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A New Crescent Moon Visibility Criteria using Circular Regression Model: A Case Study of Teluk Kemang, Malaysia

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Many astronomers have studied lunar crescent visibility throughout history. Its importance is unquestionable, especially in determining the local Islamic calendar and the dates of important Islamic events. Different criteria have been used to predict the possible visibility of the crescent moon during the sighting process. However, so far, the visibility models used are based on linear statistical theory, whereas the useful variables in this study are in the circular unit. Hence, in this paper, we propose new visibility tests using the circular regression model, which will split the data into three visibility categories; visible to the unaided eye, may need optical aid and not visible. We formulate the procedure to separate the categories using the residuals of the fitted circular regression model. We apply the model on 254 observations collected at Baitul Hilal Teluk Kemang Malaysia, starting from March 2000 to date. We show that the visibility test developed based on elongation of the moon (dependent variable) and altitude of the moon (independent variable) gives the smallest misclassification rate. From the statistical analysis, we propose the elongation of the moon 7.28°, altitude of the moon of 3.33° and arc of vision of 3.74 at sunset as the new crescent visibility criteria. The new criteria have a significant impact on improving the chance of observing the crescent moon and in producing a more accurate Islamic calendar in Malaysia. ABSTRAK Ramai ahli astronomi telah mengkaji kebolehnampakan bulan sabit sepanjang sejarah. Kepentingannya tidak dapat dipertikaikan, terutama dalam menentukan kalendar Islam tempatan dan tarikh peristiwa penting Islam. Kriteria yang berbeza telah digunakan untuk meramalkan kemungkinan kebolehnampakan bulan sabit semasa proses pencerapan. Walau bagaimanapun, setakat ini, model kebolehnampakan yang digunakan adalah berdasarkan teori statistik linear, sedangkan pemboleh ubah penting dalam kajian ini adalah dalam sukatan membulat. Oleh itu, dalam kertas ini, kami mencadangkan ujian kebolehnampakan baru menggunakan model regresi berkeliling, yang akan membahagikan data menjadi tiga kategori kebolehnampakan; dapat dilihat dengan mata kasar, mungkin memerlukan bantuan optik dan tidak kelihatan. Kami memformulasi prosedur tersebut untuk memisahkan kategori menggunakan sisa model regresi berkeliling yang sesuai. Kami mengaplikasikan model tersebut dalam 254 pemerhatian yang dikumpulkan di Baitul Hilal Teluk Kemang Malaysia, bermula dari Mac 2000 sehingga kini. Kami menunjukkan bahawa ujian kebolehnampakan dibangunkan berdasarkan pemanjangan bulan (pemboleh ubah bersandar) dan ketinggian bulan (pemboleh ubah bebas) memberikan kadar salah pengkelasan terkecil. Daripada analisis statistik, kami mencadangkan pemanjangan bulan pada 7.28°, ketinggian bulan 3.33° dan aras penglihatan 3.74° ketika matahari terbenam sebagai kriteria baharu kebolehnampakan bulan sabit. Kriteria baharu ini memberi kesan yang besar dalam meningkatkan peluang melihat bulan sabit dan menghasilkan kalendar Islam yang lebih tepat di Malaysia studied.
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Sains Malaysiana 49(4)(2020): 859-870
http://dx.doi.org/10.17576/jsm-2020-4904-15
A New Crescent Moon Visibility Criteria using Circular Regression Model: A Case
Study of Teluk Kemang, Malaysia
(Kriteria Baru Kebolehnampakan Bulan Sabit menggunakan Model Regresi Berkeliling: Suatu Kajian Kes Teluk
Kemang, Malaysia)
NAZHATULSHIMA AHMAD*, MOHD SAIFUL ANWAR MOHD NAWAWI, MOHD ZAMBRI ZAINUDDIN, ZUHAILI MOHD
NASIR, ROSSITA MOHAMAD YUNUS & IBRAHIM MOHAMED
ABSTRACT
Many astronomers have studied lunar crescent visibility throughout history. Its importance is unquestionable, especially
in determining the local Islamic calendar and the dates of important Islamic events. Dierent criteria have been used
to predict the possible visibility of the crescent moon during the sighting process. However, so far, the visibility models
used are based on linear statistical theory, whereas the useful variables in this study are in the circular unit. Hence, in
this paper, we propose new visibility tests using the circular regression model, which will split the data into three
visibility categories; visible to the unaided eye, may need optical aid and not visible. We formulate the procedure to
separate the categories using the residuals of the tted circular regression model. We apply the model on 254 observations
collected at Baitul Hilal Teluk Kemang Malaysia, starting from March 2000 to date. We show that the visibility test
developed based on elongation of the moon (dependent variable) and altitude of the moon (independent variable) gives
the smallest misclassication rate. From the statistical analysis, we propose the elongation of the moon 7.28°, altitude
of the moon of 3.33° and arc of vision of 3.74 at sunset as the new crescent visibility criteria. The new criteria have a
signicant impact on improving the chance of observing the crescent moon and in producing a more accurate Islamic
calendar in Malaysia.
Keywords: Circular regression; crescent moon; lunar month; q-test; visibility criteria
ABSTRAK
Ramai ahli astronomi telah mengkaji kebolehnampakan bulan sabit sepanjang sejarah. Kepentingannya tidak dapat
dipertikaikan, terutama dalam menentukan kalendar Islam tempatan dan tarikh peristiwa penting Islam. Kriteria yang
berbeza telah digunakan untuk meramalkan kemungkinan kebolehnampakan bulan sabit semasa proses pencerapan.
Walau bagaimanapun, setakat ini, model kebolehnampakan yang digunakan adalah berdasarkan teori statistik linear,
sedangkan pemboleh ubah penting dalam kajian ini adalah dalam sukatan membulat. Oleh itu, dalam kertas ini, kami
mencadangkan ujian kebolehnampakan baru menggunakan model regresi berkeliling, yang akan membahagikan data
menjadi tiga kategori kebolehnampakan; dapat dilihat dengan mata kasar, mungkin memerlukan bantuan optik dan
tidak kelihatan. Kami memformulasi prosedur tersebut untuk memisahkan kategori menggunakan sisa model regresi
berkeliling yang sesuai. Kami mengaplikasikan model tersebut dalam 254 pemerhatian yang dikumpulkan di Baitul Hilal
Teluk Kemang Malaysia, bermula dari Mac 2000 sehingga kini. Kami menunjukkan bahawa ujian kebolehnampakan
dibangunkan berdasarkan pemanjangan bulan (pemboleh ubah bersandar) dan ketinggian bulan (pemboleh ubah
bebas) memberikan kadar salah pengkelasan terkecil. Daripada analisis statistik, kami mencadangkan pemanjangan
bulan pada 7.28°, ketinggian bulan 3.33° dan aras penglihatan 3.74° ketika matahari terbenam sebagai kriteria
baharu kebolehnampakan bulan sabit. Kriteria baharu ini memberi kesan yang besar dalam meningkatkan peluang
melihat bulan sabit dan menghasilkan kalendar Islam yang lebih tepat di Malaysia studied.
Kata kunci: Bulan lunar; bulan sabit; kriteria kebolehnampakan; regresi berkeliling; ujian q
INTRODUCTION
Main religions in the world, including Jews, Hindu, and
Islam, have their calendars based on lunar month. They
are mainly used to determine dates of important events or
festivals in each religion. As a result, the determination of
criteria to indicate the expected visibility of the crescent
moon is of utmost importance since the time of Babylon
Era (Ilyas 1994). The criteria are mainly derived based on
the crescent moon data collected at end of the month. The
variables measured include elongation (Elon), altitude of
the moon (Alt(M)), altitude of the sun (Alt(S)), arc of vision
(ARCV), width of the crescent moon (W), lag time between
sunset and moonset (lag time), and age of the crescent
860
moon from conjunction (age). The choice of the parameters
for the criteria mainly correspond to the minimum contrast
between the brightness of the moon and the sky. That is,
we look at certain values such that the moon is bright
enough, or the sky is dark enough for the crescent moon
to be seen. For example, the Babylonians used age and lag
time as a measure of brightness of the moon and the sky,
respectively (Bruin 1977). Here the lag time is at least 48
min after sunset, and this value has changed since.
In the past, dierent studies report the criteria based
on a dierent set of variables measured in crescent moon
sighting activities. Among the early Arabic astronomers,
Al-Tabari utilized the depression angle of the sun in the
visibility of the crescent moon. The crescent would be
considered visible at the time of moonset if the altitude of
the sun was 9.5° below the horizon (Guessoum & Meziane
2001; Hogendijk 1988). In the more recent centuries, most
models were built based on the observations made by Julius
Schmidt in Athens, Greece, from 1859-1880 (Schaefer
1988). Based on 76 sets of observations of the crescent
moon, Fotheringham (1910) established necessary
specic criteria, which include the relative altitude of the
moon with respect to the sun’s altitude (known as an arc
of vision, ARCV). By placing a line of separation between
the negative and positive crescent moons, Fotheringham
(1910) gave a minimum limit of an ARCV of 12° and
a relative azimuth of 0°. Maunder (1911) formulated a
smaller minimum limit than Fotheringham (1910), which
is 11° ARCV at 0° relative azimuths, for when the crescent
moons can be seen, as he suggested there was a technical
issue with the negative data reported by Fotheringham.
Ilyas (1988) examined the ACRV and its relative azimuth
criteria and found the criteria proposed by Fotheringham
(1910) and Maunder (1911) were limited to a dierence
in azimuth of 20°. At a larger scale, these criteria cannot be
applied. Consequently, to match his criteria of elongation
of 10.4°, Ilyas (1988) concluded that the minimum limit
of ARCV was supposedly 10.5° with a relative azimuth
of 0°. Recently, Raharto et al. (2019) presented the all-
possible moon astronomical position at sun-set time in a
diagram of Alt(M) vs Elon and the azimuth dierence of
the moon and the sun. Then, by analysing the importance
of dierent variables considered in the study, they came
up with a set of new criteria based on the arc of light with
values 6° or 6.4° depending on the visibility on equatorial
and subtropical observation.
In recent years, discussion on the criteria focussed
on the Danjon limit (Danjon 1936). In the year 1931, the
French astronomer André Danjon measured 75 moon
samples observed using a theoretical approach. He
estimated the length of the crescent moon by measuring
the parts of the moon illuminated by sunlight. Crescent
is expected to be visible if the elongation is more than
7°, hereafter known as the Danjon limit (Fatoohi et al.
1998). Further improvement of the criteria was later
published. McNally (1983) suggested that atmospheric
seeing causes the crescent to be obscured when it is smaller
than the seeing disk. He concluded that the Danjon limit
is supposed to be 5° rather than 7°. However, Schaefer
(1991) explained that atmospheric seeing is not the
main factor in the deciency of the arc. He developed a
model and suggested 7° as the new Danjon limit for the
crescent to be visible. Ilyas (1983) stated that the Danjon
limit is intended to be a general guide. However, for the
formation of calendar regulation, elongation of 10.5° is
the best. Fatoohi et al. (1998) and Odeh (2004) studied
the observational reports and respectively concluded
that 7.4° and 6.4° as the estimated Danjon limits. Sultan
(2007) and Hasanzadeh (2012) developed a photometric
model of crescent visibility and re-evaluated the Danjon
limit to be 5°.
Dierent mathematical approaches have been used to
arrive at the values the criteria. McNally (1983) studied
mathematically the eect of atmosphere on the shape of
crescent moon and formulated the width as a measure of
shortening the crescent moon in terms of θ and φ, where θ
=180-χ, χbeing the elongation of the earth from the sun as
viewed from the moon centre and φ is the position of outer
terminator near the cusp. He suggested the atmospheric
factor should be considered in order to maximize the
length of the outer terminator. The poor seeing condition
will cause a shortened terminator of the crescent moon.
However, Schaefer (1991) later argued that atmospheric
factor is not important by considering the Hapke’s lunar
surface brightness measure.
Ilyas (1994) reviewed the development of criteria,
especially in producing universal international Islamic
calendar amid the challenge for quality crescent moon
data. A unied approach of ve practical considerations
is proposed to come up with a universal international
Islamic calendar in the future. Yallop (1997) introduced
the q-test as a test of the visibility of the crescent by
considering the residuals of the fitted polynomial
regression model of ARCV on the width of crescent moon.
Several dierent categories of visibility are proposed.
Homan (2003) provided a collection of crescent moon
data set observed in good weather conditions. He used
