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