如何将反锯齿函数拟合到曲线或绘图？

Question

我有一个应该是锯齿波的情节。

我正在尝试将锯齿方程（如 wiki 中给出的）拟合到数据点，但我无法这样做。

n_step_list = [-500, -400, -300, -200, -100, 0, 100, 200, 300, 400, 500]
value_list =  [-24, 73, 55, 36, 18, 0, -18, 79, 61, 43, 24]

def f(x, A, fi):
    total_sum = 0
    i = 1
    while i < 151:
        total_sum += np.power(-1, i) * np.sin(2 * np.pi * i * fi * x) / i
        i += 1

    total_sum *= 2 * A / np.pi

    return total_sum

A, fi = curve_fit(f, n_step_list, value_list, (10000000000000, 28))[0]

但我得到了荒谬的结果。最初的猜测，我使用一个值 (-100, 18) 提供给 curve_fit 并尝试计算 A 和 fi 的值。任何帮助表示赞赏。

Answer 1

这是一个不同的拟合器，使用 scipy 的锯齿波形发生器和 scipy 的差分进化遗传算法给出的初始参数估计。参数“width”特定于代码中引用的锯齿波发生器，确定波形是上升、下降还是对称的，根据 scipy 文档，范围从 0 到 1。

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


    n_step_list = [-500.0, -400.0, -300.0, -200.0, -100.0, 0.0, 100.0, 200.0, 300.0, 400.0, 500.0]
    value_list =  [-24.0, 73.0, 55.0, 36.0, 18.0, 0.0, -18.0, 79.0, 61.0, 43.0, 24.0]

    xData = numpy.array(n_step_list)
    yData = numpy.array(value_list)


    # width is from scipy docs at https://www.pydoc.io/pypi/scipy-1.0.1/autoapi/signal/waveforms/index.html#signal.waveforms.sawtooth
    def func(x, A, fi, offset, width):
        return A * scipy.signal.sawtooth(x / fi, width) + 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)

        minData = min(minY, minX)
        maxData = max(maxY, maxX)

        parameterBounds = []
        parameterBounds.append([minData, maxData]) # search bounds for A
        parameterBounds.append([minData, maxData]) # search bounds for fi
        parameterBounds.append([minData, maxData]) # search bounds for Offset
        parameterBounds.append([0, 1]) # search bounds for width, see https://www.pydoc.io/pypi/scipy-1.0.1/autoapi/signal/waveforms/index.html#signal.waveforms.sawtooth

        # "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)