to update the criteria of the q-test and further claimed
that the data could be used to validate any visibility tests.
Similarly, Odeh (2004) combined data set from dierent
studies and used them to come up with the new values of
the existing criteria. Hasanzadeh (2012) used the weighted
polynomial function of the arclength of crescent moon
against elongation and obtained a new value of the
Danjon limit by extrapolating the curves to the case of
zero arc length. Recently, Alrefay et al. (2018) analysed
the relationship between dierent pair of variables. In
particular, they proposed the hypothetical curve which can
best separated the observations with positive/negative
crescent moon visibility in terms of W and ARCV.
So far, the development of the criteria uses only
linear statistical theory. However, most of the variables in
crescent moon data are measured in degree/radians. Hence,
in this paper, we consider the circular statistical theory
to come up with new criteria for the visibility of local
861
crescent moon data. The proposed criteria follow closely
the methods adopted by Yallop (1997) for the q-test. This
paper is organized as follows: In the next section, we
present the background of the data collected at the main
observing station in Malaysia. Then, the circular statistical
theory used in this study is covered in subsequent section
while the development of the new tests is in section that
follows. In the following section, we presents the ndings
of the study. Discussion on the results and conclusion are
included in the last section.
OBSERVATIONS AND DATA COLLECTION
We carried out the crescent moon sighting activities
at Baitul Hilal Teluk Kemang (Latitude: 2° 27’ 44” N,
Longitude: 101° 51’ 21”E, height: 14 m above sea level).
Historically, this site was the rst location in Malaysia
used for sighting the crescent moon in the 1970s.
During the years, the observations were conducted only
for three lunar months (29th of Ramadhan, Syawal, and
TABLE 1. The denition of geometric variables used for the sun and moon
Variable Denition
WWidth of the crescent moon as view from the earth, measured in
arcminutes
Alt(S)Altitude of the sun
Alt(M)Altitude of the moon
ARCV Arc of vision, i.e. the geocentric dierence in the altitude between the
centre of the sun and the centre of the moon for a given latitude and
longitude with taking into account the eects of refraction
Elon Elongation, which refers to the angle between the centre of the sun
and the centre of the moon, as viewed from earth
Status Y = visible, N = not visible
Zulkaedah of Hijr) each year to determine the start of the
fasting month, Eid al-Fitr and Eid al-Adha, until 1999.
The observations were conducted using theodolites
operated by the surveyors and the committee of crescent
moon observation to validate the visibility if it is
sighted. Starting in March 2000, the observation has
been carried out consistently on the 29th and 30th of
each lunar months until now. We used various equipment
and methods in the observations, such as theodolites,
portable telescopes of 12-inch reflector and 76 mm
refractor, and the naked eye. The images of the crescent
moon were then recorded using a DLSR camera.
We have collected 254 data since 2000 (1420H)
which consists of 81 positive data, and the rest of the
data are not visible due to bad weather and severe
sky conditions, and due to the very low values of the
criteria to be observed. Table 1 and Figure 1(a) and 1(b)
respectively show the denition and diagram of the
geometric variables used for the sun and moon.
FIGURE 1 (a). Schematic diagram of geometric variables of the Sun and moon at sunset: ARCV, relative
altitude between the center of the moon and the sun; ALT , altitude of the center of the crescent moon above the
horizon; DAZ, azimuth dierence between the sun and moon; ARCL is equivalent to elongation; and Width,
the crescents width
862
FIGURE 1 (b). Global view of the geometric variables of the sun and moon after a few hours of conjunction;
Elongation is an angle between the center of the sun and the moon as seen from the Earth and the reected of
light from the moon after several hours of conjunction called the crescent
METHODS
We note that the ve variables considered in Table 1 are
circular; that they are measured in radian or degree. One
of the important properties of circular variables is the
bounded property of the variables such that the observed
values taken are within the range (0, 2π). Recent papers
in circular regression models and their diagnostic tools
include Alkasadi et al. (2018) and Kim and Rifat
(2019). Here, we intend to use the relevant theory in
circular statistics in order to come with a better visibility
test for crescent moon detection. Detail is available in
Jammalamadaka and SenGupta (2001).
MEASURES FOR CIRCULAR VARIABLES
To describe any circular data set, we need some measures
of location and dispersion. Let
n
θθ
,...,
1
be observations
in a random circular sample of size from a circular
population.
Mean direction
To summarize the circular data, we use the mean
direction as a measure of tendency. For a given circular
random sample, we consider each observation to be a
unit vector whose direction is specied by the circular
angle and nd their resultant vector. The mean direction
is dened by the angle made by the resultant vector with a
horizontal line. Specically, we have the resultant length
R
given by,
22
SCR +=
,
where
=
=
n
ii
C
1
cos
θ
and
=
=
n
ii
S
1
sin
θ
. The mean direction,
θ
, may be obtained by solving the equations,
and
R
S
=
θ
sin
, where
(1)
One of the mean direction characteristics is that
, which is analogous to the linear case.
Concentration parameter
The concentration parameter, denoted by
κ
, is a standard
measure of dispersion for c distribution. Best and Fisher
(1981) gave the maximum likelihood estimates of the
concentration parameter
κ
as follows:
(2)
where is mean resultant length and is given by
n
R
R=
.
The larger the value of concentration parameter, the more
concentrated the data towards the mean direction.
Median, quantile and percentile
Mardia and Jupp (1972) dened the median as any point
φ
, where half of the data lie in the arc
[
)
πφφ
+,
and
the other points are nearer to than to
πφ
+
. Basically,
for any circular sample, Fisher (1993) dened the
median direction as the observation which minimizes
the summation of circular distances to all observations
n
θθ
,...,
1
, that is,
. Fisher’s
.0,0ifundefined,
,0,0if,2)(tan
,0if,)(tan
,0,0if,
2
,0,0if),(tan
1-
1-
1
CS
CSCS
CCS
CS
CSCS
n
 
