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摘要: 呼吸会引起体内器官和肿瘤的运动, 这会显著影响放射治疗的过程和效果. 人体内部膈肌和胸腹部外表面是当前两种与呼吸系统高度相关的结构, 本文对其进行系统研究, 提出了一种新的分步子空间映射(Two-step subspace mapping, TSSM)算法, 通过对体外胸腹部表面的测量, 来预测体内膈肌的运动. 首先采用三维图像分割技术对4D CT图像进行分割, 在不使用标记物的情况下, 准确测量体内膈肌和体外胸腹部表面的位移. 为了解决跨空间的预测问题, TSSM首先构造特征子空间, 并将膈肌数据和胸腹外表面数据分别映射到各自的子空间中, 以减少数据的相关性和冗余信息; 然后通过线性岭回归优化过程, 对两个子空间进行二次映射, 从而有效地捕获跨空间数据之间的相关性. 根据训练得到的相关模型, 通过体外胸腹部外表面的运动情况, 对体内膈肌的运动情况进行准确的预测. 为了研究数据之间的非线性关系, 进一步将TSSM推广到了基于核的TSSM (kTSSM)算法. 实验表明, 该方法可以根据腹腔外表面的运动情况, 准确地对体内膈肌位移进行预测, 优于经典的线性模型和ANN模型. 给出了优化算法的解析解, 其运算速度快, 将有助于提高放射治疗中门控技术和跟踪技术的效率和精度.Abstract: Respiratory induced organ and tumor motion has great influence in radiation therapy. Two highly correlated structures with respiratory are comprehensively investigated including internal diaphragm and external thoracoabdominal surface in this paper. A novel two-step subspace mapping (TSSM) algorithm is proposed to predict the diaphragm displacement by markerless thoracoabdominal surface measurement. TSSM first incorporates 3D image segmentation to accurately measure the displacement of the diaphragm and the thoracoabdominal surface without markers on 4D CT images. To solve the cross-domain estimation problem, TSSM first constructs eigenspaces and projects the two organs＇ displacement data into their corresponding subspaces to reduce the redundancy and irrelevance. Then, TSSM makes a mapping between the subspaces by a Ridge regression optimization to effectively characterize their correlation. Based on the trained correlation model, the diaphragm domain can be adapted to the abdominal surface domain, and then the diaphragm displacement can be estimated by the abdominal surface without markers. In order to investigate the non-linear correlation, TSSM is further extended to kernel TSSM (kTSSM). Experiments show the proposed method can accurately predict the displacement of the internal diaphragm by the external thoracoabdominal surface, and outperform the classical linear model and ANN model. The simple closed form for the optimization algorithm leads to an extremely fast algorithm, which has potential for improving the timing accuracy of surface-guided gating and tracking in radiotherapy.
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Key words:
- Respiratory motion /
- thoracoabdominal surface /
- diaphragm /
- subspace mapping /
- regression model /
- 4D CT
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电熔镁砂是一种生产航天、航空和工业所需耐火材料的原料. 电熔镁砂生产过程采用多台电熔镁炉, 熔炼菱镁矿石, 生产电熔镁砂. 电熔镁炉是一种重大耗能设备, 耗电成本占总生产成本的60%以上[1-2]. 需量是电熔镁砂生产过程的耗电指标, 供电部门规定需量为当前时刻和过去29个采样时刻的电熔镁砂生产用电功率的均值, 其采样周期为7秒[3]. 电熔镁砂生产企业采用需量监控系统监控需量值, 当需量达到其限幅值(22 300 kW)时会切断其中一台电熔镁炉的供电, 使需量不超过限幅值, 然而切断供电会破坏炉内温度场吸热和放热平衡, 降低产品质量[4].
在电熔镁砂生产过程中, 当原料杂质成分含量增大和颗粒大小变化等造成阻抗减小, 导致熔化电流增大、功率增大, 进而需量上升. 由于熔化电流设定值不变, 因此熔化电流控制系统使熔化电流下降, 需量下降, 出现需量先升后降的尖峰现象. 当需量尖峰值达到其限幅值会造成拉闸断电, 为了避免需量尖峰造成的不必要的断电, 需要对需量尖峰进行识别, 因此需要对需量进行多步预报.
需量多步预报的研究集中在城市用电领域. 文献[5]利用采样周期为1小时的用电需量数据, 提出一种自回归模型与三次样条曲线相结合的混合负荷多步预报方法, 预报大西洋城未来一天每小时的用电需量. 文献[6]利用采样周期为0.5小时的用电需量数据, 提出基于信号滤波和季节调整的多输出前馈神经网络与经验模态分解相结合的多步预报模型, 预报澳大利亚新南威尔士州未来一天每半小时的用电需量. 文献[7]利用采样周期为1分钟的需量数据, 提出基于非线性自回归神经网络的多步预报方法, 预报未来1分钟、10分钟、1小时和2小时的住宅用电需量. 文献[8]利用采样周期为0.5小时的用电需量数据, 提出基于多目标优化算法的最小二乘支持向量机的多步预报方法, 预报电力市场未来0.5小时、1小时、2小时和3小时的用电需量. 文献[9]利用采样周期为1天的用电需量数据, 提出基于支持向量机的多步预报模型, 预报某配电公司未来30天的每天最大用电需量. 文献[10]利用采样周期为0.5小时的用电需量数据, 提出基于支持向量机、极限学习机和多层循环神经网络的用电需量多步预报方法, 预报了未来1.5小时的每半小时的城市用电需量. 文献[11]利用采样周期为1小时的用电需量数据, 提出基于自回归滑动平均模型的某地区未来一天的用电需量预报方法. 文献[12]利用采样周期为1个月的用电需量数据, 提出基于多层神经网络的未来12个月的地区用电需量预报模型. 文献[13]利用采样周期为1个月的用电需量数据, 提出基于神经网络的未来12个月的地区用电需量预报方法. 文献[14]利用采样周期为15分钟的用电需量数据, 提出基于变分模态分解的某地区未来1分钟、3分钟和5分钟的用电需量预报方法. 文献[15]利用采样周期为30秒的用电需量数据, 提出基于集成经验模态分解和卷积神经网络的某地区未来30秒、1分钟、1.5分钟和2分钟的用电需量预报方法. 电熔镁砂生产用电需量的变化是快变化的动态系统, 需量的采样周期为7秒, 发生需量尖峰的整个时间在70秒之内. 由于电熔镁砂生产过程是由多台电熔镁炉运行组成, 其用电需量变化过程是模型结构与系统阶次未知的非线性动态系统, 而城市用电需量变化是一个慢变化的系统, 可以采用上述静态建模方法[5-15]. 电熔镁砂生产用电需量的多步预报难以采用文献[5-15]方法进行需量尖峰的准确预报.
