Fits with shared parameters¶
We demonstrate how to simultaneously fit two datasets with different models that shares a common parameter.
[1]:
from iminuit import Minuit
from iminuit.cost import UnbinnedNLL
from iminuit.util import describe
from matplotlib import pyplot as plt
import numpy as np
from numba_stats import norm
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call last)
File __init__.pxd:942, in numpy.import_array()
RuntimeError: module compiled against API version 0x10 but this version of numpy is 0xf
During handling of the above exception, another exception occurred:
ImportError Traceback (most recent call last)
Input In [1], in <cell line: 6>()
4 from matplotlib import pyplot as plt
5 import numpy as np
----> 6 from numba_stats import norm
File ~/python-iminuit/src/python-iminuit/test-env/lib/python3.10/site-packages/numba_stats/norm.py:9, in <module>
1 """
2 Normal distribution.
3
(...)
6 scipy.stats.norm: Scipy equivalent.
7 """
8 import numpy as np
----> 9 from ._special import erfinv as _erfinv
10 from ._util import _jit, _trans, _generate_wrappers, _prange
11 from math import erf as _erf
File ~/python-iminuit/src/python-iminuit/test-env/lib/python3.10/site-packages/numba_stats/_special.py:7, in <module>
5 from numba.extending import get_cython_function_address
6 from numba.types import WrapperAddressProtocol, float64
----> 7 import scipy.special.cython_special as cysp
10 def get(name, signature):
11 # create new function object with correct signature that numba can call by extracting
12 # function pointer from scipy.special.cython_special; uses scipy/cython internals
13 index = 1 if signature.return_type is float64 else 0
File /usr/lib/python3.10/site-packages/scipy/special/__init__.py:649, in <module>
1 """
2 ========================================
3 Special functions (:mod:`scipy.special`)
(...)
644
645 """
647 from ._sf_error import SpecialFunctionWarning, SpecialFunctionError
--> 649 from . import _ufuncs
650 from ._ufuncs import *
652 from . import _basic
File /usr/lib/python3.10/site-packages/scipy/special/_ufuncs.pyx:1, in init scipy.special._ufuncs()
File scipy/special/_ufuncs_extra_code_common.pxi:34, in init scipy.special._ufuncs_cxx()
File __init__.pxd:944, in numpy.import_array()
ImportError: numpy.core.multiarray failed to import
[2]:
# generate two data sets which are fitted simultaneously
rng = np.random.default_rng(1)
width = 2.0
data1 = rng.normal(0, width, size=1000)
data2 = rng.normal(5, width, size=1000)
[3]:
# use two pdfs with different names for non-shared parameters,
# so that they are fitted independently
def pdf1(x, μ_1, σ):
return norm.pdf(x, μ_1, σ)
def pdf2(x, μ_2, σ):
return norm.pdf(x, μ_2, σ)
# combine two log-likelihood functions by adding them
lh = UnbinnedNLL(data1, pdf1) + UnbinnedNLL(data2, pdf2)
print(f"{describe(lh)=}")
describe(lh)=['μ_1', 'σ', 'μ_2']
The σ
parameter is shared between the data sets, while the means of the two normal distributions are independently fitted.
[4]:
def plot(cost, xe, minuit, ax, **style):
signature = describe(cost)
data = cost.data
values = minuit.values[signature]
errors = minuit.errors[signature]
cx = (xe[1:] + xe[:-1]) / 2
ym = np.diff(norm.cdf(xe, *values)) * np.sum(w)
t = []
for n, v, e in zip(signature, values, errors):
t.append(f"${n} = {v:.3f} ± {e:.3f}$")
ax.plot(cx, ym, label="\n".join(t), **style)
[5]:
m = Minuit(lh, μ_1=1, μ_2=2, σ=1)
fig, ax = plt.subplots(1, 2, figsize=(14, 5))
hists = [np.histogram(lhi.data, bins=50) for lhi in lh]
# draw data and model with initial parameters
for lhi, (w, xe), axi in zip(lh, hists, ax):
cx = (xe[1:] + xe[:-1]) / 2
axi.errorbar(cx, w, np.sqrt(w), fmt="ok", capsize=0, zorder=0)
plot(lhi, xe, m, axi, ls="--")
m.migrad()
# draw model with fitted parameters
for lhi, (w, xe), axi in zip(lh, hists, ax):
plot(lhi, xe, m, axi)
axi.legend()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Input In [5], in <cell line: 8>()
9 cx = (xe[1:] + xe[:-1]) / 2
10 axi.errorbar(cx, w, np.sqrt(w), fmt="ok", capsize=0, zorder=0)
---> 11 plot(lhi, xe, m, axi, ls="--")
13 m.migrad()
15 # draw model with fitted parameters
Input In [4], in plot(cost, xe, minuit, ax, **style)
6 errors = minuit.errors[signature]
8 cx = (xe[1:] + xe[:-1]) / 2
---> 10 ym = np.diff(norm.cdf(xe, *values)) * np.sum(w)
11 t = []
12 for n, v, e in zip(signature, values, errors):
NameError: name 'norm' is not defined
The dashed line shows the initial model before the fit, the solid line shows the model after the fit. Note that the σ parameter is shared.