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