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