THOMAS FIERING Thomas and fiering developed a model for generating a stream of monthly river. river. Implicitly, this model allows the existence of waterlessness in the monthly flow data (Clarke, 1973). In this method, the data for “n” “n” years years is divided into into 12 sections. The data data of each month is re-aggressed to the previous month, so that there are 12 linear regression equations. Seasonal variation is indicated by the use of monthly regression relationships. This model considers the existence of monthly persistence lag one. This persistence is caused by the effects of water storage as a soil moisture and groundwater. groundwater. There is also a congenital effect on seasonal weather patterns (raudkivi, 1979)
THOMAS FIERING Flow at station comprise a sample from a time series, and it is our aim to generate a sequence of numbers that are statistically indistinguishable form the observed time series or record. “Statistically Indistinguishable” as used here implies the entire phrase “indistinguishable within anticipated sampling erors” these erors vary inversely with increasing record length and directly with the sample coefficient of variation, and statistics of the generated sequences reflect these variations. A time series can be represented by several analytical functions, each of which is most useful for a particular class of series.
THE FORMULA OF THOMAS FIERING qi+!,j = Generated streams for month j and year to (i + 1) qi, j-1 = flow in the year i, in the previous month (j-1) r j = the correlation between the previous month's flow (j-1) and month j b j = regression coefficient between the previous month's flow (j-1) and month j ti = normal random numbers sd j = standard deviation month j
CALCULATION PROCEDURE 1.
Calculation of the average flow rate for each month
with: Qrerata = average discharge (m 3 / dt). n
= number of years
X i, j
= debit data in i and month j.
2. CALCULATION OF STANDARD DEVIATION
3. CALCULATION OF THE CORRELATION COEFFICIENT BETWEEN STREAMS IN TIME J AND TIME (J-1)
THE UNIFORM-DISTRIBUTED RANDOM NUMBERS •
RANDOM NUMBERS GENERATED BY COMPUTER DEVICE HAVE UNIFORM DISTRIBUTION WITH RANGE BETWEEN 0 AND 1. TABLE 1. SERIES OF RANDOM NUMBERS WITH RANGE 0 - 1 (UNIFORM DISTRIBUTION)
DISTRIBUTED NORMAL RANDOM NUMBER IN THE APPLICATION OF THE USE OF RANDOM NUMBERS, FOR EXAMPLE FOR SIMULATION SYNTHETIC DEBIT GENERATION DATA, OFTEN RANDOM NUMBERS ARE REQUIRED
THAT
HAVE
CERTAIN
DISTRIBUTIONS
DISTRIBUTION, THE NOTION OF NORMAL
OTHER
THAN
UNIFORM
Transformation of Uniform distributed random numbers into normally distributed random numbers: Box & Muller method, with transformation formula :
FORMULA 1
With : Ni and Ni + 1 : a pair of standard normal random numbers respectively. Ui and Ui + 1 : a pair of Uniform distributed random numbers with consecutive 0 - 1 ranges.
❖BY USING THE BOX & MULLER METHOD, A PAIR OF UNIFORM DISTRIBUTED RANDOM NUMBERS WILL GENERATE A PAIR OF STANDARD NORMAL RANDOM NUMBERS. E.G. 100 STANDARD NORMAL RANDOM NUMBERS MUST BE PROVIDED 100 UNIFORM DISTRIBUTED RANDOM NUMBERS . SO IF WE WANT TO GET 101 STANDARD NORMAL RANDOM NUMBERS, THEN THERE SHOULD BE 102 UNIFORM DISTRIBUTED RANDOM NUMBERS, AND THEN FROM THE NORMAL STANDARD RANDOM NUMBER GENERATED, 101 NUMBERS ARE TAKEN. SO, THE NUMBER OF UNIFORM DISTRIBUTED RANDOM NUMBERS THAT MUST BE PROVIDED IS ALWAYS EVEN
IF THE RANDOM NUMBER IS DISTRIBUTED UNIFORM WITH RANGE 0 - 1 ON TABLE 1 TRANSFORMED INTO A NORMAL STANDARD RANDOM NUMBER, WITH FORMULA 1 THEN THE RESULT IS AS FOLLOWS: TABLE 3. NORMAL RANDOM NUMBER SERIES
PROBLEMS EXAMPLE: IT IS KNOWN: MONTHLY FLOW DISCHARGE FOR 5 YEARS, AS FOLLOWS :
RESULT OF GENERATING NORMAL RANDOM NUMBER :
QUESTION : CALCULATE THE GENERATION OF DEBIT FOR JANUARY 6TH YEAR !
