Přepis řeči - PERIODIC CRB FOR NON-BAYESIAN PARAMETER ESTIMATION

0:00:13	phase estimation
0:00:15	in detail has many applications
0:00:16	such as radar are so no communications and speech and out not process
0:00:23	no let's begin it's example
0:00:25	yeah this is the example of
0:00:27	in on bayesian and phase estimation
0:00:29	in this example we demonstrate to a main problems in the general
0:00:33	periodic parameter estimation
0:00:36	so
0:00:37	in this example to consider the following more than
0:00:40	okay X and observations
0:00:42	a the amplitude
0:00:43	which is assumed to be known
0:00:45	data
0:00:46	is the unknown parameter
0:00:48	this is deterministic part are so we are and non based an estimation
0:00:52	and it is between minus and i
0:00:55	here we assume that we have a
0:00:57	so put a gaussian mean my a complex noise
0:01:00	with known fine
0:01:02	you can see that this small it is pretty a T
0:01:04	with this
0:01:05	the
0:01:05	that that's but to the uh problem
0:01:08	in this case it is a on that this is the common at on
0:01:12	okay it is proportional to the inverse
0:01:14	signal to noise ratio
0:01:16	and the maximum that the estimate that was given by the stuff
0:01:20	is
0:01:21	and that all of the
0:01:22	sample mean weight it's company
0:01:25	now and this data
0:01:27	you can see them in school uh are against the signal to noise ratio
0:01:31	the that is
0:01:32	the come but i found in and and the nine is the mse of the maximum like to estimate or
0:01:38	now it can defend that for and now
0:01:41	the common lower bound is achievable by the maximum likelihood estimate of
0:01:45	that is the common out about but it's of it where the performance
0:01:48	in the asymptotic region
0:01:51	i uh for less than all
0:01:53	you can see that the a and then estimate a close the problem
0:01:57	in fact and estimate or between minus a week was the bound because the body you know
0:02:02	it goes to infinity
0:02:05	a not designed for this phenomenon is that the comment is about that the performance of any unbiased
0:02:11	estimator
0:02:12	but in this case the maximum likelihood is biased estimate all
0:02:16	in fact there is no
0:02:17	a uniformly unbiased estimator for this case
0:02:21	so the comment on this very good asymptotic region
0:02:25	but this is not valid for low snrs
0:02:28	okay a and a
0:02:30	and to make a request
0:02:34	okay
0:02:35	so the conclusion the main purpose in the genital periodic parameter estimation
0:02:40	are different
0:02:42	first the conventional means got a good deal itself is inappropriate
0:02:46	for this estimation
0:02:48	a this is illustrated here
0:02:49	you can see that if you have to estimate and get
0:02:52	data
0:02:53	and we have a good estimate of it i
0:02:56	hmmm
0:02:56	there is this
0:02:59	and the mse use bits to use this uh
0:03:02	how important that until we discuss you know
0:03:07	so we should do that we should not let us instead of the mse
0:03:12	to make some is the prior
0:03:15	the second the second problem is that no uniformly unbiased estimator exists
0:03:21	in this case
0:03:22	and this is it right for the phase estimation problem
0:03:26	uses at and periodic parameter estimation
0:03:30	the periodic likelihood function
0:03:32	in a it's a have been proved in this too
0:03:37	no not at non bayesian estimation
0:03:39	in the minimization of the mse
0:03:42	another criterion
0:03:43	should be done under some restriction and the stuff constraint
0:03:47	the crime and unbiasedness
0:03:49	is it because no unbiased estimate like this
0:03:52	so we should find another constraint another station
0:03:57	finally
0:03:58	S was set to cram it may not be valued at low snrs and we want to find
0:04:03	to do that
0:04:04	which will do that at any snr
0:04:08	and this is exactly what we did in this right
0:04:11	we have a square periodic uh i'm this inequality on
0:04:15	instead of tennessee
0:04:18	a predefined periodic unbiasedness
0:04:21	and the constraint instead of this constraint
0:04:25	and the newer version of the content
0:04:28	which is a valid that any snr
0:04:32	no yeah can see the general what what in this work
0:04:36	yes just that our product your data
0:04:38	is that a many stick
0:04:40	and this is between minus point by
0:04:42	but is is on the for the sake of simplicity you can take any
0:04:46	time period
0:04:48	yes of the parabola space in which are made it's is the observation space
0:04:51	P
0:04:52	is the family of per emails
