0:00:22but dimensional ica its performance than of this
0:00:25applied to observation of the cost make microwave background radiation
0:00:29this is joint work with the job was like all those all model yeah and and again ms
0:00:34the motivation for this work is that's follows
0:00:37that is not or piece of the cost like drive back on radiation also known as a C M B
0:00:42consist of the small temperature fluctuations in the black body radiation
0:00:47a left over from the big bang
0:00:49the C E and is not P is depend on cosmological parameters
0:00:53and that's mapping that correctly you'll ski key quantitative information about the formation of for universe and it's structure
0:01:00line set to like of the european space agency which was launched in two thousand and nine
0:01:04is taking images of of that whole sky it's simply frequency
0:01:08we can proceed and frequency coverage and a a resolution and sensitivity
0:01:13and this dog
0:01:14we apply a second order multidimensional ica criteria and its performance and of the
0:01:19to esther physical data
0:01:20we extend form a results to deal with over to me data
0:01:24and and i don't a mention of a component
0:01:26finally we show a good match between the empirical and the predicted results
0:01:31in refuse physical terms
0:01:33what our paper is to take a set of images
0:01:35such as this one
0:01:37the the set of images into a there a to the C be the galactic emission and other component
0:01:47we use a um uh uh the the that data model which use for the multidimensional ica yeah as follows
0:01:53we use that component model in this model the observations are a sum of N C contribution which we do
0:01:59not component
0:02:00each component
0:02:01X i with index six i
0:02:03denotes component number i
0:02:06at the sample index i work side goes from one to L
0:02:09but i of each of these that component vectors to did not lead does and D where and D denotes
0:02:13the number of detector
0:02:16the components can be reconstructed from the observations by the oblique projection
0:02:20the oblique projection is a matrix which projects on the subspace
0:02:23in which a component because the is
0:02:25orthogonality to all the other components
0:02:29we use a latent model for the components
0:02:31in this model each component can be regarded as a product of a or extended matrix metrics a high
0:02:37and the short vector S i
0:02:39each each as i
0:02:40is denotes a piecewise stationary stochastic process
0:02:44with the covariance
0:02:46and i for each index I the covariance is
0:02:48indexed by Q
0:02:50for each side in main Q
0:02:52where did you do you don't the partition of the induces one to L A to Q domains where Q
0:02:57smaller than a lower or equal to a
0:03:01for each the bank you
0:03:02we can define a block diagonal matrix
0:03:05this matrix is created
0:03:07uh by taking a it's block of that gonna the conference batteries is of the process as i
0:03:14we require that the set of these block to add a no matter says
0:03:18can cannot be jointly block that can into smaller block
0:03:22the total length of the vectors as i is the dimension of the signal space which we denote by and
0:03:29uh by a um
0:03:30concatenating side by side of that tall rectangle are or says a i to construct a matrix
0:03:35which we did notice a a and required that these metrics be full column run
0:03:40the dimension mention of the signal some of the signal this space
0:03:44is larger than or equal to the number of components and smaller than or equal to the number of detectors
0:03:49where a quality on both sides texas back to the simplest ica model
0:03:54we perform component separation via maximum likelihood
0:03:58maximum likelihood can be obtained from the approximate chunk block diagonalization
0:04:03of the empirical localised covariance mattresses of the observations
0:04:07by the inverse of the matrix a
0:04:10the localized in jericho covariance recall conference mattresses of the observations
0:04:13are that
0:04:14normalized the sum of the outer product of the observations at each the
0:04:20note that because that a joint block regularization
0:04:22is possible only if you use
0:04:25exact or parametric or the X
0:04:28of the localized empirical covariance mattresses
0:04:32therefore for the maximum likelihood procedure consists of a
0:04:35joint a block diagonal i think of empirical localised covariance mattresses of the observations
0:04:40from these we can
0:04:42estimate the metrics say from this matrix
0:04:44we can construct a the set of oblique projection is
