Přepis řeči - TRANSITIONAL SURROGATES

0:00:13	or image
0:00:14	so uh
0:00:15	this fact
0:00:17	continuation of some were which chip
0:00:19	as in them
0:00:20	perform
0:00:21	uh
0:00:22	time with the people in both the you double and yeah
0:00:25	nice
0:00:26	yeah yeah
0:00:27	like a show
0:00:28	and what are we present here would be mostly a variation on a technique that we treat you
0:00:34	or the proposed
0:00:35	mostly all testing for
0:00:37	stationarity so so well
0:00:39	of course
0:00:40	uh
0:00:41	has a problem by
0:00:42	reminding some of basic yeah uh
0:00:44	relation of well uh the approach of
0:00:47	a a to for this problem
0:00:49	and
0:00:51	for a starting point to the question of testing for stationarity
0:00:55	and both as to sink a little bit again that the about what is a really stationary
0:00:59	and if we look at the theoretical definition this is something which is well known and which of course as
0:01:04	very good property
0:01:06	in
0:01:07	you
0:01:07	but from a practical point of view
0:01:10	it
0:01:11	not necessarily very here useful because we have to make some for assumptions about
0:01:16	stationarity related forums
0:01:18	some some also scale
0:01:20	and also because of what we really we
0:01:23	a stationary T which means that we compare uh
0:01:27	it won't use
0:01:28	situation in some sense
0:01:29	with a stable situation
0:01:31	needs
0:01:32	to have a
0:01:33	hand the reference to of what would be and you could nonstationary situation
0:01:38	and so what we had a a pragmatic point of you was really
0:01:41	to characterise
0:01:42	stationarity you by using
0:01:45	shouldn't the rice data
0:01:46	which are are drawn from the signal itself which means that we preserve some properties
0:01:51	which are part part is a signal be stationary or not
0:01:54	but
0:01:55	we just for from this a station or right version for when we can be a the newly but as
0:02:00	is of stationary T and uh construct test
0:02:03	and the test
0:02:04	is is based on that are
0:02:05	a features
0:02:06	which can be used either by means of this
0:02:09	which contrast the global behavior with a local good behavior
0:02:12	or or by other
0:02:14	and machine learning techniques like one class as here
0:02:18	so what else so it's
0:02:20	classical so get
0:02:21	are are constructed had of white we me a very simple way because
0:02:25	just
0:02:25	a amount to take the for transform
0:02:28	the from so signal
0:02:29	which
0:02:30	is composed of a a two in the phase
0:02:33	and just to keep and changed my magnitude
0:02:35	why replacing the face by you random one so it's a minimization of the face
0:02:40	and the but both of these in that by minimizing the phase
0:02:43	we have a strong in in fact
0:02:46	the organization in time all the frequency content of the signal which is a signature
0:02:51	the
0:02:51	a possible non
0:02:53	so a construct a we get just a
0:02:56	to do this
0:02:57	which is
0:02:58	keeping this model does
0:03:00	magnitude magnitude and replace the face by the random fake and taking inverse which one
0:03:04	and this is a they can be with had been used
0:03:07	the
0:03:08	originally originally introduced a in these things for non a test but that can be used in this context for
0:03:15	this thing for stationary so here as an example of
0:03:18	a a white uh is is a uh this makes sense so you take this signal which is a deterministic
0:03:24	signal modulated in amplitude and in frequency as revealed by a time frequency
0:03:28	a you should here
0:03:29	then we can see say that this is non-stationary stationary because we have some pollution
0:03:34	all
0:03:35	the marginal in time and we are something what we should of the local or frequency content
0:03:40	okay
0:03:41	and the little all spectrum is given here
0:03:43	and this is just a mapping
0:03:45	but now if we just keep this magnitude you'd and we run minds the say
0:03:49	and we go back to the original time domain we get a we can get something like
0:03:53	and this is what example of one sort or gate and this is a for
0:03:56	signature
0:03:57	and we clearly see that in this case we have described
0:04:00	you but i section which make that a different frequencies
0:04:03	a a of that given your in that
0:04:05	the time
0:04:06	a plane
0:04:07	and which amounts also to a the marginal which is
0:04:10	less
0:04:11	uh
0:04:13	you a let's less local
0:04:15	and if we do this for a number of different or gates force
0:04:17	averaging more of what to get here we get something
