Question

我试图使用python将一些数据拟合到幂律中。问题是我的一些要点是上限，我不知道如何包含在拟合程序中。

在数据中，我把上限作为y中的误差等于1，其余的则小得多。您可以将此错误设置为0并更改uplims列表生成器，但随后适合性很差。

代码如下：

import numpy as np
import matplotlib.pyplot as plt
from scipy.odr import *

# Initiate some data
x = [1.73e-04, 5.21e-04, 1.57e-03, 4.71e-03, 1.41e-02, 4.25e-02, 1.28e-01, 3.84e-01, 1.15e+00]
x_err = [1e-04, 1e-04, 1e-03, 1e-03, 1e-02, 1e-02, 1e-01, 1e-01, 1e-01]
y = [1.26e-05, 8.48e-07, 2.09e-08, 4.11e-09, 8.22e-10, 2.61e-10, 4.46e-11, 1.02e-11, 3.98e-12]
y_err = [1, 1, 2.06e-08, 2.5e-09, 5.21e-10, 1.38e-10, 3.21e-11, 1, 1]

# Define upper limits
uplims = np.ones(len(y_err),dtype='bool')
for i in range(len(y_err)):
    if y_err[i]<1:
        uplims[i]=0
    else:
        uplims[i]=1

# Define a function (power law in our case) to fit the data with.
def function(p, x):
     m, c = p
     return m*x**(-c)

# Create a model for fitting.
model = Model(function)

# Create a RealData object using our initiated data from above.
data = RealData(x, y, sx=x_err, sy=y_err)

# Set up ODR with the model and data.
odr = ODR(data, model, beta0=[1e-09, 2])
odr.set_job(fit_type=0)   # 0 is full ODR and 2 is least squares; AFAIK, it doesn't change within errors
# more details in https://docs.scipy.org/doc/scipy/reference/generated/scipy.odr.ODR.set_job.html

# Run the regression.
out = odr.run()


# Use the in-built pprint method to give us results.
#out.pprint()   #this prints much information, but normally we don't need it, just the parameters and errors; the residual variation is the reduced chi square estimator

print('amplitude = %5.2e +/- %5.2e \nindex = %5.2f +/- %5.2f \nchi square = %12.8f'% (out.beta[0], out.sd_beta[0], out.beta[1], out.sd_beta[1], out.res_var))

# Generate fitted data.
x_fit = np.linspace(x[0], x[-1], 1000)    #to do the fit only within the x interval; we can always extrapolate it, of course
y_fit = function(out.beta, x_fit)


# Generate a plot to show the data, errors, and fit.
fig, ax = plt.subplots()

ax.errorbar(x, y, xerr=x_err, yerr=y_err, uplims=uplims, linestyle='None', marker='x')
ax.loglog(x_fit, y_fit)
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$f(x) = m·x^{-c}$')
ax.set_title('Power Law fit')

plt.show()

拟合的结果是：

amplitude = 3.42e-12 +/- 5.32e-13
index =  1.33 +/-  0.04
chi square =   0.01484021

正如您在图中所看到的，最后两个点和最后两个点都是上限，并且拟合没有考虑到它们。而且，在倒数第二点，即使严格禁止，也会超过它。

我需要知道这个限制是非常严格的，并不是试图适应这一点本身，而只是将它们视为限制。我怎么能用odr例程（或任何其他使我成为拟合的代码并给我一个chi square-esque估计器）来做这个？

