notes: ML estimation of a model for Trial-to-Trial variability of MEG/EEG data

<div style="clear:both;"></div>Recently—well, may be not that recent—de Munck et al.
(2004)<A CLASS="sdendnoteanc" NAME="sdendnote1anc" HREF="#sdendnote1sym">i</A>
proposed the following model for MEG data
<IMG SRC="blog-deMunck04_html_2b4b6014.gif" NAME="Object1" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=246 HEIGHT=21>
where 
<IMG SRC="blog-deMunck04_html_1afd9c75.gif" NAME="Object2" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=131 HEIGHT=21>,
<IMG SRC="blog-deMunck04_html_m1bcbd9bd.gif" NAME="Object3" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=27 HEIGHT=21>
are the measurements at sensor i, at time j, in trial
k, i = 1, ..., I, j = 1, ..., J, k = 1, ..., K,
and the quantities 
<IMG SRC="blog-deMunck04_html_m5f3d149c.gif" NAME="Object5" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=100 HEIGHT=21>,
and 
<IMG SRC="blog-deMunck04_html_m1f577dcf.gif" NAME="Object4" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=163 HEIGHT=21>,
are unknown and of interest. The proposed to estimate this
model using a maximum likelihood procedure. Before specifying the
likelihood, let us first define a couple of new matrices:
<IMG SRC="blog-deMunck04_html_m4eee7d8c.gif" NAME="Object7" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=185 HEIGHT=21>
<IMG SRC="blog-deMunck04_html_719a0936.gif" NAME="Object30" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=180 HEIGHT=21>,
 
<IMG SRC="blog-deMunck04_html_544513b5.gif" NAME="Object29" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=75 HEIGHT=18>,
 and 
<IMG SRC="blog-deMunck04_html_m1750f27.gif" NAME="Object6" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=111 HEIGHT=21>
Here 
<IMG SRC="blog-deMunck04_html_1ab0f583.gif" NAME="Object11" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=47 HEIGHT=19>
is the operator that stack the columns of its argument in a single
column vector. Then 
<IMG SRC="blog-deMunck04_html_69863701.gif" NAME="Object28" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=55 HEIGHT=18>
has density
<IMG SRC="blog-deMunck04_html_e0c86.gif" NAME="Object8" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=193 HEIGHT=65>
Here 
<IMG SRC="blog-deMunck04_html_2c9ab5d8.gif" NAME="Object10" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=37 HEIGHT=19>
denote the trace of a square matrix. The log-likelihood of the model
can therefore be written (ignoring constant
<IMG SRC="blog-deMunck04_html_m59546.gif" NAME="Object9" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=380 HEIGHT=79>
To determine 
<IMG SRC="blog-deMunck04_html_19cca27d.gif" NAME="Object25" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=14 HEIGHT=18>
and 
<IMG SRC="blog-deMunck04_html_m668782b5.gif" NAME="Object26" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=17 HEIGHT=20>,
we only need to consider 
<IMG SRC="blog-deMunck04_html_m2b65c41b.gif" NAME="Object13" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=229 HEIGHT=22>.
This is, for given 
<IMG SRC="blog-deMunck04_html_m3bfc7f67.gif" NAME="Object18" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=19 HEIGHT=18>
and 
<IMG SRC="blog-deMunck04_html_62cde0b1.gif" NAME="Object19" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=17 HEIGHT=18>,
clearly a matrix approximation problem, in which the matrix R is
approximated by the rank-one matrix 
<IMG SRC="blog-deMunck04_html_m299462f1.gif" NAME="Object27" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=32 HEIGHT=22>.
The solution is well known, and was first determined by Eckart and
Young (1936)<A CLASS="sdendnoteanc" NAME="sdendnote2anc" HREF="#sdendnote2sym">ii</A>:
It is the outer product of the left and right singular vectors
associated with the largest singular value of 
<IMG SRC="blog-deMunck04_html_m1df9e679.gif" NAME="Object12" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=82 HEIGHT=21>,
multiplied by that singular value. While these singular vectors can
be obtained with a standard Singular Value Decomposition (SVD)
algorithm (e.g. DGESVD of the LAPACK library), it is quicker to first
compute 
<IMG SRC="blog-deMunck04_html_m4cb379c9.gif" NAME="Object15" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=34 HEIGHT=20>—which
yields a<IMG SRC="blog-deMunck04_html_mc6d38fb.gif" NAME="Object20" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=40 HEIGHT=18>
matrix, compute its largest eigenvalue-eigenvector pair 
<IMG SRC="blog-deMunck04_html_m4cc929d3.gif" NAME="Object14" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=51 HEIGHT=21>—the
last of which is the left singular vector of 
<IMG SRC="blog-deMunck04_html_957e6f1.gif" NAME="Object16" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=19 HEIGHT=18>,
and compute 
<IMG SRC="blog-deMunck04_html_4e49dd9e.gif" NAME="Object17" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=98 HEIGHT=23>
—which is the right singular vector. As one should have
expected from the proposed model, we are left with one indeterminacy:
we can choose 
<IMG SRC="blog-deMunck04_html_56292450.gif" NAME="Object22" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=77 HEIGHT=22>,
with 
<IMG SRC="blog-deMunck04_html_7043a971.gif" NAME="Object23" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=73 HEIGHT=20>
arbitrary, as long as we choose 
<IMG SRC="blog-deMunck04_html_709c7809.gif" NAME="Object21" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=59 HEIGHT=21>.
While they are not at all explicit about it, they seem to have chosen
<IMG SRC="blog-deMunck04_html_29768a58.gif" NAME="Object24" ALIGN=ABSMIDDLE HSPACE=8 WIDTH=61 HEIGHT=20>.
Note that, while derived completely differently (they seem to have
used the method of Lagrange multipliers), this is exactly the same
algorithm as the one proposed in de Munck et al. (2004). The
derivation here is more succinct.
 

<DIV ID="sdendnote1">
 <A CLASS="sdendnotesym" NAME="sdendnote1sym" HREF="#sdendnote1anc">i</A>de
 Munck, J. C., Bijma, F., Gaura, P., Sieluzycki, C. A., Branco, M.
 I., & Heethaar, R. M. (2004). A Maximum-Likelihood Estimator for
 Trial-to-Trial variantions in Noisy MEEG/EEG Data Sets, IEEE
 Transactions on Biomedical Engineering, 51, p. 2123-2128
</DIV>
<DIV ID="sdendnote2">
 <A CLASS="sdendnotesym" NAME="sdendnote2sym" HREF="#sdendnote2anc">ii</A>Eckart,
 C & Young, G (1936). The approximation of one matrix by another
 of lower rank, Psychometrika, 1, 211-218.
</DIV><div style="clear:both; padding-bottom:0.25em"></div>

notes

Sunday, July 24, 2005

ML estimation of a model for Trial-to-Trial variability of MEG/EEG data

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