analyzing and predicting of tianjin coastal water quality by fractional thory.pdf

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Analyzing and Predicting of Tianjin Coastal

Water Quality by Fractional Theory

LU Ren-qiang, NIU Zhi-guang, ZHANG Hong-wei

School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, China

Email: [email protected] [email protected]

Corresponding author: [email protected]

The Project is currently sponsored by the Tianjin Municipal Science and Technology Commission through the Contract # 07JCYBJC07200

Abstract. In this paper, a new method for analyzing and

predicting the pollution characteristics of Tianjin coastal

water quality was proposed through the study on the

fractional theory, and this method was based on

environmental monitoring data completely. Firstly, the

COD values of recent years from the 9 monitoring points

were gathered and normalized, and the 9 monitoring

points are distributed on the Tianjin coastal marine. The

COD values from the same monitoring point were

regarded as an interval time series strictly. Secondly, therescaled range analysis was used to analyze the pollution

characteristics of COD time series. The Hurst exponents

H of COD time series were computed and the resultsshowed that the Hurst exponents were around 0.85. It

proved that the coastal water quality pollution presented

fractional characteristics. Thirdly, according to the

periodicity and self-similarity of coastal water quality and

the fractional collage theory, the fractional interpolation

method which based on affine transform was used to find

the iterated function system, whose attractor is close to

the historical water quality data. Then the fractional

predicting model was established according to the above

iterated function system. Finally, the random iterated

algorithm was used to find the attractor of eachpredicting period which could provide the predicting data

according to the time values. The predicting results

showed that the average prediction error was 24.4%. The

fractional theory could analyze and predict the pollution

characteristics of coastal water quality deeply and

precisely. And the method proposed in this paper was

practicable and could be the decision support for

environmental management of coastal marine.

Keywords: Coastal marine; Water quality; Fractional

theory; Rescaled range analysis; Hurst exponent; Iterated

function system; Attractor

I Introduction

Coastal marine water quality system is a dynamicsystem which is exoteric, complicated and non-linear.The coastal water quality is affected by physics,chemistry, biology and human being activity et al.

Therefore it has the performance of complicated non-linear characteristics. In order to control pollution andlet environmental management departments realize thechange trend of coastal water quality effectively, itneeds to analyze and predict the pollutioncharacteristics of coastal water quality precisely. Themechanism research methods are usually used tosimulate the space-time transform characteristics ofcoastal water quality. There methods are based on two-dimensional or three-dimensional water kineticsequation combined with the physics, chemistry andbiology models [1,2]. However, because of thecomplexity in monitoring data of coastal water quality,the mechanism research method is hard to put up. So itis not suitable for coastal water environmentalmanagement. The environmental managementdepartments monitor the water quality of Tianjincoastal marine in May (dry season), August (rainyseason), and October (level season) every year. Thereis only three monitoring data of water quality in eachyear of each monitoring point, which were too little formechanism research method. In this paper, a newmethod was proposed to analyze and predict the water

quality of coastal marine for the purpose to improvethe prediction precision. This method was based onfractional theory and environmental monitoring datacompletely.

Fractional theory was original proposed byMandelbrot in the eighties of the 20th century. It hasbeen one of the most active areas in the spatialfractional and time series fractional. Much of this workhas been prompted by applications in nature sciencefield and social science field [3,4]. The theory basis isthe system has self-similarity and scale invariantbetween the components. In other words, thecharacteristics between integer and parts looked

similarly from different views. Fractional theory hasbeen expanded and developed by other scientistsafterwards. It is thought that if there is affine transformrelation between parts and integer, it also be defined asa fractional.

978-1-4244-2902-8/09/$25.00 2009 IEEE 1


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II Original data description

For China's coastal water environmentalmanagement, one of the major concerns of waterquality studies has been the chemical oxygen demand(COD). So, the COD was taken as an example to studyin this paper. The COD monitoring data used in this

paper were taken from 9 monitoring points of Tianjincoastal marine. The locations of the 9 monitoringpoints were illustrated in Figure 1. These data countedto 243, were monitored in May, August, and Octoberfrom 1996 to 2004. Among these data, the datamonitored in 2004 were used as the test data for waterquality predicting. Therefore there were 27 data from 9monitoring points were used to test. Thus, the basicdata had the characteristics of fewer samples andshorter sequence. In addition, because the COD datawere monitored in May, August and October of a year,the COD time series of a monitoring point was not aninterval time series strictly. But the monitoring valuescould reflect the change trend of coastal water quality

in three water seasons. So, the COD time series couldbe regarded as an interval time series. Let { }ix

24,,2,1 =i denoted the COD values (mg L-1). The

COD values of monitoring point 9 were illustrated inFigure 2. From Figure 2, we could find that the CODvalues changed strongly along with the time values. Soit was difficult to analyze and predict the pollutioncharacteristics of coastal water quality.

