[med-svn] [python-mne] 185/376: fixing manual
Yaroslav Halchenko
debian at onerussian.com
Fri Nov 27 17:22:33 UTC 2015
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yoh pushed a commit to annotated tag v0.1
in repository python-mne.
commit f9ce0df1264de08f71335080da3ac5483cf9be17
Author: Emily Ruzich <emilyr at nmr.mgh.harvard.edu>
Date: Wed Apr 6 11:41:49 2011 -0400
fixing manual
---
doc/source/manual/browse.rst | 2 +-
doc/source/manual/cookbook.rst | 15 +++++------
doc/source/manual/forward.rst | 59 +++++++++++++++++++++++++++++++++---------
doc/source/manual/mne.rst | 44 +++++++++++++++----------------
4 files changed, 76 insertions(+), 44 deletions(-)
diff --git a/doc/source/manual/browse.rst b/doc/source/manual/browse.rst
index 7a1a4e2..2eee0df 100755
--- a/doc/source/manual/browse.rst
+++ b/doc/source/manual/browse.rst
@@ -2515,7 +2515,7 @@ epoch.
Let the vectors
-.. math:: s_{rpj}\ ,\ p = 1\ ...\ P_r\ ,\ j = 1\ ...\ N_r\ ,\ r = 1\ ...\ R
+.. math:: s_{rpj}\ ;\ p = 1 \dotsc P_r\ ;\ j = 1 \dotsc N_r\ ;\ r = 1 \dotsc R
be the samples from all channels in the baseline corrected epochs
used to calculate the covariance matrix. In the above, INLINE_EQUATION is
diff --git a/doc/source/manual/cookbook.rst b/doc/source/manual/cookbook.rst
index 2fdad48..8164b5a 100755
--- a/doc/source/manual/cookbook.rst
+++ b/doc/source/manual/cookbook.rst
@@ -815,16 +815,13 @@ anatomy only, not on the MEG/EEG data to be analyzed.
.. note:: The MEG head to MRI transformation matrix specified with the ``--trans`` option should be a text file containing a 4-by-4 matrix:
-.. math:: \[
- T=
- \begin{matrix}
- R_{11} & R_{12} & R_{13} x_{0} \\
- R_{13} & R_{13} & R_{13} y_{0} \\
- R_{13} & R_{13} & R_{13} z_{0} \\
+.. math:: T = \begin{bmatrix}
+ R_{11} & R_{12} & R_{13} & x_0 \\
+ R_{13} & R_{13} & R_{13} & y_0 \\
+ R_{13} & R_{13} & R_{13} & z_0 \\
0 & 0 & 0 & 1
- \end{matrix}
- \]
-
+ \end{bmatrix}
+
defined so that if the augmented location vectors in MRI
head and MRI coordinate systems are denoted by :math:`r_{head}[x_{head}\ y_{head}\ z_{head}\ 1]` and :math:`r_{MRI}[x_{MRI}\ y_{MRI}\ z_{MRI}\ 1]`,
respectively,
diff --git a/doc/source/manual/forward.rst b/doc/source/manual/forward.rst
index fcb6640..20f7e12 100755
--- a/doc/source/manual/forward.rst
+++ b/doc/source/manual/forward.rst
@@ -122,7 +122,27 @@ coordinate transformation symbols (INLINE_EQUATION)
indicate the transformations actually present in the FreeSurfer
files. Generally,
-.. math:: 1 + 1 = 2
+.. math:: \begin{bmatrix}
+ x_2 \\
+ y_2 \\
+ z_2 \\
+ 1
+ \end{bmatrix} = T_{12} \begin{bmatrix}
+ x_1 \\
+ y_1 \\
+ z_1 \\
+ 1
+ \end{bmatrix} = \begin{bmatrix}
+ R_{11} & R_{12} & R_{13} & x_0 \\
+ R_{13} & R_{13} & R_{13} & y_0 \\
+ R_{13} & R_{13} & R_{13} & z_0 \\
+ 0 & 0 & 0 & 1
+ \end{bmatrix} \begin{bmatrix}
+ x_1 \\
+ y_1 \\
+ z_1 \\
+ 1
+ \end{bmatrix}\ ,
where INLINE_EQUATION are the location
coordinates in two coordinate systems, INLINE_EQUATION is
