# [Eeglablist] How phase is calculated in linear decomposition

Robert Thatcher rwthatcher2 at yahoo.com
Fri Jul 7 11:07:36 PDT 2017

```Dear Makoto,

It iscorrect that a linear decomposition where one decomposes a 19 channel timeseries into 19 ICA components then, like the inverse Fourier transform, one canrecover the phase information if one uses all 19 ICA components.   However, one cannot recover the phaseinformation in 19 channels if one uses 18 or 17 or 16 or any number less than19 components to reconstruct a new 19 channel time series.

The mathematics proving “distortion” by ICAreconstruction with a small number of components is explained by Taken’s Theorem(& other theorems in differential geometry, e.g. Shannon's) where a lower dimensional timeseries does not preserve phase when used to reconstruct a higher dimensionaltime series.  Therefore mathematically(including the inverse Fourier transform) there is always a loss of informationwhen using a smaller number of components to reconstruct a larger number ofchannels. Also, commonsense tells one that it is not possible to createsomething out of nothing, e.g., if I give you 18 Bitcoins will you give me 19Bitcoins? (credit to Georges for asking this question).

In addition to mathematics, there is empirical proofof “distortion” or “adulteration” that has been repeated over and over again onthis forum and accepted to be true by Arnaud and Stephan and Georges and RobertLarson and Ramesh but apparently not yet by you.   The empirical proof can be demonstrated by yourselfif you compute the phase differences between all channel combinations in theoriginal EEG time series vs the ICA reconstructed time series using a smallernumber of ICA components.  Anyone candemonstrate the phase difference “distortion” for themselves using the Hilberttransform for instantaneous phase differences (i.e., each time point) or theFFT for an average.

Here is a URL of a You Tube Video that providesadditional empirical proof that ICA reconstruction distorts or adulterates the physiologicallybased phase differences in the original EEG recording.

https://youtu.be/Q36ojib5OZE

Here is a url to four power points of two differenttime points that compares the phase difference before and after Arnaud’s ICAreconstruction.

http://www.appliedneuroscience.com/Phase_Differences-between-Original_&_Delorme-Post-ICA.pptx

Please download and expand the power points so thatyou can verify for yourself that the phase differences in the originalrecording have been distorted or altered. I would be happy to issue a temporary license to Neuroguide so that youcan launch two Neuroguides and import and compare the original time series andArnaud’s ICA reconstructed time series to verify for yourself that phasedifferences for all channel combinations for each and every time  point have been distorted by ICAreconstruction.

I offer this same opportunity to all of the members ofthis forum.  Download, install and launchNG-2.9.4 and paste the key A to qeeg at appliedneuroscience.com.   Here is the download url:

I will mail step by step mouse instructions so thatanyone interested can verify for themselves that ICA reconstruction distortsthe phase differences for all channel combinations and all time points.

This is important because if the primary level ofmeasurement is distorted then all subsequent analyses are invalid as a matterof verifiable fact.

Best wishes,

Robert

On Thursday, July 6, 2017, 2:31:56 PM PDT, Makoto Miyakoshi <mmiyakoshi at ucsd.edu> wrote:

Dear Robert and list,
If I remember correctly, one of the critical issue was how phase changes in 'channel data' after removing independent components. Let me focus on this point only.
1. To show my rebuttal and mathematical proof, I prepared a wiki page. In this page. I described how a phase is split/synthesized before and after linear decomposition. Note that this applies to any linear decomposition and not specific to ICA. In doing so, made it nice and educational to show my respect to Arno's effort. Hopefully this mathematical clarification helps.
https://sccn.ucsd.edu/wiki/How_phase_is_calculated_in_linear_decomposition
Robert, if you do not agree, please show the right process in a form of math.
2. Apart from this scientific/mathematical dispute, I insist that one should not use the word 'distortion' in this context because it includes subjective judgement.
Makoto

--
Makoto Miyakoshi
Swartz Center for Computational Neuroscience
Institute for Neural Computation, University of California San Diego
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