Reconstruction of dynamical forecasting model between
1 Reconstruction of dynamical forecasting model between 2 Western Pacific subtropical high area index and its 3 summer monsoon impact factors based on the improved 4 self-memorization principle 5 6 Mei Hong 1 ,Ren Zhang 1 , Xi Chen1, Shanshan Ge 1 , Chengzu Bai1, Vijay P. Singh 2, 3 7 8 1. Research Center of Ocean Environment Numerical Simulation, Institute of Meteorology 9 and oceanography, PLA University of Science and Technology, Nanjing 211101,China 10 2.Department of Biological and Agricultural Engineering, Texas A & M University, College 11 Station, TX77843, USA 12 3.Zachry Department of Civil Engineering, Texas A & M University, College Station, TX 13 77843, USA 14 15 Corresponding author address: Mei Hong 16 Institute of Meteorology, 17 PLA University of Science and Technology, 18 P.O. Box 003, No. 60,Shuanglong Road, 19 Nanjing 211101, China. 20 Telephone and Fax number: 0086-025-80831626 21 E-mail: [email protected] 22 23 24 25 1 26 Abstract. W ith the objective of tackling the problem of inaccurate long -term Western Pacific 27 Subtropical High(WPSH) forecasts，based on the concept of dynamical model reconstruction and 28 improved self-memorization principle, a new dynamical forecasting model of WPSH area (SI) index is 29 developed. To overcome the problem of single in itial pred iction value, the largest Lyap unov exponent 30 is introduced to imp rove the traditional self-memorization function, making it mo re appropriate to 31 describe the chaotic systems, such as WPSH; and the equation reconstruction by actual data is used as 32 its dynamical core to overco me the proble m of relatively simp le dynamical core. The developed 33 dynamical forecasting model of SI index is used to predict WPSH strength in long -term. Through 10 34 experiments of the WPSH abnormal years, forecast results within 25 days are found to be good, with a 35 correlation coefficient of about 0.80 and root mean square error under 8%, showing that the improved 36 model has satisfactory long-term forecasting results. Especially the aberrance of the subtropical high 37 can be draw and forecast. It is acknowledged that mechanism for the occurrence and development of 38 WPSH is co mplex, so the discussion in this paper is therefore exp loratory. 39 40 Key words: The subtropical high; Improved self-memorization function; Dynamical forecasting model 41 reconstruction; Long-term forecast; Largest Lyapunov exponent 42 43 44 45 46 47 2 48 1. Introduction 49 The Western Pacific Subtropical High (WPSH) is one of the most important components 50 of the East Asian Summer Monsoon (EASM) system [1]. The intensity and position of WPSH 51 show complex seasonal evolutions and the changes in them greatly affect the occurrence of 52 rainy season in China, including floods, droughts, heavy rains, etc[1]. For example, when 53 WPSH reaches the northernmost position, especially in summer, it significantly influences 54 rainfall over China and Japan[2]. 55 Owing to its dominance on the East Asian climate, WPSH has become one of the leading 56 topics of interest in atmospheric sciences[3-5]. Over the past decades, much effort has gone 57 into to uncovering the forecast of the WPSH[6], especially the forecast of abnormal 58 WPSH[7-8]. Current forecasts for the WPSH can be divided into two categories: numerical 59 forecasts and statistical forecasts. Numerical forecasts are widely used throughout the world; 60 examples include the numerical forecast products of the European Centre for Medium-Range 61 Weather Forecasts Model [9], the Japanese FUFE502 numerical forecast products, etc. 62 However, numerical forecasts require field boundaries and the complex computations and low 63 efficiency mean that the results are very unstable. Numerical forecast products have better 64 results for large-scale weather systems, but for mesoscale weather systems, such as the WPSH, 65 the results are less good and the forecast time is short. 66 Statistical methods, in contrast, have achieved some success in forecasting the WPSH. 67 These methods can use historical data and the computation is simple. However, there are 68 some inherent flaws in statistical methods. Using neural networks as an example, it is difficult 69 to objectively determine the number of hidden layer neurons and the training process tends to 3 70 predict a local optimum, which will limit the forecast accuracy [10]. Moreover, the reliability 71 of all these methods is gradually reduced with increasing forecast time, so the forecast results 72 and credibility become very low after two weeks [11]. Statistical forecasting products and 73 numerical forecasting products both have some degrees of bias. Especially, error is more 74 obvious in WPSH anomalies and long-term forecasting [3]. So the prediction of unusual 75 activities of WPSH within season and the long-term trend forecast of WPSH has become 76 difficult problems in recently years. 77 A statistical-dynamical model of a weather system is reconstructed from actual data and 78 can be used to describe the physical mechanisms of a complex weather system. Concerning 79 the questions of local convergence of errors, Zhang et al. [12] introduced genetic algorithms 80 (GA), which is widely used in a lot of field [13-14], to improve the determination of root 81 efficiency of model parameters. On that basis, Mei et al. [15-16] carried out research on the 82 reconstruction of a nonlinear statistical-dynamical forecast model of the WPSH, and achieved 83 good results. 84 However, the dynamical prediction equations derived by Zhang et al.[12] and Mei et 85 al.[15-16] greatly depend on the initial value, so the long-term forecast over 15 days diverged 86 significantly and the results were not satisfactory. For the long-term forecast, the model 87 should be improved. Cao [17] proposed the self-memorization principle, transforming the 88 dynamical equation into memorization equation in a 89 differential-integral equation, wherein the memory coefficients can also be determined by 90 actual data. This method has been widely used in prediction problems in meteorology, 91 hydrology and environmental field [18-20]. Because this method avoids the problem of initial broader sense, named a 4 92 condition in differential equations, it can be introduced to improve the proposed dynamical 93 forecast model. 