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 |