Question

我希望对以下数据进行曲线拟合，以使其在边缘为零斜率的情况下拟合趋势。 polyfit的输出适合该数据，但边缘不为零。

这就是我要输出的内容-赦免我的绘画工作。我需要像这样适合它，这样我才能正确消除对中心不是真实数据的正弦/余弦偏差。数据如下：

[0.23353535 0.25586247 0.26661164 0.26410896 0.24963951 0.22670266
 0.19955422 0.17190263 0.1598439  0.17351905 0.18212444 0.18438673
 0.17952432 0.18314894 0.19265689 0.19432385 0.19605163 0.20326011
 0.20890851 0.20590997 0.21856518 0.23771665 0.24530019 0.23940831
 0.22078396 0.23075128 0.2346082  0.22466281 0.24384843 0.26339594
 0.26414153 0.24664183 0.24278978 0.31023648 0.3614195  0.37773436
 0.3505998  0.28893167 0.23965877 0.24063917 0.27922502 0.32716477
 0.36553767 0.42293146 0.50968856 0.5458872  0.52192533 0.45243764
 0.36313155 0.3683921  0.40942553 0.4420537  0.46145585 0.4648034
 0.4523771  0.4272876  0.39404616 0.3570107  0.35060245 0.3860975
 0.3996996  0.44551122 0.46611032 0.45998383 0.4309985  0.38563925
 0.37105605 0.4074444  0.48815584 0.5704579  0.6448988  0.7018853
 0.73397845 0.73739105 0.7122451  0.6618154  0.591451   0.5076601
 0.48578677 0.47347385 0.4791471  0.48306277 0.47025493 0.43479836
 0.44380915 0.45868078 0.5341566  0.57549906 0.55790776 0.56244135
 0.57668275 0.561856   0.67564166 0.7512851  0.76957643 0.7266262
 0.734133   0.7231936  0.6776926  0.60511285 0.51599765 0.5579323
 0.56723005 0.5440337  0.5775593  0.5950776  0.5722321  0.57858473
 0.5652703  0.54723704 0.59561515 0.7071321  0.8169259  0.91443264
 0.9883759  1.0275097  1.0235045  0.9737119  1.029139   1.1354861
 1.1910824  1.1826864  1.1092159  0.9832138  0.9643041  0.92324203
 0.9093703  0.88915515 1.0007693  1.0542978  1.0857164  1.0211861
 0.88474303 0.8458009  0.76522666 0.7478076  0.90081936 1.0690157
 1.1569089  1.1493248  1.0622779  1.0327609  0.9805119  0.9583969
 0.8973544  0.9543319  0.9777171  0.94951093 0.97323567 1.0244237
 1.0569099  1.0951824  1.0771195  1.3078191  1.7212077  2.09409
 2.320331   2.3279085  2.125451   1.7908521  1.4180487  1.0744424
 1.0218129  1.0916439  1.1255138  1.125803   1.1139745  1.2187989
 1.300092   1.3025533  1.2312403  1.221301   1.2535597  1.2298189
 1.1458241  1.1012102  1.0889369  1.1558667  1.3051153  1.4143198
 1.6345526  1.8093723  1.9037704  1.8961821  1.7866236  1.5958548
 1.3865516  1.5308585  1.6140417  1.627337   1.5733193  1.4981418
 1.5048542  1.4935548  1.4798748  1.4131776  1.3792214  1.3728334
 1.3683671  1.3593615  1.2995907  1.2965002  1.366058   1.4795257
 1.5462885  1.61591    1.5968509  1.5222199  1.6210756  1.7074443
 1.8351102  2.3187535  2.6568012  2.7676315  2.6480794  2.3636303
 2.0673316  1.9607923  1.8074365  1.713272   1.5893831  1.4734347
 1.507817   1.5213271  1.6091452  1.7162323  1.7608733  1.7497622
 1.9187828  2.0197518  2.0487514  2.01107    1.9193696  1.7904462
 1.8558109  2.1955926  2.4700975  2.6562278  2.675197   2.6645825
 2.6295316  2.4182043  2.2114453  2.2506614  2.2086055  2.0497518
 1.9557768  1.901191   2.067513   2.1077373  2.0159333  1.8138607
 1.5413624  1.600069   1.7631899  1.9541935  1.9340311  1.805134
 2.0671906  2.2247658  2.2641945  2.3594956  2.2504601  1.9749025
 1.8905054  2.0679731  2.1193469  2.0307171  2.0717037  2.0340347
 1.925536   1.7820351  1.9467943  2.315468   2.4834185  2.3751369
 2.0240622  1.9363666  2.1732547  2.3113241  2.3264208  2.22015
 2.0187428  1.7619076  1.796859   1.8757095  2.0501778  2.44711
 2.6179967  2.508112   2.1694388  1.7242104  1.7671669  1.862043
 1.8392721  1.7120028  1.6650634  1.6319505  1.482931   1.5240219
 1.5815579  1.5691646  1.4766116  1.3731087  1.4666644  1.4061015
 1.3652745  1.425564   1.4006845  1.5000012  1.581379   1.6329607
 1.6444355  1.6098644  1.5300899  1.6876912  1.8968476  2.048039
 2.1006014  2.0271482  1.8300935  1.6986666  1.9628603  2.0521066
 1.9337255  1.6407858  1.2583638  1.2110122  1.2476432  1.2360718
 1.2886397  1.2862154  1.2343681  1.1458222  1.209224   1.2475786
 1.2353342  1.1797879  1.0963987  1.0928186  1.1553882  1.1569618
 1.1932304  1.3002363  1.3386917  1.2973225  1.1816871  1.0557054
 0.9350373  0.896656   0.8565816  0.90168726 0.9897751  1.02342
 1.0232298  1.1199353  1.1466643  1.1081418  1.0377598  1.0348651
 1.0223045  1.0607077  1.0089502  0.885213   1.023178   1.1131796
 1.1331098  1.0779471  0.9626393  0.81472665 0.85455835 0.87542623
 0.87286425 0.89130884 0.9545931  1.0355722  1.0201533  0.93568784
 0.9180018  0.8202782  0.7450139  0.72550577 0.68578506 0.6431666
 0.66193295 0.6386373  0.7060119  0.7650972  0.80093855 0.803342
 0.76590335 0.7151591  0.6946282  0.7136788  0.7714012  0.8022328
 0.79840165 0.8543819  0.8586749  0.8028453  0.7383879  0.73423904
 0.65107304 0.61139977 0.5940311  0.6151931  0.59349155 0.54995483
 0.5837645  0.5891752  0.56406695 0.5638191  0.5762535  0.58305734
 0.5830114  0.57470953 0.5568098  0.52852243 0.49031836 0.45275375
 0.47168964 0.46634504 0.4600581  0.45332378 0.41508177 0.3834329
 0.4137769  0.41392407 0.3824464  0.36310086 0.434278   0.48041886
 0.49433306 0.475708   0.43060693 0.36886734 0.34740242 0.34108457
 0.36160505 0.40907663 0.43613982 0.4394311  0.42070773 0.38575593
 0.3827834  0.4338096  0.46581286 0.45669746 0.40830874 0.3505502
 0.32584783 0.3381971  0.33949164 0.36409503 0.3759155  0.3610108
 0.37174097 0.39990777 0.38925973 0.34376588 0.32478797 0.32705626
 0.3228174  0.30941254 0.28542265 0.2687348  0.25517422 0.26127565
 0.27331188 0.3028561  0.31277937 0.29953563 0.2660389  0.27051866
 0.2913383  0.30363902 0.30684754 0.3011791  0.28737035 0.26648855
 0.26413882 0.25501928 0.23947525 0.21937743 0.19659272 0.18965112
 0.21511254 0.23329383 0.24157354 0.2391297  0.22697571 0.20739041
 0.1855308  0.18856761 0.19565174 0.20542233 0.21473111 0.22244582
 0.22726117 0.22789808 0.22336568 0.21322969 0.20314343 0.2031754
 0.19738965 0.1959791  0.20284075 0.20859875 0.21363212 0.21804498
 0.22160804 0.22381367]

