0:00:14 | okay okay welcome |
---|---|

0:00:16 | my name's eggs and a are and time |

0:00:19 | he of one from germany |

0:00:21 | and um |

0:00:22 | to they have give a talk and my based estimation of the wrong to us |

0:00:26 | a a stationary process |

0:00:28 | not |

0:00:28 | innovations |

0:00:30 | and uh and the talk |

0:00:31 | on |

0:00:33 | a |

0:00:33 | as follows five the action |

0:00:36 | or i give the motivation for |

0:00:37 | problem |

0:00:39 | um |

0:00:39 | sense of method is based on the conventional map method i will to review and |

0:00:44 | mention that that and and now explain the modifications which and necessary |

0:00:49 | to extend |

0:00:50 | the |

0:00:50 | method |

0:00:51 | for a noisy observations |

0:00:53 | talk will be computed |

0:00:55 | a summary and all |

0:00:57 | okay to start the integration |

0:00:59 | start to uh with the |

0:01:01 | process which is |

0:01:03 | described by a white caution stochastic process which is you know by your and |

0:01:08 | and uh and |

0:01:09 | a is can be interpreted as a time index and here we on the right hand |

0:01:15 | an example which years |

0:01:17 | a samples of this process a a given and dark red |

0:01:21 | and uh |

0:01:22 | what |

0:01:22 | you can see here is that the me |

0:01:24 | and the variance of this process |

0:01:25 | a a very with time |

0:01:28 | not the problem is you are not able to observe these samples of |

0:01:31 | a process but you all only you |

0:01:33 | able to to a noise samples which are denoted by the head |

0:01:38 | and uh we assume that gives a efficient error or |

0:01:40 | is a zero mean and uh |

0:01:43 | um um are only |

0:01:45 | a time varying variance of may |

0:01:47 | be strongly time-variant but you as but you know the variance |

0:01:51 | and the question is now |

0:01:53 | um um how can you um find a simple method for estimation of the time varying mean and the variance |

0:01:58 | of |

0:01:59 | process which you can |

0:02:00 | only observe zero |

0:02:01 | and noisy |

0:02:03 | which |

0:02:03 | can of the only know samples of |

0:02:06 | okay |

0:02:07 | and the yeah D is uh we assume that uh the mean and the variance are still time varying and |

0:02:12 | uh |

0:02:12 | we want to exploit the correlations between successive that's of value so use to |

0:02:17 | and uh since |

0:02:18 | this we do want to exploit a priori knowledge which we again from the previous of the divisions |

0:02:23 | and so um |

0:02:25 | for this reason we use a a maximum a posteriori approach |

0:02:28 | based approach |

0:02:30 | and uh |

0:02:31 | um |

0:02:32 | since uh |

0:02:33 | this will be the uh basic |

0:02:36 | for me |

0:02:37 | um and method which we propose i would first review on this |

0:02:40 | which everybody i think |

0:02:41 | we know here |

0:02:43 | for the first uh |

0:02:45 | first case we assume a stationary process that |

0:02:48 | parameters don't vary with time of that being set a fixed |

0:02:51 | fixed mean and variance and we assume but don't all |

0:02:54 | there's is no noise and the visions and the concept i think everybody knows |

0:02:58 | you have a some of the observations |

0:03:00 | do you want to we and and uh |

0:03:03 | you have |

0:03:03 | start with the private yeah which you gain from these of the patients and you try to prove |

0:03:08 | estimates based and new observation be of plus one |

0:03:11 | and uh |

0:03:12 | okay the concept does then you just uh compute some you estimates |

0:03:16 | and uh uh you actually structure the maximization |

0:03:20 | we have P yeah and uh i think everybody knows that the this is composed |

0:03:24 | uh |

0:03:25 | of a private yeah |

0:03:26 | uh which |

0:03:28 | um |

0:03:28 | actually uh gives information from the and you get some |

0:03:32 | and an observation actually |

0:03:34 | okay |

0:03:35 | and not what are components of |

0:03:37 | as where yeah if you have a cost an observation like of course |

0:03:40 | and then you have to assume a can get prior art |

0:03:43 | and this case |

0:03:44 | something like a product of an and inverse scale |

0:03:47 | he's square or distribution multiplied by uh |

0:03:51 | um |

0:03:52 | caution distribution and you have for like the parameters |

0:03:55 | two of them are location and scale me |

0:03:57 | actually be represented |

0:03:59 | three of uh |

0:04:01 | so do you have gained from the previous observations about the mean and |

0:04:05 | same you have for the |

0:04:07 | and for variance you have to decrease |

0:04:08 | freedom |

0:04:09 | scale which i |

0:04:10 | by |

0:04:11 | sci and |

0:04:12 | a on the square |

0:04:13 | and |

0:04:14 | now then you get some that roots for the drama us actually you increase |

0:04:18 | uh the scale |

0:04:19 | and the decrease of freedom but one means you get one observation more |

0:04:23 | and uh the |

0:04:25 | you estimate for the mean so wait a which |

0:04:28 | from the old value and a new observation |

0:04:30 | the weight |

0:04:31 | factor for you to uh for |

0:04:33 | observation is |

0:04:34 | inversely proportional to the number of observations one |

0:04:38 | and uh a similar expression for the said don't on to detect |

0:04:42 | actually |

0:04:43 | and now when you have a computer these parameters |

0:04:45 | you can uh |

0:04:47 | compute the you a maximum |

0:04:48 | the ticket and |

0:04:50 | and you get |

0:04:50 | the estimates for the mean of the variance |

0:04:53 | the standard approach okay |

0:04:54 | what happens now oh okay yeah |

0:04:56 | example |

0:04:57 | and here is um |

0:04:58 | example for so |

0:04:59 | process process the in variance a |

0:05:02 | chosen to one and you have an example of |

0:05:04 | uh a five hundred samples |

0:05:06 | and below know |

0:05:07 | there are estimates which are uh uh you um which are obtained from the |

0:05:12 | um |

0:05:12 | method |

0:05:13 | and and the right hand side you see a posterior pdf which |

0:05:16 | you're is shown after ten observations as you see a a a a lot ten observations |

0:05:21 | uh actually you |

0:05:23 | can can |

0:05:24 | so actually very flat |

0:05:26 | and uh the centre years |

0:05:28 | quite white |

0:05:29 | uh quite |

0:05:30 | uh |

0:05:30 | i i don't the since away from the the design |

0:05:33 | uh a which it should you one one |

0:05:36 | and now what happens if the observations |

0:05:38 | a increases |

0:05:39 | then uh them |

0:05:41 | distribution gets more P key and |

0:05:42 | gets closer to the |

0:05:44 | is i point |

0:05:45 | now |

0:05:46 | see that you get to much more more sure about yours |

0:05:49 | okay now what happens if you are not signal process |

0:05:52 | now uh you still have not streams nations |

0:05:55 | what uh the parent i size to be time varying and what you can do is to introduce a |

0:05:59 | for getting in |

0:06:00 | and keep the degrees of freedom |

0:06:02 | from be increased |

0:06:04 | means you can assign a constant value |

0:06:06 | to both of them |

0:06:07 | and uh a a it means that |

0:06:09 | you you it's you |

0:06:10 | actually use |

0:06:11 | information of and last observation from the past |

0:06:15 | and this value and of |

0:06:16 | of process |

0:06:17 | shows and |

0:06:18 | uh a to me to to between estimation accuracy and tracking in G |

0:06:22 | that means if you have |

0:06:23 | got a a a a high |

0:06:24 | uh value for N |

0:06:26 | you have uh |

0:06:27 | very good estimation accuracy but the tracking and you will of cost now |

0:06:31 | okay now we an example again yeah |

0:06:34 | yeah |

0:06:35 | got process of with a time varying mean and variance |

0:06:38 | a functions for that are given here |

0:06:41 | the and you are |

0:06:41 | you an example with two thousand samples |

0:06:44 | and now we can see |

0:06:46 | you low |

0:06:47 | you estimates for the meeting on the left hand side and the estimates |

0:06:50 | where on the right hand side |

0:06:51 | uh |

0:06:52 | you can see that |

0:06:53 | actually but |

0:06:53 | i go and |

0:06:55 | the the estimates what variance in fact that more course |

0:06:58 | since |

0:06:58 | second or or or uh uh statistics but you're here to be estimated |

0:07:02 | what happens now a few chris number of and then |

0:07:05 | the estimates |

0:07:06 | get most move of "'cause" but you to a since uh the tracking you |

0:07:11 | uh |

0:07:11 | not so good |

0:07:12 | as |

0:07:13 | a a variance can be no but since to |

0:07:16 | um |

0:07:17 | a function for the variance |

0:07:19 | uh |

0:07:19 | uh there is |

0:07:20 | um a very slow and time |

0:07:23 | now what happens now if of noise of the base and that is interesting case and know what oh |

0:07:28 | what kind of of modifications must |

0:07:30 | must be done |

0:07:32 | is not what what happens |

0:07:33 | oh |

0:07:34 | but |

0:07:34 | case of not it's of patients |

0:07:36 | the like to it |

0:07:37 | changes |

0:07:38 | and |

0:07:39 | you in see that you have no uh |

0:07:41 | um |

0:07:42 | at to the variance |

0:07:44 | of the you the noise |

0:07:45 | uh |

0:07:46 | at the corresponding terms of the likelihood function |

0:07:49 | and to a problem is not that a for this like a function that's of course not gonna get prior |

