示例：拟合质量

使用 polyfit 和 polyfitstat 函数执行线性回归和方差分析，以测试拟合的质量。

1. 为聚合过程定义实验数据表。反应温度 t 和催化剂进料率 fr 会影响聚合的粘度 vy。

拟合功能性

2. 调用 polyfit 函数将数据建模为线性回归。

3. 计算为各个温度设置和进料率设置预测的粘度。

4. 计算残差 (计算的模型值与测量值之差)。

5. 绘制针对观察到的粘度、温度和进料率的残差。

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残差图指出，观察到的粘度和温度方差会分别随着粘度和温度量值的增大而增大。

6. 调用 polyfitstat 函数计算线性模型的各种统计数据。在第 8 行显示由 polyfitstat 返回的 ANOVA 矩阵。

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          <real>3316.244</real>
          <real>267.604</real>
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          <real>0</real>
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          <real>151.872</real>
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          <id label="Constant">NaN</id>
          <id label="Constant">NaN</id>
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          <real>0</real>
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在 ANOVA 矩阵中，方差的源在 Regression (回归) 和 Residual (残差) 分量之间被分割。Regression (回归) 分量在各回归系数之间被进一步分割。但是，将无法针对残差来区分 Lack of Fit (失拟误差) 和 Pure Error (纯实验误差)，因为实验结果 vy 没有副本。

针对回归计算以及使用 ANOVA 表

1. 计算由误差产生的平方和 (SSE)。

SSE 等于 χ2，是拟合度的一般度量。这是计算最小二乘解时达到的最小量。误差是模型与数据拟合度的量度。它显示了尚未被回归解释的偏差量。

2. 定义误差 df_error 相对于总自由度 df_total 的自由度以及参数 df_param 的自由度。自由度是扣除拟合参数个数的数据长度。

3. 相对于总平方和 (SST) 定义由回归产生的平方和 (SSR)。

4. 定义均方误差 (MSE) 和均方回归 (MSR)。用相应的自由度除以误差。

5. 形成方差表的分析以表征拟合。

	平方和	DF	均方	F 因子
Regression	<region id="ID0EBLQW" top="3360" left="120" width="100.051666666667" height="21.2" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <id>SSR</id> <result> <real>44157.087</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>	<region id="ID0EPLQW" top="3360" left="240" width="84.0783333333333" height="21.2" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <id>df_param</id> <result> <real>2</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>	<region id="ID0E4LQW" top="3360" left="336" width="104.265" height="21.2" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <id>MSR</id> <result> <real>22078.543</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>	<region id="ID0ELMQW" top="3360" left="456" width="123.965" height="38.4" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <define> <id label="None">F_0</id> <apply> <div /> <id>MSR</id> <id>MSE</id> </apply> </define> <result> <real>82.505</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>
Error	<region id="ID0EINQW" top="3384" left="120" width="93.905" height="21.2" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <id>SSE</id> <result> <real>3478.851</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>	<region id="ID0EWNQW" top="3384" left="240" width="83.025" height="21.2" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <id>df_error</id> <result> <real>13</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>	<region id="ID0EEOQW" top="3384" left="336" width="92.135" height="21.2" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <id>MSE</id> <result> <real>267.604</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>
Total	<region id="ID0EHPQW" top="3408" left="120" width="99.4816666666667" height="21.2" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <id>SST</id> <result> <real>47635.938</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>	<region id="ID0EVPQW" top="3408" left="240" width="80.085" height="21.2" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <id>df_total</id> <result> <real>15</real> </result> </eval> <resultFormat> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>

可以将上表和 polyfitstat ANOVA 矩阵加以比较。

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          <str xml:space="preserve">      Lack of Fit</str>
          <str xml:space="preserve">      Pure Error </str>
          <str xml:space="preserve">Total         </str>
          <str xml:space="preserve">SSE</str>
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          <real>40641.418</real>
          <real>3316.244</real>
          <real>3478.851</real>
          <real>3478.851</real>
          <real>0</real>
          <real>47635.938</real>
          <str xml:space="preserve">DF</str>
          <real>2</real>
          <real>1</real>
          <real>1</real>
          <real>13</real>
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          <real>0</real>
          <real>15</real>
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          <real>22078.543</real>
          <real>40641.418</real>
          <real>3316.244</real>
          <real>267.604</real>
          <real>267.604</real>
          <real>0</real>
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          <str xml:space="preserve">F</str>
          <real>82.505</real>
          <real>151.872</real>
          <real>12.392</real>
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          <id label="Constant">NaN</id>
          <id label="Constant">NaN</id>
          <id label="Constant">NaN</id>
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          <real>0</real>
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6. 预估模型与数据的拟合度：

这说明线性回归模型解释了 92.7% 的粘度可变性。

7. 定义假设检验的显著性水平，以测试模型是否与数据拟合。

8. 计算临界的 F 值。

9. 对模型拟合数据的假设进行测试。

接受假设。可以使用此线性回归模型来预测聚合物的粘度。

参考文献

Montgomery, D.C., Design and Analysis of Experiments, 5th ed., John Wiley & Sons, New York, 2001, pp. 398