WEBVTT
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Mhm so welcome everybody to our lecture on automobile
division again. So two weeks ago since we had the last
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lecture. So let us start with one of the last lights that
we have seen two weeks ago. So we were talking about hidden
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Markov models in mark of models? where are statistical
model about systems that evolve over time, that change the
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internal state over time, and that can be observed by an
observer. An observer is a sensor say that is observing the
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system, and that is measuring some variables which depend on
the system state. And for such a system, we derived these
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equations that you can see here so that equation at
the bottom starts here with the probability of the current
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state. Given all observations up to the present point in
time. That means we assume that we have a certain observed a
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certain sequence of observations, um based on that. We have
calculated um the probability to be in a certain state of.
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And what we aim is we want to predict what is the next
state? what is a subsequent state? and for the next point in
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time. So that is what we want to derive. And that is shown
here in the blue parks. And yeah, we were using the rules for
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calculating with probabilities. And of course, the assumptions
of sarcastic independence, which we made for hidden
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Markov models to simplify the equation so that we get this
equation here and what we have here this probability of St
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plus one given as T. That is the state transition probability.
So that is a probability with which we expect that if we
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are in a certain state, the next state. Um, yeah, it is that
we achieve a certain next state and this is what we
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model in the hidden Markov model. So we can assume that we
now wait, or that we have made assumptions about that. The
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equation at the top vice versa relates the probability
for a certain state for a certain point in time,
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given only the observations up to the point in time before,
with the probability of integrating the new observation set
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tea into this reasoning. So we assume we have such a predicted
state probability for the pointed time t. Now we make the
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measurement," said T." and so we want to update our probabilities
and integrate this measurement," said tea as a new
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observed variable. And yeah, we can see again with the Beijing
formula, with the classical rules for calculating with
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probabilities. We can calculate this relationship here.
So the new probability with an integrated measurement is
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proportional to the predicted state probability times are
the probability to make a certain observation, assuming a
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certain state. Again, this probability of an observation is
something that we specify in the hid mark of models, or we
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can assume that we know it. Now we have these two equations
that actually established to steps in an algorithm with
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which we can incrementally calculate the state probabilities.
This works as shown here. So we might start here. We have
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some predicted state probability for the very first point
in time, then we make a measurement, and we integrate this
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measurement in a step that is called innovation step. And
that is just implementing the equation at the top of the last
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slide in a computer program. Yeah, that is calculating the
left hand side of this equation by evaluating the right hand
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side of this equation. By doing that, we get these estate
probability distributions, and now we can make use the other
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equation, the equation that was written at the bottom of the
last slide in order to make a prediction for the next point
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in time to predict what is a state probability distribution
for the next point of time, without yet having made the
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measurement for next point in time. This is called the
prediction step. And it yields these predicted state
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probabilities. While we again have, again, these predicted
state probabilities. And once we make a new measurement, we
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can integrate that in the next innovation step. And by
doing that, we can cycle through these two steps and
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incrementally calculate the state probabilities. And we can
either start here, or we can also start here with state
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probabilities. That is a little bit application dependent
where we start, whether we start on the left hand side and
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first apply a prediction step, or whether you start with a
innovation step. Ca so if we know the initial state very well
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% um, then % um @unoise@ and we do not make immediately a
measurement than we would start @unoise@ ah with a state
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probability on the state probability side. And first,
Ah, make a prediction @unoise@ if. We know the state very
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well then to the state very well, and we immediately make a
measurement without first having one step in time, you would
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step on the they thought on the right side with a predicted
state probability. And if we don't know anything about the
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image of state. We typically start as well. Here on the right
hand side. And we model actually in the probability
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distribution with which we start with about this inature
probability distribution that we don't know what the state is.
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So if we have two possible values for the state variable, we
would say," Okay, we do not know anything. So each of these
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two possibilities occurs with the probability of fifty
percent. That would mean we and codes are to say, in the
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probability that we are perfectly unsure about the real state.
Okay, so that means the algorithms performs like that. So
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we might start here over the probability of as one. Then we
make an innovation to integrate the first measurement, then
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we make the prediction staff to predict the probability
of the state in the second point in time, given the first
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measurement, then we make an innovation step, and so on. We
would continue like that. And as I said, we could also start
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here with the probability of as zero. And first make a
prediction step. Yeah, okay. So let us execute this
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algorithm and look at that for a very special case, namely
this case of a finite number of possible states and finite
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number of fossil observations. So we are in a discrete random
Barry Apple case where, say, we have two or three or four
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or ten or one hundred or two hundred or one thousand uh.
Possible state values or values that the state very hour can
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take on. And as well, we have a certain number of crisis
observations, or the observation is not distance that you would
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measure with a real number, but the measurement would be
just a discrete value of value from a discreet set of of
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possible values so something like I can see a car or I
can't see a car that would be a discreet, random variable,
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and that would be a measurement that we would use here. So
the hidden Markov model in this example is described by such
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a state transition graph. And what do we see? well, first of
all, here in this area. On the left we see in total three
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states, which I've named A, B, C. Here too, be able to
distinguish them, then we have these arrows, the black errors,
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which indicate which transitions might happen between the
states and the numbers next to these errors indicate the state
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transition probability. So for instance, if the system is
in state A than the probability to stay in state aid. The
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probability that at the next point in time, the state is
still a is four point eight or four point eight have this
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reflexive arrow, and the probability to make a transition to
state be can be seen. Here is O point two. And of course,
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those must add up to to one. We don't have any arrow from
state A to state see. This means the probability is zero to
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make such a transition. Now, it might also happen that some
transitions are deterministic, say, if the system is in
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state B @unoise@ then, the probability to make a transition
to state C is one that means there is no other option. We
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will always make a transition from B to see once we enter be
@unoise@ okay. So, these are the black arrows. Then we have
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the observations. I have put them here in these rectangular
blocks. Our observation, you and V, whatever you and me
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means might be. As I said, I can see a car or I can't see a
car or something like that. And then we have these dashed
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errors, and the dashed errors indicate the observation
probabilities. So that means, for instance, this. Oh point six.
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Here means, or if, or the system is in state A, then the
probability to make observation you is all and the
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probability if the system is in state A to make the observation,
the probabilities point four. Yeah, both numbers must
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add up to one um in this case. Yeah, so these are the
observation probabilities now. Let us use the basic
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algorithm to do some calculations with this very small, hidden
mark of model so we are interested in calculating
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the state transition probabilities. Sorry given
a sequence of observations for a certain observation
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sequence, say you U, U, S, or you is the first observation.
The second observation is as well view the third observation
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is we and the force observation is you again. And we have to
make an assumption about the initial state probability. So
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in this case, we assume that the probability to be an A or B
is O point five, and the probability to be in sea is zero
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now. So we do not know whether we state An, A and B, but we
are sure that we don't start and see @unoise@ okay. Now, let
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us have a look. In this case, we start with what with a
prediction step. Okay, we start with the innovation step. Okay,
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so let us do a little bit of calculation on the
blackboard. And then later on, we have a look at the table over
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there. So what do we know so we know from the text,
the image of probability p of S one and B of S one
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equals A is all point five and P of as one equals B is as
well, or point five and P of S, one equal C is equal to zero.