0:07:54 | since the the like to function |

0:07:55 | a factor |

0:07:56 | we have the variance of the observation or |

0:07:59 | is an i i is to and the spectre and there |

0:08:02 | skunk you the prior distribution |

0:08:04 | now what happens |

0:08:05 | here are just apply method |

0:08:07 | a without uh |

0:08:09 | considering that |

0:08:10 | error |

0:08:11 | and you will get a bias |

0:08:12 | and |

0:08:12 | a you few an example of a few once |

0:08:14 | mean and variance again |

0:08:16 | and uh the uh and observation or or |

0:08:19 | is is not a to be random and to |

0:08:22 | as is a uniform |

0:08:23 | draw from |

0:08:24 | this interval here in the right order or and that was a |

0:08:28 | actually a scale science crap function |

0:08:30 | and here or let's that side here |

0:08:32 | oh such a process and dark right again the noise free samples and |

0:08:35 | do not be noisy observations |

0:08:37 | and know what happens you use you what inside |

0:08:40 | uh uh what do you the algorithm actually estimates |

0:08:43 | is |

0:08:44 | um |

0:08:45 | um |

0:08:46 | and very biased since actually yeah a real tries to estimate |

0:08:49 | the |

0:08:50 | variance of the |

0:08:52 | a a couple of uh |

0:08:55 | process of means |

0:08:56 | but loose and but buttons |

0:08:57 | since |

0:08:58 | uh the variance of this process |

0:09:00 | uh make a flat rate very high |

0:09:02 | a time |

0:09:03 | the uh is |

0:09:04 | actually is not |

0:09:06 | a reasonable solution |

0:09:08 | i of the variance is high the or system |

0:09:11 | no not "'cause" all do of what's |

0:09:13 | has to be done |

0:09:14 | it to consider the observation error |

0:09:17 | oh at uh um comes as |

0:09:18 | two components |

0:09:20 | first one is uh |

0:09:22 | we proposed |

0:09:23 | first find a good approximation of the maximum |

0:09:26 | first you P yeah and the scale parameter |

0:09:29 | and the second step |

0:09:30 | we have proposed to approximate the posterior pdf |

0:09:33 | with the same shape |

0:09:35 | right I |

0:09:36 | he's that the maximum of the true posterior and the approximate |

0:09:39 | steering must match |

0:09:41 | and |

0:09:42 | we have |

0:09:43 | assume the same degrees of freedom from for the steered yeah |

0:09:47 | and the but and the approximate |

0:09:48 | posterior you have whatever that means |

0:09:51 | now a come on the first |

0:09:52 | a point |

0:09:53 | yeah i have |

0:09:54 | um |

0:09:55 | the true posterior P that looks quite complicated but |

0:09:59 | not |

0:10:00 | think you're is important i will |

0:10:02 | so you bought things here |

0:10:04 | and principle you could uh take this |

0:10:06 | as you if it happens to a local search of course |

0:10:09 | and um |

0:10:10 | about |

0:10:12 | as functions but |

0:10:12 | this would |

0:10:13 | on the one and very computationally expensive and this |

0:10:16 | point is that |

0:10:18 | a a you know |

0:10:19 | i could compute the maximal this |

0:10:20 | a it would |

0:10:22 | have no uh |

0:10:23 | clue all |

0:10:24 | escape from |

0:10:25 | now comes |

0:10:27 | a whole idea |

0:10:28 | if you look at these expressions uh which i you and colour |

0:10:31 | they were sampled you expressions |

0:10:33 | a a of the prior you have |

0:10:36 | and the prior yeah these expressions are constants and now you the expressions are |

0:10:40 | actually um |

0:10:42 | a functions of the variance |

0:10:45 | and now if you look |

0:10:46 | at these functions |

0:10:47 | for example |

0:10:48 | at the scale parameter for |

0:10:50 | for the um |

0:10:51 | for the mean |

0:10:52 | see that uh these function |

0:10:54 | they they between you probably tell |

0:10:56 | a couple of and and uh the new problem car and that's one |

0:11:00 | and uh |

0:11:01 | same same uh |

0:11:02 | holds for meeting |

0:11:04 | lies between me mean |

0:11:06 | and |

0:11:06 | you |

0:11:08 | now all idea was motivated by the fact that own |

0:11:11 | those values |

0:11:13 | uh |

0:11:13 | which are in the |

0:11:15 | vicinity of the true |

0:11:17 | uh variance variance uh since |

0:11:19 | the are |

0:11:20 | prior video will have a high values and that region |

0:11:23 | and for this reason |

0:11:25 | proposed approximate |

0:11:26 | these functions |

0:11:28 | oh |

0:11:28 | the variance by constant |

0:11:30 | by applying in the |

0:11:32 | variance estimate of the problem of the |

0:11:35 | um |

0:11:36 | process of of and |

0:11:37 | for a from the |

0:11:39 | a a time and |

0:11:40 | and |

0:11:41 | i do this we get constants |

0:11:43 | for yeah |

0:11:44 | um skate around at all in the mean out |

0:11:47 | and |

0:11:48 | first uh advantage that we uh what the maximum search |

0:11:51 | a in |

0:11:53 | and the second uh advantage |

0:11:55 | is that we |

0:11:56 | get a scale parameter |

0:11:58 | and you can see you also what happens if we do this |

0:12:02 | for example look at uh |

0:12:03 | channel |

0:12:05 | a here |

0:12:06 | uh a if the observation error is very high |

0:12:09 | and |

0:12:09 | you know it that would be done need to but this observation error or |

0:12:12 | and |

0:12:13 | the new estimate actually will |

0:12:15 | E |

0:12:16 | equal to the oldest estimate that means |

0:12:18 | that from a very no it's it's you can't learn you think that you |

0:12:21 | stick to the old value |

0:12:22 | and what happens if |

0:12:24 | it the observation are or is very low input put there as to the old to estimate here |

0:12:29 | then |

0:12:29 | uh |

0:12:30 | term maybe you can not do you get your |

0:12:33 | and |

0:12:33 | expression which is equal to one and that means that you can learn very much from this |

0:12:37 | H |

0:12:38 | okay |

0:12:40 | okay and the same of cost a holds for the mean |

0:12:43 | and now had |

0:12:45 | found that the mean |

0:12:46 | and uh the scale parameter |

0:12:48 | we in the second step um |

0:12:50 | we find the maximum of the post your pdf with |

0:12:53 | respect to the variance |

0:12:55 | and uh we have shown and all pay but that this is equivalent to finding the only root of for |

0:12:59 | for all the long you'll and known into well |

0:13:02 | and this can be uh done very easily you with a bisection method and uh |

0:13:07 | later later vacation of a new method |

0:13:09 | very |

0:13:09 | you you done |

0:13:11 | very simple and computationally efficient |

0:13:14 | on the advantage of |

0:13:15 | actually |

0:13:17 | okay and uh are now we come to |

0:13:19 | a second step now we have found the maximum of the true posterior and we have |