0:04:48on all the component
0:04:49and by applying of this
0:04:51uh estimate of the a of the oblique projections on the observations we can estimate
0:04:57but components
0:04:59we obtained component separation as desired
0:05:02note that this month multidimensional ica procedure is a generalization of the well-known ica
0:05:10in order to evaluate the performance of this a multidimensional ica criterion
0:05:14we use the mean square error
0:05:16the empirical mean square error is calculated for each component
0:05:20in each detector
0:05:21and in each domain
0:05:23and it is
0:05:24defined as that
0:05:26a normalized uh
0:05:28some of the
0:05:29squared differences of the
0:05:31empirical yeah the exact component
0:05:35note that if the model then the mean-square the testing the estimated mean square error is larger than zero only
0:05:42due to finite data that is due to the fact
0:05:45the the sound but the number of samples in each domain is finite
0:05:49if the model holds
0:05:51then the S they the
0:05:53that expectation of the empirical mean square error at each component to the detector and domain
0:05:59is given by the following expression
0:06:02this is an expression which is a function only of the exact or parametric covariance matrices of each of the
0:06:09we shall not go into details about this
0:06:11and complicated expression but we should validate
0:06:15in numerical simulations later
0:06:19we now define the ester physical data in terms of the multidimensional ica model in order to separate the cosmic
0:06:26microwave background radiation from the other components
0:06:30we use a statistical model for the stand for the C B temperature and is entropy
0:06:35according to just referred to go theory
0:06:38cosmic microwave background radiation
0:06:40is modelled as a zero-mean is zero-mean gaussian stationary process on the sphere
0:06:45with an angular power spectrum C L
0:06:47this is a a typical you much of the cosmic microwave background radiation
0:06:52and this is
0:06:53an illustration of the angular power spectrum
0:06:55where we should explain just terms in the next slide
0:06:59in terms of a lot of the C B
0:07:01is that one dimensional component we do not it and C N B Y
0:07:05the covariance mattresses of the simply at each uh in excel or angular frequency
0:07:11are given by the
0:07:12product of the angular power spectrum that these index L
0:07:16times the outer product of vectors acm we're vector A C B
0:07:20reflects the C B emission log in all the energy detectors
0:07:25the angular power spectrum is defined as follows
0:07:29two dimensional function on the surface of the sphere which is indexed by the as well
0:07:34and the polar angle
0:07:37transformed using the spherical harmonic transform into
0:07:41a set of coefficients indexed by two induces L where L is larger than or equal to zero and N
0:07:47is between minus plus and
0:07:49i of them are the coefficients of the spherical harmonic transform
0:07:53the angular power spectrum of a random stationary process on the sphere
0:07:58for each
0:07:59angular frequency
0:08:00is given by the
0:08:02average of the expectation of the
0:08:08of the uh
0:08:13of the square of the coefficients idea that
0:08:17i told em mode
0:08:18well of the uh
0:08:21all the respective a modes
0:08:22where the can mode
0:08:24in this strike a money transform the indicate directionality and therefore for their are averaged out from these
0:08:29i've i've out from this expression
0:08:33in order to model the galactic emission
0:08:37we have to consider the following properties
0:08:39the galactic emission
0:08:41a typical image of it is that a given here in one of the C be frequency
0:08:45is a superposition of several physical processes
0:08:49you just passion correlation
0:08:51this process is are cup out
0:08:53therefore for in our model we can regard this galactic emission is one multidimensional component
0:09:00we suggest to use the deterministic model for the galactic emission
0:09:04since are in in our model price and requires
0:09:08only that they
0:09:10strike a money a transform coefficients of the cosmic microwave background radiation and awfully galactic emission be uncorrelated for all
0:09:17this is
0:09:18note that if the galactic emission is deterministic and only the C is randomized
0:09:23this indeed a hold
0:09:24since there
0:09:25C and B
0:09:27is a random process with a zero zero-mean which is a from a physical re
0:09:34two we choose to tape
0:09:37it very cool
0:09:38localised covariance france mattresses of the galactic emission instead of their parametric counterpart