文献[16-17]利用采样周期为7秒的用电需量数据, 提出数据与模型驱动的电熔镁砂生产用电需量单步预报方法. 为了提高需量预报精度, 文献[18]采用用电需量大数据提出系统辨识与自适应深度学习相结合的电熔镁砂生产用电需量一步预报方法. 由于需量尖峰是需量先升高后下降, 需要对需量进行多步预报.
本文利用电熔镁砂生产过程熔化电流闭环控制系统方程, 建立了由线性模型和未知非线性动态系统组成的群炉需量多步预报模型, 将系统辨识与自适应深度学习相结合, 采用端边云协同结构, 提出了电熔镁砂生产用电需量多步智能预报方法. 采用电熔镁砂生产过程的工业大数据验证了所提方法可以准确预报用电需量的变化趋势.
1. 电熔镁砂需量监控过程描述
如图1所示, 电熔镁砂的生产过程由多台电熔镁炉控制系统, 供电系统和需量监控系统组成, 每台电熔镁炉控制系统由电熔镁炉、拉闸系统、加料系统、熔化电流控制系统组成[2]. 需量监控系统由功率采集装置和需量监控计算机组成. 电熔镁炉采用埋弧方式, 通过加料系统将原矿送入电熔镁炉内, 电流控制系统控制三相电极与原料之间的距离使产生的电弧电流达到生产工艺规定的熔化电流, 形成熔池. 边熔化边加料, 使熔池增高至炉口, 熔炼过程结束. 熔炼过程持续约10小时, 每台电熔镁炉每炉次耗电约40 MWh, 因此电熔镁炉是高耗能设备.
图 1 电熔镁砂生产用电需量监控流程图Fig. 1 An flow chart of electricity demand monitoring process for a fused magnesia production为了节能减排, 电力部门规定当前时刻和过去29个时刻的功率的平均值作为当前时刻的需量值并规定了需量的限幅值, 当需量实际值超过需量限幅值对用电企业罚款, 因此生产企业设立需量监控系统, 操作人员监视需量实际值, 当超过限幅值, 通过拉闸系统切断其中一台电熔镁炉的电源. 由于电熔镁砂生产过程常出现尖峰现象, 导致不必要的拉闸. 为避免尖峰导致的不必要的拉闸断电, 需要对尖峰进行预报, 由于尖峰现象是需量先升高后下降, 因此需要对需量进行多步预报.
2. 电熔镁砂生产用电需量多步预报模型
需量$ \bar{p}(k) $为$ k $时刻群炉功率$ p(k) $与过去29个采样周期的群炉功率的均值, 即
$$ \begin{split} \bar{p}(k)=\;& \frac{1}{30}\sum_{j = 0}^{29} p(k-j) =\\ &\bar{p}(k-1)+\frac{p(k)-p(k-30)}{30} \end{split} $$ (1) 其中, $ k = 1 $表示采样周期7秒, $ (k+n) $时刻的需量$\bar{p}(k+n)$为
$$ \begin{split} \bar{p}(k+n) = &\frac{1}{30}\sum_{j = 0}^{29} p(k+n-j)=\frac{1}{30}\sum_{i = 0}^n{p}(k+n-i)\;+\\&\frac{1}{30}\sum_{j = 0}^{29-n} p(k-j)=\bar{p}(k)\;-\\ &\frac{1}{30}\sum_{i = 0}^n{p}(k-30+i)+\frac{1}{30}\sum_{i = 1}^n p(k+i)\ \end{split} $$ (2) 其中, 群炉功率$ p(k+1), \cdots, p(k+n) $未知, 因此需要建立群炉功率多步预报模型.
2.1 群炉功率多步预报模型
首先需建立群炉功率$ p(k) $的动态模型. 电熔镁炉群炉功率$ p(k) $为 $M$个电熔镁炉功率之和$ \sum_{s = 1}^M p_s(k) $.
采用文献[18]的方法建立第$ s $个电熔镁炉的功率$ p_s(k) $的动态模型. 第$ s $个电熔镁炉的熔化电流$ y_s(k) $为电流闭环控制系统的输出, 其动态模型为
$$ \begin{align} T_s(z^{-1})y_s(k) = G_s(z^{-1})y^*+v_s(k) \end{align} $$ (3) 其中, $ y^* $是已知的熔化电流设定值, $ v_s(k) $为未知非线性项, $ T_s(z^{-1}) $ 和 $ G_s(z^{-1}) $为关于$ z^{-1} $的多项式, 即$ T_s(z^{-1}) = 1+t_{s1}(z^{-1})+t_{s2}(z^{-2})+t_{s3}(z^{-3}) $, $G_s(z^{-1}) = g_{s0}+g_{s1}(z^{-1})+g_{s2}(z^{-2})$.
第$ s $个电熔镁炉的功率动态模型为
$$ \begin{split} p_s(k) =\; &\sqrt{3}Uy_s(k)\cos \phi =-\;t_{s1}p_s(k-1)\;-\\ &t_{s2}p_s(k-2)-t_{s3}p_s(k-3)\;+\\ &d_{s0}p^*+v_s(k) \end{split} $$ (4) 其中, $ p^* $是熔化电流设定值 $ y^* $对应的功率; $p_s(k- 1) $, $p_s(k-2) $, $ p_s(k-3)$为$ (k-1) $, $ (k-2) $, $(k- 3) $时刻的第$s$个电熔镁炉功率; $d_{s0} =g_{s0}+ g_{s1}+ g_{s2} $.