00:12:21.590 --> 00:12:32.937
So now let us assume we make the first measurement. So the
first measurement said one is as it is stated here. You so
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said one is equal to you. Oh, I did like that. So now
now we have to calculate the Mhm probability of P of S. One
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is equal to a given that said one is equal to you. Mhm, this
is the innovations that we integrate the new measurement
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into our probabilities. Now that is whatever. And of course,
we also have to calculate the probability of as two being
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equal to a given. That set one is good to you, but that is not
equal to each other. And the third one is P, S, but it is
00:13:25.597 --> 00:13:37.052
one here @unoise@ as one is equal. So sorry. What if @unoise@
as one is equal to see, given that sad one is equal to you
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like that. So these three probabilities is what we have to
cake late now. Now we can go back to the slide before and
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have some slides before and have a look. What we have to
do in this innovation step. So let us go back, back, back,
00:13:54.287 --> 00:14:04.152
back, back. So the innovation step was the topmost step. And
what we see is we take the probabilities that we already
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have and multiply them with the observation probability
for our hidden Markov model So. Here we take these
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probabilities, the blue ones, these others. And I have written
on the blackboard. Yeah, at the top of the blackboard
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and then we have to multiply each one with the
probability of observing you this observation, you add the
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respective state. And this yields what the result is proportional
to the probability that we want cake in it. Hey, let
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us go back to this night. Okay, so so let us remove the
equality and say this is proportional, too proportional to,
00:14:58.096 --> 00:15:11.592
okay, the observation probability. That means the probability
to observe you in state A. And this probability, if we
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look at this diagram the probability to observe you in
state A is given by the dashed. Arrow so this arrow
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here and we can just take the probability that is written
there, or point six. So this is all point this time, the
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probability to be in state A. This is given as oh point five.
The topmost line on the blackboard four point five. This
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is equal to all point % um three, right? if you find a mistake,
please complain @unoise@ then the second probability the
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probability to be in state be given the observation. You
again, we have to take what is the probability to observe you.
00:16:10.269 --> 00:16:19.232
The observation you in state B. We can read that again from
the from the state transition plot and see it is all point
00:16:19.232 --> 00:16:31.197
too. Now, as we can see this arrow from B to you, and the
number next to it is open to. There are four point two times
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the probability to be in state B is one point five. So this
yields all point one. And finally, the third probability
00:16:42.621 --> 00:16:51.701
that we want to calculate the probability to be in state C.
Given that said one is equal to you. While we look at state
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sea, and we see the probability to observe you in state sea
is o point seven. We get that from the tenth, from the state
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transition diagram here. So we look at this arrow here at
this era. Now, with open seven written next to it. So it is
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all point seven times, while the probability to be in sea is
zero. So the result is equal to Sierra. Okay, so far. Well,
00:17:22.210 --> 00:17:34.428
now, of course, we only know it is proportional, too. And
if we sum up these numbers, we find the sum is zero point
00:17:34.428 --> 00:17:45.779
four. We expect that the sum is one, because the state can
either be A or B or C, nothing else. So the sum of all those
00:17:45.779 --> 00:17:54.584
probabilities must be equal to one. So how can we achieve
that? well, we know there is an proportionality factor
00:17:54.584 --> 00:18:02.789
hidden here somewhere. But we know that the sum of all the
probabilities must be equal to one. So we normalize them.
00:18:02.789 --> 00:18:12.724
Yeah, we divide all these numbers by the sum of those numbers.
So by for that means from that we can conclude, and
00:18:12.724 --> 00:18:25.199
from that we can conclude the P of as one being equal to a
given that said one is equal to you is equal to one point
00:18:25.199 --> 00:18:35.741
three over all point four. This is four point seven, five,
and the problem ability of as one being equal to be given,
00:18:35.741 --> 00:18:46.704
said one equal to you is equal to four point one over one
point four is equal to four point two, five, and the
00:18:46.704 --> 00:18:58.472
probability to be in the state sea, given that sad one is equal
to you is equal to zero over all point four. And this is
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zero. Okay, so. And now we are done with the first innovation
step. Well, now we have these probabilities. We know those
00:19:09.622 --> 00:19:24.197
values. And we did the first step. Okay, okay, okay. So now
we have to make the next step after the innovation step. The
00:19:24.197 --> 00:19:33.524
next step is the prediction step. And for the predictions that
we can have look back to the slide where the form line is
00:19:33.524 --> 00:19:43.044
written. That is the slide at the bottom of of this light,
we see that the rat, this red probability that we have just
00:19:43.044 --> 00:19:50.979
calculated m enters the equation, and then we have to consider
the state transition probability, which we get from the
00:19:50.979 --> 00:20:01.798
state transition diagram. And we have to sum up over all
possible prior states or previous dates, better to say, and by
00:20:01.798 --> 00:20:16.058
then we get the result. Okay, how does that look like in this
case. Okay, so now we calculate what the probability that
00:20:16.058 --> 00:20:29.025
state is two, the next state is equal to a given the first
observation. Well, given the first observation that we
00:20:29.025 --> 00:20:39.011
already observed now, by definition, or by by, by definition,
by the equation that we have found. This is nothing else
00:20:39.011 --> 00:20:48.116
then. Okay, we have to go through all possible states at the
first point in time. So for all these, we have to consider
00:20:48.116 --> 00:20:58.103
all these three cases. Mhm, we do not know what as one really
is. So we have to consider all possibilities for each of
00:20:58.103 --> 00:21:03.984
those three possibilities. We multiply this value, the
respective value in this line with a state transition
00:21:03.984 --> 00:21:13.434
probability. Okay, this means okay, as one could be a if
it is A. This happens with the probability of seven,
00:21:13.434 --> 00:21:23.293
five, all point seven, five, and the state transition
probability to make a transition from aid to A is given by this
00:21:23.293 --> 00:21:33.269
arrow here. There is transition from A to A, and it happens
with the probability of all point eight. So it is all point
00:21:33.269 --> 00:21:42.725
eight times, all point seven, five. That is one possibility
how we can enter a state a the second possibility how we
00:21:42.725 --> 00:21:53.257
can't enter state be is that we are sorry. The second possibility
to Interstate A is that we have been in state B. This
00:21:53.257 --> 00:22:02.990
happens with a probability of zero point two, five, four point
two, five, and that we make a transition from B to A. If
00:22:02.990 --> 00:22:10.948
we look at the state transition diagram. There is no arrow
from B to A. So the probability of making such a transition
00:22:10.948 --> 00:22:24.590
is zero. So zero times, all point two, five. And the third
possibility is that we have been in State C, and that may we
00:22:24.590 --> 00:22:34.243
make a transition following this arrow from C to A that
happens with probability or point five, so or point five, as
00:22:34.243 --> 00:22:44.996
transition probability times the probability to have been in
state sea. This is zero in this case. So now we add up all
00:22:44.996 --> 00:22:57.315
these three possibilities to get the result, which is, oh
yeah, okay, okay, this is zero. There is a zero, four point
00:22:57.315 --> 00:23:11.372
eight times oh point seven, five. This is oh point six. Is
it open? okay, that is the probability to be in state A. Now
00:23:11.372 --> 00:23:23.739
let us calculate the probability to the state be given save
one equals you. So the same story again. So to say, we check
00:23:23.739 --> 00:23:31.994
all possibilities how we can enter state B. And for each
possibility, we multiply the state transition probability with
00:23:31.994 --> 00:23:41.724
the probability that we have been in the respective previous
state. So okay, state A, the probability to be in state A's
00:23:41.724 --> 00:23:52.997
or point seven. Five @unoise@ the probability to make a
transition from state A to B is given in the state transition
00:23:52.997 --> 00:24:01.523
diagram. It is zero point two. So four point two times zero
point seven. Five, the probability that we are already in
00:24:01.523 --> 00:24:11.021
state B is all point to five, the probability to make a
transition from B to Be. That means to stay in B. Well, if you
00:24:11.021 --> 00:24:18.617
look in the transition diagram, there is no arrow that is
reflective that goes from be to be. So the probability is
00:24:18.617 --> 00:24:26.844
zero, zero times and the probability to have been in
state sea is zero the transition probability is given
00:24:26.844 --> 00:24:39.692
by this era it is zero point five. So it is all point
five times zero @unoise@ and we get zero zero and @unoise@
00:24:39.692 --> 00:24:53.079
we what that @unoise@ oh point two times open seven, four is
all point three, seven, five, I guess four point one, five.
