如何使曲线拟合阶跃函数？

时间：2019-06-15 04:22:01

标签： matlab curve-fitting non-linear-regression

我正在尝试使曲线适合阶跃函数。我没有尝试。诸如使用S形函数，使用多项式的比率，将高斯函数拟合到阶跃导数之类的方法，但是它们都不令人满意。现在，我想到了创建完美步长并计算完美步长到高斯函数的卷积，并使用非线性回归找到最佳拟合参数的想法。 但这也不好看。 我在这里编写Sigmoid和卷积方法的代码。 首先具有Sigmoid函数拟合：

功能：

function d=fit_sig(param,x,y)
a=param(1);
b=param(2);
d=(a./(1+exp(-b*x)))-y;
end

主要代码：

a=1, b=0.09;
p0=[a,b];
sig_coff=lsqnonlin(@fit_sig,p0,[],[],[],xavg_40s1,havg_40s1);
% Plot the original and experimental data.
sig_new = sig_coff(1)./(1+exp(-sig_coff(2)*xavg_40s1));
d= havg_40s1-step_new;
figure;
plot(xavg_40s1,havg_40s1,'.-r',xavg_40s1,sig_new,'.-b');
xlabel('x-pixel'); ylabel('dz/dx (mm/pixel)'); axis square;

这根本不起作用。我认为我最初的猜测是错误的。我尝试了多个数字，但无法正确输入。我也尝试使用曲线拟合工具，但这也无法正常工作。

创建完美步骤的代码：

h=ones(1,numel(havg_40s1)); %height=1mm
h(1:81)=-0.038;
h(82:end)=1.002; %or 1.0143
figure; 
plot(xavg_40s1,havg_40s1,'k.-', 'linewidth',1.5, 'markersize',16);
hold on
plot(xavg_40s1,h,'.-r','linewidth',1.5,'markersize',12);

使用卷积方法的代码：

功能：

function d=fit_step(param,h,x,y)
A=param(1);
mu=param(2);
sigma=param(3);
d=conv(h,A*exp(-((x-mu)/sigma).^2),'same')-y;
end

主要代码：

param1=[0.2247    8.1884    0.0802];
step_coff=lsqnonlin(@fit_step,param1,[],[],[],h,dx_40s1,havg_40s1);
% Plot the original and experimental data.
step_new = conv(h,step_coff(1)*exp(-((dx_40s1-step_coff(2))/step_coff(3)).^2),'same');

figure;
plot(xavg_40s1,havg_40s1,'.-r',xavg_40s1,step_new,'.-b');

这很近，但是台阶的边缘已经移动，并且拐角看上去比测量的台阶更锐利。

有人可以帮助我找到适合步进功能的最佳方法或任何建议来改进代码吗？

X数据：

12.6400 12.6720 12.7040 12.7360 12.7680 12.8000 12.8320 12.8640 12.8960 12.9280 12.9600 12.9920 13.0240 13.0560 13.0880 13.1200 13.1520 13.1840 13.2160 13.2480 13.2800 13.3120 13.3440 13.3760 13.4080 13.4400 13.4720 13.5040 13.5360 13.5680 13.6000 13.6.03.83.6 14.1440 14.1760 14.2080 14.2400 14.272 14.3040 14.3360 14.3680 14.4000 14.4320 14.4640 14.4960 14.5280 14.5600 14.5920 14.6240 14.6560 14.6880 14.7200 14.7520 14.7840 14.8160 14.8480 14.8800 14.9120 14.9440 14.9760 15.0080 15.0400 15.0720 15.20 15.20 15.5 15.20 15.80 15.20 15.80 15.80 15.80 15.60 15.80 15.80 15.60 15.80 15.80 15.60 15.80 15.80 15.60 15.80 15.80 15.60 15.80 15.60 15.80 15.80 15.80 15.7440 15.7760 15.8080 15.8400 15.8720 15.9040 15.9360 15.9680 16.0000 16.0320 16.0640 16.0960 16.1280 16.1600 16.1920 16.2240 16.2560 16.2880 16.3200 16.3520 16.3840 16.4160 16.4480 16.4800 16.5120 16.5440 16.5760 16.6080 16.6400 16.6720 16.7040 16.7360 16.7680 16.8000 16.8320 16.8640 16.8960 16.9160 17.20 17.21.70.2 17.20 17.21.70.2 17.20 17.20 17.21.70.2 17.20 17.21.70.2 17.20 17.20 17.21.70.2 17.60 17.21.70.2 17.60 17.21.70.2 17.60 17.21.70.2 17.60 17.21.70.2 17.60 17.21.70.2 17.60 17.21.70.2 17.20 17.20 17.20 17.20 17.20 17.21.70.0 17.20 17.20 17.20 17.20 17.20 17.20 17.20 17.20 17.20 17.20 17.21.70.0 17.20 17.20 17.20 17.20 17.5680 17.6000 17.6320 17.6640 17.6960 17.7280 17.7600

Y数据：

-0.0404 -0.0405 -0.0350 -0.0406 -0.0412 -0.0407 -0.0378 -0.0405 -0.0337 -0.0417 -0.0413 -0.0387 -0.0352 -0.0373 -0.0369 -0.0388 -0.0384 -0.0351 -0.0401 -0.0314 -0.0375 -0.0390 -0.0330- 0.0343 -0.0341 -0.0369 -0.0424 -0.0369 -0.0309 -0.0387 -0.0346 -0.0433 -0.0410 -0.0355 -0.0343 -0.0396 -0.0369 -0.0400 -0.0377 -0.0330 -0.0416 -0.0348 -0.0380 -0.0338 -0.0349 -0.0359 -0.0418 -0.0336- 0.0375 -0.0309 -0.0362 -0.0422 -0.0437 -0.0352 -0.0303 -0.0335 -0.0358 -0.0467 -0.0341 -0.0306 -0.0322 -0.0338 -0.0418 -0.0417 -0.0299 -0.0264 -0.0308 -0.0352 -0.0330 -0.0261 -0.0088 -0.0071 0.0013 0.0012 0.0151 0.0352 0.0475 0.0764 0.1423 0.2617 0.4057 0.6241 0.8076 0.8872 0.9248 0.9340 0.9395 0.9514 0.9650 0.9708 0.9875 0.9852 0.9955 0.9971 0.9971 0.9966 0.9981 0.9983 0.9932 1.0013 1.0011 0.9961 1.0044 0.9994 1.0028 1.0028 0.9996 1.0009 1.0024 1.0027 1.0075 1.0017 1.0001 1.0033 1.0062 1.0071 1.0032 1.0026 1.0027 1.0027 1.0062 1.0063 0.9981 1.0025 0.9994 1.0075 1.0026 1.0035 1.0018 0.9999 1.0045 1.0067 0.9980 1.0044 0.9976 0.9976 1.0087 1.0026 1.0010 0.9997 1.0025 0.9943 1.0098 0.9967 0.9973 0.9973 0.9974 1.0013 1.0060 1.0026 0.9960 0.9970 0.9987 1.0054 1.0048 0.9952 0.9937 0.9972

随附的是所测量台阶和拟合曲线的图像。 Fitted curve using Sigmoid Function Fitted curve using convolution approach

2 个答案:

答案 0 :(得分：1)

为什么不采用简单的方法？整理数据，计算其导数，然后找到该导数的最大值。您可以通过与容易生成的高斯导数进行卷积来完成前两个步骤。

最大值的位置是您要适合的阶跃函数的偏移。左侧的值的平均值和右侧的值的平均值是阶跃函数的下限值和上限值。

从第一原理（工具箱将使所有这些步骤更简单）开始，高斯梯度的计算如下：

x = [12.6400 12.6720 12.7040 12.7360 12.7680 12.8000 12.8320 12.8640 12.8960 12.9280 12.9600 12.9920 13.0240 13.0560 13.0880 13.1200 13.1520 13.1840 13.2160 13.2480 13.2800 13.3120 13.3440 13.3760 13.4080 13.4400 13.4720 13.5040 13.5360 13.5680 13.6000 13.6320 13.6640 13.6960 13.7280 13.7600 13.7920 13.8240 13.8560 13.8880 13.9200 13.9520 13.9840 14.0160 14.0480 14.0800 14.1120 14.1440 14.1760 14.2080 14.2400 14.272 14.3040 14.3360 14.3680 14.4000 14.4320 14.4640 14.4960 14.5280 14.5600 14.5920 14.6240 14.6560 14.6880 14.7200 14.7520 14.7840 14.8160 14.8480 14.8800 14.9120 14.9440 14.9760 15.0080 15.0400 15.0720 15.1040 15.1360 15.1680 15.2000 15.2320 15.2640 15.2960 15.3280 15.3600 15.3920 15.4240 15.4560 15.4880 15.5200 15.5520 15.5840 15.6160 15.6480 15.6800 15.7120 15.7440 15.7760 15.8080 15.8400 15.8720 15.9040 15.9360 15.9680 16.0000 16.0320 16.0640 16.0960 16.1280 16.1600 16.1920 16.2240 16.2560 16.2880 16.3200 16.3520 16.3840 16.4160 16.4480 16.4800 16.5120 16.5440 16.5760 16.6080 16.6400 16.6720 16.7040 16.7360 16.7680 16.8000 16.8320 16.8640 16.8960 16.9280 16.9600 16.9920 17.0240 17.0560 17.0880 17.1200 17.1520 17.1840 17.2160 17.2480 17.2800 17.3120 17.3440 17.3760 17.4080 17.4400 17.4720 17.5040 17.5360 17.5680 17.6000 17.6320 17.6640 17.6960 17.7280 17.7600];
y = [-0.0404 -0.0405 -0.0350 -0.0406 -0.0412 -0.0407 -0.0378 -0.0405 -0.0337 -0.0417 -0.0413 -0.0387 -0.0352 -0.0373 -0.0369 -0.0388 -0.0384 -0.0351 -0.0401 -0.0314 -0.0375 -0.0390 -0.0330 -0.0343 -0.0341 -0.0369 -0.0424 -0.0369 -0.0309 -0.0387 -0.0346 -0.0433 -0.0410 -0.0355 -0.0343 -0.0396 -0.0369 -0.0400 -0.0377 -0.0330 -0.0416 -0.0348 -0.0380 -0.0338 -0.0349 -0.0359 -0.0418 -0.0336 -0.0375 -0.0309 -0.0362 -0.0422 -0.0437 -0.0352 -0.0303 -0.0335 -0.0358 -0.0467 -0.0341 -0.0306 -0.0322 -0.0338 -0.0418 -0.0417 -0.0299 -0.0264 -0.0308 -0.0352 -0.0330 -0.0261 -0.0088 -0.0071 0.0013 0.0012 0.0151 0.0352 0.0475 0.0764 0.1423 0.2617 0.4057 0.6241 0.8076 0.8872 0.9248 0.9340 0.9395 0.9514 0.9650 0.9708 0.9875 0.9852 0.9955 0.9971 0.9966 0.9981 0.9983 0.9932 1.0013 1.0011 0.9961 1.0044 0.9994 1.0028 1.0028 0.9996 1.0009 1.0024 1.0027 1.0075 1.0017 1.0001 1.0033 1.0062 1.0071 1.0032 1.0026 1.0027 1.0062 1.0063 0.9981 1.0025 0.9994 1.0075 1.0026 1.0035 1.0018 0.9999 1.0045 1.0067 0.9980 1.0044 0.9976 0.9976 1.0087 1.0026 1.0010 0.9997 1.0025 0.9943 1.0098 0.9964 0.9994 0.9973 0.9997 1.0084 1.0035 0.9974 0.9967 0.9967 1.0013 1.0060 1.0026 0.9960 0.9970 0.9987 1.0054 1.0048 0.9952 0.9937 0.9972];

sigma = 3;
cutoff = ceil(4*sigma);
kernel = -cutoff:cutoff;
kernel = -kernel .* exp(-0.5 * kernel.^2 / sigma.^2);
grad = conv(y,kernel,'same');