@@ -134,11 +154,21 @@ matrix relating the two coordinate systems. The coordinate transformations
are present in different files produced by FreeSurfer and MNE as
summarized in :ref:`CHDJDEDJ`. The fixed transformations INLINE_EQUATION and INLINE_EQUATION are:
-.. math:: 1 + 1 = 2
+.. math:: T_{-} = \begin{bmatrix}
+ 0.99 & 0 & 0 & 0 \\
+ 0 & 0.9688 & 0.042 & 0 \\
+ 0 & -0.0485 & 0.839 & 0 \\
+ 0 & 0 & 0 & 1
+ \end{bmatrix}
and
-.. math:: 1 + 1 = 2
+.. math:: T_{+} = \begin{bmatrix}
+ 0.99 & 0 & 0 & 0 \\
+ 0 & 0.9688 & 0.046 & 0 \\
+ 0 & -0.0485 & 0.9189 & 0 \\
+ 0 & 0 & 0 & 1
+ \end{bmatrix}
.. note:: This section does not discuss the transformation between the MRI voxel indices and the different MRI coordinates. However, it is important to note that in FreeSurfer, MNE, as well as in Neuromag software an integer voxel coordinate corresponds to the location of the center of a voxel. Detailed information on the FreeSurfer MRI systems can be found at https://surfer.nmr.mgh.harvard.edu/fswiki/CoordinateSystems.
@@ -638,11 +668,16 @@ a location of a point INLINE_EQUATION in sensor coordinates
is transformed to device coordinates (INLINE_EQUATION)
by
-.. math:: 1 + 1 = 2
+.. math:: [r_D 1] = [r_c 1] T_{CD}\ ,
where
-.. math:: 1 + 1 = 2
+.. math:: T = \begin{bmatrix}
+ e_x & 0 \\
+ e_y & 0 \\
+ e_z & 0 \\
+ r_{0D} & 1
+ \end{bmatrix}\ .
Calculation of the magnetic field
=================================
@@ -661,7 +696,7 @@ field component normal to the coil plane, the output of the *k*th
MEG channel, INLINE_EQUATION, can be approximated
by:
-.. math:: 1 + 1 = 2
+.. math:: b_k = \sum_{p = 1}^{N_k} {w_{kp} B(r_{kp}) \cdot n_{kp}}
where INLINE_EQUATION are a set of INLINE_EQUATION integration
points covering the pickup coil loops of the sensor, INLINE_EQUATION is
@@ -1158,7 +1193,7 @@ al.* and references therein). mne_forward_solution approximates
the solution with three dipoles in a homogeneous sphere whose locations
and amplitudes are determined by minimizing the cost function:
-.. math:: 1 + 1 = 2
+.. math:: S(r_1,\dotsc,r_m\ ,\ \mu_1,\dotsc,\mu_m) = \int_{scalp} {(V_{true} - V_{approx})}\,dS
where INLINE_EQUATION and INLINE_EQUATION are
the locations and amplitudes of the approximating dipoles and INLINE_EQUATION and INLINE_EQUATION are
@@ -1174,26 +1209,26 @@ inner skull surface.
Field derivatives
=================
-If the --grad option is specified, mne_forward_solution includes
+If the --grad **problem - double dash shows up in html as single long dash..!** option is specified, mne_forward_solution includes
the derivatives of the forward solution with respect to the dipole
location coordinates to the output file. Let
-.. math:: 1 + 1 = 2
+.. math:: G_k = [g_{xk} g_{yk} g_{zk}]
be the INLINE_EQUATION matrix containing
the signals produced by three orthogonal dipoles at location INLINE_EQUATION making
up INLINE_EQUATIONthe gain matrix
-.. math:: 1 + 1 = 2
+.. math:: G = [G_1 \dotso G_{N_{source}}]\ .
