0:00:14okay okay welcome
0:00:16my name's eggs and a are and time
0:00:19he of one from germany
0:00:21and um
0:00:22to they have give a talk and my based estimation of the wrong to us
0:00:26a a stationary process
0:00:30and uh and the talk
0:00:33as follows five the action
0:00:36or i give the motivation for
0:00:39sense of method is based on the conventional map method i will to review and
0:00:44mention that that and and now explain the modifications which and necessary
0:00:49to extend
0:00:51for a noisy observations
0:00:53talk will be computed
0:00:55a summary and all
0:00:57okay to start the integration
0:00:59start to uh with the
0:01:01process which is
0:01:03described by a white caution stochastic process which is you know by your and
0:01:08and uh and
0:01:09a is can be interpreted as a time index and here we on the right hand
0:01:15an example which years
0:01:17a samples of this process a a given and dark red
0:01:21and uh
0:01:22you can see here is that the me
0:01:24and the variance of this process
0:01:25a a very with time
0:01:28not the problem is you are not able to observe these samples of
0:01:31a process but you all only you
0:01:33able to to a noise samples which are denoted by the head
0:01:38and uh we assume that gives a efficient error or
0:01:40is a zero mean and uh
0:01:43um um are only
0:01:45a time varying variance of may
0:01:47be strongly time-variant but you as but you know the variance
0:01:51and the question is now
0:01:53um um how can you um find a simple method for estimation of the time varying mean and the variance
0:01:59process which you can
0:02:00only observe zero
0:02:01and noisy
0:02:03can of the only know samples of
0:02:07and the yeah D is uh we assume that uh the mean and the variance are still time varying and
0:02:12we want to exploit the correlations between successive that's of value so use to
0:02:17and uh since
0:02:18this we do want to exploit a priori knowledge which we again from the previous of the divisions
0:02:23and so um
0:02:25for this reason we use a a maximum a posteriori approach
0:02:28based approach
0:02:30and uh
0:02:32since uh
0:02:33this will be the uh basic
0:02:36for me
0:02:37um and method which we propose i would first review on this
0:02:40which everybody i think
0:02:41we know here
0:02:43for the first uh
0:02:45first case we assume a stationary process that
0:02:48parameters don't vary with time of that being set a fixed
0:02:51fixed mean and variance and we assume but don't all
0:02:54there's is no noise and the visions and the concept i think everybody knows
0:02:58you have a some of the observations
0:03:00do you want to we and and uh
0:03:03you have
0:03:03start with the private yeah which you gain from these of the patients and you try to prove
0:03:08estimates based and new observation be of plus one
0:03:11and uh
0:03:12okay the concept does then you just uh compute some you estimates
0:03:16and uh uh you actually structure the maximization
0:03:20we have P yeah and uh i think everybody knows that the this is composed
0:03:25of a private yeah
0:03:26uh which
0:03:28actually uh gives information from the and you get some
0:03:32and an observation actually
0:03:35and not what are components of
0:03:37as where yeah if you have a cost an observation like of course
0:03:40and then you have to assume a can get prior art
0:03:43and this case
0:03:44something like a product of an and inverse scale
0:03:47he's square or distribution multiplied by uh
0:03:52caution distribution and you have for like the parameters
0:03:55two of them are location and scale me
0:03:57actually be represented
0:03:59three of uh
0:04:01so do you have gained from the previous observations about the mean and
0:04:05same you have for the
0:04:07and for variance you have to decrease
0:04:09scale which i
0:04:11sci and
0:04:12a on the square
0:04:14now then you get some that roots for the drama us actually you increase
0:04:18uh the scale
0:04:19and the decrease of freedom but one means you get one observation more
0:04:23and uh the
0:04:25you estimate for the mean so wait a which
0:04:28from the old value and a new observation
0:04:30the weight
0:04:31factor for you to uh for
0:04:33observation is
0:04:34inversely proportional to the number of observations one
0:04:38and uh a similar expression for the said don't on to detect
0:04:43and now when you have a computer these parameters
0:04:45you can uh
0:04:47compute the you a maximum
0:04:48the ticket and
0:04:50and you get
0:04:50the estimates for the mean of the variance
0:04:53the standard approach okay
0:04:54what happens now oh okay yeah
0:04:57and here is um
0:04:58example for so
