使用 RBFInterpolator 进行二维样条插值并外推

本文介绍了如何使用 scipy.interpolate 库中的 RBFInterpolator 类进行二维样条插值，并实现超出原始数据范围的外推。通过示例代码演示了如何利用该方法创建插值函数，并将其应用于原始数据范围之外的点，从而得到外推值。

scipy.interpolate 库提供了多种插值方法，其中 RBFInterpolator 是一种强大的工具，尤其适用于处理散乱数据。与 griddata 相比，RBFInterpolator 更直接地使用径向基函数进行插值，并且能够方便地进行外推。

使用 RBFInterpolator 进行二维插值和外推的步骤如下：

1. 导入必要的库：

   import io
   import numpy as np
   import pandas as pd
   from scipy.interpolate import RBFInterpolator
   from numpy import ma
   import matplotlib.pyplot as plt

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2. 准备数据：

   首先，需要准备包含自变量 (x, y) 和因变量 z 的数据。在这个例子中，数据是从一个 CSV 文件读取的。

   data_str = """
   dte,4350,4400,4450,4500,4550,4600,4650,4700,4750,4800,4850,4900,4950,5000,5050,5100,5150,5200,5250,5300
   0.01369863,0.19589,0.17243,0.15383,0.13883,0.12662,0.11658,0.10826,0.10134,0.09556,0.09071,0.0866,0.08308,0.08004,0.07738,0.07504,0.07296,0.07109,0.06939,0.06785
   0.02191781,0.19463,0.17149,0.15314,0.13836,0.12632,0.11644,0.10826,0.10148,0.09582,0.09099,0.08688,0.08335,0.08029,0.0776,0.07523,0.07312,0.07122,0.06949,0.06792
   0.03013699,0.1935,0.17066,0.15253,0.13794,0.12604,0.11627,0.10819,0.1015,0.0959,0.09112,0.08704,0.0835,0.08042,0.0777,0.0753,0.07316,0.07123,0.06947,0.06787
   0.04109589,0.19149,0.16901,0.15123,0.13691,0.1253,0.11576,0.10786,0.10132,0.09584,0.09117,0.08717,0.08368,0.08058,0.07783,0.07539,0.07321,0.07124,0.06945,0.06781
   0.06849315,0.18683,0.16511,0.14808,0.13434,0.12324,0.1141,0.10655,0.10033,0.09513,0.09067,0.08686,0.08352,0.08055,0.07795,0.07565,0.07359,0.07173,0.07002,0.06848
   0.09589041,0.18271,0.16178,0.14538,0.13211,0.12136,0.1125,0.10518,0.09918,0.09416,0.08984,0.08615,0.08292,0.08006,0.07755,0.07536,0.0734,0.07163,0.06999,0.06853
   0.12328767,0.17929,0.15892,0.14297,0.12999,0.1195,0.11085,0.10371,0.09788,0.09301,0.0888,0.08521,0.08207,0.07929,0.07685,0.07474,0.07285,0.07114,0.06956,0.06816
   0.15068493,0.17643,0.15643,0.14084,0.12809,0.11778,0.10929,0.10229,0.09658,0.0918,0.08767,0.08416,0.08109,0.07838,0.07599,0.07394,0.0721,0.07043,0.0689,0.06754
   0.17808219,0.17401,0.15429,0.13896,0.12642,0.11629,0.10795,0.10107,0.09547,0.09077,0.08671,0.08326,0.08025,0.0776,0.07526,0.07326,0.07146,0.06983,0.06833,0.067
   0.20547945,0.17195,0.15238,0.13719,0.12484,0.11487,0.10666,0.09989,0.09439,0.08977,0.08578,0.08238,0.07942,0.07681,0.07451,0.07255,0.07078,0.06918,0.06772,0.0664
   0.23287671,0.17014,0.15069,0.13557,0.12339,0.11356,0.10547,0.0988,0.09339,0.08885,0.08492,0.08157,0.07865,0.07608,0.07382,0.07188,0.07014,0.06856,0.06712,0.06582
   0.26027397,0.16854,0.14918,0.13414,0.1221,0.1124,0.10442,0.09785,0.09253,0.08806,0.08418,0.08087,0.07798,0.07544,0.0732,0.07128,0.06956,0.068,0.06657,0.06528
   0.28767123,0.16713,0.14784,0.13286,0.12094,0.11136,0.10348,0.09699,0.09175,0.08735,0.08352,0.08025,0.0774,0.07488,0.07266,0.07075,0.06904,0.06749,0.06607,0.0648
   0.31506849,0.16587,0.14664,0.13173,0.11994,0.11046,0.10268,0.09627,0.0911,0.08676,0.08297,0.07973,0.07691,0.07441,0.0722,0.0703,0.06861,0.06707,0.06566,0.0644
   0.34246575,0.16475,0.14557,0.13073,0.11905,0.10967,0.10198,0.09564,0.09053,0.08624,0.08249,0.07928,0.07648,0.074,0.0718,0.06991,0.06823,0.0667,0.0653,0.06405
   0.36986301,0.16375,0.14462,0.12985,0.11827,0.10897,0.10136,0.09509,0.09003,0.08578,0.08207,0.07888,0.0761,0.07364,0.07145,0.06957,0.0679,0.06638,0.06499,0.06375
   0.39726027,0.16284,0.14377,0.12907,0.11757,0.10835,0.10081,0.0946,0.08959,0.08537,0.08169,0.07852,0.07576,0.07331,0.07114,0.06927,0.06761,0.0661,0.06472,0.06349
   0.42465753,0.16203,0.14299,0.12837,0.11695,0.1078,0.10033,0.09417,0.08921,0.08502,0.08136,0.07821,0.07547,0.07303,0.07087,0.06901,0.06736,0.06586,0.06448,0.06325
   0.45205479,0.16129,0.14228,0.12773,0.11638,0.10731,0.09989,0.09378,0.08886,0.08469,0.08105,0.07792,0.07519,0.07276,0.07061,0.06876,0.06712,0.06562,0.06425,0.06303
   """
   vol = pd.read_csv(io.StringIO(data_str))
   vol.set_index('dte',inplace=True)
   
   valid_vol=ma.masked_invalid(vol).T
   
   Ti=np.linspace(float((vol.index).min()),float((vol.index).max()),len(vol.index))
   Ki=np.linspace(float((vol.columns).min()),float((vol.columns).max()),len(vol.columns))
   
