I know there is a question
Fitting only one parameter of a function with many parameters in python
but I have a little bit different situation. Problem with parameters of lambda function.
I am trying to fit Lorentz func
def lorentz(ph, A, xc, w, f0):
return f0 + A * w**2 / (w**2 + (ph - xc)**2)
if I am fitting only one parameter (xc) its working good.
p1, p2, p3, p4 = params
popt, pcov = curve_fit(lambda x, xc: lorentz(x, p1, xc, p3, p4), abjd, adata, bounds=param_bounds)
But if I try to fit only 2 parameters (a, xc) it fails
p1, p2, p3, p4 = params
popt, pcov = curve_fit(lambda x, a, xc: lorentz(x, a, xc, p3, p4), abjd, adata, bounds=param_bounds)
Error message is
Traceback (most recent call last):
File "template.py", line 294, in <module>
popt, pcov = curve_fit(lambda x, a, xc: lorentz(x, a, xc, p3, p4), abjd, adata, bounds=param_bounds)
File "/usr/local/lib/python2.7/dist-packages/scipy/optimize/minpack.py", line 683, in curve_fit
**kwargs)
File "/usr/local/lib/python2.7/dist-packages/scipy/optimize/_lsq/least_squares.py", line 769, in least_squares
f0 = fun_wrapped(x0)
File "/usr/local/lib/python2.7/dist-packages/scipy/optimize/_lsq/least_squares.py", line 764, in fun_wrapped
return np.atleast_1d(fun(x, *args, **kwargs))
File "/usr/local/lib/python2.7/dist-packages/scipy/optimize/minpack.py", line 455, in func_wrapped
return func(xdata, *params) - ydata
TypeError: <lambda>() takes exactly 3 arguments (2 given)
Here is the solution for all 4 parameters of Lorentz function
import numpy as np
from scipy.optimize import curve_fit
def lorentz(ph, A, xc, w, f0):
return f0 + A * w**2 / (w**2 + (ph - xc)**2)
A, xc, w, f0 = 2,2,2,2 # true values
ph = np.linspace(-5, 10, 100)
y = lorentz(ph, A, xc, w, f0)
ydata = y + 0.15 * np.random.normal(size=len(ph)) # sample data
popt, pcov = curve_fit(lambda x, _A, _xc: lorentz(x, _A, _xc, w, f0), ph, ydata,bounds=([1,1], [3, 3]))
A, xc = popt # fitted values (only two)
You can easily add or remove parameters by putting them under the function lorentz() or with lambda
The result looks like this
Related
I want to use scipy.optimize.minimize on a function with an array type variable input. This is what I hope to do.
I have the following signal,
import numpy as np
time = np.linspace(0, 1, 501)
data = np.cos(2 * np.pi * 4 * time) + np.cos(2 * np.pi * 9 * time) + np.cos(2 * np.pi * 20 * time)
noise = np.sqrt(1 / 25) * np.random.randn(501)
signal = data + noise
and I am hoping to find a curve fit for this signal. Since I created the data myself, I know that a sum of cosine functions will work for this. So the function that I hope to optimize is the following:
def cos_sum(x, P):
assert isinstance(P, np.ndarray)
assert P.shape[0] == P.shape[1]
sums = []
for param in P:
a, b, c = param
sums.append(a * np.cos(b * (x - c)))
sums = np.array(sums)
return np.sum(sums, axis=0)
In order to use minimize to find the correct parameters, I create this residual function.
