Statistika - Istiarto Analisis Frekuensi.pdf · UniversitasGadjahMada JurusanTeknikSipildanLingkungan* ProdiS2TeknikSipil* Statistika Analisis’Frekuensi’ 16Dec14 ’ ’ 1

Universitas Gadjah Mada Jurusan Teknik Sipil dan Lingkungan Prodi S2 Teknik Sipil

Statistika Analisis Frekuensi

16-‐Dec-‐14

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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

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2nd ANDC Yangon, Myanmar

29-‐30 September 2014

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

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

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

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

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

Normal Distribution

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yT yT

T or prob(Y < yT) KT

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

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

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

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

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

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

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

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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27 Result of AProb

_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

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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Documents

Statistika - Istiarto Analisis Frekuensi.pdf · UniversitasGadjahMada Jurusan*Teknik*Sipil*dan*Lingkungan* ProdiS2TeknikSipil* Statistika Analisis’Frekuensi’ 16Dec14 ’ ’ 1

Statistika - Istiarto Analisis Frekuensi.pdf · UniversitasGadjahMada JurusanTeknikSipildanLingkungan* ProdiS2TeknikSipil* Statistika Analisis’Frekuensi’ 16Dec14 ’ ’ 1