0:04:59process process the in variance a
0:05:02chosen to one and you have an example of
0:05:04uh a five hundred samples
0:05:06and below know
0:05:07there are estimates which are uh uh you um which are obtained from the
0:05:13and and the right hand side you see a posterior pdf which
0:05:16you're is shown after ten observations as you see a a a a lot ten observations
0:05:21uh actually you
0:05:23can can
0:05:24so actually very flat
0:05:26and uh the centre years
0:05:28quite white
0:05:29uh quite
0:05:30i i don't the since away from the the design
0:05:33uh a which it should you one one
0:05:36and now what happens if the observations
0:05:38a increases
0:05:39then uh them
0:05:41distribution gets more P key and
0:05:42gets closer to the
0:05:44is i point
0:05:46see that you get to much more more sure about yours
0:05:49okay now what happens if you are not signal process
0:05:52now uh you still have not streams nations
0:05:55what uh the parent i size to be time varying and what you can do is to introduce a
0:05:59for getting in
0:06:00and keep the degrees of freedom
0:06:02from be increased
0:06:04means you can assign a constant value
0:06:06to both of them
0:06:07and uh a a it means that
0:06:09you you it's you
0:06:10actually use
0:06:11information of and last observation from the past
0:06:15and this value and of
0:06:16of process
0:06:17shows and
0:06:18uh a to me to to between estimation accuracy and tracking in G
0:06:22that means if you have
0:06:23got a a a a high
0:06:24uh value for N
0:06:26you have uh
0:06:27very good estimation accuracy but the tracking and you will of cost now
0:06:31okay now we an example again yeah
0:06:35got process of with a time varying mean and variance
0:06:38a functions for that are given here
0:06:41the and you are
0:06:41you an example with two thousand samples
0:06:44and now we can see
0:06:46you low
0:06:47you estimates for the meeting on the left hand side and the estimates
0:06:50where on the right hand side
0:06:52you can see that
0:06:53actually but
0:06:53i go and
0:06:55the the estimates what variance in fact that more course
0:06:58second or or or uh uh statistics but you're here to be estimated
0:07:02what happens now a few chris number of and then
0:07:05the estimates
0:07:06get most move of "'cause" but you to a since uh the tracking you
0:07:11not so good
0:07:13a a variance can be no but since to
0:07:17a function for the variance
0:07:19uh there is
0:07:20um a very slow and time
0:07:23now what happens now if of noise of the base and that is interesting case and know what oh
0:07:28what kind of of modifications must
0:07:30must be done
0:07:32is not what what happens
0:07:34case of not it's of patients
0:07:36the like to it
0:07:39you in see that you have no uh
0:07:42at to the variance
0:07:44of the you the noise
0:07:46at the corresponding terms of the likelihood function
0:07:49and to a problem is not that a for this like a function that's of course not gonna get prior
0:07:54since the the like to function
0:07:55a factor
0:07:56we have the variance of the observation or
0:07:59is an i i is to and the spectre and there
0:08:02skunk you the prior distribution
0:08:04now what happens
0:08:05here are just apply method
0:08:07a without uh
0:08:09considering that
0:08:11and you will get a bias
0:08:12a you few an example of a few once
0:08:14mean and variance again
0:08:16and uh the uh and observation or or
0:08:19is is not a to be random and to
0:08:22as is a uniform
0:08:23draw from
0:08:24this interval here in the right order or and that was a
0:08:28actually a scale science crap function
0:08:30and here or let's that side here
0:08:32oh such a process and dark right again the noise free samples and
0:08:35do not be noisy observations
0:08:37and know what happens you use you what inside
0:08:40uh uh what do you the algorithm actually estimates
0:08:46and very biased since actually yeah a real tries to estimate
0:08:50variance of the
0:08:52a a couple of uh
0:08:55process of means
0:08:56but loose and but buttons
0:08:58uh the variance of this process
0:09:00uh make a flat rate very high
0:09:02a time
0:09:03the uh is
0:09:04actually is not
0:09:06a reasonable solution
0:09:08i of the variance is high the or system
0:09:11no not "'cause" all do of what's
0:09:13has to be done
0:09:14it to consider the observation error
0:09:17oh at uh um comes as
0:09:18two components
0:09:20first one is uh
0:09:22we proposed
0:09:23first find a good approximation of the maximum
0:09:26first you P yeah and the scale parameter
0:09:29and the second step
0:09:30we have proposed to approximate the posterior pdf
0:09:33with the same shape
0:09:35right I
