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Delineating soil nutrient management zones based on fuzzy clustering optimized by PSO
Mathematical and Computer Modelling 51 (2010) 1299–1305
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Delineating soil nutrient management zones based on fuzzy clustering optimized by PSO Qiang Fu ∗ , Zilong Wang, Qiuxiang Jiang College of Water Conservancy & Architecture, Northeast Agricultural University, Harbin, Heilongjiang Province, PR China
article
info
Keywords: Fuzzy clustering Management zone Particle swarm optimization Precision agriculture
abstract Delineation of soil nutrient management zones provides basis for variable fertilization technique and is the important link of variable fertilization management actualized in precision agriculture. After the spatial variability characteristics and structure of six soil nutrients were analyzed, they were taken as the variables to delineate soil nutrient management zones. The fuzzy clustering algorithm optimized by particle swarm optimization (PSO) was used to delineate management zones, and two indices were introduced to ascertain the reasonable number of management zones. According to the calculation, the reasonable number for the study area was 2. As single factor variance analysis was used to analyze soil nutrient data of the practical samples in each management zone, all soil nutrients but available phosphorus had great differences among management zones on the confidence level of 99%. Zone 1 had the higher soil fertility and zone 2 had the lower. The delineation result indicated that fuzzy clustering optimized by PSO had a good performance on delineating management zones and variable fertilization management was feasible in the study area. © 2009 Elsevier Ltd. All rights reserved.
0. Introduction Precision agriculture, firstly advanced by American agriculturalists at the beginning of 1990s [1], is a modern agricultural production mode and a technical system synthetically applying modern high and new technologies, such as 3S (RS, GIS and GPS), artificial intelligence etc., to obtain high yield, high quality and high efficiency. Variable-rate fertilization is the key technique and means to realize precision agriculture, but traditional variable-rate fertilization is mainly based on grid sampling method to obtain information, which is expensive and not practical. In recent years, many scholars put their efforts on the researches to divide a field into several relatively homogeneous sub-zones on the basis of the spatial variability and location of soil nutrient, namely, to delineate soil nutrient management zones. Scientific and rational technique for delineating soil nutrient management zones is an efficient means to carry out variable-rate fertilization for precision agriculture [2] and becomes a hotspot of precision agriculture research abroad and in home. Schepers et al. [3] studied the ability of the management zone to characterize spatial variability on the basis of soil characteristics and yield over many years. Fleming et al. [4] estimated the effect of management zone defined by farmers based on remote sensing image, terrain and production experience on variable-rate fertilization. Li et al. [5] advanced spatial continuous clustering algorithm based on K -mean algorithm to extract optimal cultivation management zone aiming at the growth vigor of wheat. Huang et al. [6] delineated soil nutrient management zones in GIS on the basis of spatial distribution of soil available nutrient in field and guided fertilization management with balanced fertilization technique. Li et al. [7] divided saline and alkaline land in the coastal area into management zones using soil electrical conductivity data and clustering
∗ Corresponding address: College of Water Conservancy & Architecture, Northeast Agricultural University, Harbin 150030, Heilongjiang Province, PR China. Tel.: +86 451 55191294. E-mail address: [email protected] (Q. Fu). 0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.10.034
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Q. Fu et al. / Mathematical and Computer Modelling 51 (2010) 1299–1305
Fig. 1. Distribution of sampling points.