0:13:22 | found an approximate of the scaling parameter |

0:13:25 | and now |

0:13:26 | we |

0:13:27 | approximate this |

0:13:28 | a with a |

0:13:29 | with a a P D F which has the same shape as a prior in order to recursively applied met |

0:13:35 | and |

0:13:35 | for this |

0:13:36 | we have to choose a hyper parameters |

0:13:38 | two |

0:13:39 | first have parameters |

0:13:40 | which are already a which referring to be in a or time and |

0:13:45 | are we have |

0:13:45 | to choose |

0:13:46 | and the |

0:13:47 | parameters |

0:13:48 | sign which once in a while |

0:13:50 | observations actually |

0:13:51 | and we set it |

0:13:53 | uh actually to the number |

0:13:54 | a couple i am plus one |

0:13:56 | and |

0:13:57 | or the setting we also get |

0:13:59 | and |

0:14:00 | this scale problem at a for the variance |

0:14:02 | no i just an example of the true posterior pdf only that and side and them |

0:14:06 | approximate posterior pdf |

0:14:08 | right hand side and |

0:14:10 | i do not know if you can see any difference |

0:14:12 | what's uh |

0:14:13 | the that yeah is |

0:14:15 | the the rotated |

0:14:17 | to the right hand side here |

0:14:19 | and |

0:14:19 | this year is actually symmetric symmetrical to this axis yeah |

0:14:23 | but uh |

0:14:24 | i want to show actually that are quite simple |

0:14:27 | now an example |

0:14:29 | um |

0:14:30 | yeah again |

0:14:31 | process with the |

0:14:32 | um a constant variance and a |

0:14:35 | the observation errors again random |

0:14:37 | and |

0:14:38 | we have a a comparison but be |

0:14:40 | a conventional method and the |

0:14:41 | proposed method |

0:14:42 | on left hand side |

0:14:44 | use you first |

0:14:45 | a comparison between the mean estimate |

0:14:48 | estimation |

0:14:49 | yeah the could mention that of course |

0:14:51 | estimates the true mean |

0:14:52 | since the bear a sense to mean of the blue samples of cost |

0:14:55 | the same |

0:14:56 | as that of the dark |

0:14:58 | right samples |

0:14:59 | since the me since the was a vision error is zero mean |

0:15:03 | but |

0:15:03 | see |

0:15:04 | that the uh propose not that estimates to be more accurate |

0:15:07 | and |

0:15:08 | same same uh for the their an system it you see that |

0:15:11 | is no why is here and that the variances |

0:15:14 | actually |

0:15:15 | estimate is quite accurate while here in the |

0:15:17 | can mention and that estimation method |

0:15:19 | C |

0:15:20 | a quite by |

0:15:22 | now an example for nonstationary process |

0:15:25 | now we have a |

0:15:27 | a time varying variance |

0:15:29 | um |

0:15:30 | we have here an example of can with two thousand observations and the observation noise is not random again |

0:15:36 | and |

0:15:36 | yeah |

0:15:37 | a the right |

0:15:38 | ball of the right well as yeah a controlled by a factor of C which controls the maximum variance |

0:15:43 | a terrible |

0:15:45 | here |

0:15:46 | a comparison of the you |

0:15:48 | performance on the that the mention a method |

0:15:51 | this see that do the estimates fact a very |

0:15:54 | hi and so on but |

0:15:55 | a method yeah |

0:15:57 | more more at here and |

0:15:59 | again here is he |

0:16:00 | for the variance estimate at very |

0:16:02 | by a very high bias for the conventional method |

0:16:05 | which is not you |

0:16:06 | true |

0:16:07 | for the |

0:16:08 | but method |

0:16:08 | proposed |

0:16:10 | um |

0:16:11 | and |

0:16:12 | i have |

0:16:12 | just to slides |

0:16:13 | i think |

0:16:14 | will |

0:16:15 | you okay |

0:16:16 | um |

0:16:17 | no what we do uh what do you that to so we measure the root mean squared error |

0:16:22 | when we better you about |

0:16:24 | right a right part of the |

0:16:26 | interval for the uh |

0:16:28 | observation or or |

0:16:29 | and what you can see is here |

0:16:31 | the um would be it's good as for the mean and the variance for |

0:16:35 | a conventional and the proposed method and |

0:16:37 | but you can see use that we always |

0:16:39 | what was that all performance is always improve compared to the dimension method |

0:16:43 | and |

0:16:44 | that to improvements |

0:16:45 | get more pronounced with increasing use of observation noise |

0:16:48 | oh come |

0:16:49 | fusion |

0:16:50 | we have |

0:16:50 | a an approximate map approach for the estimation |

0:16:53 | slowly time varying parameters of not stationary white gaussian random process |

0:16:57 | and we have shown |

0:16:58 | but in yeah |

0:17:00 | um |

0:17:01 | the case of absence of observation noise is equivalent to conventional map method |

0:17:06 | but |

0:17:06 | in |

0:17:07 | presence of observation noise |

0:17:09 | is |

0:17:09 | proved estimation accuracy |

0:17:11 | and what is important that the computation that |

0:17:14 | but the only restrict |

0:17:16 | showing this function is that |

0:17:18 | variance of the observation error has to be no |

0:17:21 | and this is you that |

0:17:23 | papers is that |

0:17:24 | we have to analyse the effects |

0:17:26 | what happens |

0:17:27 | a if you do not know you um |

0:17:29 | yeah it's of the observation are right exactly but just an estimate of |

0:17:34 | but i i i can say that uh this method will not be |

0:17:37 | that's |

0:17:38 | it sensitive to this |

0:17:40 | now |

0:17:41 | future future |

0:17:42 | thank you remote real tension |

0:17:49 | and for a couple of questions |

0:17:53 | yes one |

0:18:04 | you process |

0:18:05 | why |

0:18:06 | sure |

0:18:06 | i |

0:18:08 | you |

0:18:09 | gives |

0:18:11 | uh i suppose that this question would come |

0:18:13 | uh |

0:18:15 | um um so far we assume all the cases |

0:18:18 | is just a a just a a method uh |

0:18:20 | if |

0:18:21 | we have these assumptions and we can uh we can for a give some |

0:18:25 | uh |

0:18:26 | some method to estimate the problem |

0:18:28 | may you that might be |

0:18:30 | might be an application for example of you some uh |

0:18:33 | sensor signals which and noisy and you have |

0:18:35 | can do all the observation are a which you can expect |

0:18:38 | and then you |

0:18:39 | um |

0:18:40 | i able to estimate something like a mean |

0:18:42 | uh |

0:18:43 | like a bias in the mean or something like is |

0:18:46 | this week an application but |

0:18:47 | we do |

0:18:48 | we did not uh |

0:18:50 | find |

0:18:51 | and |

0:18:51 | a calm concrete applications |

0:18:54 | and also can i guess |

0:18:57 | i |

0:18:58 | i |

0:19:00 | like |

0:19:01 | i |

0:19:06 | oh |

0:19:08 | cover |

0:19:09 | and |

0:19:13 | oh with |

0:19:15 | oh |

0:19:16 | uh no we didn't uh |

0:19:19 | didn't |

0:19:20 | and nice |

0:19:21 | with |

0:19:22 | with connection with home more more |

0:19:24 | but |

0:19:27 | yeah but |

0:19:30 | yes |

0:19:46 | uh no we didn't |

0:19:48 | um |

0:19:50 | you mean uh |

0:19:52 | you with to the proposed to compare are you performance of all with them with |

0:19:57 | which one |

0:19:58 | which what |

0:20:08 | okay |

0:20:11 | of course |

0:20:16 | uh |

0:20:16 | no we have measure are actually you you do the true accuracy which |

0:20:20 | uh with the measure like a lot of something |

0:20:23 | like that |

0:20:24 | we just uh |

0:20:25 | so that this method works quite well and so uh happens uh and a last |

0:20:29 | yes the the performance and |

0:20:31 | this kind of a metric |

0:20:33 | i Q |

0:20:33 | post |

0:20:37 | okay thank you |

0:20:39 | a standard speaker |