00:24:53.089 --> 00:25:10.626
Okay, @unoise@ thank. You oh. Okay, okay, so. So the probability
to enter state sea. When we know that the observation,
00:25:10.626 --> 00:25:18.683
the first observation is, you is okay. Again, the three
cases, the probability to be in A and make a transition from
00:25:18.683 --> 00:25:26.486
eight to C, while the probability to be in A's again, or
point seven. Five, the probability to make a transition from
00:25:26.486 --> 00:25:34.371
aid to C. When we look into the transition diagram. There is
no error. That means it is zero. There are times the
00:25:34.371 --> 00:25:44.792
probability to be in state B is one point to five. The
probability to make a transition from B to see is one following
00:25:44.792 --> 00:25:53.257
the transition diagram, and the probability to have been
in the sea already is zero, the probability to make a
00:25:53.257 --> 00:26:04.223
transition from zero to zero is equal to Sierra . So we
get for the last case or point to five. As a result, if you
00:26:04.223 --> 00:26:11.641
like, we can check whether all our calculations are correct.
By summing up all these probabilities and check whether
00:26:11.641 --> 00:26:22.514
they sum up to one. And if it doesn't sum up to one, then
we make a mistake. If it sums up to one, then we cannot be
00:26:22.514 --> 00:26:31.001
sure that we didn't make a mistake, but at least we can
find some mistakes by doing that. Okay, so this is the next
00:26:31.001 --> 00:26:39.129
step. Now we, and this is actually the prediction step. Now,
the next step would be to consider the next observation. So
00:26:39.129 --> 00:26:48.085
the second observation in this case said to is equal to you
again. And now we would have to execute the innovation step
00:26:48.085 --> 00:26:57.566
as we did here. Yeah, @unoise@ but. Now for the second second
second observation @unoise@ yeah. For, time reasons
00:26:57.566 --> 00:27:08.744
I, leave that for you at home so that we can do it at home.
You find the result here in the table on the lights
00:27:08.744 --> 00:27:18.273
@unoise@ yeah @unoise@ and % um when we look there so
we start actually there. Here are the initial probabilities
00:27:18.273 --> 00:27:28.332
that are given when we do the first innovation step. That is
what we did at the left side of the blackboard, and we get
00:27:28.332 --> 00:27:35.453
these probabilities. This after integrating the first measurement,
then we make the prediction step. And that is what we
00:27:35.453 --> 00:27:43.184
just did on the right hand side of the backboard and to get
the predicted state probabilities. Then the next day, a step
00:27:43.184 --> 00:27:51.248
again would be to integrate the new measurement, you and we
would end up with these probabilities, which are given here
00:27:51.248 --> 00:27:59.696
round it to two decimals so the real numbers would
have more decimal, but I've rounded all the numbers to two
00:27:59.696 --> 00:28:07.545
decimals. Okay, then we would make a prediction step again
and up in these probabilities, then we would integrate the
00:28:07.545 --> 00:28:16.177
third measurement, which is V, in this case um to get these
numbers here, and then we might make a prediction step
00:28:16.177 --> 00:28:23.106
again. Then we integrate the fourth measurement you in the
next innovation step and make another prediction staff. And
00:28:23.106 --> 00:28:36.899
like that, we can go on and um continue for subsequent
observations. As we can see what we need to do with innovation
00:28:36.899 --> 00:28:44.744
and prediction steps. Um of course, the information
from the state transition diagram here for the innovation
00:28:44.744 --> 00:28:52.709
step. We need the observation probabilities for the transition
step, for the predictions that we need to stay transition
00:28:52.709 --> 00:29:00.692
probabilities, and we need only those probabilities which we
have calculated in the previous step of the algorithms. So
00:29:00.692 --> 00:29:09.276
we do not need to memorize things that we were calculating in
the very beginning to make um calculations later. And that
00:29:09.276 --> 00:29:18.012
is the big benefit advantage of this algorithm. We only need
to store the last probability state probabilities. We don't
00:29:18.012 --> 00:29:29.476
need to memorize all the past. That is not necessary. Just the
probabilities of the last steps are sufficient. So, okay,
00:29:29.476 --> 00:29:41.613
now let us go. This is so to say the simple case if you
have a finer number of possible states, values and fine at
00:29:41.613 --> 00:29:50.014
number of possible observation values. We will come back
to that later on the lecture But. Now since we were
00:29:50.014 --> 00:29:57.464
talking about tracking objects and how we can measure
velocities and estimate velocities. Of course, those are state
00:29:57.464 --> 00:30:05.308
variables which are not continue which are not discrete
with random bubbles, and where the number of possible
00:30:05.308 --> 00:30:15.798
values is in finite? well then the question is, if we have an
uncountable number of states, or and or uncountable number
00:30:15.798 --> 00:30:26.602
of observations. How can we implement this algorithm that
we develop so far for this case. So the first thing that we
00:30:26.602 --> 00:30:36.482
can do when we talk about that is we replace discreet
random variables by continuous random barriers. And that
00:30:36.482 --> 00:30:46.135
means we have to switch from discrete probability distributions,
where we can store a probability for all possible
00:30:46.135 --> 00:30:55.904
values for each value. Two probability density functions to
represent the state distribution. So that means we have to
00:30:55.904 --> 00:31:05.054
use state probability densities to all the calculations, and
at every place where we had a summation in the Formala. So
00:31:05.054 --> 00:31:14.128
far, we have to replace the sum by the integral. And if we
do so, we get. We end up with these two equations, or these
00:31:14.128 --> 00:31:21.635
two formulas that are written here, which are nothing else
than what we have just derived before, but written with
00:31:21.635 --> 00:31:29.515
density functions here at each point and replacing the
by the integration here. But indeed, it is nothing else. So
00:31:29.515 --> 00:31:41.321
we can derive it in the same way as for the discrete case.
Now, the question is, how can we implement that? we need
00:31:41.321 --> 00:31:49.792
something with which we can map, or this calculation here.
Each step maps a probably density function to another
00:31:49.792 --> 00:31:57.648
probability density function. And this only works with a
reasonable amount of work. If the probability density functions
00:31:57.648 --> 00:32:06.041
that we deal with have some nice properties, otherwise we
run into trouble. When we want to do this integration here,
00:32:06.041 --> 00:32:14.129
for instance, or when we want to do this multiplication, and
then we need to normalize everything, then we would run
00:32:14.129 --> 00:32:23.164
into trouble. But for some very nice cases of probability
distributions, things work, and we can really, explicitly and
00:32:23.164 --> 00:32:31.473
analytically, do this integration here and analytically resolve
all these calculations that are necessary here. So for
00:32:31.473 --> 00:32:41.230
which kind of probability distinct and distributions does it
apply? the answer is for so called linear gulsion models.
