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

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

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

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

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

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

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