analysis and studied the feasibility of practicing site-specific management. Fuzzy clustering is the mostly used method in delineating management zone, but its optimal problem is usually solved by iterative minimizing method which has high sensitivity to iterative initial value. Thus, particle swarm optimization was performed to optimize the objective function of fuzzy clustering to improve the disadvantage and to obtain more precise delineating result. 1. Materials and methods 1.1. Study site and data acquisition Honghe Farm of Agricultural Reclamation Jiangsanjiang branch bureau of Heilongjiang Province located at the hinterland of Sanjiang Plain was selected as the study site. The annual mean temperature of study site is 1.3 ◦ C with frostless period of 120 days and effective accumulated temperature of 2400 ◦ C. The main soil type includes planosol, meadow soil, swamp soil and peaty soil, and the main crop is paddy. The landform, soil type, parent material and land use pattern of the study site have good representation for Sanjiang Plain. A paddy field of 25 ha in Honghe Farm was studied and the data was acquired after rice harvesting. One representative sample was collected in the depth of 20 cm and geo-referenced using a Trimble global positioning system (GPS). The total number of samples is 100 and Fig. 1 shows the distribution of sampling points. Through the course of air-drying, grinding and sifting, the soil samples was measured by general soil agricultural chemistry analysis methods [8] to determine total nitrogen (TN), total phosphorus (TP) organic matter (OM), available nitrogen (AN), available phosphorus (AP) and available potassium (AK). 1.2. Fuzzy clustering algorithm Fuzzy clustering is a better classification method for things with fuzziness [9], and an unsupervised clustering with no classification criterion was used in the study. Its basic idea is to seek the minimum of objective function and the objective function applied in delineating soil nutrient management zone can be expressed as follows: K X N X
J (µ, m) =
(µnk )r kxn − mk k
(1)
k=1 n=1
where: K is the number of soil nutrient management sub-zones; N is the number of soil samples; µnk is the fuzzy memberPK ship of the nth soil sample belonging to the kth management sub-zone ( k=1 µnk = 1); r is fuzzy weighted index (1 ≤ r ≤ +∞); xn is the nth soil sample and mk is the clustering center of the kth management sub-zone. The following equation can be used to calculate clustering center for the initial fuzzy classification matrix. N P
mk =
(µnk )m xk
n=1 N P
. (µnk )m
n =1
(2)
Q. Fu et al. / Mathematical and Computer Modelling 51 (2010) 1299–1305
1301
The fuzzy classification matrix can be modified on the basis of the new clustering center by the following equation.
" µnk =
2/(m−1) K X dnk j =1
#−1
djk
.
(3)
According to the upwards equation, we optimized the objective function and obtained the corresponding fuzzy membership µ and clustering center m, then the classification of soil nutrient can be determined on the basis of fuzzy clustering theory. 1.3. Particle swarm algorithm Particle swarm optimization (PSO) is an evolutionary technique invented from the researches of predation behavior of bird group [10]. PSO has both global and local searching ability with simple parameter adjustment and fast convergence, can solve the problem of the sensitivity of iterative method to initial value and avoids that clustering algorithm gets into local optimum. For optimization problem, each particle in PSO algorithm represents a possible solution. The best position of each particle through the course of optimization is the best solution found by the particle, and then the best position experienced by the whole group is the best solution found by the whole group presently. The former is called as individual extremum (pbest) and the latter is called as global extremum (gbest). Every particle constantly updates itself through pbest and gbest to create a new population and the whole population comprehensively searches the solution region. Set the particle population size as N. The position of the ith particle (i = 1, 2, . . . , N ) can be expressed as xi , speed as vi and individual extremum as pbesti . Thus, any particle i can update its own position and speed by the following equations:
v(t + 1) = wvi (t ) + c1 r1 (t )(pbesti (t ) − xi (t )) + c2 r2 (t )(gbest (t ) − xi (t )) xi (t + 1) = xi (t ) + vi (t + 1)
(4) (5)
where: c1 and c2 are constants and called as acceleration factor; r1 and r2 are random numbers changing in the interval of (0, 1) and w is inertia weight. Then the individual extremum of each particle and global extremum of the whole particles can update by the equations as follows: pbesti (t + 1) =
xi (t + 1) pbesti (t )
xi (t + 1) ≥ pbesti (t ) xi (t + 1) < pbesti (t )
gbest (t + 1) = max(pbesti (t + 1))
i = 1, 2, . . . , N .
(6) (7)
The upwards steps make up the principal part of PSO. Soil nutrient management zone can be defined by combining PSO with fuzzy clustering to optimize the objective function and to find the global extremum gbest of particle population. 1.4. Determining rational sub-zone number Rational sub-zone number is a transparent partition that clustering algorithm gives by using the best clustering result on the basis of objective data. Two indices, separation coefficient (F ) and separation entropy (H ), were introduced in the study to determine rational sub-zone number. Separation coefficient (F ) is the approaching degree of all soil samples relative to the clustering centers of soil management sub-zones and can be expressed as: F (µ, K ) = 1 −
K
"
K −1
1−
N K 1 XX
N n=1 k=1
#
(µnk )
2
.
(8)
The value of F is ranging between 0 and 1. If the value approaches 0, clustering has less shared data and class partition is clear. If the value approaches 1, clustering has more shared data and class partition is not obvious. Separation entropy H can be defined as: H (µ, K ) = −
N K 1 XX
N n =1 k =1
µnk log(µnk ).
(9)
If all fuzzy memberships (µnk ) approach 0 or 1, the entropy is small and management zone divides the soil samples obviously. The clustering effect is good. Contrarily, if µnk approaches 0.5, the entropy is big and partition is not transparent. Clustering effect is bad.
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Table 1 Classical statistics of soil nutrient for the study site. Statistics
TN (%)
TP (%)
OM (%)
AN (mg/kg)
AP (mg/kg)
AK (mg/kg)
Mean Median Minimum Maximum SDa CV
0.24 0.22 0.10 0.57 0.08 0.35
0.09 0.08 0.04 0.17 0.02 0.25
5.30 5.11 2.37 11.14 1.72 0.32
246.15 237.95 112.70 430.90 59.84 0.24
56.84 56.60 13.00 121.00 16.48 0.29
79.53 62.60 15.40 573.00 62.83 0.79
a
SD, standard deviation; CV, coefficient of variation.
Table 2 Semi-variogram model and parameters of soil nutrient. Soil nutrient
Model
Nugget
Sill
Nugget/Sill (%)
Range (m)
R2
TN TP OM AN AP AK
Exponential Exponential Exponential Exponential Exponential Exponential
0.0277 0.0002 0.0404 0.0176 74.9 0.0475
0.0962 0.00043 0.097 0.0591 253.7 0.178
28.8 46.5 41.6 29.8 29.5 26.7
157.5 339.3 233.8 235.8 206.1 137.1
0.985 0.976 0.998 0.98 0.995 0.766
2. Results and analysis 2.1. Classical statistics Table 1 lists the classical statistics results of soil nutrient measurements for the study site. According to soil nutrient classification criteria provided by Shen [11], the mean value of soil nutrient in topsoil of study site showed high quantity, and OM had the highest content level relatively. The coefficients of variation of all soil nutrient belonged to moderate variation which CV were ranging between 10% and 100%, and AK had the biggest CV of 79%. The values of CV indicated that soil nutrient in the study site had big spatial variation, which satisfied preconditions for soil nutrient management zone and variable-rate fertilization management. 2.2. Spatial variability structure analysis and Kriging interpolation Table 2 lists the semi-variogram fitting model and parameters of soil nutrient in study site. All theoretical models of soil nutrient were exponential models and had high fitting precision (R2 ), which indicated they all had favorable spatial structure. The ratio of nugget/sill shows the variation degree of soil nutrient. The ratios of nugget/sill of six soil nutrients were ranging between 26.7% and 46.5% belonging to moderate spatial variation (25%–75%). The range of soil nutrient changing from 137.1 m to 339.3 m was larger than the smallest sampling distance (50 m), which illuminated that sampling scheme can satisfy the requirement of spatial variability structure analysis of soil nutrient in the study area. Ordinary Kriging (OK) of Geostatistics Analyst tool in the ArcGIS 9.0 software was conducted to interpolate the nonsampling locations and to delineate spatial distributions for the soil nutrient. Then the spatial distribution maps were transformed into grid images showed in Fig. 2. Through analyzing the spatial distributions of soil nutrient, we found that TN and AN had consistent spatial distribution with low content at southeast and west small region and with high content at northwest and southern middle region, the spatial distributions of TP and AP were with low content at northeast region and with high content at southwest region in the main, and the contents of OM and AK had the trend increasing from southeast to northwest gradually. From the global view, the spatial distribution of soil nutrient with block and band shapes was suitable to delineate soil nutrient management zone. 2.3. Delineating results and evaluation Fuzzy clustering optimized by PSO was programmed on the basis of MATLAB 7.0 software. The attribute data of raster images of soil nutrient spatial distributions were taken as the input of the program for clustering analysis. During the course of clustering, set the maximum updating number of particle swarm, convergence threshold and fuzzy weighted index as 100, 0.001 and 2, respectively. To obtain rational soil nutrient management zone, we divided the study area into 2, 3, 4, 5 and 6 sub-zone respectively and calculated the two indices (F and H) under different partition to determine rational partition number. The results are showed in Fig. 3. The trend graph illuminated that 2 is the rational partition number for the two indices reach the smallest value when the study area are divided into two sub-zones. Then we took the partition results and spatial locations under 2 sub-zones as data source and inputted them back into ArcGIS 9.0 software to generate soil nutrient management zone map showed in Fig. 4. The management zone map illuminated that the study area was obviously divided into two sub-zones (Zone 1 and Zone 2). Zone 1 located at the north
Q. Fu et al. / Mathematical and Computer Modelling 51 (2010) 1299–1305
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Fig. 2. Raster images of soil nutrient spatial distributions.