0:04:54	prom a tight by to by the unknown product or
0:04:59	a is the hundred observation vector and P that is an estimate of that
0:05:03	which is function from the observation space to minus by
0:05:08	now we not that even if the estimate all is restricted to the original of
0:05:12	minus by by
0:05:14	and the parameter at is also this a region
0:05:17	the is that of estimation L bit that minus the data
0:05:21	can be in general and in
0:05:23	minus two part about
0:05:24	so we should of the than the part of weight
0:05:29	and
0:05:29	yeah we use this quickly and the mean square to calculate you're
0:05:33	the S P it cost function is given here this is the square error of the preview a
0:05:38	estimation in or remote able to buy or
0:05:41	the of the estimation or
0:05:44	yeah the model but to by april or map
0:05:47	the estimation L to more by by
0:05:50	and you can see here
0:05:51	the main
0:05:52	scalability that the S P against
0:05:54	but yeah this is pretty loaded
0:05:57	a non-negative and and
0:05:59	is a better and that's a non convex
0:06:04	no of to define
0:06:05	the P the can best miss
0:06:07	and is the and the phonation for one best mess
0:06:10	and this the phonation is a a and buys this with respect to specific cost function
0:06:16	according to this definition and have to make but we said to be yeah have not by that
0:06:21	with respect to the cost function and
0:06:24	if is the expectation
0:06:26	a type like it a like to two parameter
0:06:29	of this cost function
0:06:31	if you is the true parameter but that it is there and them
0:06:35	and they have a parameter is that and the parameters
0:06:39	i the right
0:06:40	and estimator is on
0:06:42	if it was closer
0:06:43	to that's for parameter
0:06:45	but and then i mean i have a problem of in our problem space
0:06:49	the closeness
0:06:50	is a measure of using the specific cost function K
0:06:55	okay
0:06:56	the basic example sample for this a a i'm was of the best the conventional and bias net
0:07:01	and i'm that the mean square error cost function that unbiasedness is not you to
0:07:06	no no by smith
0:07:07	the expectation of the estimate the is equal to the to a parameter it's
0:07:12	so yeah no the phonation channel a i well known
0:07:15	min and by a
0:07:17	to and that's this on their
0:07:18	and apply to
0:07:20	cost function
0:07:22	as i said in this work we are interested in unbiased by under the S P cost function
0:07:28	and in this case
0:07:29	this is the
0:07:30	here and as this condition
0:07:33	in addition in this work we assume that we have continuous
0:07:36	estimator
0:07:37	that is estimate of
0:07:40	with that existing probability density function B D S
0:07:44	okay F
0:07:45	with the high
0:07:45	of the estimate parameter by paper
0:07:49	and that this assumption
0:07:51	that
0:07:51	condition
0:07:52	can
0:07:53	plus the this to conditions
0:07:55	the first condition is that the expectation of the pretty a K is the old
0:08:01	so that a and the average we have
0:08:03	it
0:08:04	the or periodic estimation all
0:08:06	and the second condition is that
0:08:08	a a in this a signal and the project of the estimate that is lower than one divided be two
0:08:15	a the form and them and i said that an estimate with periodic unbiased
0:08:19	i know that is
0:08:20	to conditions are satisfied
0:08:24	and here you can see the difference between mean and by
0:08:27	but you the combat
0:08:29	in the previous example of a phase estimation i said that no uniform an unbiased estimate legs the
0:08:35	but
0:08:36	i
0:08:37	yeah if the this set of estimate are that the can by
0:08:40	so a pretty good the mad estimator exist
0:08:43	and in particular the max some like to estimate of itself is periodic and by S
0:08:48	you can see here a bias of
0:08:51	the max like estimate the cans
0:08:53	but yeah
0:08:54	but more line
0:08:55	is the conventional and by us and
0:08:58	you can say that the max like to estimate of is by that
0:09:01	the biggest problem is that all
0:09:02	yeah
0:09:04	in big if the P L S
0:09:06	you can see that the maxima estimator is periodic unbiased estimator
0:09:11	in this case
0:09:13	no i want to do i knew
0:09:16	int
0:09:17	we bound the mspe mean-square politically or
0:09:21	of any and
0:09:22	by by put that to by an estimate of the that i
0:09:25	okay i the sound
0:09:26	i the probability condition
0:09:29	and the bound is a given here this is the preview or the calm i one
0:09:33	this of the crime that our bound apply