def resid(params, x):
assert isinstance(params, np.ndarray)
fit = cos_sum(x, params)
residual = np.sqrt(np.mean(np.abs(fit - signal)) ** 2)
return residual
I now need to create a guess. Since I already know how the signal was created, I made the following guess:
guess_A = np.random.normal(1, .2, size=3)
guess_B = 2 * np.pi * np.array([4, 9, 20], dtype=float)
guess_C = np.random.normal(0, .2, size=3)
guess = np.array([guess_A, guess_B, guess_C]).T
However, I am unable to run the following:
from scipy.optimize import minimize
optimization = minimize(resid, guess, args=(time))
and I receive the following error:
Traceback (most recent call last):
File "/Users/nickeisenberg/GitRepos/Python_Misc/Misc/minimize_curvefit_vector_variables.py", line 70, in <module>
optimization = minimize(resid, guess, args=(time))
File "/Library/Frameworks/Python.framework/Versions/3.10/lib/python3.10/site-packages/scipy/optimize/_minimize.py", line 676, in minimize
res = _minimize_bfgs(fun, x0, args, jac, callback, **options)
File "/Library/Frameworks/Python.framework/Versions/3.10/lib/python3.10/site-packages/scipy/optimize/_optimize.py", line 1296, in _minimize_bfgs
sf = _prepare_scalar_function(fun, x0, jac, args=args, epsilon=eps,
File "/Library/Frameworks/Python.framework/Versions/3.10/lib/python3.10/site-packages/scipy/optimize/_optimize.py", line 263, in _prepare_scalar_function
sf = ScalarFunction(fun, x0, args, grad, hess,
File "/Library/Frameworks/Python.framework/Versions/3.10/lib/python3.10/site-packages/scipy/optimize/_differentiable_functions.py", line 158, in __init__
self._update_fun()
File "/Library/Frameworks/Python.framework/Versions/3.10/lib/python3.10/site-packages/scipy/optimize/_differentiable_functions.py", line 251, in _update_fun
self._update_fun_impl()
File "/Library/Frameworks/Python.framework/Versions/3.10/lib/python3.10/site-packages/scipy/optimize/_differentiable_functions.py", line 155, in update_fun
self.f = fun_wrapped(self.x)
File "/Library/Frameworks/Python.framework/Versions/3.10/lib/python3.10/site-packages/scipy/optimize/_differentiable_functions.py", line 137, in fun_wrapped
fx = fun(np.copy(x), *args)
File "/Users/nickeisenberg/GitRepos/Python_Misc/Misc/minimize_curvefit_vector_variables.py", line 53, in resid
fit = cos_sum(x, params)
File "/Users/nickeisenberg/GitRepos/Python_Misc/Misc/minimize_curvefit_vector_variables.py", line 30, in cos_sum
assert P.shape[0] == P.shape[1]
IndexError: tuple index out of range
Is this possible to do?
The problem is that minimize treats parameters as 1D array. Adding print just before failing assert shows, that it reshaped guess array to 1D array.
Changing cos_sum so that it accepts 1D array of params fixes this problem. Here is the code.
def cos_sum(x, P):
assert isinstance(P, np.ndarray)
sums = []
for i in range(0, P.shape[0], 3):
a, b, c = P[i], P[i+1], P[i+2]
sums.append(a * np.cos(b * (x - c)))
sums = np.array(sums)
return np.sum(sums, axis=0)
Result:
optimization = minimize(resid, guess, args=(time))
print(optimization.x)
[ 0.96805816 25.1679919 0.25020317 1.00261543 56.44511497
0.77872223 1.00966167 125.71622787 0.55004217]
plt.figure(figsize=(8, 4))
plt.plot(data)
plt.plot([cos_sum(t, optimization.x) for t in time], 'C1--')
plt.show()
I am doing a project, where I want to use Euler's Method to show a solution to this Second Order Different equation
0=y''+y'+9.81y
So I started by changing the second-order into a system of first-order equations
y'=u, u'=f(t,y,u)
With initial condition
y(0)=180, u(0)=0
So I get two equation in the end
y[n + 1] = y[n] + u[n] * (t[n + 1] - t[n]), u[n + 1] = u[n] + f(u[0], y[n], t[0]) * (t[n + 1] - t[n])
This is my code
import numpy as np
import matplotlib.pyplot as plt
def odeEuler(f, y0, u0, t):
y = np.zeros(len(t))
u = np.zeros(len(t))
y[0] = y0
u[0] = u0
for n in range(0, len(t) - 1):
y[n + 1] = y[n] + u[n] * (t[n + 1] - t[n])
u[n + 1] = u[n] + f(u[0], y[n], t[0]) * (t[n + 1] - t[n])
return y, u
t = np.linspace(0, 100)
y0 = 180
u0 = 0
f = lambda u, y, t: -9.81 * y - u
y = odeEuler(f, y0, u0, t)
plt.plot(t, y, 'b.-')
plt.legend(['Euler'])
plt.axis([0, 100, 0, 200])
plt.grid(True)
plt.show()
However, when I run the code, it give me the error
Traceback (most recent call last):
File "/Users/huangy15/PycharmProjects/Draft/Damped Driven Pendulum/Praying this works.py", line 22, in <module>
plt.plot(t, y, 'b.-')
File "/Users/huangy15/PycharmProjects/Draft/Damped Driven Pendulum/venv/lib/python3.7/site-packages/matplotlib/pyplot.py", line 3021, in plot
**({"data": data} if data is not None else {}), **kwargs)
File "/Users/huangy15/PycharmProjects/Draft/Damped Driven Pendulum/venv/lib/python3.7/site-packages/matplotlib/axes/_axes.py", line 1605, in plot
lines = [*self._get_lines(*args, data=data, **kwargs)]
File "/Users/huangy15/PycharmProjects/Draft/Damped Driven Pendulum/venv/lib/python3.7/site-packages/matplotlib/axes/_base.py", line 315, in __call__
yield from self._plot_args(this, kwargs)
File "/Users/huangy15/PycharmProjects/Draft/Damped Driven Pendulum/venv/lib/python3.7/site-packages/matplotlib/axes/_base.py", line 501, in _plot_args
raise ValueError(f"x and y must have same first dimension, but "
ValueError: x and y must have same first dimension, but have shapes (50,) and (2, 50)
Can anyone help me check if my idea works, and if not, what other approaches I can take? Thanks!