ANSWER : THOMAS FIERING'S FORMULA :
HOW ABOUT THE GENERATION OF DATA FOR FEBRUARY 6TH YEAR ?
?
TEST OF DEBIT DATA GENERATION RESULTS Validation is the process of testing the degree of a model in making forecasting. The sample data is often obtained into two segments, one used to interpret the model parameters and the other is used as the sample reference to test the forecast made by the model. To obtain a representative model, a model that can measure its accuracy with historical data. Accuracy is a criterion used to evaluate model performance and alternative forecasting methods and can show the level of truth of the forecast measured. Accuracy can be measured using dimensions such as mean square error (MSE), mean absolute precent error (MAPE), mean percent error (MPE).
STATIONARY TEST (HOMOGENEITY TEST) •
THE STATIONARY TEST IS INTENDED TO TEST THE VALIDITY OF THE VARIANT AND MEAN VALUES OF THE DATA SET
•
TESTING IS GENERALLY DONE BY USING F-TEST AND T-TEST
•
IF THE TEST RESULT SHOWS THE NULL HYPOTHESIS IS REJECTED, IT MEANS THAT THE VALUE OF THE DATA VARIANT IS UNSTABLE OR NOT HOMOGENEOUS
STATIONARY TEST (HOMOGENEITY): 1. Test of Variant Stability (Test F) :
Value Fcritical = (dk1, dk2) = (n1-1, n2-1) Where : N1 = sample number 1; Dk = degrees of freedom N2 = sample size 2 Ho accepted if the price Fhount
2. Stability Test of Average Value :
= (Level of Significance) eg 5%. If thitung,
dk = n1 + n2 - 2
PROBLEM : PERFORM STATIONARY TEST OF DISCHARGE DATA GENERATED FROM 2001 TO 2011 WITH OBSERVATION DISCHARGE DATA FOR 1990-2000 PERIOD
1. TEST OF VARIANT STABILITY TABLE : STABILITY TEST OF VARIANT VALUES DATA (TEST F)
Stability 5% Variable Stabilization Analysis: F-Test Two-Sample for Variances
Since the value of Fcount
TABEL : NILAI FKRITIS (FC) DISTRIBUSI F ( = 5%)
2. STABILITY TEST OF AVERAGE VALUES TABLE : STABILITY TEST OF AVERAGE VALUES (T-TEST)
THE ANALYSIS OF THE ABOVE TABLE IS AS FOLLOWS :
SO, tCOUNT :
Tabel : Nilai t kritis (tc ) untuk Distribusi t uji dua sisi (α= 5%)
•
For degrees of freedom α = 5%, from the table of critical values tc (Soewarno, 1995: 77), the value of ttable is 2.086.
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Because thitung
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With respect to the results of Test F and Test T then the discharge data generated and observation discharge data is stationary where the average value and value of the variance is stable
REFERENCE Harisuseno, Donny. 2017. Teaching materials for reservoir planning, design, and operation . Malang: universitas brawijaya. Clarke, r. 1973. A Review of Some Mathematical Models Used in Hydrology with Observations on Their Calibration and Use. Raudkivi. 1978. Hydrology: An Advanced Introduction to Hydrological Processes and Modelling. London : pergamon press