0:09:36he's that the maximum of the true posterior and the approximate
0:09:39steering must match
0:09:42we have
0:09:43assume the same degrees of freedom from for the steered yeah
0:09:47and the but and the approximate
0:09:48posterior you have whatever that means
0:09:51now a come on the first
0:09:52a point
0:09:53yeah i have
0:09:55the true posterior P that looks quite complicated but
0:10:00think you're is important i will
0:10:02so you bought things here
0:10:04and principle you could uh take this
0:10:06as you if it happens to a local search of course
0:10:09and um
0:10:12as functions but
0:10:12this would
0:10:13on the one and very computationally expensive and this
0:10:16point is that
0:10:18a a you know
0:10:19i could compute the maximal this
0:10:20a it would
0:10:22have no uh
0:10:23clue all
0:10:24escape from
0:10:25now comes
0:10:27a whole idea
0:10:28if you look at these expressions uh which i you and colour
0:10:31they were sampled you expressions
0:10:33a a of the prior you have
0:10:36and the prior yeah these expressions are constants and now you the expressions are
0:10:40actually um
0:10:42a functions of the variance
0:10:45and now if you look
0:10:46at these functions
0:10:47for example
0:10:48at the scale parameter for
0:10:50for the um
0:10:51for the mean
0:10:52see that uh these function
0:10:54they they between you probably tell
0:10:56a couple of and and uh the new problem car and that's one
0:11:00and uh
0:11:01same same uh
0:11:02holds for meeting
0:11:04lies between me mean
0:11:08now all idea was motivated by the fact that own
0:11:11those values
0:11:13which are in the
0:11:15vicinity of the true
0:11:17uh variance variance uh since
0:11:19the are
0:11:20prior video will have a high values and that region
0:11:23and for this reason
0:11:25proposed approximate
0:11:26these functions
0:11:28the variance by constant
0:11:30by applying in the
0:11:32variance estimate of the problem of the
0:11:36process of of and
0:11:37for a from the
0:11:39a a time and
0:11:41i do this we get constants
0:11:43for yeah
0:11:44um skate around at all in the mean out
0:11:48first uh advantage that we uh what the maximum search
0:11:51a in
0:11:53and the second uh advantage
0:11:55is that we
0:11:56get a scale parameter
0:11:58and you can see you also what happens if we do this
0:12:02for example look at uh
0:12:05a here
0:12:06uh a if the observation error is very high
0:12:09you know it that would be done need to but this observation error or
0:12:13the new estimate actually will
0:12:16equal to the oldest estimate that means
0:12:18that from a very no it's it's you can't learn you think that you
0:12:21stick to the old value
0:12:22and what happens if
0:12:24it the observation are or is very low input put there as to the old to estimate here
0:12:30term maybe you can not do you get your
0:12:33expression which is equal to one and that means that you can learn very much from this
0:12:40okay and the same of cost a holds for the mean
0:12:43and now had
0:12:45found that the mean
0:12:46and uh the scale parameter
0:12:48we in the second step um
0:12:50we find the maximum of the post your pdf with
0:12:53respect to the variance
0:12:55and uh we have shown and all pay but that this is equivalent to finding the only root of for
0:12:59for all the long you'll and known into well
0:13:02and this can be uh done very easily you with a bisection method and uh
0:13:07later later vacation of a new method
0:13:09you you done
0:13:11very simple and computationally efficient
0:13:14on the advantage of
0:13:17okay and uh are now we come to
0:13:19a second step now we have found the maximum of the true posterior and we have
0:13:22found an approximate of the scaling parameter
0:13:25and now
0:13:27approximate this
0:13:28a with a
0:13:29with a a P D F which has the same shape as a prior in order to recursively applied met
0:13:35for this
0:13:36we have to choose a hyper parameters
0:13:39first have parameters
0:13:40which are already a which referring to be in a or time and
0:13:45are we have
0:13:45to choose
0:13:46and the
0:13:48sign which once in a while
0:13:50observations actually
0:13:51and we set it
0:13:53uh actually to the number
0:13:54a couple i am plus one
0:13:57or the setting we also get
0:14:00this scale problem at a for the variance
0:14:02no i just an example of the true posterior pdf only that and side and them
0:14:06approximate posterior pdf
0:14:08right hand side and
0:14:10i do not know if you can see any difference
0:14:12what's uh
0:14:13the that yeah is
0:14:15the the rotated
0:14:17to the right hand side here
0:14:19this year is actually symmetric symmetrical to this axis yeah
