t'
rssN 10228594
Faculty of Mathematical and Physical Sciences
Jahangirnagar UniversitY
Journal of Science
a multidisciplinary journal of sciences
Volume 39, Number 2, December 2016
ISSN 10228594
ruJS @ 2016 Jahangirnagar University Journal of Science
Vol. 39, No. 2, pP.ll22
A Study on Monthly Maximum Wind Speed Probability
Distributions at Hazrat Shahajalal and MAG Osmani
International Airport of Ban gladesh
Md. Moyazem Hossain*
Department of Statistics, Jahangirnagar university, Savar, Dhaka1342,
Bangladesh
Faruq Abdulla
Department of Statistics, Islamic University, Kushtia7003, Bangladesh
Ajit Kumar Majumder
Department of Statistics, Jahangimagar University, Savar, Dhaka1342,
Bangladesh
Abstract
wind speed is a fundamental atmospheric rate. wind speed is caused by air moving
from high pressure to low pressure, usually due to changes in temperature. Probability
densityfurrctions (PDFsf have been used in literature to describe wind speed
characteristics which include weibull, Rayleigh, bimodal weibull, Lognormal,
Gamma ard so on This paper considers a data set on maximum sustained wind speed
(Kn/h) at Hazrat Shahjalal International Arport and Osmani Intemational Airporl
Bangladesh over the period January, 1982 to June, 2015. This paper attempts to
deteimine the best frt wina speed distibution with statistical properties of the monthly
maximum sustained wind spied (Km/tr) of the Hazrat Shahjalal International Airport
and osmani International Airport. The higher value of lt'? and the lower values of KS
error, RMSE and Chisquare error indicate that GEV distribution is more accrnate than
other PDFs in modeling wind speeds of both locations.
Keywords: wind speed, Probability density functions, likelihood method, Model
selectioq Bangladesh
Introduction
Wind speed is caused by air moving from high pressule to low pfessure, usually
due to changes in temperature (h@s://en.wikipedia.org/wiki/TVindspeed).
Wind speed affects several activities like aviation; it is also the most important
parameter when examining a place's potential in generating wind power
(Zaharfu, et a1.,2009 t1]). Also, Wind speed affects weather forecasting,
aircraft and maritime operations, construction projects, growth and metabolism
rate of many plant species, and countless other implications (Michael Hogrru
20t0l2l).
Hanat Shahjalal International Airport is the largest airport in Bangladesh.
Operated and maintained by the Civil Aviation Authority, Bangladesh, it is
*Author for correspondence: email: mmhmmjustat@.emai1.com
L2 Md.Moyazznm Hossain*, Faruq Abdulla and Ajit Kumar Majumder
also used by the Bangladesh Air Force. The airport has an area of 1,98 1 acres
(802 ha). The airport is located in Kurmitola and was odginally 11 NM
(20 km; 13 mi) north of the capital Dhaka (https://en.wikipedia.orglwikil ;.
Shatrjalal_International_Alrport). On the other hand, MAG Osmani
International Airport is an international airport located 5 miles northeast of . .rr
Sylhet in Bangladesh. The vast majority of passengers using the airport are
expatriate Bangladeshis and their descendants from the Sylhet Division
living in the United Kingdom
(https ://en. wikipedia. org/wiki/O smani_International_Arport).
Probability density functions (PDFs) have been used in literature to describe
wind speed characteristics which include Weibull, Rayleigh, bimodal
Weibull, Lognormal, Gamma and so on (Ravindra Kollu, et a1.,2012 [3]).
Celik (2003) [4] made statistical analysis and summarized that Weibull
model was better than Rayleigh model. Akdag and Bagiorgas, (2010) [5]
discussed about the two component Weibull distribution and stated that
Weibull  Weibull gaye a goodfit. Tian Pau (2011) [6] used Rayleigh,
Weibull and Gamma distribution and its generalized form. Gupta and Kundu
(2010) [7] discussed about the generalized logistic distribution. Yiknaz, et
al., (2008) [8] argued that in most studies fitting of data set to Weibull
distribution was not examined even though this assumption was made.