With the --grad option, the output from mne_forward_solution also
contains the INLINE_EQUATION derivative matrix
-.. math:: 1 + 1 = 2
+.. math:: D = [D_1 \dotso D_{N_{source}}]\ ,
where
-.. math:: 1 + 1 = 2
+.. math:: D_k = [\frac{\delta g_{xk}}{\delta x_k} \frac{\delta g_{xk}}{\delta y_k} \frac{\delta g_{xk}}{\delta z_k} \frac{\delta g_{yk}}{\delta x_k} \frac{\delta g_{yk}}{\delta y_k} \frac{\delta g_{yk}}{\delta z_k} \frac{\delta g_{zk}}{\delta x_k} \frac{\delta g_{zk}}{\delta y_k} \frac{\delta g_{zk}}{\delta z_k}]\ ,
where INLINE_EQUATION are the location
coordinates of the INLINE_EQUATION dipole. If
diff --git a/doc/source/manual/mne.rst b/doc/source/manual/mne.rst
index b414492..64be025 100755
--- a/doc/source/manual/mne.rst
+++ b/doc/source/manual/mne.rst
@@ -47,7 +47,7 @@ fixed-orientation sources and M = 3P if the source orientations
are unconstrained. The regularized linear inverse operator following
from the Bayesian approach is given by the INLINE_EQUATION matrix
-.. math:: 1 + 1 = 2
+.. math:: M = R' G^T (G R' G^T + C)^{-1}\ ,
where G is the gain matrix relating the source strengths
to the measured MEG/EEG data, C is the data noise-covariance matrix
@@ -70,7 +70,7 @@ The a priori variance of the currents is, in practise, unknown.
We can express this by writing INLINE_EQUATION,
which yields the inverse operator
-.. math:: 1 + 1 = 2
+.. math:: M = R G^T (G R G^T + \lambda^2 C)^{-1}\ ,
where the unknown current amplitude is now interpreted in
terms of the regularization parameter INLINE_EQUATION.
@@ -82,7 +82,7 @@ estimate is obtained.
We can arrive in the regularized linear inverse operator
also by minimizing the cost function
-.. math:: 1 + 1 = 2
+.. math:: S = \tilde{e}^T \tilde{e} + \lambda^2 j^T R^{-1} j\ ,
where the first term consists of the difference between the
whitened measured data (see :ref:`CHDDHAGE`) and those predicted
@@ -99,7 +99,7 @@ Whitening and scaling
The MNE software employs data whitening so that a 'whitened' inverse operator
assumes the form
-.. math:: 1 + 1 = 2
+.. math:: \tilde{M} = R \tilde{G}^T (\tilde{G} R \tilde{G}^T + I)^{-1}\ ,
where INLINE_EQUATION is the spatially
whitened gain matrix. The expected current values are INLINE_EQUATION,
@@ -152,7 +152,7 @@ the noise-covariance matrix is advisable.
The MNE software accomplishes the regularization by replacing
a noise-covariance matrix estimate INLINE_EQUATION with
-.. math:: 1 + 1 = 2
+.. math:: C' = C + \sum_k {\varepsilon_k \bar{\sigma_k}^2 I^{(k)}}\ ,
where the index INLINE_EQUATION goes across
the different channel groups (MEG planar gradiometers, MEG axial
@@ -179,7 +179,7 @@ directly. However, for computational convenience we prefer to take
another route, which employs the singular-value decomposition (SVD)
of the matrix
-.. math:: 1 + 1 = 2
+.. math:: A = \tilde{G} R^{^1/_2} = U \Lambda V^T
where the superscript **INLINE_EQUATION indicates a
square root of INLINE_EQUATION. For a diagonal matrix,
@@ -189,16 +189,16 @@ thus INLINE_EQUATION.
With the above SVD it is easy to show that
-.. math:: 1 + 1 = 2
+.. math:: \tilde{M} = R^{^1/_2} V \Gamma U^T
where the elements of the diagonal matrix INLINE_EQUATION are
-.. math:: 1 + 1 = 2
+.. math:: \gamma_k = \frac{1}{\lambda_k} \frac{\lambda_k^2}{\lambda_k^2 \lambda^2}\ .
