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commit 95f5699035a914ab9f5e9672f1fe56ca76c9bf7b Author: Emily Ruzich <[email protected]> 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 -- Alioth's /usr/local/bin/git-commit-notice on /srv/git.debian.org/git/debian-med/python-mne.git _______________________________________________ debian-med-commit mailing list [email protected] http://lists.alioth.debian.org/cgi-bin/mailman/listinfo/debian-med-commit