Different probability distributions should be investigated and incorporated to
the analyses. Thus, this paper attempt to determine the best fit wind speed
distribution with statistical properties of the monthly maximum sustained
wind speed (Km/h) of Hazrat Shahjalal International Airport and Osmani
International A irport.
Methods and Materials
Data Source 
This paper considers a data set on maximum sustained wind speed (Kfi/h)
over the period January, 1982 tD June, 2015 atHanat Shahjalal International
Airport and Osmani International Airport, Bangladesh from the web address
hW : I I en.tlrtiempo. net/.
Probability Distributions
The primary tools to describe wind speed characteristics are probability
density functions. Many PDFs have been proposed in recent past, but in
present study Weibull, Lognormal, Gamma, and GEV are used to describe
A Study on Monthly Maximum Wind Speed Probability Distributions 13
wind speed characteristics. Parameters defining each distribution function are
calculated using maximum likelihood method.
Weibull (W) Distrihution
Weibul distribution is named after Waloddi Weibull who described it in
detail in 1951, although it was first identified by Fr6chet (1927) [9] and first
applied by Rosin and Rammler (1933) [10] to describe a particle size
distribution. The Weibull function is commonly used for fitting measured
wind speed probability distribution. The probability density function (PDF)
and the cumulative distribution function (CDF) of the Weibull distribution
with two parameters are given by (Weibull, 1951 [1 1]):
Probability density function (PDF):
f (v;k,c)=U"(:)r' "r[[r'] ; o<v 1@'k)o'c>o
Cumulative distribution function (CDF):
[ / \rl
F(v;k,c)=rexpl [ 1  l, o. v <a,k>o,c>0
[ \c/ .]
where, 'k' and 'c' are the shape and scale parameters respectively. The
shape and scale parameters are calculated using the maximum likelihood
method (Kececioglu,2A02 [12]) and an iterative technique such as Newton
Raphson technique and which are given by:
is the wind speed in time step i and n is the number of data
Lognormal (LN) Disrtbufion
Lognormal distribution is a probability distribution of a random variable
whose logarithm is normally distributed. The probability density function
(PDF) and the cumulative distribution function (CDF) of the Lognormal
distribution are given by (Johnson, et al.,1994ll3l):
Probability density function (PDF):
[i,rl'.
and i=l '' I
ln I
L]
where, 4
points.
14 Md.Moyazzrm Hossain', Faruq Abdulla and Ajit Kumar Majumder
f(v;p,a)=#*rl(to (u) p)'
25' ;0<y,<@,FeR,6>0
Cumulative distribution function (CDF):
where, 'p' and 'd' are the mean and standard deviation of the normal
random variable ln(v) respectively and '(D' is the standard cumulative
distribution function. The mean and standard deviation are calculated using
the maximum likelihood method and which do not need an iterative
procedure are given by (Carta et a1.,2009 ll4l):
F (v; 1t,d)= t(ry)' o 1v,1@, tten,d >o
Im(q)
ft=i=r and 6=
,n
where, 4 is the wind speed in time step i and n is the number of data
points.
Gamma (G) Distribution
Lancaster (1966) [15] quotes from Laplace (1836) U6l in which latter obtain
a Gamma distribution. The probability density function (PDF) and the
cumulative distribution function (CDF) of the gamma distribution are given
by:
Probability density function (PDF):
f (v;a,n) = ft1"*n(;)' v > o,a> o,b > o
Cumulative distribution function (CDF) :
;v>0,a>0,0>0
I
1
?
A Study on Monthly Maximum Wind Speed Probability Distributions 15
where, ,(r,;) is the lower incomplete gamma function. The parameters
^ ta' arrd'b' are the shape and scaleparameters respectively. The shape and
scale parameters are calculated using the maximum likelihood method and
. which are given by:
A * ]l:; log r > Iogx *a t, =*
logx logx a
Generalized Extreme Value (GEV) distribution
GEV distribution is a flexible model that combines the Gumbel, Frechet and
Weibull maximum extreme vilue distributions (Ying and Pandey, 2007
[17]). For, GEV
Probability density function (PDF):
r (v; p, o, 6) = Il,
* e(" ')l( zF *, { [' . t ( 7))r1,, .,
/\
asain for v > (r linthe case C >o,andfor , .(r?)tthe case
€ <0.
