continuous two line segment with smooth fitted break point

[bytelinker]

Hello,

this example shows a two line segment with smooth fitted break point. It is a reduced example, the target functions are more complex so the available multi line fitting packages are not a solution.
I would like to set an constraint that the line segments are continuous at x0.
x0 * m1 + a1 = x0 * m2 + a2

Which is the correct syntax to realize this, please?

When setting additional parameters (https://lmfit.github.io/lmfit-py/constraints.html), theses parameters are also send to the called functions, is it possible to suppress this?

Thanyk you
bytelinker

code with format

import matplotlib.pyplot as plt
import lmfit
import numpy as np
from scipy.special import erf
plt.ion()

def plot_02(xx1, xx2, yy3):
    fig, axs = plt.subplots(1, squeeze=False, figsize=(1.5 * 5, 1.5 * 4))
    axs = axs.flat
    axs[0].plot(np.append(xx1, xx2), result_fit.best_fit, label='fit')
    axs[0].plot(np.append(xx1, xx2), yy3, label='data')
    axs[0].legend(loc='upper center', bbox_to_anchor=(0.5, -0.35), \
                  fancybox=True, shadow=False, ncol=7, fontsize=6)
    plt.tight_layout()
    plt.show(block=False)

def linear(m, a, x):
    answer = m * x + a
    return answer

def fit_line(xx, yy):
    params = lmfit.Parameters()
    params.add('a', value=1, vary=True)
    params.add('m', value=1, vary=True)
    model = lmfit.Model(func=linear, independent_vars=['x'], param_names=['a', 'm'], nan_policy='omit')
    result = model.fit(yy, params=params, x=xx)
    return result

def two_lines_smooth(x, x0, m1, a1, m2, a2):
    # https://lmfit.github.io/lmfit-py/faq.html#faq-discrete-params
    m1 = linear(m=m1, a=a1, x=x)
    m2 = linear(m=m2, a=a2, x=x)
    # step up from 0 to 1 at x0: (erf(x-x0)+1)/2
    # step down from 1 to 0 at x0: (1-erf(x-x0))/2
    model = m2 * (erf(x - x0) + 1) / 2 + m1 * (1 - erf(x - x0)) / 2
    return model

def two_with_border(m1, a1, m2, a2, x0, x):
    result = np.piecewise(x,
                          [x < x0, x >= x0],
                          [lambda x: linear(m=m1, a=a1, x=x),
                           lambda x: linear(m=m2, a=a2, x=x)])
    return result

def fit_multi_line_smooth(xx, yy):
    params = lmfit.Parameters()
    params.add('a1', value=1.1, vary=True)
    params.add('m1', value=1.1, vary=True)
    params.add('a2', value=-2.5, vary=True)
    params.add('m2', value=19.0, vary=True)
    params.add('x0', value=5.5, vary=True)
    model_linear_lm = lmfit.Model(func=two_lines_smooth, independent_vars=['x'],
                                  param_names=['a1', 'm1', 'a2', 'm2', 'x0'], nan_policy='omit')
    result_linear_10 = model_linear_lm.fit(yy, params=params, x=xx, method='least_squares')
    return result_linear_10

#%%
np.random.seed(1001)
n = 50
xx1 = np.linspace(0, 5, n, endpoint=False)
yy1 = linear(m=1, a=1, x=xx1) + np.random.randn(n)

xx2 = np.linspace(5, 10, n, endpoint=True)
yy2 = linear(m=-2, a=18, x=xx2) + np.random.randn(n)

result_fit = fit_multi_line_smooth(xx=np.append(xx1, xx2), yy=np.append(yy1, yy2))
lmfit.report_fit(result_fit)
yy3 = two_with_border(x=np.append(xx1, xx2), x0=result_fit.params['x0'].value,
                      m1=result_fit.params['m1'].value,
                      a1=result_fit.params['a1'].value,
                      m2=result_fit.params['m2'].value,
                      a2=result_fit.params['a2'].value)

plot_02(xx1=xx1, xx2=xx2, yy3=yy3)

[newville]

@bytelinker

this example shows a two line segment with smooth fitted break point.

Please confirm that you read https://lmfit.github.io/lmfit-py/faq.html#can-parameters-be-used-for-array-indices-or-discrete-values

When setting additional parameters (https://lmfit.github.io/lmfit-py/constraints.html), theses parameters are also send to the called functions, is it possible to suppress this?

No.

When using lmfit.Model, the values of the parameters named in the model function's signature (and matching any prefix) are sent.

When using the more general-purpose minimize, all the parameters are sent to the objective function.

I don't really understand your code or what point it is trying to make. It seems like you're trying something two different ways. I don't know why. If you want us to understand something about your code, tell us what that is.

And, if you want us to understand your code, format it correctly.

[bytelinker]

I tried different solutions. But I didn't find a solution to include the constraint in the code.

[newville]

@bytelinker
For sure, your two_with_border function uses x0 as a discrete variable. As you have read in the FAQ, that will not work.

For the constraint to make "x0 * m1 + a1 = x0 * m2 + a2", the answer is right there. Rearrange that to

   a2  = x0 * m1 + a1 - x0 * m2 = a1 + x0 * (m1-m2)

so use

  params.Add('a2', expr='a1 + x0 * (m1-m2)')

[bytelinker]

This solution is doing exactly what was mathematically expected.

Thank you very much.