0:09:46we now use
0:09:47a simplified yet close to realistic setup for our a very close study
0:09:53we consider only two components the C B of course
0:09:56and that that i think condition
0:09:58this and
0:09:59options are possible from desperate
0:10:01from the ester physical perspective in the range of in this is a between two and nine hundred
0:10:07since our model is free of noise
0:10:09we all to noise our simulation
0:10:13we now a partition the range of in this is between two and nine hundred in two consecutive non overlapping
0:10:19each of length five L note
0:10:22we take the number of detectors to be nine
0:10:24which is the same as in the plant experiment
0:10:27our simulations are based on the plants sky model
0:10:29which is a soft or back and we can buy the black
0:10:32uh component separation working group
0:10:35i it creates a realistic images of this kind emissions missions at C
0:10:38i the the C and the frequencies
0:10:43the regions in the maps where the C B is to compare related are method out
0:10:47therefore for the set of for data can it's of nine images is more or less like
0:10:52this one
0:10:57it turns out that number the multi dimensional i think yeah
0:11:01but a model which we have just present that can to be applied directly to the data
0:11:06this happens
0:11:07uh due to the following point
0:11:09are are as this requires that the the signal space they mentioned be what a number of detectors
0:11:15since and they mention of the C B is one which is a
0:11:18well known from mister physical
0:11:20and know this
0:11:21this means that we for the dimension of the galactic emission to be and
0:11:25since the eigenvalues of the empirical localised go fast as of the galactic emission with a channel orders of magnitude
0:11:34that the joint loved the notation of that
0:11:37observations of the it trickle a guys covariance friends mattresses of the observation
0:11:41with the with block men's is one and eight
0:11:44is in ill conditioned
0:11:46this means that in practice
0:11:48the the dimension of the signal space is smaller than the number of detectors
0:11:51and and are were determined problem
0:11:54since we do not want to ignore detectors
0:11:56then we have to answer the following questions
0:11:59given the correct dimension of the galactic emission how can we applied the maximum likelihood
0:12:04and the joint block localisation
0:12:06this set and what is the correct them mention for the galactic emission
0:12:13given the right that they mentioned for the galactic emission we want to turn the over determined problem into that
0:12:18to one
0:12:19we do this
0:12:20uh V a man's production using a
0:12:22principal component analysis
0:12:25first dimension mention of the signal space is the dimension of the galactic emission last one
0:12:30we now take to the first and S singular vectors of the empirical conference the of all the observations
0:12:36in two
0:12:37uh the full rate then a matrix U S
0:12:40and the transpose of this metrics
0:12:43the N D dimensional observations
0:12:45onto to a reduced
0:12:46and S dimensional vector space
0:12:50we don't take in period a local go fast mattresses as
0:12:53of the observations in there were just the vector space
0:12:57and used them as the input to the joint block to globalisation
0:13:00which now works
0:13:01and and the output is an invertible matrix which we note by a
0:13:05not a a and this one has indeed rank and is
0:13:09we are used
0:13:09the inverse projection to expand this matrix back to the and D dimensional observation space
0:13:15and again we have obtained obtained an estimate of the mixing a sort of
0:13:20and mixing matrix eight
0:13:22metrics a a then from this matrix again we can
0:13:25an estimate the oblique projection my says
0:13:27and apply and all to on the observations
0:13:30and obtain estimates of the components
0:13:33a a to obtained
0:13:34component separation as desired
0:13:37we now discuss
0:13:39the problem of
0:13:40selecting the order or the they dimension of the galactic emission
0:13:44in general
0:13:45the considerations are that it's this they it should be the smallest one for which the model hold
0:13:50with a certain tolerated error
0:13:53if we choose a dimension which is too small this means that we give a wrong model for the data
0:13:58and therefore are
0:14:00you model our separation will not work
0:14:02if we choose a no it in which is too large
0:14:05then again to go back to the bad condition problem
0:14:08and also we shall have a redundant parameters
0:14:13so in order to select that the base and for the galactic emission
0:14:16we use
0:14:16no of the
0:14:19and this figure
0:14:20some this experiment
0:14:23for each candidate they mention of the galactic emission
0:14:26we can relate to the empirical
0:14:28many square error twice
0:14:30in this example we
0:14:32uh is for the reconstruction of the cosmic microwave background background uh
0:14:38once we got calculate the empirical mean square error
0:14:41well the finite data or that is the normal way
0:14:44and second we calculated it without out to find a day errors
0:14:47this is obtained by performing the joint block globalisation
0:14:51on the exact or
0:14:53that that the uh the exact parameters which are the expectation of that
0:14:57empirical localised conference and
0:15:00but off the observations
0:15:02this is a a this evening page
0:15:04i think but in condition
0:15:07in each case
0:15:08the empirical mean square error is it is
0:15:11averaged over forty but want to colour trials
0:15:14so the results
0:15:15for to um
0:15:18the results are summarised in the following figure
0:15:21uh a for that it can lead a mention of four
0:15:24we see that the the graph the blue line and the red circles design
0:15:29are separate
0:15:32a this means that indeed
0:15:34the the reason um
0:15:36for this they for this and that for this dimension of the galactic emission
0:15:40in the the a data error or is that is the dominant factor in the mean square or
0:15:45which is the uh you is that the
0:15:48seems that the model is okay
0:15:50we compared it to the to be compared this to a they mention of three
0:15:54in this case we see that the both results
0:15:56that it's a blue line and the line was uh
0:16:00it's a it's a a green line and a good and a with the red is
0:16:04black circles
0:16:05we that is lies all over overlap
0:16:08so first we see that
0:16:09the mean square error
0:16:10for dimensional three is significantly larger of that the error for the mint for four which is it's not a
0:16:16very favourable
0:16:17and second this implies
0:16:19that that for the dimension of three the model error is done in
0:16:24we have also run this experiment with a the think they mentioned candidate of five which is not to the
0:16:29and the results were very very similar to the results in the of four
0:16:34we have also a similar trend
0:16:36with there
0:16:38running the send experiment on uh the mean square error in either uh
0:16:43as well as a with the reconstruction of the galactic emission component
0:16:47and the results uh we're always so a similar
0:16:50so we conclude
0:16:51that the best uh they mention to describe the galactic emission
0:16:55is a for
0:16:58now out using a a a can i of four
0:17:01we would like to compare our
0:17:04that's your right decoder prediction of the mean square error in which was the complicated expression which we have shown
0:17:10several slides before
0:17:12we want to compare to the empirical results
0:17:14so in this figure
0:17:17the blue colours
0:17:18do not that the empirical mean square error again calculated for the C B component in one of the detector
0:17:25there are that blue line denotes the average of the empirical mean square error or over the of trials
0:17:32and the for called the lines and do not to how the standard deviation of of is fourteen want to
0:17:37color trials
0:17:38the red line the big
0:17:40the predicted as you're code mean square
0:17:44we normalize normalized
0:17:46this don't mean square errors as the precoder then that your a by one they are normalized
0:17:50for each index L for each and a lower frequency yeah
0:17:53by the end what our power spectrum
0:17:55all that C M B component
0:17:58so first we see that the normalized mean square error
0:18:01is more or less
0:18:02and at
0:18:03at to the value of ten to the minus four
0:18:07this means that the error is relatively small and it indicates one separation
0:18:12second we see that the predicted value
0:18:15is with that standard deviation margin
0:18:19we we again
0:18:20we obtain seem not trends at uh and the scent experience with different a
0:18:25detector frequencies and also for the reconstruction of the galactic emission component
0:18:30so these results
0:18:31validate date
0:18:32our theoretical prediction of the mean square error
0:18:36for this one and for this data
0:18:40so to summarise
0:18:41we applied to multidimensional ica
0:18:44uh method and its performance and was this
0:18:46in terms of components
0:18:48first or physical data