采用未知常数$ t_{1}, t_{2}, t_{3}, d_{0} $代替式(4)中的$ t_{s1}, t_{s2}, t_{s3}, d_{s0} $, 群炉功率$ p(k) $为
$$ \begin{split} p(k) =\; &\sum_{s = 1}^{M} p_s(k)= -\;t_{1}p(k-1)-t_{2}p(k-2)\;-\\ &t_{3}p(k-3)+d_{0}p^*+v(k) \end{split} $$ (5) 其中, $ v(k) $为$v(k) = \sum_{s = 1}^{M}\left((t_1-t_{s1})p(k-1)+\right. (t_2-$ $t_{s2})p(k-2) $ $+\;(t_3\;-\;t_{s3})p(k-3)\;-\;(d_0\;-\;d_{s0})p^*\;+$ $v_s(k) ). $
由式(5)可得$ (k+1) $时刻的群炉功率$ p(k+1) $为
$$ \begin{split} {p}(k+1) =\; &{\boldsymbol{\varphi}}(k){{\boldsymbol{\theta}}}^1+{v}(k+1)=\\ &{\boldsymbol{\varphi}}(k){{\boldsymbol{\theta}}}^1+{r}(k+1) \end{split} $$ (6) 其中, ${\boldsymbol{\varphi}}(k) \,= \,( p(k),\, p(k-1),\, p(k-2), p(k-3) )$, $ {{\boldsymbol{\theta}}}^1 = ( {\theta}_1^1, {\theta}_2^1, {\theta}_3^1, {\theta}_4^1)^{\rm{T}} = (-t_{1}, -t_{2}, -t_{3}, d_{0})^{\rm{T}} $, ${r}(k\;+ 1) = {v}(k+1).$ $ (k+2) $时刻的群炉功率 $ p(k+2) $为
$$ \begin{split} {p}(k+2) =\; &{\boldsymbol{\varphi}}(k+1){\boldsymbol{\theta}}^1+v(k+2)=\\ &({\boldsymbol{\varphi}}(k){{\boldsymbol{\theta}}}^1+{v}(k+1), p(k), p(k-1),\\ & p^* ){\boldsymbol{\theta}}^1+v(k+2)=\\ &p(k)((\theta_1^1)^2+\theta_2^1)+p(k-1)(\theta_1^1\theta_2^1+\theta_3^1)\;+\\ &p(k-2)\theta_3^1\theta_1^1+p^*(\theta_1^1\theta_4^1+\theta_4^1)\;+\\ &r(k+1)\theta_1^1+v(k+2)=\\ &{\boldsymbol{\varphi}}(k){{\boldsymbol{\theta}}}^2(k)+r(k+2) \\[-10pt]\end{split} $$ (7) 其中, 线性模型参数${{\boldsymbol{\theta}}}^2 = ({\theta}_1^2, {\theta}^2_{2}, {\theta}^2_{3}, {\theta}^2_{4})^{\rm{T}},{\theta}_1^2 = (\theta_1^1)^2\, +$ $\theta_2^1, {\theta}^2_{2} = \theta_1^1\theta_2^1+\theta_3^1, {\theta}^2_{3} = \theta_3^1\theta_1^1, {\theta}^2_{4} = \theta_1^1\theta_4^1+\theta_4^1$, 未知非线性项$ r(k+2) $ 为 ${r}(k+2) \;=\; {r}(k+1)\theta_1^1(k+1)\;+ v(k+2)$.
采用数学归纳法可以证明假设$ (k+n-1) $时刻的群炉功率$ p(k+n-1) $为式(8)成立, 则$ (k+n) $时刻的群炉功率$ p(k+n) $表达式(9)成立
$$ {p}(k+n-1) = {\boldsymbol{\varphi}}(k){{\boldsymbol{\theta}}}^{n-1}+ {r}(k+n-1) $$ (8) $$ {p}(k+n) = {\boldsymbol{\varphi}}(k){{\boldsymbol{\theta}}}^n+ {r}(k+n) $$ (9) 其中, ${{\boldsymbol{\theta}}}^{n-1} = ({\theta}_1^{n-1}, {\theta}^{n-1}_{2}, {\theta}^{n-1}_{3}, {\theta}^{n-1}_{4})^{\rm{T}}$, ${{\boldsymbol{\theta}}}^n = ({\theta}_1^n,$${\theta}_2^n, {\theta}^n_{3}, {\theta}^n_{4})^{\rm{T}}, {\theta}_1^n = {\theta}_1^{n-1}\theta_1^1+{\theta}_1^{n-2}\theta_2^1+{\theta}_1^{n-3}\theta_3^1$, ${\theta}_2^n = $ ${\theta}_2^{n-1}\theta_1^1+{\theta}_2^{n-2}\theta_2^1+{\theta}_2^{n-3}\theta_3^1$, $ {\theta}_3^n = {\theta}_3^{n-1}\theta_1^1+{\theta}_3^{n-2}\theta_2^1\;+ $ ${\theta}_3^{n-3}\theta_3^1 $, $ {\theta}_4^n = {\theta}_4^{n-1}\theta_1^1+{\theta}_4^{n-2}\theta_4^1+{\theta}_4^{n-3}\theta_3^1+\theta_4^1$, 未知非线性项$ {r}(k+n) $为${r}(k+n) = {r}(k+n- 1)\theta_1^1+{r}(k\;+ n-2)\theta_2^1+{r}(k+n-3)\theta_3^1+{v}(k+n)$. 因此群炉功率多步预报模型为
$$ \begin{align} {P}(k+n) = &\Theta^{\rm{T}}\varphi^{\rm{T}}(k)+ {R}(k+n) \end{align} $$ (10) 其中, $ {P}(k+n) = (p(k+1), \cdots, p(k+n))^{\rm{T}} $, 模型参数$ \Theta = ({\boldsymbol{\theta}}^1, \cdots, {\boldsymbol{\theta}}^n) $, $R(k+n) = (r_1(k+1), \cdots, r_i(k\,+ i), \cdots, r_n(k+n))^{\rm{T}}$.
由式(10)可知模型参数$ \Theta $的辨识方程为
$$ \begin{align} P_s(k) = \begin{pmatrix} {\boldsymbol{\varphi}}(k-1){\boldsymbol{\theta}}^1\\ \vdots\\ {\boldsymbol{\varphi}}(k-i){\boldsymbol{\theta}}^i\\ \vdots\\ {\boldsymbol{\varphi}}(k-n){\boldsymbol{\theta}}^n \end{pmatrix} + {R_s}(k) \end{align} $$ (11) 其中, $ n\times 1 $的向量$ P_s(k+n) $为
$$ P_s(k+n) = (p(k), \cdots, p(k))^{\rm{T}} $$ $$ {R_s}(k) = (r_1(k), \cdots, r_i(k), \cdots, r_n(k))^{\rm{T}} $$ 采用最小二乘算法[19]离线辨识模型参数$ \Theta $, 由式(10)和式(11)可得群炉功率预报模型为
$$ \begin{align} {P}(k+n) = \hat{\Theta}^{\rm{T}}{\boldsymbol{\varphi}}^{\rm{T}}(k)+ \bar{R}(k+n) \end{align} $$ (12) 其中, $\hat{\Theta}^{\rm{T}} = ({\hat{\boldsymbol{\theta}}}^1, \cdots, {\hat{\boldsymbol{\theta}}}^n), \bar{R}(k+n) = (\Theta-\hat{\Theta}^{\rm{T}})\varphi^{\rm{T}}(k)\,+$ $R(k+n) = (\bar{r}_1(k+1), \cdots, \bar{r}_i(k+i), \cdots, \bar{r}_n(k + n))^{\rm{T}}$,
$$ \begin{split} \bar{r}_i(k+i) =\;& f_i(p(k), p(k-1), \cdots, \\ &\bar{r}_i(k), \bar{r}_i(k-1), \cdots) \end{split} $$ (13) 其中, $ f(\cdot) $是模型结构与阶次未知的未知非线性函数, $i = 1, \cdots, n$. 由式(11)知
$$ \begin{align} {\bar{R}_s}(k) = P_s(k)- \begin{pmatrix} {\boldsymbol{\varphi}}(k-1){\hat{\boldsymbol{\theta}}}^1\\ \vdots\\ {\boldsymbol{\varphi}}(k-i){\hat{\boldsymbol{\theta}}}^i\\ \vdots\\ {\boldsymbol{\varphi}}(k-n){\hat{\boldsymbol{\theta}}}^n \end{pmatrix} \end{align} $$ (14) 其中, $ \bar{R}_s(k) = \left(\bar{r}_{1}(k), \cdots, \bar{r}_{i}(k), \cdots, \bar{r}_{n}(k)\right) $.
2.2 需量多步预报模型
由式(2)和式(12)可以得到需量多步预报模型为
$$ \begin{split} \bar{P}\left( k+n \right) =\; &\left( \begin{matrix} \bar{p}(k) \\ \bar{p}(k) \\ \vdots \\ \bar{p}(k) \\ \end{matrix} \right)-\frac{1}{30}\left( \begin{matrix} p(k-29) \\ \displaystyle\sum\limits_{i = 1}^{2}{p(k-30+i)} \\ \vdots \\ \displaystyle\sum\limits_{i = 1}^{n}{p(k-30+i)} \\ \end{matrix} \right)+\\ & \frac{1}{30}\left( \begin{matrix} p(k\text+1) \\ \displaystyle\sum\limits_{i = 1}^{2}{p(k+i)} \\ \vdots \\ \displaystyle\sum\limits_{i = 1}^{n}{p(k+i)} \end{matrix} \right) =\\ &\left( \begin{matrix} \bar{p}(k) \\ \bar{p}(k) \\ \vdots \\ \bar{p}(k) \end{matrix} \right)-\frac{1}{30}\left( \begin{matrix} p(k-29) \\ \displaystyle\sum\limits_{i = 1}^{2}{p(k-30+i)} \\ \vdots \\ \displaystyle\sum\limits_{i = 1}^{n}{p(k-30+i)} \end{matrix} \right)+\\ &\frac{1}{30}\left( \begin{matrix} {\boldsymbol{\varphi}} (k){{{\hat{\theta }}}^{1}} \\ \displaystyle\sum\limits_{i = 1}^{2}{{\boldsymbol{\varphi}} (k){{{\hat{\theta }}}^{i}}} \\ \vdots \\ \displaystyle\sum\limits_{i = 1}^{n}{{\boldsymbol{\varphi}} (k){{{\hat{\theta }}}^{i}}} \\ \end{matrix} \right)+\frac{1}{30}\left( \begin{matrix} \bar{r}(k) \\ \displaystyle\sum\limits_{i = 1}^{2}{\bar{r}(k+i)} \\ \vdots \\ \displaystyle\sum\limits_{i = 1}^{n}{\bar{r}(k+i)} \\ \end{matrix} \right) =\\ &{\bar{P}_S(k)}+\frac{1}{30}AP(k+n-30)\;+\\ &\frac{1}{30}A\hat{\Theta}^{\rm{T}}{\boldsymbol{\varphi}}^{\rm{T}}(k)+ \frac{1}{30}A\bar{R}(k+n) \\[-10pt]\end{split} $$ (5) 其中, $\bar{P}(k+n) = (\bar{p}(k+1), \cdots, \bar{p}(k+n))^{\rm{T}}$, ${\bar{P}_S(k)} = (\bar{p}(k), \cdots, \bar{p}(k))^{\rm{T}}$, $P(k+n-30) = (p(k-29), \cdots, p(k-n+30))^{\rm{T}}$, $ \bar{R}(k+n) $ 为 $(\bar{r}_1(k), \cdots, \bar{r}_i(k+ i), \cdots, \bar{r}_n(k+n))^{\rm{T}}$, $ A $为非0元素为1的$ n\times n $维下三角阵. 式(15)中$ {\bar{P}_S(k)}, P(k+n-30), \hat{\Theta}^{\rm{T}}{\boldsymbol{\varphi}}^{\rm{T}}(k) $已知, 向量$ \bar{R}(k+n) $是模型结构阶次与参数未知的非线性动态系统.
3. 需量多步智能预报方法
3.1 需量多步预报策略
采用文献[18]方法建立式(15)的未知非线性动态系统的由在线深度学习多步预报模型、自校正深度学习多步预报模型和自校正机制组成的自适应深度学习多步预报模型.
采用图2所示的端边云协同结构实现电熔镁砂生产用电需量多步智能预报算法, 其中端−需量监控系统实时采集电熔镁砂生产过程中的群炉功率$ p(k) $与需量$ \bar{p}(k) $, 边−需量预报计算机执行数据处理、在线计算$ \hat{\Theta}^{\rm{T}}\varphi^{\rm{T}}(k) $, 采用窗口长度为$ N $的数据和$ \bar{R}(k\;+ n) $的在线深度学习多步预报模型获得$ \hat{\bar{R}}(k+n) $, 采用式(15)求取需量多步预报值$ \hat{\bar{P}}(k+n) $, 云−数据服务器和人工智能计算平台采用$ k $时刻以及以前所有时刻的输入输出数据和$ \bar{R}(k+n) $的自校正深度学习多步预报模型更新自校正深度学习多步预报模型的全部权值参数和偏置参数. 自校正机制在线监控$ \bar{R}(k+n) $的在线深度学习多步预报模型的预报精度, 当不满足精度要求时, 采用自校正深度学习多步预报模型的参数校正在线深度学习多步预报模型的参数, 从而保证需量的预报精度.
图 2 端边云协同需量多步智能预报结构图Fig. 2 Schematic of multi-step intelligent forecasting of demand with edge-cloud coordination3.2 $ \bar{R}(k+n) $的自适应深度学习多步预报算法
首先建立$ \bar{R}(k+n) $的自适应深度学习多步预报算法. 采用长短周期记忆(Long short-term memory, LSTM)[20-22]的网络架构提出如图3所示的需量第$ i $步预报$ \bar{r}(k+i) $的深度学习预报模型结构. 结合$ \bar{r}(k+i) $动态特性, 将其输入变量作为单个神经元的输入, 未知阶次用神经元个数$ t $来表示, 由式(13)和式(14)可知$ \bar{r}(k+i) $的深度学习网络的第$ j $个神经元的输入为 ${\boldsymbol{x}}_i(k-j+1) = (p(k- j\;+ 1), \bar{r}_i(k-j+1))^{\rm{T}}$, 其中 $\bar{r}_i(k-j+1) = p(k- j\;+ 1)-{\boldsymbol{\varphi}}(k- i-j+1){\hat{\boldsymbol{\theta}}}^i$, $ j = 1, \cdots, t $. 采用文献[18]的训练方法, 利用30个炉次(150000组)的群炉功率数据离线训练$ \bar{r}(k+i) $的深度学习模型结构, 确定神经元个数$ t = 25 $、神经元的节点个数$ \bar{h} = 200 $、隐藏层数$ L = 3 $、在线训练数据窗口长度$ N = 2\,000 $. 采用该深度学习预报模型结构, 提出由在线深度学习预报模型、自校正深度学习预报模型和自校正机制组成的$ \bar{r}(k+i) $的自适应深度学习预报算法.