请注意，我需要轻松地将功能更改为其他概括，因此不需要使用powerlaw模块。

谢谢！

Answer 1

此回答与this帖子有关，我在此讨论是否符合x和y错误。因此，不需要ODR模块，但可以手动完成。因此，可以使用leastsq或minimize。关于限制因素，我在其他帖子中明确表示，如果可能，我会尽量避免这些限制。这也可以在这里完成，虽然编程和数学的细节有点麻烦，特别是如果它应该是稳定和万无一失的。我只想粗略一点。假设我们想要y0 > m * x0**(-c)。在日志形式中，我们可以将其写为eta0 > mu - c * xeta0。即有一个alpha，eta0 = mu - c * xeta0 + alpha**2。其他不平等也是如此。对于第二个上限，您可以获得beta**2，但您可以决定哪个是较小的一个，因此您可以自动满足其他条件。同样的事情适用于gamma**2和delta**2的下限。假设我们可以使用alpha和gamma。我们可以结合不平等条件来将这两者联系起来。最后，我们可以使用sigma和alpha = sqrt(s-t)* sigma / sqrt( sigma**2 + 1 )，其中s和t来自不等式。 sigma / sqrt( sigma**2 + 1 )函数只是让alpha在某个范围内变化的一个选项，即alpha**2 < s-t根域可能变为负数这一事实表明存在无解的情况。知道alpha后，会计算mu，因此m。因此，拟合参数为c和sigma，它将不等式考虑在内并使m依赖。我厌倦了它的确有效，但手头的版本并不是最稳定的版本。我可以根据要求发布。

由于我们已经拥有手工制作的剩余功能，我们有第二种选择。我们只介绍我们自己的chi**2函数并使用minimize，它允许约束。由于minimize和constraints关键字解决方案非常灵活，剩余功能很容易针对其他功能进行修改，而且m * x**( -c )整体结构非常灵活。它看起来如下：

import matplotlib.pyplot as plt
import numpy as np
from random import random, seed
from scipy.optimize import minimize,leastsq

seed(7563)
fig1 = plt.figure(1)


###for gaussion distributed errors
def boxmuller(x0,sigma):
    u1=random()
    u2=random()
    ll=np.sqrt(-2*np.log(u1))
    z0=ll*np.cos(2*np.pi*u2)
    z1=ll*np.cos(2*np.pi*u2)
    return sigma*z0+x0, sigma*z1+x0


###for plotting ellipses
def ell_data(a,b,x0=0,y0=0):
    tList=np.linspace(0,2*np.pi,150)
    k=float(a)/float(b)
    rList=[a/np.sqrt((np.cos(t))**2+(k*np.sin(t))**2) for t in tList]
    xyList=np.array([[x0+r*np.cos(t),y0+r*np.sin(t)] for t,r in zip(tList,rList)])
    return xyList

###function to fit
def f(x,m,c):
    y = abs(m) * abs(x)**(-abs(c))
    #~ print y,x,m,c
    return y


###how to rescale the ellipse to make fitfunction a tangent
def elliptic_rescale(x, m, c, x0, y0, sa, sb):
    #~ print "e,r",x,m,c
    y=f( x, m, c ) 
    #~ print "e,r",y
    r=np.sqrt( ( x - x0 )**2 + ( y - y0 )**2 )
    kappa=float( sa ) / float( sb )
    tau=np.arctan2( y - y0, x - x0 )
    new_a=r*np.sqrt( np.cos( tau )**2 + ( kappa * np.sin( tau ) )**2 )
    return new_a

###residual function to calculate chi-square
def residuals(parameters,dataPoint):#data point is (x,y,sx,sy)
    m, c = parameters
    #~ print "m c", m, c
    theData = np.array(dataPoint)
    best_t_List=[]
    for i in range(len(dataPoint)):
        x, y, sx, sy = dataPoint[i][0], dataPoint[i][1], dataPoint[i][2], dataPoint[i][3]
        #~ print "x, y, sx, sy",x, y, sx, sy
        ###getthe point on the graph where it is tangent to an error-ellipse
        ed_fit = minimize( elliptic_rescale, x , args = ( m, c, x, y, sx, sy ) )
        best_t = ed_fit['x'][0]
        best_t_List += [best_t]
        #~ exit(0)
    best_y_List=[ f( t, m, c ) for t in best_t_List ]
    ##weighted distance not squared yet, as this is done by scipy.optimize.leastsq
    wighted_dx_List = [ ( x_b - x_f ) / sx for x_b, x_f, sx in zip( best_t_List,theData[:,0], theData[:,2] ) ]
    wighted_dy_List = [ ( x_b - x_f ) / sx for x_b, x_f, sx in zip( best_y_List,theData[:,1], theData[:,3] ) ]
    return wighted_dx_List + wighted_dy_List