这很接近，但由于边缘不是零斜率而并非完全正确：How do I fit a sine curve to my data with pylab and numpy?

是否有任何可用的方法可以让我执行此操作而无需编写自定义算法来处理？谢谢。

Answer 1

这是适合您的数据的洛伦兹型峰方程，对于“ x”值，我使用的索引与我在您的示例输出图中看到的相似。我还放大了峰中心，以更好地显示您提到的正弦曲线形状。讨论时，您也许可以从该峰值方程式中减去预测值以调节或预处理原始数据。

a =  1.7056067124682076E+02
b =  7.2900803359572393E+01
c =  2.5047064423525464E+02
d =  1.4184767800540945E+01
Offset = -2.4940994412221318E-01

y = a/ (b + pow((x-c)/d, 2.0)) + Offset

Answer 2

Starting from you own example based on a sine fit, I added the constraints such that the derivatives of the model have to be zero at the end points. I did this using symfit, a package I wrote to make this kind of thing easier. If you prefer to do this using scipy you can adapt the example to that syntax if you want, symfit is just a wrapper around their minimizers that adds symbolical manipulations using sympy.

# Make variables and parameters
x, y = variables('x, y')
a, b, c, d = parameters('a, b, c, d')
# Initial guesses
b.value = 1e-2
c.value = 100

# Create a model object
model = Model({y: a * sin(b * x + c) + d})

# Take the derivative and constrain the end-points to be equal to zero.
dydx = D(model[y], x).doit()
constraints = [Eq(dydx.subs(x, xdata[0]), 0),
               Eq(dydx.subs(x, xdata[-1]), 0)]

# Do the fit!
fit = Fit(model, x=xdata, y=ydata, constraints=constraints)
fit_result = fit.execute()
print(fit_result)

plt.plot(xdata, ydata)
plt.plot(xdata, model(x=xdata, **fit_result.params).y)
plt.show()

This prints: (from current symfit PR#221, which has better reporting of the results.)

Parameter Value        Standard Deviation
a         8.790393e-01 1.879788e-02
b         1.229586e-02 3.824249e-04
c         9.896017e+01 1.011472e-01
d         1.001717e+00 2.928506e-02
Status message         Optimization terminated successfully.
Number of iterations   10
Objective              <symfit.core.objectives.LeastSquares object at 0x0000016F670DF080>
Minimizer              <symfit.core.minimizers.SLSQP object at 0x0000016F78057A58>

Goodness of fit qualifiers:
chi_squared            29.72125657199736
objective_value        14.86062828599868
r_squared              0.8695978050586373

Constraints:
--------------------
Question: a*b*cos(c) == 0?
Answer:   1.5904051811454707e-17

Question: a*b*cos(511*b + c) == 0?
Answer:   -6.354261416082215e-17

Python中的曲线拟合，边沿的斜率为零

2 个答案: