	O
determining	O
the	O
hierarchies	O
present	O
because	O
of	O
the	O
scaling	O
nature	O
of	O
urban	O
morphology	O
and	O
could	O
be	O
of	O
further	O
use	O
when	O
studying	O
urban	O
street	O
network	O
applications	O
.	O
</s>
<s>
This	O
is	O
especially	O
the	O
case	O
for	O
accessibility	O
analysis	O
,	O
combined	O
with	O
space	O
syntax	O
head/tail	B-Application
breaks	I-Application
allow	O
for	O
an	O
in	O
depth	O
understanding	O
of	O
street	O
network	O
structure	O
.	O
</s>
<s>
Urban	O
structures	O
,	O
like	O
street	O
networks	O
have	O
been	O
proven	O
to	O
be	O
fractal	B-Application
in	O
nature	O
.	O
</s>
<s>
An	O
important	O
point	O
to	O
note	O
is	O
that	O
this	O
structure	O
is	O
not	O
consisting	O
of	O
only	O
one	O
defined	O
fractal	B-Application
,	O
it	O
is	O
characterized	O
by	O
a	O
multifractal	O
complex	O
network	O
.	O
</s>
<s>
This	O
means	O
that	O
on	O
different	O
scales	O
,	O
the	O
defined	O
fractal	B-Application
can	O
change	O
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
can	O
be	O
used	O
to	O
determine	O
the	O
structure	O
of	O
the	O
complex	O
network	O
over	O
different	O
scales	O
,	O
as	O
it	O
adjusts	O
based	O
on	O
the	O
data	O
with	O
each	O
new	O
hierarchy	O
.	O
</s>
<s>
Head/tail	B-Application
breaks	I-Application
as	O
a	O
classification	O
method	O
can	O
be	O
used	O
to	O
visualize	O
growth	O
or	O
spread	O
patterns	O
in	O
for	O
example	O
a	O
global	O
pandemic	O
,	O
like	O
the	O
Covid-19	O
one	O
.	O
</s>
<s>
By	O
using	O
head/tail	B-Application
breaks	I-Application
,	O
main	O
spread	O
events	O
can	O
be	O
effectively	O
mapped	O
and	O
visualized	O
where	O
locations	O
with	O
a	O
high	O
infection	O
rate	O
are	O
highlighted	O
specifically	O
due	O
to	O
them	O
being	O
in	O
the	O
highest	O
class	O
.	O
</s>
<s>
Because	O
of	O
the	O
self-similarity	O
characteristics	O
of	O
these	O
fractures	O
combined	O
with	O
the	O
fractal	B-Application
nature	O
they	O
inhibit	O
,	O
head/tail	B-Application
breaks	I-Application
provide	O
accurate	O
measurements	O
and	O
analysis	O
into	O
these	O
rock	O
fracture	O
networks	O
.	O
</s>
<s>
The	O
following	O
implementations	O
are	O
available	O
under	O
Free/Open	B-License
Source	I-License
Software	I-License
licenses	O
.	O
</s>
<s>
HT	O
calculator	O
:	O
a	O
winform	O
application	O
for	O
obtaining	O
related	O
metrics	O
of	O
head/tail	B-Application
breaks	I-Application
applying	O
on	O
a	O
single	O
data	O
array	O
.	O
</s>
<s>
HT	O
in	O
JavaScript	O
:	O
a	O
JavaScript	O
implementation	O
for	O
applying	O
head/tail	B-Application
breaks	I-Application
on	O
a	O
single	O
data	O
array	O
.	O
</s>
<s>
HT	O
Mapping	O
tool	O
:	O
a	O
function	O
in	O
the	O
free	O
plug-in	O
Axwoman	O
6.3	O
to	O
ArcMap	B-Application
10.2	O
that	O
conducts	O
geo-data	O
symbolization	O
automatically	O
based	O
on	O
the	O
head/tail	B-Application
breaks	I-Application
classification	O
.	O
</s>
<s>
HT	O
in	O
Python	O
:	O
Python	O
and	O
JavaScript	O
code	O
for	O
the	O
head/tail	B-Application
breaks	I-Application
algorithm	O
.	O
</s>
<s>
pysal.esda.mapclassify	O
:	O
Python	O
classification	O
schemes	O
for	O
choropleth	O
mapping	O
,	O
including	O
head/tail	B-Application
breaks	I-Application
map	O
classification	O
.	O
</s>
<s>
:	O
An	O
excel	O
sheet	O
calculator	O
which	O
calculates	O
both	O
head/tail	B-Application
breaks	I-Application
1.0	O
and	O
head/tail	B-Application
breaks	I-Application
2.0	O
with	O
a	O
version	O
for	O
smaller	O
datasets	O
and	O
a	O
version	O
for	O
very	O
large	O
(	O
binned	O
)	O
datasets	O
.	O
</s>
<s>
classInt	O
:	O
R	B-Language
package	O
that	O
implements	O
several	O
methods	O
for	O
choosing	O
univariate	O
class	O
intervals	O
for	O
mapping	O
or	O
other	O
graphic	O
purposes	O
also	O
includes	O
head/tail	B-Application
breaks	I-Application
map	O
classification	O
.	O
</s>
