0:00:22 | but dimensional ica its performance than of this |
---|---|

0:00:25 | applied to observation of the cost make microwave background radiation |

0:00:29 | this is joint work with the job was like all those all model yeah and and again ms |

0:00:34 | the motivation for this work is that's follows |

0:00:37 | that is not or piece of the cost like drive back on radiation also known as a C M B |

0:00:42 | consist of the small temperature fluctuations in the black body radiation |

0:00:47 | a left over from the big bang |

0:00:49 | the C E and is not P is depend on cosmological parameters |

0:00:53 | and that's mapping that correctly you'll ski key quantitative information about the formation of for universe and it's structure |

0:01:00 | line set to like of the european space agency which was launched in two thousand and nine |

0:01:04 | is taking images of of that whole sky it's simply frequency |

0:01:08 | we can proceed and frequency coverage and a a resolution and sensitivity |

0:01:13 | and this dog |

0:01:14 | we apply a second order multidimensional ica criteria and its performance and of the |

0:01:19 | to esther physical data |

0:01:20 | we extend form a results to deal with over to me data |

0:01:24 | and and i don't a mention of a component |

0:01:26 | finally we show a good match between the empirical and the predicted results |

0:01:31 | in refuse physical terms |

0:01:33 | what our paper is to take a set of images |

0:01:35 | such as this one |

0:01:37 | and |

0:01:37 | the the set of images into a there a to the C be the galactic emission and other component |

0:01:47 | we use a um uh uh the the that data model which use for the multidimensional ica yeah as follows |

0:01:53 | we use that component model in this model the observations are a sum of N C contribution which we do |

0:01:59 | not component |

0:02:00 | each component |

0:02:01 | X i with index six i |

0:02:03 | denotes component number i |

0:02:06 | at the sample index i work side goes from one to L |

0:02:09 | but i of each of these that component vectors to did not lead does and D where and D denotes |

0:02:13 | the number of detector |

0:02:16 | the components can be reconstructed from the observations by the oblique projection |

0:02:20 | the oblique projection is a matrix which projects on the subspace |

0:02:23 | in which a component because the is |

0:02:25 | orthogonality to all the other components |

0:02:29 | we use a latent model for the components |

0:02:31 | in this model each component can be regarded as a product of a or extended matrix metrics a high |

0:02:37 | and the short vector S i |

0:02:39 | each each as i |

0:02:40 | is denotes a piecewise stationary stochastic process |

0:02:44 | with the covariance |

0:02:46 | and i for each index I the covariance is |

0:02:48 | indexed by Q |

0:02:50 | for each side in main Q |

0:02:52 | where did you do you don't the partition of the induces one to L A to Q domains where Q |

0:02:57 | smaller than a lower or equal to a |

0:03:01 | for each the bank you |

0:03:02 | we can define a block diagonal matrix |

0:03:05 | this matrix is created |

0:03:07 | uh by taking a it's block of that gonna the conference batteries is of the process as i |

0:03:14 | we require that the set of these block to add a no matter says |

0:03:18 | can cannot be jointly block that can into smaller block |

0:03:22 | the total length of the vectors as i is the dimension of the signal space which we denote by and |

0:03:29 | uh by a um |

0:03:30 | concatenating side by side of that tall rectangle are or says a i to construct a matrix |

0:03:35 | which we did notice a a and required that these metrics be full column run |

0:03:40 | the dimension mention of the signal some of the signal this space |

0:03:44 | is larger than or equal to the number of components and smaller than or equal to the number of detectors |

0:03:49 | where a quality on both sides texas back to the simplest ica model |

0:03:54 | we perform component separation via maximum likelihood |

0:03:58 | maximum likelihood can be obtained from the approximate chunk block diagonalization |

0:04:03 | of the empirical localised covariance mattresses of the observations |

0:04:07 | by the inverse of the matrix a |

0:04:10 | the localized in jericho covariance recall conference mattresses of the observations |

0:04:13 | are that |

0:04:14 | normalized the sum of the outer product of the observations at each the |

0:04:20 | note that because that a joint block regularization |

0:04:22 | is possible only if you use |

0:04:24 | that |

0:04:25 | exact or parametric or the X |

0:04:27 | expectation |

0:04:28 | of the localized empirical covariance mattresses |

0:04:32 | therefore for the maximum likelihood procedure consists of a |

0:04:35 | first |

0:04:35 | joint a block diagonal i think of empirical localised covariance mattresses of the observations |

0:04:40 | from these we can |

0:04:42 | estimate the metrics say from this matrix |

0:04:44 | we can construct a the set of oblique projection is |

0:04:48 | on all the component |

0:04:49 | and by applying of this |

0:04:51 | uh estimate of the a of the oblique projections on the observations we can estimate |

0:04:57 | but components |

0:04:59 | we obtained component separation as desired |

0:05:02 | note that this month multidimensional ica procedure is a generalization of the well-known ica |

0:05:10 | in order to evaluate the performance of this a multidimensional ica criterion |

0:05:14 | we use the mean square error |

0:05:16 | the empirical mean square error is calculated for each component |

0:05:20 | in each detector |

0:05:21 | and in each domain |

0:05:23 | and it is |

0:05:24 | defined as that |

0:05:26 | a normalized uh |

0:05:28 | some of the |

0:05:29 | squared differences of the |

0:05:31 | empirical yeah the exact component |

0:05:35 | note that if the model then the mean-square the testing the estimated mean square error is larger than zero only |

0:05:42 | due to finite data that is due to the fact |

0:05:45 | the the sound but the number of samples in each domain is finite |

0:05:49 | if the model holds |

0:05:51 | then the S they the |

0:05:53 | for |

0:05:53 | that expectation of the empirical mean square error at each component to the detector and domain |

0:05:59 | is given by the following expression |

0:06:02 | this is an expression which is a function only of the exact or parametric covariance matrices of each of the |

0:06:08 | component |

0:06:09 | we shall not go into details about this |

0:06:11 | and complicated expression but we should validate |

0:06:14 | it |

0:06:15 | in numerical simulations later |

0:06:19 | we now define the ester physical data in terms of the multidimensional ica model in order to separate the cosmic |

0:06:26 | microwave background radiation from the other components |

0:06:30 | we use a statistical model for the stand for the C B temperature and is entropy |

0:06:35 | according to just referred to go theory |

0:06:37 | the |

0:06:38 | cosmic microwave background radiation |

0:06:40 | is modelled as a zero-mean is zero-mean gaussian stationary process on the sphere |

0:06:45 | with an angular power spectrum C L |

0:06:47 | this is a a typical you much of the cosmic microwave background radiation |

0:06:52 | and this is |

0:06:53 | an illustration of the angular power spectrum |

0:06:55 | where we should explain just terms in the next slide |

0:06:59 | in terms of a lot of the C B |

0:07:01 | is that one dimensional component we do not it and C N B Y |

0:07:05 | yeah |

0:07:05 | the covariance mattresses of the simply at each uh in excel or angular frequency |

0:07:11 | are given by the |

0:07:12 | product of the angular power spectrum that these index L |

0:07:16 | times the outer product of vectors acm we're vector A C B |

0:07:20 | reflects the C B emission log in all the energy detectors |

0:07:25 | the angular power spectrum is defined as follows |

0:07:29 | two dimensional function on the surface of the sphere which is indexed by the as well |

0:07:34 | and the polar angle |

0:07:36 | i |

0:07:37 | transformed using the spherical harmonic transform into |

0:07:41 | a set of coefficients indexed by two induces L where L is larger than or equal to zero and N |

0:07:47 | is between minus plus and |

0:07:49 | i of them are the coefficients of the spherical harmonic transform |

0:07:52 | therefore |

0:07:53 | the angular power spectrum of a random stationary process on the sphere |

0:07:58 | for each |

0:07:59 | angular frequency |

0:08:00 | is given by the |

0:08:02 | average of the expectation of the |

0:08:06 | um |

0:08:08 | of the uh |

0:08:10 | uh |

0:08:11 | sorry |

0:08:13 | of the square of the coefficients idea that |

0:08:17 | i told em mode |

0:08:18 | well of the uh |

0:08:20 | uh |

0:08:21 | all the respective a modes |

0:08:22 | where the can mode |

0:08:24 | in this strike a money transform the indicate directionality and therefore for their are averaged out from these |

0:08:29 | i've i've out from this expression |

0:08:33 | in order to model the galactic emission |

0:08:36 | uh |

0:08:37 | we have to consider the following properties |

0:08:39 | the galactic emission |

0:08:41 | which |

0:08:41 | a typical image of it is that a given here in one of the C be frequency |

0:08:45 | is a superposition of several physical processes |

0:08:49 | you just passion correlation |

0:08:51 | this process is are cup out |

0:08:53 | therefore for in our model we can regard this galactic emission is one multidimensional component |

0:09:00 | we suggest to use the deterministic model for the galactic emission |

0:09:04 | since are in in our model price and requires |

0:09:08 | only that they |

0:09:09 | uh |

0:09:10 | strike a money a transform coefficients of the cosmic microwave background radiation and awfully galactic emission be uncorrelated for all |

0:09:17 | this is |

0:09:18 | then |

0:09:18 | note that if the galactic emission is deterministic and only the C is randomized |

0:09:23 | this indeed a hold |

0:09:24 | since there |

0:09:25 | C and B |

0:09:27 | is a random process with a zero zero-mean which is a from a physical re |

0:09:33 | therefore |

0:09:33 | we |

0:09:34 | two we choose to tape |

0:09:36 | that |

0:09:37 | it very cool |

0:09:38 | localised covariance france mattresses of the galactic emission instead of their parametric counterpart |

0:09:46 | we now use |

0:09:47 | a simplified yet close to realistic setup for our a very close study |

0:09:52 | first |

0:09:53 | we consider only two components the C B of course |

0:09:56 | and that that i think condition |

0:09:58 | this and |

0:09:59 | options are possible from desperate |

0:10:01 | from the ester physical perspective in the range of in this is a between two and nine hundred |

0:10:07 | since our model is free of noise |

0:10:09 | we all to noise our simulation |

0:10:13 | we now a partition the range of in this is between two and nine hundred in two consecutive non overlapping |

0:10:19 | is |

0:10:19 | each of length five L note |

0:10:22 | we take the number of detectors to be nine |

0:10:24 | which is the same as in the plant experiment |

0:10:27 | our simulations are based on the plants sky model |

0:10:29 | which is a soft or back and we can buy the black |

0:10:32 | uh component separation working group |

0:10:35 | i it creates a realistic images of this kind emissions missions at C |

0:10:38 | i the the C and the frequencies |

0:10:41 | further |

0:10:43 | the regions in the maps where the C B is to compare related are method out |

0:10:47 | therefore for the set of for data can it's of nine images is more or less like |

0:10:52 | this one |

0:10:56 | now |

0:10:57 | it turns out that number the multi dimensional i think yeah |

0:11:01 | but a model which we have just present that can to be applied directly to the data |

0:11:06 | this happens |

0:11:07 | uh due to the following point |

0:11:09 | are are as this requires that the the signal space they mentioned be what a number of detectors |

0:11:15 | since and they mention of the C B is one which is a |

0:11:18 | well known from mister physical |

0:11:20 | and know this |

0:11:21 | this means that we for the dimension of the galactic emission to be and |

0:11:24 | however |

0:11:25 | since the eigenvalues of the empirical localised go fast as of the galactic emission with a channel orders of magnitude |

0:11:34 | that the joint loved the notation of that |

0:11:37 | observations of the it trickle a guys covariance friends mattresses of the observation |

0:11:41 | with the with block men's is one and eight |

0:11:44 | is in ill conditioned |

0:11:46 | this means that in practice |

0:11:48 | the the dimension of the signal space is smaller than the number of detectors |

0:11:51 | and and are were determined problem |

0:11:54 | since we do not want to ignore detectors |

0:11:56 | then we have to answer the following questions |

0:11:59 | first |

0:11:59 | given the correct dimension of the galactic emission how can we applied the maximum likelihood |

0:12:04 | and the joint block localisation |

0:12:06 | this set and what is the correct them mention for the galactic emission |

0:12:11 | so |

0:12:12 | for |

0:12:13 | given the right that they mentioned for the galactic emission we want to turn the over determined problem into that |

0:12:18 | to one |

0:12:19 | we do this |

0:12:20 | uh V a man's production using a |

0:12:22 | principal component analysis |

0:12:25 | first dimension mention of the signal space is the dimension of the galactic emission last one |

0:12:30 | we now take to the first and S singular vectors of the empirical conference the of all the observations |

0:12:36 | in two |

0:12:37 | uh the full rate then a matrix U S |

0:12:40 | and the transpose of this metrics |

0:12:42 | project |

0:12:43 | the N D dimensional observations |

0:12:45 | onto to a reduced |

0:12:46 | and S dimensional vector space |

0:12:50 | we don't take in period a local go fast mattresses as |

0:12:53 | of the observations in there were just the vector space |

0:12:57 | and used them as the input to the joint block to globalisation |

0:13:00 | which now works |

0:13:01 | and and the output is an invertible matrix which we note by a |

0:13:05 | not a a and this one has indeed rank and is |

0:13:09 | we are used |

0:13:09 | the inverse projection to expand this matrix back to the and D dimensional observation space |

0:13:15 | and again we have obtained obtained an estimate of the mixing a sort of |

0:13:20 | and mixing matrix eight |

0:13:22 | metrics a a then from this matrix again we can |

0:13:25 | an estimate the oblique projection my says |

0:13:27 | and apply and all to on the observations |

0:13:30 | and obtain estimates of the components |

0:13:33 | a a to obtained |

0:13:34 | component separation as desired |

0:13:37 | we now discuss |

0:13:39 | the problem of |

0:13:40 | selecting the order or the they dimension of the galactic emission |

0:13:44 | in general |

0:13:45 | the considerations are that it's this they it should be the smallest one for which the model hold |

0:13:50 | with a certain tolerated error |

0:13:53 | if we choose a dimension which is too small this means that we give a wrong model for the data |

0:13:58 | and therefore are |

0:14:00 | you model our separation will not work |

0:14:02 | if we choose a no it in which is too large |

0:14:05 | then again to go back to the bad condition problem |

0:14:08 | and also we shall have a redundant parameters |

0:14:13 | so in order to select that the base and for the galactic emission |

0:14:16 | we use |

0:14:16 | no of the |

0:14:18 | experiments |

0:14:19 | and this figure |

0:14:20 | some this experiment |

0:14:23 | for each candidate they mention of the galactic emission |

0:14:26 | we can relate to the empirical |

0:14:28 | many square error twice |

0:14:30 | in this example we |

0:14:32 | uh is for the reconstruction of the cosmic microwave background background uh |

0:14:36 | component |

0:14:38 | once we got calculate the empirical mean square error |

0:14:41 | well the finite data or that is the normal way |

0:14:44 | and second we calculated it without out to find a day errors |

0:14:47 | this is obtained by performing the joint block globalisation |

0:14:51 | on the exact or |

0:14:53 | that that the uh the exact parameters which are the expectation of that |

0:14:57 | empirical localised conference and |

0:15:00 | but off the observations |

0:15:02 | this is a a this evening page |

0:15:04 | i think but in condition |

0:15:07 | in each case |

0:15:08 | the empirical mean square error is it is |

0:15:11 | averaged over forty but want to colour trials |

0:15:14 | so the results |

0:15:15 | for to um |

0:15:17 | first |

0:15:18 | the results are summarised in the following figure |

0:15:20 | first |

0:15:21 | uh a for that it can lead a mention of four |

0:15:24 | we see that the the graph the blue line and the red circles design |

0:15:29 | are separate |

0:15:32 | a this means that indeed |

0:15:34 | the the reason um |

0:15:36 | for this they for this and that for this dimension of the galactic emission |

0:15:40 | in the the a data error or is that is the dominant factor in the mean square or |

0:15:45 | which is the uh you is that the |

0:15:48 | seems that the model is okay |

0:15:50 | we compared it to the to be compared this to a they mention of three |

0:15:54 | in this case we see that the both results |

0:15:56 | that it's a blue line and the line was uh |

0:16:00 | it's a it's a a green line and a good and a with the red is |

0:16:04 | black circles |

0:16:05 | we that is lies all over overlap |

0:16:08 | so first we see that |

0:16:09 | the mean square error |

0:16:10 | for dimensional three is significantly larger of that the error for the mint for four which is it's not a |

0:16:16 | very favourable |

0:16:17 | and second this implies |

0:16:19 | that that for the dimension of three the model error is done in |

0:16:24 | we have also run this experiment with a the think they mentioned candidate of five which is not to the |

0:16:28 | big |

0:16:29 | and the results were very very similar to the results in the of four |

0:16:34 | we have also a similar trend |

0:16:36 | with there |

0:16:37 | uh |

0:16:38 | running the send experiment on uh the mean square error in either uh |

0:16:42 | detectors |

0:16:43 | as well as a with the reconstruction of the galactic emission component |

0:16:47 | and the results uh we're always so a similar |

0:16:50 | so we conclude |

0:16:51 | that the best uh they mention to describe the galactic emission |

0:16:55 | is a for |

0:16:58 | now out using a a a can i of four |

0:17:01 | we would like to compare our |

0:17:03 | um |

0:17:04 | that's your right decoder prediction of the mean square error in which was the complicated expression which we have shown |

0:17:10 | several slides before |

0:17:12 | we want to compare to the empirical results |

0:17:14 | so in this figure |

0:17:17 | the blue colours |

0:17:18 | do not that the empirical mean square error again calculated for the C B component in one of the detector |

0:17:23 | channel |

0:17:25 | there are that blue line denotes the average of the empirical mean square error or over the of trials |

0:17:32 | and the for called the lines and do not to how the standard deviation of of is fourteen want to |

0:17:37 | color trials |

0:17:38 | the red line the big |

0:17:40 | the predicted as you're code mean square |

0:17:43 | so |

0:17:44 | and |

0:17:44 | we normalize normalized |

0:17:46 | this don't mean square errors as the precoder then that your a by one they are normalized |

0:17:50 | for each index L for each and a lower frequency yeah |

0:17:53 | by the end what our power spectrum |

0:17:55 | all that C M B component |

0:17:58 | so first we see that the normalized mean square error |

0:18:01 | is more or less |

0:18:02 | and at |

0:18:03 | at to the value of ten to the minus four |

0:18:07 | this means that the error is relatively small and it indicates one separation |

0:18:12 | second we see that the predicted value |

0:18:15 | is with that standard deviation margin |

0:18:19 | we we again |

0:18:20 | we obtain seem not trends at uh and the scent experience with different a |

0:18:25 | detector frequencies and also for the reconstruction of the galactic emission component |

0:18:30 | so these results |

0:18:31 | validate date |

0:18:32 | our theoretical prediction of the mean square error |

0:18:36 | for this one and for this data |

0:18:40 | so to summarise |

0:18:41 | we applied to multidimensional ica |

0:18:44 | uh method and its performance and was this |

0:18:46 | in terms of components |

0:18:48 | first or physical data |

0:18:49 | we extend and form results |

0:18:51 | to do with over the it's data and and and and a mention of a component |

0:18:55 | and we have shown good match between the pure ego and the predicted result |

0:19:00 | finally we we acknowledge to use of the plans a model developed by the components suppression working group of the |

0:19:05 | plan collaboration |

0:19:07 | and Q |

0:19:15 | oh |

0:19:16 | um |

0:19:17 | my first question |

0:19:19 | what do you |

0:19:21 | or |

0:19:23 | so with that might you have any performed |

0:19:31 | the horse yes |

0:19:32 | there is a and we have shown this in paper which we showed in and |

0:19:36 | a conference |

0:19:38 | as as be conference |

0:19:39 | yeah you're right there is a |

0:19:41 | then |

0:19:42 | i mean |

0:19:42 | a what in this in this data that which does not where the model does not hold exactly and we |

0:19:47 | have actually ignored the fact |

0:19:49 | we assume that the the |

0:19:51 | power spectral of are piecewise stationary but in the in in practice they or not |

0:19:56 | so there |

0:19:57 | the model does not hold exactly so in practice we compared our results to also to classical ica |

0:20:03 | with a rank of the data |

0:20:04 | and in fact the results are very similar |

0:20:08 | but |

0:20:09 | for a tech |

0:20:12 | and a |

0:20:14 | what |

0:20:15 | actually like |

0:20:17 | i |

0:20:18 | how the pen |

0:20:20 | that |

0:20:21 | for |

0:20:23 | yeah but this method does not separate the pair that components it is only in tandem |

