2018 Scienceline Publication  
Journal of Civil Engineering and Urbanism  
Volume 8, Issue 2: 12-16; Mar 25, 2018  
ISSN-2252-0430  
Extreme Value Analysis of Wind Speed Data using Maximum  
Likelihood Method of Six Probability Distributions  
Vivekanandan N   
Central Water and Power Research Station, Pune, Maharashtra, India  
Corresponding author’s E-mail: anandaan@rediffmail.com  
ABSTRACT: Assessment of wind speed at a region is a pre-requisite while designing tall structures viz. cooling  
towers, stacks, transmission line towers, etc. This can be achieved through Extreme Value Analysis (EVA) by  
fitting of probability distributions to the annual series of extreme wind speed (EWS) data that is derived from hourly  
maximum wind speed. This paper details the study on EVA of wind speed data recorded at India Meteorological  
Department Observatories of Delhi and Kanyakumari adopting six probability distributions such as Normal, Log  
Normal, Gamma, Pearson Type-3, Log Pearson Type-3 (LP3) and Extreme Value Type-1. Maximum likelihood  
method is applied for determination of parameters of the distributions. The adequacy of fitting of probability  
distributions to the series of recorded EWS data is evaluated by Goodness-of-Fit tests viz., Anderson-Darling and  
Kolmogorov-Smirnov and diagnostic test using D-index. Based on GoF and diagnostic tests results, the study  
suggests the LP3 distribution is better suited amongst six probability distributions adopted for EVA of wind speed  
data for Delhi ad Kanyakumari.  
Keywords: Anderson-Darling test, D-index, Kolmogorov-Smirnov test, Log Pearson Type-3, Maximum likelihood  
method, Wind speed  
INTRODUCTION  
Technical and engineering appraisal of large  
estimation of extreme events such as rainfall, stream flow  
and wind speed (Bivona et al., 2003; Della-Marta et al.,  
2009). Number of studies has been carried out by different  
researchers on adoption of different probability  
distributions for Extreme Value Analysis (EVA) of wind  
speed. Palutikof et al. (1999) expressed that the  
Generalized Extreme Value (GEV) distribution is better  
suited for EVA of wind speed for Sumburgh (Shetland).  
Pandey et al. (2001) applied GEV and GAM distributions  
for estimation of EWS for Helena, Boise and Duluth  
stations in United States of America. Topaloglu (2002)  
reported that the frequency analysis of the largest, or the  
smallest, of a sequence of hydrologic events has long been  
an essential part of the design of hydraulic structures.  
Guevara (2003) carried out hydrologic analysis using  
probabilistic approach to estimate the design parameters  
of storms in Venezuela.  
infrastructure projects such as nuclear, hydro and thermal  
power plants, dams, bridges and flood control measures  
needs to be carried out during the planning and  
formulation stages of such projects. In a hydrological  
context, it is well recognized that whatsoever extreme the  
design-loading, more severe conditions are likely to be  
encountered in nature. Therefore, the accurate estimation  
of the occurrence of extreme wind speed (EWS) is an  
important factor in achieving the correct balance. Such  
estimates are commonly expressed in terms of the quantile  
value (  
), i.e., the EWS which is exceeded, on average,  
xT  
once every T-year, the return period. For this situation, the  
annual series of EWS data derived from hourly maximum  
wind speed is generally fitted to a theoretical distribution  
in order to calculate the quantiles.  
Probability distributions (PDs) such as Normal  
(NOR), 2-parameter Log Normal (LN2), Gamma (GAM),  
Pearson Type-3 (PR3), Log-Pearson Type-3 (LP3) and  
Extreme Value Type-1 (EV1) are commonly used for  
Lee (2005) studied the rainfall distribution  
characteristics of Chia-Nan plain area using six PDs.  
Kunz et al. (2010) compared the GAM and Generalized  
Pareto (GP) distributions for estimation of EWS and  
concluded that the GP provides better estimates than  
To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions. J. Civil Eng.  
Urban., 8 (2): 12-16. www.ojceu.ir  
12  
GAM distribution. El-Shanshoury and Ramadan (2012)  
applied EV1 distribution to estimate the EWS for Dabaa  
area in the north-western coast of Egypt. Escalante-  
Sandoval (2013) applied five mixed extreme value  
distributions to estimate the EWS at 45 locations of the  
Netherlands. He also expressed that the mixed reverse  
Weibull and the mixture Gumbel-reverse Weibull  
distributions are better suited for estimation of EWS at 34  
locations. Ahmed (2013) expressed that the rank  
regression method is the best suited amongst four methods  
studied for determination of parameters of Weibull  
distribution for estimation of EWS for Halabja region.  
