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<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622b83ae3d2fbb00079f38b0/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">全国高等学校电力系统及其自动化专业第十八届学术年会论文集<span class="ff2"> </span></div><div class="t m0 x2 h3 y2 ff3 fs1 fc0 sc0 ls0 ws0">基于小波原理的短期负荷预测<span class="ff2"> </span></div><div class="t m0 x3 h4 y3 ff4 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 x4 h5 y4 ff1 fs2 fc0 sc0 ls0 ws0">宋<span class="ff4"> </span>毅<span class="ff4"> </span>孙雅明<span class="ff4"> </span></div><div class="t m0 x5 h4 y5 ff1 fs0 fc0 sc0 ls0 ws0">天津大学电气自动化学院电力系<span class="ff4"> 300072 </span>天津<span class="_"> </span><span class="ff4"> </span></div><div class="t m0 x3 h6 y6 ff2 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x6 h2 y7 ff3 fs0 fc0 sc0 ls0 ws0">摘<span class="ff2 ls1"> </span>要<span class="_ _0"> </span><span class="ff1">随着新的数学工具小波分析的实用化<span class="_ _0"> </span>为基于<span class="_ _1"> </span><span class="ff2 ls2">NN<span class="_"> </span></span>负荷预测模型性能的改善提供了理论依据<span class="_ _0"> </span>对</span></div><div class="t m0 x6 h7 y8 ff1 fs0 fc0 sc0 ls0 ws0">于电力系统负荷非线性时间序列的辨识</div><div class="t m0 x7 h7 y9 ff1 fs0 fc0 sc0 ls0 ws0">在预测方法研究中应给予重视<span class="_"> </span>在本文所用的基于小波原理和</div><div class="t m0 x6 h2 ya ff2 fs0 fc0 sc0 ls2 ws0">NN<span class="_"> </span><span class="ff1 ls0">融合的预测原理是具有强的非线性时间序列的辩能力</span></div><div class="t m0 x8 h7 yb ff1 fs0 fc0 sc0 ls0 ws0">由研究和仿真表明它能有效提高预测的精度</div><div class="t m0 x9 h2 yc ff2 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 x6 h7 yd ff3 fs0 fc0 sc0 ls0 ws0">关键词</div><div class="t m0 xa h2 ye ff2 fs0 fc0 sc0 ls0 ws0"> <span class="_"> </span><span class="ff1">小波分析</span><span class="ls1"> </span><span class="ff1">小波神经网络</span><span class="ls1"> </span><span class="ff1">电力系统</span><span class="ls1"> </span><span class="ff1">短期负荷预测</span> </div><div class="t m0 x6 h8 yf ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 x6 h9 y10 ff5 fs4 fc0 sc0 ls3 ws0">0 <span class="ff3 ls0">引言<span class="ff5"> </span></span></div><div class="t m0 x6 h8 y11 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb ha y12 ff1 fs3 fc0 sc0 ls4 ws0">在七十年代中期以后<span class="_"> </span>由于智能<span class="_ _2"></span>技术和高深数学的发展为<span class="_ _2"></span>预测研究开辟了新方法<span class="_"> </span><span class="ls0">如</span></div><div class="t m0 x6 ha y13 ff1 fs3 fc0 sc0 ls4 ws0">基于专家系统</div><div class="t m0 xc ha y14 ff1 fs3 fc0 sc0 ls0 ws0">模糊理论<span class="_"> </span>神经网络<span class="_ _3"> </span>灰色理论<span class="_"> </span>混沌理论<span class="_ _3"> </span><span class="ls4">小波分析等基础上的预测方</span></div><div class="t m0 x6 ha y15 ff1 fs3 fc0 sc0 ls0 ws0">法</div><div class="t m0 xb ha y16 ff1 fs3 fc0 sc0 ls4 ws0">如基于神经网络模型的短期<span class="_ _2"></span>负荷预测受到世界各国的<span class="_ _2"></span>广泛关注<span class="_"> </span>该模型具有复杂的非</div><div class="t m0 x6 ha y17 ff1 fs3 fc0 sc0 ls4 ws0">线性映射性能<span class="_"> </span>从理论上讲它<span class="_ _4"> </span>可以将气候因素与负荷变<span class="_ _2"></span>化有机的结合起来<span class="_"> </span>具有较高的</div><div class="t m0 x6 ha y18 ff1 fs3 fc0 sc0 ls4 ws0">预报精度</div><div class="t m0 xd ha y19 ff1 fs3 fc0 sc0 ls4 ws0">可解决传统预测方法难以解决的<span class="_ _2"></span>问题<span class="_"> </span>但就国内目前而言<span class="_ _4"> </span>由于提供的历史负</div><div class="t m0 x6 ha y1a ff1 fs3 fc0 sc0 ls0 ws0">荷资料中缺乏气候资料</div><div class="t m0 xe h8 y1b ff1 fs3 fc0 sc0 ls0 ws0">因此基于<span class="_ _5"> </span><span class="ff2 ls5">NN<span class="_ _5"> </span></span>模型的预测也难于完善得到解决<span class="_"> </span>另一方面<span class="_"> </span>由于</div><div class="t m0 x6 h8 y1c ff2 fs3 fc0 sc0 ls5 ws0">NN<span class="_ _6"> </span><span class="ff1 ls4">的学习算法也存在局限性</span></div><div class="t m0 xf ha y1d ff1 fs3 fc0 sc0 ls4 ws0">使该预测模型不能实现预期目标<span class="_"> </span>从而影响短期负荷预测</div><div class="t m0 x6 ha y1e ff1 fs3 fc0 sc0 ls0 ws0">的可靠性和准确性</div><div class="t m0 x10 h8 y1f ff1 fs3 fc0 sc0 ls0 ws0">随着新的数学工具小波分析的实用化<span class="_"> </span>为基于<span class="_ _5"> </span><span class="ff2 ls5">NN<span class="_ _5"> </span></span>负荷预测模型性能</div><div class="t m0 x6 ha y20 ff1 fs3 fc0 sc0 ls0 ws0">的改善提供了理论依据</div><div class="t m0 x11 hb y21 ff2 fs5 fc0 sc0 ls0 ws0">[<span class="_ _7"> </span>]<span class="_ _8"></span>2<span class="_ _9"></span>,<span class="_ _a"></span>1</div><div class="t m0 x12 h8 y22 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb ha y23 ff1 fs3 fc0 sc0 ls4 ws0">负荷数据是随机数据时间序列</div><div class="t m0 x13 ha y24 ff1 fs3 fc0 sc0 ls4 ws0">即是按照时间顺序排列的<span class="_ _2"></span>一组数据<span class="_"> </span>在广义上是指一</div><div class="t m0 x6 ha y25 ff1 fs3 fc0 sc0 ls4 ws0">组有序的随机数据</div><div class="t m0 x10 ha y26 ff1 fs3 fc0 sc0 ls4 ws0">该时间序列的数据性质不<span class="_ _2"></span>仅是数据的大小而且还具有前后<span class="_ _2"></span>继承顺序的</div><div class="t m0 x6 h8 y27 ff1 fs3 fc0 sc0 ls0 ws0">性质<span class="ff2">.</span>即蕴含了数据顺序的相关性</div><div class="t m0 x14 ha y28 ff1 fs3 fc0 sc0 ls0 ws0">它也表征了变化的动态过程<span class="_ _b"> </span>所以对时间序列也称为<span class="_ _b"> </span>动</div><div class="t m0 x6 ha y29 ff1 fs3 fc0 sc0 ls0 ws0">态数据<span class="_ _c"> </span>且大多数时间序列都属于非线性时间序列<span class="_ _d"> </span>如电力系统的日负荷时间序列就是非</div><div class="t m0 x6 ha y2a ff1 fs3 fc0 sc0 ls4 ws0">线性的</div><div class="t m0 x15 ha y2b ff1 fs3 fc0 sc0 ls0 ws0">因此<span class="_"> </span><span class="ls4">对于非线性时间序列的辨识和预测是系<span class="_ _2"></span>统工程理论研究的重要内<span class="_ _2"></span>容<span class="_"> </span><span class="ls6">在预</span></span></div><div class="t m0 x6 ha y2c ff1 fs3 fc0 sc0 ls0 ws0">测方法研究中应给予重视</div><div class="t m0 x16 h8 y2d ff1 fs3 fc0 sc0 ls0 ws0">在本文所用的基于小波原理和<span class="_ _5"> </span><span class="ff2 ls5">NN<span class="_ _5"> </span></span>融合的预测原理是具有强的</div><div class="t m0 x6 ha y2e ff1 fs3 fc0 sc0 ls0 ws0">非线性时间序列的辩能力</div><div class="t m0 x16 ha y2f ff1 fs3 fc0 sc0 ls0 ws0">由研究和仿真表明它能有效提高预测的精度</div><div class="t m0 x17 h8 y30 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb h8 y31 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 x6 h9 y32 ff5 fs4 fc0 sc0 ls3 ws0">1 <span class="ff3 ls0">小波原理及小波神经网络简述<span class="ff5"> </span></span></div><div class="t m0 x6 h8 y33 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb ha y34 ff1 fs3 fc0 sc0 ls0 ws0">小波变换</div><div class="t m0 xc h8 y35 ff2 fs3 fc0 sc0 ls7 ws1">wavelet transform</div><div class="t m0 x7 h8 y34 ff1 fs3 fc0 sc0 ls0 ws0">是<span class="_ _5"> </span><span class="ff2 ls8">80<span class="_ _e"> </span></span>年代后期发展起来的应用数学分支<span class="_"> </span>小波分析的</div><div class="t m0 x6 ha y36 ff1 fs3 fc0 sc0 ls4 ws0">出现被认为是傅立叶分析的突破<span class="_ _2"></span>性进展</div><div class="t m0 x18 ha y37 ff1 fs3 fc0 sc0 ls4 ws0">它具有伸缩<span class="_"> </span><span class="ls0">平移<span class="_"> </span>放大的功能<span class="_"> </span></span>可以对信号进</div><div class="t m0 x6 ha y38 ff1 fs3 fc0 sc0 ls4 ws0">行多尺度分析</div><div class="t m0 xc ha y39 ff1 fs3 fc0 sc0 ls4 ws0">有效的从信号中提取所需的<span class="_ _2"></span>特征信息<span class="_"> </span>实现在时域和<span class="_ _2"></span>频域的高分辨局部定</div><div class="t m0 x6 ha y3a ff1 fs3 fc0 sc0 ls0 ws0">位</div><div class="t m0 xb ha y3b ff1 fs3 fc0 sc0 ls4 ws0">它在电力系统中也得到了成<span class="_ _2"></span>功的应用<span class="_"> </span>如在故障诊断<span class="_ _4"> </span>故障定位<span class="_"> </span>继电保护和谐波分</div><div class="t m0 x6 ha y3c ff1 fs3 fc0 sc0 ls0 ws0">析等方面</div><div class="t m0 xd h8 y3d ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb ha y3e ff1 fs3 fc0 sc0 ls0 ws0">小波神经网络</div><div class="t m0 x10 h8 y3f ff2 fs3 fc0 sc0 ls9 ws2">W<span class="_ _f"></span>avelet Ne<span class="_ _10"></span>ural Net<span class="_ _10"></span>work<span class="_ _11"> </span><span class="lsa ws0">WNN</span></div><div class="t m0 x19 ha y3e ff1 fs3 fc0 sc0 ls0 ws0">它起源于小波分解<span class="_"> </span>是一种新型</div><div class="t m0 x1a hc y40 ff4 fs3 fc0 sc0 ls0 ws0"> <span class="ff2 ls8">184 <span class="_ _12"> </span></span> </div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>
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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/622b83ae3d2fbb00079f38b0/bg2.jpg"><div class="t m0 xf h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">基于小波原理的短期负荷预测<span class="ff2"> </span></div><div class="t m0 x6 ha y41 ff1 fs3 fc0 sc0 ls0 ws0">的前馈型网络</div><div class="t m0 x1b h8 y42 ff1 fs3 fc0 sc0 ls0 ws0">最早由法国的<span class="_ _13"> </span><span class="ff2 lsb ws3">Qinghua Zhan<span class="_ _10"></span>g<span class="_"> </span></span><span class="lsc">等于<span class="_ _13"> </span><span class="ff2 ls9">1992<span class="_ _13"> </span></span></span>年提出的<span class="_ _4"> </span>小波神经网络引入了两</div><div class="t m0 x6 ha y43 ff1 fs3 fc0 sc0 ls4 ws0">个新的参变量<span class="_"> </span><span class="ls0">即伸缩因子和平移因子<span class="_ _3"> </span></span>所以具有比一般<span class="_ _2"></span>的神经网络更多的自由度<span class="_"> </span><span class="ls6">从而</span></div><div class="t m0 x6 ha y44 ff1 fs3 fc0 sc0 ls4 ws0">使之具有更灵活有效的函数逼近<span class="_ _2"></span>能力</div><div class="t m0 x1c ha y45 ff1 fs3 fc0 sc0 ls4 ws0">经过分析比较<span class="_"> </span><span class="ls0">得到合适的参数<span class="_"> </span></span>通过较少的级数</div><div class="t m0 x6 ha y46 ff1 fs3 fc0 sc0 ls0 ws0">项组成小波神经网络</div><div class="t m0 x1d ha y47 ff1 fs3 fc0 sc0 ls0 ws0">就能达到良好的逼近效果<span class="_"> </span>并且具有较快的学习速度</div><div class="t m0 x1e h8 y48 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 x6 h5 y49 ff4 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x6 h5 y4a ff4 fs2 fc0 sc0 ls0 ws0">1.1<span class="_ _e"> </span><span class="ff1">小波变换的数学描述</span> </div><div class="t m0 xb ha y4b ff1 fs3 fc0 sc0 lsd ws0">我们称满足以下条件</div><div class="t m0 x1f h6 y4c ff2 fs2 fc0 sc0 ls0 ws0">(</div><div class="t m0 x20 ha y4b ff1 fs3 fc0 sc0 lsd ws0">的平方可积函数<span class="_ _14"> </span>为基本小波或者母波</div><div class="t m0 x21 h6 y4c ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _15"></span>1</div><div class="t m0 x22 h6 y4d ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _16"></span>(<span class="_ _17"></span>)<span class="_ _18"></span>(</div><div class="t m0 x23 hb y4e ff2 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x24 hd y4c ff6 fs2 fc0 sc0 ls0 ws0">R<span class="_ _19"></span>L<span class="_ _1a"></span>t<span class="_ _1b"> </span><span class="ff7">∈</span></div><div class="t m1 x25 he y4c ff7 fs6 fc0 sc0 ls0 ws0">ψ</div><div class="t m0 x26 ha y4b ff1 fs3 fc0 sc0 lsd ws0">其中</div><div class="t m0 x27 ha y4f ff1 fs3 fc0 sc0 ls0 ws0">称为</div><div class="t m0 x28 h6 y50 ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _1c"></span>(</div><div class="t m1 x29 he y51 ff7 fs6 fc0 sc0 ls0 ws0">ω<span class="_ _1d"></span>ψ</div><div class="t m0 x2a hf y52 ff7 fs5 fc0 sc0 ls0 ws0">∧</div><div class="t m0 x1b h6 y51 ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _1c"></span>(</div><div class="t m1 x2b he y51 ff7 fs6 fc0 sc0 ls0 ws0">ω</div><div class="c x2c y53 w2 h10"><div class="t m1 x0 he y54 ff7 fs6 fc0 sc0 ls0 ws0">ψ</div></div><div class="t m0 x1 ha y4f ff1 fs3 fc0 sc0 ls0 ws0">的<span class="_ _1e"> </span>变换</div><div class="t m0 x2d hd y51 ff6 fs2 fc0 sc0 ls0 ws0">Fourier</div><div class="t m0 x4 h8 y55 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb h8 y56 ff2 fs3 fc0 sc0 lse ws0"> <span class="_ _10"></span> <span class="_ _10"></span> </div><div class="t m0 x2e h11 y57 ff7 fs7 fc0 sc0 ls0 ws0">∞<span class="_ _1f"></span><</div><div class="t m0 x2f h12 y58 ff7 fs8 fc0 sc0 ls0 ws0">∫</div><div class="t m0 x30 h13 y59 ff7 fs9 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x31 h13 y5a ff7 fs9 fc0 sc0 ls0 ws0">∞<span class="_ _20"></span>−</div><div class="t m0 x32 h13 y5b ff7 fs9 fc0 sc0 ls0 ws0">∧</div><div class="t m2 x33 h14 y5c ff7 fsa fc0 sc0 ls0 ws0">ω</div><div class="t m2 x14 h14 y5d ff7 fsa fc0 sc0 ls0 ws0">ω</div><div class="t m2 x34 h14 y5e ff7 fsa fc0 sc0 ls0 ws0">ω<span class="_ _21"></span>ψ</div><div class="t m0 x1c h15 y5f ff6 fs7 fc0 sc0 ls0 ws0">d</div><div class="t m0 x35 h16 y60 ff2 fs9 fc0 sc0 ls0 ws0">2</div><div class="t m0 x36 h17 y61 ff2 fs7 fc0 sc0 ls0 ws0">)<span class="_ _1f"></span>(</div><div class="t m0 x37 h8 y62 ff2 fs3 fc0 sc0 lse ws0"> <span class="_ _10"></span> <span class="_ _10"></span> <span class="_ _10"></span> <span class="_ _12"> </span><span class="ls0">1<span class="_ _22"> </span> </span></div><div class="t m0 x38 h6 y63 ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _18"></span>(<span class="ff6">t</span></div><div class="c xb y64 w3 h18"><div class="t m1 x0 he y54 ff7 fs6 fc0 sc0 ls0 ws0">ψ</div></div><div class="t m0 x39 ha y65 ff1 fs3 fc0 sc0 ls0 ws0">经过伸缩平移可以产生一组小波函数基</div><div class="t m0 x3a h8 y66 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb h8 y67 ff2 fs3 fc0 sc0 lse ws0"> <span class="_ _10"></span> <span class="_ _10"></span> </div><div class="t m0 x3b h19 y68 ff2 fsb fc0 sc0 ls0 ws0">)<span class="_ _23"></span>(</div><div class="t m0 x33 h19 y69 ff2 fsb fc0 sc0 ls0 ws0">1</div><div class="t m0 x4 h19 y68 ff2 fsb fc0 sc0 ls0 ws0">)<span class="_ _24"></span>(</div><div class="t m0 x3c h1a y6a ff2 fsc fc0 sc0 ls0 ws0">,</div><div