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