Cumulative distribution function (CDF):
F(v;p,o,g):*r{[,* ,(t:\1%i, r *o
'r."ar '1. L '( o )) ) '
for r*5(54)r o, *h"t ,u eR is the location parameter, o > 0the scale
parameter and (e R the shape parameter.
Generalized Gamma (3P G) Distribufion
The generalized gamnn also known as three parameters gafirma distribution
is a continuous probability dishibution with three parameters. For non
negative ,v, the probability density function of the generalized gamma is
(Stacy, 1962 [18]):
16 Md. Moyazzem Hossain., Faruq Abdulla and Ajit Kumar Majumder
Probability density function (PDF) :
(r/,\u'',(%)'
f (v;a,d,p)=" o 't ,;\ ;a>0,d >0,p>0
'l/o)
where, f (.)denotes the gamma function.
cumulative dishibution tunction (cDF): ct... ^ s r'(%'('o,where
F \v;a,d, p) = __W, w,srs
f (.) denotes the lower incomplete gamma function.
Three Parameters Lognormal (3P LI{) Distribution
The lognormal distribution derives its name from the relationship that exists
between random variables T and y =ln(V  a).lt I is distributed nornally
(b,"), then V is lognormal (a,b,c). Accordingly, the probability density
function of V maybe written as (Cohen and Whitten, 1980 [19]):
Probability density function (PDF) :
f (v; a,b, c) = c;ffi, [*#t*f, ;, r,,< v < oo .
Cumulative distribution function (CDF):
F (v; a, b, c) =! O ;ffi.., [t*?']r,
Goodnessoffit Tests "
Goodnessoffit tests are used to check the accuracy of the predicted data
using theoretical probability function. To evaluate the goodnessoffit of the i
PDFs to the wind speed data, the KS, the R2, the 262 and the RMSE were
used.
KolmogorovSmirnov (KS) Ewor Test
The KS test computes the largest difference between the cumulative
distribution function of the model and the empirical distribution function.
The KS test statistic is defined as:
€
A Study on Monthly Maximum Wind Speed Probability Distributions
o=*e*lr _el
where, .i is the predicted cumulative probability of the i'h observation
obtained with the theoretical cdf and {. is the empirical probability of the
i" observation are obtained with the Cunnane (1978) [20] formula:
 i0.4
'''  n+02
where, i =1,... ,n is the rank for ascending ordered observations.
R2 Test
The ,R2 test is used widely for goodnessoffit comparisons and hypothesis
testing because it quantifies the correlation between the observed cumulative
probabilities and the predicted cumulative probabilities of a wind speed
distribution. A larger value of R2 indicates a better fit of the model
cumulative probabilities .F to the observed cumulative probabilities F . The
R2 is defined as:
t(F,F)'
R2= i(r, p)' *i(o,  n)'
ChiSquare Error Test
ChiSquare test is used to assess whether the observed probability differs
from the predicted probability. ChiSquare test statistic is defined as
t ^ t2
, +(44)
,L /r :
i=lfi
Root Mean Squared Enor (RMSE) Test
Root mean square error (RMSE) provides a termbyterm comparison of the
actual deviation between observed probabilities and predicted probabilities.
A lower value of RMSE indicates a better distribution function model. Root
mean squue error (RMSE) is defined as
l7
rl
n
ZF,
.fr_ i=tn
18 Md.Moyazznm Hossain', Faruq Abdulla and {it Kumar Majumder
Results and Discussion
The mean and standard deviation of observed maximum sustained wind
speed for Station 4lg23} (Dhaka) are 25.96484 Wnlh and.22.25632 lrintin,
respectively with minimum 3.5 kn/h and maximum 94.3 Wfl/h. Whereas, for
the station 418910 (Sylhet), the mean and standard deviation of observed
maximum sustained wind speed 'are 27.40224 l.l.nlh and 21.55088 lcnlh,
respectively. The minimum and maximum values of observed maximum
sustained wind speed are 5.4 km/h and 103.5 km/h respectively for the
station 418910 (sylhet). That is the average maximum sustained wind speed
in Sylhet is higher than Dhaka station.