Your odeEuler function is returning two variables: y and u, you need to select just one of them to plot.
Store the output of the function in two different variables will solve the problem:
y, u = odeEuler(f, y0, u0, t)
Here is a part of a code but not working, what might be the reason for it?
population = float(46750238)
country_df = pd.DataFrame()
country_df['ConfirmedCases'] = train.loc[train['Country_Region']=='Spain'].ConfirmedCases.diff().fillna(0)
country_df = country_df[10:]
country_df['day_count'] = list(range(1,len(country_df)+1))
ydata = [i for i in country_df.ConfirmedCases]
xdata = country_df.day_count
ydata = np.array(ydata, dtype=float)
xdata = np.array(xdata, dtype=float)
N = population
inf0 = ydata[0]
sus0 = N - inf0
rec0 = 0.0
def sir_model(y, x, beta, gamma):
sus = -beta * y[0] * y[1] / N
rec = gamma * y[1]
inf = -(sus + rec)
return sus, inf, rec
def fit_odeint(x, beta, gamma):
return integrate.odeint(sir_model, (sus0, inf0, rec0), x, args=(beta, gamma))[:,1]
popt, pcov = optimize.curve_fit(fit_odeint, xdata, ydata)
fitted = fit_odeint(xdata, *popt)
plt.plot(xdata, ydata, 'o')
plt.plot(xdata, fitted)
plt.title("Fit of SIR model for Spain infected cases")
plt.ylabel("Population infected")
plt.xlabel("Days")
plt.show()
print("Optimal parameters: beta =", popt[0], " and gamma = ", popt[1])
gives an error saying: RuntimeError Traceback (most recent call last)
in
24 return integrate.odeint(sir_model, (sus0, inf0, rec0), x, args=(beta, gamma))[:,1]
25
---> 26 popt, pcov = optimize.curve_fit(fit_odeint, xdata, ydata)
27 fitted = fit_odeint(xdata, *popt)
28
C:\ProgramData\Anaconda3\lib\site-packages\scipy\optimize\minpack.py in curve_fit(f, xdata, ydata, p0, sigma, absolute_sigma, check_finite, bounds, method, jac, **kwargs)
746 cost = np.sum(infodict['fvec'] ** 2)
747 if ier not in [1, 2, 3, 4]:
--> 748 raise RuntimeError("Optimal parameters not found: " + errmsg)
749 else:
750 # Rename maxfev (leastsq) to max_nfev (least_squares), if specified.
RuntimeError: Optimal parameters not found: Number of calls to function has reached maxfev = 600.
It could be because you are feeding curve_fit less than 3 points of data. Although there are other reasons I am looking into.
I am trying to do a curve fit for a transient thermal data. I have equations which will calculate the delta temperature in every timepoint. This delta has to be added to the temperature from previous timepoint to get the temperature at any given timepoint.
ie; Tn = Tn-1 + delta.