我们可以用max找到最大样本：

[~,ii] = max(grad);

这是最靠近过渡点中间的样本。我们可以通过将抛物线拟合峰附近的3个样本来优化此位置：

px = x(ii-1:ii+1).';
py = grad(ii-1:ii+1).';
% solve the equation: py = [px.*px, px, ones(3,1)] * params;
params = [px.*px, px, ones(3,1)] \ py;
x_max = -params(2)/(2*params(1));

最后，我们可能希望在过渡到拟合之前和之后包括y值：

left = median(y(x<x_max));
right = median(y(x>x_max));

（尽管我们可能要假设left=0和right=1。）

绘图：

plot(x,y)
hold on
plot([x(1),x_max,x_max,x(end)],[left,left,right,right])

data + fitted step function

要拟合一个完整的误差函数（它是高斯积分的整数），我们只需要再做一步：上面我们在最大附近的三个样本上拟合了一个抛物线，现在我们对y的对数拟合了一个抛物线。值（有关说明，请参见this other Q&A），并为此选择大于峰值的0.2倍的所有y值，以免产生噪声。这约为2 sigma，应足以获得对高斯峰的准确估计。高斯峰的参数也是平滑误差函数的参数，我们可以为此额外的平滑校正估计的sigma：

% using grad from the code above (as well as x and y)
[m,ii] = max(grad);
w = sum(grad > m * 0.2) / 2;
px = x(ii-w:ii+w).';
py = log(grad(ii-w:ii+w)).';
% solve the equation: py = [px.*px, px, ones(3,1)] * params;
params = [px.*px, px, ones(size(px))] \ py;
% obtain Gaussian parameters 
fitted_mu = -params(2)/(2*params(1));
fitted_sigma = sqrt(-0.5/params(1));
% correct for smoothing applied
fitted_sigma = sqrt(fitted_sigma^2 - (sigma*mean(diff(x)))^2);
% evaluated fitted function
fitted_y = (erf((x-fitted_mu)/fitted_sigma) + 1) / 2 * (right-left) + left;

clf
plot(x,y)
hold on
plot(x,fitted_y)

data + fitted erf function

答案 1 :(得分：1)

这是使用您的数据和我的方程式搜索中的S型曲线的图形拟合器示例。此示例使用标准的scipydifferential_evolution遗传算法模块来确定用于曲线拟合的初始参数估计，并且该模块使用Latin Hypercube算法来确保对需要搜索范围的参数空间进行彻底搜索。在此示例中，搜索范围是从数据得出的。请注意，估计初始参数估计值的范围要比指定值容易得多。

plot 

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([12.6400, 12.6720, 12.7040, 12.7360, 12.7680, 12.8000, 12.8320, 12.8640, 12.8960, 12.9280, 12.9600, 12.9920, 13.0240, 13.0560, 13.0880, 13.1200, 13.1520, 13.1840, 13.2160, 13.2480, 13.2800, 13.3120, 13.3440, 13.3760, 13.4080, 13.4400, 13.4720, 13.5040, 13.5360, 13.5680, 13.6000, 13.6320, 13.6640, 13.6960, 13.7280, 13.7600, 13.7920, 13.8240, 13.8560, 13.8880, 13.9200, 13.9520, 13.9840, 14.0160, 14.0480, 14.0800, 14.1120, 14.1440, 14.1760, 14.2080, 14.2400, 14.272, 14.3040, 14.3360, 14.3680, 14.4000, 14.4320, 14.4640, 14.4960, 14.5280, 14.5600, 14.5920, 14.6240, 14.6560, 14.6880, 14.7200, 14.7520, 14.7840, 14.8160, 14.8480, 14.8800, 14.9120, 14.9440, 14.9760, 15.0080, 15.0400, 15.0720, 15.1040, 15.1360, 15.1680, 15.2000, 15.2320, 15.2640, 15.2960, 15.3280, 15.3600, 15.3920, 15.4240, 15.4560, 15.4880, 15.5200, 15.5520, 15.5840, 15.6160, 15.6480, 15.6800, 15.7120, 15.7440, 15.7760, 15.8080, 15.8400, 15.8720, 15.9040, 15.9360, 15.9680, 16.0000, 16.0320, 16.0640, 16.0960, 16.1280, 16.1600, 16.1920, 16.2240, 16.2560, 16.2880, 16.3200, 16.3520, 16.3840, 16.4160, 16.4480, 16.4800, 16.5120, 16.5440, 16.5760, 16.6080, 16.6400, 16.6720, 16.7040, 16.7360, 16.7680, 16.8000, 16.8320, 16.8640, 16.8960, 16.9280, 16.9600, 16.9920, 17.0240, 17.0560, 17.0880, 17.1200, 17.1520, 17.1840, 17.2160, 17.2480, 17.2800, 17.3120, 17.3440, 17.3760, 17.4080, 17.4400, 17.4720, 17.5040, 17.5360, 17.5680, 17.6000, 17.6320, 17.6640, 17.6960, 17.7280, 17.7600])

yData = numpy.array([-0.0404, -0.0405, -0.0350, -0.0406, -0.0412, -0.0407, -0.0378, -0.0405, -0.0337, -0.0417, -0.0413, -0.0387, -0.0352, -0.0373, -0.0369, -0.0388, -0.0384, -0.0351, -0.0401, -0.0314, -0.0375, -0.0390, -0.0330, -0.0343, -0.0341, -0.0369, -0.0424, -0.0369, -0.0309, -0.0387, -0.0346, -0.0433, -0.0410, -0.0355, -0.0343, -0.0396, -0.0369, -0.0400, -0.0377, -0.0330, -0.0416, -0.0348, -0.0380, -0.0338, -0.0349, -0.0359, -0.0418, -0.0336, -0.0375, -0.0309, -0.0362, -0.0422, -0.0437, -0.0352, -0.0303, -0.0335, -0.0358, -0.0467, -0.0341, -0.0306, -0.0322, -0.0338, -0.0418, -0.0417, -0.0299, -0.0264, -0.0308, -0.0352, -0.0330, -0.0261, -0.0088, -0.0071, 0.0013, 0.0012, 0.0151, 0.0352, 0.0475, 0.0764, 0.1423, 0.2617, 0.4057, 0.6241, 0.8076, 0.8872, 0.9248, 0.9340, 0.9395, 0.9514, 0.9650, 0.9708, 0.9875, 0.9852, 0.9955, 0.9971, 0.9966, 0.9981, 0.9983, 0.9932, 1.0013, 1.0011, 0.9961, 1.0044, 0.9994, 1.0028, 1.0028, 0.9996, 1.0009, 1.0024, 1.0027, 1.0075, 1.0017, 1.0001, 1.0033, 1.0062, 1.0071, 1.0032, 1.0026, 1.0027, 1.0062, 1.0063, 0.9981, 1.0025, 0.9994, 1.0075, 1.0026, 1.0035, 1.0018, 0.9999, 1.0045, 1.0067, 0.9980, 1.0044, 0.9976, 0.9976, 1.0087, 1.0026, 1.0010, 0.9997, 1.0025, 0.9943, 1.0098, 0.9964, 0.9994, 0.9973, 0.9997, 1.0084, 1.0035, 0.9974, 0.9967, 0.9967, 1.0013, 1.0060, 1.0026, 0.9960, 0.9970, 0.9987, 1.0054, 1.0048, 0.9952, 0.9937, 0.9972])

def func(x, a, b, Offset): # Sigmoid A with Offset from zunzun.com
    return 1.0 / (1.0 + numpy.exp(-1.0 * a * (x - b))) + 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([minX, maxX]) # search bounds for a
    parameterBounds.append([minX, maxX]) # search bounds for b
    parameterBounds.append([minY, 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()

# 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)
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