polynomial.polynomial.
polyfit
多项式与数据的最小二乘拟合。
返回多项式的系数 deg 这是符合数据值的最小二乘法 y 给定点 x .如果 y 是1-d，返回系数也将是1-d。如果 y 是否进行了二维多次拟合，每列一次 y ，生成的系数存储在二维返回的相应列中。拟合多项式的形式为
System Message: WARNING/2 (P（x）=C_0+C_1 * x + ... + c_n * X^ n)
latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2018-12-01> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Package inputenc Error: Unicode character （ (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...=C_0+C_1 * x + ... + c_n * X^ n\end{split} ! Package inputenc Error: Unicode character ） (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...=C_0+C_1 * x + ... + c_n * X^ n\end{split} ! Package inputenc Error: Unicode character （ (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...=C_0+C_1 * x + ... + c_n * X^ n\end{split} ! Package inputenc Error: Unicode character ） (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...=C_0+C_1 * x + ... + c_n * X^ n\end{split} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 496 bytes). Transcript written on math.log.
在哪里？ n 是 deg .
X坐标 M 样本（数据）点 (x[i], y[i]) .
(x[i], y[i])
采样点的Y坐标。多组共享相同X坐标的采样点可以（独立）与一个调用匹配 polyfit 通过传递 y 每列包含一个数据集的二维数组。
拟合多项式的度数。如果 deg 是一个整数，所有项都小于或等于 deg “这一术语包含在拟合中。对于numpy版本>=1.11.0，可以使用指定要包含项的度数的整数列表。
拟合的相对条件数。奇异值小于 rcond 相对于最大奇异值，将被忽略。默认值为 len(x)*eps 在哪里 eps 是平台浮动类型的相对精度，在大多数情况下约为2E-16。
len(x)*eps
开关确定返回值的性质。什么时候？ False （默认值）只返回系数；当 True 同时返回奇异值分解（用于求解拟合矩阵方程）的诊断信息。
False
True
砝码。如果不是没有，每个点的贡献 (x[i],y[i]) 拟合的权重为 w[i] . 理想情况下，选择权重是为了使产品的误差 w[i]*y[i] 都有相同的差异。默认值为无。
(x[i],y[i])
w[i]*y[i]
1.5.0 新版功能.
多项式系数由低到高排序。如果 y 是二维的，列中的系数 k 属于 coef 表示符合数据的多项式 y 的 k 第四列。
只有当 full =真
resid——最小二乘拟合秩的平方残差之和——标度范德蒙矩阵Sv的数值秩——标度范德蒙矩阵Rcond的奇异值——标度范德蒙矩阵Rcond的值。 rcond .
有关详细信息，请参阅 numpy.linalg.lstsq .
numpy.linalg.lstsq
如果最小二乘拟合中的矩阵是秩亏的，则引发。只有在以下情况下才会发出警告： full =假。警告可通过以下方式关闭：
>>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning)
参见
numpy.polynomial.chebyshev.chebfit
numpy.polynomial.legendre.legfit
numpy.polynomial.laguerre.lagfit
numpy.polynomial.hermite.hermfit
numpy.polynomial.hermite_e.hermefit
polyval
计算多项式。
polyvander
范德蒙功率矩阵。
从矩阵计算最小二乘拟合。
scipy.interpolate.UnivariateSpline
计算样条曲线拟合。
笔记
解是多项式的系数。 p 使加权平方误差之和最小化
System Message: WARNING/2 (E=\sum j w_j^2*y_j-p（x_j）^2，)
latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2018-12-01> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Package inputenc Error: Unicode character （ (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...sum j w_j^2*y_j-p（x_j）^2，\end{split} ! Package inputenc Error: Unicode character ） (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...sum j w_j^2*y_j-p（x_j）^2，\end{split} ! Package inputenc Error: Unicode character ， (U+FF0C) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...sum j w_j^2*y_j-p（x_j）^2，\end{split} ! Package inputenc Error: Unicode character （ (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...sum j w_j^2*y_j-p（x_j）^2，\end{split} ! Package inputenc Error: Unicode character ） (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...sum j w_j^2*y_j-p（x_j）^2，\end{split} ! Package inputenc Error: Unicode character ， (U+FF0C) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 ...sum j w_j^2*y_j-p（x_j）^2，\end{split} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 548 bytes). Transcript written on math.log.
何处 是重量。通过建立（通常）超定矩阵方程来解决这个问题：
System Message: WARNING/2 (V（x） * c = w * Y)
latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2018-12-01> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Package inputenc Error: Unicode character （ (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 \begin{split}V（x） * c = w * Y\end{split} ! Package inputenc Error: Unicode character ） (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 \begin{split}V（x） * c = w * Y\end{split} ! Package inputenc Error: Unicode character （ (U+FF08) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 \begin{split}V（x） * c = w * Y\end{split} ! Package inputenc Error: Unicode character ） (U+FF09) (inputenc) not set up for use with LaTeX. See the inputenc package documentation for explanation. Type H <return> for immediate help. ... l.14 \begin{split}V（x） * c = w * Y\end{split} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 356 bytes). Transcript written on math.log.
在哪里？ V 是的加权伪范德蒙矩阵 x ， c 是要求解的系数， w 是重量，和 y 是观察值。然后用奇异值分解法求解该方程。 V .
如果某些奇异值 V 太小以致被忽视（和 full = False a） RankWarning 将被提升。这意味着系数值的确定可能很差。拟合到低阶多项式通常会消除警告（当然，可能不是您想要的；如果您有独立的理由选择不起作用的程度，则可能必须：a）重新考虑这些原因，和/或b）重新考虑数据的质量）。这个 rcond 也可以将参数设置为小于其默认值的值，但由此产生的拟合可能是虚假的，并且具有来自舍入误差的较大贡献。
使用双精度的多项式拟合往往在（多项式）阶数20时“失败”。使用切比雪夫或勒让德级数的拟合通常条件较好，但很大程度上仍然取决于采样点的分布和数据的平滑度。如果配合质量不够，样条曲线可能是一个很好的选择。
实例
>>> np.random.seed(123) >>> from numpy.polynomial import polynomial as P >>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1] >>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + N(0,1) "noise" >>> c, stats = P.polyfit(x,y,3,full=True) >>> np.random.seed(123) >>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1 array([ 0.01909725, -1.30598256, -0.00577963, 1.02644286]) # may vary >>> stats # note the large SSR, explaining the rather poor results [array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, # may vary 0.28853036]), 1.1324274851176597e-014]
同样的东西没有额外的噪音
>>> y = x**3 - x >>> c, stats = P.polyfit(x,y,3,full=True) >>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1 array([-6.36925336e-18, -1.00000000e+00, -4.08053781e-16, 1.00000000e+00]) >>> stats # note the minuscule SSR [array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, # may vary 0.50443316, 0.28853036]), 1.1324274851176597e-014]