class="t m0 x3d h1b y6b ff6 fsb fc0 sc0 ls0 ws0">a</div><div class="t m0 x8 h1b y6c ff6 fsb fc0 sc0 ls0 ws0">b<span class="_ _1d"></span>x</div><div class="t m0 x3e h1b y6d ff6 fsb fc0 sc0 ls0 ws0">a</div><div class="t m0 x34 h1b y6e ff6 fsb fc0 sc0 ls0 ws0">x</div><div class="t m0 xf h1c y6f ff6 fsc fc0 sc0 ls0 ws0">b<span class="_ _15"></span>a</div><div class="t m0 x3d h1d y70 ff7 fsb fc0 sc0 ls0 ws0">−</div><div class="t m0 x3f h1d y71 ff7 fsb fc0 sc0 ls0 ws0">=</div><div class="t m3 x40 h1e y72 ff7 fsd fc0 sc0 ls0 ws0">ψ</div><div class="t m3 x41 h1e y68 ff7 fsd fc0 sc0 ls0 ws0">ψ</div><div class="t m0 x42 h8 y73 ff2 fs3 fc0 sc0 lse ws0"> <span class="_ _10"></span> <span class="_ _10"></span> <span class="_ _12"> </span><span class="ls0">2<span class="_ _7"> </span> </span></div><div class="t m0 xb ha y74 ff1 fs3 fc0 sc0 ls0 ws0">其中</div><div class="t m0 xd hd y75 ff6 fs2 fc0 sc0 ls0 ws0">a</div><div class="t m0 x43 h8 y76 ff2 fs3 fc0 sc0 ls0 ws0">,</div><div class="t m0 x44 hd y75 ff6 fs2 fc0 sc0 ls0 ws0">b</div><div class="t m0 xc ha y77 ff1 fs3 fc0 sc0 ls0 ws0">分别为伸缩<span class="_ _7"> </span>平移尺度因子<span class="_ _7"> </span>对于函数</div><div class="t m0 x45 h6 y75 ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _24"></span>(<span class="_ _25"></span><span class="ff6">x<span class="_ _16"></span>f</span></div><div class="t m0 x46 h6 y78 ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _16"></span>(</div><div class="t m0 x47 hb y79 ff2 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x48 hd y75 ff6 fs2 fc0 sc0 ls0 ws0">R<span class="_ _26"></span>L</div><div class="t m0 x49 h1f y7a ff7 fs2 fc0 sc0 ls0 ws0">∈</div><div class="t m0 x4a ha y77 ff1 fs3 fc0 sc0 ls0 ws0">其连续小波变换表</div><div class="t m0 xb h8 y7b ff1 fs3 fc0 sc0 ls0 ws0">示为<span class="ff2"> </span></div><div class="t m0 x4b h8 y7c ff2 fs3 fc0 sc0 lse ws0"> <span class="_ _10"></span> </div><div class="t m0 x4c hd y7d ff6 fs2 fc0 sc0 ls0 ws0">Wf</div><div class="t m0 x24 h8 y7e ff2 fs3 fc0 sc0 lse ws0"> <span class="_ _10"></span> <span class="_ _10"></span> </div><div class="t m0 x4d h20 y7f ff7 fs1 fc0 sc0 ls0 ws0">∫</div><div class="t m0 x1c hf y80 ff7 fs5 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x4e hf y81 ff7 fs5 fc0 sc0 ls0 ws0">∞<span class="_ _15"></span>−</div><div class="t m0 x4 h1f y82 ff7 fs2 fc0 sc0 ls0 ws0">=</div><div class="t m0 x4f hd y7d ff6 fs2 fc0 sc0 ls0 ws0">dx<span class="_ _27"></span>x<span class="_ _16"></span>f<span class="_ _1f"></span>x<span class="_ _28"></span>b<span class="_ _1c"></span>a</div><div class="t m0 x50 h21 y83 ff6 fs5 fc0 sc0 ls0 ws0">b<span class="_ _29"></span>a</div><div class="t m0 x51 h6 y7d ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _24"></span>(<span class="_ _1c"></span>)<span class="_ _2a"></span>(<span class="_ _2b"></span>)<span class="_ _2a"></span>,<span class="_ _2c"></span>(</div><div class="t m0 x2e hb y83 ff2 fs5 fc0 sc0 ls0 ws0">,</div><div class="t m0 x40 hb y84 ff2 fs5 fc0 sc0 ls0 ws0">*</div><div class="t m1 x52 he y82 ff7 fs6 fc0 sc0 ls0 ws0">ψ</div><div class="t m0 x53 h8 y7e ff2 fs3 fc0 sc0 ls0 ws0">3<span class="_ _7"> </span> </div><div class="t m0 x54 ha y85 ff1 fs3 fc0 sc0 ls0 ws0">当小波函数满足条件</div><div class="t m0 x55 h1f y86 ff7 fs2 fc0 sc0 ls0 ws0">∞<span class="_ _2d"></span><<span class="_ _2e"></span>=</div><div class="t m0 x56 h20 y87 ff7 fs1 fc0 sc0 ls0 ws0">∫</div><div class="t m0 x57 hf y88 ff7 fs5 fc0 sc0 ls0 ws0">∧</div><div class="t m0 x3f hf y89 ff7 fs5 fc0 sc0 ls0 ws0">−</div><div class="t m0 x58 hd y8a ff6 fs2 fc0 sc0 ls0 ws0">dy<span class="_ _2f"></span>y<span class="_ _30"></span>y<span class="_ _31"></span>C</div><div class="t m0 x59 h21 y8b ff6 fs5 fc0 sc0 ls0 ws0">R</div><div class="t m0 x50 hb y8c ff2 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x35 hb y8d ff2 fs5 fc0 sc0 ls0 ws0">1</div><div class="t m0 x5a h6 y8e ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _32"></span>(</div><div class="t m1 x5b he y86 ff7 fs6 fc0 sc0 ls0 ws0">ψ</div><div class="t m0 x5c ha y85 ff1 fs3 fc0 sc0 ls0 ws0">时<span class="_"> </span>我们有<span class="_ _33"> </span>的小波反演公式</div><div class="t m0 x5d h6 y8a ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _2a"></span>(<span class="_ _34"></span><span class="ff6">x<span class="_ _16"></span>f</span></div><div class="t m0 x5e h8 y8f ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 x31 h6 y90 ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _2a"></span>(<span class="_ _34"></span><span class="ff6">x<span class="_ _16"></span>f</span></div><div class="t m0 x5f h8 y91 ff2 fs3 fc0 sc0 ls0 ws0">=</div><div class="t m0 x49 hd y90 ff6 fs2 fc0 sc0 ls0 ws0">db<span class="_ _35"></span>b<span class="_ _16"></span>a<span class="_ _36"></span>w<span class="_ _1c"></span>t</div><div class="t m0 x1c hd y92 ff6 fs2 fc0 sc0 ls0 ws0">a</div><div class="t m0 x5b hd y93 ff6 fs2 fc0 sc0 ls0 ws0">da</div><div class="t m0 x60 hd y94 ff6 fs2 fc0 sc0 ls0 ws0">C</div><div class="t m0 x40 h21 y95 ff6 fs5 fc0 sc0 ls0 ws0">R</div><div class="t m0 x61 h21 y96 ff6 fs5 fc0 sc0 ls0 ws0">f<span class="_ _37"></span>b<span class="_ _29"></span>a</div><div class="t m0 x62 h21 y95 ff6 fs5 fc0 sc0 ls0 ws0">R</div><div class="t m0 x22 h6 y90 ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _2a"></span>,<span class="_ _2c"></span>(<span class="_ _1d"></span>)<span class="_ _18"></span>(</div><div class="t m0 x63 hb y97 ff2 fs5 fc0 sc0 ls0 ws0">,</div><div class="t m0 x52 hb y98 ff2 fs5 fc0 sc0 ls0 ws0">2</div><div class="t m0 x4 hb y99 ff2 fs5 fc0 sc0 ls0 ws0">1</div><div class="t m0 x64 h20 y9a ff7 fs1 fc0 sc0 ls0 ws0">∫<span class="_ _38"></span>∫</div><div class="t m0 x32 hf y9b ff7 fs5 fc0 sc0 ls0 ws0">−</div><div class="t m1 x65 he y9c ff7 fs6 fc0 sc0 ls0 ws0">ψ</div><div class="t m0 x66 h8 y91 ff2 fs3 fc0 sc0 lsf ws0"> <span class="_ _2"></span> <span class="_ _2"></span> <span class="_ _4"> </span><span class="ls0">4<span class="_ _6"> </span> </span></div><div class="t m0 x6 h5 y9d ff4 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x6 h5 y9e ff4 fs2 fc0 sc0 ls0 ws0">1.2 <span class="ff1">小波神经网络</span></div><div class="t m0 x67 hc y9f ff4 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb h8 ya0 ff2 fs3 fc0 sc0 lsa ws0">WNN<span class="_ _39"> </span><span class="ff1 ls0">是一种以小波基函数代替通常的非线性<span class="_ _39"> </span></span><span class="ls10">Sigmoid<span class="_ _39"> </span><span class="ff1 ls0">功能函数作为神经元的激励函</span></span></div><div class="t m0 x6 h8 ya1 ff1 fs3 fc0 sc0 ls0 