0:09:36	by this fact or but this nonnegative
0:09:39	the come our boundaries
0:09:40	of course that this is the best of the fisher information
0:09:44	okay and this fact
0:09:46	and have applied the common how
0:09:49	this is a new bound
0:09:53	let's see some of its but what D
0:09:55	but but is and the first property is
0:09:57	that the new about the period of and this valid that any signal-to-noise ratio
0:10:02	well i is three style the come at all about may not be it
0:10:07	the second part of the is that the you bound is always lower will it but to the problem of
0:10:11	our bound
0:10:13	for unbiased estimate
0:10:15	and
0:10:16	this can be seen here yeah
0:10:18	we have the kind of a applied to this fact
0:10:21	and this um
0:10:23	according to the set condition and the
0:10:25	but you the can but this condition
0:10:28	this them should be lower than one divided be two but
0:10:31	and of course this is a non-negative them
0:10:33	so all this fact there is between zero and one
0:10:37	so i are is that was level
0:10:40	however i remember that the common are bound is not to provide bad bound
0:10:43	for political estimation
0:10:45	so actually this factor keeps
0:10:47	are are bound to paint to permit it
0:10:49	permit of the region
0:10:50	in divided region
0:10:54	"'kay" that that but is that's the con that our bound so mean biased estimate of the have does it
0:11:00	of the common are bound to in with a bound for periodic a and by if a all
0:11:06	and you can see here
0:11:08	the by a
0:11:09	a of a a a a high about this specific by a
0:11:12	is that then they got to i'm about with the a constrained of periodic and my
0:11:17	this
0:11:18	to can so it
0:11:20	and this is a surprising
0:11:22	because our bounds a bound of the pretty good performance
0:11:25	on the mspe
0:11:27	and the kind of a bound is about bounded the non periodic performance of the M E
0:11:31	so this is not a trivial
0:11:36	finally a
0:11:37	in a similar manner
0:11:39	we can do that the bound for a vector
0:11:41	parameter estimation
0:11:43	and also for weeks
0:11:45	the all parameter estimation in which
0:11:47	part of the product or a periodic and part
0:11:50	of the parameters are not but
0:11:52	you can see have for example if we have
0:11:54	to parameters
0:11:55	one of them is
0:11:57	in you your that can run as well L
0:11:59	and the estimate are also with the same
0:12:02	nature
0:12:04	yeah the following to
0:12:05	constraint was the prove you the can best that's constraint
0:12:08	for a are one
0:12:10	and the main by the school constraint for pick up to
0:12:13	and and of this constraint
0:12:16	are a matrix bound
0:12:17	is the from nine
0:12:18	the covariance matrix
0:12:20	or of the
0:12:21	and a a spectral okay the but that is that a a a a big error
0:12:25	for the periodic part of of people one
0:12:28	and the
0:12:29	yeah irregular four
0:12:30	non part department
0:12:33	so the covariance matrix of
0:12:35	the aspect of is
0:12:36	where or equal to this data
0:12:39	a image which J
0:12:40	is the fisher information matrix
0:12:42	use the inverse
0:12:44	of this matrix
0:12:45	and you have a is that they have an automatic
0:12:48	in H
0:12:49	a a for a not of the parameter
0:12:52	we have one
0:12:53	and for a and the P from P test we have this data
0:12:58	okay so we can use
0:12:59	i the
0:13:01	for any
0:13:02	well i'm not a not base some parameter estimation problems to
0:13:05	but got the call non the parameters
0:13:08	and for a vector or a scalar estimation
0:13:12	okay okay
0:13:14	okay
0:13:15	yeah can see an example and and
0:13:17	this example use
0:13:19	example of of that to a parameter estimation
0:13:22	we want to estimate
0:13:23	i a
0:13:25	and
0:13:26	the phase fee
0:13:28	we have a known frequency on as they are
0:13:31	and
0:13:31	a a gaussian noise
0:13:34	in in this case again we have very low
0:13:37	i'm by april
0:13:39	but we don't have
0:13:41	conventional mean unbiased biased estimators
0:13:44	the crown a lower bound to metrics
0:13:47	is given here
0:13:48	and this is they have a not matrix in this case
0:13:51	and there are
0:13:52	bound is given a yeah
0:13:54	you can say this fact or C N
0:13:57	and for um is calculated using the
0:14:00	but it of the function of the maximum likelihood estimate them
0:14:03	which is that the
0:14:05	or are known for this
0:14:11	so here is a yeah can see
0:14:14	E
0:14:15	the uh is that
0:14:17	for
0:14:17	the phillies estimation
0:14:19	okay that really bothered