00:32:41.240 --> 00:32:51.784
What does it mean? it means that the relationship between
the state and the successor state should be described by a
00:32:51.784 --> 00:33:01.742
linear mapping plus an additive galishing noise, a small
amount of noise that is randomly chosen from from a
00:33:01.742 --> 00:33:10.438
distribution. Now that is one piece. And the second step is
that also the observation is assumed to depend on the state
00:33:10.438 --> 00:33:20.195
by a linear relationship. So the observation depends linearly
on the state, plus an additive, small, random noise that
00:33:20.195 --> 00:33:30.122
is taken from a Gaugeian distribution. So this is shown
here in formulas in terms of formula. So we assume that the
00:33:30.122 --> 00:33:39.985
successor state S, T plus one depends on the present state
as T, by a linear A by a linear funct function and a leader
00:33:39.985 --> 00:33:48.548
function. If we deal with vectors or as tea, as a vector,
as deep as one might be a vector, a linear function is
00:33:48.548 --> 00:33:56.649
implemented by a matrix multiplication us, plus an additional
offset that we might add up. So as one equals a
00:33:56.649 --> 00:34:06.806
matrix that described the system behavior times the
present state plus ut and constant offset that we add up to to
00:34:06.806 --> 00:34:17.520
the state. So that is the general form of a linear function,
depending on his tea. And then we have this additional
00:34:17.520 --> 00:34:29.862
exile on T. This Epsilon T is assumed to be a random variable
that is true, chosen from a Gauchian distribution with
00:34:29.862 --> 00:34:41.911
Cyril Mien and a certain matrix um. This is said
to be often in literature. This is known as white noise. A
00:34:41.911 --> 00:34:50.229
white noise means a random variable absolent T, which has
zero expectation value. Therefore, it is called white .
00:34:50.239 --> 00:35:00.662
And your noise means just some random dissipation I saw.
This is Eclipse. So as I said, is a a matrix. We
00:35:00.662 --> 00:35:08.186
assume that we know that we can describe the system
property, the randomness that is in the system and that we
00:35:08.186 --> 00:35:17.422
know it. And guilt can describe it by this matter
executive. So, um, yeah, you might see that these very others,
00:35:17.422 --> 00:35:30.299
A, T, U, T, Um. I put always this index T to this means,
in theory, they can differ from time step to time step,
00:35:30.299 --> 00:35:42.059
but they must be known in advance, and they must not depend
on the state, on the current state, Mhm, but they might be
00:35:42.059 --> 00:35:51.815
different from state to state. If you go to literature,
people often don't write this lower index team and make it a
00:35:51.815 --> 00:36:00.815
little bit simpler by saying, well, we assume that those state
transition mattresses and so on are always the same for
00:36:00.815 --> 00:36:09.989
all, for all steps. That is what you typically find in
literature but indeed, what you really could do is you
00:36:09.989 --> 00:36:18.234
could change these variables for each time step. And in
practice, this becomes relevant when we deal with real
00:36:18.234 --> 00:36:25.759
measurement processes. Now, when we make real measurement
processes, it might happen that the measurements do not come
00:36:25.759 --> 00:36:33.620
in at equally space, time intervals, but at unequally space
time intervals, though, it might happen that between the
00:36:33.620 --> 00:36:42.219
first and the third second measurement may be. There is one
second time in between and between the second and the third
00:36:42.219 --> 00:36:50.904
measurement, there is one point five seconds time in between.
And of course, if you think about a car that is moving the
00:36:50.904 --> 00:36:59.582
distance that is covered in one second, or in one point five
seconds, that is different, and therefore a T needs to be
00:36:59.582 --> 00:37:08.935
different. In these cases, it may be also ut, and it depends
on what you want. And maybe if there is more time between
00:37:08.935 --> 00:37:17.007
two measurements and two, two points in time @unoise@ um,
then maybe also this noise is is larger, because the driver
00:37:17.007 --> 00:37:24.825
might have accelerated more, whatever. Now, therefore, in
practice, when you implement something and you have unequally
00:37:24.825 --> 00:37:36.360
spaced point in time, then it becomes relevant. And to make
these % um system @unoise@ and and this offset and the
00:37:36.360 --> 00:37:46.125
noise vector, depending on the length of the time interval,
for instance. Yeah, but if you go to literature for the
00:37:46.125 --> 00:37:55.926
simplest case, um um you might first assume that um
the and U t and eighty @unoise@ % um that those are
00:37:55.926 --> 00:38:04.130
independent of the time. Step Okay. So, that is
about the state transition model. And now we look at the
00:38:04.130 --> 00:38:11.271
measurement model, we see the same, a very similar
structure, so sati. The measurement depends on sty by a
00:38:11.271 --> 00:38:18.250
multiplication with the matrix Ht, a measurement metrics Ht
that relates in which way the measurement depends on esteem.
00:38:18.250 --> 00:38:25.860
That is also a linear relationship, as we can see. The only
difference is that we don't have such an offset duty here.
00:38:25.869 --> 00:38:35.494
That is more for practical reasons. You might also add an
offset here in this equation would work in the same way,
00:38:35.494 --> 00:38:44.613
actually, though, however in practice Ah, this
kind of technique often is is Ah implemented, shown in the
00:38:44.613 --> 00:38:53.732
simplified version, because what we treat as measurement, we
are free in defining what we treat as measurement. Yes. So
00:38:53.732 --> 00:39:03.230
if the sense are, say, outputs the measurement three, and we
assume that there is some offset here in the measurement,
00:39:03.230 --> 00:39:12.728
then we can subject that offset from the measurement and deal
this modified measurement as a measurement that we use for
00:39:12.728 --> 00:39:20.594
the description of this relationship. So actually, yeah, if
you like to write an offset here in additional that here you
00:39:20.594 --> 00:39:28.796
are afraid to do that. But we don't need it. @unoise@ however,
a, linear relationship given by this measurement matrix,
00:39:28.796 --> 00:39:36.526
each team. And again, we have this delta here. That, again,
is some random variable, which describes the randomness in
00:39:36.526 --> 00:39:44.042
the measurement. Again, some noise, caution, noise, white
Gaunt noise. We assume that it is taken from a gauge and
00:39:44.042 --> 00:39:51.971
distribution with zero mean and with a certain culverance
our team that we again assume to be known. And again,
00:39:51.971 --> 00:40:03.743
you see these indie says here for T, R, T here and H T here
in literature, this % um @unoise@ entities are usually not
00:40:03.743 --> 00:40:12.808
indexed by the time point point in time. But again, theoretically,
you could do that. You can could say," Okay, for each
00:40:12.808 --> 00:40:19.603
point in time. The uncertainty, for instance, of the measurement
is different, though. This coherence matrix, or this
00:40:19.603 --> 00:40:25.469
measurement equation is different. Now, this might be relevant.
For instance, if you have two different senses, and you
00:40:25.469 --> 00:40:32.129
want to integrate them. And at some point in time, you give
measurements from one sensor, and for at other points in
00:40:32.129 --> 00:40:39.071
time, you get measurements from the other sensor. And at some
point in time, you get measurements from both sensors, and
00:40:39.071 --> 00:40:46.728
actually you have to update the H, D matrix depending
on which measurements are available. Yeah, so that would be
00:40:46.728 --> 00:40:54.779
a case in which that is necessary, but for simplicity, for
understanding the principle, assume that the measurements are
00:40:54.779 --> 00:41:04.806
all of the same time type every time. And that H, T and also
R, T are the same every time, but theoretically, for some,
00:41:04.806 --> 00:41:13.380
some in some cases, it might be relevant to modify them
from time to time, but actually, of course, they also might
00:41:13.380 --> 00:41:24.440
these choice of art. And H, D must not depend on the present
state, definitely not. Or our guess what the present state
00:41:24.440 --> 00:41:33.026
is, or something like that. Also theoretically is
not allowed to make. For instance, rit depending on the
00:41:33.026 --> 00:41:42.103
measurement that we currently made, although people do it
sometimes in practice. Theoretically, this is not sound. Yeah,
00:41:42.103 --> 00:41:53.870
this is what I said. So let us have a look at a very simple
example of how we can model a real a problem like that. So
00:41:53.870 --> 00:42:03.569
we assume we have a car that is moving along a road. Now we
assume that as a one dimensional problem. In the case that
00:42:03.569 --> 00:42:12.232
we only consider the long internal position of the car
are not the lateral position within the lane, but only a
00:42:12.232 --> 00:42:20.195
longitudinal um position. So we have a coordinate system, acts
the the we um measure the position of the car, or want to
00:42:20.195 --> 00:42:27.378
model the position of the cars then the car has a
velocity. We assume that it is driving with constant, more
00:42:27.378 --> 00:42:35.519
or less constant velocity. So then we could say, okay, the
state reactor consists out of X, T and meet the position and
00:42:35.519 --> 00:42:43.779
the velocity. So and we know that one the position at
the next point in time is actually the present position,
00:42:43.779 --> 00:42:51.562
plus while the time, the length of the time interval.
