Parameter estimation of S-shaped growth model: A modified particle swarm algorithm

2012, Vol.17 No.2, 137-143

Article ID 1007-1202(2012)02-0137-07

DOI 10.1007/s11859-012-0818-3

Parameter Estimation of S-Shaped Growth Model: A Modified Particle Swarm Algorithm

□ XU Xing1, WEI Bo2, WU Yu2, LIU Bingxiang1,

LI Yuanxiang2

1. College of Information and Engineering, Jingdezhen Ceramic

Institute, Jingdezhen 333403, Jiangxi, China;

2. School of Computer, Wuhan University, Wuhan 430072,

Hubei, China

© Wuhan University and Springer-Verlag Berlin Heidelberg 2012

Abstract: Parameter estimation plays a critical role for the appli-cation and development of S-shaped growth model in the agricul-tural sciences and others. In this paper, a modified particle swarm optimization algorithm based on the diffusion phenomenon (DPPSO) was employed to estimate the parameters for this model. Under the sense of least squares, the parameter estimation problem of S-shaped growth model, taking the Gompertz and Logistic models for example, is transformed into a multi-dimensional func-tion optimization problem. The results show that the DPPSO algo-rithm can effectively estimate the parameters of the S-shaped growth model. Key words: particle swarm optimization; diffusion phenomenon; parameter estimation; S-shaped growth model CLC number: TP 301.6 Received date: 2011-07-12 Foundation item: Supported by the National Natural Science Foundation of China (61070009), the National Science and Technology Support Plan (2012BAH25F02), the Project of Jingdezhen Science and Technology Bureau (2011-1-47), the National Natural Science Foundation of Jiangxi Province (2009GZS0065), and the Youth Science Foundation of Jiangxi Provincial De-partment of Education (GJJ12514) Biography: XU Xing, male, Lecturer, research direction: evolutionary com-putation and swarm intelligence. E-mail: [email protected]

0 Introduction

The S-shaped growth model is a very commonly used nonlinear model in sociology, biostatistics, educa-tional measurement, economy and other areas. The pa-rameter of the S-shaped growth model is the key factor for the application and development of the model. Wang et al[1] estimated the parameters of the S-curve by the opti-mization regression combing method, and this method re-quires calculating partial derivatives. However, for the pa-rameter estimation of nonlinear models, traditional algo-rithms are usually effective only for a particular type of problem, and more or less dependent on the expressions of models.

Particle swarm optimization (PSO), which is enlight-ened by artificial life and social behavior of bird flocking or fish schooling, was initiated and developed by Kennedy et al [2,3]. There is no requirement about the concrete expres-sion of the model in the running process of PSO algorithm. Therefore, PSO is also used for parameter estimation of nonlinear models. PSO effectively estimated the immeas-urable parameters of chaotic systems[4,5]. Gill et al[6] pro-posed a multi-objective PSO algorithm and estimated a hydrology parameter by this algorithm. Awadallah[7] ap-plied hybrid PSO algorithm combined with genetic algo-rithm (GA) to estimate the six equivalent circuit parame-ters of three-phase induction machines from the nameplate data for steady-state analysis. Pagano et al[8] obtained the optimal parameters of the autocatalytic model by PSO. Schwaab et al[9] introduced PSO to solve parameter esti-mation problems in chemical engineering and optimize the

Wuhan University Journal of Natural Sciences 2012, Vol.17 No.2 138

confidence region of parameter estimation. There were also some researches on estimating parameters of the S-shaped growth model by particle swarm optimization. Liu et al [10] estimated the parameters of Logistic curve based on improved PSO method, and the method revised the change strategy of the inertia weight. They did not analyze and discuss the factors that affect the perform-ance of the algorithm in this paper, and experimental data only contains the actual data. Xu et al[11] adopted the standard PSO algorithm to obtain parameters estimation of the Logistic model; however, the convergence speed of the standard PSO algorithm is relatively slow and only the Logistic model is considered in Refs. [10,11].