图 3 $ \bar{r}(k+i)$的深度学习预报模型结构Fig. 3 Structure of deep learning prediction model of $ \bar{r}(k+i)$采用窗口长度$ N $的实时数据在线校正$ \bar{r}(k+i) $的在线深度学习预报模型的输出层权值与偏置参数. 校正算法为
$$ \begin{align} \hat{\bar{r}}_i(k+i) = {\hat{\boldsymbol{\beta}}}_i(k){\boldsymbol{h}}_i^3(k)+\hat{b}_i(k) \end{align} $$ (16) 其中,
$$ {\hat{\boldsymbol{\beta}}}_i(k) = {\hat{\boldsymbol{\beta}}}_i(k-1)-\alpha\frac{\partial L_i(k)}{\partial {{\boldsymbol{\beta}}}_i(k-1)} $$ (17) $$ \hat{b}_i(k) = \hat{b}_i(k-1)-\alpha\frac{\partial L_i(k)}{\partial{b}_i(k-1)} $$ (18) $$ L_i(k) =\frac{1}{N}\sum_{m = 0}^{N-1}||\Delta \bar{r}_i(k-m)||_2 $$ (19) 其中, $ \Delta\bar{r}_i(k-m) = \bar{r}_i(k-m)-\hat{\bar{r}}_i(k-m) $.
采用当前时刻$ k $以及以前所有时刻的实时数据校正自校正深度学习预报模型的全部权值和偏置参数. 其输出层的权值与偏置参数采用式(16) ~ (19)校正. 第$ l $层第$ j $个神经元的输出
$$ \begin{split} {\boldsymbol{h}}_i^l(k-j+1) =\;& {\boldsymbol{o}}_i^l(k-j+1) \;\odot\\ &\tanh( C_i^l(k-j+1)) \end{split} $$ (20) 其中, $ \odot $为哈达玛积[22], $ \tanh(\cdot) $为双曲正切函数, $ ( C_i^l(k-j+1)) $和$ {\boldsymbol{o}}_i^l(k-j+1) $表示第$ l $层第$ j $个神经元的状态和输出门的输出, 采用文献[12]方法计算, 其中, 第$ l $层第$i $个神经元的输出门的权值与偏置参数的校正算法为
$$ W_i^l(k-j+1) = W_i^l(k-j)-\alpha\frac{\partial L_i(k)}{\partial W_i^l(k-j)} $$ (21) $$ b_i^l(k-j+1) = b_i^l(k-j)-\alpha\frac{\partial L_i(k)}{\partial b_i^l(k-j)} $$ (22) $$L_i(k) =\frac{1}{k}\sum_{m = 0}^{k-1}||\Delta \bar{r}_i(k-m)||_2 $$ (23) 其中, $ \Delta\bar{r}_i(k-m) = \bar{r}_i(k-m)-\tilde{r}_i(k-m) $.
为了准确预报需量尖峰, 需要保证需量预报误差精度和需量变化趋势预报精度, 由式(15)知需量预报精度取决于$ \bar{r}(k+i) $的预报精度. 采用自校正机制监控$ \bar{r}(k+i) $的在线深度学习预报模型的预报误差和变化趋势的预报精度, 当不满足精度要求时, 采用自校正深度学习预报模型的各层权值参数与偏置参数校正在线深度学习预报模型的权值参数与偏置参数.
自校正机制采用$ \bar{r}(k+i) $的预报误差$ \Delta \bar{r}_i(k) $, 未知非线性系统第$ i $步上升趋势预报准确率$ { TPR_i(k)} $和第$ i $步下降趋势预报准确率$ { TNR_i(k)} $三项指标, 即
$$ |\Delta\bar{r}_i(k)| = |\bar{r}_i(k)-\hat{\bar{r}}_i(k)| $$ (24) $$ { TPR_i(k)} = \frac{\sum\limits_{m = 0}\limits^{N-1}{ TP}_i(k-m)}{\sum\limits_{m = 0}\limits^{N-1}{ TP}_i(k-m)+\sum\limits_{m = 0}\limits^{N-1}{ FP}_i(k-m)} $$ (25) $$ { TNR_i(k)} =\frac{\sum\limits_{m = 0}\limits^{N-1}{ TN}_i(k-m)}{\sum\limits_{m = 0}\limits^{N-1}{ TN}_i(k-m)+\sum\limits_{m = 0}\limits^{N-1}{ FN}_i(k-m)} $$ (26) 其中, $ {TP}_i(k), { FP}_i(k), { TN}_i(k), { FN}_i(k) $的计算方式如表1所示.
表 1 $ { TP}_i(k), { FP}_i(k), { TN}_i(k), {FN}_i(k)$的计算方式Table 1 Formula mode of ${ TP}_i(k), { FP}_i(k), $ ${ TN}_i(k), {FN}_i(k) $$\hat{\bar{r} }_i(k)-\hat{\bar{r} }_i(k-1)\geq 0$ $\hat{\bar{r} }_i(k)-\hat{\bar{r} }_i(k-1) < 0$ $\bar{r}_i(k)-\bar{r}_i(k-1)\geq 0$ ${TP}_i(k)=1$ ${FP}_i(k)=1$ $\bar{r}_i(k)-\bar{r}_i(k-1)< 0$ ${FN}_i(k)=1$ ${TN}_i(k)=1$ $ k $时刻在线深度学习预报模型的预报误差$ |\Delta\bar{r}_i(k)|\geq \delta_i $ ($ \delta_i $为预报误差上界), 自校正深度学习预报模型的预报误差$ |\Delta\bar{r}_i(k)|< \delta_i $, 且在线深度学习预报模型的上升趋势和下降趋势预报准确率均小于自校正深度学习预报模型的上升趋势和下降趋势预报准确率, 采用自校正深度学习预报模型的全部权值参数和偏置参数校正在线深度学习预报模型的全部权值参数和偏置参数.