If I express using the example from scipy's documentation for curve_fit. It would be something similar.
def func(x, a, b, c):
return a * np.exp(-b * x) + c #+ func(x[n-1], a, b, c) <<< need help here
xdata = np.linspace(0, 4, 50)
y = func(xdata, 2.5, 1.3, 0.5)
np.random.seed(1729)
y_noise = 0.2 * np.random.normal(size=xdata.size)
ydata = y + y_noise
popt, pcov = curve_fit(func, xdata, ydata)
print(popt)
plt.plot(xdata, ydata, 'b-', label='data')
plt.plot(xdata, func(xdata, *popt), 'r-',
label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))
Any lead on how to achive this is really appreciated. Thanks!
If you have a differential equation you need to find the integral before fitting it to data. Unless the data is also differential, in which case you can just fit.
The question seems to imply that in your case delta is given by a * np.exp(-b * x) + c, which makes the resulting y values easy to compute because curve_fit passes all the x values to func and expects it to return all the y values anyway.
def delta_func(x, a, b, c):
return a * np.exp(-b * x) + c
def func(x, a, b, c):
y = np.empty(x.shape)
y[0] = delta_func(0, a, b, c)
for i in range(1, len(x)):
y[i] = y[i-1] + delta_func(x[i], a, b, c)
return y
This was for illustration. You can obtain the same result with np.cumsum:
def func(x, a, b, c):
return np.cumsum(a * np.exp(-b * x) + c)
I am trying to plot streamlines and velocity potential for basic potential flows (uniform, source/sink, vortex, etc.)
I am just starting out with python and so I'm a little confused. I am following this guide..
I can plot the streamlines for the flow around a cylinder using this function
def cylinder_stream_function(U=1, R=1):
r = sympy.sqrt(x**2 + y**2)
theta = sympy.atan2(y, x)
return U * (r - R**2 / r) * sympy.sin(theta)
and it works. But when I change the return statement to
return U * r * sympy.cos(theta)
for uniform flow I get the following error
Traceback (most recent call last):
File "test.py", line 42, in
<module>
plot_streamlines(ax, u, v)
File "test.py", line 32, in plot_streamlines
ax.streamplot(X, Y, u(X, Y), v(X, Y), color='cornflowerblue')
File "/usr/local/lib/python3.6/site-packages/matplotlib/__init__.py",
line 1710, in inner
return func(ax, *args, **kwargs)
File "/usr/local/lib/python3.6/site-packages/matplotlib/axes/_axes.py",
line 4688, in streamplot
integration_direction=integration_direction)
File "/usr/local/lib/python3.6/site-packages/matplotlib/streamplot.py",
line 136, in streamplot
if (u.shape != grid.shape) or (v.shape != grid.shape):
AttributeError: 'int' object has no attribute 'shape'
I checked the type for the return object and it is <class 'sympy.core.mul.Mul'> with the first return statement and <class 'sympy.core.symbol.Symbol'> with the second one. Maybe this is related to why it doesn't work but I'm not sure how?
I plot the streamlines with the following
import numpy as np
import matplotlib.pyplot as plt
import sympy
from sympy.abc import x, y
def uniform_flow_stream_function(U=1):
r = sympy.sqrt(x**2 + y**2)
theta = sympy.atan2(y, x)
return U * r * sympy.sin(theta)
def velocity_field(psi):
u = sympy.lambdify((x, y), psi.diff(y), 'numpy')
v = sympy.lambdify((x, y), -psi.diff(x), 'numpy')
return u, v
def plot_streamlines(ax, u, v, xlim=(-4, 4), ylim=(-4, 4)):
x0, x1 = xlim
y0, y1 = ylim
# create a grid of values
Y, X = np.ogrid[y0:y1:100j, x0:x1:100j]
ax.streamplot(X, Y, u(X, Y), v(X, Y), color='cornflowerblue')
psi = uniform_flow_stream_function()
u, v = velocity_field(psi)
fig, ax = plt.subplots(figsize=(5, 5))
plot_streamlines(ax, u, v)
plt.show()
Can someone please help me understand why this doesn't work and how I can get it to work? Thank you!
The reason that this does not work is because of the class difference. Your function U * r * sympy.cos(theta)=y. This means that you are returning a function of only y. Therefore your -psi.diff(x)=0 and you get a integer for v.
It is impossible to plot streamlines with 1D data. So in order to plot streamlines in 2D you have to have both x and y in your uniform_flow_stream_function.