Diffusion and migration phenomena are very com-mon in nature, and there are also similar migration phe-nomena in human society. Simulating diffusion and mi-gration mechanism, many researchers proposed a list of improved algorithms, and used these algorithms to deal with biological sequence motif detection, single-modal and multi-modal function optimization problems [12-14]. The experimental results show that the diffusion mecha-nism can significantly speed up the convergence rate of the algorithm, improve the global search capability of the algorithm, or effectively solve the premature convergence problem. Moreover, the modified algorithms based on diffusion mechanism have good performance in function optimization or practical problems, for example, the PSO algorithm based on diffusion phenomenon (DPPSO) in the statistical physics and thermodynamics field is proposed and preliminarily applied to multi-modal, high-dimen-sional numerical optimization problems[15]. The DPPSO algorithm basically obtained the optimal value of the four test functions. In order to broaden the application scope of the DPPSO algorithm and to verify its effectiveness and feasibility for parameter estimation, the DPPSO algorithm is applied to obtain the optimal parameter combination of S-shaped growth model. Experimental results indicate that DPPSO is an effective approach for parameter estimation of S-shaped growth model.

1 Problem Description and DPPSO Algorithm

1.1 S-shaped Growth Model In the field of agriculture, biology, engineering and

economics, the S-shaped growth curve is very common. For example, the growth process of the onion, forest and pasture has shown S-shaped behavior. In order to build

an S-shaped curve model, a number of mathematical functions are proposed. These are Gompertz, Logistic, Richards, Morgan-Mercer-Flodin, and Weibull models[16]. There are three parameters in the previous two models and four parameters in the other three models. Only Gompertz and Logistic models are taken into considera-tion in this paper because of the limitation of space and the unified treatment.

The expressions of Gompertz and Logistic are de-scribed as follows:

Gompertz: ( , ) exp( exp( ))f x xθ α β γ= − − Logistic: ( , ) / (1 exp( ))f x xθ α β γ= + −

where (0,1)γ ∈ , ( , , )θ α β γ= , and θ is the unknown parameter combination to be estimated. When the in-dependent variable x increases and θ is set to a cer-tain value, the two function curves will present S-shaped, and when x exceeds a certain value, the func-tion curve will get close to a certain asymptote y α= . For comparison, the characteristics of Gompertz and Logistic model is shown in Fig. 1, in which α , β , and γ is set to 48, 5, and 0.2, respectively. Each of them has the inflection point. Moreover, the inflection point of Logistic is also its symmetry center; however, this is not the case for the inflection point of Gompertz. They both effectively describe the phenomenon of bounded growth. In practice, the model choice depends on the specific issues and the forecaster experiences.

Fig. 1 Schematic illustration of the Gompertz and Logistic model

In the present work, least squares parameter estima-tion is just considered. Suppose that there is a set of obser-vation data{( , ) : 1,2, , }i ix y i n= . The parameter estima-tion of the two models is transformed into the following function optimization problem:

2

1

min ( ) ( ( , ))n

i ii

J y f xθ θ=

= −

The DPPSO algorithm is introduced to optimize the

XU Xing et al : Parameter Estimation of S-Shaped Growth Model: A Modified …

139

least squares function, that is, search suitable * *( ,θ α= * * T, )β γ to make *, ( ) ( )J Jθ θ θ∀ ∈ R ≤ (J is the abbre-

viation of ( )J θ in the subsequent section). 1.2 Particle Swarm Optimization Algorithm Based on Diffusion Phenomenon

The diffusion phenomenon of thermodynamics was applied to improve particle swarm optimization, and the modified PSO algorithm based on diffusion phenomenon (DPPSO) was recently proposed in our previous work[15]. The core ideas of DPPSO algorithm and the algorithm process can be embodied on the ba-sis of three definitions, which include the diffusion energy Q of the particle, the population temperature T of the swarm, and the diffusion probability P of the particle.

In our definitions[15], the diffusion energy Qi of the

i-th particle is defined as its kinetic energy assuming that

the mass of each particle is the unit.

The population temperature is defined by

1

1

M

ii

T M Q−

=

=

where M is the number of the particles, that is, popula-tion size. From the above-mentioned equation, the tem-perature can be considered as the arithmetic mean of all particles’ diffusion energy without considering the di-mension.

The diffusion probability of the particle is defined

by

10 0

0

1 1 e 1 ei iQ Q

T Ti

DP D D

D

− −−= − = − = −

where Qi is the i-th particle diffusion energy, T is the population temperature, the gas constant R is set to 1, D0 is the diffusion constant, D is the diffusion coefficient computed by D = D0 e iQ RT− .

In DPPSO algorithm, double populations (POP1 and POP2) are used to simulate the diffusion mechanism. During the evolution of DPPSO, the particle of each swarm is chosen into the diffusion pool of each swarm according to diffusion probability. The diffusion pools of both swarms exchange and share information. The pro-cedure of the modified particle swarm optimization based on diffusion mechanism is as follows:

Step 1 Initialize velocity and position of the parti-cles in POP1 and POP2, respectively.

Step 2 Evaluate the fitness value of each particle in POP1 and POP2, and then update the global best among all particles and the best previous position of each

particle. Step 3 Compute the diffusion energy of all parti-

cles in POP1 and POP2. Step 4 Compute the population temperature of

POP1 and POP2. Step 5 Compute the diffusion probability of all

particles in POP1 and POP2. Step 6 For population POP1: for (i=0; i ∧M; i++) { if (rand( ) ∧Pi)

The i-th particle is chosen into the diffusion pool DP1;

} Step 7 For population POP2: for (j=0; j ∧M; j++) { if (rand( ) ∧Pj)

The j-th particle is chosen into the diffusion pool DP2;

} Step 8 Two particles are randomly selected in the

diffusion pool DP1 (DP2) and used to generate a differ-ence vector, which is the disturbance of the global mini-mum of population POP1 (POP2). If the global minimum with the disturbance vector is better than the global minimum of other population POP2 (POP1) then replace it. That is, the similar operations are implemented to POP1 and POP2.

Step 9 Adjust the velocity and position according to the method given in Refs.[2, 3].

Step 10 If the convergence criteria is not met, go to Step 2, else output the global optimal solution and end the algorithm.

2 Experiment Results and Analysis

2.1 Parameter Estimation Using Actual Data In the current sub-section, a group of data about the

relationship between forage production and growth time is presented in Table 1 and used to test the performance of DPPSO. The parameters setting of DPPSO are as fol-lows: the population size and maximum number of gen-erations are set to 20 and 500; c1=2, c2=2; ω linearly decreases from 1.0 to 0. For comparison, the data (Table 1) we used are the same as that of Ref. [17], in which the evolutionary algorithm (EA) is adopted to estimate pa-rameters of the S-shaped growth model. Therefore, the


results obtained by EA, PSO and DPPSO are compared as shown in Table 2. Here, the residual variance ψ is defined as 2 ( ) / ( 3)J nψ = −θ . Standard deviations (SDs) of , , , Jα β γ obtained by PSO and DPPSO are shown in Table 3. From Table 2, it can be found that parameters estimated by DPPSO are almost the same with the results obtained by the EA algorithm. However, the parameters achieved by DPPSO are averages of the ten repeated runs; results obtained by EA are just the best of five repeated runs. In Table 3, the magnitude of standard deviation (SD) is almost under 10E−5. These results show good stability of the DPPSO algorithm. Simultaneously, the target value of J obtained by the DPPSO for Gompertz and Logistic model is 6.06E−10 and 1.18E−14, respectively. It is very close to zero, indicating that parameters estimated by the DPPSO are very close to the true values. The results ob-tained by PSO are from Ref. [11]. In this paper, the pa-rameters setting of PSO are as follows: the maximum number of iterations is set to 1000, and the population size is 20. In the DPPSO algorithm, the two parameters are 500 and 20, respectively. From Table 2 and 3, the re-sults obtained by the PSO and DPPSO algorithm are al-most the same. However, the number of iterations

Table 1 The actual data

No. Growth time Forage production

1 9 8.93

2 14 10.80

3 21 18.59

4 28 22.33

5 42 39.35

6 57 56.11

7 63 61.73

8 70 64.62

9 79 67.08

Table 2 Comparison between the results of EA, PSO, and DPPSO method

Model Parameter Results by

EA[17]

Results by

PSO[11]

Results by

DPPSO

α 82.821014 82.832170 82.832170

β 1.223820 1.223714 1.223714

γ 0.037083 0.037075 0.037075 Gompertz

2ψ 3.632329 3.632332 3.632332

α 72.462005 72.462240 72.462240

β 2.618009 2.618077 2.618077

γ 0.067357 0.067359 0.067359 Logistic

2ψ 1.342751 1.342754 1.342754

that the DPPSO algorithm needs is only half of what the PSO algorithm needs.

Table 3 Standard deviation of α, β, γ, J

Model Parameter SD by PSO[11] SD by DPPSO

α 4.62E−05 4.78E−05

β 4.38E−07 4.53E−07

γ 3.41E−08 3.52E−08 Gompertz

J 2.57E−10 6.06E−10

α 1.14E−07 1.17E−07

β 5.29E−09 6.34E−09

γ 2.48E−10 2.62E−10 Logistic

J 7.56E−15 1.18E−14

In summary, DPPSO can effectively estimate pa-rameters using the actual data, and the DPPSO’s stability is better than the EA’s just from this set of data, and the convergence speed of DPPSO is faster than the PSO al-gorithm.

2.2 Parameter Estimation Using Simulation Data

In the above discussion, results obtained by DPPSO and EA are compared, but the algorithm performance and the influence factors are not discussed. In this sub-section, the DPPSO algorithm using simulated data is analyzed. The specific content contains: the influence of sampling range on performance of DPPSO by Gompertz model and the affect of noise on convergence and stability of DPPSO by Logistic model. 2.2.1 Influence of sampling range on DPPSO by Gompertz model

Eight parameter combinations are used to generate sample data randomly and are listed in Table 4. For each parameter combination, 100 points are stochastically sam-pled from sampling range [0,5], [0,10], [0,100], respec-tively. Table 5 lists the final estimated parameter combina-tions and their SD. All the results presented in this table are the mean values of five repeated runs.

There are some meaningful phenomena that can

Table 4 The parameter combinations

No. α β γ

1 9.0 5.0 0.2

2 9.0 8.0 0.2

3 9.0 5.0 0.6

4 9.0 8.0 0.6

5 48.0 5.0 0.2

6 48.0 8.0 0.2

7 48.0 5.0 0.6

8 48.0 8.0 0.6


141

be observed based on the experiments. The sampling points cannot reflect the real function curve of the Gom-pertz model when these points are randomly sampled in range [0,5] and [0,10]. Therefore, the estimated parame-ters deviate seriously from the parameters listed in Table 4, and the standard deviations of these two cases are also larger than the case of range [0,100]. We also conducted the experiments for other sampling intervals, such as [0, 3], [5, 15], and [80,100]. The results obtained by the DPPSO algorithm are unstable because the growth curve in these ranges is smooth. Therefore, the algorithm stabil-ity of DPPSO is affected and the final parameters cannot be estimated. The points sampled in range [0,100] may completely reflect the property of Gompertz model. Thus, the actual parameters are fully identified by DPPSO in the eight different parameter combinations. The more the sampling interval reflects the curve shape, the better the performance of the DPPSO will be. That is, the perform-ance of DPPSO is influenced by sampling range. To sum up, sampling is the key step in the model construction and parameter estimation based on the above observations and analysis. We should consider the model characteristic and experimental data should be sampled in the interval that

adequately reflect the property of the model. 2.2.2 Influence of noise on DPPSO by Logistic model

In this part, the effect of noisy data on performance of DPPSO is investigated. According to the conclusions of the previous section, we stochastically sample 100 points {( , ) : 1,2, ,100}i ix y i = from the sampling range [0,100] using the eight parameter combinations listed in Table 4. Then random noise is added to these points. Assuming the noise satisfies Gaussian distribu-tion 2(0, ),N δ therefore the points become {( ix +

2 2(0, ), (0, )) : 1,2, ,100}.iN y N iδ δ+ = Finally, these data with noises are applied to estimate parameters of the Logistic model.

If δ is assigned at a large value, the sampling error would be rather big and experiments would become use-less. δ is restricted in the range of [1E−4,1E−1] andδ is equal to 1E−1, 1E−2, 1E−3, and 1E−4, respectively. The sampling error will fall into the interval [− δ , δ ], [−2δ ,2δ ], [−3δ ,3δ ], and [−4δ ,4δ ]with probability 68%, 95%, 99.7%, and 99.9%, respectively. Table 6 pre-sents the final estimated parameters , , ,α β γ J as well as their SD of five runs, when δ is assigned four kinds of values. From Table 6, we can see that the final estimated

Table 5 α, β, γ, J and their SD obtained by DPPSO for different ranges

No. Range α SD(α) β SD(β) γ SD(γ) J SD(J)

1 [0,5] 11.500 724 8.51 5.192 045 3.73E−01 0.238 937 7.64E−02 0.62 2.26

1 [0,10] 15.254 609 5.41 5.003 705 1.54E−01 0.198 111 1.74E−02 0.17 0.34

1 [0,100] 9.000 003 4.16E−06 4.999 954 6.81E−05 0.199 998 2.64E−06 6.35E−26 9.25E−25

2 [0,5] 6.628 193 5.75 7.920 164 1.91 0.266 781 1.93E−01 1.27 3.86

2 [0,10] 8.466 079 9.17 7.289 406 2.62 0.130 293 1.55E−01 0.41 0.73

2 [0,100] 8.999 995 1.43E−05 7.999 982 1.06E−04 0.200 000 2.62E−06 1.29E−27 5.82E−26

3 [0,5] 14.391 724 7.98 5.173 209 6.90E−01 0.644 246 1.99E−01 0.89 2.97

3 [0,10] 9.025 145 1.60E−01 4.989 682 8.22E−02 0.598 418 1.23E−02 0.14 0.26

3 [0,100] 8.999 999 1.32E−06 5.000 049 1.04E−04 0.600 005 9.66E−06 3.23E−25 4.43E−24

4 [0,5] 19.127 839 1.95 9.530 121 1.02 0.907 703 2.06E−01 1.54 3.91

4 [0,10] 7.216 951 2.55 8.179 333 2.52E−01 0.621 880 3.08E−02 0.35 0.67

4 [0,100] 9.000 000 7.60E−07 8.000 028 6.48E−05 0.600 002 4.63E−06 2.66E−28 8.45E−28

5 [0,5] 42.774 847 6.04 5.754 335 1.65 0.353 846 3.35E−01 1.10 3.21

5 [0,10] 51.113 051 6.75 5.019 508 5.50E−02 0.201 693 6.22E−03 0.57 1.16

5 [0,100] 48.000 000 5.40E−07 5.000 000 5.61E−07 0.200 000 2.39E−08 4.63E−27 7.44E−27

6 [0,5] 50.198 798 3.56 8.629 597 7.29E−01 0.170 708 1.09E−01 0.87 2.06

6 [0,10] 53.820 917 4.57 8.245 216 1.58 0.147 987 1.44E−01 0.71 1.67

6 [0,100] 47.999 919 8.89E−05 8.000 431 4.87E−04 0.200 010 1.18E−05 2.88E−25 8.56E−25

7 [0,5] 49.861 367 9.96 4.996 474 4.77E−02 0.598 768 1.50E−02 1.78 4.02

7 [0,10] 47.725 550 1.11 5.032 505 1.20E−01 0.604 816 1.78E−02 0.27 0.55

7 [0,100] 48.000 000 3.02E−09 5.000 000 3.64E−08 0.600 000 4.14E−09 3.23E−30 6.63E−29

8 [0,5] 53.496 540 9.29 9.300 183 9.55E−01 0.861 441 1.92E−01 1.67 3.80

8 [0,10] 45.647 882 8.47 8.422 883 8.87E−01 0.643 571 9.07E−02 0.73 1.57

8 [0,100] 47.999 999 5.53E−06 7.999 958 1.90E−04 0.599 997 1.38E−05 5.42E−26 6.93E−25


Table 6 α, β, γ, J and their SD Obtained by DPPSO for different noises

No. Noise α SD(α) β SD(β) γ SD(γ) J SD(J)

1 1E−1 8.988 521 2.05E−09 5.044 570 1.72E−08 0.200 987 6.51E−10 1.182 263 1.39E−151 1E −2 9.002 403 8.29E−10 4.988 448 4.43E−09 0.199 487 2.00E−10 1.059 390E−02 1.87E−161 1E −3 8.999 854 5.32E−09 5.000 107 3.98E−08 0.200 006 1.64E−09 1.259 016E−04 9.54E−151 1E−4 9.000 003 1.79E−09 4.999 996 1.46E−08 0.199 999 5.76E−10 8.664 811E−07 1.28E−152 1E−1 8.997 037 2.87E−09 7.993 589 7.97E−09 0.199 874 1.80E−10 0.924 799 1.14E−152 1E−2 8.999 835 9.82E−10 8.014 132 8.79E−09 0.200 369 2.41E−10 1.075 633E−02 1.64E−162 1E−3 8.999 604 4.70E−10 7.999 331 2.13E−09 0.199 987 5.32E−11 8.727 847E−05 1.08E−172 1E−4 8.999 978 1.03E−10 8.000 103 6.63E−10 0.200 002 1.71E−11 9.316 066E−07 4.32E−193 1E−1 9.000 456 5.40E−10 5.114 805 2.37E−08 0.610 266 2.96E−09 1.121 866 6.47E−163 1E−2 8.999 596 2.30E−10 4.993 541 6.95E−10 0.599 541 7.04E−11 1.071 565E−02 2.24E−173 1E−3 9.000 027 4.95E−11 5.002 266 1.07E−09 0.600 309 1.17E−10 1.197 281E−04 2.67E−183 1E−4 8.999 988 4.65E−11 4.999 940 5.30E−10 0.599 997 5.45E−11 1.083 024E−06 4.36E−194 1E−1 9.010 718 1.16E−09 7.662 189 2.55E−08 0.576 484 1.66E−09 0.928 6880 1.29E−154 1E−2 8.999 625 4.55E−10 7.986 365 3.55E−09 0.598 722 2.54E−10 1.206 675E−02 6.51E−174 1E−3 9.000 030 1.38E−10 8.000 365 3.35E−09 0.600 051 2.35E−10 1.068 286E−04 7.65E−184 1E−4 9.000 002 3.47E−11 8.000 202 1.15E−09 0.600 017 8.48E−11 9.707 919E−07 2.79E−195 1E−1 47.997 124 4.37E−09 4.988 044 5.92E−09 0.199 318 2.44E−10 1.749 668 5.77E−155 1E−2 48.000 070 5.81E−10 4.998 251 2.07E−09 0.199 913 8.14E−11 1.634 568E−02 4.23E−165 1E−3 47.999 906 6.24E−10 5.000 139 3.83E−10 0.200 005 1.41E−11 1.695 597E−04 5.13E−175 1E−4 47.999 990 1.66E−10 5.000 010 2.00E−10 0.200 000 7.69E−12 1.508 694E−06 3.88E−186 1E−1 48.030 666 7.22E−09 7.996 233 1.03E−08 0.199 919 2.78E−10 1.435 793 6.92E−156 1E−2 47.999 283 1.18E−09 8.000 969 1.95E−09 0.200 019 5.24E−11 1.459 929E−02 2.66E−166 1E−3 48.000 017 8.84E−10 7.999 981 9.90E−10 0.200 000 2.73E−11 1.908 862E−04 7.49E−176 1E−4 47.999 991 2.18E−10 8.000 056 5.20E−10 0.200 001 1.36E−11 1.892 415E−06 2.00E−177 1E−1 47.991 037 4.90E−09 5.007 779 1.07E−08 0.602 984 1.31E−09 4.375 171 8.58E−157 1E−2 48.000 502 1.07E−09 5.003 467 1.60E−09 0.600 300 1.77E−10 1.462 022E−02 1.68E−167 1E−3 48.000 190 2.64E−10 4.999 474 6.61E−10 0.599 896 8.58E−11 2.098 901E−04 1.91E−177 1E−4 48.000 001 8.84E−11 4.999 975 1.73E−10 0.599 997 2.02E−11 3.833 929E−06 3.14E−188 1E−1 48.012 570 4.52E−09 7.914 444 3.37E−08 0.596 072 2.19E−09 2.413 561 1.04E−148 1E−2 47.999 078 1.33E−09 7.988 278 2.18E−09 0.599 756 1.81E−10 2.476 873E−02 1.61E−168 1E−3 48.000 055 5.31E−10 7.998 041 1.88E−09 0.599 854 1.39E−10 3.325 410E−04 9.43E−178 1E−4 47.999 971 8.07E−11 8.000 300 1.30E−09 0.600 018 8.99E−11 1.746 113E−06 5.10E−18

parameters are almost equal to the original parameters listed in Table 4. When δ =1E−1, the sampling error is relatively larger, the parameters estimated by the DPPSO are worse than the situations that δ is smaller than 1E−2. In all the circumstances, the standard deviation of

,α β , γ , and J is less than 1E−8. It demonstrates that the stability of the DPPSO is not affected. ,α β , and γ can converge more and more closely to the real values, when δ decreases from 1E−1 to 1E−4. J decreases from about 1 to the magnitude of 1E−6 and converges more and more closely to zero, when the value of δ is getting smaller and smaller. In conclusion, when sam-pling error is small, the impact of noise on DPPSO algo-rithm performance is not significant.

3 Conclusion

Because of its good performance, the modified parti-cle swarm optimization (DPPSO) based on the diffusion

law and the Arrhenius rate law is introduced to demonstrate the feasibility for estimating the parameter of the S-shaped growth model by the DPPSO algorithm. Experimental data consist of the real observation data, randomly sampling data, and the noisy data. It is shown through experiments that DPPSO is an effective and robust method to estimate the parameters of S-shaped growth model with a global search capability. We can draw a conclusion that DPPSO may be regarded as an option to solve the parameter estima-tion of the nonlinear model. Future work will be focused on applying DPPSO to estimate parameters for other S-shaped growth model, such as the Richards, Morgan-Mercer- Flodin, and Weibull models.

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