数学建模----插值的实现

• 数据插值的方法
  • 代数多项式插值
  • 样条插值

• 选取多项式函数的理由
  • 多项式构造简单
  • 性质良好，求导和积分简单
  • 代数多项式可一致逼近闭区间上的任意连续的函数

• 样条插值
  • 在两个点之间确定一个函数，这个函数就是一个样条，然后把所有样条分段结合成一个函数（插值函数）
• 在Scipy的样条内插的系数
  • 内插是在样本数据内进行预测
  • 外插是利用其他方法预测如微分方程
• 两种插值方法
  • 待定系数插值法
    • 待定系数法插值在我们拥有n个插值节点时构造一个n次多项式
    • 构造一个非齐次线性方程组
    • 解线性方程组，得到数据
  • 拉格朗日插值法
    • 复杂的操作

    • x = np.linspace(-10,10,100)
      y = np.sin(x) + np.cos(x)
      scipy.interpolate.lagrange(x,y)
      

          

• 一维插值

  • 通过定义"kind"的类型来进行不同类别的插值

  • 三次插值 kind = "cubic"

  • 二次插值 kind = "quadratic"

  • 一次插值 kind = "inear"

  • 零阶插值 kind = "zero"

  • nearest 插值法

  • from scipy import interpolate
     
    x = np.linsapce(-1,1,20)
    y = np.sin(x)
     
    f = interpolate.interp1d(x,y,kind="?")
    

• 二维插值
  • 相关代码

  • def z(x,y):
        return x+y
     
    x,y = np.mgrid[-1:1:20j,-1:1:20j]
     
    fun = interpolate.interp2d(x,y,z(x,y),kind="cubic")
    xnew = np.linspace(-1,1,200)
    ynew = np.linspace(-1,1,200)
    fnew = fun(xnew,ynew)
    

• 实例  根据已有数据推测中间值
• 导入相关包

import matplotlib.pyplot as plt
import scipy as sp
import numpy as np
from scipy.interpolate import interp1d
from scipy.interpolate import CubicSpline
import xlsxwriter as xls
import random
random.seed(666)

        

导入相关数据

water = [1800,1900,2100,2200,2300,2400,2500,2600,2650,
         2700,2720]
 
water2 = sorted([2650,2600,2500,2300,2200,2000,1850,
         1820,1800,1755,1500,1000,900],reverse=False)
 
sand = [32,60,75,85,90,98,100,102,108,
        112,115,116,118,120,118,105,80,
        60,50,30,26,20,8,5][:len(water)]
 
sand2 = sorted([116,118,120,118,105,80,60,50,30,26,20,8,5],reverse=False)

插值

cubic_interp = interp1d(water,sand,kind='cubic')
cs1 = CubicSpline(water,sand,bc_type="natural")
cs2 = CubicSpline(water2,sand2,bc_type="natural")
 
#两个数据点之间
#f0 = a0 + b0*(x-x0) + c0*(x-x0)**2 + d0*(x-x0)**3
#.
#.
#.
#fn = an + bn*(x-x0) + cn*(x-x0)**2 + dn*(x-x0)**3

写入文件

workbook1 = xls.Workbook("上升期数据.xlsx")
workbook2 = xls.Workbook("下降期数据.xlsx")
 
worksheet1 = workbook1.add_worksheet("Sheet1")
worksheet2 = workbook2.add_worksheet("Sheet2")
 
#两个数据点之间
#f0 = a0 + b0*(x-x0) + c0*(x-x0)**2 + d0*(x-x0)**3
#.
#.
#.
#fn = an + bn*(x-x0) + cn*(x-x0)**2 + dn*(x-x0)**3
headings = ["a0","b0","c0","d0"]
worksheet1.write_row("A1",headings)
worksheet2.write_row("A1",headings)
 
a1 = [cs1.c.item(3,i) for i in range(len(water)-1)]
b1 = [cs1.c.item(2,i) for i in range(len(water)-1)]
c1 = [cs1.c.item(1,i) for i in range(len(water)-1)]
d1 = [cs1.c.item(0,i) for i in range(len(water)-1)]
a2 = [cs2.c.item(3,i) for i in range(len(water2)-1)]
b2 = [cs2.c.item(2,i) for i in range(len(water2)-1)]
c2 = [cs2.c.item(1,i) for i in range(len(water2)-1)]
d2 = [cs2.c.item(0,i) for i in range(len(water2)-1)]
 
worksheet1.write_column("A2",a1)
worksheet1.write_column("B2",b1)
worksheet1.write_column("C2",c1)
worksheet1.write_column("D2",d1)
 
worksheet2.write_column("A2",a2)
worksheet2.write_column("B2",b2)
worksheet2.write_column("C2",c2)
worksheet2.write_column("D2",d2)
 
workbook1.close()
workbook2.close()