94 The set of self-memory function is relatively simple [17] and is suitable for cyclical 95 and linear systems. For nonlinear systems, especially chaotic systems, forecast results are 96 unsatisfactory [20].As the atmosphere and ocean are nonlinear systems, the self-memory 97 function is needed to be modified for nonlinear-system, modeling. The largest Lyapunov 98 exponent is introduced to improve the traditional self-memorization function. Finally, the 99 improved dynamical forecasting model of WPSH with a new self-memorization function is 100 developed. The improved function not only takes into account the chaotic characteristics of 101 the nonlinear system, but also absorbs the information of past observations. 102 In our study, we firstly define the WPSH area index (SI) as a measurement of the 103 scope and form of the WPSH. The rest of this paper is organized as follows: Section 2 104 introduces data and factors. Three summer monsoon factors which have higher correlation 105 with SI are chosen. The dynamical model of SI and three factors is reconstructed in section 3. 106 In section 4, self-memorization dynamics is introduced to improve the reconstructed model. 107 The improved self-memorization functions for chaotic systems are investigated and discussed 108 in section 5. Forecast experiment of 2010 and further forecast tests of an additional nine 109 WPSH abnormal years are described in section 6, and results are summarized in section 7. 110 2. Research data and factors 111 2.1.Data 112 The daily data from May to October of the past 30 years from 1982 to 2011 were 113 obtained from the National Centers for Environmental Prediction (NECP) Climate Forecast 5 114 System Reanalysis (CFSR), including: horizontal wind field and geopotential height field at 115 850 hPa and 200 hPa; geopotential height field at 500 hPa; sea level pressure field with a 116 resolution of 0.5° × 0.5°; and sensible heat flux, latent heat flux, and precipitation rate in the 117 Gaussian grid. 118 Observed long-wave radiation (OLR) from May to October of the 30 years from 1982 to 119 2011 of National Oceanic and Atmospheric Administration (NOAA) satellites, with a 120 resolution of 0.5° × 0.5°. Unit is W m−2 . 121 These data are primarily used to calculate the SI and summer monsoon impact factors in 122 section 2.3. 123 2.2 The basic facts of WPSH activity in the summer of 2010 124 Change of WPSH within seasons in various years is very different compared with the 125 average, especially the “abnormal” activities of the WPSH, which often lead to East Asia 126 subtropical circulation anomalies and extreme weather events in China in some years, such as 127 1998, 2003, 2006 and 2010 [21-22]. 128 In order to better describe the changes in the WPSH, SI is used, which has been defined 129 by the Central Meteorological Observatory [23] to characterize the WPSH range and intensity, 130 i.e. the average grid points of 500 hPa geopotential height greater than 588 gpm in the range 131 [10 ° -90° N, 110 ° E -180 ° E]. The higher the value, the wider is the range represented, or 132 the greater is the intensity. 133 SI of summer half-year (from May to October) from 1982 to 2011 can be calculated, and 134 changes of SI in different years are shown in Fig. 1. In Fig. 1, the straight line represents the 135 average SI in the observed years. SI varied widely in different years, representing irregular 6 136 activities of the WPSH; particularly, in some years, SI showed dramatic changes. For example, 137 in 2010 SI is 32, while in 1984 SI is just 6, which indicates that WPSH activities behaved 138 abnormally in these particular years. If we use the average data of several years, these 139 exceptions will not be apparent. So we should choose an abnormal SI year to study. 140 From Fig. 1, 2010 is the most prominent year of WPSH anomalies as shown by the 141 arrow. From May to October in 2010, SI is above the mean and it is the peak of nearly 30 142 years. The anomaly of WPSH strength resulted in a very unusual climate in China in 2010. 143 Extreme heat and heavy precipitation events occurred frequently. The strength and wide range 144 of these events are rarely seen in history. Especially, the most powerful WPSH happened 145 from June to August and is to date the strongest one in the meteorological records (Fig. 2). 146 This most powerful WPSH directly resulted in rare flood disasters in the eastern part of south 147 China, the Yangtze River and northeast and northwest of China. Therefore, we selected the 148 summer of 2010 as a typical case to analyze the relevance between the enhanced WPSH and 149 the members of the monsoon system. 150 2.3. Selection of three factors 151 According to previous studies [21-22], there are many members of the summer monsoon 152 system, 21 of which are closely related to the WPSH. If all are used for modeling, the 153 equation would be too complex. Therefore, the correlation analysis method is used to analyze 154 the correlation between these factors and SI. (Specific definition of each factor can be seen in 155 Xue et al. [24] and Yu et al. [22]. Values of these factors can be calculated from CFSR data as 156 in section 2 and are not described in detail here.) 157 Based on the above correlation analysis, three factors with the best correlation with the 7 158 SI are filtered out for further study, which are: 159 (1) Mascarene cold high strength index (MH): the average grid points of sea level pressure 160 within [40–-60° E, 20–10° S] region; 161 (2) Tibetan high (eastern type) activity index (TH): the average grid points of 200 hPa 162 geopotential height within [95–105° E, 25–30° N] range; 163 (3) Monsoon circulation index at Bay of Bengal (J1V): the average grid point J1V = 164 V850-V200 within [80–100° E, 0–20° N] region. 