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评估曲线拟合

打开实时脚本

此示例说明如何使用曲线拟合。

加载数据并拟合多项式曲线

load census
curvefit = fit(cdate,pop,'poly3','normalize','on')
curvefit = 
     Linear model Poly3:
     curvefit(x) = p1*x^3 + p2*x^2 + p3*x + p4
       where x is normalized by mean 1890 and std 62.05
     Coefficients (with 95% confidence bounds):
       p1 =       0.921  (-0.9743, 2.816)
       p2 =       25.18  (23.57, 26.79)
       p3 =       73.86  (70.33, 77.39)
       p4 =       61.74  (59.69, 63.8)

输出显示拟合的模型方程、拟合的系数以及拟合系数的置信边界。

绘制拟合、数据、残差和预测边界

plot(curvefit,cdate,pop)

Figure contains an axes object. The axes object with xlabel x, ylabel y contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Data, Fitted curve.

绘制残差拟合图。

plot(curvefit,cdate,pop,'Residuals')

Figure contains an axes object. The axes object with xlabel x contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Data, Zero line.

绘制拟合的预测边界。

plot(curvefit,cdate,pop,'predfunc')

Figure contains an axes object. The axes object with xlabel x, ylabel y contains 4 objects of type line. One or more of the lines displays its values using only markers These objects represent Data, Fitted curve, Prediction bounds - 95%.

评估指定点处的拟合

使用以下格式通过指定 x 的一个值来计算特定点处的拟合：y = fittedmodel(x)。

curvefit(1991)
ans = 
252.6690

评估多个点处的拟合值

计算模型在值向量处的值，将值外插到 2050 年。

xi = (2000:10:2050).';
curvefit(xi)
ans = 6×1

  276.9632
  305.4420
  335.5066
  367.1802
  400.4859
  435.4468

获取这些值的预测边界。

ci = predint(curvefit,xi)
ci = 6×2

  267.8589  286.0674
  294.3070  316.5770
  321.5924  349.4208
  349.7275  384.6329
  378.7255  422.2462
  408.5919  462.3017

基于经外插后的拟合范围绘制拟合和预测区间图。默认情况下，系统会基于数据范围拟合图。要查看拟合的外插值，请在绘制拟合图之前将坐标区的 x 上限设置为 2050。要绘制预测区间图，请使用 predobs 或 predfun 作为绘图类型。

plot(cdate,pop,'o')
xlim([1900,2050])
hold on
plot(curvefit,'predobs')
hold off

Figure contains an axes object. The axes object with xlabel x, ylabel y contains 4 objects of type line. One or more of the lines displays its values using only markers These objects represent Fitted curve, Prediction bounds - 95%.

获取模型方程

输入拟合名称以显示模型方程、拟合系数和拟合系数的置信边界。

curvefit
curvefit = 
     Linear model Poly3:
     curvefit(x) = p1*x^3 + p2*x^2 + p3*x + p4
       where x is normalized by mean 1890 and std 62.05
     Coefficients (with 95% confidence bounds):
       p1 =       0.921  (-0.9743, 2.816)
       p2 =       25.18  (23.57, 26.79)
       p3 =       73.86  (70.33, 77.39)
       p4 =       61.74  (59.69, 63.8)

要仅获得模型方程，请使用 formula。

formula(curvefit)
ans = 
'p1*x^3 + p2*x^2 + p3*x + p4'

获取系数名称和值

通过名称指定一个系数。

p1 = curvefit.p1
p1 = 
0.9210

p2 = curvefit.p2
p2 = 
25.1834

获取所有系数名称。查看拟合方程（例如 f(x) = p1*x^3+...）以查看每个系数的模型项。

coeffnames(curvefit)
ans = 4×1 cell
    {'p1'}
    {'p2'}
    {'p3'}
    {'p4'}

获取所有系数值。

coeffvalues(curvefit)
ans = 1×4

    0.9210   25.1834   73.8598   61.7444

获得系数的置信边界

使用系数的置信边界来帮助您评估和比较拟合情况。系数的置信边界确定其准确性。相距很远的边界表示存在不确定性。如果线性系数的置信边界跨越了零点，这意味着您无法确定这些系数是否与零有差异。如果一些模型项的系数为零，则它们对拟合没有影响。

confint(curvefit)
ans = 2×4

   -0.9743   23.5736   70.3308   59.6907
    2.8163   26.7931   77.3888   63.7981

检查拟合优度统计量

要在命令行中获得拟合优度统计量，您可以采用以下任一方法：

  • 打开曲线拟合器。在曲线拟合器选项卡的导出部分中，点击导出并选择导出到工作区，以将您的拟合和拟合优度导出到工作区。

  • 使用 fit 函数指定 gof 输出参量。

通过指定 gof 和输出参量重新创建拟合，以获得拟合优度统计量和拟合算法信息。

[curvefit,gof,output] = fit(cdate,pop,'poly3','normalize','on')
curvefit = 
     Linear model Poly3:
     curvefit(x) = p1*x^3 + p2*x^2 + p3*x + p4
       where x is normalized by mean 1890 and std 62.05
     Coefficients (with 95% confidence bounds):
       p1 =       0.921  (-0.9743, 2.816)
       p2 =       25.18  (23.57, 26.79)
       p3 =       73.86  (70.33, 77.39)
       p4 =       61.74  (59.69, 63.8)

gof = struct with fields:
           sse: 149.7687
       rsquare: 0.9988
           dfe: 17
    adjrsquare: 0.9986
          rmse: 2.9682


output = struct with fields:
        numobs: 21
      numparam: 4
     residuals: [21×1 double]
      Jacobian: [21×4 double]
      exitflag: 1
     algorithm: 'QR factorization and solve'
    iterations: 1

绘制残差直方图，寻找大致正态的分布。

histogram(output.residuals,10)

Figure contains an axes object. The axes object contains an object of type histogram.

绘制拟合、数据和残差的图

plot(curvefit,cdate,pop,'fit','residuals')
legend Location SouthWest

Figure contains 2 axes objects. Axes object 1 with xlabel x, ylabel y contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Data, Fitted curve. Axes object 2 with xlabel x contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Data, Zero line.

subplot(2,1,1)
legend Location NorthWest

Figure contains an axes object. The axes object is empty.

查找方法

列出您可以用于拟合的每种方法。

methods(curvefit)
Methods for class cfit:

argnames                category                cfit                    coeffnames              coeffvalues             confint                 dependnames             differentiate           feval                   fitoptions              formula                 indepnames              integrate               islinear                numargs                 numcoeffs               plot                    predint                 probnames               probvalues              setoptions              solveModelNonseparably  type

有关如何使用拟合方法的详细信息，请参阅 cfit。

另请参阅

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