3.3 需量多步智能预报算法
由式(15)可得需量多步预报模型为
$$ \begin{split} \hat{\bar{P}}(k+n) =\; &{\bar{P}_S(k)}-\frac{1}{30}AP(k+n-30)\;+\\ &\frac{1}{30}A\hat{\Theta}^{\rm{T}}\varphi^{\rm{T}}(k)+ \frac{1}{30}A\hat{\bar{R}}(k+n) \end{split} $$ (27) 其中,
$$ \begin{split} \hat{\bar{P}}(k+n) =\;& (\hat{\bar{p}}_1(k+1), \cdots, \hat{\bar{p}}_i(k+i), \cdots,\\ &\hat{\bar{p}}_n(k+n))^{\rm{T}}\\ \hat{\bar{R}}(k+n) =\;&(\hat{\bar{r}}_1(k+1), \cdots, \hat{\bar{r}}_i(k+i), \cdots,\\ &\hat{\bar{r}}_n(k+n))^{\rm{T}} \end{split} $$ 端边云协同的需量多步智能预报算法:
1)端−需量监控系统实时采集电熔镁砂生产过程中的群炉功率$ p(k) $与需量$ \bar{p}(k) $;
2) 边−需量预报计算机执行数据处理和需量在线多步预报模型. 在线计算$ \hat{\Theta}^{\rm{T}}\varphi^{\rm{T}}(k) $, 由式(14)计算$ \hat{\bar{R}}(k+n) $. 采用窗口长度为$ N $的输入输出数据由$ \bar{R}(k+n) $的在线深度学习多步预报模型得其预报值$ \hat{\bar{R}}(k+n) $, 由需量多步预报模型式(27)得需量多步预报值$ \hat{\bar{P}}(k+n) $;
3)云−数据服务器和人工智能计算平台采用$ k $时刻以及以前所有时刻的输入输出数据和$ \bar{R}(k+n) $的自校正深度学习多步预报模型得其预报值$ \hat{\bar{R}}(k+ n) $. 采用自校正机制的三项指标式(24) ~ (26), 当$ \bar{R}(k+n) $的在线深度学习多步预报模型的预报误差超过预报精度上界时, 采用自校正深度学习多步预报模型的权值参数和偏置参数校正在线深度学习多步预报模型的权值参数和偏置参数.
4. 实验结果
采用某电熔镁砂生产企业的实际功率和需量数据进行了本文提出的需量多步智能预报方法的实验, 并与文献[9]提出的基于支持向量机、极限学习机和循环神经网络的用电需量多步预报方法进行了对比实验.
采用30个炉次的150000组功率数据离线确定$ \bar{r}(k+i) $的深度学习预报模型结构, 神经元个数$ t = 25 $、神经元的节点个数$ \bar{h} = 200 $、隐藏层数$ L = 3 $. 校正算法中的式(17)、式(18)、式(21)和式(22)中的学习率为$ \alpha = 0.1 $, 在线训练数据窗口长度$ N = 2\,000 $. 自校正机制中的预报误差上界$ \delta_i = 100 $.
由于发生需量尖峰的整个时间小于70秒, 而需量的采样周期为7秒, 因此选择需量预报步数$ i = 1, \cdots, 10 $. 采用线性模型参数辨识方程式(11)得模型参数$ \hat{\Theta} $为
$$ \begin{split} &{{{\hat{\boldsymbol{\theta}} }}^{1}} = {{\left( {0}{.140, 0}{.075, -0}{.157, 0}{.869} \right)}^{\rm T}}\\ & {{{\hat{\boldsymbol{\theta}} }}^{2}} = {{\left( {0}{.201, 0}{.135, -0}{.042, 0}{.587} \right)}^{\rm T}} \\ & {{{\hat{\boldsymbol{\theta}} }}^{3}} = {{\left( {0}{.204, 0}{.146, 0}{.061, 0}{.431} \right)}^{\rm T}}\\ &{{{\hat{\boldsymbol{\theta}} }}^{4}} = {{\left( {0}{.201, 0}{.109, 0}{.103, 0}{.411} \right)}^{\rm T}} \\ & {{{\hat{\boldsymbol{\theta}} }}^{5}} = {{\left( {0}{.203, 0}{.103, 0}{.068, 0}{.427} \right)}^{\rm T}}\\ &{{{\hat{\boldsymbol{\theta}} }}^{6}} = {{\left( {0}{.207, 0}{.103, 0}{.063, 0}{.409} \right)}^{\rm T}} \\ & {{{\hat{\boldsymbol{\theta}} }}^{7}} = {{\left( {0}{.208 , 0}{.103 , 0}{.066, 0}{.389} \right)}^{\rm T}} \\ &{{{\hat{\boldsymbol{\theta}} }}^{8}} = {{\left( {0}{.204, 0}{.103, 0}{.069, 0}{.374} \right)}^{\rm T}}\\ &{{{\hat{\boldsymbol{\theta}} }}^{9}} = {{\left( {0}{.203, 0}{.097, 0}{.071, 0}{.364} \right)}^{\rm T}}\\ &{{{\hat{\boldsymbol{\theta}} }}^{10}} = {{\left( {0}{.202, 0}{.095, 0}{.067, 0}{.358} \right)}^{\rm T}}\end{split} $$ 采用上述30个炉次的150000组需量数据离线建立文献[9]的用电需量多步预报模型:
$$ \begin{split} \hat{\bar{p}}(k+i) =\; &{{W}_{1}}{{\hat{\bar{p}}}_{svm}}(k+i)+{{W}_{2}}{{\hat{\bar{p}}}_{elm}}(k+i)\;+\\ &{{W}_{3}}{{\hat{\bar{p}}}_{rnn}}(k+i) \end{split} $$ 其中, $ \hat{\bar{p}}_{svm}, \hat{\bar{p}}_{elm}, \hat{\bar{p}}_{rnn} $分别为支持向量机、极限学习机和循环神经网络的输出, 上述模型的阶次$ t = 25 $, 支持向量机中径向基函数的期望为150、方差为0.08, 循环神经网络的每层节点数为$ \bar{h} = 150 $, 隐藏层数为$ L = 2 $, 权参数$W_1 = 0.16, W_2 = 0.34, W_3 = 0.71$.
采用实时采集的70个炉次的350000组的需量与功率数据对本文所提需量多步预报算法与文献[9]多步预报算法进行了预报步数$ i = 1, \cdots, 10 $的对比实验. 采用文献[23]的均方根误差(Root mean square error, RMSE)式(28), 文献[24]的拟合优度$ R^2 $式(29), 文献[17]的平均绝对百分比误差(Mean absolute percentage error, MAPE)式(30), 需量第$ i $步预报的上升趋势预报准确率$ {TPR}_i $式(25)和下降趋势预报准确率$ { TNR}_i $式(26)指标对本文所提需量多步预报算法和文献[9]多步预报算法的实验结果进行评估. 实验结果见表2.