0:04:20	which is more or stationary in the standard that local behavior is similar global if here
0:04:26	and it as a reading to be an side that either for this signal of for this one was
0:04:31	we have a
0:04:32	exactly the say
0:04:33	global all you know in frequency which is a signature of
0:04:37	all the uh a stationary stationarity this would be very
0:04:40	so
0:04:41	okay okay
0:04:41	so classically that's been shown that
0:04:44	a small
0:04:45	number because if
0:04:47	five
0:04:48	independently drawn sort kits
0:04:50	can be enough to characterise a that the new it is of stationarity
0:04:54	and it has also be shown that
0:04:56	we get a strict stationarity by doing this
0:04:59	strong get
0:05:01	however it has been also shown that a strict stationarity can be a too strong
0:05:06	or commitment to we want to compare
0:05:08	and observed signal with the station rice
0:05:11	a count that
0:05:12	especially because the rejection rate of the hypotheses
0:05:15	he's
0:05:16	higher than the prescribed confidence level and also because when which
0:05:20	a with the signal which is deterministic for which also we want to have an interpretation of uh
0:05:25	stationarity he of course because of the minimization we get some more what is order
0:05:30	by means of the randomizations and uh we we get a exactly
0:05:34	uh uh we get we met
0:05:35	make this little bit
0:05:36	see
0:05:38	so what we propose here
0:05:40	a some kind of of to get between the situation of
0:05:44	nonstationary Q as such signal that to say and that
0:05:47	strictly stationary version of thing but the or okay
0:05:51	instead of replacing the phase
0:05:54	we just
0:05:55	proposed to modify the phase
0:05:57	so that here
0:05:58	if the original phase
0:06:01	so X we just at
0:06:03	so
0:06:03	random phase
0:06:05	he which is some random phase noise
0:06:07	so that
0:06:08	the frequency for
0:06:09	distribution of the straw gate which is a two D fourier transform of the covariance function which can be nonstationary
0:06:16	can be factored factor like this
0:06:18	two terms one which
0:06:20	really is related
0:06:21	to of the structure of the original signal and you are the one which plays a role of a weighting
0:06:26	fact
0:06:27	here
0:06:28	a on this
0:06:29	additional phase
0:06:30	for which
0:06:31	stationary you which is well known to be
0:06:35	to have a main diagonal only and not a you in at the frequency of
0:06:40	can see plain will be use a signature of stationary
0:06:44	so a first possibility for a this
0:06:46	is
0:06:47	the simplest one just to take for see that the additional a white gaussian noise of a given the variance
0:06:52	and in this case and can be easy to prove that the the weighting factor takes on these four
0:06:57	when the sequence is your mean stationary you with these correlation function
0:07:01	uh
0:07:02	like this
0:07:03	and with this model we have a
0:07:05	this expression
0:07:06	and this expression of course
0:07:08	that
0:07:09	to these
0:07:10	that right
0:07:10	limit mate
0:07:11	in the lead of
0:07:12	the variance of the noise but could in which is exactly the situation of stationary
0:07:17	and so what we see that we increase the my level of course we get to transition
0:07:21	for a non if
0:07:23	signal signal is non-stationary two
0:07:25	uh stationarity what's the name of transitional
0:07:29	several
0:07:30	so as an illustration here is a case of a
0:07:32	P tone them for which you and for
0:07:34	supposed to be one
0:07:36	everywhere
0:07:37	here and here is the evolution of the envelope
0:07:41	"'kay" of the sort gate
0:07:42	when the level of the noise is pretty increase
0:07:46	here
0:07:46	and what we see that ultimately we get something extremely erratic Q
0:07:51	but
0:07:51	with this transition we get set
0:07:54	get some soft and transition
0:07:56	which make
0:07:57	more sense
0:07:58	the sense of uh comparing a uh with the station wise
0:08:01	situation
0:08:03	uh the second option to we consider is
0:08:06	two
0:08:06	re think about the weighting factor and just we remark that this has something to do with a characteristic function
0:08:12	of of the uh increments
0:08:14	process of the face
0:08:16	and so if
0:08:17	we write these
0:08:19	these these way for at at phase which as stationary increments
0:08:23	then
0:08:24	a weighting factor takes a very simple form which is just late to to the second was structure function of
0:08:30	of these uh
0:08:32	noise
0:08:33	and for instance if we take the example of fractional the ocean noise
