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Hydrologic Statistics. ANALISIS FREKUENSI DAN PROBABILITAS. DISTRIBUSI NORMAL DISTRIBUSI LOG NORMAL - DISTRIBUSI LOG-PERSON III DISTRIBUSI GUMBEL. PARAMETER STATISTIK. Rata – rata - Simpangan Baku Koefisien Variasi Koefisien skewness. DISTRIBUSI NORMAL. Contoh : - PowerPoint PPT Presentation
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Hydrologic Statistics
ANALISIS FREKUENSI DAN PROBABILITAS
-DISTRIBUSI NORMAL-DISTRIBUSI LOG NORMAL
-- DISTRIBUSI LOG-PERSON III-DISTRIBUSI GUMBEL
PARAMETER STATISTIK
• Rata – rata -
• Simpangan Baku
• Koefisien Variasi
• Koefisien skewness
31
3
2
1
2
1
21
1
1
1
snn
xxnG
x
sCV
xxn
s
xn
x
n
ii
n
ii
n
ii
DISTRIBUSI NORMAL
Gauss) reduksi variabelnilai tabeldari ( frekuensifaktor K
variannilaistandar deviasi S
varianhitung rata-rata nilaix
tahunan T ulang periode
dengan terjadidiharapkan yang nilai
T
perkiraanx
sKxx
T
TT
• Contoh :Dari data debit puncak tahunan Sungai di Jawa
Timur seperti pada tabel , hitunglah debit puncak pada periode ulang 2, 5, 20, 50 tahunan dengan menggunakan distribusi normal
Data Debit Puncak TahunanNo Tahun Debit (m3/detik )1 1960 345.072 1961 511.473 1962 270.424 1963 903.725 1964 180.836 1965 294.627 1966 398.108 1967 482.359 1968 319.51
DISTRIBUSI LOG NORMAL
Gauss) reduksi variabelnilai tabeldari ( frekuensifaktor K
variannilaistandar deviasi S
varianhitung rata-rata nilaiy
tahunan T ulang periode
dengan terjadidiharapkan yang nilai
xlog
T
perkiraany
y
sKyy
T
TT
DISTRIBUSI LOG PERSON
sKxx
snn
xxnG
n
xxs
x
T
n
ii
n
ii
n
i
.loglog
21
loglog
1
loglog
n
xlogx log
xlog
31
3
5.0
1
2
1i
• Nilai K untuk Distribusi Log Person
• Hitung dengan menggunakan metoda Log Person
- Log x-
3
2
loglog
loglog
log
xx
xx
x
DISTRIBUSI GUMBEL
• Reduced Mean, Yn• Reduced Standard Deviasi, Sn• Reduced Varian , YTr sebagai fungsi periode
ulang
n
n
n
TrTr
s
syxb
s
sa
ya
bx
1
16
Hydrologic Models
• Deterministic (eg. Rainfall runoff analysis)– Analysis of hydrological processes using deterministic
approaches – Hydrological parameters are based on physical relations of
the various components of the hydrologic cycle. – Do not consider randomness; a given input produces the
same output. • Stochastic (eg. flood frequency analysis)
– Probabilistic description and modeling of hydrologic phenomena
– Statistical analysis of hydrologic data.
Classification based on randomness.
17
Probability
• A measure of how likely an event will occur• A number expressing the ratio of favorable
outcome to the all possible outcomes • Probability is usually represented as P(.)
– P (getting a club from a deck of playing cards) = 13/52 = 0.25 = 25 %– P (getting a 3 after rolling a dice) = 1/6
18
Random Variable
• Random variable: a quantity used to represent probabilistic uncertainty– Incremental precipitation – Instantaneous streamflow– Wind velocity
• Random variable (X) is described by a probability distribution
• Probability distribution is a set of probabilities associated with the values in a random variable’s sample space
20
Sampling terminology• Sample: a finite set of observations x1, x2,….., xn of the random
variable• A sample comes from a hypothetical infinite population
possessing constant statistical properties• Sample space: set of possible samples that can be drawn from a
population• Event: subset of a sample space ExampleExample
Population: streamflowPopulation: streamflow Sample space: instantaneous streamflow, annual Sample space: instantaneous streamflow, annual
maximum streamflow, daily average streamflow maximum streamflow, daily average streamflow Sample: 100 observations of annual max. streamflowSample: 100 observations of annual max. streamflow Event: daily average streamflow > 100 cfsEvent: daily average streamflow > 100 cfs
21
Types of sampling• Random sampling: the likelihood of selection of each member of the
population is equal – Pick any streamflow value from a population
• Stratified sampling: Population is divided into groups, and then a random sampling is used– Pick a streamflow value from annual maximum series.