def chi2(params, pnts):  
    r = np.array( residuals( params, pnts ) )
    s = sum( [ x**2 for x in  r]  )
    #~ print params,s,r
    return s


def myUpperIneq(params,pnt):
    m, c = params
    x,y=pnt
    return y - f( x, m, c )


def myLowerIneq(params,pnt):
    m, c = params
    x,y=pnt
    return f( x, m, c ) - y


###to create some test data
def test_data(m,c, xList,const_sx,rel_sx,const_sy,rel_sy):
    yList=[f(x,m,c) for x in xList]
    xErrList=[ boxmuller(x,const_sx+x*rel_sx)[0] for x in xList]
    yErrList=[ boxmuller(y,const_sy+y*rel_sy)[0] for y in yList]
    return xErrList,yErrList


###some start values
mm_0=2.3511
expo_0=.3588
csx,rsx=.01,.07
csy,rsy=.04,.09,

limitingPoints=dict()
limitingPoints[0]=np.array([[.2,5.4],[.5,5.0],[5.1,.9],[5.7,.9]])
limitingPoints[1]=np.array([[.2,5.4],[.5,5.0],[5.1,1.5],[5.7,1.2]])
limitingPoints[2]=np.array([[.2,3.4],[.5,5.0],[5.1,1.1],[5.7,1.2]])
limitingPoints[3]=np.array([[.2,3.4],[.5,5.0],[5.1,1.7],[5.7,1.2]])

####some data
xThData=np.linspace(.2,5,15)
yThData=[ f(x, mm_0, expo_0) for x in xThData]

#~ ###some noisy data
xNoiseData,yNoiseData=test_data(mm_0,  expo_0, xThData, csx,rsx, csy,rsy)
xGuessdError=[csx+rsx*x for x in xNoiseData]
yGuessdError=[csy+rsy*y for y in yNoiseData]



for testing in range(4):
    ###Now fitting with limits
    zipData=zip(xNoiseData,yNoiseData, xGuessdError, yGuessdError)    
    estimate = [ 2.4, .3 ]
    con0={'type': 'ineq', 'fun': myUpperIneq, 'args': (limitingPoints[testing][0],)}
    con1={'type': 'ineq', 'fun': myUpperIneq, 'args': (limitingPoints[testing][1],)}
    con2={'type': 'ineq', 'fun': myLowerIneq, 'args': (limitingPoints[testing][2],)}
    con3={'type': 'ineq', 'fun': myLowerIneq, 'args': (limitingPoints[testing][3],)}
    myResult = minimize( chi2 , estimate , args=( zipData, ), constraints=[ con0, con1, con2, con3 ]  )
    print "############"
    print myResult


    ###plot that
    ax=fig1.add_subplot(4,2,2*testing+1)
    ax.plot(xThData,yThData)
    ax.errorbar(xNoiseData,yNoiseData, xerr=xGuessdError, yerr=yGuessdError, fmt='none',ecolor='r')


    testX = np.linspace(.2,6,25)
    testY = np.fromiter( ( f( x, myResult.x[0], myResult.x[1] ) for x in testX ), np.float)

    bx=fig1.add_subplot(4,2,2*testing+2)
    bx.plot(xThData,yThData)
    bx.errorbar(xNoiseData,yNoiseData, xerr=xGuessdError, yerr=yGuessdError, fmt='none',ecolor='r')
    ax.plot(limitingPoints[testing][:,0],limitingPoints[testing][:,1],marker='x', linestyle='')
    bx.plot(limitingPoints[testing][:,0],limitingPoints[testing][:,1],marker='x', linestyle='')
    ax.plot(testX, testY, linestyle='--')
    bx.plot(testX, testY, linestyle='--')

    bx.set_xscale('log')
    bx.set_yscale('log')

plt.show()