表 2 需量预报精度对比Table 2 Precision comparison of demand forecast预报步数$i$ 1 2 3 4 5 6 7 8 9 10 $R^2_i\;(\%)$ 本文 99.96 99.62 99.59 99.47 99.39 99.31 98.99 98.51 98.03 97.95 文献[9] 90.34 90.05 89.77 89.54 88.73 88.48 88.01 87.76 87.33 86.94 ${{RMSE} }_i$ 本文 9.93 11.06 11.99 13.03 13.89 14.73 16.05 16.83 17.93 18.78 文献[9] 24.92 30.01 34.49 39.99 44.79 50.23 54.93 60.05 65.32 70.64 ${{MAPE} }_i\;(\%)$ 本文 0.04 0.05 0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.08 文献[9] 0.11 0.13 0.15 0.18 0.20 0.22 0.24 0.27 0.29 0.32 $TPR_i\;(\%)$ 本文 94.88 93.21 92.19 91.42 90.17 89.77 88.21 90.05 91.55 89.66 文献[9] 86.12 82.11 80.05 80.11 78.94 79.33 79.11 77.06 80.15 79.02 $TNR_i\;(\%)$ 本文 93.22 94.67 92.19 92.01 94.21 93.18 90.96 89.99 88.12 90.01 文献[9] 81.12 80.04 80.67 83.72 79.99 80.15 77.56 86.77 80.15 76.91 $$ {{RMSE}}_i = \sqrt{\frac{1}{N_t}\sum\limits_{k = 1}^{N_t} \left( \bar{p}(k)-\hat{\bar{p}}_i(k)\right)^2} $$ (28) $$ { R}_i^2 = \frac{\sum\limits_{k = 1}\limits^{N_t}\left( \hat{\bar{p}}_i(k)-{\tilde{p}}_i\right)^2 }{\sum\limits_{k = 1}\limits^{N_t}\left(\bar{p}_i(k)-\tilde{p}_i \right)^2 }\times 100{\text{%}} $$ (29) 其中, $ \tilde{p}_i = \frac{1}{N_t}\sum_{k = 1}^{N_t} \bar{p}(k+i-1) $.
$$ {{MAPE}}_i = \frac{1}{N_t}\sum\limits_{k = 1}^{N_t}\left|\frac{\bar{p}(k)-\hat{\bar{p}}_i(k)}{\bar{p}(k)}\right|\times 100{\text{%}} $$ (30) 为了能清楚地对比实验结果, 采用图4 ~ 图6表示从在线的350000组数据的实验结果中随机抽取100组$ i = 1, 5, 10 $步需量预报结果, 虚线为本文需量预报结果, 实线为文献[9]需量预报结果, 点线为需量真实值. 可以看出本文需量预报方法与文献[9]方法相比, 预报精度明显提高.
从表2可以看出, 本文的方法与文献[9]方法的$ i = 1, 5, 10 $步预报结果相比, $ R^2 $提高$ 11.01{\text{%}} $, $ {{RMSE}}_i $降低51.86, $ {{MAPE}}_i $降低$ 0.24{\text{%}} $, 上升趋势$ { TPR}_i $提高$ 10.23{\text{%}} $, 下降趋势$ { TNR}_i $提高$ 14.22{\text{%}} $. 对比实验结果表明本文方法比文献[9]方法的需量多步预报精度和需量变化趋势的预报精度明显提高.
5. 结论
本文通过电熔镁砂生产过程熔化电流闭环控制系统方程建立需量动态模型, 在此基础上建立了由线性模型和未知非线性动态系统组成的需量多步预报模型, 采用文献[18]方法建立了未知非线性动态系统的自适应深度学习多步预报模型, 在此基础上提出了端边云协同的电熔镁砂生产过程需量多步智能预报方法. 采用70个炉次的电熔镁砂生产过程的实际数据的实验结果表明本文的方法与文献[9]方法的$ i = 1, 5, 10 $ 步预报结果相比, $ R^2 $提高$ 11.01{\text{%}} $, $ {{RMSE}}_i $降低51.86, $ {{MAPE}}_i $降低$ 0.24{\text{%}} $, 上升趋势$ { TPR}_i $提高$ 10.23{\text{%}} $, 下降趋势$ { TNR}_i $提高$ 14.22{\text{%}} $, 验证了本文所提的预报方法可以准确预报需量的变化趋势, 为实现需量尖峰的准确预报和控制创造了条件.
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图 1 三维图像分割结果横断平面展示图, 背景体素值为0, 身体区域的体素值为1, 肺部体素值为3
Fig. 1 3D segmentation images shown on the axial plane, where the background voxels is set to 0, the voxels of body area to 1, and the voxels of lungs to 3
图 2 三维图像分割结果, 及其对应的肺部区域和身体区域
Fig. 2 A 3D segmentation result and the corresponding separated 3D lungs and body
图 6 膈肌和胸腹部表面的位移, 每一种颜色的曲线对应一个患者的数据
Fig. 6 The displacement of diaphragm and thoracoabdominal surface, where each color corresponds to a specific patient＇s data
图 7 归一化后的膈肌和胸腹部表面的位移, 每一种颜色的曲线对应一个患者的数据
Fig. 7 The displacement of diaphragm and thoracoabdominal surface after normalization, where each color corresponds to a specific patient＇s data
图 8 三个方向的预测结果. 该结果对应于100次运行MSE的中位数值
Fig. 8 Prediction results corresponding the median MSE value in three directions after 100 independent runs
图 9 三个方向各相位预测误差的统计箱图
Fig. 9 Boxplot of the prediction performance on every phase in three directions
图 10 主方向的预测结果, 该结果对应于100次运行MSE的中位数值
Fig. 10 Prediction results corresponding the median MSE values in the principal direction after 100 independent runs
图 11 主方向各相位预测误差的统计箱图
Fig. 11 Boxplot of the prediction performance on every phase in the principal direction
图 13 噪声对主方向预测性能的影响
Fig. 13 Influence of noise on the prediction in principal direction
图 14 PCA中的阈值
$\varepsilon $ 对TSSM预测模型的影响Fig. 14 The influence of
$\varepsilon $ in PCA on the prediction performance of TSSM表 1 在三个方向上的预测结果
Table 1 Prediction results in three directions
