Question

我想找到适合these curves的函数，而不必猜测它们的基本形式和，添加θ-> 0（渐近）的边界条件< / p>

没有提供基本功能作为拟合形式，optimize_curve_fit无效。

Answer 1

这是一个图形多项式拟合器，您可以使用自己的数据并指定不同的多项式阶数，以查看拟合是否足以满足您的建模要求。

import numpy, matplotlib
import matplotlib.pyplot as plt

polynomialOrder = 2 # example quadratic

xData = numpy.array([1.1, 2.2, 3.3, 4.4, 5.0, 6.6, 7.7, 0.0])
yData = numpy.array([1.1, 20.2, 30.3, 40.4, 50.0, 60.6, 70.7, 0.1])

# curve fit the test data
fittedParameters = numpy.polyfit(xData, yData, polynomialOrder)
print('Fitted Parameters:', fittedParameters)

modelPredictions = numpy.polyval(fittedParameters, xData)
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('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 = numpy.polyval(fittedParameters, xModel)

    # 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 2

在注释中，您引用了参数范围。尽管在我之前的示例中numpy的线性拟合器polyfit不直接支持参数范围，但是scipy的非线性拟合器curve_fit确实允许参数范围，尽管非线性拟合器需要初始参数估计。此示例具有参数范围，并使用scipy的differential_evolution遗传算法模块来估计初始参数值，并且该模块中的scipy实现使用Latin Hypercube算法来确保对参数空间进行彻底搜索，要求搜索范围在此处-这些范围是从数据的最大值和最小值中获取，其中一个参数最小为硬编码，偏移最小为零。提供搜索范围而不是初始参数估计的特定值要容易得多。

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

    xData = numpy.array([19.1647, 18.0189, 16.9550, 15.7683, 14.7044, 13.6269, 12.6040, 11.4309, 10.2987, 9.23465, 8.18440, 7.89789, 7.62498, 7.36571, 7.01106, 6.71094, 6.46548, 6.27436, 6.16543, 6.05569, 5.91904, 5.78247, 5.53661, 4.85425, 4.29468, 3.74888, 3.16206, 2.58882, 1.93371, 1.52426, 1.14211, 0.719035, 0.377708, 0.0226971, -0.223181, -0.537231, -0.878491, -1.27484, -1.45266, -1.57583, -1.61717])
    yData = numpy.array([0.644557, 0.641059, 0.637555, 0.634059, 0.634135, 0.631825, 0.631899, 0.627209, 0.622516, 0.617818, 0.616103, 0.613736, 0.610175, 0.606613, 0.605445, 0.603676, 0.604887, 0.600127, 0.604909, 0.588207, 0.581056, 0.576292, 0.566761, 0.555472, 0.545367, 0.538842, 0.529336, 0.518635, 0.506747, 0.499018, 0.491885, 0.484754, 0.475230, 0.464514, 0.454387, 0.444861, 0.437128, 0.415076, 0.401363, 0.390034, 0.378698])


    def func(x, a, b, offset): #exponential curve fitting function
        return a * numpy.exp(-b*x) + offset


    # 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([-0.185, maxX]) # search bounds for a
        parameterBounds.append([minX, maxX]) # search bounds for b
        parameterBounds.append([0.0, maxY]) # search bounds for Offset

        # "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()
    print('fit with parameter bounds (note the -0.185)')
    print(geneticParameters)
    print()

    # second call to curve_fit made with no bounds for comparison
    fittedParameters, pcov = curve_fit(func, xData, yData, geneticParameters)

    print('re-fit with no parameter bounds')
    print(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)

没有已知函数的数据拟合到曲线

2 个答案: