How to use leastsq function from scipy.optimize in

问题:

How would i fit a straight line and a quadratic to the data set below using the leastsq function from scipy.optimize? I know how to use polyfit to do it. But i need to use leastsq function.

Here are the x and y data sets:

x: 1.0,2.5,3.5,4.0,1.1,1.8,2.2,3.7

y: 6.008,15.722,27.130,33.772,5.257,9.549,11.098,28.828

Can someone help me out please?

回答1:

The leastsq() method finds the set of parameters that minimize the error function ( difference between yExperimental and yFit). I used a tuple to pass the parameters and lambda functions for the linear and quadratic fits.

leastsq starts from a first guess ( initial Tuple of parameters) and tries to minimize the error function. At the end, if leastsq succeeds, it returns the list of parameters that best fit the data. ( I printed to see it). I hope it works best regards

from scipy.optimize import leastsq
import numpy as np
import matplotlib.pyplot as plt


def main():
   # data provided
   x=np.array([1.0,2.5,3.5,4.0,1.1,1.8,2.2,3.7])
   y=np.array([6.008,15.722,27.130,33.772,5.257,9.549,11.098,28.828])
   # here, create lambda functions for Line, Quadratic fit
   # tpl is a tuple that contains the parameters of the fit
   funcLine=lambda tpl,x : tpl[0]*x+tpl[1]
   funcQuad=lambda tpl,x : tpl[0]*x**2+tpl[1]*x+tpl[2]
   # func is going to be a placeholder for funcLine,funcQuad or whatever 
   # function we would like to fit
   func=funcLine
   # ErrorFunc is the diference between the func and the y "experimental" data
   ErrorFunc=lambda tpl,x,y: func(tpl,x)-y
   #tplInitial contains the "first guess" of the parameters 
   tplInitial1=(1.0,2.0)
   # leastsq finds the set of parameters in the tuple tpl that minimizes
   # ErrorFunc=yfit-yExperimental
   tplFinal1,success=leastsq(ErrorFunc,tplInitial1[:],args=(x,y))
   print " linear fit ",tplFinal1
   xx1=np.linspace(x.min(),x.max(),50)
   yy1=func(tplFinal1,xx1)
   #------------------------------------------------
   # now the quadratic fit
   #-------------------------------------------------
   func=funcQuad
   tplInitial2=(1.0,2.0,3.0)

   tplFinal2,success=leastsq(ErrorFunc,tplInitial2[:],args=(x,y))
   print "quadratic fit" ,tplFinal2
   xx2=xx1

   yy2=func(tplFinal2,xx2)
   plt.plot(xx1,yy1,'r-',x,y,'bo',xx2,yy2,'g-')
   plt.show()

if __name__=="__main__":
   main()

回答2:

from scipy.optimize import leastsq
import scipy as sc
import numpy as np
import matplotlib.pyplot as plt

with optimize.curve_fit the code is simpler, there is no need to define the residual(error) function.

fig, ax = plt.subplots ()
# data
x=np.array([1.0,2.5,3.5,4.0,1.1,1.8,2.2,3.7])
y=np.array([6.008,15.722,27.130,33.772,5.257,9.549,11.098,28.828])

# modeling functions
def funcLine(x, a,b):
    return a*x+b
def funcQuad(x, a, b, c):
    return a*x**2+b*x+c

# optimize constants for the linear function
constantsLine, _ = sc.optimize.curve_fit (funcLine, x, y)

X=np.linspace(x.min(),x.max(),50)
Y1=funcLine(X, *constantsLine)

# optimize constants for the quadratic function
constantsQuad, _ = sc.optimize.curve_fit (funcQuad, x, y)

Y2=funcQuad(X,*constantsQuad)
plt.plot(X,Y1,'r-',label='linear approximation')
plt.plot(x,y,'bo',label='data points')
plt.plot(X,Y2,'g-', label='quadratic approximation')
matplotlib.pylab.legend ()
ax.set_title("Nonlinear Least Square Problems", fontsize=18)
plt.show()

回答3:

Here's a super simple example. Picture a paraboloid, so like a bowl with sides growing like a parabola. If we put the bottom at coordinates (x, y) = (a, b) and then minimize the height of the paraboloid over all values of x and y - we would expect the minimum to be x=a and y=b. Here's code that would do this.

import random

from scipy.optimize import least_squares


a, b = random.randint(1, 1000), random.randint(1, 1000)
print("Expect", [a, b])

def f(args):
    x, y = args
    return (x-a)**2 + (y-b)**2

x0 = [-1, -3]

result = least_squares(fun=f, x0=x0)

print(result.x)