[med-svn] [python-mne] 187/376: fixing manual

Fri Nov 27 17:22:34 UTC 2015

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yoh pushed a commit to annotated tag v0.1
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commit 95f5699035a914ab9f5e9672f1fe56ca76c9bf7b
Author: Emily Ruzich <emilyr at nmr.mgh.harvard.edu>
Date:   Fri Apr 8 09:57:27 2011 -0400

    fixing manual
---
 doc/source/manual/analyze.rst | 31 +++++++++++++++++--------------
 doc/source/manual/morph.rst   | 18 +++++++++---------
 2 files changed, 26 insertions(+), 23 deletions(-)

diff --git a/doc/source/manual/analyze.rst b/doc/source/manual/analyze.rst
index 82e14d5..98c171b 100755
--- a/doc/source/manual/analyze.rst
+++ b/doc/source/manual/analyze.rst
@@ -1405,7 +1405,7 @@ signal at channel INLINE_EQUATION. This signal
 is related to the primary current distribution INLINE_EQUATIONthrough
 the lead field INLINE_EQUATION:
 
-.. math::    1 + 1 = 2
+.. math::    x_k = \int_G {L_k(r) \cdot J^p(r)}\,dG\ ,
 
 where the integration space INLINE_EQUATION in
 our case is a spherical surface. The oblique boldface characters
@@ -1413,48 +1413,51 @@ denote three-component locations vectors and vector fields.
 
 The inner product of two leadfields is defined as:
 
-.. math::    1 + 1 = 2
+.. math::    \langle L_j \mid L_k \rangle = \int_G {L_j(r) \cdot L_k(r)}\,dG\ ,
 
 These products constitute the Gram matrix INLINE_EQUATION.
 The minimum -norm estimate can be expressed as a weighted sum of
 the lead fields:
 
-.. math::    1 + 1 = 2
+.. math::    J^* = w^T L\ ,
 
 where INLINE_EQUATION is a weight vector
 and INLINE_EQUATION is a vector composed of the
 continuous lead-field functions. The weights are determined by the
 requirement
 
-.. math::    1 + 1 = 2
+.. math::    x = \langle L \mid J^* \rangle = \Gamma w\ ,
 
 i.e., the estimate must predict the measured signals. Hence,
 
-.. math::    1 + 1 = 2
+.. math::    w = \Gamma^{-1} x\ .
 
 However, the Gram matrix is ill conditioned and regularization
 must be employed to yield a stable solution. With help of the SVD
 
-.. math::    1 + 1 = 2
+.. math::    \Gamma = U \Lambda V^T
 
 a regularized minimum-norm can now found by replacing the
 data matching condition by
 
-.. math::    1 + 1 = 2
+.. math::    x^{(p)} = \Gamma^{(p)} w^{(p)}\ ,
 
 where
 
-.. math::    1 + 1 = 2
+.. math::    x^{(p)} = (U^{(p)})^T x \text{  and  } \Gamma^{(p)} = (U^{(p)})^T \Gamma\ ,
 
 respectively. In the above, the columns of INLINE_EQUATION are
 the first *k* left singular vectors of INLINE_EQUATION.
 The weights of the regularized estimate are
 
-.. math::    1 + 1 = 2
+.. math::    w^{(p)} = V \Lambda^{(p)} U^T x\ ,
 
 where INLINE_EQUATION is diagonal with
 
-.. math::    1 + 1 = 2
+.. math::    \Lambda_{jj}^{(p)} = \Bigg\{ \begin{array}{l}
+		 1/{\lambda_j},j \leq p\\
+		 \text{otherwise}
+	     \end{array}
 
 INLINE_EQUATION being the INLINE_EQUATION singular
 value of INLINE_EQUATION. The truncation point INLINE_EQUATION is
@@ -1462,19 +1465,19 @@ selected in mne_analyze by specifying
 a tolerance INLINE_EQUATION, which is used to
 determine INLINE_EQUATION such that
 
-.. math::    1 + 1 = 2
+.. math::    1 - \frac{\sum_{j = 1}^p {\lambda_j}}{\sum_{j = 1}^N {\lambda_j}} < \varepsilon
 
 The extrapolated and interpolated magnetic field or potential
 distribution estimates INLINE_EQUATION in a virtual
 grid of sensors can be now easily computed from the regularized
 minimum-norm estimate. With
 
-.. math::    1 + 1 = 2
+.. math::    \Gamma_{jk}' = \langle L_j' \mid L_k \rangle\ ,
 
 where INLINE_EQUATION are the lead fields
 of the virtual sensors,
 
-.. math::    1 + 1 = 2
+.. math::    \hat{x'} = \Gamma' w^{(k)}\ .
 
