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找到带Python Scipy的曲线方程

发布于 2025-02-09 17:23:22 字数 5228 浏览 2 评论 0原文

我试图找到具有Scipy的曲线方程，但我会遇到错误。我很确定正确的方程式为1-（exp（-a x+b）/（1+exp（-a x+b））。这是为表示logit函数

。 试图返回系数的logit函数（A和B）。

X轴和Y轴编号实际上是我

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


# extracted from plot
xData = numpy.array([-1500,
-992.407809110628,
-507.59219088937,
-449.023861171366,
-377.440347071583,
-335.140997830802,
-299.349240780911,
-263.557483731019,
-218.004338394793,
-178.958785249457,
-152.9284164859,
-123.644251626898,
-110.629067245119,
-91.1062906724512,
-74.8373101952279,
-61.822125813449,
-55.3145336225593,
-22.7765726681127,
0,
16.2689804772235,
39.0455531453362,
58.5683297180039,
74.8373101952274,
100.867678958785,
123.644251626898,
139.913232104121,
159.436008676789,
182.212581344902,
191.973969631236,
221.258134490238,
237.527114967462,
247.288503253796,
276.572668112798,
299.349240780911,
315.618221258134,
325.379609544468,
344.902386117136,
357.917570498915,
380.694143167027,
406.724511930585,
432.754880694143,
475.054229934924,
510.845986984815,
553.145336225596,
601.952277657267,
644.251626898047,
702.819956616052,
761.388286334056,
823.210412147505,
888.286334056399,
979.39262472885,
1037.96095444685,
1086.76789587852,
1138.82863340563,
1210.41214750542,
1252.7114967462,
1301.51843817787,
1363.34056399132])
yData = numpy.array([0.00160513643659698,
0.00372906968241948,
-0.00372906968241992,
-0.00706468943569538,
-0.00240248186822578,
0.00071726270268746,
0.00545607114131807,
0.00858974314335125,
0.0181230697450929,
0.0292754602145519,
0.0452711148560425,
0.0564443964721814,
0.0692576331027179,
0.0852672151753288,
0.102888897400096,
0.11730727046723,
0.131739570965484,
0.160562389668631,
0.197431781701444,
0.234315101165377,
0.28242044825437,
0.330532759058923,
0.388282852198618,
0.436381235572051,
0.500537947027015,
0.551867494420322,
0.61924144246404,
0.660926243806645,
0.710664582194475,
0.760361138288945,
0.795639321316281,
0.834141704647931,
0.858156077756847,
0.891815196916466,
0.90622660626804,
0.923862215923928,
0.93345125225015,
0.943054216007492,
0.955846561491349,
0.963816533949854,
0.974996779281553,
0.982931933162257,
0.990881014474082,
0.995605895481593,
0.997106576184788,
1.00183145719229,
1.00170611031221,
1.00158076343213,
1.00305358927309,
1.00291431496189,
1.007534740236,
1.00580425691932,
1.00730493762251,
1.00879865461015,
1.00864545286783,
1.00534465169235,
1.00684533239555,
1.0115284311097])


def func(x, a, b): # from the zunzun.com "function finder"
    return (1-((numpy.exp(-a*x+b))/(1+numpy.exp(-a*x+b))))


# 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(minX, minY)
    maxData = max(maxX, maxY)

    parameterBounds = []
    parameterBounds.append([minData, maxData]) # search bounds for a
    parameterBounds.append([minData, maxData]) # search bounds for b
    #parameterBounds.append([minData, maxData]) # search bounds for Offset

    # "seed" the numpy random number generator for repeatable results
    result = differential_evolution(sumOfSquaredError, parameterBounds, seed=2)
    print(parameterBounds)
    return result.x

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

# 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, maxfev=5000000)
print('Fitted parameters:', f' a: {fittedParameters[0]}', f' b: {fittedParameters[1]}')
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)
原文

I am trying to find equation of a curve with Scipy but I get an error. I am pretty sure the correct equation is 1-(exp(-ax+b)/(1+exp(-ax+b)). This is meant to represent a logit function.