Fig. 3. Changes in indices with increasing number of management zones.
side and south middle small region, while Zone 2 at most region of south side. Each sub-zone is concentrative, so easy for variable-rate fertilization machine to work. To estimate if the partition result can effectively characterize the spatial variability of soil nutrient in the study area, mean statistics and single factor variance analysis were conducted on 100 measurements of the study site and Table 3 lists the results. The analysis results indicated that measurements of all soil nutrients, except for AP, had significant different between sub-zones under a confidence level of 99%, so soil management zone can characterize spatial variability of soil nutrient effectively. Zone 1 has high soil fertility level while Zone 2 has low soil fertility level. In a word, after partition,
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Q. Fu et al. / Mathematical and Computer Modelling 51 (2010) 1299–1305
Fig. 4. Soil nutrient management zone map.
Table 3 Mean and single factor variance analysis of soil nutrient. Sub-zone
Zone 1 Zone 2 Significance
Soil nutrient TN (%)
TP (%)
OM (%)
AN (mg/kg)
AP (mg/kg)
AK (mg/kg)
0.34 0.21 Very
0.10 0.08 Very
7.44 4.72 Very
328.18 224.28 Very
48.28 58.92 Not
114.61 66.34 Very
spatial distribution difference of soil nutrient is small in the same sub-zone and is significant between sub-zones, namely, the same fertilization is feasible in a sub-zone for the study site and variable-rate fertilization management can be carried out between sub-zones. The mean values of soil nutrient in each sub-zone can be used as a reference for variable-rate fertilization. 3. Conclusions The spatial variability and spatial variable structure of six soil nutrients shows moderate spatial variability and good spatial structure. The spatial distributions of soil nutrient generated by OK method reveal block and band shapes. The study area is feasible to delineate management zone. Through the procedure of fuzzy clustering optimized by PSO, the study area was divided into several sub-zones and the two performance indices determined the rational partition number was 2. The mean statistics and single factor variance analysis indicated that management zone characterized the spatial variability of soil nutrient effectively and differences between sub-zones were significant, except for AP. The fuzzy clustering algorithm is stable and PSO is simple with easy parameter adjustment and both global and local searching ability, so we combine the two methods to delineate soil nutrient management zone for the purposes of providing methods for variable-rate fertilization and agricultural management zone partition, which has important theoretical and practical significance for precision agriculture. Acknowledgement The project was supported by the Science & Technology Tackle Key Problem Program of Heilongjiang (No. GB06B106-7). References [1] W.L. Peng, P. Robert, H.X. Cheng, Development of agricultural information technology and precision agriculture, Transactions of the Chinese Society of Agricultural Engineering 17 (2) (2001) 9–11. [2] R. Khosla, K. Fleming, J.A. Delgado, et al., Use of site-specific management zones to improve nitrogen management for precision agriculture, Journal of Soil and Water Conservation 57 (6) (2000) 513–518. [3] A.R. Schepers, J.F. Shanahan, M.A. Liebig, et al., Appropriateness of management zones for characterizing spatial variability of soil properties and irrigated corn yields across years, Agronomy Journal 96 (1) (2004) 195–203. [4] K.L. Fleming, D.G. Westfall, D.W. Wiens, et al., Evaluating farmer defined management zone maps for variable rate fertilizer application, Precision Agriculture 2 (2000) 201–215. [5] X. Li, Y.C. Pan, C.J. Zhao, et al., Delineating precision agriculture management zones based on spatial contiguous clustering algorithm, Transactions of the Chinese Society of Agricultural Engineering 21 (8) (2005) 78–82. [6] S.W. Huang, J.Y. Jin, L.P. Yang, et al., Spatial variability and regionalized management of soil nutrients in the grain crop region in Yutian County, Acta Pedologica Sinica 40 (1) (2003) 79–88. [7] Y. Li, Z. Shi, F. Li, Delineation of site-specific management zones based on temporal and spatial variability of soil, Pedosphere 17 (2) (2007) 156–164.
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[8] S.D. Bao, Soil Agricultural Chemistry Analysis, Agriculture Press, Beijing, China, 2000. [9] J.J. Xie, C.P. Liu, Fuzzy Mathematics Method and its Application, Huazhong University of Science and Technology Press, Wuhan, 2000. [10] J. Kennedy, R.C. Eberhart, Particle swarm optimization, in: Proceedings of IEEE International Conference on Neural Networks, IEEE Press, Piscataway, NJ, 1995, pp. 1942–1948. [11] H. Shen, The selection of participating factors in soil evaluation and the determination of gradation indices, Acta Agriculturae Boreali-Sinica 5 (3) (1990) 63–69.
