没有已知函数的数据拟合曲线

Question

我想找到适合 these curves 的函数，而不用猜测它们的基本形式，和添加边界条件θ->0（渐近）

optimize_curve_fit不给一个基本函数作为拟合形式是行不通的

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来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

在您引用参数范围的评论中。虽然在我之前的示例中 numpy 的线性拟合器 polyfit 不直接支持参数边界，但 scipy 的非线性拟合器 curve_fit 确实允许参数边界，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 实现使用拉丁超立方体算法来确保彻底搜索参数 space，需要搜索范围 - 此处这些范围取自数据最大值和最小值，其中一个参数最小值是硬编码的，偏移量最小值是零。提供搜索范围比初始参数估计值的具体值要容易得多。

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

    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曲线

Data Fit to a Curve without a known Function

python

curve-fitting

data-fitting