With INLINE_EQUATION the expression for
the expected current is
-.. math:: 1 + 1 = 2
+.. math:: \hat{j}(t) = R^C V \Gamma w(t) = \sum_k {\bar{v_k} \gamma_k w_k(t)}\ ,
where INLINE_EQUATION, INLINE_EQUATION being
the kth column of V. It is thus seen that the current estimate is
@@ -237,12 +237,12 @@ variance. Noise normalization serves three purposes:
In practice, noise normalization requires the computation
of the diagonal elements of the matrix
-.. math:: 1 + 1 = 2
+.. math:: M C M^T = \tilde{M} \tilde{M}^T\ .
With help of the singular-value decomposition approach we
see directly that
-.. math:: 1 + 1 = 2
+.. math:: \tilde{M} \tilde{M}^T\ = \bar{V} \Gamma^2 \bar{V}^T\ .
Under the conditions expressed at the end of :ref:`CHDBEHBC`, it follows that the t-statistic values associated
with fixed-orientation sources) are thus proportional to INLINE_EQUATION while
@@ -269,7 +269,7 @@ the regularization applied.
In the SVD approach we easily find
-.. math:: 1 + 1 = 2
+.. math:: \hat{x}(t) = G \hat{j}(t) = C^{^1/_2} U \Pi w(t)\ ,
where the diagonal matrix INLINE_EQUATION has
elements INLINE_EQUATION The predicted data is
@@ -350,7 +350,7 @@ weighting scheme employed in MNE analyze, the elements of R corresponding
to the INLINE_EQUATION source location are be
scaled by a factor
-.. math:: 1 + 1 = 2
+.. math:: f_p = (g_{1p}^T g_{1p} + g_{2p}^T g_{2p} + g_{3p}^T g_{3p})^{-\gamma}\ ,
where INLINE_EQUATION are the three colums
of INLINE_EQUATION corresponding to source location INLINE_EQUATION and INLINE_EQUATION is
@@ -387,47 +387,47 @@ is originally one corresponding to raw data. Therefore, it has to
be scaled correctly to correspond to the actual or effective number
of epochs in the condition to be analyzed. In general, we have
-.. math:: 1 + 1 = 2
+.. math:: C = C_0 / L_{eff}
where INLINE_EQUATION is the effective
number of averages. To calculate INLINE_EQUATION for
an arbitrary linear combination of conditions
-.. math:: 1 + 1 = 2
+.. math:: y(t) = \sum_{i = 1}^n {w_i x_i(t)}
we make use of the the fact that the noise-covariance matrix
-.. math:: 1 + 1 = 2
+.. math:: C_y = \sum_{i = 1}^n {w_i^2 C_{x_i}} = C_0 \sum_{i = 1}^n {w_i^2 / L_i}
which leads to
-.. math:: 1 + 1 = 2
+.. math:: 1 / L_{eff} = \sum_{i = 1}^n {w_i^2 / L_i}
An important special case of the above is a weighted average,
where
-.. math:: 1 + 1 = 2
+.. math:: w_i = L_i / \sum_{i = 1}^n {L_i}
and, therefore
-.. math:: 1 + 1 = 2
+.. math:: L_{eff} = \sum_{i = 1}^n {L_i}
Instead of a weighted average, one often computes a weighted
sum, a simplest case being a difference or sum of two categories.
For a difference INLINE_EQUATION and INLINE_EQUATION and
thus
-.. math:: 1 + 1 = 2
+.. math:: 1 / L_{eff} = 1 / L_1 + 1 / L_2
or
-.. math:: 1 + 1 = 2
+.. math:: L_{eff} = \frac{L_1 L_2}{L_1 + L_2}
Interestingly, the same holds for a sum, where INLINE_EQUATION.
Generalizing, for any combination of sums and differences, where INLINE_EQUATION or INLINE_EQUATION , INLINE_EQUATION,
we have
-.. math:: 1 + 1 = 2
+.. math:: 1 / L_{eff} = \sum_{i = 1}^n {1/{L_i}}
.. _CBBDDBGF:
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