 
85.0if
,34
85.053.0if
,
1
43.0
39.14.0
53.0if
,
6
5
2
ˆ
1
23
53
R
RRR
R
R
R
R
RRR
 
 
85.0if
,34
85.053.0if
,
1
43.0
39.14.0
53.0if
,
6
5
2
ˆ
1
23
53
R
RRR
R
R
R
R
RRR
( )
=
=
n
ii
d
1
φθππφ
R
C
=
θ
cos
R
φ
863
denition is used to obtain the circular median in the
Oriana statistical software package. On the other hand,
the rst and third quantile directions
1
Q
and
3
Q
are any
solution of
(3)
respectively.
1
Q
can be considered as the median of the
rst half of the ordered data and
3
Q
as the median of the
second. The percentiles can then be obtained by further
splitting the ordered sample.
CIRCULAR CORRELATION
Special measure of correlation has been developed for
any two circular variables. Given is
a random sample of observations measured as angles.
As for measuring the correlation between two circular
variables, we use the sample circular correlation given by
(4)
where and are sample mean directions. As in the linear
case, takes values in the range and the closer to 1 or -1
indicates the stronger relationship between the variables.
The relationship for circular variables can also be
described using spoke plot (Zubairi et al. 2008).
CIRCULAR REGRESSION MODEL
Due the bounded property of circular variables, various
circular regression models have been proposed to model
the relationship between 2 circular variables, see for
example in Hussin et al. (2004). Here, due to its simple
property and possibility to be extended to a general
case, we consider the regression model proposed by
Jammalamadaka and Sarma (1993), (JS hereafter) for
two circular random variables U and V in terms of the
conditional expectation of e (iv) given u given by,
(5)
where e iv = cos v + i sin v, represents the conditional
mean direction of v given u and p(u) the conditional
concentration parameter for some periodic function g1(u)
and g2(u). Equivalently, we may write
(6)
We may then predict v such that
The diculty of non-parametrically estimating
g1(u) and g2(u) leads us to approximate them by using
suitable functions, taking into account they are both
periodic with period 2π. The approximations used are
the trigonometric polynomials of suitable degree m of the
form
(8)
We therefore have the following models:
(9)
where is the vector of random errors following
the normal distribution with mean vector 0 and
unknown dispersion matrix Σ. The parameters Ak, Bk, Ck,
and Dk, where k = 0, 1, …, m, the standard errors as well
as the dispersion matrix Σ can then be estimated using
the generalized least squares estimation method. As for
the errors, we use the denition of circular distance as
given Jammalamadaka and SenGupta (2001) such that,
(10)
where is the estimated value of v.
THE CRESCENT MOON VISIBILITY TESTS
In this section, we revisit the q-test proposed by Yallop
(1997) and propose new tests based on circular regression
model.
THE Q-TEST
In developing new crescent moon visibility tests using
circular regression model, we follow closely the q-test
of Yallop (1997). The test is based on the topocentric
crescent width, W, and geocentric ARCV. Yallop’s
algorithm computed crescent visibility based on the
residuals resulting from the tted polynomial regression
= 11.8371 - 6.3226W + 0.7319W 2 - 0.1018W 3
on 295 observations compiled by Schaefer (1996). The
residuals are then divided by 10 giving the q-statistic,
(11)
1
25.0)(
Q
df
and
3
25.0)(
Q
df
(3)
(
1
,
1
), … , (
,
)
 