The x-axis and y-axis numbers are actually from a logit function for which I am trying to back solve for coefficients (a and b).

the error I get suggests this is not workable?

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


# extracted from plot
xData = numpy.array([-1500,
-992.407809110628,
-507.59219088937,
-449.023861171366,
-377.440347071583,
-335.140997830802,
-299.349240780911,
-263.557483731019,
-218.004338394793,
-178.958785249457,
-152.9284164859,
-123.644251626898,
-110.629067245119,
-91.1062906724512,
-74.8373101952279,
-61.822125813449,
-55.3145336225593,
-22.7765726681127,
0,
16.2689804772235,
39.0455531453362,
58.5683297180039,
74.8373101952274,
100.867678958785,
123.644251626898,
139.913232104121,
159.436008676789,
182.212581344902,
191.973969631236,
221.258134490238,
237.527114967462,
247.288503253796,
276.572668112798,
299.349240780911,
315.618221258134,
325.379609544468,
344.902386117136,
357.917570498915,
380.694143167027,
406.724511930585,
432.754880694143,
475.054229934924,
510.845986984815,
553.145336225596,
601.952277657267,
644.251626898047,
702.819956616052,
761.388286334056,
823.210412147505,
888.286334056399,
979.39262472885,
1037.96095444685,
1086.76789587852,
1138.82863340563,
1210.41214750542,
1252.7114967462,
1301.51843817787,
1363.34056399132])
yData = numpy.array([0.00160513643659698,
0.00372906968241948,
-0.00372906968241992,
-0.00706468943569538,
-0.00240248186822578,
0.00071726270268746,
0.00545607114131807,
0.00858974314335125,
0.0181230697450929,
0.0292754602145519,
0.0452711148560425,
0.0564443964721814,
0.0692576331027179,
0.0852672151753288,
0.102888897400096,
0.11730727046723,
0.131739570965484,
0.160562389668631,
0.197431781701444,
0.234315101165377,
0.28242044825437,
0.330532759058923,
0.388282852198618,
0.436381235572051,
0.500537947027015,
0.551867494420322,
0.61924144246404,
0.660926243806645,
0.710664582194475,
0.760361138288945,
0.795639321316281,
0.834141704647931,
0.858156077756847,
0.891815196916466,
0.90622660626804,
0.923862215923928,
0.93345125225015,
0.943054216007492,
0.955846561491349,
0.963816533949854,
0.974996779281553,
0.982931933162257,
0.990881014474082,
0.995605895481593,
0.997106576184788,
1.00183145719229,
1.00170611031221,
1.00158076343213,
1.00305358927309,
1.00291431496189,
1.007534740236,
1.00580425691932,
1.00730493762251,
1.00879865461015,
1.00864545286783,
1.00534465169235,
1.00684533239555,
1.0115284311097])


def func(x, a, b): # from the zunzun.com "function finder"
    return (1-((numpy.exp(-a*x+b))/(1+numpy.exp(-a*x+b))))


# 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(minX, minY)
    maxData = max(maxX, maxY)

    parameterBounds = []
    parameterBounds.append([minData, maxData]) # search bounds for a
    parameterBounds.append([minData, maxData]) # search bounds for b
    #parameterBounds.append([minData, maxData]) # search bounds for Offset

    # "seed" the numpy random number generator for repeatable results
    result = differential_evolution(sumOfSquaredError, parameterBounds, seed=2)
    print(parameterBounds)
    return result.x

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

# 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, maxfev=5000000)
print('Fitted parameters:', f' a: {fittedParameters[0]}', f' b: {fittedParameters[1]}')
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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去了角落 2025-02-16 17:23:22

您的代码实际上几乎可以正常工作。两次更改后，它可以完美地工作：

  1. 我将logistic函数的定义更改为lambda（a*x+b），lambda（y）= 1/（1+np.exp（-y））。在保留解释的同时，这需要更少的计算（一些代数表明，这两个定义与从B到-b的变化相似）。
  2. 您仍面临非线性优化问题。因此，起始值很重要。我刚刚从curve_fit（）删除了输入“遗传参数”，从而强迫该算法从默认的初始参数开始。结果拟合看起来不错，RMSE为0.0066，R平方为0.9998。为了进行比较：根据拟合的最佳参数为A：0.0115和B：-1.3748，而您的遗传参数分别选择了10.23和-1431.41。因此，遗传参数的参数猜测很远，这解释了收敛问题。