0:14:23but uh
0:14:24i want to show actually that are quite simple
0:14:27now an example
0:14:30yeah again
0:14:31process with the
0:14:32um a constant variance and a
0:14:35the observation errors again random
0:14:38we have a a comparison but be
0:14:40a conventional method and the
0:14:41proposed method
0:14:42on left hand side
0:14:44use you first
0:14:45a comparison between the mean estimate
0:14:49yeah the could mention that of course
0:14:51estimates the true mean
0:14:52since the bear a sense to mean of the blue samples of cost
0:14:55the same
0:14:56as that of the dark
0:14:58right samples
0:14:59since the me since the was a vision error is zero mean
0:15:04that the uh propose not that estimates to be more accurate
0:15:08same same uh for the their an system it you see that
0:15:11is no why is here and that the variances
0:15:15estimate is quite accurate while here in the
0:15:17can mention and that estimation method
0:15:20a quite by
0:15:22now an example for nonstationary process
0:15:25now we have a
0:15:27a time varying variance
0:15:30we have here an example of can with two thousand observations and the observation noise is not random again
0:15:37a the right
0:15:38ball of the right well as yeah a controlled by a factor of C which controls the maximum variance
0:15:43a terrible
0:15:46a comparison of the you
0:15:48performance on the that the mention a method
0:15:51this see that do the estimates fact a very
0:15:54hi and so on but
0:15:55a method yeah
0:15:57more more at here and
0:15:59again here is he
0:16:00for the variance estimate at very
0:16:02by a very high bias for the conventional method
0:16:05which is not you
0:16:07for the
0:16:08but method
0:16:12i have
0:16:12just to slides
0:16:13i think
0:16:15you okay
0:16:17no what we do uh what do you that to so we measure the root mean squared error
0:16:22when we better you about
0:16:24right a right part of the
0:16:26interval for the uh
0:16:28observation or or
0:16:29and what you can see is here
0:16:31the um would be it's good as for the mean and the variance for
0:16:35a conventional and the proposed method and
0:16:37but you can see use that we always
0:16:39what was that all performance is always improve compared to the dimension method
0:16:44that to improvements
0:16:45get more pronounced with increasing use of observation noise
0:16:48oh come
0:16:50we have
0:16:50a an approximate map approach for the estimation
0:16:53slowly time varying parameters of not stationary white gaussian random process
0:16:57and we have shown
0:16:58but in yeah
0:17:01the case of absence of observation noise is equivalent to conventional map method
0:17:07presence of observation noise
0:17:09proved estimation accuracy
0:17:11and what is important that the computation that
0:17:14but the only restrict
0:17:16showing this function is that
0:17:18variance of the observation error has to be no
0:17:21and this is you that
0:17:23papers is that
0:17:24we have to analyse the effects
0:17:26what happens
0:17:27a if you do not know you um
0:17:29yeah it's of the observation are right exactly but just an estimate of
0:17:34but i i i can say that uh this method will not be
0:17:38it sensitive to this
0:17:41future future
0:17:42thank you remote real tension
0:17:49and for a couple of questions
0:17:53yes one
0:18:04you process
0:18:11uh i suppose that this question would come
0:18:15um um so far we assume all the cases
0:18:18is just a a just a a method uh
0:18:21we have these assumptions and we can uh we can for a give some
0:18:26some method to estimate the problem
0:18:28may you that might be
0:18:30might be an application for example of you some uh
0:18:33sensor signals which and noisy and you have
0:18:35can do all the observation are a which you can expect
0:18:38and then you
0:18:40i able to estimate something like a mean
0:18:43like a bias in the mean or something like is
0:18:46this week an application but
0:18:47we do
0:18:48we did not uh
0:18:51a calm concrete applications
0:18:54and also can i guess
0:19:13oh with
0:19:16uh no we didn't uh
0:19:20and nice
0:19:22with connection with home more more
0:19:27yeah but
0:19:46uh no we didn't
0:19:50you mean uh
0:19:52you with to the proposed to compare are you performance of all with them with
0:19:57which one
0:19:58which what
0:20:11of course
0:20:16no we have measure are actually you you do the true accuracy which
0:20:20uh with the measure like a lot of something
0:20:23like that
0:20:24we just uh
0:20:25so that this method works quite well and so uh happens uh and a last
0:20:29yes the the performance and
0:20:31this kind of a metric
0:20:33i Q
0:20:37okay thank you
0:20:39a standard speaker