我有两个变量(x和y)彼此之间具有某种S型关系，并且我需要找到某种预测方程，使我能够在给定任何x值的情况下预测y的值。我的预测方程式需要显示两个变量之间的某种S形关系。因此，我无法解决产生一条线的线性回归方程。我需要看到在两个变量的曲线图的左右两侧都发生了斜率的逐渐曲线变化。

在谷歌搜索曲线回归和python之后，我开始使用numpy.polyfit，但这给了我可怕的结果，如果您运行下面的代码，您可以看到。 谁能告诉我如何重新编写以下代码以获得所需的S型回归方程式？

如果运行下面的代码，则可以看到它具有向下的抛物线，这与变量之间的关系不一样。相反，我的两个变量之间应该有更多的S型关系，但与下面代码中使用的数据紧密匹配。以下代码中的数据是来自大型样本研究的数据，因此它们具有比其五个数据点所暗示的更多的统计能力。我没有来自大样本研究的实际数据，但确实有以下均值及其标准偏差(未显示)。我宁愿只用下面列出的均值数据绘制一个简单的函数，但是如果复杂度可以提供实质性的改进，则代码可能会变得更加复杂。

如何更改代码以显示最合适的S型函数，最好使用scipy，numpy和python？ 这是我的代码的当前版本，需要修复:

import numpy as np
import matplotlib.pyplot as plt

# Create numpy data arrays
x = np.array([821,576,473,377,326])
y = np.array([255,235,208,166,157])

# Use polyfit and poly1d to create the regression equation
z = np.polyfit(x, y, 3)
p = np.poly1d(z)
xp = np.linspace(100, 1600, 1500)
pxp=p(xp)

# Plot the results
plt.plot(x, y, '.', xp, pxp, '-')
plt.ylim(140,310)
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()

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

# Create numpy data arrays 
'''
# Comment out original data
#x = np.array([821,576,473,377,326]) 
#y = np.array([255,235,208,166,157]) 
'''

# Re-calculate x values as a percentage of the first (maximum)
# original x value above
x = np.array([1.000,0.702,0.576,0.459,0.397])

# Recalculate y values as a percentage of their respective x values
# from original data above
y = np.array([0.311,0.408,0.440,0.440,0.482])

def sigmoid(p,x): 
    x0,y0,c,k=p 
    y = c / (1 + np.exp(-k*(x-x0))) + y0 
    return y 

def residuals(p,x,y): 
    return y - sigmoid(p,x) 

p_guess=(600,200,100,0.01) 
(p,  
 cov,  
 infodict,  
 mesg,  
 ier)=scipy.optimize.leastsq(residuals,p_guess,args=(x,y),full_output=1,warning=True)  

'''
# comment out original xp to allow for better scaling of
# new values
#xp = np.linspace(100, 1600, 1500) 
'''

xp = np.linspace(0, 1.1, 1100) 
pxp=sigmoid(p,xp) 

x0,y0,c,k=p 
print('''\ 
x0 = {x0}
y0 = {y0}
c = {c}
k = {k}
'''.format(x0=x0,y0=y0,c=c,k=k)) 

# Plot the results 
plt.plot(x, y, '.', xp, pxp, '-') 
plt.ylim(0,1) 
plt.xlabel('x') 
plt.ylabel('y') 
plt.grid(True) 
plt.show()

最佳答案

使用scipy.optimize.leastsq:

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

def sigmoid(p,x):
    x0,y0,c,k=p
    y = c / (1 + np.exp(-k*(x-x0))) + y0
    return y

def residuals(p,x,y):
    return y - sigmoid(p,x)

def resize(arr,lower=0.0,upper=1.0):
    arr=arr.copy()
    if lower>upper: lower,upper=upper,lower
    arr -= arr.min()
    arr *= (upper-lower)/arr.max()
    arr += lower
    return arr

# raw data
x = np.array([821,576,473,377,326],dtype='float')
y = np.array([255,235,208,166,157],dtype='float')

x=resize(-x,lower=0.3)
y=resize(y,lower=0.3)
print(x)
print(y)
p_guess=(np.median(x),np.median(y),1.0,1.0)
p, cov, infodict, mesg, ier = scipy.optimize.leastsq(
    residuals,p_guess,args=(x,y),full_output=1,warning=True)  

x0,y0,c,k=p
print('''\
x0 = {x0}
y0 = {y0}
c = {c}
k = {k}
'''.format(x0=x0,y0=y0,c=c,k=k))

xp = np.linspace(0, 1.1, 1500)
pxp=sigmoid(p,xp)

# Plot the results
plt.plot(x, y, '.', xp, pxp, '-')
plt.xlabel('x')
plt.ylabel('y',rotation='horizontal') 
plt.grid(True)
plt.show()

x0 = 0.826964424481
y0 = 0.151506745435
c = 0.848564826467
k = -9.54442292022

"your name seems to pop up as a writer of the scipy documentation"

1.) Does leastsq() call residuals(), which then returns the difference between the input y-vector and the y-vector returned by the sigmoid() function?

If so, how does it account for the difference in the lengths of the input y-vector and the y-vector returned by the sigmoid() function?

In [138]: x
Out[138]: array([821, 576, 473, 377, 326])

In [139]: y
Out[139]: array([255, 235, 208, 166, 157])

In [140]: p=(600,200,100,0.01)

In [141]: sigmoid(p,x)
Out[141]: 
array([ 290.11439268,  244.02863507,  221.92572521,  209.7088641 ,
        206.06539033])

y = c / (1 + np.exp(-k*(x-x0))) + y0

2.) It looks like I can call leastsq() for any math equation, as long as I access that math equation through a residuals function, which in turn calls the math function. Is this true?

3.) Also, I notice that p_guess has the same number of elements as p. Does this mean that the four elements of p_guess correspond in order, respectively, with the values returned by x0,y0,c, and k?

scipy.optimize.leastsq(residuals,p_guess,args=(x,y))

(residuals,p_guess,args=(x,y))

residuals(p_guess,x,y)

4.) Is the p that is sent as an argument to the residuals() and sigmoid() functions the same p that will be output by leastsq(), and the leastsq() function is using that p internally before returning it?

5.) Can p and p_guess have any number of elements, depending on the complexity of the equation being used as a model, as long as the number of elements in p is equal to the number of elements in p_guess?

关于python - 使用scipy，numpy，python等进行S型回归，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/4308168/