M00474

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CROWN DIAMETER OF PHYTHOREMEDIATION AGENT BY LOGISTIC MODEL

By : H.A Parhusip

Science and Mathematics Faculty, Satya Wacana Christian University

Jl. Diponegoro 52-60 Salatiga, 50711, Central Java,Indonesia

Abstract : The logistic model of crown diameter of Phythoremediation Agent is discussed here. Crown diameter of Kailan is best to be modeled by logistic model. Since 6 Kailans are measured in 9 weeks for each Kailan, then there are 6 pairs of parameters in the model.

The best parameters are approximated by least square method and averaged by geometric mean and harmonic mean. The harmonic mean present better approximation compared to the geometric mean of parameters.

We also observe by principal component analysis that the data set of Kailan present the largest eigenvalue compared the other three tested plants.

Keywords : logistic model, crown diameter, geometric mean, harmonic mean 1. Introduction

One of mechanisms by certain plants to take up essential mineral nutrients and toxic heavy metals from soils is called phytoremediation. Phytoremediation is the use of green plants to remove pollutants from the environment. The process is illustrated in Figure 1.

Figure 1. The process of phytoremidiation

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Suresh dan Ravishankar (2004) stated that phytoremidiation do not harm environment. The used plants as phytoremidiation agencies possess genes that regulate the amount of metals taken up from the soils by roots and deposited at other location within the plants.

Kasmiyati, et.all (2008) have studied 4 types of plants as phytoremidiation agencies. These are 2 plants in Brassicaceae i.e: Brassica oleracea and Kailan and 2 plants from Asteraceae, i.e Chrysanthemum and sunflower (Helianthus annus). These plants grow in soils which are contaminated by textile manufacturing process which contains heavy metals. The heavy metals (Zn, Cd, Pb, Cu) are bound to soil components in varying degrees depending on soil conditions such as pH, clay content and organic matter (Hinchman, et.all 1998).

Figure 1. Crown diameter of tested plants (by Kasmiyati, 2008)

The growth rates of the tested plants are analyzed statistically (Kasmiyati, et.all (2008)). The tested plants are evaluated in 9 weeks. The crown of diameters, the number of leaves, the dried weights are measured. By Tukey test, it was concluded that the Chrysanthemum has the smallest crown diameter and dried weight compared to other three. The measurements of dried weights are illustrated in Figure 2.

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Dried weights are obtained by drying the tested plants in an oven (with 105 oC ) after 3 months growing. Dried weights illustrate an ability of plant in photosynthesis. As a result, the number of leaves of Chrisanthemum is the smallest compared the other three plants. As a conclusion of this research that Chrisanthemum is the worst agent used in phytoremidiation.

This paper presents the growth rate of the tested plants as a logistic function. Kailan is the studied plant though several tested plants are available. The model is shown on the section 2.

2.1 Logistic model of crown diameter

We actually idealize the situation by assuming that the diameter of crown is a logistic function. The used data is shown in Table 1.

Tabel 1. Data of crown diameter (cm) of 6 Kailans (by Kasmiyati, 2008)

-th Week

(t) KAILAN

I II III IV V VI

1 13.7 9.5 16.0 10.8 11.8 10.3

2 18.5 14.3 16.5 10.7 14.5 19.7

3 20.7 18.3 22.0 24.7 23.8 20.8

4 23.7 19.3 24.3 28.7 20.0 16.7

5 25.3 22.3 23.7 30.0 20.3 19.3

6 28.0 23.7 18.7 34.0 18.7 20.3

7 24.0 20.3 27.0 30.0 19.7 20.0

8 27.0 24.7 30.3 29.3 23.3 25.7

9 25.7 24.0 30.7 30.0 24.7 25.3

By introducing variable t is the time variable, and D(t) is the diameter at time t, we have (Stewart,1999)

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) 0 ( ) 0 ( , ) ( D D K A Ae K K t

D _kt = −

= ₋ (1a)

where D(0) represents the initial diameter, and parameter K and k must be determined based on the given data.

Before using the model, one has to present the given data in dimensionless form by dividing the number on each coloum by the maximum number on each coloum. This is shown in Table 2.