OLS 模型 ANN TSSM (线性) kTSSM (多项式) kTSSM (高斯) 没有噪声 MSE 0.45 (0.23) 0.30 (0.16) 0.20 (0.09) 0.18 (0.08) 0.20 (0.10) R2 0.70 (0.14) 0.81 (0.07) 0.87 (0.05) 0.88 (0.04) 0.88 (0.04) MAPE 232.11 (156.44) 159.60 (112.37) 150.82 (83.98) 116.57 (71.28) 130.69 (87.26) 噪声 ($\sigma =0.1$) MSE 0.46 (0.22) 0.29 (0.15) 0.22 (0.10) 0.19 (0.09) 0.22 (0.09) R2 0.70 (0.15) 0.82 (0.07) 0.86 (0.05) 0.88 (0.04) 0.86 (0.05) MAPE 276.86 (551.10) 214.82 (257.92) 157.28 (117.30) 137.25 (92.36) 170.47 (204.00) 噪声 ($\sigma =0.2$) MSE 0.54 (0.24) 0.39 (0.18) 0.25 (0.11) 0.23 (0.10) 0.24 (0.10) R2 0.69 (0.13) 0.76 (0.10) 0.84 (0.05) 0.86 (0.04) 0.85 (0.06) MAPE 277.42 (348.60) 211.14 (196.70) 172.08 (171.03) 131.98 (78.23) 166.12 (156.08) 噪声 ($\sigma =0.3$) MSE 0.65 (0.25) 0.42 (0.21) 0.30 (0.11) 0.28 (0.11) 0.29 (0.12) R2 0.59 (0.20) 0.75 (0.11) 0.82 (0.05) 0.82 (0.06) 0.82 (0.06) MAPE 426.45 (1371.37) 211.37 (228.00) 242.75 (274.21) 157.12 (94.87) 179.81 (115.10) 噪声 ($\sigma =0.4$) MSE 0.80 (0.28) 0.52 (0.33) 0.37 (0.13) 0.33 (0.13) 0.38 (0.13) R2 0.53 (0.21) 0.69 (0.19) 0.79 (0.07) 0.80 (0.06) 0.78 (0.07) MAPE 592.47 (1687.33) 262.36 (405.00) 198.89 (112.67) 174.14 (116.58) 185.34 (133.87) 噪声 ($\sigma =0.5$) MSE 0.98 (0.33) 0.65 (0.29) 0.46 (0.16) 0.41 (0.15) 0.45 (0.17) R2 0.47 (0.22) 0.64 (0.15) 0.75 (0.08) 0.76 (0.08) 0.76 (0.09) MAPE 365.33 (399.98) 254.35 (291.97) 234.93 (185.28) 194.18 (150.11) 214.82 (260.37) 
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表 2 在主方向上的预测结果
Table 2 Prediction results in principal direction
OLS 模型 ANN TSSM (线性) kTSSM (多项式) kTSSM (高斯) 没有噪声 MSE 0.47 (0.28) 0.27 (0.18) 0.14 (0.07) 0.13 (0.06) 0.15 (0.06) R2 0.73 (0.20) 0.86 (0.08) 0.93 (0.03) 0.93 (0.03) 0.92 (0.03) MAPE 91.07 (38.34) 63.62 (23.72) 47.85 (15.31) 45.17 (14.94) 46.47 (14.39) 噪声 ($\sigma =0.1$) MSE 0.40 (0.26) 0.25 (0.18) 0.15 (0.07) 0.14 (0.06) 0.15 (0.07) R2 0.78 (0.16) 0.86 (0.08) 0.92 (0.04) 0.92 (0.03) 0.91 (0.03) MAPE 143.90 (173.93) 100.38 (136.34) 116.46 (198.17) 68.29 (66.63) 107.08 (145.85) 噪声 ($\sigma =0.2$) MSE 0.43 (0.23) 0.29 (0.12) 0.18 (0.07) 0.17 (0.06) 0.18 (0.07) R2 0.76 (0.14) 0.84 (0.07) 0.90 (0.04) 0.91 (0.04) 0.90 (0.04) MAPE 188.52 (476.86) 138.61 (386.48) 133.32 (129.00) 79.96 (53.09) 82.96 (79.07) 噪声 ($\sigma =0.3$) MSE 0.61 (0.33) 0.38 (0.18) 0.23 (0.10) 0.20 (0.08) 0.24 (0.09) R2 0.64 (0.20) 0.79 (0.11) 0.87 (0.06) 0.87 (0.06) 0.88 (0.05) MAPE 192.18 (223.53) 136.84 (116.74) 134.21 (151.02) 86.07 (64.90) 101.09 (88.78) 噪声 ($\sigma =0.4$) MSE 0.70 (0.31) 0.45 (0.36) 0.30 (0.12) 0.26 (0.09) 0.29 (0.11) R2 0.65 (0.17) 0.77 (0.21) 0.83 (0.07) 0.87 (0.06) 0.85 (0.07) MAPE 221.12 (417.10) 175.31 (342.96) 142.09 (168.43) 94.42 (64.70) 109.11 (116.15) 噪声 ($\sigma =0.5$) MSE 1.01 (0.50) 0.52 (0.28) 0.39 (0.13) 0.32 (0.13) 0.36 (0.14) R2 0.47 (0.35) 0.75 (0.15) 0.80 (0.09) 0.84 (0.07) 0.81 (0.09) MAPE 173.44 (98.15) 193.44 (455.90) 146.14 (212.00) 102.30 (130.34) 116.74 (94.49)
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表 3 TSSM和kTSSM取得最优预测结果时的参数设置
Table 3 Parameter setting of TSSM and kTSSM for optimal results
模型 TSSM (线性) kTSSM (高斯) kTSSM (多项式) 参数 λ λ σkernel λ d c 三个方向的预测 没有噪声 1.0 0.1 0.2 100.0 6.0 9 噪声 ($\sigma =0.1$) 1.0 0.1 0.2 1000.0 3.5 9 噪声 ($\sigma =0.2$) 1.0 0.1 0.1 100.0 2.5 9 噪声 ($\sigma =0.3$) 1.0 0.1 0.1 1.0 0.5 4 噪声 ($\sigma =0.4$) 1.0 0.1 0.1 10.0 1.5 7 噪声 ($\sigma =0.5$) 1.0 1.0 0.1 10.0 1.0 4 主方向的预测 没有噪声 0.1 0.1 0.2 1.0 1.0 5 噪声 ($\sigma =0.1$) 0.1 0.1 0.2 0.1 1.0 7 噪声 ($\sigma =0.2$) 1.0 0.1 0.2 0.1 0.5 4 噪声 ($\sigma =0.3$) 1.0 0.1 0.1 1.0 0.5 3 噪声 ($\sigma =0.4$) 1.0 0.1 0.2 1.0 1.0 4 噪声 ($\sigma =0.5$) 1.0 0.1 0.1 0.1 0.5 7
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