A PROFILE OF MULTIPLICATION FREQUENCY CURVE TYPE III (X-III)

Yang Lixing  Zheng Zuguo

College of Water Conservancy, Xinjiang Agricultural University

Nanchang Road No.42, Urumqi, 830052, China

Li Xiangyun

The institute of Geographic Science and Natural Resources Research,

China Academy of Sciences, Datun Road No.3, Beijing, 100101, China

Abstract: Based on the multiplication theorem of probability and the attribute of median, this paper summarizes the theory, characteristics, method and example concerned the multiplication frequency curve type III(X-III，for short).The P(K≥k)=1-0.5^D,  D=(a-1)/(a^(k^c)-1), 0.5 is frequency of median. Its inverse of probability distribution function is a algebraic expression. The method of fitting curve by density parameters a and c is very simple and easy. This frequency curve advantage is resistance and robustness.

Keywords: flood frequency analysis, multiplication frequency curve type III, median, resistance, robustness

1  INTRODUCTION

The multiplication theorem of probability usually is neglected in the theoretical distribution of flood or frequency curve of flood, which often applies the mean value that has neither resistance nor robustness.

Considered that the flood is a random variable, it should follow the multiplication theorem of probability. Meanwhile, the median has the different attribute from the mean value, it is both resistance and  robustness [Hoaglin et al.1983].Based the multiplication theorem of probability and the attribute of the median, this paper briefly introduces the theory, the characteristic, the application method and example about the  multiplication  frequency curve type III(X-III，for short).This simple probability distribution is available for the flood occurred in the kinds of rivers and streams.

2  THEORY

Based the multiplication theorem of probability and the attribute of the median, and assumed D=(a-1)/(a^(k^c)-1), then theX-III expresses as follows:

The probability distribution function:

P(K≥k)=1-0.5^D                                                                                              (1)

The probability density function:

f(k)=-ln 0.5 *0.5^D *(a-1) *a^(k^c) *c *ln a *k^(C-1)/(a^(k^c)-1)^2                                        (2)

The inverse of the probability distribution function:

k={ln [(a-1)*ln 0.5/ln (1-P)+1]/ln a}^(1/c)                                                                      (3)

Where k=X/X0, X0= median of the sample, P(k≥1)=0.5  is frequency of median. A and c are two parameters of probability density function, which restrict to a>1,c>0.

3  CHARACTERISTICS

3.1  The clearness and lucidity of the theory

The multiplication theorem of probability is as follows:

                   K=1(median)      P(K≥1)=0.5^1

                   K=2             P(K≥2)=0.5^2

                   K=0.1          P(K≥0.1)=0.5^0.1

                   K=0             P(K≥0)=0.5^0=1

                   K=∞           P(K≥∞)=0.5^∞=0

                    .                     .

                   K=k          P(K≥k)=0.5^k                                                                                                                                    (4)

The expressions(4) names X-I, and is a single curve, Cv=0.999, Cs=1.98. It is neither flexibility nor adaptability. When two density parameters(a and c) of probability are added, this curve calls X-III as the expressions(1).It is not only flexibility but also adaptability. The expressions (1) is the extension of the multiplication theorem of probability.

Resistant and robust, good flexibility and little error

X-III is based on the multiplication theorem of probability, and the median represents location of distribution, so X-III is very resistant and robust.

Moreover, the probability density function as the expressions (2) includes exponential function and power function, and can be effectively applied to the upper tail and lower tail distribution. Therefore, X-III has the good flexibility and the small error in the fitting curve.

3.2  The simple usage

The expressions (3), the inverse function of the probability distribution, is algebraic. Thus, known P, a and c, k can be calculated by using expressions (3) as table 1 and table 2. When c=1,the curve calls X-III₁;when c=2, the curve calls X-III₂.

X-III₁ is available for a instantaneous value such as a flood peak;

X-III₂ is available for a non-instantaneous value, such as flood volume and  precipitation.

A curve can be directly fitted by using a and c, not using Cv, Cs and Φ. The application method is very simple.

3.3  No J-Shaped distribution and no appearance k<0

Both table 1 and table 2 show: when k<1,tail ofX-III is not level, namely not J-shaped distribution. And there is no appearance of k<0 which is the disadvantage of P-III.

4  METHODS OF FITTING CURVE

4.1  Preliminaries: calculation of median x0 and k=x/x0

After sorted samples, while n=odd, median X0 is the central value of sort order; while n=even, X0 is the mean value of two central value in the order. Thus, k=X/X0, empirical frequency P=m/(n+1).

4.2  Eye estimate curve fitting

By selecting k value in table 1 and table 2,the curve fitting process is simple and easy by using parameters a and c, completely not using Φ,Cv and Cs.

4.3  Optimizing curve fitting

Considering the expression (3) is algebraic, a optimizing process of fitting curve can be programmed in the computer. This method gets rid of the fitting trouble of using Φ,Cv and Cs.

5  OTHER USAGE: CALCULATION Cv and Cs UNDER A KNOWN A AND C

Sampling more than 10000 data in X-III curve using the expressions (3) is necessary, thus n=10000, p=m/(n+1). Through a and c obtained using the methods above-mentioned, k and X=k*X0 can be calculated. The Cv and Cs values can be got using these sampling data and the formula of matrix method.

They can be also obtained by infixing method in the contrast table. Table 1 and Table 2 list the partial relationship of contrast a and c with Cv and Cs.

6  EXAMPLE: FLOOD FREQUENCY ANALYSIS

The observed data from three hydrological stations is used inX-III, and doesn't include historical flood and paleo-flood; the flood data which includes historical flood and paleo-flood in P-III.The results (Table 3) of these two methods show the same effect.

Table 3 results verify: the analysis results of X-III frequency without paleo-floods are the same as the results of P-III with paleo-floods.

7  CONCLUSIONS

(1)X-III is based on the multiplication theorem of probability and median attribute, resistibility and robust. The theory of probability statistical distribution of three parameters is clear.

(2)X-III is available for the upper tail and lower tail distribution, its flexibility is wide, and can fit the distributions of kinds of rivers and streams.

(3)X-III can be fitted direct by two density parameters a and c, and is simple. The Cv and Cs value can be calculated by using parameters a and c.

References

Yang Lixing. (1985) The multiplication frequency curve type III₁, The proceedings of a symposium on the applied probability statistics of the first national water resource and hydropower field. Nanjing

Hoaglin.C.，Chen Zhonglian, etc Translation. (1998) The probing data analysis. The statistical press of China. Beijing.

Jin Guangyan. (1999) The review on hydrological frequency analysis, Advances in water science. 10(3), 319-327. Beijing.

Zhan Daojiang, etc. (1997) The new progress in the flood estimation—A study on paleo-flood. Hydrology. 1,1-5. Beijing.

Xie Yuebo, etc. (2000) The rationality analysis on the result of designed flood added a paleo-flood data, Journal of He-Hai University. 28(4),8-12. Nanjing

Table1  The K_p Value Table Of The Multiplication Frequency Curve Type Ⅲ₁（C=1）

Table2  The K_p Value Talbe Of The Multiplication Frequency Curve Type Ⅲ₂（C=2）