提供结果

############
  status: 0
 success: True
    njev: 8
    nfev: 36
     fun: 13.782127248002116
       x: array([ 2.15043226,  0.35646436])
 message: 'Optimization terminated successfully.'
     jac: array([-0.00377715,  0.00350225,  0.        ])
     nit: 8
############
  status: 0
 success: True
    njev: 7
    nfev: 32
     fun: 41.372277637885716
       x: array([ 2.19005695,  0.23229378])
 message: 'Optimization terminated successfully.'
     jac: array([ 123.95069313, -442.27114677,    0.        ])
     nit: 7
############
  status: 0
 success: True
    njev: 5
    nfev: 23
     fun: 15.946621924326545
       x: array([ 2.06146362,  0.31089065])
 message: 'Optimization terminated successfully.'
     jac: array([-14.39131606, -65.44189298,   0.        ])
     nit: 5
############
  status: 0
 success: True
    njev: 7
    nfev: 34
     fun: 88.306027468763432
       x: array([ 2.16834392,  0.14935514])
 message: 'Optimization terminated successfully.'
     jac: array([ 224.11848736, -791.75553417,    0.        ])
     nit: 7

我检查了四个不同的限制点（行）。结果正常显示并以对数刻度（列）显示。通过一些额外的工作，你也可能会遇到错误。

非对称错误更新 说实话，目前我不知道如何处理这个属性。天真地，我定义了我自己的非对称损失函数，类似于this post。如果出现x和y错误，我会通过象限而不是仅检查正面或负面。因此，我的错误椭圆变为四个连接的部分。不过，这有点合理。为了测试并展示它是如何工作的，我用一个线性函数做了一个例子。我想OP可以根据他的要求组合两段代码。

如果线性拟合，它看起来像这样：

import matplotlib.pyplot as plt
import numpy as np
from random import random, seed
from scipy.optimize import minimize,leastsq

#~ seed(7563)
fig1 = plt.figure(1)
ax=fig1.add_subplot(2,1,1)
bx=fig1.add_subplot(2,1,2)

###function to fit, here only linear for testing.
def f(x,m,y0):
    y = m * x +y0
    return y

###for gaussion distributed errors
def boxmuller(x0,sigma):
    u1=random()
    u2=random()
    ll=np.sqrt(-2*np.log(u1))
    z0=ll*np.cos(2*np.pi*u2)
    z1=ll*np.cos(2*np.pi*u2)
    return sigma*z0+x0, sigma*z1+x0


###for plotting ellipse quadrants
def ell_data(aN,aP,bN,bP,x0=0,y0=0):
    tPPList=np.linspace(0, 0.5 * np.pi, 50)
    kPP=float(aP)/float(bP)
    rPPList=[aP/np.sqrt((np.cos(t))**2+(kPP*np.sin(t))**2) for t in tPPList]

    tNPList=np.linspace( 0.5 * np.pi, 1.0 * np.pi, 50)
    kNP=float(aN)/float(bP)
    rNPList=[aN/np.sqrt((np.cos(t))**2+(kNP*np.sin(t))**2) for t in tNPList]

    tNNList=np.linspace( 1.0 * np.pi, 1.5 * np.pi, 50)
    kNN=float(aN)/float(bN)
    rNNList=[aN/np.sqrt((np.cos(t))**2+(kNN*np.sin(t))**2) for t in tNNList]

    tPNList = np.linspace( 1.5 * np.pi, 2.0 * np.pi, 50)
    kPN = float(aP)/float(bN)
    rPNList = [aP/np.sqrt((np.cos(t))**2+(kPN*np.sin(t))**2) for t in tPNList]

    tList = np.concatenate( [ tPPList, tNPList, tNNList, tPNList] )
    rList = rPPList + rNPList+ rNNList + rPNList

    xyList=np.array([[x0+r*np.cos(t),y0+r*np.sin(t)] for t,r in zip(tList,rList)])
    return xyList


###how to rescale the ellipse to touch fitfunction at point (x,y)
def elliptic_rescale_asymmetric(x, m, c, x0, y0, saN, saP, sbN, sbP , getQuadrant=False):
    y=f( x, m, c ) 
    ###distance to function
    r=np.sqrt( ( x - x0 )**2 + ( y - y0 )**2 )
    ###angle to function
    tau=np.arctan2( y - y0, x - x0 )
    quadrant=0
    if tau >0:
        if tau < 0.5 * np.pi: ## PP
            kappa=float( saP ) / float( sbP )
            quadrant=1
        else:
            kappa=float( saN ) / float( sbP )
            quadrant=2
    else:
        if tau < -0.5 * np.pi: ## PP
            kappa=float( saN ) / float( sbN)
            quadrant=3
        else:
            kappa=float( saP ) / float( sbN )
            quadrant=4
    new_a=r*np.sqrt( np.cos( tau )**2 + ( kappa * np.sin( tau ) )**2 )
    if quadrant == 1 or quadrant == 4:
        rel_a=new_a/saP
    else:
        rel_a=new_a/saN
    if getQuadrant:
        return rel_a, quadrant, tau
    else:
        return rel_a

### residual function to calculate chi-square
def residuals(parameters,dataPoint):#data point is (x,y,sxN,sxP,syN,syP)
    m, c = parameters
    theData = np.array(dataPoint)
    bestTList=[]
    qqList=[]
    weightedDistanceList = []
    for i in range(len(dataPoint)):
        x, y, sxN, sxP, syN, syP = dataPoint[i][0], dataPoint[i][1], dataPoint[i][2], dataPoint[i][3], dataPoint[i][4], dataPoint[i][5]
        ### get the point on the graph where it is tangent to an error-ellipse
        ### i.e. smallest ellipse touching the graph
        edFit = minimize(  elliptic_rescale_asymmetric, x , args = ( m, c, x, y, sxN, sxP, syN, syP ) )
        bestT = edFit['x'][0]
        bestTList += [ bestT ]
        bestA,qq , tau= elliptic_rescale_asymmetric( bestT, m, c , x, y, aN, aP, bN, bP , True)
        qqList += [ qq ]
    bestYList=[ f( t, m, c ) for t in bestTList ]
    ### weighted distance not squared yet, as this is done by scipy.optimize.leastsq or manual chi2 function
    for counter in range(len(dataPoint)):
        xb=bestTList[counter]
        xf=dataPoint[counter][0]
        yb=bestYList[counter]
        yf=dataPoint[counter][1]
        quadrant=qqList[counter]
        if quadrant == 1:
            sx, sy = sxP, syP
        elif quadrant == 2:
            sx, sy = sxN, syP
        elif quadrant == 3:
            sx, sy = sxN, syN
        elif quadrant == 4:
            sx, sy = sxP, syN
        else:
            assert 0
        weightedDistanceList += [ ( xb - xf ) / sx, ( yb - yf ) / sy ]
    return weightedDistanceList


def chi2(params, pnts):  
    r = np.array( residuals( params, pnts ) )
    s = np.fromiter( ( x**2 for x in  r), np.float ).sum()
    return s

####...to make data with asymmetric error (fixed); for testing only
def noisy_data(xList,m0,y0, sxN,sxP,syN,syP):
    yList=[ f(x, m0, y0) for x in xList]
    gNList=[boxmuller(0,1)[0] for dummy in range(len(xList))]
    xerrList=[]
    for x,err in zip(xList,gNList):
        if err < 0:
            xerrList += [ sxP * err + x ]
        else:
            xerrList += [ sxN * err + x ]
    gNList=[boxmuller(0,1)[0] for dummy in range(len(xList))]
    yerrList=[]
    for y,err in zip(yList,gNList):
        if err < 0:
            yerrList += [ syP * err + y ]
        else:
            yerrList += [ syN * err + y ]
    return xerrList, yerrList


