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