Indhumathy et al. (2014) applied four parameter  
estimation methods of Weibull distribution and found that  
the energy pattern factor method is the best method to  
estimate the EWS for Kanyakumari region. Generally,  
when different distributional models are used for  
modelling EWS, a common problem that arises is how to  
determine which model fits best for a given set of data.  
This can be answered by formal statistical procedures  
involving Goodness-of-Fit (GoF) and diagnostic tests; and  
the results are quantifiable and reliable than those from  
the empirical procedures.  
Qualitative assessment was made from the plot of  
the recorded and estimated EWS. For the quantitative  
assessment on EWS within in the recorded range, GoF  
tests viz., Anderson-Darling (AD) and Kolmogorov-  
Smirnov (KS) are applied. A diagnostic test of D-index is  
used for the selection of most suitable probability  
distribution for EVA of wind speed. In this paper, study  
on EVA of wind speed data adopting six PDs is presented.  
The applicability of GoF and diagnostic tests procedures  
in identifying which distribution is best for EVA of wind  
speed is also presented with illustrative example.  
Here, Zi=F(xi), for i=1,2,3,…,N with x1<x2< ….xN ,  
F(xi) is the Cumulative Distribution Function (CDF) of ith  
sample (  
x
) and N is the sample size (Zhang, 2002). The  
i
critical value (ADC) of AD test statistic for different  
sample size (N) at 5% significance level is computed  
from:  
(2)  
ADC 0.757 (1(0.2/ N)  
Similarly, the critical value (KSC) of KS test statistic  
for different sample size (N) at 5% significance level is  
computed from:  
N
(3)  
KSC Max ( F (x i ) FD (x i))  
e
i1  
Here, Fe(xi)=(i-0.44)/(N+0.12) is the empirical CDF  
of xi and Fe(xi) is the computed CDF of xi.  
Test criteria. If the computed value of GoF tests  
statistics given by the distribution is less than that of  
critical values at the desired significance level, then the  
distribution is considered to be acceptable for EVA of  
wind speed.  
Table 1. Quantile estimator of six PDs  
Distribution  
PDF  
xT  
2
1
xm  
xT mKT  
NOR  
2
f(x;,m)   
1 2  
e
,x,  0  
2
1
ln(x)m  
xT emK  
T
LN2  
2
f(x;,m) 1 x 2  
e
, x,  0  
ex (x)1  
()  
1
XT  
K
    
f(x;,)   
,x,  0  
GAM  
T
f(x;,,m)   
e(xm)  
x m  
  
1,x,  0  
  KP   
XT m   
PR3  
()  
m  
e
1  
)/)  
P
f(x;,,m)   
  
lnx m  
  
,x,  0  
LP3  
XT em((K  
1  
()  
x
(xm)/  
e(xm)/ee  
, x, >0  
xT mYT  
EV1  
f(x : ,m)   
MATERIALS AND METHODS  
In Table 1, α, and m are the scale, shape and  
location parameters respectively. For NOR, the values of  
m and α are computed from mean and standard deviation  
of the series of EWS. Similarly, for LN2, the values of m  
and α are computed from the mean and standard deviation  
of the log-transformed series of EWS. For EV1  
distribution, the reduced variate (YT) corresponding to  
The effort made in this study is to assess the  
applicability of PDs adopted in EVA of wind speed. For  
this, it is required to carry out various steps, which  
include: (i) Select six PDs such us NOR, LN2, GAM,  
PR3, LP3 and EV1 for EVA; (ii) select maximum  
likelihood method (MLM) for estimation of parameters of  
the distributions; (iii) select GoF and diagnostic tests and  
(iv) conduct EVA and analyse the results obtained thereof.  
return period (T) is defined by YT=-ln(-ln(1-(1/T))).  
K
T is  
the frequency factor corresponding to return period and  
Coefficient of Skewness (CS) [CS= 2/ for GAM,  
Table 1 gives the quantile estimator (  
are used in EVA of wind speed.  
) of six PDs that  
xT  
CS=0.0 for NOR and LN2].  
KP is the frequency factor  
corresponding to CS of the original and log-transformed  
series of EWS for PR3 and LP3 distributions respectively  
(Rao and Hameed, 2000). The parameters of PDs are  
computed by MLM and used in estimation of wind speed.  
The theoretical descriptions of MLM of GAM, PR3, LP3  
and EV1 are briefly described in the text book titled  
‘Flood Frequency Analysis’ published by Rao and  
Hameed (2000).  
Goodness-of-Fit tests  
GoF tests viz., Anderson-Darling (AD) and  
Kolmogorov-Smirnov (KS) are applied for checking the  
adequacy of fitting of PDs to the recorded EWS data. The  
AD test statistic is defined by:  
   
1 NN  
(2i 1) ln(Z )   
i
(1)  
AD   
N  
2N 12i ln(1Zi )  
i1  
To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions. J. Civil Eng.  
Urban., 8 (2): 12-16. www.ojceu.ir  
13  
Table 4. Estimates of EWS given by six PDs for Kanyakumari  
Diagnostic test  
Return  
period  
(year)  
2
5
10  
20  
50  
100  
200  
500  
1000  
Estimated EWS (km/hr) using  
The selection of most suitable probability  
distribution for EVA of wind speed is performed through  
D-index, which is defined by:  
NOR  
LN2 GAM  
PR3  
LP3  
EV1  
42.3  
51.5  
56.3  
60.3  
64.7  
67.7  
70.5  
73.8  
76.1  
41.1  
49.6  
54.6  
59.2  
64.8  
68.9  
72.8  
77.8  
81.5  
41.4  
50.5  
55.7  
60.2  
65.6  
69.4  
73.0  
77.5  
80.7  
40.1  
49.4  
55.7  
61.6  
69.2  
74.7  
80.1  
87.2  
92.4  
39.3  
48.3  
55.3  
62.8  
73.8  
83.1  
93.3  
108.4  
121.3  
40.4  
48.1  
53.3  
58.2  
64.6  
69.3  
74.1  
80.4  
85.1  
6
x x*  
(4)  
D index   
Here,  
1 x  
i
i
i1  
x
is the mean value of the recorded EWS.  
Also, xi is the ith sample of the first six highest values in  
the series of recorded EWS and x*i is the corresponding  
estimated value by PDs. The distribution having the least  
D-index is considered as better suited distribution for  
EVA of wind speed (USWRC, 1981).  
From Table 3, it may be noted that the estimated  
EWS given by PR3 distribution are higher than the  
corresponding values of other five PDs for return period  
of 10-year and above for Delhi. Also, from Table 4, it may  
be noted that the LP3 distribution gave higher estimates  
for return period of 20-year and above when compared to  
the corresponding values of other five PDs for  
Kanyakumari. For qualitative assessment, the plots of  
recorded and estimated EWS were developed and  
presented in Figure 1.  
Application  
In this paper, a study on EVA of wind speed  
adopting six probability distributions (using MLM) was  
carried out. HMWS data recorded at Delhi for the period  
1969 to 2007 and Kanyakumari for the period 1970 to  
2008 is used. The annual series of EWS is extracted from  
hourly wind speed data and further used for EVA. Table 2  
gives the descriptive statistics of annual EWS for the  
regions under study.  
Table 2. Descriptive statistics of annual EWS  
Statistical parameters  
Mean  
(km/hr)  
Standard  
deviation of Skewness of kurtosis  
(km/hr)  
Coefficient Coefficient  
Region  
Delhi  
Kanyakumari  
66.1  
42.3  
261.1  
123.0  
0.047  
2.219  
-1.709  
6.848  
RESULTS AND DISCUSSIONS  
By applying the procedures, as described above,  
computer program through R-package was developed and  
used for EVA of wind speed. The program computes the  
parameters of six PDs, GoF (AD and KS) tests statistic  
and D-index values for Delhi and Kanyakumari.  
Estimation of EWS using six PDs  
The parameters obtained from MLM were used for  
estimation of EWS for Delhi and Kanyakumari through  
quantile functions of the respective PDs and presented in  
Tables 3 and 4 respectively.  
Table 3. Estimates of EWS given by six PDs for Delhi  
Return  
period  
(year)  
2
Estimated EWS (km/hr) using  
NOR  
LN2 GAM  
PR3  
LP3  
EV1  
Figure 1. Plots of recorded and estimated EWS for different  
66.1  
79.6  
64.2  
79.0  
64.7  
79.7  
61.2  
80.0  
65.0  
80.2  
63.3  
78.9  
return periods by six PDs for Delhi and Kanyakumari  
5
10  
86.6  
88.1  
88.4  
93.1  
87.6  
89.2  
20  
50  
100  
200  
500  
1000  
92.4  
98.9  
103.2  
107.2  
112.0  
115.4  
96.4  
96.0  
105.7  
121.8  
133.8  
145.6  
161.1  
172.7  
93.4  
99.6  
103.5  
106.9  
110.8  
113.4  
99.1  
From Figure 1, it can be seen that there is no  
significant difference between the frequency curves of  
LN2 and GAM distributions for Kanyakumari. Similarly,  
for Delhi, it can be seen that the frequency curves of NOR  
and LP3 distributions are very close to each other.  
106.7  
114.1  
121.4  
130.8  
137.9  
105.0  
111.4  
117.4  
124.9  
130.4  
111.9  
121.5  
131.1  
143.7  
153.3  
To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions. J. Civil Eng.  