###some start values
m0=1.3511
y0=-2.2
aN, aP, bN, bP=.2,.5, 0.9, 1.6

#### some data
xThData=np.linspace(.2,5,15)
yThData=[ f(x, m0, y0) for x in xThData]
xThData0=np.linspace(-1.2,7,3)
yThData0=[ f(x, m0, y0) for x in xThData0]

### some noisy data
xErrList,yErrList = noisy_data(xThData, m0, y0, aN, aP, bN, bP)

###...and the fit
dataToFit=zip(xErrList,yErrList,  len(xThData)*[aN], len(xThData)*[aP], len(xThData)*[bN], len(xThData)*[bP])
fitResult = minimize(chi2, (m0,y0) , args=(dataToFit,) )
fittedM, fittedY=fitResult.x
yThDataF=[ f(x, fittedM, fittedY) for x in xThData0]


### plot that
for cx in [ax,bx]:
    cx.plot([-2,7], [f(x, m0, y0 ) for x in [-2,7]])

ax.errorbar(xErrList,yErrList, xerr=[ len(xThData)*[aN],len(xThData)*[aP] ], yerr=[ len(xThData)*[bN],len(xThData)*[bP] ], fmt='ro')

for x,y in zip(xErrList,yErrList)[:]:
    xEllList,yEllList = zip( *ell_data(aN,aP,bN,bP,x,y) )
    ax.plot(xEllList,yEllList ,c='#808080')
    ### rescaled
    ### ...as well as a scaled version that touches the original graph. This gives the error shortest distance to that graph
    ed_fit = minimize( elliptic_rescale_asymmetric, 0 ,args=(m0, y0, x, y, aN, aP, bN, bP ) )
    best_t = ed_fit['x'][0]
    best_a,qq , tau= elliptic_rescale_asymmetric( best_t, m0, y0 , x, y, aN, aP, bN, bP , True)
    xEllList,yEllList = zip( *ell_data( aN * best_a, aP * best_a, bN * best_a, bP * best_a, x, y) )
    ax.plot( xEllList, yEllList, c='#4040a0' )

###plot the fit

bx.plot(xThData0,yThDataF)
bx.errorbar(xErrList,yErrList, xerr=[ len(xThData)*[aN],len(xThData)*[aP] ], yerr=[ len(xThData)*[bN],len(xThData)*[bP] ], fmt='ro')
for x,y in zip(xErrList,yErrList)[:]:
    xEllList,yEllList = zip( *ell_data(aN,aP,bN,bP,x,y) )
    bx.plot(xEllList,yEllList ,c='#808080')
    ####rescaled
    ####...as well as a scaled version that touches the original graph. This gives the error shortest distance to that graph
    ed_fit = minimize( elliptic_rescale_asymmetric, 0 ,args=(fittedM, fittedY, x, y, aN, aP, bN, bP ) )
    best_t = ed_fit['x'][0]
    #~ print best_t
    best_a,qq , tau= elliptic_rescale_asymmetric( best_t, fittedM, fittedY , x, y, aN, aP, bN, bP , True)
    xEllList,yEllList = zip( *ell_data( aN * best_a, aP * best_a, bN * best_a, bP * best_a, x, y) )
    bx.plot( xEllList, yEllList, c='#4040a0' )

plt.show()

哪些情节

上图显示了原始线性函数以及使用非对称高斯误差从中生成的一些数据。绘制误差条，以及分段误差椭圆（灰色...并重新调整以触摸线性函数，蓝色）。下图还显示了拟合函数以及重新缩放的分段椭圆，触及拟合函数。

Python幂律符合上限和＆amp;使用ODR的数据中的非对称错误

1 个答案: