基于pso-svm的sci文章

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1. Lou et aL. Applied Mathematical Modelling 39(2015)5907-5916 5909 In this paper, a novel method of phytoplankton abundance prediction and forecast are presented by svr based on pso algorithms. This approach is the use of pso to determine free parameters of SvR, i.e., PSo-SVR, which optimize all svr's parameters simultaneously from the training data. Then the monthly phytoplankton abundance data in MSR was used as a case study for the development of the prediction and forecast model to simulate the dynamic change of phytoplankton abundance in Macau Reservoir given a variety of water variables. The measured data from 2001 to 2011 were used to train and test the model. Present study will lead to better understanding of the algal problems in Macau, which will help to develop later guidelines for forecasting the onset of algae blooms in raw water resources The rest of this paper was organized as follows: The study area and data source of msr were first introduced concisely, followed by modeling approaches(SvR and pso)formulation, and the performance indicators used for evaluation in Section 2. Section 3 presented and discussed the modeling results performed by pso and svr respectively, and made com parisons between both methods. The conclusion was drawn finally in Section 4. 2. Materials and methods 2. 1. MSR and water parameters measurement Macau is located at south of China with a subtropical seasonal climate that is greatly influenced by the monsoons. As the biggest reservoir in Macau, Macau Main Storage Reservoir (msr)(11333/12"E longitude and 22 12/12"N latitude)(as shown in Fig. 1)is located in the east of Macau peninsula with the capacity of about 1.9 million m and the water surface area of 0.35 km In recent years, algal blooms and the situation appeared to be worsening were reported(macao Water Co. Ltd, unpublished data). Macau Water Supply Co. Ltd is responsible for water-quality monitoring and management. Locations in the inlet, outlet and mid of the reservoir was selected for sampling. Samples were collected in duplicate monthly from May 2001 to February 2011 at 0.5 m from the water surface. There are 23 water quality parameters, including hydrological, physical, chemical and biological parameters, were monitored monthly and these 23 parameters were measured according to the standard methods [28, 29]. Imported volume, exported volume and water level were recorded by the inlet and outlet flow meters, based on which the hydraulic retention time(hrt) can be calculated The phytoplankton samples were fixed using 5% formaldehyde and transported to laboratory for microscopic counting In order to identify the water parameters that were significantly correlated with phytoplankton abundance, correlation analysis was conducted firstly The criteria for selecting parameters as the inputs to the pso-svr models is that correlation coefficients for selected parameters needs to be greater than 0.3. It should be pointed out that in forecast models the param eters are different from those in the prediction models, because the water parameters in previous data were also used in the correlation analysis. In this study the forecast model was based on the last three months data. Including the three month data in this forecast model is to adopt the historical effect of the last year that have similar environmental conditions such as temperature influence the growth of phytoplankton 2. 2. Support Vector Regression (SVR) The svr is to map the feature vector x into a higher dimensional space by using a nonlinear mapping (x). Assume the f(x) takes the linear form f(X,)=(·x)+ba→(),b), Chi s 十 Macau 1. Location of the msr 5910 L. Lou et aL. / Applied Mathematical Modelling 39(2015)5907-5916 Where x is the independent variables, y is the dependent one. o refers to a vector for regression coefficients, b is model error value. The coefficients(w and b) are estimated by minimizing the following regularized risk function Rm(x)=1∑C1f(xk,m)+C.Sf),x∈A 2) n∑>|(yk-f(x,x)l+c·S(f) where is a function of parameters a in domain n (0, c)where Su) is the smoothness of functions f and c is the trade-off coef- ficient between the empirical risk and smoothness of functions f. SVM is designed to find the a that minimizes the risk Function Rsvm(o) Ce=lyk -f(xk, ae= yk -f(xk, a) lYk-f(Xk, a)I<& otherwise (3 Ca is called o-insensitive Cost Function, C(y, f(x, a)), is define to evaluate the discrepancy between an observation yk and the response of learning machine f(xk, a). Determining the tradeoff between the training error and the model flatness, n is the number of training samples, a is a prescribed parameter controlling the tolerance to error Both C and 1 are user-determined parameters. Two positive slack variables s and s*, which represent the distance from actual values to the corresponding boundary values of 1-tube, are introduced. Then, (4) is transformed into the following convex optimization problem minimIze 2|o2+C∑(1+) y1-()·x1)-b≤E+ (4 subject to(Oxi)+b-y;<&+5 By introducing lagrange multipliers and exploiting the optimality constraints the decision function given by (1) has the following explicit Start Set scope of three parameters of SVM and Stopping condition of the whole program Initialize Pso particles and velocities of a population Evaluate the fitness value Update particle best and global best Update particle velocity and position Does it meet the stopping condition Yes Output optimal parameters of SVM Fig. 2. A flow diagram of the proposed nonlinear ensemble forecasting model 1. Lou et aL. Applied Mathematical Modelling 39(2015)5907-5916 5911 f(x)=(O(x)+b=∑(1-1)(x)x)+b=∑(1-)k(xx)+b, Where, K(xi, xi)is called the kernel function, li and ii are the So-called Lagrange multipliers. Kernel function is an inner pro duct of two transformed feature vectors and it is written by k(xi xi)=p(xi).p(x). In the literature, there are several kernel functions, namely linear, polynomial, and Gaussian kernels. Generally, using Gaussian function will yield better prediction performance. Thus, in this work, the Gaussian function K(xi, xi)=exp(x;-x l2/(202)), is used in the SVR. Where, o2 repre sents the bandwidth of gaussian kernel To build a svr model efficiently we need to select three positive parameters, o, s and C, of a SVr model These parameters must be selected accurately, since they determine the structure of high dimensional feature space and govern the complexity of the final solution. However, structural methods of selecting parameters efficiently are lacking Thus, despite its superior features, SVR is limited in some research and applications, because the user has to define various parameters appropriately. An excessively large value for parameters in SVR leads to over-fitting or while a disproportionately small value leads to under-fitting [18, 20 Different parameter settings can cause significant differences in performance Therefore, selecting the optimal hyper-parameter is an important step in svr design [ 20, 30. 2.3. Particle Swarm Optimization(PSO) In PSO, a particle's neighborhood is the subset of particles with which it is able to communicate. Depending on how the neighborhood is determined, the pso algorithm may embody the gbest and lbest models. In the former, each particle is con- nected to every other in the swarm, thus obtaining information from the whole group In the latter a particle is not able to communicate with the entire swarm, but only with some selected particles. The particle Swarm Optimization concept consists of, at each time step, changing the velocity of (accelerating) each par ticle toward its pbest and lbest locations (local version of PSO). Acceleration is weighted by a random term, with separate ran dom numbers being generated for acceleration toward pbest and lbest locations. In the standard PSo model, with M particles in D-dimensional problem space, each particle is denoted as Xi=( xil, Xi,.., Xp,i=1, 2, ., M which is represented as a potential solution the velocity of each particle along with each dimension is denoted as vi=(vil, v2,.. vip The velocity and position of particle i at(t+ 1) iteration are updated by the fol lowing equations vi(t+1)=wvi(t)+Ciri(t)P-Cu(t)-Xi(t)]+C2r2(t)[P, (t)-Xu(t)], X(t+1)=X(t)+v(t+1), Table 1 Correlation analysis of prediction and forecast model Parameters Prediction model Forecast model Time lagged(month) t t-3 Turbidity 0.03 0.01 0.06 Temperature 0.19 0.19 0.14 PH 0.49 0.42 0.38 0.33 0.08 0.01 0.14 0.21 0.01 0.22 0.28 -0.03 0.03 0.14 0.22 Sio 0.3 0.31 0.16 0.04 Alkalinity 0.34 0.3 0.2 0.12 HCO 0.4 0.24 0.39 0.34 0.29 0.22 -0.22 0.15 0.1 0.08 -0.02 0.03 NH 0.11 0.08 0.25 TN 0.68 0.6 0.53 0.46 254 0.56 0.55 0. 0.47 0.14 -0.08 0.02 0.08 0.02 Suspended solid 0.3 OC 0.38 0.33 0.29 0.35 HRT 0.12 0.13 0.16 Water level 0.13 0.05 0.01 0.02 Precipitation 0.09 0.05 0.11 0.06 Phytoplankton abundance 0.82 0.71 0.62 5912 L. Lou et aL. / Applied Mathematical Modelling 39(2015)5907-5916 where vi is velocity measures for particles; w is inertial weight controlling velocity direction(a measure of how much the particle"trusts"its own exploration ); C1 and C2 are acceleration coefficients; ri and r2 are random numbers uniformly dis tributed between [0, 1]. Thus C,r represents a uniformly distributed random number from 0 to C1. It is a measure of how much a particle"trusts"its neighborhood best velocity, and c2r2 represents a uniformly distributed random number from 0 to C2. Independent from Cirl, it is a measure of how much a particle"trusts"the global velocity; p_ci is the neighborhood best position. "-"represents the difference of two positions is the velocity that will transform the second position into the first position p- g; is the global best position. " represents the transformation of a position using the velocity(yields a posi- tion). x is position measure of for particles Thus, the change of position of each particle from one iteration to another can be computed according to the distance between the current position and its previous best position, and the distance between the current position and the best posi tion of swarm. The particle flies through potential solutions towardsp_cit)andp(t in a navigated way while still exploring new areas by the stochastic mechanism to escape from local optima. Since there was no actual mechanism for controlling the velocity of a particle, it is necessary to impose a maximum value vmax on it. If the velocity exceeds the threshold, it is set equal to vmax, which controls the maximum travel distance at each iteration to avoid this particle flying past good solutions. The PSO algorithm is terminated with a maximal number of generations or the best particle position of the entire swarm cannot be improved further after a sufficiently large number of generations. The pso algorithm has shown its robustness and effi cacy in solving function value optimization problems in real number spaces 2. 4. The proposed hybrid model integrating SVM and pSo The research scheme of the proposed methodology is shown in Fig. 2 First, the population of particles is initialized, each particles is represented as(τ1,τ2,τ3), whereτ1,τ2andτ3 denote the regularization parameters C, 02 and 8, respectively. The initial group is randomly generated, each group having a random position and a random velocity for each dimension. Second, each particle fitness function of pso (fit- ness=MAPe=2(yi-Yi/Yi where nMApe is the mean absolute percentage error; Yi and Yi are the observation data and the modeling results, respectively) for the SvM is evaluated. The each particles fitness in this study is the regression accuracy. If the fitness is better than the particle's best fitness, then the position vector is saved for the particle. If the par- ticle's fitness is better than the global best fitness, then the position vector is saved for the global best. Finally the particles velocity and position are updated until the termination condition is satisfied PSO-SVR is applied in the proposed svr software reliability prediction model to optimize parameter selection 2.5 Performance indicators The performance of models was evaluated using the following indicators: coefficient of determination(r)that provides the variability measure for the data reproduced in the model Prediction R is a good measure for both comparison and seeing the model's prediction capability The calculation method is also known as the cross validation, in which we exclude the first observation and build the model with the remaining ones, use this model to predict the excluded observation and repeat for all observations. It is a good measure for out-of-sample accuracy. As this test cannot give the accuracy of the model, other statistical parameters should be reported. Mean absolute error(Mae) and root mean square error(rmse) that measure residual errors, providing a global idea of the difference between the observation and modeling. the indicators were defined as below by Eqs. 8-12 F (8) F 9) ∑(Y1-Y) (10) Table 2 Performance indexes of the prediction and forecast models Prediction model Forecast model Performance index Accuracy performance Generalization Accuracy performance Generalization (Training set) performance(Testing set) (Training set) performance(Testing SVM 31 SVM+ PSo SVM[31 SVM+ PSo SVM 31 SVM+ PSO SVM[31 SVM+PSO R2 0.760 0.764 0.758 0.767 0.863 0.875 0.863 0.876 RMSE 0.307 0.291 0.351 0.343 0.229 0.219 0.264 0.258 MAE 0.243 0.236 0.274 0.267 0.127 0.120 0.226 0.224 1. Lou et aL. Applied Mathematical Modelling 39(2015)5907-5916 5913 8.5 Observed PSO-SVM 87.5 6.5 20012002 2003 2004 2005 2006 20072008 2009 Fig 3 Observed and predicted phytoplankton level for the training and validation data set of the prediction models 8.5 8 7.5 百65 Observed PSO-SVM 2009 2010 Fig. 4. Observed and predicted phytoplankton level for the testing data set of the prediction models 8. R2=0.747239 R2=0.734257 y=0.596*X+3.012 y=0.675x+2309 7.5 7.5 65 6.5 6.5 7.5 8.5 6.5 7.5 8.5 Observed Observed ig. 5. PSo-SVM results for the training and validation (a)and testing(b) data set of the prediction model MAE= RMSE n∑(YY), 5914 L. Lou et aL. / Applied Mathematical Modelling 39(2015)5907-5916 8.5 8 PSO-SVM 87.5 6.5 6 Year Fig. 6. Observed and predicted phytoplankton level for the training and validation data set of the forecast model 1 that based on the previous 1st, 2nd and 3rd months data where n is the number of data; Yi and Yi are observation data and the mean of observation data, respectively, and Yi is the modeling results(see Table 1). 3. Results and discussion To illustrate this hybrid prediction and forecast methods, the estimation of phytoplankton abundances based on the water parameters and environmental variables were studied in this work. The correlation of log1o phytoplankton abun- dances and all variables for prediction model and forecast model were shown in Table 2. It can be seen that parameters with correlation coefficients greater than 0.3 are highlighted in bold, and these highlighted parameters will be kept in the models Because the water parameters in previous data were also used in the correlation analysis, the parameters selected in forecast models are different from those in the prediction models. In the forecast models of svr, phytoplankton abundance(t) is a function in terms of water parameter(t-1), water parameter(t-2)and water parameters(t-3). At here t-1, t-2 and t-3 represent the 1 month, 2 months and 3 months before the time t. So, in this way only 9 parameters used in the prediction models while 23 time-lagged parameters selected for the forecast models When the correlation analysis is done it needs to test the models which cover two parts, the accuracy performance and the generalization performance. Accuracy performance is to check how capable the model is to predict the output for the given input set that originally used to train the model For generalization performance is to test if the model is capable of predicting the output for the given input sets that were not used to train the model. It is not good if model that is memorizing the inputs instead of generalized learning, so in order avoid this problem both performance checks should be covered In this work, 50 runs were conducted to average the performance indexes for Svr-based models. So for the predication models in the application of Svr in this work, after the correlation analysis, as the independent variables, there are 9 parameters including pH, SiO 2 are selected, and phytoplankton abundance are selected as the induced variable to set a target. And then, the data process for both the prediction and forecast model follows the same steps as illus- trated in 31. After the data were processed, the performance of prediction and forecast models were shown in Table 2. The results indicated that the svr were successful in the prediction and forecast phytoplankton abundance in msr, with the r greater than 0.87 for both training and testing data sets. Compared to the prediction model with the r2 of 0.764(0.767) 8.5 8 8 6.5 Observed PSO-SVM 2009 2010 2011 Yea Fig. 7. Observed and predicted phytoplankton level for the testing data set of the forecast model 1 that based on the previous 1 st, 2nd and 3rd months data 1. Lou et aL. Applied Mathematical Modelling 39(2015)5907-5916 5915 8.5 8.5 R4=0.874729 R2=0.876301 y=0.818*×+1.290 8 0.869*x+0.809 7.5 6.5 a 6.5 7.5 8.5 Observed 7.5 Observed Fig 8. PSo-SVM results for the training and validation(a)and testing(b)data set of the forecast model 1 that based on the previous 1st, 2nd and 3rd months data RMSE of 0. 291(0.343), and MAE of 0. 236(0.267)for training(testing), the forecast model had better performance with the r2 of 0. 875(0.876), RMSE of 0. 219(0.258), and MAe of 0. 120(0.224)for training(testing), suggesting that the historical water parameters including the phytoplankton abundance has effect on the prediction, which can improve the prediction powers These results further confirmed the historical effects on the model accuracy and generalization performance, and also implied that take the latest 3 months data as memorizing learning can improve the prediction power in the forecast model Besides, further compared with our previous study [31] for the prediction and forecast of phytoplankton abundance using SVR only with R of 0.863, the present study integrating SVR and PSo as the optimization algorithm have better prediction power with r, RMse and mae The observed data versus the modeling data were shown in Fig. 5 (prediction model), and Fig 8(forecast model 1), and the observed and modeling phytoplankton abundance change over time were listed in Figs. 3 and 4 (prediction model) and Figs 6 and 7(forecast model 1). These results confirmed that integrating SVM and pSo can handle well the non-linear rela tionship between water parameters and phytoplankton abundance From the modeling point of view, a disadvantage of learning machine method is the mechanisms of the inner signal processing are not unknown. However, it has provided enough information for msr to prevent the algal bloom problems, for controlling the algal bloom problem occurrence, the engineers can only control the predicted or forecast phytoplankton by adjusting the operational variables, such as reduc- ing the nitrogen and phosphorus concentrations without understanding the complete mechanisms and the relationships among the variables. The forecast model integrating PSO and SvR showed high values of RMSe and R and simple algorithm which is highly probably due to the more complete algal bloom causative variables involved, thus providing more information in explaining the algal bloom phenomena, despite that the complete mechanisms causing algal bloom and the relationships among vari- ables are still unclear 4. Conclusions To avoid algal blooms in drinking water lakes or reservoirs, it is very crucial to have accurate the phytoplankton abun- dance prediction and forecast. This study proposes the pso-svm based prediction and forecast models for monthly phyto- plankton abundance time series in MSr pSo was applied to obtain the optimal parameters of svr. It can be seen that the forecast model have better performance with the r of up to 0.876 than prediction model with the r of 0.767. This means the algal bloom problem is a complicated non-linear dynamic system that is affected not only by the water variables in cur- rent month but also by those in a couple of previous months. Furthermore, SvR in the study showed superior forecast power and root mean square errors with the latest 3 months data in the forecast model, and this indicates that the historical water parameters and phytoplankton abundance have impact on the phytoplankton dynamics of the reservoir. From numerical results, it can be concluded that this work proposes an effective approach for water system modeling and management, and it can also be noted that this approach can be further improved by combining other numerical approaches and optimiza on algorithms such as genetic algorithms [32-34 Acknowledgments We thank macao Water Co Ltd. for providing historical data of water quality parameters and phytoplankton abundances The financial support from the fundo para o desenvolvimento das Ciencias e da Tecnologia(fDct)(grant FDCT/069/2014 5916 L. Lou et aL. / Applied Mathematical Modelling 39(2015)5907-5916 A2)and Research Committee at University of Macau are gratefully acknowledged. This project was also supported by University of Macau MYRG2014-00074-F References [1Z. Selman, S Greenhalgh, R Diaz, Eutrophication and Hypoxia in Coastal Areas: A Global Assessment of the State of Knowledge, World Resources Inst Washington D.C., 2008 [2]S W. Fei, M. Wang, Y.-B. Miao, J. 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