165 166 The correlation coefficients between the above three factors and SI are 0.77, -0.80 and 0.86, all of which can reach above 0.75. 167 The Mascarene high in the southern hemisphere enhances the WPSH early in the year, 168 which is a positive correlation and a very close relationship, consistent with previous research 169 [24]. The close relationships among TH, J1V and SI are basically consistent with previous 170 research [22,25]. 171 3. Reconstruction of the dynamical model based on GA 172 Takens [26] set forth and tested the basic idea of reconstructing the dynamical system 173 from the time series of observed data in his phase space reconstruction theory. Hence, 174 nonlinear dynamics research has entered a new stage, where the evolution information of the 175 dynamical system can be extracted from the time series, such as the calculation of fractal 176 dimension and Lyapunov exponent [27]. However, these studies help understand what a 177 chaotic system is and what the periodic system is. Dynamical models for practical problems 178 have not yet been developed fully. Huang and Yi [28] proposed a method of nonlinear 179 dynamic model reconstruction from the actual data and tested it with the Lorenz system, the 8 180 result of which was satisfactory. Therefore, we introduce this idea and improve it for 181 reconstructing a dynamical model of the SI and three factors. 182 The principle of dynamical model reconstruction has been introduced in the 183 previous studies [12,15-16] and is not described in detail here but can be found in Appendix 184 A. 185 The existing parameter estimation methods (such as neighborhood search method and 186 least-square estimation) are mostly one-way search which needs to travel the entire parameter 187 space, so the searching efficiency is low. Because of the limitation of the error gradient 188 convergence and its dependence on initial solution, parameter estimation is prone to fall into 189 local optimum, rather than global optimum. GA is a method extensively used as a global 190 optimization method which has developed in recent years. GA has been found to be excellent 191 in global search and parallel computing, and error convergence rate is greatly improved, so it 192 is helpful in parameter optimization. 193 We can use a simplified second-order nonlinear dynamic model to depict the basic 194 characteristics of the atmosphere and the ocean interactions [29]. In our paper, 195 x 1 ,x 2 ,x 3 ,x 4 is used to represent time coefficient series of SI, MH, TH and J1V, then GA 196 is introduced to reconstruct the dynamical model and for parameter optimization. 197 198 T With S ( D GP) ( D GP) minimum as a restriction, an optimization solution searching is made in the model parameter space by GA. 199 Suppose that the following nonlinear second-order ordinary differentia l equations are 200 taken as the dynamical model of reconstruction and reconstruction, we choose the time 201 coefficient series of SI, MH, TH and J1V from May 1 to July 31 in 2010 as the expected 9 202 data to optimize and retrieve model parameters. dx1 dt dx2 203 dt dx3 dt dx4 dt a1 x1 a2 x2 a3 x3 a4 x4 a5 x1 a6 x2 a7 x3 a8 x4 a9 x1 x2 a10 x1 x3 a11 x1 x4 a12 x 2 x3 a13 x 2 x 4 a14 x3 x 4 2 2 2 2 b1 x1 b2 x2 b3 x3 b4 x4 b5 x1 b6 x2 b7 x3 b8 x4 b9 x1 x2 b10 x1 x3 b11 x1 x4 b12 x2 x3 b13 x2 x4 b14 x3 x4 2 2 2 2 c1 x1 c2 x2 c3 x3 c4 x4 c5 x1 c6 x2 c7 x3 c8 x4 c9 x1 x2 c10 x1 x3 c11 x1 x4 c12 x2 x3 c13 x2 x4 c14 x3 x4 2 2 2 2 d1 x1 d 2 x2 d 3 x3 d 4 x4 d 5 x1 d 6 x2 d 7 x3 d 8 x 4 d 9 x1 x 2 d 10 x1 x3 d 11 x1 x 4 d 12 x 2 x3 d 13 x 2 x 4 d 14 x3 x 4 2 2 2 2 204 (1) Suppose that the parameter matrix P [a1 , a2 ,......a9 ; b1 , b2 ,......b9 ; c1 , c2 ,......c9 ] in 205 206 the 207 S ( D GP)T ( D GP) is the objective function, the individual fitness value is li 208 and the total fitness value is L above equation is the population, the minimal residual sum of squares 1 , S n l . The idiographic operating steps include: coding and i 1 i 209 creating of initial population, calculation of fitness, male individual choice, crossover and 210 variation, and so on. For a complete theoretical explanation, one can refer to the literature 211 [30]. During calculation, the step length is one day. After about 35 times of genetic operations 212 and optimization search, it’s possible to converge to the target adaptive value rapidly and 213 retrieve each optimized parameter of the dynamical equations. After eliminating weak items 214 with little dimension coefficient, the nonlinear dynamic model of SI and three factors are 215 inverted as follows: 216 217 dx1 dt dx2 dt dx3 dt dx 4 dt F1 15.86 x 0.0332 x 838.0874 x 49.2840 x 3.259 10 x 1.5532 10 x x 0.0083 x x 4.9142 10 x x 7 1 2 3 4 4 2 2 4 1 2 2 3 2 4 F2 25.4550 x1 2.245 10 x3 194.127 x4 2.4791 10 x1 x2 0.0222 x2 x3 0.0019 x2 x4 4 3 8 5 4 F3 2.1993 x1 0.0032 x2 73.1122 x3 3.1485 10 x2 2.1743 10 x1 x2 7.2315 10 x2 x3 2 F4 5.7855 x1 0.0225 x2 11.8816 x4 0.3044 x3 0.0054 x4 0.0092 x1 x3 0.0975 x3 x4 2 2 (2) 10 218 The dynamical reconstruction model should be tested. So we chose the data of August 1 219 in 2010 of SI, MH, TH and J1V which do not participate in the modeling as the forecast 220 initial data. Then the Longkuta method is used to carry out numerical integration of the above 221 equations and every step of integration can be regarded as a day forecasting result. The 222 forecast results of four time series within 25 days can be obtained. The correlation 223 coefficients between forecast results and actual results of SI, MH, TH and J1V within 25 days 224 are respectively, 0.5659，0.5746，0.6091，0.5023 and root mean square errors (RMSE) is 225 36.22%, 34.54% , 32.71% and 36.78%. This indicates that the results of long-term forecast 226 within 25 days are unsatisfactory for the dynamical reconstruction model. That is because the 227 integration results may diverge significantly with time, as well as our initial data for 228 integration is relatively simple. Therefore, we should improve the model to get better 229 long-term forecast results. 