Let us say delta teaser. This should be the length of the
00:42:51.562 --> 00:42:58.294
time and novel time, speedy the present velocity not to be
as if you assume concert velocity, when we can conclude that
00:42:58.294 --> 00:43:05.862
this is the expected position for the next point in time. And
of course, there is some noise and some random noise. This
00:43:05.862 --> 00:43:13.689
white gauge noise that occurs because the car is not perfect
in its motion. The driver might exhilarate, accelerate a
00:43:13.689 --> 00:43:21.900
little bit whatever so there is always some imposition
in this movement and for the velocity we conclude? well,
00:43:21.900 --> 00:43:29.161
the new velocity is equal to the old velocity, plus some
randomly here. Again, the driver might accelerate exhilarate a
00:43:29.161 --> 00:43:37.285
little bit @unoise@ um that can be modelled by this uncertainty
@unoise@ so. Now we see that this actually okay, this is
00:43:37.285 --> 00:43:44.911
Tim interval Ah. It is already what I said @unoise@
so. Now we can these two equations and grind it
00:43:44.911 --> 00:43:52.872
into a metric spectrum multiplication equation. So if he
summarize as teeth of a baptisty, the state factor that
00:43:52.872 --> 00:44:02.810
contains when we can say that the new state, etcetera,
contains one and one is equal to this matrix.
00:44:02.829 --> 00:44:13.148
One delta T, zero one times S, T Plus these uncertainty
the vector of uncertainties @unoise@ and. If we do this
00:44:13.148 --> 00:44:21.706
multiplication then we see what we get is exactly the right
hand side of these two equations above so we are able
00:44:21.706 --> 00:44:29.957
to rewrite these two system equations in the appropriate way
that we need to treat that as a linear caution model. Well,
00:44:29.957 --> 00:44:39.492
so now, eighty is this matrix here. And U, T is a vector
that just contains zero here. So we don't have a constant
00:44:39.492 --> 00:44:47.668
offset for this problem and the factor of uncertainties
is the vector that contains a position and velocity
00:44:47.668 --> 00:44:55.638
uncertainty @unoise@ so. Now the measurement let us assume
we have a binocular camera system like that. And with a
00:44:55.638 --> 00:45:03.555
binoculars system, we can measure the position of the vehicle.
So we say the measurement, the measured position. Our
00:45:03.555 --> 00:45:13.125
measurement said tea is equal to, et cetera, et cetera, et
cetera, etc.. Is a real position, which we do not know, but
00:45:13.125 --> 00:45:22.479
which somehow exists, but we don't know it." and said," to
is a sense position that what we get from the camera system.
00:45:22.489 --> 00:45:30.623
Both are similar, but not the same, because there is always
some imposition in the center and in the center. Egaliation
00:45:30.623 --> 00:45:38.964
with the camera image, an evaluation process. So therefore
we this uncertainty, this little bit of randomness, which
00:45:38.964 --> 00:45:48.074
exists is covered by this delta tea by this measurement
uncertainty. So if we have this, this is our only measurement
00:45:48.074 --> 00:45:56.494
that we get with the camera so we can't measure
anything else. We assume we just may the position. So let us
00:45:56.494 --> 00:46:04.863
rewrite this measurement equation in the form that we need to
say that it is a linear gauge model. So we says that he is
00:46:04.863 --> 00:46:14.140
equal to a matrix with one row and two columns, so that
contains one zero times Sd. If we do mold out this
00:46:14.140 --> 00:46:22.851
multiplication here we get exactly x sixty one star one times,
et cetera, plus delta tea, the measurement noise. So this
00:46:22.851 --> 00:46:31.983
means also here this equation fits to our needs of a linear
cultural model. So for linear Gulf models, like the ones
00:46:31.983 --> 00:46:42.058
that we just introduced. It holds that if we choose a caution
distribution, a caution probability density function
00:46:42.058 --> 00:46:52.490
to model um. These kind of distributions, the distribution
of the current state. Given all observations up to that
00:46:52.490 --> 00:47:02.686
state, then it can be found analytically by a lot of annoying
calculations that then this predicted state probability
00:47:02.686 --> 00:47:12.386
can also be described by a gauge and probability density
function. So that means we start with the Gauchian distribution
00:47:12.386 --> 00:47:23.552
and the " What is it?" the prediction step also provides a
caution vision, and that means. And furthermore, if these
00:47:23.552 --> 00:47:33.284
predicted state probabilities can be described by a caution
distribution, then the next time state distribution
00:47:33.284 --> 00:47:42.678
probabilities can also be described by a caution. And that
means the innovation step also Mets caution distributions to
00:47:42.678 --> 00:47:51.547
caution distributions. That means if we have a linear gauge
model, and we use garage and distributions to model the
00:47:51.547 --> 00:47:58.692
state distribution probabilities, then all the distributions
which occur throughout all calculations are and that
00:47:58.692 --> 00:48:07.585
is nice because to represent a Gaussian distribution, we
only need to know two things. The expectation value of this
00:48:07.585 --> 00:48:18.575
mule vector. And the matrix is Sigma matrix. So that
means, in this case, a prediction step takes an expectation
00:48:18.575 --> 00:48:29.142
value vector and a province matrix for negotiation, distribution
and yields Mhm, an expectation value vector, and and
00:48:29.142 --> 00:48:40.884
other matrix so and the innovation step takes again
such an expectation, value vector and a coherence metrics and
00:48:40.884 --> 00:48:51.012
a measurement and yields an expectation value vector and
a coherence metric. So we can only calculate with these
00:48:51.012 --> 00:48:59.352
expectation value vectors @unoise@ and the covance mattresses
and map those two new ones and all everything can be
00:48:59.352 --> 00:49:06.565
calculated analytically. So we don't need. We don't need
numerical integration and all that annoying stuff. Okay, this
00:49:06.565 --> 00:49:16.283
is shown here. So we stay Say we start here with
an initial estimate of the of the predicted state
00:49:16.283 --> 00:49:23.955
distribution @unoise@ using a density function to represent
this distribution. That means we provide this value of
00:49:23.955 --> 00:49:31.913
expectation with this expectation value of value, which is
written here as a one predicted and the matrix,
00:49:31.913 --> 00:49:40.268
which is given by a Sigma T plus one predicted, then we
execute an innovation step that integrates a new measurement
00:49:40.268 --> 00:49:47.431
vector at T, plus one and provides another caution distribution
represented by the beauty, the new expectation value
00:49:47.431 --> 00:49:56.779
vector and the new coverance about matrix seek my tea. So
what we do is we take these, we take these two things here.