Table 2. Data Table 1 in dimensionless form

plant/

week I II III IV V VI 1 0.4893 0.3846 0.5212 0.3176 0.4777 0.4008 2 0.6607 0.5789 0.5375 0.3147 0.5870 0.7665 3 0.7393 0.7409 0.7166 0.7265 0.9636 0.8093 4 0.8464 0.7814 0.7915 0.8441 0.8097 0.6498 5 0.9036 0.9028 0.7720 0.8824 0.8219 0.7510 6 1.0000 0.9595 0.6091 1.0000 0.7571 0.7899 7 0.8571 0.8219 0.8795 0.8824 0.7976 0.7782 8 0.9643 1.0000 0.9870 0.8618 0.9433 1.0000 9 0.9179 0.9717 1.0000 0.8824 1.0000 0.9844

The parameter k and K can be obtained by least square method (Parhusip,2009) (Stoer dan Bulirsch,1993,hal.217) and we get K=0.9844 dan k=3.6688. Therefore the logistic model for the first plant (denoted by I) becomes

4893 . 0 4893 . 0 9844 . 0 , 9844 . 0 9844 . 0 )

( ₃_.₆₆₈₈ = −

= ₋ A

Ae t

D _t . (1b)

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Figure 3. Comparison logistic model and data of crown diameter of the first Kai-lan (2-nd coloum of Table 2).

With the same procedure as before, we may compute the values of parameter K and k for each Kailan and the results are listed in Table 3.

Table 3. The values of parameters K and k for each Kailan

No K k

I 0.9844 3.6688

II 1.0064 3.9832

III 1567.1 0.7

IV 0.9610 4.5827

V 0.9266 4.0359

VI 0.9444 3.7926

The error is defined to guarantee how good the logistic model approximates the crown diameter as a function of time. The error is denoted by e and is defined by

% 100 x D

D D e

d l d −

= (1c)

where || .|| reads as ’norm’ and D_d := crown diameter given by data and D_l:= crown diameter given by logistic model. The Euclidian norm is used here (Stoer and Bulirsch, 1993).

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Using equation (1c), we obtain the error of each plant (I-VI) is respectively given by 6.7675%, 7.5598%, 10.6854%, 14.0065%, 12.7693%, 13.2521%. Computation of parameter for each plant becomes time consuming. Therefore we introduce the geometric mean and harmonic mean to find the best estimation of parameters.

2.2 Geometric mean and Harmonic mean

Kasmiyati (2008) has used the arithmetic mean to analyze her data by Tukey test. In this paper we introduce geometric mean and harmonic mean to find the best value of parameters of the logistic models.

Suppose we have data x₁,x₂,...,x_n and the geometric mean is denoted by xr_G which is defined by

n n i i G x x / 1

1 ⎟⎟⎠ ⎞ ⎜⎜ ⎝ ⎛ =

∏

= r (2a)

where _n

i xx x x = ₁ ₂...

∏

. If we define

∑

= n

i xi n h 1 1 1 1 where h

xr_H = 1 (2b)

then xr_H is called harmonic mean. The relation of arithmetic mean, geometric mean and harmonic mean is known (Peressini, et.all, 1988) as

H G

A x x

xr ≥ r ≥ r . (3)

The equal sign is satisfied if each value of the given data is the same. Equations (2a-2b) will be used to find the mean values of K and k.

3. Research Method

1. Data presented into dimensionless

2. By assuming the crown diameter is a logistic function then the parameter K and k must be determined by using least square method.

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3. Using geometric mean and harmonic mean, find the best values of K and k for all plants.

4. Using the best parameter in step 3 to present the crown diameter as a logistic function.

5. Construct the covariance matrix of each data set of plant

6. Find the largest eigenvalue of each data set of plant and compute its corresponding eigenvector.

4. Result and Discussion

Using equation (2a) and (2b), we may obtain the mean values of parameters in Table 3 are also computed and shown in Table 4.

Table 4. Geometric mean and harmonic mean of parameter K and k.

Parameter Geometric mean

Harmonic mean

K 4.2891 1.3741

k 2.9554 2.1492

The mean values of parameters are used in the logistic model to present the crown diameter as a function of time. Using geometric mean, we get 135.8004% and 10.9801% using harmonic mean. These results are illustrated in Figure 4a-4b. We conclude that the harmonic mean is the best approximation of parameters for all tested plants.

What about other tested plants ?

The logistic model is also tried to model the crown diameter of other tested plants such as Brassica oleracea and 2 plants from Asteraceae, i.e Chrysanthemum and sunflower (Helianthus annus). In the case of Chryanthemum, the model does not fit to the obtained data. One example is shown by Figure 5. We have also studied the sun flower. As a result, the logistic model is not a good model for its crown diameter. Therefore we can not proceed as in the case of Kailan.

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Figure 4a. Logistic model of Crown Diameter using geometric mean

Figure 4b. Logistic model of Crown Diameter using harmonic mean

1 2 3 4 5 6 7 8 9 0.6

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

i−th week

Crown diameter (no dimension)

data model

Figure 5. The logistic model compared to the crown diameter data of the first tested

Chrysanthemum.

