﻿ 快速双非凸回归算法及其电力数据预测应用
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 智能系统学报  2018, Vol. 13 Issue (4): 665-672  DOI: 10.11992/tis.201708033 0

### 引用本文

WANG Fenghua, CHENG Jingzhou, WEN Fan. Fast double nonconvex regression algorithm for forecast of electric power data[J]. CAAI Transactions on Intelligent Systems, 2018, 13(4): 665-672. DOI: 10.11992/tis.201708033.

### 文章历史

1. 国网浙江省电力公司，浙江 杭州 310000;
2. 国网浙江省电力公司 经济技术研究院，浙江 杭州 310000

Fast double nonconvex regression algorithm for forecast of electric power data
WANG Fenghua1, CHENG Jingzhou1, WEN Fan2
1. State Grid Zhejiang Electric Power Company, Hangzhou 310000, China;
2. Economic Research Institute, State Grid Zhejiang Electric Power Company, Hangzhou 310000, China
Abstract: In this paper, we propose a new forecasting algorithm called double nonconvex regression (DNR) for the fast forecast of electricity power data such as the outputs of production ability and operational benefit. First, we reinterpret the typical sparse coding classification method as a regression model for forecasting, and further divide the model into training and testing phases to fit scalar-quantity forecasts. Next, we transform the constraints of representation residuals and coefficient regularization into a nonconvex lp norm for better approximation and broader application. Lastly, we adopt the alternating direction method of multipliers algorithm to optimize the formulated forecast problem. To achieve a fast update rule for lp norm constrained subproblems, we propose a new iterative threshold method that avoids the local minimum issue. Compared with typical methods such as the SVM, BP neural network, and nonconvex regularization methods, the proposed algorithm achieves surprisingly good experimental results for electricity power data.
Key words: alternating direction method of multiplier (ADMM)    forecast of electric power data    lp norm constraint    iterative threshold method

1 双非凸回归算法

 $\min \left\| {{x} - {{A\alpha }}} \right\|_2^2 + \lambda {\left\| {{\alpha }} \right\|_0}$ (1)

 $\min \left\| {{x} - {A\alpha }} \right\|_2^2 + \lambda {\left\| {\alpha } \right\|_1}$ (2)

 $\min \left\| {{x} - {A \alpha }} \right\|_1^{} + \lambda {\left\| {\alpha } \right\|_1}$ (3)

 $\min \left\| {{x} - {A \alpha }} \right\|_q^q + \lambda \left\| {\alpha } \right\|_p^p$ (4)

2 模型优化求解

 $\begin{array}{c}\min \left\| {{{x}} - {{A}}{{\beta} }} \right\|_q^q + \lambda \left\| {{\beta} } \right\|_p^p\\{\rm{s.t.}}\;{x} - {A\alpha } = {e},{\alpha } = {\beta }\end{array}$ (5)

 $\begin{array}{c}{L_{{\mu _e},{\mu _\beta }}}({{e}},{{\alpha }},{{\beta }}) = ||{{e}}||_q^q + \lambda ||{{\beta }}||_p^p + \displaystyle\frac{\mu _e}{2}||{{x}} - {{A\alpha }} - {{e}}||_2^2 + \displaystyle\frac{\mu _\beta }{2}||{{\alpha }} - {{\beta} }||_2^2 + {{\gamma} }_e^{\rm{T}}({{x}} -{ {A\alpha} } - {{e}}) + {{\gamma }}_\beta ^{\rm{T}}({{\alpha} } - {{\beta }})\end{array}$ (6)

1) 固定αγββk+1的更新子问题为

 $\begin{array}{c}{{{\beta }}_{k + 1}} = \arg \mathop {\min }\limits_{{\beta }} {L_{{\mu _\beta }}}({{\beta} },{{\alpha }},{{{\gamma} }_\beta }) = \\\displaystyle\frac{\lambda }{{{\mu _\beta }}}||{{\beta} }||_p^p + \frac{1}{2}||{{\beta} } - {{\alpha} } - \frac{{{{{\gamma} }_\beta }}}{{{\mu _\beta }}}||_2^2\end{array}$ (7)

2) 固定αγeek+1的更新子问题为

 $\begin{array}{c}{{{e}}_{k + 1}} = \arg \mathop {\min }\limits_{{e}} {L_{{\mu _e}}}({{e}},{\alpha },{{{\gamma} }_e}) = \displaystyle\frac{1}{{{\mu _e}}}||{{e}}||_q^q + \frac{1}{2}||{{e}} - {{x + A\alpha }} - \frac{{{{{\gamma} }_e}}}{{{{\mu} _e}}}||_2^2\end{array}$ (8)

3) 固定eβγeγβαk+1的更新子问题为

 $\begin{array}{c}{{{\alpha} }_{k + 1}} = \arg \mathop {\min }\limits_{{e}} {L_{{\mu _e},{\mu _\beta }}}({{e}},{{\beta} },{{{\gamma }}_e},{{{\gamma }}_\beta }) = \\\displaystyle\frac{{{{\mu} _e}}}{2}||{{A\alpha }} -{ {x + e}} - \frac{{{{{\gamma} }_e}}}{{{{\mu} _e}}}||_2^2 + \frac{{{{\mu} _\beta }}}{2}||{{\alpha }} - {{\beta}}+\frac{{{{{\gamma} }_\beta }}}{{{{\mu} _\beta }}}||_2^2\end{array}$ (9)

4) 根据计算所得的βeα1，更新γeγβ

 $\begin{array}{c}{\gamma }_e^{k + 1} = {\gamma }_e^k + {{\mu} _e}({x} - {A\alpha } - {e})\\{\gamma }_\beta ^{k + 1} = {\gamma }_\beta ^k + {{\mu} _\beta }({\alpha } - {\beta })\end{array}$ (10)

 ${{\alpha }^{k + 1}} = {C}({\mu _e}{{A}^{\text{T}}}({x} - {e}) + {{A}^{\text{T}}}{{\gamma }_e} + {\mu _\beta }{\beta } - {{\gamma }_\beta })$ (11)

 $\min f({\delta} ) = \frac{1}{2}\left\| {{\delta} - {\sigma} } \right\|_2^2 + \lambda {\left\| {\delta} \right\|^p}$ (12)

p=1时，可由经典的软阈值算法[14]进行有效求解。针对本文的非凸情况(0<p<1)，IRLS、IRL1、IST等求解算法都存在局部次优解的缺陷。如图1所示，当σ=0.9，p=0.2且λ=1时，IRLS、IRL1和IST都陷入了局部最小值。为解决该问题，本文提出一种改进的阈值迭代方法，在保证高效求解的同时能够获得全局最优值。

 Download: 图 1 几种算法对典型非凸问题式(12)的最优解 Fig. 1 Several algorithms for the optimal solution to the typical nonconvex problem in formula (12)
2.2 改进的迭代阈值优化算法

 Download: 图 2 不同σ值下非凸问题f (δ)的最优解 Fig. 2 The optimal solution of non-convex problem f (δ) under different σ values

 $f'(\delta ) = \delta - \sigma + \lambda p{\delta ^{p - 1}}$ (13)
 $f''(\delta ) = 1 + \lambda p(p - 1){\delta ^{p - 2}}$ (14)

f"(δ(λ, p))=0，可得δ(λ, p)=(λp(1-p))1/(2-p)。结合图2可知，当δ∈(0, δ(λ, p))时，f(δ)是凹函数；当δ∈(δ(λ, p), +∞)时，则f(δ)是凸函数。进一步，为保证f(δ)在(δ(λ, p), +∞)具有最小值，需满足f'(δ(λ, p))≤0，文献[13]令f'(δ(λ, p))=0并计算出τσIST用于迭代阈值求解。然而，该阈值设法存在问题，如图1所示，IST计算所得的解满足上述所有规则，且στσIST时保证

 ${\delta ^*} - \sigma + \lambda p{\delta ^*}^{p - 1} = 0$ (15)

 $\frac{1}{2}{({\delta ^*} - \tau _\sigma ^{})^2} + \lambda {({\delta ^*})^p} = \frac{1}{2}{(\tau _\sigma ^{})^2}$ (16)
 ${\delta ^*} - \tau _\sigma ^{} + \lambda p{({\delta ^*})^{p - 1}} = 0$ (17)

 ${\delta ^{*p}}(2\lambda (1 - p) - {({\delta ^*})^{2 - p}}) = 0$ (18)

 ${\tau _\sigma } = {[2\lambda (1 - p)]^{\frac{1}{{2 - p}}}} + \lambda p{[2\lambda (1 - p)]^{\frac{{p - 1}}{{2 - p}}}}$ (19)

1) 按式(19)计算τσ值；

2) 如|σ|<τσ；则令δ*=0；

3) f反之，令k=0，δk=σ

4) for k=1, 2, $\cdots$ , J

5) ${\delta ^k}^{ + 1} = |\sigma | - \lambda p{({\delta ^k})^p}^{ - 1}$

6) End

7) δ*=sgn(σ)δJ

3 实验分析

3.1 电能输出预测

 ${\text{MAE}} = (|{p_1} - {r_1}| + |{p_2} - {r_2}| + \cdots + |{p_n} - {r_n}|)/n$
 ${\text{RMSE}} = \sqrt {({{({p_1} - {r_1})}^2} + {{({p_2} - {r_2})}^2} + \cdots + {{({p_n} - {r_n})}^2})/n}$

3.2 运营数据预测

 Download: 图 3 DNR对训练样本的拟合效果及对测试样本的预测效果对比 Fig. 3 Comparison of the fitting effect of DNR on training samples and the prediction effect of test samples
 Download: 图 4 SVM对训练样本的拟合效果及对测试样本的预测效果对比 Fig. 4 Comparison of test samples and training samples’s fitting effect by SVM
 Download: 图 5 BP神经网络对训练样本的拟合效果及对测试样本的预测效果对比 Fig. 5 Comparison of test samples and training samples’s fitting effect by BP neural network
 Download: 图 6 NNR对训练样本的拟合效果及对测试样本的预测效果对比 Fig. 6 Comparison of test samples and training samples’s fitting effect by NNR

4 结束语

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