 
 
 
()
(cos |)=
1
()
(sin |)=
2
()
tan

()
()

()> 0
()== arctan
()
()
= + tan

()
()

()0
 
()=
()= 0
tan

()
()

()> 0
()== arctan
()
()
= + tan

()
()

()0
 
()=
()= 0
tan

()
()

()> 0
()== arctan
()
()
= + tan

()
()

()0
 
()=
()= 0
tan

()
()

()> 0
()== arctan
()
()
= + tan

()
()

()0
 
()=
()= 0
tan

()
()

()> 0
()== arctan
()
()
= + tan

()
()

()0
 
()=
()= 0
(7)
1() (
=0 cos  +sin )
2() (
=0 cos  +sin )
cos = (
=0 cos  +sin ) + 1
sin = (
=0 cos  +sin ) + 1
=π − | − | − ̂||
),(
21
ε

̂=11.8371 6.3226+ 0.7319 20.10183
=[−
̂]
10
.
=π − | − | − ̂||
 


 





(4)
864
He further dened ve dierent categories depending on
the visibility of crescent moon using various instruments
and non-visible categories for q < 0.293 as described in
Table 2.
TABLE 2. The q-test types by Yallop (1997)
Types q-test value Justication
Aq > +0.216 Easily visible to the unaided eye (≥ 12ARCV)
B–0.014 < q < +0.216 Visible under certain atmospheric conditions
C–0.160 < q < –0.014 May need optical aid to nd the thin crescent
moon before it can be seen with the unaided eye
D–0.232 < q < –0.160 Can only be seen with binoculars or a telescope
E–0.293 < q < –0.232 Below the normal limit for detection with a
telescope
Fq < –0.293 Not visible below the Danjon limit
Homan (2003) investigated the validity of
the Yallop (1997) criteria using the results of 539
observations of the moon made over several years by
many experienced observers in good weather conditions.
The data were selected from 1047 reports. He proposed
a three-category of visibility type namely; the crescent
moon is visible if q is higher than 0.43 and not visible
if q is less than −0.06. These suggest that dierent data
set may give dierent ranges of the categories.
We also applied the same q-test on our 254 data,
and we found that the lowest value of the q-test with
a positive crescent sighting is 0.347. This value is
signicantly lower than the minimum limit of the q-test
by Yallop (1997). Though the q-test value is lower, the
elongation of the crescent moon is 11.33°, which is
higher than the Danjon limit. The inconsistent values of
the test make this q-test very subjective, and the value
cannot be used as a standard limit for crescent moon
visibility.
THE NEW CRESCENT MOON VISIBILITY TEST
The main interest of this work is to nd alternative
crescent visibility tests besides the q-test.The new test is
developed by generalizing the derivation of the q-test by
Yallop (1997) based on the circular regression model.
The new crescent moon visibility test utilizes two
circular variables, say U and V only. We t the variable
U on V using the JS circular regression model, as
described in Circular Regression Model section, giving
the tted values of . Hence, we dene the UV-statistic
as
(12)
As in the q-test, we use the resulting residuals from the
JS circular regression model on two circular variables to
categorize the visibility of the crescent moon. We then
attempt to classify the residuals into dierent groups and
relate the groups according to the visibility of the crescent
moon. We achieve that by taking the following steps:
First, nd the 99%, 95%, and 90% condence
intervals of the residuals, namely [L99, U99], [L95, U95]
and [L90, U90], respectively. Next, form 7 categories,
namely (−∞, L99], (L99, L95], (L95, L90], (L90, U90],
(U90, U95], (U95, U99], (U99, ). After that, tabulate the
frequency of crescent moon visibility/non-visibility for
each category. Lastly, reduce the number of categories
based on the tabulated frequencies.
In cases considered, we reduced the categories into
three groups associated with ‘Visible to the unaided
eye’, ‘May need optical aid’, and ‘Not visible’. These
categories will then be nalized, and the performance of
the test is investigated.
RESULTS AND ANALYSIS
CIRCULAR DESCRIPTIVE ANALYSIS
The distribution of the values of the circular variables
can be described by a rose diagram, as depicted in Figure
2. The data are mainly concentrated and close to zero.
These show the condition of an early phase of the moon
after the sunset. Table 3 provides the mean, minimum/
maximum, and the 95% condence interval (CI) for the
circular variables. As expected, the width of the crescent
moon during sighting is generally minimal, and the
altitude of the moon is mostly above the horizon though
ARCV is more substantial than the altitude of the moon
because ARCV considers the position of the sun below the
horizon.
 =π − | − | − ̂||
.
̂,
̂,
865
TABLE 3. Summary statistics for linear/circular variables
Variable Mean direction
(degree)
(min, max)
(degree)
95% CI
(degree)
