📄 第三章 模糊神经网络.htm
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<TR>
<TD width="80%" colSpan=2 height=13><IMG height=26
src="第三章 模糊神经网络.files/6.htm84.gif" width=169 border=0></TD>
<TD width="20%" height=13>(3-64)</TD></TR>
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<TD width="80%" colSpan=2 height=15><FONT
size=2>其中：D是距离，如式(3—59)所示。<BR>则说明HFNN可以逼近映射F；也即是说F是一个通用逼近器。<BR>下面给出证明过程：<BR>(1)设对于给定<SPAN
style="FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">ε</SPAN>>0，令</FONT></TD>
<TD width="20%" height=15></TD></TR>
<TR>
<TD width="80%" colSpan=2 height=9><IMG height=28
src="第三章 模糊神经网络.files/6.htm85.gif" width=263 border=0></TD>
<TD width="20%" height=9>(3-65)</TD></TR>
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<TD width="80%" colSpan=2 height=21><FONT size=2>则对应存在<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">δ</SPAN>>o，令</FONT></TD>
<TD width="20%" height=21></TD></TR>
<TR>
<TD width="80%" colSpan=2 height=15><IMG height=31
src="第三章 模糊神经网络.files/6.htm86.gif" width=256 border=0></TD>
<TD width="20%" height=15>(3-66)</TD></TR>
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<TD width="80%" colSpan=2 height=18><FONT
size=2>其中：D(N<SUB>1</SUB>．M<SUB>1</SUB>)，(N<SUB>2</SUB>，M<SUB>2</SUB>)属于U。<BR>对U中的每一个(N，M)，令</FONT></TD>
<TD width="20%" height=18></TD></TR>
<TR>
<TD width="80%" colSpan=2 height=21><IMG height=30
src="第三章 模糊神经网络.files/6.htm87.gif" width=542 border=0></TD>
<TD width="20%" height=21>(3-67)</TD></TR>
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<TD width="80%" colSpan=2 height=11><FONT size=2>这即是紧集合U的开复盖(open
cover)。因此，存在有限子复盖(subcover):</FONT></TD>
<TD width="20%" height=11></TD></TR>
<TR>
<TD width="80%" colSpan=2 height=15><IMG height=28
src="第三章 模糊神经网络.files/6.htm88.gif" width=246 border=0></TD>
<TD width="20%" height=15>(3-68)</TD></TR>
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<TD width="80%" colSpan=2 height=15><FONT
size=2>令Nk＝Ak，Mk＝Bk，以及映射F为</FONT></TD>
<TD width="20%" height=15></TD></TR>
<TR>
<TD width="80%" colSpan=2 height=20><IMG height=31
src="第三章 模糊神经网络.files/6.htm89.gif" width=256 border=0></TD>
<TD width="20%" height=20>(3-69)</TD></TR>
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<TD width="80%" colSpan=2 height=13><FONT size=2>其中：1<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>k<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>K。<BR>考虑U中任意给出的(N，M)，对于映射F有</FONT></TD>
<TD width="20%" height=13></TD></TR>
<TR>
<TD width="80%" colSpan=2 height=21><IMG height=30
src="第三章 模糊神经网络.files/6.htm90.gif" width=141 border=0></TD>
<TD width="20%" height=21>(3-70)</TD></TR>
<TR>
<TD width="80%" colSpan=2 height=11><FONT
size=2>由于在式(3—69)，(3—70)中的映射F相同；同时从式(3—68)可知</FONT>
<P><IMG height=28 src="第三章 模糊神经网络.files/6.htm91.gif" width=241
border=0></P>
<P>故而必定有：</P>
<P><IMG height=30 src="第三章 模糊神经网络.files/6.htm92.gif" width=260
border=0></P></TD>
<TD width="20%" height=11></TD></TR>
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<TD width="12%" height=12><FONT size=2>也即是有</FONT></TD>
<TD width="68%" height=12><IMG height=29
src="第三章 模糊神经网络.files/6.htm93.gif" width=130 border=0></TD>
<TD width="20%" height=12>(3-71)</TD></TR>
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<TD width="12%" height=15><FONT size=2>或者</FONT></TD>
<TD width="68%" height=15><IMG height=30
src="第三章 模糊神经网络.files/6.htm94.gif" width=128 border=0></TD>
<TD width="20%"
height=15>(3-72)</TD></TR></TBODY></TABLE></CENTER></DIV></TD></TR>
<TR>
<TD width="100%" height=12>
<P>(2)对于图3-6所示的混合模糊神经网络，由于其输入是(N，M)；同时，(N，M) 属于紧集合U，而且必须至少是
<P><IMG height=30 src="第三章 模糊神经网络.files/6.htm95.gif" width=245
border=0></P>
<P>集合中的元素之一，亦即N＝A<SUB>k</SUB>，M＝B<SUB>k</SUB>，1<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>k<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>K。令</P>
<P>G={k|<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">λ</SPAN><SUB>k</SUB>=1,1<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>k<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">≤</SPAN>K}