The estimation of parameters of all the PDFs considered in this study were
carried out using maximum likelihood method and computed parameter
values of different PDFs used for all the two stations are presented in Table
1.
Table I Computed parameter values of different PDFs considered in this
study.
PDF' Parameters Estimates
HSIA. Dhaka OIA. Svlhet
wshape (r) 1.29640s t.453077
Scale (c) 28.281896 30.028883
LN vtean (r) 2.947s227 3.0663842
Standard Deviation
(6) 0.7673038 0.6313268
GShape (a) t.780474 2.3742s9
Scale (b) 14.s41727 tt.322937
GEV snape (6) 0.5166852 04991436
Scale (o) 9.2498732 8.0903s76
f , ^av2
I r(q i)' I
RMSE =l ir I
lnl
L]
A Study on Monthly Maximum Wind Speed Probability Distributions
Location (;z) t3.8419043 16.061 1 1 18
3.P G
Shape (d) 1.160726 r.426t74
Scale (a) 19.331254 15.092712
rhreshold (r) 3.452904 5.358748
3P LN
Shape (c) 0.9073547 0.83270t4
Scale (a) 2.7629273 2.7341814
Threshold (b) 2.3200703 4.82s3r93
* HSIA : }J.azrat Shahjalal International Airport and OIA : Osmani
International Airport.
The statistical parameters for fitness evaluation of PDFs currently analyzed
are presented in Table 2. Considering KS error, 7' error and RMSE, the
distribution functions Weibull, Lognormal, Gamma, three parameters
Gamma and three parameters Lognormal have large effors indicating their
inadequacy in modeling wind speeds of the both stations considered in this
study. The higher value of R2 andthe lower values of KS error, RMSE and
Chisquare error indicate that GEV distribution is more accurate than other
PDFs in modeling wind speeds of both locations.
Table 2 Values of Statistical tests for different distribution functions of
OSIA, Dhaka and OIA, Sylhet.
19
PDF
Values of Different Statistical Tests
IISIA, Dhaka OIA. Svlhet
KS
Error R2 x'
Error RMSE
Error K ,S
Error R2 T,
Error RMSE
Error
w0.13094 0.94173 s.27091 0.06800 0.143m 0.89413 9.29634 0.08882
LN 0.08927 0.98469 1.13112 0.03657 0. I 0855 0.95927 3.86296 0.0s932
G0.13425 0.94975 4.43101 0.06593 0.13922 0.91579 7.98422 0.08429
GEV 0.04868 0.99470 0.35711 0.u145 0.0576s 0.99313 0.s9696 0.02423
3P
G0.10084 0.97292 2.17107 0.04720 0.1 r468 0.94517 4.82677 0.06735
3P
LN 0.06356 0.99239 0.50s61 0.02540 0.0724 0.9826s 1.533 18 0.03814
20 Md. Moyazzem Hossain*, Faruq Abdulla and Ajit Kumar Majumder
The graphical comparisons of different distribution functions considered in
this study and the histogram of the observed maximum sustained wind speed
for HSIA, Dhaka and OIA, Sylhet are presented in Figure 1. As seen from
Figure 1 and statistical parameters from Table 2, Generalaed Extreme value
distribution (GED provided the best fit for the observed wind data for both
locations.
o
q
o
q
Zc
e
o
q
o
e
3
o
o
e
q
'o c
o
Crapft lcd Comparlsqn ol glfrerent DlBtibdlon3
for tfie Dhaka sldion Graphical Comparison of Ditrerent Diaflbutohs
for rle Sylhet Station
N4aximum Vvind Speed (Km/h)
(b)
distribution functions of (a) HSIA,
40 60
Maimum Wnd Speed (Km/ir)
(a)
Figure 1: Graphical Comparison for different
Dhaka and (b) OIA, Sylhet.
Conclusion
In recent years it has been investigated that the fitting of specific distribution
to wind speed is required for use in practical application as air pollution
modeling, estimation of wind loads on building and wind power analysis. So
a model is required for wind speed distribution. Extensive literature search
indicates that various parametric distribution models have been presented to
estimate wind speed distributions. wind speed data at HSIA, Dhaka and
OIA, Sylhet were used in evaluating different pdfs to access their suitability.
The Weibull, Lognormal, Gamma, three parameters Gamma and three
parameters lognormal distribution functions have large errors indicating their
inadequacy in modeling wind speeds of the both stations considered in this
study. on the other hand, the higher value of R2 and the lower values of KS
error, RMSE and chi square error indicate that GEV distribution is more
A Study on Monthly Maximum Wind Speed probability Distributions
accurate than other PDFs in modeling wind speeds of both Dhaka an Sylhet
locations.
Reference
tll Zahelrinl A., Najid, S. K., Razali, A. M. and Sopian, K., Analysing Malaysian wind
speed data using statistical distribution, Proceedings of the 4th IASME/WES
International Conference on Energt & Environment (EE'09), Universiti Kebngsaan
Malaysia, 2009, 36337 0
121 Michael Hogan, c., Abiotic factor, Encyclopedia of Earth, eds Emily Monosson and
c. cleveland, National council for Science and the Environment, washingon DC,
2010.
t3l Ravindra Kollu, Srinivasa Rao Rayapudi, SVL Narasimham and Krishna Mohan
Paklarrthi, Mixture probability distribution functions to model wind speed
distributions, International Journal of Energ,t and Environmental Engineering, i}l2,
3, 27 . doi:10.1186/225 l6832327 .
l4l Celik, A. N., A statistical analysis of wind power density based on the Weibull and
Rayleigh models at the southern region of Turkey, Renou Energ1,,,2003,29,593604.
t5l Akdag, S. A. and Bagiorgas , H. s., Use of two component weibull mixtures in the
analysis of wind speed in the Eastern Mediterranean , Applied Energt, 2010,97 ,2566
2573.
t6l Tian Pau, C., Estimation of wind energy potential using different probability density
tunctions, Applied Energt" 2011, 88(5), 18481856.
Ul Gupta, R. D. and Kundu, D., Generalized logistic distribution, Journal of Applied
Statistical Scimces, 2010, 18, 5 166.
t8l Yilmaz, v. and celik, H. E., A statistical Approach to Estimate The wind Speed
Distribution: The case of Gelibolu Region, Do{us Aniversitesi Dergisi,200g,9(l),
122132.
t9] Fr6chet, Maurice, Sur la loi de probabilit6 de l'6cart maximunl Annales de la Societi
Po I onais e de Mathemat ique, Cr aootie, I 927, 6, 93_l I 6.
tlO] Rosin, P. and Rammler, E., The Laws Goveming the Fineness of powdered coal,
Journal of the Institute of Fue,1933,7,2936.
tl1l weibull, w., A statistical distribution function of wide applicability, J. Appl. Mech.
Trans, ASME, 1951, 18(3), 293197.
1121 Kececioglu, D., Reliability mgineering handbook, vol.l and 2, Destech
Pubblications, Pennsylvania, 2002.
[3] Johnson, Norman L.o Kotz, Samuel, and Balakrishnan, N., 14: Lognormal
Distributions, Continuous univariate distributions. Vol. l,Wiley Series in Probability
and Mathematical Statistics: Applied Probability and statistics (2nd ed.), New york:
John Wiley & Sons, 1994.
2t
t
:
i,'11',
22 Md. Moyazzem Hossain', F'aruq Abdulla and Ajit lfumar Mq1'r,*rder
t14l 9y+ 1.A., Ramlrez.pl and, yari,,qu*,,, s., A review of wind speed profubility
distributions used inwind enerry analysis case studies i" tn" "*url, islands, Aenew
Sustain Energt Rev,2009, l3(5) 933_i55.
tl5l LaplacgP.s.,ThaieAnalyiquedesprobabilities,lg36. ;: r
tl6] Lancaster, H. O., Forerunners of the Pearson Chisquare, Australian Jownal of
Statistics, 1966, 8, 117126.  14' "we" u
llTl Ying A. and Pandey, M. D., The r largest order statistics model for extreme wind
' speed estimation, J Mnd Eng lyd Aerodlm, 2007, gle), I 6!l gt.
tlsl Stacy, E. W., A Generalization of the Gamma Distribution, Annals of Mathematical
Stafistics, 1962, 33(3), llg7 l lg2.
t19l cohen, A. c. and lftjT"l B J., Estimation in the Threeparameter Lognormal
Distribution, Jownar of.the American statisticar Association, rgg0;ri, 3gg 404.
t20l curnang c., U:rbiased protting positions  a review, I Hydror, rg7g,37(34):20r222.