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Delineating soil nutrient management zones based on fuzzy clustering optimized by PSO Qiang Fu ∗ , Zilong Wang, Qiuxiang Jiang College of Water Conservancy & Architecture, Northeast Agricultural University, Harbin, Heilongjiang Province, PR China
article
info
Keywords: Fuzzy clustering Management zone Particle swarm optimization Precision agriculture
abstract Delineation of soil nutrient management zones provides basis for variable fertilization technique and is the important link of variable fertilization management actualized in precision agriculture. After the spatial variability characteristics and structure of six soil nutrients were analyzed, they were taken as the variables to delineate soil nutrient management zones. The fuzzy clustering algorithm optimized by particle swarm optimization (PSO) was used to delineate management zones, and two indices were introduced to ascertain the reasonable number of management zones. According to the calculation, the reasonable number for the study area was 2. As single factor variance analysis was used to analyze soil nutrient data of the practical samples in each management zone, all soil nutrients but available phosphorus had great differences among management zones on the confidence level of 99%. Zone 1 had the higher soil fertility and zone 2 had the lower. The delineation result indicated that fuzzy clustering optimized by PSO had a good performance on delineating management zones and variable fertilization management was feasible in the study area. © 2009 Elsevier Ltd. All rights reserved.
0. Introduction Precision agriculture, firstly advanced by American agriculturalists at the beginning of 1990s [1], is a modern agricultural production mode and a technical system synthetically applying modern high and new technologies, such as 3S (RS, GIS and GPS), artificial intelligence etc., to obtain high yield, high quality and high efficiency. Variable-rate fertilization is the key technique and means to realize precision agriculture, but traditional variable-rate fertilization is mainly based on grid sampling method to obtain information, which is expensive and not practical. In recent years, many scholars put their efforts on the researches to divide a field into several relatively homogeneous sub-zones on the basis of the spatial variability and location of soil nutrient, namely, to delineate soil nutrient management zones. Scientific and rational technique for delineating soil nutrient management zones is an efficient means to carry out variable-rate fertilization for precision agriculture [2] and becomes a hotspot of precision agriculture research abroad and in home. Schepers et al. [3] studied the ability of the management zone to characterize spatial variability on the basis of soil characteristics and yield over many years. Fleming et al. [4] estimated the effect of management zone defined by farmers based on remote sensing image, terrain and production experience on variable-rate fertilization. Li et al. [5] advanced spatial continuous clustering algorithm based on K -mean algorithm to extract optimal cultivation management zone aiming at the growth vigor of wheat. Huang et al. [6] delineated soil nutrient management zones in GIS on the basis of spatial distribution of soil available nutrient in field and guided fertilization management with balanced fertilization technique. Li et al. [7] divided saline and alkaline land in the coastal area into management zones using soil electrical conductivity data and clustering
∗ Corresponding address: College of Water Conservancy & Architecture, Northeast Agricultural University, Harbin 150030, Heilongjiang Province, PR China. Tel.: +86 451 55191294. E-mail address: [email protected] (Q. Fu). 0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.10.034
1300
Q. Fu et al. / Mathematical and Computer Modelling 51 (2010) 1299–1305
Fig. 1. Distribution of sampling points.