00:49:56.789 --> 00:50:04.454
Mule t plus one predicted signalty was one predicted. And the
measurement and we provide mutual insignity. And for the
00:50:04.454 --> 00:50:12.401
next point in time. So in between the increment, the time
index, and then for the prediction stuff, we take these two
00:50:12.401 --> 00:50:19.496
here, Mutt and , and we care great mutations, one
predicted, and Sigmund cheaper, one protected. And like that, we
00:50:19.496 --> 00:50:28.835
can iterate. We can cycle through that loop and integrate
all the measurements incrementally @unoise@ okay. How, does
00:50:28.835 --> 00:50:38.123
that look like so if we do all the calculations analytically,
we find out that the steps are implemented like that. So
00:50:38.123 --> 00:50:47.009
in the direction step here we have Mutt and Sigmati given, and
we want to calculate these @unoise@ a predicted one
00:50:47.009 --> 00:50:56.490
and will predict one and the formulas to do
without them. The equations which we used to to implement it.
00:50:56.500 --> 00:51:05.980
Look like that. So then you, you value the new expectation
value vector is equal to eighty. This state transition matrix
00:51:05.980 --> 00:51:15.079
a times beauty plus beauty will offset. So actually, what
we do is we take the beauty factor. We treat that as if it
00:51:15.079 --> 00:51:23.786
would be the two state of the system. And we apply this linear
dynamics with linear transition. And to this state vector
00:51:23.786 --> 00:51:34.217
beauty, or to this expectation value of activity @unoise@
and. For the matrix the equation looks like that. Well,
00:51:34.217 --> 00:51:45.024
how to explain it. So actually, the offset, it actually doesn't
matter for the matrix is if I take a random ,
00:51:45.024 --> 00:51:54.600
and I just shifted by a non offset. When the matrix,
the sprat of the of the random of the random variable
00:51:54.600 --> 00:52:01.534
doesn't, doesn't change, and only its location changes.
Therefore, this beauty doesn't occur here in this second
00:52:01.534 --> 00:52:08.774
equation, while when this multiplication. Eighty six.
Ninety eighty transpose is actually something that we can
00:52:08.774 --> 00:52:18.147
interpret it as we apply this linear Trump mapping given
by eighty to the matrix. Now that is actually what
00:52:18.147 --> 00:52:26.827
happens here and then what we do here, we add up the uncertainty
which comes in from the transition noise from the form
00:52:26.827 --> 00:52:37.099
of an impression in the state transition. And this is described
by this matrix beauty. So we add up this as an
00:52:37.099 --> 00:52:45.774
additional source of uncertainty to our knowledge. Yeah,
okay. That is a predictive prediction step where innovation
00:52:45.774 --> 00:52:54.209
step looks a little bit more difficult, as though we start
with these predicted and with a measurement. And we
00:52:54.209 --> 00:53:03.932
want to get there is a beauty and sigmundity. So here is
the calculation looks a little bit more difficult. So
00:53:03.932 --> 00:53:12.919
typically, one intermediate matrix is calculated called
Kalaman Gain. This is this K variable here. Well, Yeah.
00:53:12.929 --> 00:53:21.851
The interpretation of these formulas is actually a
little bit more difficult. And it is not that easy. But what I
00:53:21.851 --> 00:53:31.622
can see is following for the beauty value. What do we do
while we compare the measurement that we have made with H T
00:53:31.622 --> 00:53:40.604
time, some new beauty predicted. So this is the most probable
predicted state in which we are multiplying it with H. Tea
00:53:40.604 --> 00:53:49.973
means that we cake late here. In this part, the most probable
observation that we expect to be faced with. So we compare
00:53:49.973 --> 00:54:00.330
the true observation that we made with the observation that
we expect to make. And so this term in brackets, so to say
00:54:00.330 --> 00:54:11.897
tells us something about how good our expectation fits to the
measurement that we made. And based on that, our predicted
00:54:11.897 --> 00:54:20.673
state, the predicted state, or this expectation value for the
predicted state is modified a little bit and this k
00:54:20.673 --> 00:54:28.930
miss can be interpreted as a kind of waiting factor.
Ah, that weighed somehow the uncertainty in the measurement
00:54:28.930 --> 00:54:37.923
and the some uncertainty in the knowledge of the present state
with each other. So this is this term here @unoise@ Ah,
00:54:37.923 --> 00:54:47.307
where we can see that here is a vizier. This uncertainty in
the measurement enters this equation in its inverse form to
00:54:47.307 --> 00:54:57.340
the power minus one and the uncertainty of our knowledge of
the previous of the present state enters here. So to say in
00:54:57.340 --> 00:55:07.884
the non inverted way. So it is a kind of ratio that is
calculated here between our half horn. Well, we know how sure we
00:55:07.884 --> 00:55:19.812
are about the the state so far from our calculations so far
and how reliable the measurement is. Mhm yeah, so that is
00:55:19.812 --> 00:55:31.922
actually this part. And the update of the uncertainty metrics.
Uh, signity goes on like that here. I never found any
00:55:31.922 --> 00:55:43.676
intuitive um idea to explain that. So for simple cases,
if the state vector just contains one variable. And if the
00:55:43.676 --> 00:55:53.074
measurement just contains one variable, the measurement
vector, then try it out, then things simplify, and it becomes
00:55:53.074 --> 00:56:03.478
clearer what these formulas do in this general case of matrix
multiplications. So at least I didn't find any intuitive
00:56:03.478 --> 00:56:11.767
explanation, or the last equation okay. So, that is the
prediction and the innovation step. Once we implement them
00:56:11.767 --> 00:56:20.333
we implemented something that has a name attacking,
that has a name that is called a going back to a
00:56:20.333 --> 00:56:27.980
researcher researcher named So. And this comment
filter is quite popular and used in many applications.
00:56:27.989 --> 00:56:38.728
Yeah, to show it up. So where is it? okay, so let us apply
it to this example of observing a car that is driving is
00:56:38.728 --> 00:56:46.921
Consumm so. We assume a frame rate of one second. That
means that the time interval between two points in time is
00:56:46.921 --> 00:56:57.774
assumed to be one second. So now, based on what we have
derived so far, we can say that this matrix Ave has the shave
00:56:57.774 --> 00:57:08.900
won one, zero one, and offset you is zero, zero, and the
matrix age is one zero. That is what we already derived
00:57:08.900 --> 00:57:17.706
@unoise@ now. Of, course we, need these % um coverance
mattresses to describe the uncertainty um of the state
00:57:17.706 --> 00:57:26.654
transitions cue, and the uncertainties of the measurements.
And of course, that is always a little bit a tricky story
00:57:26.654 --> 00:57:36.826
how to get them. I personally prefer to make some rough
guesses, yeah? and to simplify things and not to run into too
00:57:36.826 --> 00:57:47.347
much trouble. Choose those mattresses normally as diagonal
mattresses so actually these mattresses must be
00:57:47.347 --> 00:57:57.820
symmetric, positive, definite mattresses. Hmm, so you can put
different values and zeros to the non diagonal elements,
00:57:57.820 --> 00:58:09.580
but then managing to get a positive, definite matrix is tricky.