 Field mapping preferences
 =========================
@@ -1678,7 +1681,7 @@ data in green.
 The SNR estimate is computed from the whitened data INLINE_EQUATION,
 related to the measured data INLINE_EQUATION by
 
-.. math::    1 + 1 = 2
+.. math::    \tilde{x}(t) = C^{-^1/_2} x(t)\ ,
 
 where INLINE_EQUATION is the whitening
 operator, introduced in :ref:`CHDDHAGE`.
diff --git a/doc/source/manual/morph.rst b/doc/source/manual/morph.rst
index 2ce04a8..cc67b2e 100755
--- a/doc/source/manual/morph.rst
+++ b/doc/source/manual/morph.rst
@@ -33,7 +33,7 @@ A morphing map is a linear mapping froprem cortical surface values
 in subject A (INLINE_EQUATION) to those in another
 subject B (INLINE_EQUATION)
 
-.. math::    1 + 1 = 2
+.. math::    x^{(B)} = M^{(AB)} x^{(A)}\ ,
 
 where INLINE_EQUATION is a sparse matrix
 with at most three nonzero elements on each row. These elements
@@ -47,15 +47,15 @@ the location INLINE_EQUATION within the triangle INLINE_EQUATION.
 
 It follows from the above definition that in general
 
-.. math::    1 + 1 = 2
+.. math::    M^{(AB)} \neq (M^{(BA)})^{-1}\ ,
 
 *i.e.*,
 
-.. math::    1 + 1 = 2
+.. math::    x_{(A)} \neq M^{(BA)} M^{(AB)} x^{(A)}\ ,
 
 even if
 
-.. math::    1 + 1 = 2
+.. math::    x^{(A)} \approx M^{(BA)} M^{(AB)} x^{(A)}\ ,
 
 *i.e.*, the mapping is *almost* a
 bijection.
@@ -79,7 +79,7 @@ iterative procedure, which produces a blurred image INLINE_EQUATIONfrom
 the original sparse image INLINE_EQUATION by applying
 in each iteration step a sparse blurring matrix:
 
-.. math::    1 + 1 = 2
+.. math::    x^{(p)} = S^{(p)} x^{(p - 1)}\ .
 
 On each row INLINE_EQUATIONof the matrix INLINE_EQUATIONthere
 are INLINE_EQUATION nonzero entries whose values
@@ -96,7 +96,7 @@ the topology of the triangulation are fixed the matrices INLINE_EQUATION are
 fixed and independent of the data. Therefore, it would be in principle
 possible to construct a composite blurring matrix
 
-.. math::    1 + 1 = 2
+.. math::    S^{(N)} = \prod_{p = 1}^N {S^{(p)}}\ .
 
 However, it turns out to be computationally more effective
 to do blurring with an iteration. The above formula for INLINE_EQUATION also
@@ -387,15 +387,15 @@ the rows are the signals at different vertices of the cortical surface.
 The average computed by mne_average_estimates is
 then:
 
-.. math::    1 + 1 = 2
+.. math::    A_{jk} = |w[\newcommand\sgn{\mathop{\mathrm{sgn}}\nolimits}\sgn(B_{jk})]^{\alpha}|B_{jk}|^{\beta}
 
 with
 
-.. math::    1 + 1 = 2
+.. math::    B_{jk} = \sum_{p = 1}^p {\bar{w_p}[\newcommand\sgn{\mathop{\mathrm{sgn}}\nolimits}\sgn(S_{jk}^{(p)})^{\alpha_p}|S_{jk}^{(p)}|^{\beta_p}}
 
 and
 
-.. math::    1 + 1 = 2
+.. math::    \bar{w_p} = w_p / \sum_{p = 1}^p {|w_p|}\ .
 
 In the above, INLINE_EQUATION and INLINE_EQUATION are
 the powers and weights assigned to each of the subjects whereas INLINE_EQUATION and INLINE_EQUATION are

-- 
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