• Uniform sampling: Data are selected such that the points are uniformly far apart in time or space– Pick steamflow values measured on Monday midnight
• Convenience sampling: Data are collected according to the convenience of experimenter.– Pick streamflow during summer
22
Summary statistics• Also called descriptive statistics
– If x1, x2, …xn is a sample then
n
iixn
X1
1
2
1
2
1
1
n
ii Xx
nS
2SS
X
SCV
Mean,
Variance,
Standard deviation,
Coeff. of variation,
for continuous data
for continuous data
for continuous data
Also included in summary statistics are median, skewness, correlation coefficient,
24
Graphical display
• Time Series plots• Histograms/Frequency distribution• Cumulative distribution functions• Flow duration curve
25
Time series plot• Plot of variable versus time (bar/line/points)• Example. Annual maximum flow series
0
100
200
300
400
500
600
1905 1908 1918 1927 1938 1948 1958 1968 1978 1988 1998
Year
An
nu
al M
ax F
low
(10
3 c
fs)
Colorado River near Austin
0
100
200
300
400
500
600
1900 1900 1900 1900 1900 1900 1900
Year
An
nu
al M
ax F
low
(10
3 c
fs)
26
Histogram• Plots of bars whose height is the number ni, or fraction
(ni/N), of data falling into one of several intervals of equal width
0
10
20
30
40
50
60
70
80
90
100
0 50 100 150 200 250 300 350 400 450 500
Annual max flow (103 cfs)
No
. of
occ
ure
nce
s Interval = 50,000 cfs
0
10
20
30
40
50
60
Annual max flow (103 cfs)
No
. of
occ
ure
nce
s
Interval = 25,000 cfs
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350 400 450 500
Annual max flow (103 cfs)
No
. of
occ
ure
nce
s
Interval = 10,000 cfs
Dividing the number of occurrences with the total number of points will give Probability Mass Function
28
Using Excel to plot histograms
1) Make sure Analysis Tookpak is added in Tools.
This will add data analysis command in Tools
2) Fill one column with the data, and another with the intervals (eg. for 50 cfs interval, fill 0,50,100,…)3) Go to ToolsData AnalysisHistogram
4) Organize the plot in a presentable form (change fonts, scale, color, etc.)
29
Probability density function• Continuous form of probability mass function is probability
density function
0
10
20
30
40
50
60
70
80
90
100
0 50 100 150 200 250 300 350 400 450 500
Annual max flow (103 cfs)
No
. of
occ
ure
nce
s
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 100 200 300 400 500 600
Annual max flow (103 cfs)
Pro
bab
ility
pdf is the first derivative of a cumulative distribution function
31
Cumulative distribution function• Cumulate the pdf to produce a cdf• Cdf describes the probability that a random variable is less
than or equal to specified value of x
0
0.2
0.4
0.6
0.8
1
0 100 200 300 400 500 600
Annual max flow (103 cfs)
Pro
bab
ility
P (Q ≤ 50000) = 0.8
P (Q ≤ 25000) = 0.4
36
Flow duration curve
• A cumulative frequency curve that shows the percentage of time that specified discharges are equaled or exceeded.
StepsSteps Arrange flows in chronological order Arrange flows in chronological order Find the number of records (N)Find the number of records (N) Sort the data from highest to lowest Sort the data from highest to lowest Rank the data (m=1 for the highest value and m=N for the lowest value)Rank the data (m=1 for the highest value and m=N for the lowest value) Compute exceedance probability for each value using the following Compute exceedance probability for each value using the following
formulaformula
Plot p on x axis and Q (sorted) on y axisPlot p on x axis and Q (sorted) on y axis
1100
N
mp
37
Flow duration curve in Excel
0
100
200
300
400
500
600
0 20 40 60 80 100
% of time Q will be exceeded
Q (
1000
cfs
) Median flow
38
Statistical analysis
• Regression analysis• Mass curve analysis• Flood frequency analysis• Many more which are beyond the scope of
this class!
39
Linear Regression
• A technique to determine the relationship between two random variables.– Relationship between discharge and velocity in a stream– Relationship between discharge and water quality constituents
A regression model is given by :A regression model is given by :
yi = ith observation of the response (dependent variable)
xi = ith observation of the explanatory (independent) variable
0 = intercept
1 = slope
i = random error or residual for the ith observation
n = sample size
nixy iii ,...,2,110
40
Least square regression
• We have x1, x2, …, xn and y1,y2, …, yn observations of independent and dependent variables, respectively.
• Define a linear model for yi,
• Fit the model (find b0 and b1) such at the sum of the squares of the vertical deviations is minimum– Minimize
nixy ii ,...,2,1ˆ 10
nixyyy iiii ,...,2,1)(ˆ 210
2
Regression applet: http://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html
41
Linear Regression in Excel
• Steps: – Prepare a scatter plot– Fit a trend line
TDS = 0.5946(sp. Cond) - 15.709R2 = 0.9903
0
300
600
900
1200
1500
1800
0 500 1000 1500 2000 2500 3000
Specific Conductance ( S/cm)
TD
S (
mg
/L)
Alternatively, one can use ToolsAlternatively, one can use ToolsData Data AnalysisAnalysisRegressionRegression
Data are for Brazos River near Highbank, TX
42
Coefficient of determination (R2)
• It is the proportion of observed y variation that can be explained by the simple linear regression model
SST
SSER 12
2)( yySST i Total sum of squares, Ybar is the mean of yi
2)ˆ( ii yySSE Error sum of squares
The higher the value of RThe higher the value of R22, the more successful is the model in explaining y , the more successful is the model in explaining y variation.variation.
If RIf R22 is small, search for an alternative model (non linear or multiple is small, search for an alternative model (non linear or multiple regression model) that can more effectively explain y variationregression model) that can more effectively explain y variation