範例：函數的線性組合

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範例：函數的線性組合

使用 linfit 函數以項數的線性總和，設定數據集模型。每個項數都是由要擬合之函數與 linfit 計算所得的係數所構成。

1. 定義以下矩陣。

按一下以複製此運算式

<region id="ID0EGTLT" top="441.6" left="38.4" width="130.883333333333" height="583.6" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="24">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <id label="None" xml:space="preserve">data</id>
      <matrix rows="31" cols="2">
        <real>1</real>
        <real>2</real>
        <real>3</real>
        <real>4</real>
        <real>5</real>
        <real>6</real>
        <real>7</real>
        <real>8</real>
        <real>9</real>
        <real>10</real>
        <real>11</real>
        <real>12</real>
        <real>13</real>
        <real>14</real>
        <real>15</real>
        <real>16</real>
        <real>17</real>
        <real>18</real>
        <real>19</real>
        <real>20</real>
        <real>21</real>
        <real>22</real>
        <real>23</real>
        <real>24</real>
        <real>25</real>
        <real>26</real>
        <real>27</real>
        <real>28</real>
        <real>29</real>
        <real>30</real>
        <real>31</real>
        <real>4.181</real>
        <real>4.665</real>
        <real>5.296</real>
        <real>5.365</real>
        <real>5.448</real>
        <real>5.744</real>
        <real>5.653</real>
        <real>5.844</real>
        <real>6.362</real>
        <real>6.38</real>
        <real>6.311</real>
        <real>6.457</real>
        <real>6.479</real>
        <real>6.59</real>
        <real>6.74</real>
        <real>6.58</real>
        <real>6.852</real>
        <real>6.531</real>
        <real>6.682</real>
        <real>7.013</real>
        <real>6.82</real>
        <real>6.647</real>
        <real>6.951</real>
        <real>7.121</real>
        <real>7.143</real>
        <real>6.914</real>
        <real>6.941</real>
        <real>7.226</real>
        <real>6.898</real>
        <real>7.392</real>
        <real>6.938</real>
      </matrix>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

2. 定義要擬合的函數向量。

按一下以複製此運算式

<region id="ID0EWULT" top="825.6" left="38.4" width="113.743333333333" height="69.0453337043524" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="25">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id label="None" xml:space="preserve">vf</id>
        <boundVars>
          <id label="None" xml:space="preserve">x</id>
        </boundVars>
      </function>
      <matrix rows="3" cols="1">
        <apply>
          <id label="Function" xml:space="preserve" label-is-contextual="true">ln</id>
          <id label="Variable" xml:space="preserve" label-is-contextual="true">x</id>
        </apply>
        <apply>
          <nthRoot />
          <placeholder />
          <id label="Variable" xml:space="preserve" label-is-contextual="true">x</id>
        </apply>
        <real>1</real>
      </matrix>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

模型函數是 y(x) = c0 * ln(x) + c1 * sqrt x +c3。

3. 呼叫 linfit 函數計算每個函數的係數。以向量 c 記錄係數。

4. 定義擬合函數 f 使其成為 vf 中函數線性組合乘以 c 中的係數。

5. 繪製數據點與擬合函數。

按一下以複製此運算式

<region id="ID0EZ1LT" top="1248" left="19.2" width="600" height="277" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot origin-positioning="true" background-type="white" show-axis-expressions="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace resultRef="30">
          <traceStyle color="#FF00008B" symbol="x" line-weight="1" line-style="None">線條</traceStyle>
        </trace>
        <trace resultRef="31">
          <traceStyle color="#FFFF0000" symbol="none" line-weight="1" line-style="Solid">線條</traceStyle>
        </trace>
      </traces>
      <graph-size width="484.8" height="219.4" />
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="1" end="31">
          <axisLine position="bottom" positionticmark="-1" legendWidth="64.2033333333333" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="32">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">X</id>
              </math>
              <math resultRef="33">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="4.05" end="8.05">
          <axisLine position="left" positionticmark="-1" legendWidth="85.5766666666667" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <plotEquations>
            <plotEquation>
              <math resultRef="36">
                <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">Y</id>
              </math>
              <math resultRef="37">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math resultRef="38">
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <vectorize />
                  <apply>
                    <id xml:space="preserve">f</id>
                    <id xml:space="preserve">X</id>
                  </apply>
                </apply>
              </math>
              <math resultRef="39">
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue resultRef="35">
              <real xmlns="http://schemas.mathsoft.com/math50">8</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

函數的線性組合對數據而言是良好的擬合。經由相關係數值近似於 1 即可證實：

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函數的線性組合

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