0:14:21	and
0:14:22	yeah this is the N S P in this paper you are to get against snr
0:14:26	the
0:14:27	but black time is our uh about
0:14:29	and the part line is the common out
0:14:32	this is the line
0:14:34	oh a
0:14:35	the performance of
0:14:36	the unbiased put the to combat the estimator
0:14:40	but and i mean
0:14:41	in the maximum an accurate estimate of
0:14:44	and can can that a a other bound to about it
0:14:47	if any snr
0:14:49	well i the kind of lower bound is not valid
0:14:52	hmmm know with an all
0:14:53	yeah hmmm in the so that the bound and the and the speed of the estimate of the of the
0:14:58	pen of the ferry
0:15:00	and
0:15:00	it a can the crime a bound and
0:15:03	are about
0:15:04	a a of a then by the that code estimate of for a high snrs
0:15:08	okay but
0:15:09	i about you was
0:15:11	fit of added
0:15:12	here
0:15:16	to can close in this all the concept of non bayesian periodic parameter estimation was it introduce
0:15:23	the periodic unbiased and that i
0:15:25	S P E square periodic it'll cost function
0:15:28	well as defined here as in the lemon definition for one best man
0:15:32	the project S and as of the common are bound for P you gotta parameter
0:15:36	for a mixed
0:15:38	the periodic and non
0:15:40	the vector parameter estimation were developed
0:15:44	a S we said that
0:15:45	put a to come as i don't provided that at lower bound in political mention
0:15:49	and uh a a at this so that
0:15:52	it but i as some periodic unbiased estimate all's and in a bound for phase estimation
0:15:57	a
0:15:59	from and different phase estimation
0:16:01	and a kind of yeah king
0:16:03	on
0:16:04	the relation of the but i can like
0:16:06	periodic bound
0:16:07	which supposed to be tried to of and the common are about
0:16:11	a a and also on a hybrid balance
0:16:14	based on in on a bayesian
0:16:16	parameter
0:16:17	and they finally about the periodic minimax estimation
0:16:21	thank you
0:16:23	but
0:16:29	we have sorry for a cushion
0:16:43	a
0:16:44	the bound is not a function of
0:16:46	the estimate but it is a function of its statistics
0:16:49	of
0:16:50	but you in specific point
0:16:53	this is a of it's a statistic
0:16:55	properties
0:17:02	uh_huh
0:17:10	hmmm
0:17:24	uh_huh
0:17:34	it to about it yes i
0:17:36	a a a a the bounds of the bounds as function of it also the colour of are bound not
0:17:41	only input public bounded estimation
0:17:43	of the parameter i yeah
0:17:44	cool
0:17:46	but uh at
0:17:47	it's not a function the estimate of okay you this is on the function of
0:17:51	it it's that is its properties
0:17:56	yeah
0:18:03	this this you know we use
0:18:04	but is a small or not
0:18:06	um
0:18:07	that's what is it for a new and all of a only use like on the
0:18:12	since you bound to used used
0:18:16	soon as much
0:18:18	but you know that you're is most useful
0:18:25	i say that is the L B is not used for because this is the bottle and biased estimate
0:18:30	but we don't have any unbiased estimate of
0:18:33	no not okay use is some small uh our knowledge
0:18:36	uh know about the the problem is that the
0:18:39	mse S them it's got a quite it says is inappropriate
0:18:42	okay if i
0:18:46	you are i'm pretty good
0:18:48	you have periodic parameter like
0:18:50	and again like do you way like
0:18:52	and this periodic parameter estimation
0:18:55	and we want to estimate to
0:18:56	okay
0:18:57	the ms E is "'cause" yeah could you on that
0:19:00	min makes you is this uh
0:19:02	okay
0:19:03	and this is not
0:19:04	a a a a a perfect for prosodic parameters mention
0:19:08	okay yeah
0:19:09	we really it you are
0:19:11	it and but with them and of and the printer to care of them are denoted by all
0:19:16	and also that is it the chi bounds and all the bounds our bounds on the mse
0:19:50	yeah
0:19:51	uh_huh
0:19:53	okay
0:19:53	so so the problem only not only low snrs
0:19:57	but that's
0:19:58	the problem is that the common a bound is not a and the problem is not that the bound is
0:20:02	more tight
0:20:03	okay you we can see
0:20:06	our example
0:20:08	okay that the that is not that type of the bar
0:20:12	okay the problem is that
0:20:13	and an estimate of request the bound
0:20:16	"'cause" so this is not a valid bound that all
0:20:22	okay
0:20:25	okay
0:20:26	but you again
0:20:27	you

PERIODIC CRB FOR NON-BAYESIAN PARAMETER ESTIMATION

Detection and Estimation

Přednášející: Tirza Routtenberg, Autoři: Tirza Routtenberg, Joseph Tabrikian, Ben-Gurion University of the Negev, Israel