Width 0.006 (0,0. 0.032) (0.005, 0.007)
Alt(S) -1.652 (-19.934, 4.019) (-2.094, -1.274)
Alt(M) 7.498 (-5.460, 27.835) (6.791, 8.204)
ARCV 9.159 (-5.165, 26.405) (8.440, 9.957)
Elon 10.779 (0.563, 28.244) (10.099, 11.527)
TABLE 4. Circular correlation between circular variables
Variable W Alt(S) Alt(M) ARCV Elon
W
Alt(S) -0.233
Alt(M) 0.827 0.212
ARCV 0.924 -0.334 0.850
Elon 0.962 -0.287 0.841 0.966
FIGURE 2. Rose diagrams of the circular variables
866
We also calculated the correlation values between
the variables, which are tabulated in Table 4. We found
that the correlation values of Alt(M)-W, ARCV-W, Elon-
Width, ARCV-Alt(M), Elon-Alt(M), and Elon-ARCV are
high. Nevertheless, not all the highly correlated variables
are suitable to be used as the parameters for crescent
visibility. Based on the description of variables as given
in Figure 1, ARCV-Alt(M) and Elon-W are expected to
be highly correlated as they are collinear to each other.
Thus, they are not a good combination of variables for
the visibility criteria. As for width, the variation of the
values is too small and might aect the relationship with
other variables (Homan 2003). Hence, in this paper,
we use the combination of Elon-Alt(M), and Elon-ARCV.
Figures 3 and 4 give the plots of Elon against Alt(M)
and ARCV, respectively. The altitude of the moon for
Y visibility is recorded at the time of the moon being
sighted, which occurs a few minutes after the sunset. In
this case, the Sun’s altitude is at several degrees below
the horizon. Whereas for N visibility, the altitude of the
moon is calculated at sunset. Therefore, the distribution
of data for Y is expected to be more scattered than N cases.
From both plots, we observed that the larger the values
of Elon, ARCV, and Alt(M), the higher the possibility of
sighting the crescent moons.
FIGURE 3. Elon versus the Alt(M) variables FIGURE 4. Elon versus the ARCV
THE NEW VISIBILITY TEST
For these new visibility test, we use the combination of
variables as discussed in Circular Descriptive Analysis
section. They are Elon-Alt(M), and Elon-ARCV. We
compare the performance of the two new visibility tests
by the misclassication percentage of the data.
EA-test
The new crescent moon visibility test, called EA-test,
utilizes two circular variables Elon, E, and Alt(M), A.
The best tted JS circular regression model with m=1
is given by
(13)
Using the approach adopted by Yallop (1997), we
dene the EA-test, which takes values of residuals of the
tted JS circular regression model. We then attempt to
categorize EA using the procedure described in EA-test
section as tabulated in Table 5. The second column gives
the intervals of the categories based on the EA-test; for
example, Category A consists of observations with EA
greater than 0.0086. The third and fourth columns give
the frequency of crescent moon non-visibility (N) and
visibility (Y). For Category A, the number of Y is greater
than N, while for Category C, more N compared to Y.
Hence, we label Category A as ‘Visible to the unaided
eye’ while Category C as ‘Not visible’. As for Group
B, the number of N and Y are fewer with smaller Y
compared to N. This might be due to many reasons
including the condition of the sky, and hence labelled
as ‘May need optical aid’. The percentage of correct
classication for the EA-test is 70.10%.
TABLE 5. Distribution of moon visibility based on three categories
Category EA-test value N Y Total % Interpretation
A[0.0086, ∞) 21 52 73 (29) Visible to the unaided eye
B [-0.00516, 0.0086) 26 9 35 (14) May need optical aid
C(-∞, -0.0052) 126 20 146 (57) Not visible
cos
̂= 0.1832 + 0.8104 cos  − 0.0372 sin
sin
̂= 1.0108 0.9178 cos + 0.6172 sin .
867
FIGURE 5. EA-test versus the Alt(M)FIGURE 6. EV-test versus the ARCV
DISCUSSION
The results in Results and Analysis section indicate
that EA-test provides the best indicator of visibility of
the crescent moon because of the higher percentage of
correct detection. Hence, we attempt to come up with
the new visibility criteria based on the EA-test. The lower
limit of the variables is then used as the criteria. In this
work, the criteria will be based on Category B of Table 5
as nowadays, telescopes or other optical aid systems are
used in the observations. As Elon is taken as the dependent
variable in the EA-test, we first estimate the criteria
value of Elon by its percentile values. The 5th, 10th, and
15th percentile mean 5%, 10%, and 15% of the ordered
observations will be smaller than the percentile values,
respectively. That corresponds to 1, 3, and 5 observations
and, hence, the choice of 15% percentile seems to be
adequate for this data. The ve observations are listed in
Table 7. Most of them have rather low values of Alt(M)
and ARCV, which makes it rather dicult to sight the
crescent moon after the sunset. Hence, the value of the
criteria for Elon is estimated at 7.28°.
As for Alt(M) and ARCV, we consider the plot of
Elon-Alt(M) and Elon-ARCV, as shown in Figures 7 and
8, respectively. We then estimate the corresponding values
of Alt(M) and ARCV given that Elon = 7.28°. Hence,
the values are Alt(M) and ARCV are 3.03° and 3.74°,
respectively. Consequently, by denition, the estimated
Alt(S) is taken as the difference between ARCV and
Alt(M), that is -0.71°. As for W, the observed values are
consistently small and we use the 15th percentile as its
estimate, which is 0.1°. We note that the sun’s altitude
Figure 5 gives the plot of EA versus Alt(M). It shows
that the residuals separate the Y/N values quite well.
Observations with low residuals and small Alt(M) are
largely categorized as non-visible, which is below
-0.00156. For EA above 0.0086, the moon can be observed
by unaided eyes. Otherwise, an optical aid may be need-
ed in the sightings.
EV-test
We repeat the process using the Elon and ARCV, V,
denoted as EV-test. The best tted JS regression model
with m=1 is given by
(14)
Using the same approach as for the EA-test, the nal
categories are given in Table 6. The EV-test does not
give a good result, with the percentage of correct clas-
sication is only 43.7%. This low performance is sup-
ported by the plot of EV versus ARCV, as given in Figure
6. The distribution of the residual values of Y and N data
are more scattered than that of the EA-test; thus, it fails
to separate the Y and N data very well.
cos
̂=0.0480 + 1.0452 cos  − 0.0021 sin
sin
̂= 1.5536 1.4884 cos + 0.5877 sin
TABLE 6. Distribution of moon visibility based on three categories
Category EV-test value N Y Total % Interpretation
A[0.0039, ∞) 59 21 80 (31) Visible to the unaided eye
B [-0.0022, 0.0039) 24 16 40 (16) May need optical aid
C(-∞, -0.0022) 90 44 135 (53) Not visible
868
of 0.71° below the horizon has considered the eect of
refraction near the horizon and semi-diameter of the sun.
During sunset, the centre of the sun is estimated at 0.35°
below the horizon, and hence the estimated time taken for
the sun to the altitude -0.71° is 1.4 min after it sets.
In determining the nal value for the crescent visibility
criteria, we use the elongation and altitude of the crescent
moon at sunset. We note that the duration of 1.4 min after
sunset is considered negligible to elongation as the average
rotation rate of the moon surrounding the earth takes
about 0.008°/min. Hence, the adjusted values of criteria
for Elon and Alt(M) are 7.34° and 3.33°, respectively, and
the corresponding values for ARCV = 3.74° and Alt(S) =
-0.35. The nal new values of the crescent moon visibility
criteria are as listed in Table 8.
TABLE 7. Observations with Elon less than the 15th percentile value
Date of
moon
sighting
Date of moon
sighting (Hijr)
Elon
(°)
Alt(M)
(°)
ARCV (°)Alt(S)
(°)
Width
(°)
Visibility
(Y/N)
27.07.2014 29 Ramadan
1435
7.042 2.577 2.912 -0.335 0.11 N
10.11.2007 29 Syawal
1428
6.898 1.963 2.314 -0.351 0.11 N
16.09.2012 29 Syawal
1433
6.286 0.945 1.361 -0.416 0.1 N
25.04.2009 29
Rabiulakhir
1430
6.276 1.286 1.495 -0.209 0.1 N
27.06.2014 29 Syaaban
1435
4.888 -0.319 0.114 -0.433 0.05 N
TABLE 8. The values of new criteria of variables for Category B of the EA-test at sunset
Variables Value of criteria (°)
Elon 7.28
Alt(M)3.33
ARCV 3.74
Alt(S) -0.35
Width 0.10
FIGURE 7. Elon vs Alt(M) for observations Category B of the
EA-test
FIGURE 8. Elon vs ARCV for observations in Category B of the
EA-test
869
CONCLUSION
We consider 254 observations collected consistently
every month at Baitul Hilal Teluk Kemang Malaysia
for the past 19 years. We derive two new visibility tests
based on elongation, the altitude of moon and ARCV
of the crescent moon. We divide the test values into
three categories, namely, crescent moon visible by the
naked eye, visible with optical aid, and not visible.
The new criteria are dened based on the observation
of the second category, visible with optical aid. We use
the 15th percentile value to be the value of criteria for
elongation and width. We then estimate the criteria values
of the altitude of the moon and ARCV by utilizing the
relationship between Elon-Alt(M) and Elon-ARCV. The
criteria are further adjusted so that the elongation and
altitude of the crescent moon are measured at sunset. We
use the 15th percentile of the elongation and width and