Compared to the other three tested plants, Kailan has shown that its diameter crown can be modeled by the logistic function. We will give some more general comments on this. This is done by considering the principal component of the covariance of given data. One may refer to (Parhusip and Siska, 2009) to have the idea of using this method.

By principal component here means that we construct the covariance matrix of each set of data. By computing its eigenvalues and selecting the largest eigenvalue, we may have correlation between the first principal component and each tested plant in the

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same set. One may observe that the largest eigenvalue is given by the set diameter crown of Kailan. The largest eigenvalue determines its greatest variance (Johnson and Wichern, 2002). We can conclude that the Kailan is the most dominant plant to present its diameter crown compared to the other three plants.

Conclusion and Remark

This paper presents a logistic model of crown diameter of Kailan. Since there are several observations, we need to determine the best parameters which describe the growth rate (k) of plant and the maximum capacity of crown diameter to grow (K). The best parameters are chosen by using geometric mean and harmonic mean. The result shows that the harmonic mean of parameters are the best aproximation.

Instead of crown diameter, dried weight and the number of leaves are also measured to study the influence of polluted soils. We have not concluded which plant is considered as the best phytoremediation agent.

Aknowledgment : I thank to Kasmiyati, MSc (Biology Faculty, UKSW) who has provided me her data for this paper.

References

Aiyen. 2005. Ilmu Remediasi untuk Atasi Pencemaran Tanah di Aceh dan Sumatera Utara. http://www.kompas.com/kompas-cetak/0503/04/ilpeng/1592821.htm

Hinchman, R.R, Negri M. C., and Gatliff, E. G., 1998. Phytoremediation : Using Green Plants to Clean Up Contaminated Soil, Ground Water and Waste Water, Applied Natural Science, Inc, Argonne National Laboratory, Illionis,

Johnson ,R.A., and Wichern, D.W. 2002. Applied Multivariate Statistical Analysis, 6th ed. Prentice Hall, ISBN 0-13-187715-1.

Kasmiyati, S., E.B.E. Kristiani, M.M. Herawati, Potensi Tanaman Anggota Brassicaceae sebagai Agen Fitoekstraksi Logam Berat Krom pada Limbah Padat Tekstil. Laporan Hasil Penelitian Program Penelitian Dosen Muda, Program Fasilitasi Perguruan Tinggi Kopertis Wilayah VI Jawa Tengah, Tahun 2008.

Parhusip H. A., Determination Parameter by Nonlinear Least Square, Proceedings of 4th International Conference on Mathematics and Statistics (ICOMS 2009) , Universitas Malahayati Badar Lampung @MSMSSEA and Univ.Malahayati, 13-14 August 2009,page 295-304, ISSN 2085-7748.

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Parhusip H. A., dan Siska A. 2009. Principal Component Analysis (PCA) untuk Analisis Perlakukan Pemberian Pakan dan Mineral terhadap Produksi Susu Sapi, Prosiding Seminar Nasioanal Matematika UNPAR, ISSN 1907-3909, Vol 4, hal.AA 42-51.

Stoer, J and Bulirsch, R.,1993. Introduction to Numerical Analysis, Second Edition, Text in Applied Mathematics, 12, Springer-Verlag,New-York,Inc.

Suresh. B. and Ravishankar. G.A. 2004. Phytoremediation--a novel and promising

approach for environmental clean-up. Crit Rev Biotechnol 24(2-3):97-124.

Peressini, A.L., Sullivan, F.E.,Uhl, J.J., 1988. The Mathematics of Nonlinear Programming, Springer-Verlag, New-York.

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Figure 3. Comparison logistic model and data of crown diameter of the first Kai-lan (2-nd coloum of Table 2).

With the same procedure as before, we may compute the values of parameter K and k for each Kailan and the results are listed in Table 3.

Table 3. The values of parameters K and k for each Kailan

No K k

I 0.9844 3.6688

II 1.0064 3.9832

III 1567.1 0.7

IV 0.9610 4.5827

V 0.9266 4.0359

VI 0.9444 3.7926

The error is defined to guarantee how good the logistic model approximates the crown diameter as a function of time. The error is denoted by e and is defined by

% 100 x D

D D e

d l d −

= (1c)

where || .|| reads as ’norm’ and D_d := crown diameter given by data and D_l:= crown diameter given by logistic model. The Euclidian norm is used here (Stoer and Bulirsch, 1993).