',:,i,:
'.9,:il:1r
l.i:r:arlj"is
. ..r":. X!ri:
lrlrr €r a
1 . ';..,. .
i#.
..:':nil
5..1
''*;
Y
.rj
a,l
:t4
.'.:,i
L'.i
1l
rI
.I
,l
"c
.l
t'
Jahangirnagar University Journal of
Science  a multidisciplinary joumal of
sciences is a publication ofthe Faculty of
Mathematical and Physical Sciences,
Jahangimagar University. It is published
twice a yeaq in June and in December, by
the Faculty of Mathematical and Physical
Sciences.
2016 Jahangimagar University Journal of
Science. Al1 rights reserved. However,
permission is granted to quote from any
afiicle of the journal, to photocopy any
part or full of a arlicle for education
and/or research purpose to individuals,
institutions, and libraries with an
appropriate citation in the reference
and/or customary acknowledgement of
the joumal.
The subscription lor each volume is
For institutions
For individuals
For Bangladesh
: US $25 (postage included)
: US $20 (postage included)
(lnstitution & Individual) : Tk. 250 (postage included)
Al1 conespondence with regard to the joumal
should be addressed to The Editor,
Jahangimagar University Journal of Science,
Faculty of Mathematics and Physical sciences,
Jahangirnagar University, Savar, Dhaka1342,
Bangaldesh
(email : mazibju@yahoo.com)
Jahangirnagar University Journal of Science
Volume 39. Number 2. Jarutu12077
SineCosine Method lor the Soliton Solutions to the KdV and 1
rnKdV Equations M Abdur Rab' Md. Monirul Islatn
A Study on Monthly Maximum Wind Speed Probability 11
Distributions at Hazrat Shahajalal and MAG Osmani
Intemational Airport of Bangladesh
Mrl. Moyazzem Hossain, Faruq Abdulla, Ajit Kumar Majumder
Spectroscopic and Computational Investigation of the 23
Molecular Interaction of Iodine with Benzaldehyde &
Substituted Benzaldehydes
Sh ar iftn' R a h.m an, Mo h am m a d Kh ait'u l I s l a m, Mo h amme d
Delwar Hossain, Shahed Rana, Md. Anamul Hoque
A Comparative Study on Risks of the Estimators for Shape 39
Parameter of a Power Function Distribution
Chandan Kumer Podder
Evaluation of Some Geomechanical Parameters of the Soil 49
Samples of Muktepur area, Munshiganj, Bangladesh
Md. Emdaful Haque, Md. Hasan Imam and Mahmuda khatun
PhysicoChemical study of the Interaction of Flucloxacillin 65
Sodium with Cetyltrimethylammonium Bromide in Aqueous
Medium and in Aqueous Solution of Salts
M. A. Islam, M. A. Hoque, M. Kabir
Ion Association and Solvation Behavior of Some Syrnmetrical 77
Electrolytes in AcetonitrileWater Mixed Solvents. A
Conductance Study Md. Rashidul lslam and Md Minantl lslam
Synthesis of Some 2Adamantane Derivatives as Cytotoxic 91
Agents Md. Razzak and Md. Rabittl Islam
Atlsorption Kinetics for the Removal of Remazol Brilliant 103
Violet in Aqueous Solution using Puffed Rice
Subarna Karmaker and Md. Amirul Islam
Population Projection of Bangladesh: Using a fbur parameter 115
logistic growth model
Mohfulur Rahman, Sohel Rana and Mijanttr Rahman
e+
#
+i
z
{il
z
(i_l
z

an
(u
Fl
Li

z
ri
o
\
a
o
rn
+
o
lrl
o
z
5
d
o
N
l
o
o
o
o
o
E
IJ
C)
(:\
t'
ffimB
smlr
'h* F63
.IAHANGIRNAGAR UNIVERSITY