Yeah, so I would suggest if there is no reason to to do
00:58:09.580 --> 00:58:18.789
it differently and do it like that. Um, if choose the non
diagonal elements, a zero @unoise@ and for the diagonal
00:58:18.789 --> 00:58:26.985
elements you have to find reasonable values. So for cue, of
course, this models the imposition in the state transition,
00:58:26.985 --> 00:58:35.592
each of the diagonal elements refers to the variants, to the
spread, to the uncertainty for one and state very album. So
00:58:35.592 --> 00:58:42.718
in this case, this number here refers to the uncertainty
in the position of the vehicle. And this one refers to the
00:58:42.718 --> 00:58:49.842
uncertainty in the velocity of the vehicle. Now think a little
bit what are typical values that we expect. So how much
00:58:49.842 --> 00:58:58.739
of imposition do we expect from a car to observe when it is
driving for one second. How much meters of errors do you
00:58:58.739 --> 00:59:09.595
expect? and if you say," Well, I expect maybe I rose up to
say ten centimeters or four point one meter. If that is your
00:59:09.595 --> 00:59:19.231
guess, then there is a rule of some that says tests take
this value divided by two, three or four doesn't matter so
00:59:19.231 --> 00:59:27.220
much. Now, I prefer to be divided by two, and then take the
square of it. And that yields a reasonable number. So here
00:59:27.220 --> 00:59:34.884
in this case, I was assuming an error of one meter. So a
maximum error in this position of one met him. So I took one
00:59:34.884 --> 00:59:42.294
meter, divided it by a half by a by two. So I get a half. And
then I took the square of it. So therefore all point five
00:59:42.294 --> 00:59:50.246
square here. And for the velocity calculating velocities in
meter per second. I thought, okay, maybe the error could be
00:59:50.246 --> 00:59:59.327
opened to meet up a second that occurs in one second. I
am using some assumptions about typical accelerations and
00:59:59.327 --> 01:00:09.307
acceleration values occur. And okay, then I get open two,
divided by two yields, or point one squared. And I was
01:00:09.307 --> 01:00:18.379
choosing that as the second entry in this matrix. And for the
measurement noise the same story. So we have one variable,
01:00:18.379 --> 01:00:27.248
one measurement, very upper said. So the matrix R becomes
one by one matrix. That is very simple. If two
01:00:27.248 --> 01:00:34.904
independent measurements say if you would be able to measure
the position and the velocity of the car directly, because
01:00:34.904 --> 01:00:43.068
we have another kind of center, then we would. It would have
a two by two metrics. If we would have three independent
01:00:43.068 --> 01:00:49.466
measurements, because we have a very strange, whatever sensor
that is also able to measure accelerations. When we would
01:00:49.466 --> 01:00:57.233
have a three by three metrics as well. Positive define it,
and symmetry again, the same trick. So choose all. Not
01:00:57.233 --> 01:01:04.828
meant to be zero @unoise@ um at least the first gas
@unoise@ and uh choose appropriate values for the diagonal
01:01:04.828 --> 01:01:12.875
elements. Here I would say @unoise@ a camera a stereo camera
is not that accurate in measuring things @unoise@ so. I Was
01:01:12.875 --> 01:01:21.844
saying okay, let us say the measure and accuracy would be
something like four meters, which would have really be a very
01:01:21.844 --> 01:01:33.087
large error. So then take the half of it and take a square of
it, and then you end up with a two square here. Then later
01:01:33.087 --> 01:01:42.139
on, when you apply this carbon filter and you test the system
and you observe what is going on in the common filter.
01:01:42.150 --> 01:01:48.508
Please vary these numbers and check whether things
become better, the performance of the system becomes better
01:01:48.508 --> 01:01:55.574
with other values, because sometimes you are wrong in your
first guess what could be reasonable values, then try out
01:01:55.574 --> 01:02:03.449
what happens if you change the values and whether the system
behavior improves okay, so that is the thing. Now we
01:02:03.449 --> 01:02:11.840
finally need the initial guests for the state and the initial
guests for the Coverence Metrics to express how sure we
01:02:11.840 --> 01:02:19.851
are. And here in this case, maybe we do not know anything.
We don't have any prior knowledge about where the car could
01:02:19.851 --> 01:02:28.770
be and how fast it could be if that happens and choose just
an arbitrary number. So for instance, zero um as a initial
01:02:28.770 --> 01:02:37.245
expectation value and choose a corverance might exist. Very
large entries in the diagonal. This expresses wow. It could
01:02:37.245 --> 01:02:49.419
be zero, zero, the initial state, but I'm very, very sure about
it. And so you don't you. Don't disturb the common
01:02:49.419 --> 01:02:57.283
filter by providing some bad initial guess. Of course, if
you know something, if you know that the initial situation
01:02:57.283 --> 01:03:05.479
always starts with zero velocity, because you know that
whatever you track a car, that well that is standing at the
01:03:05.479 --> 01:03:13.797
beginning. Then, of course, you can choose other values here.
And then you can also express how sure you are about this
01:03:13.797 --> 01:03:23.204
initial situation. But here, for this example, let us say we
choose. We prefer to express that. We don't know anything
01:03:23.204 --> 01:03:32.397
in the beginning. Now, let us run the carbon shelter and see
what the output is. So here I am plotting the result of
01:03:32.397 --> 01:03:41.226
after each innovation step so. In rat we see actually
the measured position. So the horizontal axis here is a
01:03:41.226 --> 01:03:50.257
time, the point in time. The vertical axis is the position
of the vehicle. The rat cross is the measurement. Yeah, the
01:03:50.257 --> 01:04:01.012
blue cross is actually the estimated position. So this new
value that we calculate in the Carmen filter. And you see,
01:04:01.012 --> 01:04:09.642
Yeh, it fits somehow. So it is not exactly the same as though
they in the measurement, which is filtered out by
01:04:09.642 --> 01:04:17.374
the by the common filter and the vertical bars indicates
actually the uncertainty that exists. So it expresses somehow
01:04:17.374 --> 01:04:25.400
the shape of the matrix. The larger this bar is, the
more uncertain the common filter is about the real position.
