Python SciPy库——插值与拟合

插值与拟合

原文链接：https://zhuanlan.zhihu.com/p/28149195

1.最小二乘拟合

实例1

# -*- coding: utf-8 -*-    import numpy as np  import matplotlib.pyplot as plt  from scipy.optimize import leastsq  ## 设置字符集，防止中文乱码  import matplotlib  matplotlib.rcParams['font.sans-serif']=[u'simHei']  matplotlib.rcParams['axes.unicode_minus']=False    plt.figure(figsize=(9,9))  x=np.linspace(0,10,1000)  X = np.array([8.19, 2.72, 6.39, 8.71, 4.7, 2.66, 3.78])  Y = np.array([7.01, 2.78, 6.47, 6.71, 4.1, 4.23, 4.05])  #计算以p为参数的直线和原始数据之间的误差  def f(p):      k, b = p      return(Y-(k*X+b))  #leastsq使得f的输出数组的平方和最小，参数初始值为[1,0]  r = leastsq(f, [1,0])  k, b = r[0]  print("k=",k,"b=",b)    plt.scatter(X,Y, s=100, alpha=1.0, marker='o',label=u'数据点')    y=k*x+b    ax = plt.gca()    ax.set_xlabel(..., fontsize=20)  ax.set_ylabel(..., fontsize=20)  #设置坐标轴标签字体大小    plt.plot(x, y, color='r',linewidth=5, linestyle=":",markersize=20, label=u'拟合曲线')    plt.legend(loc=0, numpoints=1)  leg = plt.gca().get_legend()  ltext  = leg.get_texts()  plt.setp(ltext, fontsize='xx-large')    plt.xlabel(u'安培/A')  plt.ylabel(u'伏特/V')    plt.xlim(0, x.max() * 1.1)  plt.ylim(0, y.max() * 1.1)    plt.xticks(fontsize=20)  plt.yticks(fontsize=20)  #刻度字体大小      plt.legend(loc='upper left')    plt.show()

k= 0.6134953491930442 b= 1.794092543259387

实例2

# -*- coding: utf-8 -*-    #最小二乘拟合实例  import numpy as np  from scipy.optimize import leastsq  import pylab as pl  ## 设置字符集，防止中文乱码  import matplotlib  matplotlib.rcParams['font.sans-serif']=[u'simHei']  matplotlib.rcParams['axes.unicode_minus']=False    def func(x, p):      """      数据拟合所用的函数: A*cos(2*pi*k*x + theta)      """      A, k, theta = p      return A*np.sin(k*x+theta)       def residuals(p, y, x):      """      实验数据x, y和拟合函数之间的差，p为拟合需要找到的系数      """      return y - func(x, p)    x = np.linspace(0, 20, 100)  A, k, theta = 10, 3, 6 # 真实数据的函数参数  y0 = func(x, [A, k, theta]) # 真实数据  y1 = y0 + 2 * np.random.randn(len(x)) # 加入噪声之后的实验数据        p0 = [10, 0.2, 0] # 第一次猜测的函数拟合参数    # 调用leastsq进行数据拟合  # residuals为计算误差的函数  # p0为拟合参数的初始值  # args为需要拟合的实验数据  plsq = leastsq(residuals, p0, args=(y1, x))    print (u"真实参数:", [A, k, theta] )  print (u"拟合参数", plsq[0]) # 实验数据拟合后的参数    pl.plot(x, y0, color='r',label=u"真实数据")  pl.plot(x, y1, color='b',label=u"带噪声的实验数据")  pl.plot(x, func(x, plsq[0]), color='g', label=u"拟合数据")  pl.legend()  pl.show()

真实参数: [10, 3, 6]  拟合参数 [-1.16428658  0.24215786 -0.794681  ]

2.插值

实例1

# -*- coding: utf-8 -*-    # -*- coding: utf-8 -*-    import numpy as np  import pylab as pl  from scipy import interpolate   import matplotlib.pyplot as plt  ## 设置字符集，防止中文乱码  import matplotlib  matplotlib.rcParams['font.sans-serif']=[u'simHei']  matplotlib.rcParams['axes.unicode_minus']=False    x = np.linspace(0, 2*np.pi+np.pi/4, 10)  y = np.sin(x)    x_new = np.linspace(0, 2*np.pi+np.pi/4, 100)  f_linear = interpolate.interp1d(x, y)  tck = interpolate.splrep(x, y)  y_bspline = interpolate.splev(x_new, tck)    plt.xlabel(u'安培/A')  plt.ylabel(u'伏特/V')    plt.plot(x, y, "o",  label=u"原始数据")  plt.plot(x_new, f_linear(x_new), label=u"线性插值")  plt.plot(x_new, y_bspline, label=u"B-spline插值")    pl.legend()  pl.show()

实例2

# -*- coding: utf-8 -*-    import numpy as np  from scipy import interpolate  import pylab as pl  ## 设置字符集，防止中文乱码  import matplotlib  matplotlib.rcParams['font.sans-serif']=[u'simHei']  matplotlib.rcParams['axes.unicode_minus']=False    #创建数据点集并绘制  pl.figure(figsize=(12,9))  x = np.linspace(0, 10, 11)  y = np.sin(x)  ax=pl.plot()    pl.plot(x,y,'ro')  #建立插值数据点  xnew = np.linspace(0, 10, 101)  for kind in ['nearest', 'zero','linear','quadratic']:      #根据kind创建插值对象interp1d      f = interpolate.interp1d(x, y, kind = kind)      ynew = f(xnew)#计算插值结果      pl.plot(xnew, ynew, label = str(kind))    pl.xticks(fontsize=20)  pl.yticks(fontsize=20)    pl.legend(loc = 'lower right')  pl.show()

B样条曲线插值
一维数据的插值运算可以通过 interp1d()实现。
其调用形式为：
Interp1d可以计算x的取值范围之内任意点的函数值，并返回新的数组。
interp1d(x, y, kind=‘linear’, …)
参数 x和y是一系列已知的数据点
参数kind是插值类型，可以是字符串或整数

B样条曲线插值
Kind给出了B样条曲线的阶数：
 ‘
zero‘ ‘nearest’ :0阶梯插值，相当于0阶B样条曲线
 ‘slinear’‘linear’ :线性插值，相当于1阶B样条曲线
 ‘quadratic’‘cubic’:2阶和3阶B样条曲线，更高阶的曲线可以直接使用整数值来指定

（1）#创建数据点集：

import numpy as np

x = np.linspace(0, 10, 11)

y = np.sin(x)

（2）#绘制数据点集：

import pylab as pl

pl.plot(x,y,'ro')

创建interp1d对象f、计算插值结果：
xnew = np.linspace(0, 10, 11)
from scipy import interpolate
f = interpolate.interp1d(x, y, kind = kind)
ynew = f(xnew)

根据kind类型创建interp1d对象f、计算并绘制插值结果：
xnew = np.linspace(0, 10, 11)
for kind in ['nearest', 'zero','linear','quadratic']:
#根据kind创建插值对象interp1d
f = interpolate.interp1d(x, y, kind = kind)
ynew = f(xnew)#计算插值结果
pl.plot(xnew, ynew, label = str(kind))#绘制插值结果

如果我们将代码稍作修改增加一个5阶插值

import numpy as np  from scipy import interpolate  import pylab as pl  #创建数据点集并绘制  pl.figure(figsize=(12,9))  x = np.linspace(0, 10, 11)  y = np.sin(x)  ax=pl.plot()    pl.plot(x,y,'ro')  #建立插值数据点  xnew = np.linspace(0, 10, 101)  for kind in ['nearest', 'zero','linear','quadratic',5]:      #根据kind创建插值对象interp1d      f = interpolate.interp1d(x, y, kind = kind)      ynew = f(xnew)#计算插值结果      pl.plot(xnew, ynew, label = str(kind))    pl.xticks(fontsize=20)  pl.yticks(fontsize=20)    pl.legend(loc = 'lower right')  pl.show()  运行得到