0:18:49we extend and form results
0:18:51to do with over the it's data and and and and a mention of a component
0:18:55and we have shown good match between the pure ego and the predicted result
0:19:00finally we we acknowledge to use of the plans a model developed by the components suppression working group of the
0:19:05plan collaboration
0:19:07and Q
0:19:17my first question
0:19:19what do you
0:19:23so with that might you have any performed
0:19:31the horse yes
0:19:32there is a and we have shown this in paper which we showed in and
0:19:36a conference
0:19:38as as be conference
0:19:39yeah you're right there is a
0:19:42i mean
0:19:42a what in this in this data that which does not where the model does not hold exactly and we
0:19:47have actually ignored the fact
0:19:49we assume that the the
0:19:51power spectral of are piecewise stationary but in the in in practice they or not
0:19:56so there
0:19:57the model does not hold exactly so in practice we compared our results to also to classical ica
0:20:03with a rank of the data
0:20:04and in fact the results are very similar
0:20:09for a tech
0:20:12and a
0:20:15actually like
0:20:18how the pen
0:20:23yeah but this method does not separate the pair that components it is only in tandem
0:20:27to a to money
0:20:32independent components
0:20:34it does not separate to and compose in
0:20:37them so that is can not
0:20:43oh okay you're basic to to the band
0:20:47got like to call home
0:20:51uh_huh so
0:20:52i cannot see that
0:20:55first uh that one advantage is that
0:20:58a theoretical performance and all of this
0:21:00which is derived from that multi dimensional model
0:21:03i do not now a performance was which is drive
0:21:07from the uh
0:21:08classical ica
0:21:11this is a set of rules
0:21:13there are some
0:21:16working on the
0:21:19think all source
0:21:21and it is also a
0:21:29this your L
0:21:33so one i mean
0:21:35right now nice
0:21:39what should be
0:21:41what you actually
0:21:43we first you're a okay first this is a in this article working the very preliminary work using this um
0:21:50it is not so a it does not intend or uh
0:21:55say that is going to
0:21:56to be better than at there
0:21:58in a method especially does not in it may be used
0:22:01it does not intend to separate the
0:22:03uh uh components but it may be used as a a perhaps a better to like to estimate
0:22:09i separate the C M D from the galactic emission
0:22:13using a blind some some which has an advantage at least in some parts of the data it is not
0:22:17intended to be a better solution for
0:22:21instead of all the other uh
0:22:23uh methods which are applied to ica
0:22:25i don't mess may be better mean
0:22:27uh separating the
0:22:29for is that may be better separate the components and and the dependent components but they may be more sensitive
0:22:36to some
0:22:37and their properties of the data which this but that is not since it is long
0:22:42one but we know that yes
0:22:44only well
0:22:45you know it's a maximum for forty
0:22:50for ecological
0:22:53what we actually or oh encroach on their mission
0:22:57collect all conditions
0:23:01what can be
0:23:02for all
0:23:04it is not right for you get it is not for one probably if we talk about
0:23:08which was asking more of the galactic emission
0:23:13we would have time to a more i is um
0:23:17then the um
0:23:19the coupling between is uh
0:23:21this is really
0:23:23galactic emission a sources which you have mentioned
0:23:25maybe a
0:23:26more negligible and
0:23:28very likely if we had um
0:23:30why mess
0:23:32and we would of work on the of the um
0:23:34higher are
0:23:35and all of the higher
0:23:37alright of of this not and we would have obtained to collecting the middle from this dimension is only intended
0:23:43to run the separation it is not
0:23:45it does not have a store physical meaning by itself
0:23:48it is say a parameter
0:23:50for this operation
0:23:53but really to to what you
0:23:57but can think that the part in the middle
0:24:00you're basically here so
0:24:03are writers or you
0:24:07do like to compose or or the colour
0:24:10joe are or like to say
0:24:13you all there i guess is this is a very simple example we can also choose
0:24:17different patches of the sky perhaps
0:24:20only the galactic uh
0:24:21play or
0:24:22after their as which contain only part of the a a use of the sky which i have
0:24:27i and use again this algorithm one these patches and you probably have
0:24:31different results
0:24:33this again this is just a a simplified example
0:24:35okay thank you
0:24:36thank you