(3-73)</P>
<P>由于<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">λ</SPAN><SUB>k</SUB>不可能全部为0，故G是非空集。所以，有<SPAN
style="FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: Times New Roman; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">θ</SPAN>>0。<BR>用m表示G所含的基本元素；为简单起见，设G＝{1，2，…，m}，则有</P>
<DIV align=center>
<CENTER>
<TABLE height=449 cellSpacing=0 cellPadding=0 width="80%" border=0>
<TBODY>
<TR>
<TD width="80%" height=43><IMG height=41
src="第三章 模糊神经网络.files/6.htm96.gif" width=160 border=0></TD>
<TD width="20%" height=43>(3-74)</TD></TR>
<TR>
<TD width="80%" height=41><IMG height=39
src="第三章 模糊神经网络.files/6.htm97.gif" width=227 border=0></TD>
<TD width="20%" height=41>(3-75)</TD></TR>
<TR>
<TD width="80%" height=18><FONT size=2>从而有</FONT></TD>
<TD width="20%" height=18></TD></TR>
<TR>
<TD width="80%" height=42><IMG height=40
src="第三章 模糊神经网络.files/6.htm98.gif" width=179 border=0></TD>
<TD width="20%" height=42>(3-76)</TD></TR>
<TR>
<TD width="80%" height=98><FONT size=2>(3)考虑距离d</FONT>
<P><IMG height=26 src="第三章 模糊神经网络.files/6.htm99.gif" width=97
border=0></P>
<P>可写成</P></TD>
<TD width="20%" height=98></TD></TR>
<TR>
<TD width="80%" height=45><IMG height=43
src="第三章 模糊神经网络.files/6.htm100.gif" width=160 border=0></TD>
<TD width="20%" height=45>(3-77)</TD></TR>
<TR>
<TD width="80%" height=102><FONT size=2>对于d(A，B)，如果<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">β</SPAN>>0，A，B<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">∈</SPAN>L。则有</FONT>
<P><IMG height=29 src="第三章 模糊神经网络.files/6.htm101.gif" width=248
border=0></P>
<P>同理，对于式(3—77)就有</P></TD>
<TD width="20%" height=102></TD></TR>
<TR>
<TD width="80%" height=42><IMG height=40
src="第三章 模糊神经网络.files/6.htm102.gif" width=325 border=0></TD>
<TD width="20%" height=42>(3-78)</TD></TR>
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<TD width="80%" height=1><FONT size=2>(4)对于L中的A，B，C，有</FONT></TD>
<TD width="20%" height=1></TD></TR>
<TR>
<TD width="80%" height=18><IMG height=32
src="第三章 模糊神经网络.files/6.htm103.gif" width=416 border=0></TD>
<TD width="20%"
height=18>(3-79)</TD></TR></TBODY></TABLE></CENTER></DIV></TD></TR>
<TR>
<TD width="100%" height=7>
<P>首先，对于A，B，C的<SPAN
style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-bidi-font-size: 10.0pt; mso-bidi-font-family: 'Times New Roman'; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA">α</SPAN>截集有
<P><IMG height=26 src="第三章 模糊神经网络.files/6.htm104.gif" width=203
border=0></P>
<P><IMG height=27 src="第三章 模糊神经网络.files/6.htm105.gif" width=204
border=0></P>
<P><IMG height=27 src="第三章 模糊神经网络.files/6.htm106.gif" width=201
border=0></P>
<P>这都是闭区间。故有</P>
<P><IMG height=157 src="第三章 模糊神经网络.files/6.htm107.gif" width=628
border=0></P>
<P>从式(3—59)，可知式(3—79)成立。</P>
<P>(5)最后考虑d(Q’，Q)</P>
<P>从式(3—76)．(3—77)，(3—78)可知</P>
<DIV align=center>
<CENTER>
<TABLE cellSpacing=0 cellPadding=0 width="80%" border=0>
<TBODY>
<TR>
<TD width="78%"><IMG height=40 src="第三章 模糊神经网络.files/6.htm108.gif"
width=288 border=0></TD>
<TD width="22%">(3-81)</TD></TR></TBODY></TABLE></CENTER></DIV></TD></TR>
<TR>
<TD width="100%" height=11>
<P>从中有 </P>
<P><IMG height=43 src="第三章 模糊神经网络.files/6.htm109.gif" width=319
border=0></P>
<P>也可写成</P>
<P><IMG height=38 src="第三章 模糊神经网络.files/6.htm110.gif" width=360
border=0></P>
<P>故从式(3—79)，则有</P>
<TABLE cellSpacing=0 cellPadding=0 width="80%" align=center border=0>
<TBODY>
<TR>
<TD width="74%"><IMG height=41 src="第三章 模糊神经网络.files/6.htm111.gif"
width=289 border=0></TD>
<TD width="26%">(3-82)</TD></TR>
<TR>
<TD width="74%"><FONT size=2>从式(3—72)，则有</FONT></TD>
<TD width="26%"></TD></TR>
<TR>
<TD width="74%"><IMG height=39 src="第三章 模糊神经网络.files/6.htm112.gif"
width=198 border=0></TD>
<TD width="26%">(3-83)</TD></TR></TBODY></TABLE></TD></TR>
<TR>
<TD width="100%" height=52>
<P>再从式(3—81)有 </P>
<P><IMG height=30 src="第三章 模糊神经网络.files/6.htm113⌨️ 快捷键说明
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