Figure 1. Locations of the 9 monitoring points

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 5 10 15 20 25

Water Seasons

CO

D

Values(mg/L

Figure 2. COD data of monitoring point 9

III Analyzing the fractional characteristics of coastal

water quality

R/S analysis is an important part of fractionaltheory. It was created by H. E. Hurst when he workedon the Nile River Dam Project in the early 20thcentury[5], and later refined by Mandelbrot. It is a new

statistical method for distinguishing completely randomtime series from correlated time series, and mainly

used to estimate the Hurst exponent H and analyzethe fractional characteristics of time series. The Hurstexponent H can help to judge whether a time series isa fractional distributing model. In this paper, the R/Sanalysis was used to analyze the fractionalcharacteristics of coastal water quality time series.Taking monitoring point 9 of Tianjin coastal marine as

an example, the Hurst exponent H can be calculatedbased on monitoring data as follows [6,7].

Firstly, the COD time series{ }ix , 24,,2,1 =i , isdivided into a sub-series with length of n , satisfyingthat 24=an . Each sub-series is labeled as

kI ( ak ,,2,1 = ). And each element of kI is labeled

as kjx , ( nj ,,2,1 = ).Then, the average value of kjx ,

is computed as

=

=

n

j

kjk xn

e1

,

1( ak ,,2,1 = ) (1)

Secondly, the time series kjy , of accumulated

distances from the average value is computed as

=

=

j

t

kktkj exy1

,, )( ( nj ,,2,1 = ) (2)

Hence, the range that the COD time series coversrelative to the average value within each sub-series isdefined as

kjnj

kjnj

k yyR ,1

,1

minmax

= (3)

Thirdly, the standard deviationkS of each sub-

series is defined as

( )2

12

1

,1

1

=

=

n

j

kkjkex

nS (4)

From the definition ofkR , we can find that the kR

depends on the time length of n , and has positive

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relativity with n . kS in the R/S analysis method is

used to standardize the rangekR to allow comparisons

of different data sets. The average value of the

kk SR / for each sub-series with the length of n is

computed as

( ) ( )=

=

a

k

kknSR

aSR

1

/1

/

(5)

Finally, in order to investigate various time series,the equation (5) is transformed by Hurst as follows

bnHSR n lnln)/ln( += (6)

In equation (6), b is a constant, and H is calledHurst exponent.

The R/S analysis characterizes correlations in time

series data in terms of Hurst exponent H [8]. Thevalues of the Hurst exponent range between 0 and 1.

ForH>0.5 the time series is called persistent, i.e. adecreasing trend in the past implies a decrease in thefuture, or an increasing trend in the past implies a

continued increasing trend in the future. If, a H


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N

i

i BWBW1

)()(=

= )(XHB (8)

is a contraction mapping on the complete metric space

))(),(( dhXH , with contraction factor S, in otherwords,

),())(),(( BAShBWAWh )(, XHBA (9)

where h is known as the Hausdorff metric.

Then, there exists a unique fixed point

)(XHP which satisfied

N

i

i PWPWP1

)()(=

== (10)

The fixed point P may be obtained by an iterativescheme as follows

)(lim BWPn

n

= , )(XHB (11)

where )))((()( BWWWBWn = , and the

fixed point P is the attractor of the

IFS{ }NiWX i ,,2,1,; = .

B. The collage theorem

Let ( dX, ) be a complete metric space, and anIFS { }NWWWX ,,,; 21 with contractionfactor 10 S , also let 0> . Supposing any

)(XHL , we have

N

i

i LWLh1

))(,(=

(12)

Then SPLh 1),( (13)

where P is the attractor of theIFS { }NWWWX ,,,; 21 , and ),( PLh is theHausdorff distance.

The collage theorem tells us that there exists an

IFS { }NWWWX ,,,; 21 whose attractor is similar

with a given set )(XHL . The fixed point P is

obtained from the self-transformation )(PW . So, the

contraction transformed can be used to reconstruct a

new set L based on the given set )(XHL .

C. Fractional interpolation method based on affine

transform

The IFS { }N

WWWX ,,,;21

is the basis of

fractional interpolation method, where each

NiWi ,,2,1, = is defined as the affine transform as

follows

+

=

i

i

ii

i

ih

e

y

x

dc

a

y

xW

0(14)

D. Predicting coastal water quality by fractional

interpolation method

It was proved the COD time series was a fractional

time series. In this section, according to the periodicity,self-similarity and self-affinity of fractional time series,the fractional interpolation method was used to predictthe water quality COD of Tianjin coastal marine. Andalso taking the monitoring point 9 as an example, theprocess was as follows [10,11].

(1) The COD values from 1996 to 2003 weredivided into four time stage according to thechronological order. So each time stage composed oftwo years and contained six COD monitoring data.

(2) Obtained the COD values as the sample data todetermine the interpolation points set of each timestage. Here took the water quality monitoring times ofeach time stage as X-axis, and expressed by 0-5respectively. And the COD values were taken as Y-axis. All the points constituted the interpolation pointsset.

(3) Standardization of COD sample data

The COD sample data could be standardizedaccording to the equation (15) as follows

minmax

min

XX

XXx ii

=

minmax

min

YY

YYy ii

= (15)

WhereiX was the monitoring time, iY was the

COD monitoring data (mg L-1). minX maxX were

the minimal and maximal values of the time sample

data respectively. minY maxY were the minimum and

maximal values of the COD sample data respectively

(mg L-1). ix , iy were the standardization values of

iX , iY respectively.

