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Universitas Gadjah Mada Jurusan Teknik Sipil dan Lingkungan Prodi Magister Teknik Pengelolaan Bencana Alam
Teknik Pengolahan Data Analisis Frekuensi
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Analisis Frekuensi • Diambilkan dari paparan pada • The Second ASEAN Natural Disaster Conference
• University of Yangon, Myanmar • 29-‐30 September 2014
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Frequency analysis on extreme hydrologic data I s t i a r t o Master of Engineering in Natural Disaster Management Universitas Gadjah Mada, INDONESIA
29-‐Sep
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2nd ANDC Yangon, Myanmar
29-‐30 September 2014
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Needs • Engineers (hydraulics, hydrology) are frequently required to derive design values on hydrologic events, such as flow discharge or rainfall depth (precipita8on) • Sta8s8cal method applied to 8me series data is a common prac8ce
• Fit those data to theore8cal probability distribu8ons • Use cumula8ve distribu8on func8on of the selected probability distribu8on to predict magnitude of hydrologic events
• Define the design value(s) based on the magnitude of hydrologic event having par8cular probability of occurence
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4 this method is known as frequency analysis
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Frequency Analysis Time series (annual or par8al 8me series of hydrologic event data)
Fi[ng of the 8me series to theore8cal distribu8ons
Predic8ng probability of occurrence of the hydrologic events
Define design values of the hydrologic events
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Probability Distributions
Theo
re8cal Probability
Distrib
u8on
s
Gumbel
Log Normal
Log Pearson Type III
Normal
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Common probability distribu8ons applicable to
hydrologic events
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Probability Distributions • The theore8cal probability distribu8ons • their pdf and cdf have complex expression so that they are not readily solvable
• graphical method is a prac8cal solu8on to fit data to the theore8cal distribu8ons
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Ven te Chow et al. (1988) developed a transforma8on coordinate such that the cdf of a theore8cal distribu8on shows as a straight line
fi[ng data to theore8cal distribu8on
and predic8ng probability of occurrence can be
easily carried out
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Transformed probability coordinate • Chow et al. (1988)
• yT is the hydrologic magnitude at T-‐year return period, • y is the average value of the data, • KT is a frequency factor, and • s is the standard devia8on of the data.
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!yT = y +KT ⋅ s
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Transformed probability coordinate • Chow et al. (1988)
•
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!yT = y +KT ⋅ s
!
ys
constant plot of (yT vs KT) is a straight line
defined by data
!!prob Y < yT( ) =1− 1T ⇔ T = 1
1−prob Y < yT( )
KT is func8on of T and distribu8on
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Normal Distribution
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yT yT
T or prob(Y < yT) KT
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Normal Distribution
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!KT = z
!!
w =
ln 1p2
!
"#
$
%&
'
()
*
+,
1 2
0< p=1 T <0.5
ln 1
1−p( )2
!
"
###
$
%
&&&
'
(
)))
*
+
,,,
1 2
p=1 T >0.5
.
/
000
1
000
!!z=w− 2.515517+0.802853w+0.010328w2
1+1.432788w+0.189269w2 +0.001308w3
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Gumbel probability paper
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Normal probability paper
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Probability Paper • In the past • Probability papers were available in the market
• Normal, Log Normal, Gumbel distribu8ons • Not available for Log Pearson Type III since KT changes with data • Log Pearson Type III was plo4ed on Log Normal paper • It shows as a curve, not a straight line
• Data were manually plo4ed on the paper • Sca4er data plot and theore8cal line give good insight into the distribu8on of the data
• Fi[ng the data to the theore8cal line can easily be performed
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Probability Paper • Nowadays • Frequency analysis can easily be carried out by spreadsheet computer applica8on • Unfortunately, spreadsheet cannot provide data plot the way probability paper does
• Most of frequency analysis applica8on program do not provide data plot at all
• Lost of insight into data pa4ern (distribu8on) • Lost of ‘physical’ meaning of the computed values
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Example of frequency analysis performed by using spreadsheet applica8on/computer program
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17 Example of frequency analysis performed by using spreadsheet applica8on/computer program
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Probability Paper • Nowadays • There is no (or at least difficult to find) computer applica8on on frequency analysis having data plot capability • Commercial applica8on program may exist (?) • Prac8sing engineers and students might be not able to afford for commercial applica8on program
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Need for frequency analysis applica8on program, which is a free-‐ware and capable to produce graphical plot of the data on probability paper
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Probability Paper • AProb_4E • A computer applica8on on frequency analysis that has data plo[ng capability
• Can be freely downloaded from website h4p://is8arto.staff.ugm.ac.id/
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AProb_4E • Plo[ng posi8on of data values on probability paper
• m is the rank of the data being sorted in ascending order • n is the number of data
• This is known as Weibull formula of plo[ng 8me series data
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!!prob Y < ym( ) = m
n+1
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AProb_4E • Confidence interval (1 – α) • An interval within which the true value (which is unkonwn) can reasonably be expected to lie.
• The size of the interval depends on the confidence level (1 – α) • The es8mate of event value for a par8cular return period, yT • An upper and lower limits are specified by adjustment of KT
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!!UT ,α =Y + sy ⋅KT ,α
U
!!LT ,α =Y + sy ⋅KT ,α
L
upper and lower confidence limit factors for normally distributed data; these are determined using the noncentral t distribu8on
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AProb_4E
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• Goodness of fit test • Smirnov-‐Kolmogorov test • Chi-‐square test
• Both tests are applied with confidence level of (1 – α) = 0.90
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Goodness of Fit Test
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• Smirnov-‐Kolmogorov Test
!!Δmax =max prob(Y < y)−prob(Y < y) !!Δmax <Dcà rejected if
Dc is cri8cal value according to the Smirnov-‐Kolmogorov table
• according to the observed data
• according to the distribu8on being tested
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Goodness of Fit Test
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• Chi-‐square Test
!!χc2 =
Oi −EiEi
#
$%%
&
'((
i=1
k
∑
Oi the observed rela8ve frequency in the ith class interval Ei the expected rela8ve frequency, according to the distribu8on being
tested, in the ith class interval k the number of class intervals
!!χc2 > χ1−α 2,k−p−1
2à rejected if
p is the number of parameters es8mated from the data
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_4E in te
xt fo
rmat
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AProb_4E • Improvement • Feature that will be added: test for outliers
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!
yH =Y +Kn ⋅ sYyL =Y −Kn ⋅ sY
high and low outlier thresholds
in log unit if logarithms of the data are greater than yH or less than yL à they are considered as outliers
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References • Chow, Ven Te, Maidment, David R., Mays, Larry W., 1988, Applied Hydrology, McGraw-‐Hill, Inc., New York.
• Haan, C.T., 1982, Sta=s=cal Methods in Hydrology, 1st Ed., 3rd Prin8ng, The Iowa State Univ. Press, Ames, Iowa, USA.
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