0:20:27 | to a to money |

0:20:29 | separate |

0:20:31 | but |

0:20:32 | independent components |

0:20:34 | and |

0:20:34 | it does not separate to and compose in |

0:20:37 | them so that is can not |

0:20:41 | what |

0:20:43 | oh okay you're basic to to the band |

0:20:47 | got like to call home |

0:20:49 | yeah |

0:20:51 | uh_huh so |

0:20:52 | i cannot see that |

0:20:55 | first uh that one advantage is that |

0:20:58 | a theoretical performance and all of this |

0:21:00 | which is derived from that multi dimensional model |

0:21:03 | i do not now a performance was which is drive |

0:21:07 | from the uh |

0:21:08 | classical ica |

0:21:10 | so |

0:21:11 | this is a set of rules |

0:21:13 | there are some |

0:21:16 | working on the |

0:21:19 | think all source |

0:21:21 | and it is also a |

0:21:23 | try |

0:21:24 | so |

0:21:25 | and |

0:21:27 | i |

0:21:29 | this your L |

0:21:31 | i |

0:21:33 | so one i mean |

0:21:35 | right now nice |

0:21:36 | okay |

0:21:36 | right |

0:21:38 | all |

0:21:39 | what should be |

0:21:41 | what you actually |

0:21:43 | we first you're a okay first this is a in this article working the very preliminary work using this um |

0:21:49 | algorithm |

0:21:50 | it is not so a it does not intend or uh |

0:21:55 | say that is going to |

0:21:56 | to be better than at there |

0:21:58 | in a method especially does not in it may be used |

0:22:01 | it does not intend to separate the |

0:22:03 | uh uh components but it may be used as a a perhaps a better to like to estimate |

0:22:09 | i separate the C M D from the galactic emission |

0:22:13 | using a blind some some which has an advantage at least in some parts of the data it is not |

0:22:17 | intended to be a better solution for |

0:22:20 | uh |

0:22:21 | instead of all the other uh |

0:22:23 | uh methods which are applied to ica |

0:22:25 | i don't mess may be better mean |

0:22:27 | uh separating the |

0:22:29 | for is that may be better separate the components and and the dependent components but they may be more sensitive |

0:22:36 | to some |

0:22:37 | and their properties of the data which this but that is not since it is long |

0:22:42 | one but we know that yes |

0:22:44 | only well |

0:22:45 | you know it's a maximum for forty |

0:22:49 | um |

0:22:50 | for ecological |

0:22:52 | but |

0:22:53 | what we actually or oh encroach on their mission |

0:22:57 | collect all conditions |

0:22:59 | here |

0:23:01 | what can be |

0:23:02 | for all |

0:23:04 | it is not right for you get it is not for one probably if we talk about |

0:23:08 | which was asking more of the galactic emission |

0:23:12 | then |

0:23:12 | yeah |

0:23:13 | we would have time to a more i is um |

0:23:17 | then the um |

0:23:19 | the coupling between is uh |

0:23:21 | this is really |

0:23:23 | galactic emission a sources which you have mentioned |

0:23:25 | maybe a |

0:23:26 | more negligible and |

0:23:28 | very likely if we had um |

0:23:30 | why mess |

0:23:32 | and we would of work on the of the um |

0:23:34 | higher are |

0:23:35 | and all of the higher |

0:23:37 | alright of of this not and we would have obtained to collecting the middle from this dimension is only intended |

0:23:43 | to run the separation it is not |

0:23:45 | it does not have a store physical meaning by itself |

0:23:48 | it is say a parameter |

0:23:50 | for this operation |

0:23:52 | i |

0:23:53 | but really to to what you |

0:23:55 | okay |

0:23:57 | but can think that the part in the middle |

0:24:00 | you're basically here so |

0:24:03 | are writers or you |

0:24:06 | okay |

0:24:07 | do like to compose or or the colour |

0:24:09 | way |

0:24:10 | joe are or like to say |

0:24:13 | you all there i guess is this is a very simple example we can also choose |

0:24:17 | different patches of the sky perhaps |

0:24:20 | only the galactic uh |

0:24:21 | play or |

0:24:22 | after their as which contain only part of the a a use of the sky which i have |

0:24:27 | i and use again this algorithm one these patches and you probably have |

0:24:31 | different results |

0:24:33 | this again this is just a a simplified example |

0:24:35 | okay thank you |

0:24:36 | thank you |