拟合由两种不同状态组成的数据的曲线 [英] Fit a curve for data made up of two distinct regimes

问题描述

我正在寻找一种通过一些实验数据绘制曲线的方法.数据显示了一个小线性区域，梯度较浅，在阈值之后是陡峭的线性区域.

I'm looking for a way to plot a curve through some experimental data. The data shows a small linear regime with a shallow gradient, followed by a steep linear regime after a threshold value.

我的数据在这里:http://pastebin.com/H4NSbxqr

My data is here: http://pastebin.com/H4NSbxqr

我可以相对容易地用两条线拟合数据，但我想用一条连续线理想地拟合 - 它应该看起来像两条线，在阈值附近有一条平滑的曲线将它们连接起来(数据中的~5000，如图所示)以上).

I could fit the data with two lines relatively easily, but I'd like to fit with a continuous line ideally - which should look like two lines with a smooth curve joining them around the threshold (~5000 in the data, shown above).

我尝试使用 scipy.optimize curve_fit 并尝试了一个包含直线和指数之和的函数:

I attempted this using scipy.optimize curve_fit and trying a function which included the sum of a straight line and an exponential:

y = a*x + b + c*np.exp((x-d)/e)

尽管尝试了很多次，也没有找到解决方案.

although despite numerous attempts, it didn't find a solution.

如果有人对拟合分布/方法的选择或curve_fit实现有任何建议，我们将不胜感激.

If anyone has any suggestions please, either on the choice of fitting distribution / method or the curve_fit implementation, they would be greatly appreciated.

推荐答案

如果您没有特别的理由相信线性 + 指数是数据的真正根本原因，那么我认为适合两条线最有意义.您可以通过将拟合函数设为两行的最大值来实现，例如:

If you don't have a particular reason to believe that linear + exponential is the true underlying cause of your data, then I think a fit to two lines makes the most sense. You can do this by making your fitting function the maximum of two lines, for example:

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

def two_lines(x, a, b, c, d):
    one = a*x + b
    two = c*x + d
    return np.maximum(one, two)

那么，

x, y = np.genfromtxt('tmp.txt', unpack=True, delimiter=',')

pw0 = (.02, 30, .2, -2000) # a guess for slope, intercept, slope, intercept
pw, cov = curve_fit(two_lines, x, y, pw0)
crossover = (pw[3] - pw[1]) / (pw[0] - pw[2])

plt.plot(x, y, 'o', x, two_lines(x, *pw), '-')

如果你真的想要一个连续且可微的解，我突然想到双曲线有一个急弯，但它必须旋转.实现起来有点困难(也许有更简单的方法)，但这里有一个方法:

If you really want a continuous and differentiable solution, it occurred to me that a hyperbola has a sharp bend to it, but it has to be rotated. It was a bit difficult to implement (maybe there's an easier way), but here's a go:

def hyperbola(x, a, b, c, d, e):
    """ hyperbola(x) with parameters
        a/b = asymptotic slope
         c  = curvature at vertex
         d  = offset to vertex
         e  = vertical offset
    """
    return a*np.sqrt((b*c)**2 + (x-d)**2)/b + e

def rot_hyperbola(x, a, b, c, d, e, th):
    pars = a, b, c, 0, 0 # do the shifting after rotation
    xd = x - d
    hsin = hyperbola(xd, *pars)*np.sin(th)
    xcos = xd*np.cos(th)
    return e + hyperbola(xcos - hsin, *pars)*np.cos(th) + xcos - hsin

运行它

h0 = 1.1, 1, 0, 5000, 100, .5
h, hcov = curve_fit(rot_hyperbola, x, y, h0)
plt.plot(x, y, 'o', x, two_lines(x, *pw), '-', x, rot_hyperbola(x, *h), '-')
plt.legend(['data', 'piecewise linear', 'rotated hyperbola'], loc='upper left')
plt.show()

我也能够使线 + 指数收敛，但看起来很糟糕.这是因为它不是您数据的良好描述符，它是线性的，而指数与线性相去甚远！

I was also able to get the line + exponential to converge, but it looks terrible. This is because it's not a good descriptor of your data, which is linear and an exponential is very far from linear!

def line_exp(x, a, b, c, d, e):
    return a*x + b + c*np.exp((x-d)/e)

e0 = .1, 20., .01, 1000., 2000.
e, ecov = curve_fit(line_exp, x, y, e0)

如果你想保持简单，总会有多项式或样条(分段多项式)

If you want to keep it simple, there's always a polynomial or spline (piecewise polynomials)

from scipy.interpolate import UnivariateSpline
s = UnivariateSpline(x, y, s=x.size)  #larger s-value has fewer "knots"
plt.plot(x, s(x))

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