analysis and studied the feasibility of practicing site-specific management. Fuzzy clustering is the mostly used method in delineating management zone, but its optimal problem is usually solved by iterative minimizing method which has high sensitivity to iterative initial value. Thus, particle swarm optimization was performed to optimize the objective function of fuzzy clustering to improve the disadvantage and to obtain more precise delineating result. 1. Materials and methods 1.1. Study site and data acquisition Honghe Farm of Agricultural Reclamation Jiangsanjiang branch bureau of Heilongjiang Province located at the hinterland of Sanjiang Plain was selected as the study site. The annual mean temperature of study site is 1.3 ◦ C with frostless period of 120 days and effective accumulated temperature of 2400 ◦ C. The main soil type includes planosol, meadow soil, swamp soil and peaty soil, and the main crop is paddy. The landform, soil type, parent material and land use pattern of the study site have good representation for Sanjiang Plain. A paddy field of 25 ha in Honghe Farm was studied and the data was acquired after rice harvesting. One representative sample was collected in the depth of 20 cm and geo-referenced using a Trimble global positioning system (GPS). The total number of samples is 100 and Fig. 1 shows the distribution of sampling points. Through the course of air-drying, grinding and sifting, the soil samples was measured by general soil agricultural chemistry analysis methods [8] to determine total nitrogen (TN), total phosphorus (TP) organic matter (OM), available nitrogen (AN), available phosphorus (AP) and available potassium (AK). 1.2. Fuzzy clustering algorithm Fuzzy clustering is a better classification method for things with fuzziness [9], and an unsupervised clustering with no classification criterion was used in the study. Its basic idea is to seek the minimum of objective function and the objective function applied in delineating soil nutrient management zone can be expressed as follows: K X N X
J (µ, m) =
(µnk )r kxn − mk k
(1)
k=1 n=1
where: K is the number of soil nutrient management sub-zones; N is the number of soil samples; µnk is the fuzzy memberPK ship of the nth soil sample belonging to the kth management sub-zone ( k=1 µnk = 1); r is fuzzy weighted index (1 ≤ r ≤ +∞); xn is the nth soil sample and mk is the clustering center of the kth management sub-zone. The following equation can be used to calculate clustering center for the initial fuzzy classification matrix. N P
mk =
(µnk )m xk
n=1 N P
. (µnk )m
n =1
(2)
Q. Fu et al. / Mathematical and Computer Modelling 51 (2010) 1299–1305
1301
The fuzzy classification matrix can be modified on the basis of the new clustering center by the following equation.
" µnk =
2/(m−1) K X dnk j =1
#−1
djk
.
(3)
According to the upwards equation, we optimized the objective function and obtained the corresponding fuzzy membership µ and clustering center m, then the classification of soil nutrient can be determined on the basis of fuzzy clustering theory. 1.3. Particle swarm algorithm Particle swarm optimization (PSO) is an evolutionary technique invented from the researches of predation behavior of bird group [10]. PSO has both global and local searching ability with simple parameter adjustment and fast convergence, can solve the problem of the sensitivity of iterative method to initial value and avoids that clustering algorithm gets into local optimum. For optimization problem, each particle in PSO algorithm represents a possible solution. The best position of each particle through the course of optimization is the best solution found by the particle, and then the best position experienced by the whole group is the best solution found by the whole group presently. The former is called as individual extremum (pbest) and the latter is called as global extremum (gbest). Every particle constantly updates itself through pbest and gbest to create a new population and the whole population comprehensively searches the solution region. Set the particle population size as N. The position of the ith particle (i = 1, 2, . . . , N ) can be expressed as xi , speed as vi and individual extremum as pbesti . Thus, any particle i can update its own position and speed by the following equations:
v(t + 1) = wvi (t ) + c1 r1 (t )(pbesti (t ) − xi (t )) + c2 r2 (t )(gbest (t ) − xi (t )) xi (t + 1) = xi (t ) + vi (t + 1)
(4) (5)
where: c1 and c2 are constants and called as acceleration factor; r1 and r2 are random numbers changing in the interval of (0, 1) and w is inertia weight. Then the individual extremum of each particle and global extremum of the whole particles can update by the equations as follows: pbesti (t + 1) =
xi (t + 1) pbesti (t )
xi (t + 1) ≥ pbesti (t ) xi (t + 1) < pbesti (t )
gbest (t + 1) = max(pbesti (t + 1))
i = 1, 2, . . . , N .
(6) (7)
The upwards steps make up the principal part of PSO. Soil nutrient management zone can be defined by combining PSO with fuzzy clustering to optimize the objective function and to find the global extremum gbest of particle population. 1.4. Determining rational sub-zone number Rational sub-zone number is a transparent partition that clustering algorithm gives by using the best clustering result on the basis of objective data. Two indices, separation coefficient (F ) and separation entropy (H ), were introduced in the study to determine rational sub-zone number. Separation coefficient (F ) is the approaching degree of all soil samples relative to the clustering centers of soil management sub-zones and can be expressed as: F (µ, K ) = 1 −
K
"
K −1
1−
N K 1 XX
N n=1 k=1
#
(µnk )
2
.
(8)
The value of F is ranging between 0 and 1. If the value approaches 0, clustering has less shared data and class partition is clear. If the value approaches 1, clustering has more shared data and class partition is not obvious. Separation entropy H can be defined as: H (µ, K ) = −
N K 1 XX
N n =1 k =1
µnk log(µnk ).
(9)
If all fuzzy memberships (µnk ) approach 0 or 1, the entropy is small and management zone divides the soil samples obviously. The clustering effect is good. Contrarily, if µnk approaches 0.5, the entropy is big and partition is not transparent. Clustering effect is bad.
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Q. Fu et al. / Mathematical and Computer Modelling 51 (2010) 1299–1305
Table 1 Classical statistics of soil nutrient for the study site. Statistics
TN (%)
TP (%)
OM (%)
AN (mg/kg)
AP (mg/kg)
AK (mg/kg)
Mean Median Minimum Maximum SDa CV
0.24 0.22 0.10 0.57 0.08 0.35
0.09 0.08 0.04 0.17 0.02 0.25
5.30 5.11 2.37 11.14 1.72 0.32
246.15 237.95 112.70 430.90 59.84 0.24
56.84 56.60 13.00 121.00 16.48 0.29
79.53 62.60 15.40 573.00 62.83 0.79
a
SD, standard deviation; CV, coefficient of variation.
Table 2 Semi-variogram model and parameters of soil nutrient. Soil nutrient
Model
Nugget
Sill
Nugget/Sill (%)
Range (m)
R2
TN TP OM AN AP AK
Exponential Exponential Exponential Exponential Exponential Exponential
0.0277 0.0002 0.0404 0.0176 74.9 0.0475
0.0962 0.00043 0.097 0.0591 253.7 0.178
28.8 46.5 41.6 29.8 29.5 26.7
157.5 339.3 233.8 235.8 206.1 137.1
0.985 0.976 0.998 0.98 0.995 0.766
2. Results and analysis 2.1. Classical statistics Table 1 lists the classical statistics results of soil nutrient measurements for the study site. According to soil nutrient classification criteria provided by Shen [11], the mean value of soil nutrient in topsoil of study site showed high quantity, and OM had the highest content level relatively. The coefficients of variation of all soil nutrient belonged to moderate variation which CV were ranging between 10% and 100%, and AK had the biggest CV of 79%. The values of CV indicated that soil nutrient in the study site had big spatial variation, which satisfied preconditions for soil nutrient management zone and variable-rate fertilization management. 2.2. Spatial variability structure analysis and Kriging interpolation Table 2 lists the semi-variogram fitting model and parameters of soil nutrient in study site. All theoretical models of soil nutrient were exponential models and had high fitting precision (R2 ), which indicated they all had favorable spatial structure. The ratio of nugget/sill shows the variation degree of soil nutrient. The ratios of nugget/sill of six soil nutrients were ranging between 26.7% and 46.5% belonging to moderate spatial variation (25%–75%). The range of soil nutrient changing from 137.1 m to 339.3 m was larger than the smallest sampling distance (50 m), which illuminated that sampling scheme can satisfy the requirement of spatial variability structure analysis of soil nutrient in the study area. Ordinary Kriging (OK) of Geostatistics Analyst tool in the ArcGIS 9.0 software was conducted to interpolate the nonsampling locations and to delineate spatial distributions for the soil nutrient. Then the spatial distribution maps were transformed into grid images showed in Fig. 2. Through analyzing the spatial distributions of soil nutrient, we found that TN and AN had consistent spatial distribution with low content at southeast and west small region and with high content at northwest and southern middle region, the spatial distributions of TP and AP were with low content at northeast region and with high content at southwest region in the main, and the contents of OM and AK had the trend increasing from southeast to northwest gradually. From the global view, the spatial distribution of soil nutrient with block and band shapes was suitable to delineate soil nutrient management zone. 2.3. Delineating results and evaluation Fuzzy clustering optimized by PSO was programmed on the basis of MATLAB 7.0 software. The attribute data of raster images of soil nutrient spatial distributions were taken as the input of the program for clustering analysis. During the course of clustering, set the maximum updating number of particle swarm, convergence threshold and fuzzy weighted index as 100, 0.001 and 2, respectively. To obtain rational soil nutrient management zone, we divided the study area into 2, 3, 4, 5 and 6 sub-zone respectively and calculated the two indices (F and H) under different partition to determine rational partition number. The results are showed in Fig. 3. The trend graph illuminated that 2 is the rational partition number for the two indices reach the smallest value when the study area are divided into two sub-zones. Then we took the partition results and spatial locations under 2 sub-zones as data source and inputted them back into ArcGIS 9.0 software to generate soil nutrient management zone map showed in Fig. 4. The management zone map illuminated that the study area was obviously divided into two sub-zones (Zone 1 and Zone 2). Zone 1 located at the north
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Fig. 2. Raster images of soil nutrient spatial distributions.
Fig. 3. Changes in indices with increasing number of management zones.
side and south middle small region, while Zone 2 at most region of south side. Each sub-zone is concentrative, so easy for variable-rate fertilization machine to work. To estimate if the partition result can effectively characterize the spatial variability of soil nutrient in the study area, mean statistics and single factor variance analysis were conducted on 100 measurements of the study site and Table 3 lists the results. The analysis results indicated that measurements of all soil nutrients, except for AP, had significant different between sub-zones under a confidence level of 99%, so soil management zone can characterize spatial variability of soil nutrient effectively. Zone 1 has high soil fertility level while Zone 2 has low soil fertility level. In a word, after partition,
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Fig. 4. Soil nutrient management zone map.
Table 3 Mean and single factor variance analysis of soil nutrient. Sub-zone
Zone 1 Zone 2 Significance
Soil nutrient TN (%)
TP (%)
OM (%)
AN (mg/kg)
AP (mg/kg)
AK (mg/kg)
0.34 0.21 Very
0.10 0.08 Very
7.44 4.72 Very
328.18 224.28 Very
48.28 58.92 Not
114.61 66.34 Very
spatial distribution difference of soil nutrient is small in the same sub-zone and is significant between sub-zones, namely, the same fertilization is feasible in a sub-zone for the study site and variable-rate fertilization management can be carried out between sub-zones. The mean values of soil nutrient in each sub-zone can be used as a reference for variable-rate fertilization. 3. Conclusions The spatial variability and spatial variable structure of six soil nutrients shows moderate spatial variability and good spatial structure. The spatial distributions of soil nutrient generated by OK method reveal block and band shapes. The study area is feasible to delineate management zone. Through the procedure of fuzzy clustering optimized by PSO, the study area was divided into several sub-zones and the two performance indices determined the rational partition number was 2. The mean statistics and single factor variance analysis indicated that management zone characterized the spatial variability of soil nutrient effectively and differences between sub-zones were significant, except for AP. The fuzzy clustering algorithm is stable and PSO is simple with easy parameter adjustment and both global and local searching ability, so we combine the two methods to delineate soil nutrient management zone for the purposes of providing methods for variable-rate fertilization and agricultural management zone partition, which has important theoretical and practical significance for precision agriculture. Acknowledgement The project was supported by the Science & Technology Tackle Key Problem Program of Heilongjiang (No. GB06B106-7). References [1] W.L. Peng, P. Robert, H.X. Cheng, Development of agricultural information technology and precision agriculture, Transactions of the Chinese Society of Agricultural Engineering 17 (2) (2001) 9–11. [2] R. Khosla, K. Fleming, J.A. Delgado, et al., Use of site-specific management zones to improve nitrogen management for precision agriculture, Journal of Soil and Water Conservation 57 (6) (2000) 513–518. [3] A.R. Schepers, J.F. Shanahan, M.A. Liebig, et al., Appropriateness of management zones for characterizing spatial variability of soil properties and irrigated corn yields across years, Agronomy Journal 96 (1) (2004) 195–203. [4] K.L. Fleming, D.G. Westfall, D.W. Wiens, et al., Evaluating farmer defined management zone maps for variable rate fertilizer application, Precision Agriculture 2 (2000) 201–215. [5] X. Li, Y.C. Pan, C.J. Zhao, et al., Delineating precision agriculture management zones based on spatial contiguous clustering algorithm, Transactions of the Chinese Society of Agricultural Engineering 21 (8) (2005) 78–82. [6] S.W. Huang, J.Y. Jin, L.P. Yang, et al., Spatial variability and regionalized management of soil nutrients in the grain crop region in Yutian County, Acta Pedologica Sinica 40 (1) (2003) 79–88. [7] Y. Li, Z. Shi, F. Li, Delineation of site-specific management zones based on temporal and spatial variability of soil, Pedosphere 17 (2) (2007) 156–164.
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[8] S.D. Bao, Soil Agricultural Chemistry Analysis, Agriculture Press, Beijing, China, 2000. [9] J.J. Xie, C.P. Liu, Fuzzy Mathematics Method and its Application, Huazhong University of Science and Technology Press, Wuhan, 2000. [10] J. Kennedy, R.C. Eberhart, Particle swarm optimization, in: Proceedings of IEEE International Conference on Neural Networks, IEEE Press, Piscataway, NJ, 1995, pp. 1942–1948. [11] H. Shen, The selection of participating factors in soil evaluation and the determination of gradation indices, Acta Agriculturae Boreali-Sinica 5 (3) (1990) 63–69.