propose the elongation of 7.28 and the width of 7.1 as
the new criteria for new crescent moon visibility. Then,
we obtain another two criteria values, the altitude of the
moon of 3.38, and ARCV of 3.74 using the relationship
between Elon-Alt(M) and Elon-ARCV, measured at sunset.
This new criteria of crescent moon visibility will give
an alternative to the authorities in Malaysia to consider
the possibility of using them in developing the Islamic
calendar.
ACKNOWLEDGEMENTS
This research was supported by the Department of
Islamic Development Malaysia (JAKIM) with co-
operation of the Islamic Religious Department of
Negeri Sembilan (JAINS) and University Malaya
Research Grant IIRG002B-19FNW and IIRG002A-19FNW.
We thank our colleagues from Space Physics Laboratory,
Department of Physics, Faculty of Science, University of
Malaya, especially to Miss Nurhidayah Ismail, Madam
Saedah Haron, Mr. Joko Satria Ardianto, Mr. Wei Loon
and Mr. Muhammad Shamim as well as the students
and academic stas of Islamic Astronomy Program of
Academy of Islamic Studies, University of Malaya,
who have willingly helped us out with their abilities in
developing the project. Finally, many thanks go to the
management of Telok Kemang Observatory and Klana
Beach Resort for their kind co-operation and technical
support throughout the observations being carried out.
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Nazhatulshima Ahmad*
Space Physics Laboratory
Department of Physics
Faculty of Science
University of Malaya
50603 Kuala Lumpur, Federal Territory
Malaysia
Mohd Saiful Anwar Mohd Nawawi & Mohd Zambri Zainuddin
Islamic Astronomy Programme
Department of Fiqh and Usul
Academy of Islamic Studies
University of Malaya
50603 Kuala Lumpur, Federal Territory
Malaysia
Zuhaili Mohd Nasir, Rossita Mohamad Yunus & Ibrahim
Mohamed
Institute of Mathematical Sciences
University of Malaya
50603 Kuala Lumpur, Federal Territory
Malaysia
*Corresponding author; email: n_ahmad@um.edu.my
Received: 22 October 2019
Accepted: 13 January 2020
... They may arise in many areas of scientific experimentation such as biological, geological and environmental sciences. For example, the wind direction measured by two different equipments (see [1]) or the altitudes of the moon and the sun observed at the beginning of the lunar month (see [2]) form spherical data. The analysis of spherical data generally concentrates on the directional vector of the auditory object and, in most cases, ignores the distance effects. ...
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Book
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Book
Preface; 1. The purpose of the book; 2. Survey of contents; 3. How to use the book; 4. Notation, terminology and conventions; 5. Acknowledgements; Part I. Introduction: Part II. Descriptive Methods: 2.1. Introduction; 2.2. Data display; 2.3. Simple summary quantities; 2.4. Modifications for axial data; Part III. Models: 3.1. Introduction; 3.2. Notation; trigonometric moments; 3.3. Probability distributions on the circle; Part IV. Analysis of a Single Sample of Data: 4.1. Introduction; 4.2. Exploratory analysis; 4.3. Testing a sample of unit vectors for uniformity; 4.4. Nonparametric methods for unimodal data; 4.5. Statistical analysis of a random sample of unit vectors from a von Mises distribution; 4.6. Statistical analysis of a random sample of unit vectors from a multimodal distribution; 4.7. Other topics; Part V. Analysis of Two or More Samples, and of Other Experimental Layouts: 5.1. Introduction; 5.2. Exploratory analysis; 5.3. Nonparametric methods for analysing two or more samples of unimodal data; 5.4. Analysis of two or more samples from von Mises distributions; 5.5. Analysis of data from more complicated experimental designs; Part VI. Correlation and Regression: 6.1. Introduction; 6.2. Linear-circular association and circular-linear association; 6.3. Circular-circular association; 6.4. Regression models for a circular response variable; Part VII. Analysis of Data with Temporal or Spatial Structure: 7.1. Introduction; 7.2. Analysis of temporal data; 7.3. Spatial analysis; Part VIII. Some Modern Statistical Techniques for Testing and Estimation: 8.1. Introduction; 8.2. Bootstrap methods for confidence intervals and hypothesis tests: general description; 8.3. Bootstrap methods for circular data: confidence regions for the mean direction; 8.4. Bootstrap methods for circular data: hypothesis tests for mean directions; 8.5. Randomisation, or permutation, tests; Appendix A. Tables; Appendix B. Data sets; References; Index.