範例：二階回應曲面

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範例：二階回應曲面

使用 boxwilson、polyfitc 與 polyfit 函數可設計實驗，以及建立所研究之流程的模型。使用解題指令群求解模型所預測的最大產出。

1. 定義 boxwilson 設計矩陣。

2. 定義 A (以分鐘表示的反應時間) 與 B (以華氏表示的溫度) 的實數，這兩者都是所研究之化學流程的自然變數。

3. 呼叫 doelabel 函數，將設計矩陣的編碼值轉換為自然變數的實數。

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4. 記錄每次執行的化學流程產出。

5. 呼叫 polyfitc，使用編碼值執行二階多項式遞歸。

6. 從 polyfitc 的輸出萃取遞歸係數。

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    <define xmlns="http://schemas.mathsoft.com/math50">
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          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">polyfitc</id>
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            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">X</id>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Y</id>
            <real>2</real>
          </sequence>
        </apply>
        <real>1</real>
      </apply>
    </define>
    <resultFormat>
      <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
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7. 建立形成化學反應模型之編碼值的函數。使用此函數可預測在 A=-1 且 B=1 時的產出。

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    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">yield</id>
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          <id labels="VARIABLE" xml:space="preserve">B</id>
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        <plus />
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                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c</id>
                  <real>1</real>
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                <apply>
                  <mult />
                  <apply>
                    <indexer />
                    <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c</id>
                    <real>2</real>
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                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id>
                </apply>
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              <apply>
                <mult />
                <apply>
                  <indexer />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c</id>
                  <real>3</real>
                </apply>
                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">B</id>
              </apply>
            </apply>
            <apply>
              <mult />
              <apply>
                <mult />
                <apply>
                  <indexer />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c</id>
                  <real>4</real>
                </apply>
                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id>
              </apply>
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">B</id>
            </apply>
          </apply>
          <apply>
            <mult />
            <apply>
              <indexer />
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c</id>
              <real>5</real>
            </apply>
            <apply>
              <pow />
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">A</id>
              <real>2</real>
            </apply>
          </apply>
        </apply>
        <apply>
          <mult />
          <apply>
            <indexer />
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">c</id>
            <real>6</real>
          </apply>
          <apply>
            <pow />
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">B</id>
            <real>2</real>
          </apply>
        </apply>
      </apply>
    </define>
    <resultFormat>
      <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
      <matrix size="14,8" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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8. 呼叫 polyfit，建立形成化學反應模型之實數的函數。使用此函數可預測在 t=80 分鐘且 T=170F 時的產出。

一如預期，計算的產出與步驟 7 中的相同。

9. 使用解題指令群求解最大產出。

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                <id labels="VARIABLE" xml:space="preserve">b</id>
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            <apply>
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Yield</id>
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                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">a</id>
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          <define xmlns="http://schemas.mathsoft.com/math50">
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              <id labels="VARIABLE" xml:space="preserve">t_max</id>
              <id labels="VARIABLE" xml:space="preserve">T_max</id>
            </matrix>
            <apply>
              <id labels="KEYWORD" xml:space="preserve">maximize</id>
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                <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">b</id>
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            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
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          </resultFormat>
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10. 使用等高線圖顯現產出數在 t_max 及 T_max 附近變更的情況。

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          <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">Yield</id>
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參照

Montgomery, D.C., Design and Analysis of Experiments, 5th ed., John Wiley & Sons, New York, 2001, pp. 442.

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