r"""
Gaussian probability density fit.
The function in this module is normalized to one, as a PDF should be:
.. math::
f(x) = \frac{1}{\sigma \sqrt{2 \pi}}
e^{-\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^2} \quad
\int_{-\infty}^{+\infty} f(t)dt = 1
For for the unnormalized Gaussian (with an additional amplitude
parameter), see :mod:`~skg.gauss`. For the CDF, see
:mod:`~skg.gauss_cdf`.
.. todo::
Add proper handling of colinear inputs (and other singular matrix cases).
.. todo::
Add tests.
.. todo::
Add nan_policy argument.
.. todo::
Add axis parameter. Figure out how to do it properly.
.. todo::
Add PEP8 check to formal tests.
.. todo::
Include amplitude in integrals.
.. todo::
Allow broadcasting of x and y, not necessarily identical size
"""
from numpy import array, cumsum, diff, empty, exp, pi, sqrt
from scipy.linalg import lstsq
from .util import preprocess_pair
__all__ = ['gauss_pdf_fit']
[docs]def gauss_pdf_fit(x, y, sorted=True):
r"""
Gaussian PDF fit of the form
:math:`\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}`.
This implementation is based on the approximate solution to integral
equation :eq:`gauss-pdf-eq`, presented in :ref:`ref-reei`.
Parameters
----------
x : array-like
The x-values of the data points. The fit will be performed on a
raveled version of this array.
y : array-like
The y-values of the data points corresponding to `x`. Must be
the same size as `x`. The fit will be performed on a raveled
version of this array.
sorted : bool
Set to True if `x` is already monotonically increasing or
decreasing. If False, `x` will be sorted into increasing order,
and `y` will be sorted along with it.
Return
------
mu, sigma : ~numpy.ndarray
A two-element array containing the estimated mean and standard
deviation, in that order.
References
----------
- [Jacquelin]_ "\ :ref:`ref-reei`\ ", :ref:`pp. 6-8. <reei1-sec3>`
"""
x, y = preprocess_pair(x, y, sorted)
d = 0.5 * diff(x)
xy = x * y
M = empty(xy.shape + (2,), dtype=xy.dtype)
M[0, :] = 0
M[1:, 0] = cumsum((y[1:] + y[:-1]) * d)
M[1:, 1] = cumsum((xy[1:] + xy[:-1]) * d)
Y = y - y[0]
(A, B), *_ = lstsq(M, Y, overwrite_a=True, overwrite_b=True)
out = array([-A / B, sqrt(-1.0 / B)])
return out
[docs]def model(x, mu, sigma):
r"""
Compute :math:`y = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}`.
Parameters
----------
x : array-like
The value of the model will be the same shape as the input.
mu : float
The mean.
sigma : float
The standard deviation.
Return
------
y : array-like
An array of the same shape as `x`, containing the model
computed for the given parameters.
"""
return exp(-0.5 * ((x - mu) / sigma)**2) / (sigma * sqrt(2 * pi))
gauss_pdf_fit.model = model