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

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

2009). Number of studies has been carried out by different

researchers on adoption of different probability

distributions for Extreme Value Analysis (EVA) of wind

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,

x_{T }

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

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, Z_{i}=F(x_{i}), for i=1,2,3,…,N with x_{1}<x_{2}< ….x_{N },

F(x_{i}) is the Cumulative Distribution Function (CDF) of i^{th }

sample (

x

) and N is the sample size (Zhang, 2002). The

i

critical value (AD_{C}) of AD test statistic for different

sample size (N) at 5% significance level is computed

from:

(2)

AD_{C } 0.757 (1 (0.2/ N)

Similarly, the critical value (KS_{C}) of KS test statistic

for different sample size (N) at 5% significance level is

computed from:

N

(3)

KS_{C } Max ( F (x _{i }) F_{D }(x _{i}))

e

i1

Here, F_{e}(x_{i})=(i-0.44)/(N+0.12) is the empirical CDF

of x_{i }and F_{e}(x_{i}) is the computed CDF of x_{i}.

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

x_{T }

2

1

xm

x_{T } mK_{T }

NOR

2

f(x;,m)

1 2

e

,x, 0

2

1

ln(x)m

x_{T } e^{mK }

T

LN2

2

f(x;,m) 1 x 2

e

, x, 0

e^{x }(x)^{1 }

()

1

X_{T }

K

f(x;,)

,x, 0

GAM

T

f(x;,,m)

e^{(xm) }

x m

K_{P }

X_{T } m

PR3

()

m

e

1

)/)

P

f(x;,,m)

lnx m

,x, 0

LP3

X_{T } e^{m((K }

1

()

x

(xm)/

e^{(xm)/}e^{e }

, x, >0

x_{T } mY_{T }

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 (Y_{T}) 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 Y_{T}=-ln(-ln(1-(1/T))).

K

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

x_{T }

CS=0.0 for NOR and LN2].

K_{P }is the frequency factor

corresponding to CS of the original and log-transformed

series of EWS for PR3 and LP3 distributions respectively

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

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 N ^{N }

(2i 1) ln(Z )

i

(1)

AD

N

2N 1 2i ln(1 Z_{i })

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, x_{i }is the i^{th }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

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.

REFERENCES

Bivona S, Burlon R, Leone C. (2003). Hourly wind speed

analysis in Sicily. Journal of Renewable Energy,

28(9): 1371-1385.

Della-Marta PM, Mathis H, Frei C, Liniger MA, Kleinn J,

Appenzeller C. (2009). The return period of wind

storms over Europe. Journal of Climatology, 29(3):

437-459.

Palutikof JP, Brabson BB, Lister DH, Adcock ST. (1999).

A review of methods to calculate extreme wind

speeds. Journal of Meteorological Applications, 6(2):

119-132.

Urban., 8 (2): 12-16. www.ojceu.ir

15

Pandey MD, Van Gelder PHAJM, Vrijling JK. (2001).

The estimation of extreme quantiles of wind velocity

using L-moments in the peaks-over-threshold

approach. Journal of Structural Safety, 23(2): 179-

192.

Topaloglu F. (2002). Determining suitable probability

distribution models for flow and precipitation series

of the Seyhan River Basin. Turkish Journal of

Agriculture and Forestry, 26(4): 187-194.

Guevara E. (2003). Engineering design parameters of

storms in Venezuela. Hydrology Days, 80-91.

Lee C. (2005). Application of rainfall frequency analysis

on studying rainfall distribution characteristics of

Chia-Nan Plain Area in Southern Taiwan. Journal of

Crop, Environment and Bioinformatics, 2(1): 31-38.

Kunz M, Mohr S, Rauthe M, Lux R, Kottmeier Ch.

(2010). Assessment of extreme wind speeds from

regional climate models – Part 1: Estimation of

return values and their evaluation. Natural Hazards

Earth System Sciences, 10(4): 907-922.

El-Shanshoury I, Ramadan AA. (2012). Estimation of

Extreme Value Analysis of Wind Speed in the

North-Western Coast of Egypt. Arab Journal of

Nuclear Science and Applications, 45(4): 265-274.

Escalante-Sandoval CA. (2013). Estimation of extreme

wind speeds by using mixed distributions. Journal of

Engineering Investigation and Technology,14(2):

153-162.

Ahmed SA. (2013). Comparative study of four methods

for estimating Weibull parameters for Halabja, Iraq.

Journal of Physical Sciences. 8(5): 186-192.

Indhumathy D, Seshaiah CV, Sukkiramathi K. (2014).

Estimation of Weibull parameters for wind speed

calculation at Kanyakumari in India. Journal of

Innovative Research in Science, Engineering and

Technology, 3(1): 8340-8345.

Rao AR, Hameed KH. (2000). Flood frequency analysis.

CRC Publications, Washington D.C., New York.

Zhang J. (2002). Powerful goodness-of-fit tests based on

the likelihood ratio. Journal of Royal Statistical

Society, Series B, 64(2): 281-294.

USWRC. (1981). Guidelines for Determining Flood Flow

Frequency. United States Water Resources Council

(USWRC) Bulletin No. 17B, 15-19.

Urban., 8 (2): 12-16. www.ojceu.ir

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