Urban., 8 (2): 12-16. www.ojceu.ir  
14  
Analysis based on GoF tests  
CONCLUSIONS  
The paper presented the study on EVA of wind  
speed adopting six PDs (using MLM). Based on the  
results of EVA of wind speed, GoF and diagnostic tests,  
the following conclusions were drawn from the study:  
a) AD test results confirmed the applicability of  
PR3, LP3 and EV1 distributions for EVA of wind speed  
for Kanyakumari.  
By applying the procedures of GoF tests,  
quantitative assessment on fitting of PDs to the series of  
EWS was carried out; and the results are given in Table 5.  
Table 5. Computed values of GoF tests statistics by six PDs  
Computed values of GoF tests statistics for  
Probability  
Delhi  
Kanyakumari  
distribution  
AD  
KS  
AD  
KS  
NOR  
2.541  
0.215  
1.992  
0.197  
b) AD test results didn’t support the use of all six  
PDs for EVA of wind speed for Delhi.  
LN2  
GAM  
PR3  
LP3  
2.181  
2.078  
1.666  
2.192  
2.027  
0.205  
0.201  
0.179  
0.203  
0.200  
0.946  
1.279  
0.523  
0.540  
0.542  
0.166  
0.173  
0.120  
0.108  
0.136  
c) KS test results supported the use of all six PDs  
for EVA of wind speed for Delhi and Kanyakumari.  
d) D-index value of LP3 is found as minimum for  
Kanyakumari whereas the D-index value of LP3 is the  
second minimum for Delhi.  
e) LP3 distribution is identified as better suited  
amongst six distributions adopted for estimation of  
extreme wind speed for Delhi and Kanyakumari.  
The study suggested that the 1000-year return period  
EWS of 113.4 km/hr (for Delhi) and 121.3 km/hr (for  
Kanyakumari) adopting LP3 distribution could be used as  
the design parameters for planning and design of  
hydraulic structures in the regions.  
EV1  
From Table 5, it may be noted that the computed  
values of AD test statistic by six PDs are greater than the  
theoretical value of 0.781 at 5% significance level, and at  
this level, all six PDs are not acceptable for EVA of wind  
speed for Delhi. For Kanyakumari, it may be noted that  
the computed values of AD test statistic by PR3, LP3 and  
EV1 distributions are not greater than the theoretical value  
of 0.781 and therefore these three distributions are  
acceptable for EVA of wind speed. Also, from Table 5, it  
may be noted that the computed values of KS tests  
statistic by six PDs are not greater than the theoretical  
value of 0.218 at 5% significance level, and at this level,  
all six PDs are found to be acceptable for EVA of wind  
speed for Delhi and Kanyakumari.  
DECLARATIONS  
Acknowledgements  
The author is grateful to Dr. (Mrs.) V.V. Bhosekar,  
Additional Director, Central Water and Power Research  
Station, Pune, for providing the research facilities to carry  
out the study. The author is thankful to M/s Nuclear  
Power Corporation of India Limited, Mumbai, for the  
supply of wind speed data.  
Analysis based on diagnostic test  
For the selection of most suitable PD for estimation  
of EWS, the D-index values of six PDs were computed  
and presented in Table 6.  
Author’s contribution  
Table 6. Indices of D-index for six PDs  
Shri N. Vivekanandan, Scientist-B, Central Water  
and Power Research Station, Pune, carried out the data  
analysis and prepared the manuscript.  
D-index  
Region  
NOR  
LN2  
GAM PR3  
LP3  
EV1  
Delhi  
Kanyakumari 0.857  
0.428  
0.646  
0.918  
0.622  
0.826  
1.203  
0.707  
0.471  
0.600  
0.805  
1.014  
Competing interests  
The author declares that he has no competing  
interests.  
From Table 6, it may be noted that the indices of  
D-index given by NOR and LP3 distributions are  
minimum when compared to the corresponding indices of  
other distributions for Delhi and Kanyakumari  
respectively. But, the AD test results showed that the  
NOR distribution is not acceptable for EVA of wind speed  
for Delhi. After eliminating the NOR distribution from the  
group of six PDs, it may be noted that the D-index value  
of LP3 is the second minimum next to NOR; and therefore  
LP3 is considered as most appropriate PD for estimation  
of wind speed for Delhi. On the basis of GoF and  
diagnostic test results, LP3 distribution is identified as  
better suited for estimation of EWS for Delhi and  
Kanyakumari.  
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To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions. J. Civil Eng.  
Urban., 8 (2): 12-16. www.ojceu.ir  
15  
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To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions. J. Civil Eng.  
Urban., 8 (2): 12-16. www.ojceu.ir  
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