0:08:36	with the hurst exponent
0:08:38	as you one one with
0:08:39	one half
0:08:40	just a class
0:08:43	white gaussian noise
0:08:44	there we get an excuse
0:08:45	for what this weighting factor and again we get something which is a weighting
0:08:49	all the make a a a a the main diagonal which
0:08:52	makes
0:08:53	the process convert stationarity when a
0:08:56	i sorry when either
0:08:58	or
0:08:59	uh a square and uh
0:09:02	greece or
0:09:03	a a correlation
0:09:04	good for
0:09:06	a a a a a the
0:09:08	correlation
0:09:09	but if
0:09:10	and so here we just
0:09:11	focus on the that of be not to be process which is
0:09:14	they with it
0:09:16	or one half
0:09:17	and we only make use of the variance as a or by
0:09:21	for
0:09:21	H
0:09:22	so how framework
0:09:24	is is getting
0:09:25	for a given signal to make use of a given time-frequency distribution which acts as the spectral it's time for
0:09:31	a spectral estimate of a time-varying spectrum it can be
0:09:34	a spectral or multi window spectrogram as well
0:09:38	and the i if
0:09:39	it
0:09:40	process is stationary than the local spec
0:09:42	should you don't you five mobile spectrum for at time
0:09:46	and
0:09:46	the test
0:09:47	to compare
0:09:48	how how much these two quantities is local and global uh identical or not
0:09:52	and the reference
0:09:55	which is used for the newly but is a stationary is constructed
0:09:58	on the sort gates for which we can draw a so uh uh distribution of the problem
0:10:04	and so here a uh more precisely
0:10:07	we do this
0:10:08	been a two parameters
0:10:09	uh
0:10:10	plane
0:10:11	but i know but i think that the local uh
0:10:15	i frequency suspect a given time uh this way
0:10:18	and then i extracting something which is a signature nature of the time of variation of the much you know
0:10:23	sense and he
0:10:25	signature nature of the frequent
0:10:26	essentially centroid uh
0:10:28	which is
0:10:29	also
0:10:30	a signature in the the free
0:10:31	see a
0:10:32	along the frequency axis
0:10:33	and here we are interested
0:10:36	in how this quantity
0:10:37	to rates
0:10:38	for for different
0:10:40	for the different
0:10:41	issue
0:10:42	and uh uh uh a for making the test uh fact you we do not to make use of the
0:10:46	distance here we just to use
0:10:48	a something you in spite from mentioned in and technique what class
0:10:52	spoke
0:10:52	machines
0:10:53	we to try to do is
0:10:55	to and close
0:10:56	as much as possible
0:10:58	off
0:10:59	is uh to by teachers
0:11:01	with a given
0:11:02	so go here was some slack able
0:11:04	so that
0:11:06	there is
0:11:06	in with the it is so circle we can give
0:11:09	a given confidence
0:11:10	for a uh T a situation of stationarity which
0:11:15	for which led to say that
0:11:17	this so it's plays the role of of uh learning set
0:11:20	for a for the machine learning a
0:11:23	so here is an example here
0:11:25	where we have a
0:11:26	uh uh get a class with a migration from that the situation and he of the observation which is
0:11:33	what it is
0:11:34	okay
0:11:34	and he which progressively goes to more and more stationary situation by mean of the increase label of of the
0:11:42	not
0:11:43	and here is
0:11:44	the asymptotic situation which would be a day
0:11:47	with the classical uh will get technique
0:11:50	what
0:11:51	he of course
0:11:51	it that are in this by leader
0:11:54	we can make
0:11:55	the effect situation more or less distant
0:11:57	for
0:11:58	yeah
0:11:59	the the from
0:12:01	the observation from
0:12:02	the station or right
0:12:04	situation
0:12:05	okay so to uh
0:12:07	a exam two two ways of measuring the efficiency of the approach first under H zero which is supposed to
0:12:13	be a a a a a little i was it
0:12:15	is of stationary T
0:12:16	so we
0:12:17	i interested in how the approach
0:12:19	allows to work but use the new it was
0:12:21	each night so we start as a stationary process which is just a given by a ar processing
0:12:26	case
0:12:27	and we look at different transition so we get
0:12:30	one variance the signal
0:12:33	and here this has been done with a a one house realisation of the process and in each case with
0:12:38	a
0:12:38	this work
0:12:39	and here is a function
0:12:41	all the false alarm rate i it is observed
0:12:43	as
0:12:44	a a function of the variance
0:12:46	a standard deviation
0:12:47	phase noise
0:12:48	okay
0:12:49	with three different
0:12:50	that's not here which goes from a five
0:12:53	uh
0:12:54	percentage the percentage per
0:12:56	and here are in blue the situation of the white gaussian noise uh
0:13:01	a transitional surrogates and right
0:13:03	the we and should sort of it and you are the its course
0:13:07	duration given
0:13:08	three like H
0:13:09	and what we see here is yes um
0:13:11	class
0:13:12	uh so gates
0:13:13	and with
0:13:14	that that be a slightly
0:13:16	over
0:13:16	uh
0:13:18	missed
0:13:19	a to what
0:13:20	scribe label
0:13:21	and all
0:13:22	let's
0:13:23	true and
0:13:24	say
0:13:25	and here we see how we control this variation and of course
0:13:28	here we go to a digital
0:13:30	additionally
0:13:31	not do anything in terms
0:13:32	station right
0:13:34	and he would sir
0:13:35	that that is controlled by the which allows
0:13:37	i taking a C which is about one
0:13:40	to be in uh in a
0:13:42	should
0:13:42	for a rubber using single the it's you
0:13:46	and has as a H one hypothesis we get the same kind of the a you're in this case would
0:13:51	take for the signal
0:13:53	something which is supposed to be
0:13:55	and increasing a nonstationary uh signal because it just
0:13:59	the modulation of a white gaussian noise
0:14:02	i something which is one plus a cosine function
0:14:05	okay and we take this
0:14:07	number five which is the ratio between the lights and the period
0:14:10	of the uh
0:14:12	a
0:14:12	uh
0:14:14	see here
0:14:16	and we very here and which control the amplitude modulation and this is a function
0:14:21	this function
0:14:22	i'll think should modulation here
0:14:25	control product
0:14:26	of peaks stationary as it is observed
0:14:28	with here
0:14:29	friend
0:14:30	label
0:14:33	noise
0:14:34	which is
0:14:35	which
0:14:36	and what we see a that you observe
0:14:38	for S your contacts that's and
0:14:40	i
0:14:41	as
0:14:41	a again i two
0:14:43	why
0:14:44	that we get collapsed
0:14:46	for the detection of to say of a a a a a a all the different curves
0:14:50	on the uh what is expected
0:14:52	the right
0:14:54	so as a summary uh E
0:14:56	in these uh paper we
0:14:58	proposed to introduce a new so gates
0:15:00	which is
0:15:01	basically basic something which
0:15:03	control control transition
0:15:05	between
0:15:05	a nonstationary situation in a stationary situation which which is aimed at
0:15:10	at improving
0:15:11	the
0:15:13	let the same more classical tech
0:15:14	now that the be introduced before
0:15:16	of a station raising by means of uh a a strong it was just
0:15:20	a random the phase
0:15:22	and uh what we showed them those example it that this
0:15:25	as opposed to uh
0:15:28	or to a to a the law for that to tuning of the stationary that's specially because of so i'm
0:15:33	rate can be set of the prescribed label with north
0:15:36	section
0:15:37	of course this
0:15:38	a maybe be uh
0:15:40	uh
0:15:41	improvement of nation's the can be jean
0:15:44	for the control for is
0:15:46	uh uh here we just
0:15:48	can can something in which we very
0:15:51	the
0:15:52	five you of the signal of of the noise but we can
0:15:55	i of something which which
0:15:57	much
0:15:58	much more had
0:15:59	the flavour of of a cumulative function will we would have
0:16:03	the distribution of the signal so as to mean
0:16:06	to more and more of the situation of increasing uh
0:16:10	levels of noise
0:16:12	and also we hear chose
0:16:14	not to play with the second by means or in the uh uh the all U T S uh if
0:16:19	i mean D which is a hurst exponent and the hurst exponent
0:16:22	of course
0:16:23	these is to control the correlation of the increment
0:16:26	process
0:16:27	a stationary
0:16:28	stationary increments
0:16:30	and of course adding some correlation in the phase
0:16:33	one oh is used to weight
0:16:36	to uh uh do something which is a less
0:16:39	stationary
0:16:40	then is classical so we get and so we can also imagine to play with these uh extract time there
0:16:45	we all we have to be a a
0:16:48	a a a a time do which is the level of
0:16:50	fractional gaussian noise for
0:16:52	uh improving
0:16:54	in
0:16:55	thank you very much
0:17:05	the the a much or
0:17:07	oh
0:17:07	oh
0:17:08	so additional some but to might the original used for testing for modeling acting data
0:17:15	meaning you the fusion from gaussian
0:17:17	and and are all hypothesis testing or are you that the statistics like don't a slow or correlation integral
0:17:24	see
0:17:25	are
0:17:26	or
0:17:27	time to her so or you
0:17:29	and
0:17:30	and those does to clue what mister mistake stationary
0:17:34	for for non linearity
0:17:35	so we could not differentiate
0:17:37	what are the signal was