修改后的代码可以在下面找到。

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


# extracted from plot
xData = numpy.array([-1500,
-992.407809110628,
-507.59219088937,
-449.023861171366,
-377.440347071583,
-335.140997830802,
-299.349240780911,
-263.557483731019,
-218.004338394793,
-178.958785249457,
-152.9284164859,
-123.644251626898,
-110.629067245119,
-91.1062906724512,
-74.8373101952279,
-61.822125813449,
-55.3145336225593,
-22.7765726681127,
0,
16.2689804772235,
39.0455531453362,
58.5683297180039,
74.8373101952274,
100.867678958785,
123.644251626898,
139.913232104121,
159.436008676789,
182.212581344902,
191.973969631236,
221.258134490238,
237.527114967462,
247.288503253796,
276.572668112798,
299.349240780911,
315.618221258134,
325.379609544468,
344.902386117136,
357.917570498915,
380.694143167027,
406.724511930585,
432.754880694143,
475.054229934924,
510.845986984815,
553.145336225596,
601.952277657267,
644.251626898047,
702.819956616052,
761.388286334056,
823.210412147505,
888.286334056399,
979.39262472885,
1037.96095444685,
1086.76789587852,
1138.82863340563,
1210.41214750542,
1252.7114967462,
1301.51843817787,
1363.34056399132])
yData = numpy.array([0.00160513643659698,
0.00372906968241948,
-0.00372906968241992,
-0.00706468943569538,
-0.00240248186822578,
0.00071726270268746,
0.00545607114131807,
0.00858974314335125,
0.0181230697450929,
0.0292754602145519,
0.0452711148560425,
0.0564443964721814,
0.0692576331027179,
0.0852672151753288,
0.102888897400096,
0.11730727046723,
0.131739570965484,
0.160562389668631,
0.197431781701444,
0.234315101165377,
0.28242044825437,
0.330532759058923,
0.388282852198618,
0.436381235572051,
0.500537947027015,
0.551867494420322,
0.61924144246404,
0.660926243806645,
0.710664582194475,
0.760361138288945,
0.795639321316281,
0.834141704647931,
0.858156077756847,
0.891815196916466,
0.90622660626804,
0.923862215923928,
0.93345125225015,
0.943054216007492,
0.955846561491349,
0.963816533949854,
0.974996779281553,
0.982931933162257,
0.990881014474082,
0.995605895481593,
0.997106576184788,
1.00183145719229,
1.00170611031221,
1.00158076343213,
1.00305358927309,
1.00291431496189,
1.007534740236,
1.00580425691932,
1.00730493762251,
1.00879865461015,
1.00864545286783,
1.00534465169235,
1.00684533239555,
1.0115284311097])


def func(x, a, b): # from the zunzun.com "function finder"
    #return (1-((numpy.exp(-a*x+b))/(1+numpy.exp(-a*x+b))))
    # => Rephrase logit function
    return (1/(1+numpy.exp(- (a*x+b) )))


# 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(minX, minY)
    maxData = max(maxX, maxY)

    parameterBounds = []
    parameterBounds.append([minData, maxData]) # search bounds for a
    parameterBounds.append([minData, maxData]) # search bounds for b
    #parameterBounds.append([minData, maxData]) # search bounds for Offset

    # "seed" the numpy random number generator for repeatable results
    result = differential_evolution(sumOfSquaredError, parameterBounds, seed=2)
    print(parameterBounds)
    return result.x

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

# 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, maxfev=5000000)
print('Fitted parameters:', f' a: {fittedParameters[0]}', f' b: {fittedParameters[1]}')
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)
    print(yModel)

    # 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)

Your code is actually almost working. It works perfectly after two changes:

  1. I changed the definition of the logistic function to Lambda(a*x+b) with Lambda(y) = 1/( 1+np.exp(-y) ). This requires less computations while preserving interpretation (some algebra shows that that these two definitions are similar up to a change from b to -b).
  2. You are still facing a nonlinear optimisation problem. The starting values are thus important. I just removed the input "geneticParameters" from curve_fit() thereby forcing this algorithm to start with default initial parameters. The resulting fit looks good with an RMSE of 0.0066 and an R-squared of 0.9998. For comparison: the optimal parameters according to the fit are a: 0.0115 and b: -1.3748 whereas your geneticParameters selected 10.23 and -1431.41, respectively. The parameter guess by geneticParameters is thus pretty far off and this explains the convergence problems.

