From 4d5d49aa4c59ae8d8a4b769992e4c1602ee68bf1 Mon Sep 17 00:00:00 2001
From: "Olivier J.N. Bertrand" <olivier.bertrand@uni-bielefeld.de>
Date: Wed, 20 Jun 2018 19:40:55 +0200
Subject: [PATCH] Adapt error prop doc in tutorials ipython notebook
---
.../error_propagation/error_propagation.rst | 1 -
.../examples/error_propagation_univariate.py | 37 -----
doc/source/tutorials/index.rst | 4 +-
navipy/errorprop/__init__.py | 150 ++----------------
4 files changed, 12 insertions(+), 180 deletions(-)
delete mode 100644 doc/source/error_propagation/error_propagation.rst
delete mode 100644 doc/source/error_propagation/examples/error_propagation_univariate.py
diff --git a/doc/source/error_propagation/error_propagation.rst b/doc/source/error_propagation/error_propagation.rst
deleted file mode 100644
index c4bc28a..0000000
--- a/doc/source/error_propagation/error_propagation.rst
+++ /dev/null
@@ -1 +0,0 @@
-.. automodule:: bodorientpy.errorprop
diff --git a/doc/source/error_propagation/examples/error_propagation_univariate.py b/doc/source/error_propagation/examples/error_propagation_univariate.py
deleted file mode 100644
index 37b7329..0000000
--- a/doc/source/error_propagation/examples/error_propagation_univariate.py
+++ /dev/null
@@ -1,37 +0,0 @@
-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()
diff --git a/doc/source/tutorials/index.rst b/doc/source/tutorials/index.rst
index 2cb79fe..bcf445d 100644
--- a/doc/source/tutorials/index.rst
+++ b/doc/source/tutorials/index.rst
@@ -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
--------------------------------
diff --git a/navipy/errorprop/__init__.py b/navipy/errorprop/__init__.py
index 06af5bb..6e87962 100644
--- a/navipy/errorprop/__init__.py
+++ b/navipy/errorprop/__init__.py
@@ -1,140 +1,8 @@
"""
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
--
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