   Ti,Ki = np.meshgrid(Ti,Ki)
   
   valid_Ti = Ti[~valid_vol.mask]
   valid_Ki = Ki[~valid_vol.mask]
   valid_vol = valid_vol[~valid_vol.mask]
   
   points = np.column_stack((valid_Ti.ravel(), valid_Ki.ravel()))
   values = valid_vol.ravel()
   

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3. 创建 RBFInterpolator 对象：

   

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   使用 RBFInterpolator 类创建一个插值对象。需要传入自变量和因变量的数据。 smoothing 参数可以调整插值的平滑程度。

   rbf = RBFInterpolator(points, values, smoothing=0)

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4. 进行插值和外推：

   使用创建的 RBFInterpolator 对象进行插值和外推。 可以传入任意的 (x, y) 坐标，包括原始数据范围之外的坐标。

   # 在原始数据范围内插值
   interp_value = rbf(np.array([0.015, 4545]))
   print(f"Interpolated value at (0.015, 4545): {interp_value}")
   
   # 在原始数据范围外外推
   extrapolated_value = rbf(np.array([0, 4500]))
   print(f"Extrapolated value at (0, 4500): {extrapolated_value}")

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5. 可视化结果（可选）：

   可以使用 matplotlib 库将插值结果可视化，以便更直观地了解插值效果。

   fig = plt.figure()
   ax = fig.add_subplot(111, projection='3d')
   x = np.linspace(Ti.min(),Ti.max(),100)
   
   y = np.linspace(Ki.min(),Ki.max(),100)
   x,y=np.meshgrid(x,y)
   z = rbf(np.column_stack((x.ravel(), y.ravel()))).reshape(x.shape)
   ax.plot_surface(x, y, z, cmap='viridis')
   plt.show()

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完整代码示例：

import io
import numpy as np
import pandas as pd
from scipy.interpolate import RBFInterpolator
from numpy import ma
import matplotlib.pyplot as plt


data_str = """
dte,4350,4400,4450,4500,4550,4600,4650,4700,4750,4800,4850,4900,4950,5000,5050,5100,5150,5200,5250,5300
0.01369863,0.19589,0.17243,0.15383,0.13883,0.12662,0.11658,0.10826,0.10134,0.09556,0.09071,0.0866,0.08308,0.08004,0.07738,0.07504,0.07296,0.07109,0.06939,0.06785
0.02191781,0.19463,0.17149,0.15314,0.13836,0.12632,0.11644,0.10826,0.10148,0.09582,0.09099,0.08688,0.08335,0.08029,0.0776,0.07523,0.07312,0.07122,0.06949,0.06792
0.03013699,0.1935,0.17066,0.15253,0.13794,0.12604,0.11627,0.10819,0.1015,0.0959,0.09112,0.08704,0.0835,0.08042,0.0777,0.0753,0.07316,0.07123,0.06947,0.06787
0.04109589,0.19149,0.16901,0.15123,0.13691,0.1253,0.11576,0.10786,0.10132,0.09584,0.09117,0.08717,0.08368,0.08058,0.07783,0.07539,0.07321,0.07124,0.06945,0.06781
0.06849315,0.18683,0.16511,0.14808,0.13434,0.12324,0.1141,0.10655,0.10033,0.09513,0.09067,0.08686,0.08352,0.08055,0.07795,0.07565,0.07359,0.07173,0.07002,0.06848
0.09589041,0.18271,0.16178,0.14538,0.13211,0.12136,0.1125,0.10518,0.09918,0.09416,0.08984,0.08615,0.08292,0.08006,0.07755,0.07536,0.0734,0.07163,0.06999,0.06853
0.12328767,0.17929,0.15892,0.14297,0.12999,0.1195,0.11085,0.10371,0.09788,0.09301,0.0888,0.08521,0.08207,0.07929,0.07685,0.07474,0.07285,0.07114,0.06956,0.06816
0.15068493,0.17643,0.15643,0.14084,0.12809,0.11778,0.10929,0.10229,0.09658,0.0918,0.08767,0.08416,0.08109,0.07838,0.07599,0.07394,0.0721,0.07043,0.0689,0.06754
0.17808219,0.17401,0.15429,0.13896,0.12642,0.11629,0.10795,0.10107,0.09547,0.09077,0.08671,0.08326,0.08025,0.0776,0.07526,0.07326,0.07146,0.06983,0.06833,0.067
0.20547945,0.17195,0.15238,0.13719,0.12484,0.11487,0.10666,0.09989,0.09439,0.08977,0.08578,0.08238,0.07942,0.07681,0.07451,0.07255,0.07078,0.06918,0.06772,0.0664
0.23287671,0.17014,0.15069,0.13557,0.12339,0.11356,0.10547,0.0988,0.09339,0.

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