230 Following these analyses, correlation coefficients ( r ) were evaluated using the t-test 231 ( 0.05 ,). 232 confidence level and were thus deemed to be statistically acceptable. 233 4. Introduction of self-memorization dynamics to improve the reconstructed model All resulting coefficients were found to be statistically significant at the 95% 234 From the above discussion, we know that the accuracy of forecast results of the 235 dynamical reconstruction model within 25 days is unsatisfactory. Literature suggests that 236 introducing the principle of self- memorization into the mature model can improve long-term 237 forecasting results [18,20]. So the self- memorization principle is introduced to improve the 238 model. 239 Following the study of Cao [17] and Feng et al. [19], the mathematical principle of 11 240 241 242 243 244 245 246 247 248 self-memorization dynamics of systems is shown in Appendix B. Using equations (2) as the dynamical kernel, the improved model based on the self-memorization equation (B10) in Appendix B can be expressed as: 0 0 x1t a1i y1i 1i F1 ( x1i , x2i , x3i , x4i ) i p i p 0 0 x2t a2i y2i 2i F2 ( x1i , x2i , x3i , x4i ) i p i p 0 0 x a3i y3i 3i F3 ( x1i , x2i , x3i , x4i ) 3t i p i p 0 0 x a y 4i F4 ( x1i , x2i , x3i , x4i ) 4 t 4 i 4 i i p i p (4) Because we predict the value of SI，the first formula in equation (4) are used to get results of our final prediction. That is: x1t 0 0 i p i p a1i y1i 1i F1 ( x1i , x2i , x3i , x4i ) (5) If we can get the value of a, , Eq. (5) can be used for prediction. Values of a, are associated with the self-memorization function , which is defined in the next section. 249 We make a prediction with the self- memorization equation (5), the model uses the p 250 values before t0 , therefore (t ) in the self- memorization equation(5) in memorizing the 251 p 1 values of x . This explains the reason why we call (t ) the self-memorization function. 252 This is the mathematical basis for introducing the self-memorization principle. 253 The physical basis for introducing the self-memorization principle is that the 254 thermodynamic equation is one of atmospheric motion equations, which implies that the 255 atmospheric motion is an irreversible process. An important contribution to the study of 256 irreversible process is to introduce the memory concept to physics. So the atmospheric 12 257 development in the future is not only related to the state at the initial time, but also to states in 258 the past, which means the atmosphere does not forget its past. 259 5. Improved self-memorization functions for chaotic systems 260 Cao [17] referred to the problem of self-memorization function assignment. Based on 261 experience and tests, he found the memory level gradually decreased with the distance of 262 initial prediction time, so the memory function can be expressed as: 263 0 (i) e a (tN ti ) 1 ti t N P t N P ti t N (6) ti t N 264 It shows that the self-memorization function (6) indicated by Cao only considers the 265 characteristics of memory level gradually decreasing with the distance of initial prediction 266 time, and ignores nonlinear characteristics of the self-memorization function. So we modify 267 the self-memorization function as: 268 0 a (tN ti ) (1 r ) (i ) e ti 1 ti t N P t N P ti t N (7) ti t N 269 Term e a (tN ti ) represents the characteristics of memory level gradually decreasing with the 270 distance of initial prediction time; while ti (1 r ) reflects the nonlinear characteristics of 271 self-memorization function. r and a are the parameters to be determined, and they are 272 important for the specification of the assigned self-memorization function (i ) . In Cao’s 273 study, he found no good way to determine parameters r and a , only believing that they 274 had powerful connection with the past observational data. 275 The Lyapunov exponent(LE) has long been used to study atmospheric and oceanic 276 predictability [31]. In chaotic dynamical systems theory, the largest Lyapunov exponent can 13 277 be portrayed as a whole (long-term) average forecast error growth rate, which generally 278 describes the divergence of nonlinear chaotic systems. So it is often introduced in the forecast 279 study of chaotic systems such as atmospheric and oceanic systems [32-33]. The traditional 280 self-memorization function is useful in the linear periodic system forecast, but not appropriate 281 in the forecast of nonlinear chaotic systems. The largest Lyapunov exponent (LLE) is one of 282 the indexes which can represent the characteristics of chaotic systems. So here r can be 283 considered as LLE of the system, which can be calculated from the reconstructed dynamical 284 equation in section 3. And a can be obtained through GA from historical data. 285 5.1. Determination of Parameter r 286 Based on the above analysis, the solution of r requires the solution of the LLE in 287 dynamical equation (2). 288 5.1.1 Definition of LLE 289 The definition of the Lyapunov spectrum and the largest Lyapunov exponent (LLE) can 290 be referred to the literature [34] and are not described in detail here. The LLE determines the 291 notion of predictability for a dynamical system. A positive LLE is usually taken as an 292 indication that the system is chaotic (provided some other conditions are met, e.g., phase 293 space compactness). 294 5.1.2 The principle of Lyapunov exponent spectrum calculations 295 If the dynamic differential equations are known, through theoretical derivation or using 296 a numerical iterative algorithm to discretize differential equations, the exact Lyapunov 14 297 exponents of known dynamical systems can be obtained. First, we work out the solution of 298 the ordinary differential equations and get the Jacobi matrix. Then, we decompose the Jacobi 299 matrix by QR while necessary orthogonal reformings in a multiple of small time intervals are 300 performed. Finally, the Lyapunov spectrum of the system can be obtained by iterative 301 calculations. The specific calculation principle of the the Lyapunov exponents spectrum are 302 not shown here, which can be referred to Hubertus et al. [35]. 303 5.1.3 Lyapunov spectrum of dynamical systems and determination of r 304 According to the above calculation method, the Lyapunov spectrum of the reconstruction 305 dynamical model of SI and three factors can be obtained as in Fig. 2 which shows that 306 convergence speed and volatility are not too large and they are relatively stable. Final 307 Lyapunov exponents are [0.3816，0.0016，-0.5715], containing both negative Lyapunov 308 exponent and two positive Lyapunov exponents, which demonstrate our dynamical system is 309 indeed a chaotic system. 310 311 Finally, r is taken as the LLE, and r =0.3816. 5.2. Determination of Parameters a 312 Not only the nonlinear characteristics of chaotic system should be considered, but also 313 the impact of the past actual data on self-memorization function should be took into account. 314 So parameter a should be optimized. 315 The equation (5) is written as equation (8): 15 x(t ) f (t , x, ) 316 317 318 319 The initial value is x(ti ) xi , i N , N 1,..., N p . Our goal is to make the smallest error between the fitting values and the observed values of p periods before, i.e., secondary quality index reaches a minimum. J (u ) 320 N ( xˆ(t ) x(t )) i p 1 321 322 325 i i 2 min (9) where xˆ is the estimated value of the prediction model. x is the actual value. When further constraint is added, the self-memorization function should be taken as: (ti ) 1 323 324 (8) (10) Parameter r can be obtained by solving for the optimal solution of equation (9) in the constraint condition of equation (10). Setting J (u ) N ( xˆ (t ) x (t )) i p 1 i i 2 as the objective 326 function of GA, putting (ti ) 1 as a constraint, the specific process of GA is discussed in 327 section 3. 328 We can see the value of parameter a is related with the p data before forecasting time t . 329 The forecasting value is preserved as the early data for the next prediction. With the change of 330 data, the value of parameter a is varied. The first calculation is a1 0.244 through 39 331 iterations of GA, then in next prediction step, parameter a is varied, which we do not list 332 one by one here. In fact, the continuous adjustment of parameter a is more accurate. 333 Self-memorization function is determined, and then it is substituted into equation (5), 334 which can be used to do prediction. 335 6. Model Prediction experiments 16 336 6.1. Effective steps and maximum effective computation time 337 In order to further study the improved model, the algorithm to solve for effective steps 338 and maximum effective computation time should be given, based on the dynamic core 339 (equation(2)) of the improved model. Currently the optimal search method is used, which is 340 achieved based on the size of the differences among many numerical solutions. Its principles 341 can be seen in David [36]. 342 In step interval [ hmin , hmax ], ( hmin is a very small number), n steps are selected, 343 hi hmin i * dtt hmax (i 1, 2,..., n) , dtt is known as the time increment. The n steps are 344 integral to t at the same time, getting the n number of numerical solutions, denoted 345 by y t ( n ) . If their differences Vs (t ) are less than the given tolerance limit , it means that 346 they are relatively close to the true solution at t time. At that time the effective time step 347 interval is [ hmin , hmax ] and the width is Wh (t ) lg hmax lg hmin . Otherwise, if Vs (t ) , 348 that means values between some step appear larger deviations, so removing these step from 349 y t ( n ) will make the difference of the remaining n1 (n1 n) solutions less than . The 350 optimum is to exclude the least number. 351 The effective step intervals which meet such conditions can be gotten. Given integral 352 step, we continue to do integral calculation to t Nd t , here, N is an integer. Integral 353 operation continues until 354 interval [ h j (t1 ), h j 1 (t1 ) ] is divided into m parts for a another round of integral operation, 355 until the difference of the solution in m 1 step is smaller than . In this case, the maximum 356 effective computation time of the dynamical kernel of the improved model (equation (2)) can 357 be obtained, as shown in Fig. 4. there are only two adjacent steps h j (t1 ), h j 1 (t1 ) left. Then step 17 358 From Figure 4, when is certain, for our system, the maximum effective computation 359 time increases with the increment of dtt , and finally remains at 28.7, which is the maximum 360 effective computation time(Figure 4a). When dtt remains constant, the maximum effective 361 computation time oscillates with (Figure 4b). 362 6.2. Prediction experiment of 2010 363 The period from 2 August to 5 September in 2010 is selected to do predictions (35 days in 364 total). August contains abnormal activities of WPSH. Here, the forecasting result is obtained by 365 the sum of equation (5), which is called as step-by-step forecast. The specific procedure is shown 366 as follows: 367 Step-by-step forecasts are made after retrospective order p is fixed. That means when we 368 forecast the SI value of 2 August, we must get y i based on the previous p +1 time SI data 369 and F (x 1i ,x 2i ,x 3i，x 4i ) based on the previous p time SI, MH, TH and J1V data 370 (because x 2 , x 3 and x 4 are MH, TH and J1V). Then using them in equation (5) can obtain 371 the SI value of 2 August. Then, taking the SI value of 2 August as the initial value for the next 372 prediction step, the SI value of 3 August can be generated, and so on. 373 6.2.1 Determination of p 374 When the principle of self-memorization is introduced, the retrospective order p is 375 related with self-memorization of the system [17]. If the system “forgets” slowly, which 376 means parameters a and r are smaller, a high level of p should be used. WPSH abnormal 377 activities are on ten days scale[22,24], which is a slow process in contrast to the large-scale 378 atmospheric motion. So parameters a and r are smaller, and generally p is in the range 379 of 5 to 15 . 18 380 381 382 383 The correlation coefficients and root mean square errors(RMSE) between forecast value and real value can be calculated with different retrospective order, as shown in Table 1. From the table, we can see when p 8 , the correlation coefficient is the largest and the root mean square error is the smallest. So we choose p 8 . 384 After p is determined, the improved self-memorization equation (5) can be used for 385 prediction experiments. Short, medium and long-term integration forecasts of 15 days, 25 386 days and 35days are carried out. Retrospective order p 8 , it means earlier observational 387 data ( p 1 9 ) are used for integration began. The integral result per day is preserved as 388 preliminary information, and will be used for integration in next period. 389 6.2.2 Prediction results of 2010 390 391 392 The forecast results within 35 days are shown in Fig. 5, which show that the forecast result of SI is better. As can be seen in Fig. 5, forecast performance of the first 15 days is better. The 393 correlation coefficient has reached 0.9542 and the root mean square errors(RMSE) is 2.45%. 394 Two peaks and one valley of SI are also forecast very accurate. The forecast time series from 395 15 days to 25 days gradually diverges, but the trend is accurate. The correlation coefficient 396 reaches 0.9254 and the RMSE is 6.37%. Forecast rate is still accurate. The peak of SI in 18 397 August is also forecasts accurately. 398 After nearly 25 days, the RMSE begins to increase: the forecast trend is still accurate 399 with correlation coefficient of 0.8136. Forecast curve from 25 days to 35 days is accurate 400 compared with the actual situation. But the peaks and valleys are not forecast accurately. 401 Especially after 28 days, the error has started to increase significantly. For example, the 19 402 forecast value of August 28 is nearly 100 more than the actual value, resulting a false peak. 403 The root mean square error increases to 19.18%, but it is still controlled within 20%. Fig.5 404 shows the forecast results after 28 days are obviously unsatisfactory with the occurrence of 405 greater oscillations. The effective forecast period (within 28 days) happens to be in accord 406 with the maximum integration time (28.7 units) calculated in section 6.1. 407 In brief, from Fig. 5, we can see that the short-term forecast within 28 days is accurate, 408 and the average RMSEs are less than 8%, which indicates that the reconstruction model can 409 perform accurate predictions of the changing trends of indices. 410 6.3 More forecasting experiments of WPSH abnormal years 411 To further test the forecasting performance of the improved model, cross testing of more 412 experiments is performed. From Fig. 2 discussed in section 2.2, we choose another four years 413 (1998, 2006, 1987 and 1983) in which WPSH intensity is abnormally strong (bigger SI) and 414 five years (1984, 2000, 1994, 1999 and 1985) in which WPSH intensity is abnormally weak 415 (smaller SI) to carry out integral forecast experiments of SI. In accordance with the previous 416 idea, the common 2010 model which is reconstructed in section 4 is used to carry out the 417 cross testing. Forecast results of different time periods (1–15 days as short term, 16–25 days 418 as medium term and 26–35 days as long term) are compared with the actual situation. Cross 419 test results are displayed in Table 2. From the table, we can see the forecast results of the short 420 term and the medium term are accurate, and those of the long term (>26 days) can also be 421 accepted. The results of MH, FLH and TH are similar to those of SI. 422 7. Summary and discussion 423 7.1 Summary 20 424 Although in recent years the WPSH has become one of the leading topics of 425 atmospheric sciences, it is still difficult to be forecast. Combining dynamical system 426 reconstruction idea with the improved self-memorization principle, a new dynamical 427 forecasting model of SI index is developed. The approach of this paper consists of the 428 following steps: 429 (1) We use correlation analysis method to discuss contributions and impact of the key 430 members of EASM to the abnormal WPSH in 2010 and identify the three most significant 431 factors: the Mascarene high, the Tibetan high (eastern type) and Monsoon circulation index at 432 Bay of Bengal. 433 (2) Take the SI, MH, XZ and J1V time series and consider them as trajectories of a set 434 of 4 coupled quadratic differential equations based on the dynamic system reconstruction idea. 435 Parameters of this dynamical model are estimated by a genetic algorithm (GA). 436 (3) Apply the "principle of self-memorization" in order to improve the forecasting 437 results of the above dynamical model. This involves a manipulation of the time series using 438 an exponentially decaying (in time) function (r ) , which also depends on the largest 439 Lyapunov exponent of the above dynamical system. A second free parameter of (a) is 440 determined by minimizing the distance of the modified reconstructed time series. 441 (4)The improved model is used to forecast the SI index. Through 2010 experiments and 442 other 9 experiments of WPSH abnormal years, we find forecast results within 25 days are 443 good, which proves that our improved model has better long-term forecasting results. Given 444 the complexity of the mechanism of the WPSH [3], the new dynamical forecasting model 445 has some scientific significance and practical value. 