【正文】 goal is to estimate a model of the form is the mapping to a high dimensional (and possibly infinite dimensional) featurespace, and the residuals e are assumed to be . with zero mean and constant (and finite) variance. The following optimization problem with a regularized cost function is formulated:where a is a given constant which can take either 1 or 1. The first restriction is the standard modelformulation in the LSSVM framework. The second restriction is a shorthand for the cases where we want to impose the nonlinear function to be even (resp. odd) by using a = 1 (resp. a = ?1). Thesolution is formalized in the KKT lemma. 3. Application to Chaotic Time Series In this section, the effects of imposing symmetry to the LSSVM are presented for two cases of chaotic time series. On each example, an RBF kernel is used and the parameters and are found by 10fold cross validation over the corresponding training sample. The results using the standard LSSVM are pared to those obtained with the symmetryconstrained LSSVM (SLSSVM) from (2). The examples are defined in such a way that there are not enough training datapoints on every region of the relevant space。 thus, it is very difficult for a blackbox model to ”learn” the symmetry just by using the available information. The examples are pared in terms of the performance in the training sample (crossvalidation mean squared error, MSECV) and the generalization performance (MSE out of sample, MSEOUT). For each case, a Nonlinear AutoRegressive (NAR) blackbox model is formulated: where g is to be identified by LSSVM and SLSSVM. The order p is selected during the crossvalidation process as an extra parameter. After each model is estimated, they are used in simulation mode, where the future predictions are puted with the estimated model using past predictions: . Lorenz attractor This example is taken from [1]. The x?coordinate of the Lorenz attractor is used as an example of a time series generated by a dynamical system. A sample of 1000 datapoints is used for training, which corresponds to an unbalanced sample over the evolution of the system, shown on Figure 1 as a timedelay 2 (top) shows the training sequence (thick line) and the future evolution of the series (test zone). Figure 2 (bottom) shows the simulations obtained from both models on the test zone. Results are presented on Table 1. Clearly the SLSSVM can simulate the system for the next 500 timesteps, far beyond the 100 points that can be simulated by the LSSVM. . Multiscroll attractors This dataset was used for the Time Series Prediction Competition . The series was generated by where h is the multiscroll equation, x is the 3dimensional coordinate vector, and W,V are the interconnection matrices of the nonlinear function (a 3units multilayer perceptron, MLP). This MLP function hides the underlying structure of the attractor .A training set of 2,000 points was available for model estimation, shown on Figure 3, and the goal was to predict the next 200 points out of sample. The winner of the petition followed a plete methodology involving local modelling, specialized manysteps ahead crossvalidation parameters tuning, and the exploitation of the symmetry properties of the series (which he did by flipping the series around the time axis). Following the winner approach, both LSSVM and SLSSVM are trained using 10stepahead crossvalidation for hyperparameters selection. To illustrate the difference between both models, the out of sample MSE is puted considering only the first n simulation points, where n = 20, 50, 100, 200. It is important to emphasize that both models are trained using exactly the same methodology for order and hyperparameter selection。 the only difference is the symmetry constraint for the SLSSVM case. Results are reported on Table 2. The simulations from both models are shown on Figure 4. Figure 1: The training (left) and test (right) series from the x?coordinate of the Lorenz attractor Figure 2: (Top) The series from the x?coordinate of the Lorenz attractor, part of which is used for training (thick line). (Bottom) Simulations with LSSVM (dashed line), SLSSVM (thick line) pared to the actual values (thin line). Figure 3: The training sample (thick line) and future evolution (thin line) of the series from the Time Series Competition Figure 4: Simulations with LSSVM (dashed line), SLSSVM (thick line) pared to the actual values (thin line) for the next 200 points of the data. Table 1: Performance of LSSVM and SLSSVM on the Lorenz data. Table 2: Performance of LSSVM and SLSSVM on the data. 4. Conclusions For the task of chaotic time series prediction, we have illustrated how to use LSSVM regression with symmetry constraints to improve the simulation performance for the cases of series generated by Lorenz attractor and multiscroll attractors. By adding symmetry constraints to the LSSVM formulation, it is possible to embed the information about symmetry into the kernel level. This translates not only in better predictions for a given time horizon, but also on a larger forecast horizon in which the model can track the time series into the future. 基于最小二乘支持向量回歸的短期混沌時(shí)間序列預(yù)測(cè)摘要：本文解釋了在混沌序列預(yù)測(cè)范圍內(nèi)，先驗(yàn)知識(shí)到模型階段任意使用對(duì)稱性的作用效果。以洛倫茲吸引器和多渦卷吸引子為例，表明了使用具有對(duì)稱約束的最小二乘支持向量機(jī)能夠提高仿真性能，其不僅能夠獲得的預(yù)測(cè)結(jié)果精確，而且所獲得的預(yù)測(cè)范圍能夠得到擴(kuò)展。應(yīng)用非線性時(shí)間序列分析，對(duì)一個(gè)非線性的黑箱模型的預(yù)測(cè)，要得到準(zhǔn)確的預(yù)測(cè)，從一開始的一系列觀測(cè)是比較常見的做法。通常一個(gè)時(shí)間序列模型是根據(jù)可用的時(shí)間t數(shù)據(jù)預(yù)測(cè)，他的最后評(píng)估時(shí)基于t+1時(shí)刻起的仿真性能。由于時(shí)間序列的本質(zhì)產(chǎn)生于混沌系統(tǒng)，那里的時(shí)間序列不僅顯示非線性行為而且激烈的范圍變化引起了吸引子的局部不穩(wěn)定，這是一個(gè)非常具有挑戰(zhàn)性的任務(wù)。因此，混沌時(shí)間序列已經(jīng)在幾次時(shí)間序列的競(jìng)爭(zhēng)中被看做基準(zhǔn)?；煦鐣r(shí)間序列的模型可以利用它的一些屬性來(lái)改善。如果真正的底層系統(tǒng)是對(duì)稱的，那么這個(gè)信息可以強(qiáng)加到模型中作為先驗(yàn)知識(shí)。在這篇文章中，短期預(yù)測(cè)混沌時(shí)間序列是利用最小二乘支持向量機(jī)（LSSVM）的回歸來(lái)產(chǎn)生。我們的分析表明，與無(wú)約束的最小二乘支持向量機(jī)算法相比，利用對(duì)稱約束的最小二乘支持向量機(jī)可以產(chǎn)生精確的預(yù)測(cè)結(jié)果：不僅得到了精確的預(yù)測(cè)結(jié)果，而且這些預(yù)測(cè)的預(yù)測(cè)界限還得到了擴(kuò)展。這篇文章的摘要組織如下。第二部分描述了最小二乘支持向量回歸的技術(shù)，以及對(duì)稱可以以一種簡(jiǎn)便的方法實(shí)行。第三部分描述了以x坐標(biāo)下的洛倫茲吸引器為例的應(yīng)用，以及多軸吸引子的非線性轉(zhuǎn)變所產(chǎn)生的數(shù)據(jù)。最小二乘支持向量機(jī)（LSSVM）是一個(gè)強(qiáng)大的非線性黑箱模型的回歸方法，它在所謂的輸入是一系列非線性映射