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(4) Computed the parameters of the IFS of eachtime stage. According to the interpolation points setstandardized in step (3), the equation (14) was used tocompute the parameters of the IFS. Because there wereonly six data in each time stage, the contraction

operators id couldnt be computed by analytical

method. In this paper, chosen an appropriately valuethat id =0.2( 5,,2,1 =i ) by repeated calculation.

Then, the other parameters were computed, which were

ia , ic , ie , ih , to determine the affine transform iW .

(5) The weighted summation method was used toobtain a statistical IFS of the predicting periodaccording to the four IFS which were obtained in step(4). The weight values could be given on experienceaccording to the chronological order. In this paper, theweight values 0.2, 0.2, 0.3, 0.3 were taken here. Thestatistical IFS of monitoring point 9s COD changecurve in 2004 were illustrated in Table 3.

TABLE 3. The IFS of the COD change curve of monitoring point 9in 2004

IFS a c d e h

1 0.2000 -0.2183 0.2000 0.0000 0.4966

2 0.2000 -0.0394 0.2000 0.2000 0.2301

3 0.2000 0.1325 0.2000 0.4000 0.1425

4 0.2000 0.1994 0.2000 0.6000 0.2268

5 0.2000 -0.0742 0.2000 0.8000 0.3780

(6) Starting at any point, the random iteratedalgorithm was used to find the attractor of the IFSwhich illustrated in Table 3. This attractor could beregarded as the COD load curve of 2004 which wasobtained by extrapolation based on historical waterquality data. The attractor of monitoring point 9s CODcurve obtained by random iterated algorithm wasillustrated in Figure 4.

Figure 4. The attractor of the COD change curve of monitoring point

9 in 2004

(7) The attractor which illustrated in Figure 4 could

provide the predicting data iy according to the time

values ix . And we could estimate the change range of

the COD according to the historical water quality data.

Then we could obtain the predicting valueiY

of COD

in 2004 according to the equation (15). The CODpredicting values of monitoring point 9 in May, August

and October in 2004 were 0.970.78 and 0.71. The

prediction errors were 49.4% 25.0% and 11.2%

respectively. And the average prediction error was28.5%.

The COD values of other 8 monitoring points werepredicted by the fractional predicting method describedfrom steps (1) to (7). The predicting results were

illustrated in Table 4, wheremC denoted the

monitoring values (mg L-1), pC denoted the predicting

values (mg L-1). From Table 4, we could find that themaximal prediction error was 49.4%, the minimalprediction error was 2.3% and the average predictionerror was 24.4%. The prediction results could satisfythe needs of coastal water environmental management.

TABLE 4. Predicting results of coastal water quality based on fractional theory

Monitoring pointsMay, 2004 August, 2004 October, 2004

Cm Cp Error (%) Cm Cp Error (%) Cm Cp Error (%)

01 0.84 0.91 8.3 0.48 0.69 43.7 0.64 0.79 23.4

02 2.40 1.63 32.1 0.76 0.89 17.1 1.04 1.22 17.3

03 0.64 0.73 14.1 0.52 0.66 26.9 1.12 1.06 5.4

06 0.56 0.71 26.8 1.84 1.22 33.7 1.00 1.09 9.0

08 2.96 1.58 46.6 1.36 0.93 31.6 1.28 1.31 2.3

09 1.92 0.97 49.4 1.04 0.78 25.0 0.80 0.71 11.2

10 2.08 1.66 20.2 1.12 1.35 20.5 0.80 1.04 30.0

12 1.58 0.95 39.8 1.48 1.29 12.8 1.00 1.32 32.0

15 1.28 0.91 28.9 1.56 0.99 36.5 1.20 1.03 14.2

Average Error (%) 29.6 27.5 16.1

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V Conclusion

In this paper, a new method was proposed based onfractional theory, which was used to analyze andpredict coastal water quality. Taking the coastal marineof Tianjin as an example, firstly, the R/S analysis wasused to analyze the pollution characteristics of CODtime series. The Hurst exponents of COD time serieswere computed. The results showed that the Hurstexponents were around 0.85, which proved that thecoastal water quality pollution presented fractionalcharacteristics. Secondly, according to the fractionalcollage theory, the fractional interpolation method wasused to find the IFS of historical water quality. Thenthe fractional predicting model was established.Finally, the random iterated algorithm was used to findthe attractor of each predicting period which couldprovide the predicting data according to the timevalues. The predicting results showed that the averagepredicting error was 24.4%. Although the prediction

accuracy was not high, it could meet the needs ofcoastal water environmental management. Through thestudy, the fractional theory was introduced to analyzeand predict the pollution characteristics of coastalwater quality for the first time. And it also expandedthe application fields of fractional theory.

Acknowledgment

The Project is currently sponsored by the TianjinMunicipal Science and Technology Commissionthrough the Contract # 07JCYBJC07200.

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