ws0">数的<span class="_ _13"> </span><span class="ff2 ls11">FNN<span class="_"> </span></span><span class="ls4">模型</span></div><div class="t m0 x68 ha ya2 ff1 fs3 fc0 sc0 ls0 ws0">它既可看作是以小波函数为基底的函数连接型网络<span class="_ _d"> </span>也可看作是径向基函</div><div class="t m0 x6 h8 ya3 ff1 fs3 fc0 sc0 ls0 ws0">数网络的推广<span class="ff2 ls12">,WNN<span class="_ _6"> </span></span>采用单隐层结构</div><div class="t m0 x5b ha ya4 ff1 fs3 fc0 sc0 ls0 ws0">并不影响它的逼近能力</div><div class="t m0 x69 hb ya5 ff2 fs5 fc0 sc0 ls0 ws0">]<span class="_ _8"></span>3<span class="_ _9"></span>[</div><div class="t m0 x6a h8 ya6 ff2 fs3 fc0 sc0 ls13 ws0">WNN<span class="_ _6"> </span><span class="ff1 ls0">在逼近单变量函</span></div><div class="t m0 x6 ha ya7 ff1 fs3 fc0 sc0 ls0 ws0">数时<span class="_"> </span>是最优渐进的逼近器</div><div class="t m0 x2f hb ya8 ff2 fs5 fc0 sc0 ls0 ws0">[<span class="_ _3a"> </span>]<span class="_ _8"></span>4</div><div class="t m0 x6b h8 ya7 ff1 fs3 fc0 sc0 ls0 ws0">如图<span class="_ _13"> </span><span class="ff2">1<span class="_"> </span></span>所示<span class="_"> </span>为一个多输入多输出的<span class="_ _13"> </span><span class="ff2 ls11">FNN<span class="_"> </span></span>结构<span class="ff2">,</span>设<span class="_ _13"> </span><span class="ff2">x</span></div><div class="t m0 x6c hb ya9 ff2 fs5 fc0 sc0 ls0 ws0">k</div><div class="t m0 x26 ha ya7 ff1 fs3 fc0 sc0 lsc ws0">为输</div><div class="t m0 x6 h8 yaa ff1 fs3 fc0 sc0 ls0 ws0">入层第<span class="_ _13"> </span><span class="ff2">k<span class="_"> </span></span>个输入变量</div><div class="t m0 x6d h8 yab ff2 fs3 fc0 sc0 ls0 ws0">y</div><div class="t m0 x6e hb yac ff2 fs5 fc0 sc0 ls0 ws0">i</div><div class="t m0 x6f h8 yad ff1 fs3 fc0 sc0 ls0 ws0">为输出层第<span class="_ _13"> </span><span class="ff2">i<span class="_"> </span></span>个输出值</div><div class="t m0 x70 h8 yab ff2 fs3 fc0 sc0 ls0 ws0">w</div><div class="t m0 x8 hb yac ff2 fs5 fc0 sc0 ls14 ws0">ij</div><div class="t m0 x71 h8 yad ff1 fs3 fc0 sc0 ls0 ws0">为连接输出层节点<span class="_ _13"> </span><span class="ff2">i<span class="_"> </span></span>和隐层节点<span class="_ _13"> </span><span class="ff2">j<span class="_"> </span></span>之间</div><div class="t m0 x6 ha yae ff1 fs3 fc0 sc0 ls0 ws0">的权值</div><div class="t m0 x38 h8 yaf ff2 fs3 fc0 sc0 ls0 ws0">w</div><div class="t m0 x72 hb yb0 ff2 fs5 fc0 sc0 ls14 ws0">jk</div><div class="t m0 x73 h8 yb1 ff1 fs3 fc0 sc0 ls0 ws0">为连接隐层节点<span class="_ _13"> </span><span class="ff2">j<span class="_"> </span></span>和输入层节点<span class="_ _13"> </span><span class="ff2">k<span class="_"> </span></span>之间的权重<span class="_ _0"> </span>假定<span class="_ _1"> </span><span class="ff2">w</span></div><div class="t m0 x74 hb yb0 ff2 fs5 fc0 sc0 ls15 ws0">io</div><div class="t m0 x75 h8 yb1 ff1 fs3 fc0 sc0 ls0 ws0">是第<span class="_ _13"> </span><span class="ff2">i<span class="_"> </span></span>个输出层节点的</div><div class="t m0 x6 ha yb2 ff1 fs3 fc0 sc0 ls0 ws0">阈值</div><div class="t m0 x76 h8 yb3 ff2 fs3 fc0 sc0 ls0 ws0">a</div><div class="t m0 xa hb yb4 ff2 fs5 fc0 sc0 ls0 ws0">j</div><div class="t m0 x77 h8 yb3 ff2 fs3 fc0 sc0 ls0 ws0">b</div><div class="t m0 x73 hb yb4 ff2 fs5 fc0 sc0 ls0 ws0">j</div><div class="t m0 x78 h8 yb5 ff1 fs3 fc0 sc0 ls0 ws0">分别是第<span class="_ _5"> </span><span class="ff2">j<span class="_ _5"> </span></span>个隐层节点的伸缩和平移系数</div><div class="t m0 x79 h8 yb3 ff2 fs3 fc0 sc0 ls0 ws0">P<span class="_ _4"> </span><span class="ls16">p=1,2…..P</span></div><div class="t m0 x7a ha yb5 ff1 fs3 fc0 sc0 ls0 ws0">为样本的总数</div><div class="t m0 x7b h8 yb3 ff2 fs3 fc0 sc0 ls0 ws0">m</div><div class="t m0 x6 ha yb6 ff1 fs3 fc0 sc0 ls0 ws0">为输入层节点个数</div><div class="t m0 x10 h8 yb7 ff2 fs3 fc0 sc0 ls16 ws0">k=1,2…..m<span class="_ _3b"> </span><span class="ls0">n<span class="_"> </span><span class="ff1">为隐含层节点个数<span class="_"> </span></span></span>j=1,2….n<span class="_ _3b"> </span><span class="ls0">N<span class="_"> </span><span class="ff1">为输出层节点的个</span></span></div><div class="t m0 x6 ha yb8 ff1 fs3 fc0 sc0 ls0 ws0">数</div><div class="t m0 xb h8 yb9 ff2 fs3 fc0 sc0 ls17 ws0">i=1,2….N</div><div class="t m0 x1 h8 yba ff1 fs3 fc0 sc0 ls0 ws0">则<span class="_ _13"> </span><span class="ff2 lsa">WNN<span class="_"> </span></span>的输出可表示为</div><div class="t m0 x7c h8 yb9 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 x6 h8 ybb ff2 fs3 fc0 sc0 lse ws0"> <span class="_ _10"></span> </div><div class="t m0 x79 h8 ybc ff2 fs3 fc0 sc0 ls0 ws0"> <span class="_ _3c"> </span><span class="lsf"> <span class="_ _2"></span> </span></div><div class="t m0 x7d h6 ybd ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _20"></span>)<span class="_ _3d"></span>)<span class="_ _18"></span>(<span class="_ _3e"></span>(<span class="_ _3f"></span>(<span class="_ _37"></span>)<span class="_ _18"></span>(</div><div class="t m0 x6b hb ybe ff2 fs5 fc0 sc0 ls18 ws0">00</div><div class="t m0 x1c hb ybf ff2 fs5 fc0 sc0 ls0 ws0">,</div><div class="t m0 x7e h20 yc0 ff7 fs1 fc0 sc0 ls19 ws0">∑∑</div><div class="t m0 x7f hf yc1 ff7 fs5 fc0 sc0 ls1a ws0">==</div><div class="t m0 x80 h1f yc2 ff7 fs2 fc0 sc0 ls0 ws0">=</div><div class="t m0 x7f h21 yc3 ff6 fs5 fc0 sc0 ls0 ws0">n</div><div class="t m0 x81 h21 yc4 ff6 fs5 fc0 sc0 ls0 ws0">j</div><div class="t m0 x82 h21 yc3 ff6 fs5 fc0 sc0 ls0 ws0">m</div><div class="t m0 x57 h21 yc4 ff6 fs5 fc0 sc0 ls0 ws0">k</div><div class="t m0 x83 h21 yc5 ff6 fs5 fc0 sc0 ls0 ws0">k<span class="_ _40"></span>jk<span class="_ _30"></span>b<span class="_ _29"></span>a<span class="_ _41"></span>ij<span class="_ _42"></span>i</div><div class="t m0 x71 hd ybd ff6 fs2 fc0 sc0 ls0 ws0">t<span class="_ _43"></span>x<span class="_ _44"></span>w<span class="_ _45"></span>w<span class="_ _46"></span>f<span class="_ _1a"></span>t<span class="_ _1c"></span>y</div><div class="t m1 x84 he yc2 ff7 fs6 fc0 sc0 ls0 ws0">ψ</div><div class="t m0 x85 h6 ybd ff2 fs2 fc0 sc0 ls0 ws0">)<span class="_ _2f"></span>.....<span class="_ _21"></span>2<span class="_ _15"></span>,<span class="_ _20"></span>1<span class="_ _36"></span>(<span class="_ _47"> </span><span class="ff6">N<span class="_ _48"></span>i</span></div><div class="t m0 x86 h1f yc6 ff7 fs2 fc0 sc0 ls0 ws0">=</div><div class="t m0 x87 h8 ybc ff2 fs3 fc0 sc0 ls0 ws0">5<span class="_ _6"> </span> </div><div class="t m0 xb ha yc7 ff1 fs3 fc0 sc0 ls0 ws0">最小均方误差能量函数为</div><div class="t m0 xf h8 yc8 ff2 fs3 fc0 sc0 ls0 ws0"> </div><div class="t m0 x88 hc y40 ff4 fs3 fc0 sc0 ls0 ws0"> <span class="ff2 ls8">185 <span class="_ _49"> </span></span> </div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>