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2.2 Geometric mean and Harmonic mean

Suppose we have data x₁,x₂,...,x_n and the geometric mean is denoted by xr_G which is defined by

n n i i G x x / 1

1 ⎟⎟⎠ ⎞ ⎜⎜ ⎝ ⎛ =

∏

= r (2a)

where _n

i xx x

x = ₁ ₂...

∏

. If we define

∑

= n

i xi

n h 1 1 1 1 where h

xr_H = 1 (2b)

then xr_H is called harmonic mean. The relation of arithmetic mean, geometric mean and harmonic mean is known (Peressini, et.all, 1988) as

H G

A x x

xr ≥ r ≥ r . (3)

The equal sign is satisfied if each value of the given data is the same. Equations (2a-2b) will be used to find the mean values of K and k.

3. Research Method

1. Data presented into dimensionless

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3. Using geometric mean and harmonic mean, find the best values of K and k for all plants.

4. Using the best parameter in step 3 to present the crown diameter as a logistic function.

5. Construct the covariance matrix of each data set of plant

6. Find the largest eigenvalue of each data set of plant and compute its corresponding eigenvector.

4. Result and Discussion

Using equation (2a) and (2b), we may obtain the mean values of parameters in Table 3 are also computed and shown in Table 4.

Table 4. Geometric mean and harmonic mean of parameter K and k. Parameter Geometric

mean

Harmonic mean

K 4.2891 1.3741

k 2.9554 2.1492

What about other tested plants ?

The logistic model is also tried to model the crown diameter of other tested plants such as Brassica oleracea and 2 plants from Asteraceae, i.e Chrysanthemum and sunflower (Helianthus annus). In the case of Chryanthemum, the model does not fit to the obtained data. One example is shown by Figure 5. We have also studied the sun flower. As a result, the logistic model is not a good model for its crown diameter. Therefore we can not proceed as in the case of Kailan.

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Figure 4a. Logistic model of Crown Diameter using geometric mean

Figure 4b. Logistic model of Crown Diameter using harmonic mean

1 2 3 4 5 6 7 8 9

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

i−th week

Crown diameter (no dimension)

data model

Figure 5. The logistic model compared to the crown diameter data of the first tested Chrysanthemum.

By principal component here means that we construct the covariance matrix of each set of data. By computing its eigenvalues and selecting the largest eigenvalue, we

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Conclusion and Remark

Aknowledgment : I thank to Kasmiyati, MSc (Biology Faculty, UKSW) who has provided me her data for this paper.

References

Aiyen. 2005. Ilmu Remediasi untuk Atasi Pencemaran Tanah di Aceh dan Sumatera Utara. http://www.kompas.com/kompas-cetak/0503/04/ilpeng/1592821.htm

Hinchman, R.R, Negri M. C., and Gatliff, E. G., 1998. Phytoremediation : Using Green Plants to Clean Up Contaminated Soil, Ground Water and Waste Water, Applied Natural Science, Inc, Argonne National Laboratory, Illionis,

Johnson ,R.A., and Wichern, D.W. 2002. Applied Multivariate Statistical Analysis, 6th ed. Prentice Hall, ISBN 0-13-187715-1.

Kasmiyati, S., E.B.E. Kristiani, M.M. Herawati, Potensi Tanaman Anggota Brassicaceae sebagai Agen Fitoekstraksi Logam Berat Krom pada Limbah Padat Tekstil. Laporan Hasil Penelitian Program Penelitian Dosen Muda, Program Fasilitasi Perguruan Tinggi Kopertis Wilayah VI Jawa Tengah, Tahun 2008.

Parhusip H. A., Determination Parameter by Nonlinear Least Square, Proceedings of 4th International Conference on Mathematics and Statistics (ICOMS 2009) , Universitas Malahayati Badar Lampung @MSMSSEA and Univ.Malahayati, 13-14 August 2009,page 295-304, ISSN 2085-7748.

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Parhusip H. A., dan Siska A. 2009. Principal Component Analysis (PCA) untuk Analisis Perlakukan Pemberian Pakan dan Mineral terhadap Produksi Susu Sapi, Prosiding Seminar Nasioanal Matematika UNPAR, ISSN 1907-3909, Vol 4, hal.AA 42-51.

Stoer, J and Bulirsch, R.,1993. Introduction to Numerical Analysis, Second Edition, Text in Applied Mathematics, 12, Springer-Verlag,New-York,Inc.

Suresh. B. and Ravishankar. G.A. 2004. Phytoremediation--a novel and promising approach for environmental clean-up. Crit Rev Biotechnol 24(2-3):97-124.

Peressini, A.L., Sullivan, F.E.,Uhl, J.J., 1988. The Mathematics of Nonlinear Programming, Springer-Verlag, New-York.

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