0:17:39	stationary but not linear or or non stationary whatever else
0:17:43	no i'm no be pleased that we have a test for nonstationary
0:17:46	but of course my question be done how well normally have it can so about
0:17:53	to question first because uh uh well
0:17:56	alright right
0:17:57	not easy to to to to speak about the
0:18:00	station what what would like to do was to uh
0:18:03	uh
0:18:04	rephrase
0:18:06	all the different P what is stationary for signal
0:18:09	linear or T on you know it is not ready for signal but for system general the a signal
0:18:14	and that's not bad to you either how we can mix to be so here we really want to
0:18:20	to stick to
0:18:21	stationary or or non stationarity
0:18:24	and for a point of view which are avoided all source
0:18:27	some of the that people thought to be they should for classical so we get the than any
0:18:32	and for as you mention the gaussian at and it's important because
0:18:37	in our case we do
0:18:39	so many thing on the data and especially we going in transform plane and then we square and then we
0:18:44	have a rate and so on and so forth
0:18:46	that we are not really uh face with the problem of uh can preserving but
0:18:51	storing not lee the as is
0:18:53	in
0:18:53	uh
0:18:54	something else in
0:18:55	as
0:18:56	in uh
0:18:56	in the
0:18:58	what would say that a at this moment we
0:19:01	didn't think of of like the same kind of thing for uh
0:19:04	for a in i D because i
0:19:06	i i think most
0:19:07	think that been done with so it's for a nonlinear
0:19:10	but not for
0:19:12	a let's say that
0:19:13	non was sold
0:19:15	as a problem for a uh that's or a data and it is for people interested in non linearity
0:19:20	and for us it's not a problem it is not an edge
0:19:24	so
0:19:46	yeah
0:19:52	yes
0:19:54	i i will just distinguish between the
0:19:56	a from type of how split to say
0:19:58	principle the idea of of uh using a so it the data for constructing your reference
0:20:05	station or as the reference for the north i it
0:20:07	stationary stationary you which would be a much in at a to say the feel or something like is this
0:20:11	has been done
0:20:12	we did that
0:20:13	a clean not
0:20:14	not fully but we did that and some images
0:20:17	but for a variation uh i present a to day of course we can imagine to do this in this
0:20:22	has not been done with these transitional sir
0:20:25	but otherwise for teenagers
0:20:26	has been considered
0:20:28	and the idea was a
0:20:29	specially to to think about a or you of texture
0:20:33	and uh
0:20:34	global more structure the
0:20:35	can exist in so that
0:20:36	principal leagues
0:20:37	is that
0:20:38	this has not been
0:20:39	solving in this to give a no one applied to many application
0:20:49	um
0:20:50	and don't
0:20:51	if
0:20:51	um
0:20:52	that's use
0:20:53	because no
0:20:54	traditionally you would use
0:20:55	a good in just for
0:20:57	a a year or not
0:20:59	what you propose is that
0:21:01	you have a
0:21:01	circuits that you can you
0:21:04	well yeah "'cause"
0:21:05	for which you can student demand
0:21:07	oh
0:21:08	non
0:21:09	if any
0:21:11	how does that with data
0:21:13	to test
0:21:14	for uh
0:21:15	station
0:21:15	yeah
0:21:16	well okay so let's to say that what we really is
0:21:19	first
0:21:20	to use so okay
0:21:21	or testing for stationarity
0:21:23	by saying that
0:21:25	so again station or rise of they that so we can
0:21:27	be some distribution of features
0:21:30	thanks
0:21:31	is the set
0:21:31	so
0:21:32	get construct this
0:21:34	and what we observe that this that's
0:21:36	that's
0:21:37	not to work with
0:21:38	probably for instance and what was is is when you are in a situation which is supposed to be a
0:21:43	stationary situation
0:21:45	and what we see with these at point to that when we have this
0:21:49	extract we you freedom
0:21:50	which is
0:21:51	given
0:21:52	i i these sigma square square to stick by me to for the transition that can choose a as were
0:21:57	used
0:21:57	this plot
0:21:58	which is
0:21:59	mostly
0:22:00	control the domain of stationary you which corresponds
0:22:04	more precisely to what we expect the stationary situation
0:22:08	mostly mostly that
0:22:09	but i can see that
0:22:10	sorry we have to
0:22:12	a i so to get off fine
0:22:13	so sorry

TRANSITIONAL SURROGATES

Non-Stationary Signal Analysis

Přednášející: Patrick Flandrin, Autoři: Pierre Borgnat, Patrick Flandrin, CNRS - ENS Lyon, France; André Ferrari, Cédric Richard, Université de Nice Sophia-Antipolis, France