Skip to content
Snippets Groups Projects
Commit 4d5d49aa authored by Olivier Bertrand's avatar Olivier Bertrand
Browse files

Adapt error prop doc in tutorials ipython notebook

parent a5dced31
No related branches found
No related tags found
No related merge requests found
Hide whitespace changes
Inline Side-by-side
doc/source/error_propagation/error_propagation.rst deleted 100644 → 0
+ 0
− 1
View file @ a5dced31
.. automodule:: bodorientpy.errorprop
doc/source/error_propagation/examples/error_propagation_univariate.py deleted 100644 → 0
+ 0
− 37
View file @ a5dced31
import numpy as np
import matplotlib.pyplot as plt
from bodorientpy.errorprop import propagate_error
def plot_error_f(ax, func_f, x, delta_x, delta_y, x0):
y0 = func_f(x0)
xdelta = np.linspace(x0-delta_x, x0+delta_x, len(x))
ydelta = np.linspace(y0-delta_y, y0+delta_y, len(x))
y = func_f(x)
ax.fill_between(xdelta, min(y), max(y), facecolor='k',
alpha=0.3, edgecolor='None')
ax.fill_between(x, min(ydelta), max(ydelta),
facecolor='r', alpha=0.5, edgecolor='None')
ax.plot(x, y, 'k', linewidth=2)
ax.set_xlim([min(x), max(x)])
ax.set_ylim([min(y), max(y)])
ax.set_xlabel('x')
ax.set_ylabel('y=f(x)')
def func_f(x):
return (x+1)*(x-2)*(x+3)
delta_x = 0.5
x = np.linspace(-5, 4.5, 100)
fig, axarr = plt.subplots(1, 2, figsize=(15, 5))
x0 = -1
delta_y = propagate_error(func_f, x0, delta_x)
plot_error_f(axarr[0], func_f, x, delta_x, delta_y, x0=x0)
x0 = 3
delta_y = propagate_error(func_f, x0, delta_x)
plot_error_f(axarr[1], func_f, x, delta_x, delta_y, x0=x0)
fig.show()
doc/source/tutorials/index.rst
+ 3
− 1
View file @ 4d5d49aa
......@@ -2,10 +2,12 @@ Tutorials
=========
.. toctree::
:maxdepth: 1
01-building-arena.ipynb
02-recording-animal-trajectory.ipynb
03-rendering-along-trajectory.ipynb
04-error-propagation.ipynb
Average place-code vector homing
--------------------------------
......
navipy/errorprop/__init__.py
+ 9
− 141
View file @ 4d5d49aa
"""
Numerical error propagation
===========================
Sources of observational errors
-------------------------------
Observational error (or measurement error) is the difference between \
a measured value of quantity and its true value. An error is not \
necessarly a "mistake" (e.g. in Statistics). Variability is an \
inherent part of things being measured and of the measurement process.
Measurement errors can be divided into two components:
* random error
* systematic error (bias).
Types of bias (non exhaustiv):
* *Selection bias* (Berksonian bias) involves individuals being more \
likely to be selected for study than others.
* *The bias of an estimator* is the difference between an estimator's \
expectations and the true value of the parameter being estimated.
* *Omitted-variable bias* is the bias that appears in estimates of \
parameters in a regression analysis when the assumed specification \
omits an independent variable that should be in the model.
* *Detection bias* occurs when a phenomenon is more likely to be \
observed for a particular set of study subjects.
* *Funding bias* may lead to selection of outcomes, test samples, \
or test procedures that favor a study's financial sponsor.
* *Reporting bias* involves a skew in the availability of data, such \
that observations of a certain kind are more likely to be reported.
* *Analytical bias* arise due to the way that the results are evaluated.
* *Exclusion bias* arise due to the systematic exclusion of certain \
individuals from the study.
* *Observer bias* arises when the researcher unconsciously influences \
the experiment due to cognitive bias where judgement may alter how an \
experiment is carried out / how results are recorded.
Propagation of uncertainty
--------------------------
In statistics, propagation of uncertainty (or propagation of error) \
is the effect of variables' uncertainties (or errors) on the \
uncertainty of a function based on them. When the variables are \
the values of experimental measurements they have uncertainties \
due to measurement limitations (e.g., instrument precision) which \
propagate to the combination of variables in the function.
Non-linear function of one variable
-----------------------------------
Let's be :math:`f(x)` a non linear function of variables :math:`x`, and \
:math:`\Delta x` the measurement error of :math:`x`.
We see that the measurement error of $x$ (gray) propagate on $y$ \
(red), via $f(x)$ (black line). The propagation depends on the \
measured value $x_{o}$, the slope of $f(x)$, the curvature of \
$f(x)$, ... , i.e. the derivatives of $f(x)$ relativ to $x$.
The propagated error $\Delta y$ can therefore be express as a \
Taylor-expansion of $f(x)$ around $x_0$ the measured value, i.e. \
calculated the value $f(x_{o}+\Delta x)$ for $\Delta x \approx 0$
.. math::
f(x_{o}+\Delta x) = \sum_{n=0}^{\inf} \
\frac{f^{(n)}(x_{o})}{n!}(\Delta x)^{n}.
Using only the first-order of the equation:
.. math::
f(x_{o}+\Delta x) \approx f(x_{o})+\frac{df}{dx}(x_{o})\Delta x
The propagated error is defined has \
$\Delta y=\left|f(x_{o}+\Delta x)-f(x_{o})\right|$. From the \
equation above we find that \
$\Delta y=\left|\frac{df}{dx}(x_{o})\right|\Delta x $
Instead of deriving the equation, one can approximate the derivitive \
of the function numerically. For example, to propagate the error on \
the function :math:`f(x)=(x+1)(x-2)(x+3)`, one need to \
define the function, the error :math:`\delta_x`, and the point of \
interest :math:`x_0`.
.. literalinclude:: examples/error_propagation_univariate.py
:lines: 23-27, 31
The the error can be calculated as follow:
.. literalinclude:: examples/error_propagation_univariate.py
:lines: 32
In the following figure, the of error on x has been propagated at two \
different locations:
.. plot:: error_propagation/examples/error_propagation_univariate.py
Non-linear function of multiple variables
-----------------------------------------
We can extend the framework developped above to function $f$ of \
multiple variables, $x_0,x_1,...x_i,...x_n$. When f is a set of \
non-linear combination of the variables x, an interval propagation \
could be performed in order to compute intervals which contain all \
consistent values for the variables. In a probabilistic approach, \
the function f must usually be linearized by approximation to a \
first-order Taylor series expansion, though in some cases, exact \
formulas can be derived that do not depend on the expansion as is \
the case for the exact variance of products. The Taylor expansion \
would be:
.. math::
f_k \approx f^0_k+ \sum_i^n \frac{\partial f_k}{\partial {x_i}} x_i
where \partial f_k/\partial x_i denotes the partial derivative of fk \
with respect to the i-th variable, evaluated at the mean value of \
all components of vector x. Or in matrix notation,
.. math::
\mathrm{f} \approx \mathrm{f}^0 + \mathrm{J} \mathrm{x}\
where J is the Jacobian matrix. Since f_0 is a constant it does not \
contribute to the error on f. Therefore, the propagation of error \
follows the linear case, above, but replacing the linear \
coefficients, Aik and Ajk by the partial derivatives, \
$\frac{\partial f_k}{\partial x_i}$ and \
$\frac{\partial f_k}{\partial x_j}$. In matrix notation,
.. math::
\mathrm{\Sigma}^\mathrm{f} = \mathrm{J}
\mathrm{\Sigma}^\mathrm{x} \mathrm{J}^\top
That is, the Jacobian of the function is used to transform the rows \
and columns of the variance-covariance matrix of the argument.
tools to propogate the error by using an estimate
of the jacobian matrix (matrix of derivatives)
"""
import numpy as np
......@@ -152,7 +20,7 @@ distance between two markers
markers.loc[:, rigid_marker_id[1]])**2).sum(
axis=1, skipna=False))
median_dist = np.nanmedian(dist)
return np.abs(dist-median_dist)
return np.abs(dist - median_dist)
def estimate_jacobian(fun, x, args=None, epsilon=1e-6):
......@@ -184,16 +52,16 @@ fixed parameters needed to completely specify the function.
dfs2 = fun(x_plus, args)
dfs1 = np.array(dfs1)
dfs2 = np.array(dfs2)
deriv = (dfs2-dfs1)/(2*epsilon)
deriv = (dfs2 - dfs1) / (2 * epsilon)
jacobian_matrix.append(deriv)
return np.array(jacobian_matrix).transpose()
else:
x_minus = x-epsilon
x_plus = x+epsilon
x_minus = x - epsilon
x_plus = x + epsilon
if args is None:
deriv = (fun(x_plus) - fun(x_minus))/(2*epsilon)
deriv = (fun(x_plus) - fun(x_minus)) / (2 * epsilon)
else:
deriv = (fun(x_plus, args) - fun(x_minus, args))/(2*epsilon)
deriv = (fun(x_plus, args) - fun(x_minus, args)) / (2 * epsilon)
return deriv
......@@ -215,4 +83,4 @@ fixed parameters needed to completely specify the function.
if isinstance(x, (list, tuple, np.ndarray)):
return jacobian_matrix.dot(covar.dot(jacobian_matrix.transpose()))
else:
return np.abs(jacobian_matrix)*covar
return np.abs(jacobian_matrix) * covar
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Please register or to comment