Python scipy.optimise.curve_fit 提供线性拟合

Question

我在使用 scipy 中的 curve_fit 参数时遇到了一个问题。我最初复制了文档建议的代码。然后我稍微改变了方程，这很好，但是增加了 np.linspace，整个预测最终变成了一条直线。有什么想法吗？

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


def f(x, a, b, c):
    # This works fine on smaller numbers
    return (a - c) * np.exp(-x / b) + c


xdata = np.linspace(60, 3060, 200)
ydata = f(xdata, 100, 400, 20)

# noise
np.random.seed(1729)
ydata = ydata + np.random.normal(size=xdata.size) * 0.2

# graph
fig, ax = plt.subplots()
plt.plot(xdata, ydata, marker="o")
pred, covar = curve_fit(f, xdata, ydata)
plt.plot(xdata, f(xdata, *pred), label="prediciton")
plt.show()

Answer 1

这是由于Levenberg–Marquardt algorithm 默认使用curve_fit 的限制。使用它的好方法是在优化之前为参数提供一些不错的初始猜测。根据我的经验，在优化像您的示例这样的指数函数时，这一点尤为重要。使用像 LM 这样的迭代算法，起点的质量决定了结果的收敛点。您拥有的参数越多，最终结果就越有可能收敛到完全不需要的曲线。总体而言，解决方案以某种方式找到了一个很好的初始猜测，就像其他答案一样。

Answer 2

这是使用您的数据和方程的示例代码，初始参数估计由 scipy 的差分进化遗传算法模块给出。该模块使用拉丁超立方体算法来确保对参数空间的彻底搜索，这需要搜索范围。在此示例中，这些界限取自数据最大值和最小值。为初始参数估计提供范围比提供具体值要容易得多。

import numpy, scipy, matplotlib
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import differential_evolution
import warnings


def func(x, a, b, c):
    return (a - c) * numpy.exp(-x / b) + c


xData = numpy.linspace(60, 3060, 200)
yData = func(xData, 100, 400, 20)

# noise
numpy.random.seed(1729)
yData = yData + numpy.random.normal(size=xData.size) * 0.2


# function for genetic algorithm to minimize (sum of squared error)
def sumOfSquaredError(parameterTuple):
    warnings.filterwarnings("ignore") # do not print warnings by genetic algorithm
    val = func(xData, *parameterTuple)
    return numpy.sum((yData - val) ** 2.0)


def generate_Initial_Parameters():
    # min and max used for bounds
    maxX = max(xData)
    minX = min(xData)
    maxY = max(yData)
    minY = min(yData)

    parameterBounds = []
    parameterBounds.append([minY, maxY]) # search bounds for a
    parameterBounds.append([minX, maxX]) # search bounds for b
    parameterBounds.append([minY, maxY]) # search bounds for c

    # "seed" the numpy random number generator for repeatable results
    result = differential_evolution(sumOfSquaredError, parameterBounds, seed=3)
    return result.x

# by default, differential_evolution completes by calling curve_fit() using parameter bounds
geneticParameters = generate_Initial_Parameters()

# now call curve_fit without passing bounds from the genetic algorithm,
# just in case the best fit parameters are aoutside those bounds
fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)
print('Fitted parameters:', fittedParameters)
print()

modelPredictions = func(xData, *fittedParameters) 

absError = modelPredictions - yData

SE = numpy.square(absError) # squared errors
MSE = numpy.mean(SE) # mean squared errors
RMSE = numpy.sqrt(MSE) # Root Mean Squared Error, RMSE
Rsquared = 1.0 - (numpy.var(absError) / numpy.var(yData))

print()
print('RMSE:', RMSE)
print('R-squared:', Rsquared)

print()


##########################################################
# graphics output section
def ModelAndScatterPlot(graphWidth, graphHeight):
    f = plt.figure(figsize=(graphWidth/100.0, graphHeight/100.0), dpi=100)
    axes = f.add_subplot(111)

    # first the raw data as a scatter plot
    axes.plot(xData, yData,  'D')

    # create data for the fitted equation plot
    xModel = numpy.linspace(min(xData), max(xData))
    yModel = func(xModel, *fittedParameters)

    # now the model as a line plot
    axes.plot(xModel, yModel)

    axes.set_xlabel('X Data') # X axis data label
    axes.set_ylabel('Y Data') # Y axis data label

    plt.show()
    plt.close('all') # clean up after using pyplot

graphWidth = 800
graphHeight = 600
ModelAndScatterPlot(graphWidth, graphHeight)

Answer 3

您可能需要从更好的猜测开始，默认初始猜测(1.0, 1.0, 1.0) 似乎在发散区域。

我使用最初的猜测 p0 = (50,200,100) 并且它有效

fig, ax = plt.subplots()
plt.plot(xdata, ydata, marker="o")
pred, covar = curve_fit(f, xdata, ydata, p0 = (50,200,100))
plt.plot(xdata, f(xdata, *pred), label="prediciton")
plt.show()