21 446 Based on the discussion in section 1, we know that the forecast results and credibility of 447 the previous methods are very low after two weeks [11]. So our improved 448 statistical-dynamical model represents an exploration of and supplement to the traditional 449 numerical forecasting and statistical forecasting methods, and also extends the prediction 450 time. 451 7.2 Discussion 452 Why the forecasting results of our improved model are especially good? (1) Previous 453 studies have showed that the long-term forecasting results of dynamical system reconstructed 454 model are unsatisfactory, thus, we introduce the self-memorization principle to improve the 455 long-term forecasting results. Previous studies have proved that this approach is feasible. For 456 example, Gu et al.[18] used the self-memorization principle to improve the traditional T42 457 model. And Wang et al. [25] also carried out dynamical prediction of building subsidence 458 deformation with self-memorization model. Long-term forecasting results of their models 459 were very good. Our study also shows that introducing the self-memorization principle to 460 improve a mature model can bring better long-term forecasting results. (2)The largest 461 Lyapunov exponent is firstly introduced to improve the traditional self-memorization 462 function, making it more appropriate to describe the chaotic systems, such as WPSH. (3) In 463 developing the improved model, the parameters are obtained from the historical data, which 464 contain sufficient information of WPSH abnormal process. Statistical data combining with 465 improved dynamical model makes forecasting results more reliable. 466 467 Although the forecast results of improved model are better, there are still some issues which are needed for further research: 22 468 469 (1) The physical meaning of the factors in the model is not clear, so its dynamical characteristics should be further analyzed. 470 (2) The forecast accuracy has a great relationship with self-memorization functions. 471 Maybe we can find a better self-memorization function to improve forecast accuracy for long 472 term in the future. 473 These two items are the focus of our next work. 474 Acknowledgments.This study is supported by the Chinese National Natural Science Fund for 475 young scholars (41005025/D0505), the Chinese National Natural Science Fund (41375002; 476 41075045) and the Chinese National Natural Science Fund (BK2011123) of Jiangsu Province. 477 478 Conflict of Interests: The authors declare that there is no conflict of interests regarding the 479 publication of this article. 480 481 482 483 484 485 486 487 488 489 APPENDIX A: Principle of dynamical model reconstruction Suppose that the finite difference form of the physical law of any nonlinear system evolving with time can be expressed as follows: qi( j 1) t qi( j 1) t f i (q1jt , q 2jt ,..., qijt ,..., q Njt ) 2t where fi is the generalized nonlinear function of j 2,3,......M 1 q1, q2 ,...,qi ,...,qN (A1) , N is the number of state variables and M is the length of time series of observed data. We assume that fi (q1jt , q2jt ,..., qijt ,..., qNjt ) variable qi and Pi k contains two parts: G jk representing the expanding items containing just representing the corresponding parameters which are real numbers 23 490 491 ( i 1, 2,...N , j 1,2,...M , k 1,2,..., K ). It can be supposed that: K 492 f i ( q1 , q2 ,..., qn ) G jk Pik (A2) k 1 493 The matrix form of equation (A2) is D GP , in which 494 qi3t qit d1 2t d q 4 t q 2 t i i D 2 2t ... ... d M qiMt qi( M 2) t 2t G11 , G12 ,......G1K Pi 1 G , G ,......G P i 21 22 2, K P 2 G ... ... Pi K GM 1 , GM 2 ,.....GM , K ， ， (A3) 495 Coefficients of the above generalized unknown equation can be identified through 496 inverting the observed data. Given a vector D, vector P can be solved to satisfy the above 497 equation. It is a nonlinear system with respect to q , however it is a linear system with respect 498 to P (assume P is unknown). So the classical least square method can be introduced to 499 T T estimate the equation and the regular equation G GP G D can be derived by making the 500 T residual sum of squares S ( D GP) ( D GP) minimum. 501 502 Based on the above approach, coefficients of the nonlinear dynamical systems can be determined and the nonlinear dynamical equations of observed data can be established. 503 504 505 506 APPENDIX B: Mathematical Principle of self-memorization dynamics of systems In general, dynamical equations of a system can be written as: xi Fi ( x, , t ) t i 1, 2,..., J (B1) 507 where J is an integer，xi the ith variable of the system state, parameter. Eq. (B1) 508 expresses the relationship between a local change of x and a source function F . Obviously, 24 is scalar function at the space r0 509 x 510 time T [t p ...t0 ...tq ] 511 space R [ra ...ri ...r ] ,where ri is a spatial point considered. An inner product in space 512 L2 : T R is defined by where t0 is an initial t . Consider a set of time, b ( f , g ) f ( ) g ( )d , f , g L2 513 514 , and the time a and a set of (B2) Accordingly, define a norm b f [ ( f ( ) 2 d ] 515 1 2 a 516 Making a completion for L2 , it becomes a Hilbert space H . A solution of the 517 multi-time model can be regarded as a generalized one in H . Dropping i in Eq. (B1) and 518 applying an operation of the inner product defined in Eq.(B2), by introducing a memorial 519 function (r , t ) , we obtain 520 t t0 ( ) t x d ( ) F ( x, )d t0 (B3) 521 where r in (r , t ) is dropped because of fixing on the spatial point r0 . Supposing 522 variable x and function (r , t ) etc. are all continuous, differentiable and integrable, 523 following the calculus, making an integration by parts for the left of q. (B3) yields 524 525 526 527 t t0 ( ) t x d (t ) x(t ) (t0 ) x(t0 ) x( ) ( )d t 0 (B4) where (t ) (t ) / t . Applying the median theorem in calculus to the third term in the right-hand side of Eq. (B4). i.e. is obtained. t x( ) ( )d xm (t0 )[ (t ) (t0 )] t0 (B5) 25 528 529 530 where the median x m (t0 ) x(tm ), t0 tm t . Substituting Eqs. (B4) and (B5) for (B3) and performing algebraic operation, we get x(t ) (t0 ) (t ) (t0 ) m 1 t x(t0 ) x (t0 ) ( ) F ( x, )d (t ) (t ) (t ) t (B6) 0 531 As the first and second term in Eq. (B6) relate to the x value only on the fixed point 532 r0 itself at the initial time t0 and the middle time t m , therefore they are called a 533 self-memory term. Naturally, call the third term an exogenous effect, i.e., total effect 534 contributed by other spatial points to point r0 in an interval t0 , t . 