The modified code can be found below.

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


# extracted from plot
xData = numpy.array([-1500,
-992.407809110628,
-507.59219088937,
-449.023861171366,
-377.440347071583,
-335.140997830802,
-299.349240780911,
-263.557483731019,
-218.004338394793,
-178.958785249457,
-152.9284164859,
-123.644251626898,
-110.629067245119,
-91.1062906724512,
-74.8373101952279,
-61.822125813449,
-55.3145336225593,
-22.7765726681127,
0,
16.2689804772235,
39.0455531453362,
58.5683297180039,
74.8373101952274,
100.867678958785,
123.644251626898,
139.913232104121,
159.436008676789,
182.212581344902,
191.973969631236,
221.258134490238,
237.527114967462,
247.288503253796,
276.572668112798,
299.349240780911,
315.618221258134,
325.379609544468,
344.902386117136,
357.917570498915,
380.694143167027,
406.724511930585,
432.754880694143,
475.054229934924,
510.845986984815,
553.145336225596,
601.952277657267,
644.251626898047,
702.819956616052,
761.388286334056,
823.210412147505,
888.286334056399,
979.39262472885,
1037.96095444685,
1086.76789587852,
1138.82863340563,
1210.41214750542,
1252.7114967462,
1301.51843817787,
1363.34056399132])
yData = numpy.array([0.00160513643659698,
0.00372906968241948,
-0.00372906968241992,
-0.00706468943569538,
-0.00240248186822578,
0.00071726270268746,
0.00545607114131807,
0.00858974314335125,
0.0181230697450929,
0.0292754602145519,
0.0452711148560425,
0.0564443964721814,
0.0692576331027179,
0.0852672151753288,
0.102888897400096,
0.11730727046723,
0.131739570965484,
0.160562389668631,
0.197431781701444,
0.234315101165377,
0.28242044825437,
0.330532759058923,
0.388282852198618,
0.436381235572051,
0.500537947027015,
0.551867494420322,
0.61924144246404,
0.660926243806645,
0.710664582194475,
0.760361138288945,
0.795639321316281,
0.834141704647931,
0.858156077756847,
0.891815196916466,
0.90622660626804,
0.923862215923928,
0.93345125225015,
0.943054216007492,
0.955846561491349,
0.963816533949854,
0.974996779281553,
0.982931933162257,
0.990881014474082,
0.995605895481593,
0.997106576184788,
1.00183145719229,
1.00170611031221,
1.00158076343213,
1.00305358927309,
1.00291431496189,
1.007534740236,
1.00580425691932,
1.00730493762251,
1.00879865461015,
1.00864545286783,
1.00534465169235,
1.00684533239555,
1.0115284311097])


def func(x, a, b): # from the zunzun.com "function finder"
    #return (1-((numpy.exp(-a*x+b))/(1+numpy.exp(-a*x+b))))
    # => Rephrase logit function
    return (1/(1+numpy.exp(- (a*x+b) )))


# 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(minX, minY)
    maxData = max(maxX, maxY)

    parameterBounds = []
    parameterBounds.append([minData, maxData]) # search bounds for a
    parameterBounds.append([minData, maxData]) # search bounds for b
    #parameterBounds.append([minData, maxData]) # search bounds for Offset

    # "seed" the numpy random number generator for repeatable results
    result = differential_evolution(sumOfSquaredError, parameterBounds, seed=2)
    print(parameterBounds)
    return result.x

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

# 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, maxfev=5000000)
print('Fitted parameters:', f' a: {fittedParameters[0]}', f' b: {fittedParameters[1]}')
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)
    print(yModel)

    # 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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