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