535 For multi-time ti , i p, p 1..., t0 , t , similar to Eq. (B4), we have t p1 t p 2 t t x x x d ( ) d ... ( ) d ( ) F ( x, )d t p1 t0 t p 536 537 By eliminating the same term (ti ) x(ti ), i p 1, p 2,..., 0 , it gives 538 (t ) x(t ) (t p ) x(t p ) [ (ti 1 ) (ti )]x m (ti ) ( ) F ( x, ) d 0 539 540 t p ( ) 543 t i p t p are used in the following context. Then, equation(B7) can be written as: 0 t i p t p t xt p x p xim ( i 1 i ) ( ) F ( x, )d 0 (B8) m Setting x p x p1 , p1 0 , we rewrite Eq. (B8) as: xt 1 t 0 i p 1 xim ( i 1 i ) 1 t t t p ( ) F ( x, )d S1 S 2 (B9) 544 We call S 1 a self-memory term, S 2 an exogenous effect term. 545 For the convenience of calculations, to discretize self-memorization equation, the 546 (B7) For simplicity, setting t (t ), 0 (t 0 ), xt x(t ), x0 x(t 0 ) , similar notations 541 542 0 integration in Eq. (B9) in Appendix B is replaced by summation and the differential by 26 547 difference, and the median xim is simply replaced by the mean of two values at adjoining 548 times, i.e., xim 1 ( xi 1 xi ) yi 2 By sampling an equal time interval t , i.e., ti t0 it , i 1, 0, 1, 2,..., p , taking an 549 550 equal time interval t i t i 1 t i 1 , where t0 is initial time, t0 t is forecast time, p 551 is retrospective order, and incorporating i and t , a discretized self-memorization 552 equation is obtained: xt 553 1 i p 1 i yi 0 F ( x, i ) i p 554 where F is the dynamic core of the self-memorization equation; i 555 i 556 557 (B10) i ( i 1 i ) t ; i . t We call a technique with which one make forecast and computation based on Eq.(B10) a self-memorization principle. 558 559 560 561 References: 562 [1] Miyasaka, T., and H. Nakamura. Structure and formation mechanisms of the Northern 563 Hemisphere summertime subtropicalhighs. J. Climate, 18 (2005) 5046–5065. 564 [2] Kurihara, K.. 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Local Lyapunov exponents of the quasi–geostrophic ocean dynamics. Appl Math Comput. 104 (1999) 217—257. [34] Firdaus E. Udwadia, Huertus F.von Bremen. Computation of Lyapunov 649 characteristic exponents for continuous dynamical systems. Z. angew. Math. Phys. 650 (2001) 123-128. 651 652 653 654 53 [35] Hubertus F. von Bremen, Firdaus E. Udwadia, Wlodek Proskurowski. An efficient QR based method for the computation of Lyapunov exponents. Physica D. 101 (1997) 1-16. [36] DAVID KIRKPATRICK. OPTIMAL SEARCH IN PLANAR SUBDIVISIONS. SIAM J. COMPUT. 12 (1983) 28-35. 655 656 657 31 658 List of Figures: 659 Fig.1 SI of summer half-year (from May to October) from 1982 to 2011 660 Fig.2. The average monthly geopotential height field at 500 hPa of July and August in 2010 661 Fig.3 the Lyapunov spectrum of the reconstructed dynamical model 662 Fig.4. the maximu m effect ive computation time changing with time increment 663 limit (b) 664 Fig.5 dtt (a) and tolerance The 35 day forecast results of the subtropical high area index 665 666 667 Table captions: 668 Table1. The correlation analysis between SI and three factors 669 Table2. The correlation coefficient and Root mean square errors when the retrospective order 670 different 671 Table3. Co rrelation coefficients and Root mean square errors between forecast value and real value of 672 different events of WPSH abnormal years p is 673 674 675 676 677 678 679 680 681 682 32 683 Figure: 684 685 686 Fig.1 SI of summer half-year (from May to October) from 1982 to 2011 687 688 689 Fig.2. The average monthly geopotential height field at 500 hPa of July and August in 2010 33 690 691 Fig.3 the Lyapunov spectrum of the reconstructed dynamical model 692 693 34 694 695 (a) 696 697 (b) 35 698 Fig.4. the maximu m effect ive computation time changing with time increment 699 limit (b) dtt (a) and tolerance 700 701 Fig.5 The 35 day forecast results of the subtropical high area index 702 703 Table: 704 Table1. The correlat ion coefficient (C.C.) and Root mean square errors (RM SE) when the retrospective 705 order p p is different 1 5 6 7 8 9 10 11 12 C.C. 0.38 0.72 0.78 0.81 0.94 0.92 0.87 0.86 0.84 RMSE 27.53 % 14.33 % 13.76 % 13.06 % 3.72% 4.63% 5.01% 6.77% 7.01% 13 0.83 14 0.71 15 0.67 20 0.44 30 0.36 40 0.32 50 0.28 60 0.21 70 0.17 10.98 % 13.24 % 15.50 % 24.13 % 28.91 % 30.22 % 39.11 % 42.18 % 50.24 % 1 5 6 7 8 9 10 11 12 C.C. 0.38 0.72 0.78 0.81 0.94 0.92 0.87 0.86 0.84 RMSE 27.53 14.33 13.76 13.06 3.72% 4.63% 5.01% 6.77% 7.01% % % % % 13 14 15 20 30 40 50 60 70 0.83 10.98 0.71 13.24 0.67 15.50 0.44 24.13 0.36 28.91 0.32 30.22 0.28 39.11 0.21 42.18 0.17 50.24 p C.C. RMSE p p C.C. RMSE 36 % % % % % % % % % 706 707 708 709 710 Table2. The correlation coefficients (C.C.) and Root mean square errors (RMSE) 711 between forecast value and real value of different events of WPSH abnormal years 712 Sta tisti cal tests Short term medium term long term (1~15days) (16~25 days) (26~35 days) Forecast events C.C. RM SE C.C. RM SE C.C. RM SE 0.957 2.92% 0.812 4.51% 0.723 11.91% 0.936 2.88% 0.877 4.72% 0.776 10.98% 0.942 3.16% 0.881 3.97% 0.718 11.48% 0.958 3.40% 0.820 4.06% 0.729 10.90% SI bigger event1(1998. 06.21 as initial values to forecast) SI bigger event2(2006. 07.18 as initial values to forecast) SI bigger event3(1987. 07.08 as initial values to forecast) SI bigger event4(1983. 08.05 as initial values to forecast) 37 SI smaller event1(1984. 07.28 as initial values to 0.951 3.12% 0.815 3.43% 0.789 11.21% 0.892 3.47% 0.825 4.97% 0.708 10.76% 0.914 4.52% 0.755 3.83% 0.698 11.87% 0.809 3.97% 0.873 3.89% 0.777 11.70% 0.926 2.07% 0.882 4.85% 0.737 10.17% 0.921 3.28% 0.838 4.25% 0.739 11.22% forecast) SI smaller event2(2000. 06.29 as initial values to forecast) SI smaller event3(1994. 08.17 as initial values to forecast) SI smaller event4(1999. 06.12 as initial values to forecast) SI smaller event5(1985. 07.11 as initial values to forecast) The average 713 714 715 38
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