01:04:25.420 --> 01:04:33.522
So this means the two position might be some way in this
interval. We do not exactly know where it is, but the most
01:04:33.522 --> 01:04:41.196
probable position is at the blue cross here. So for the estimated
position, this looks very much like just smoothing. So
01:04:41.196 --> 01:04:49.158
to say the measurements that we get for the . It may
be a little bit more interesting. Of course, we don't have
01:04:49.158 --> 01:04:56.335
velocity measurements, but we can have a look at what the
Carmen Filter um provides us output in the first measurement,
01:04:56.335 --> 01:05:03.512
we can see the output is zero, which is not really surprising,
because if you make just one measurement, one position
01:05:03.512 --> 01:05:11.238
measurement of a vehicle. We cannot conclude anything about
its, about its velocity. So therefore, there is no reason to
01:05:11.238 --> 01:05:21.077
change the value of sear to any other value. But we also see
that this arrow bar here is rather large. Yeah, so it is
01:05:21.077 --> 01:05:29.256
still very unsure what the real velocity of the car is because,
of course, the information was not sufficient to make a
01:05:29.256 --> 01:05:38.151
good guess. In the second, at the second point in time,
when the second measurement was integrated, we see that the
01:05:38.151 --> 01:05:46.721
method was already able to estimate that there is some non
zero velocity, and the uncertainty is decreasing, and the
01:05:46.721 --> 01:05:54.862
more measurements we get, the smaller the uncertainty becomes,
and the more stable also the estimate becomes, and the
01:05:54.862 --> 01:06:03.528
more sure we are that the true velocity is something like
one meter per second. However, this uncertainty will never
01:06:03.528 --> 01:06:13.033
become zero, and it does not converge to zero. Why? because
in the state transition step and the prediction step, we
01:06:13.033 --> 01:06:23.945
always add some uncertainty to the to the measurement. And
therefore this converges but it does not become
01:06:23.945 --> 01:06:34.684
zero mhm nasal it. The a behavior is not converging
to zero @unoise@ ok there is a slide that we already
01:06:34.684 --> 01:06:44.719
have. Here is another example. The same story, the same
modeling. But here I was somehow mimicking that the car is
01:06:44.719 --> 01:06:51.908
moving forward. And at a certain point in time, it is
immediately moving backward. So that is physically impossible
01:06:51.908 --> 01:07:00.098
here. It is more like you observe a marble or a ball. Now
that is bouncing against the wall and then bouncing back
01:07:00.098 --> 01:07:08.661
something like that here you might imagine. Of course, this
is a behavior that is not modeled in our state transition
01:07:08.661 --> 01:07:16.724
modeling in the state transition modeling, we assume that the
object is moving all the time. Straightforward. But here
01:07:16.724 --> 01:07:24.171
something happens, which contradicts the state transition
model. So what happens so in the beginning, in this period of
01:07:24.171 --> 01:07:31.931
time. And the same story happens as we have seen before, the
Carmen filter becomes more and more sure about the real
01:07:31.931 --> 01:07:40.182
velocity of one meter per second. And so it becomes better
and better. But at that point in time when this velocity is
01:07:40.182 --> 01:07:46.068
changing, we see over the measurements, the position
measurements are decreasing again, but the common filter still
01:07:46.068 --> 01:07:54.703
says," Okay, but I am. I am sure that I move that the object
is moving forward with one meter per second. So still, I
01:07:54.703 --> 01:08:02.426
expect that it is moving for what was on me to a second, and
therefore they expected," Ah, this. This calculated the
01:08:02.426 --> 01:08:10.638
estimated positions where the object will be deviate very
much from the measurements at this point in time, the
01:08:10.638 --> 01:08:18.851
assumption of constant velocity is violated, and the common
field has really big problems to follow this change, to
01:08:18.851 --> 01:08:27.280
adapt to this change. And it takes long time until the
common filter learns or adapt to the new situation that the
01:08:27.280 --> 01:08:36.351
velocity is not plus one meat of a second anymore, but minus
one meat of a second. We see when we observe the velocity,
01:08:36.351 --> 01:08:45.687
but at the end here, after thirty, thirty, forty points in
time. It was able to adapt to the new velocity. But we see
01:08:45.687 --> 01:08:53.697
that it took long time. And what this may be surprising is
when we are in this situation here, where obviously the
01:08:53.697 --> 01:09:01.705
measurements and the the state of the system that is estimated
by the common field that do not fit together any more
01:09:01.705 --> 01:09:09.627
still. The uncertainties are not increasing, but decreasing.
That is a little bit surprising. That comes from the fact
01:09:09.627 --> 01:09:19.524
that we modeled the system as a hidden mark of model. And
from that comes that the common filter will not react on such
01:09:19.524 --> 01:09:28.227
a violation of the assumption of the basic assumptions of
constant velocity by increasing the uncertainty, but it still
01:09:28.227 --> 01:09:37.283
will decrease the uncertainty. Oh, sorry. And Sir Wolfe,
so it will not increase the uncertainty but it will
01:09:37.283 --> 01:09:45.173
decrease the uncertainty of the estimation of the Internet.
Yeah, so that is a little bit counterintuitive, but this
01:09:45.173 --> 01:09:54.788
comes from the simplifications which we made with the hidden
mark of model. And with the fact that we did not consider
01:09:54.788 --> 01:10:05.205
violations of of the assumptions that we made @unoise@ so.
Now we have two methods with which we can track objects like
01:10:05.205 --> 01:10:12.920
cars or so, and estimate velocities of object. We started with
a linear regression in the beginning of this chapter. And
01:10:12.920 --> 01:10:20.899
now we have seen the and you might ask, how do we come
with these two methods? compare with each other. Both can
01:10:20.899 --> 01:10:30.778
be used. Both are used in practice. What are the advantages
of disadvantages of both. So the first thing is, if we
01:10:30.778 --> 01:10:39.463
consider the models that are on which those methods are
based. Then, of course, in bows cases, we assume some linear
01:10:39.463 --> 01:10:46.996
system behavior, a linear dependencies and Gauchian noise
that applies to both miles for aggression. I didn't make it
01:10:46.996 --> 01:10:54.962
explicit, but implicitly it is assumed Gauchian noise. And
for a common filter as far. But there is a small difference
01:10:54.962 --> 01:11:02.004
that the difference is that in the common filter, we assume
that the state changes over time, at least slightly, while
01:11:02.004 --> 01:11:08.426
in the regression model, we assume that this state does
not change over time. Not only their observation depends
01:11:08.426 --> 01:11:17.586
linearly on the state. So what is the state vector in the
Carlman filter is the vector of regression coefficients. In
01:11:17.586 --> 01:11:27.183
the regression case that can be seen as then which
linear independence assumption did we do well in the comment
01:11:27.183 --> 01:11:35.515
filter. We did this assumption of my coffeeian independence.
That means here the subsequent state is depends only on the
01:11:35.515 --> 01:11:43.854
present state, and not from past the past and the present.
Observation only depends on the present state and not on the
01:11:43.854 --> 01:11:51.491
pass in the linear regression case, we also make an assumption.
Let me so called I Id identically and independently
01:11:51.491 --> 01:11:58.836
distributed measurements. Now we assume also that the
measurements are independent of each other. That is very similar
01:11:58.836 --> 01:12:06.894
to the @unoise@ assumption @unoise@ and that they follow
the same distribution that they are all distributed with
01:12:06.894 --> 01:12:15.844
respect to the same @unoise@ to the same glacier @unoise@ so
how do we calculate it? we have seen in the common filter,
01:12:15.844 --> 01:12:21.693
we do some incremental calculation with each measurement that
we get, we calculate a new prediction and new innovations
01:12:21.693 --> 01:12:30.322
that, based on the result that we have calculated so far. In
the regression case, it is not that that easy we have
01:12:30.322 --> 01:12:39.004
more or less do a repeated calculation. So for each measurement
that we want to add to our calculation, we have to
01:12:39.004 --> 01:12:46.299
calculate the full regression, again, sort of thing that
is meant with repeated calculation. Of course, if we are
01:12:46.299 --> 01:12:55.008
clever, we can save a little bit of time and so on, by storing
some intermediate results. But in general, we would say
01:12:55.008 --> 01:13:03.140
it is a repeated calculation, while common filter is incremental
calculation @unoise@ so what. Do we have to store
01:13:03.140 --> 01:13:11.538
when we want to calculate these filters in an incremental or
repeated way in the common theater. We only have to store
01:13:11.538 --> 01:13:18.501
this, the you value, this expectated state value and the
corverance matrix, nothing else. You can forget everything
01:13:18.501 --> 01:13:26.223
else. And once we know these two things, once we know the
present caution that describes the state distribution that is
01:13:26.223 --> 01:13:33.230
sufficient to do all future calculations. And for linear
regressions, we need to stay memorize all the measurements that
01:13:33.230 --> 01:13:40.992
we made so far, and that we want to incorporate in future
cake relations. Though the memory requirements are a little
01:13:40.992 --> 01:13:50.580
bit different. So then we can ask how much influence do the
measurements have so assume we made one hundred measurements
01:13:50.580 --> 01:14:01.281
and now we calculate the present state in the cut with
a common filter, or we do a regression approach, we might
01:14:01.281 --> 01:14:10.899
ask how much influence does each measurement have on the
result. And then we find for the linear regression model. All
01:14:10.899 --> 01:14:18.460
the measurements have the same influence doesn't matter
whether they were sensed just now, or ten seconds before,
01:14:18.460 --> 01:14:25.745
doesn't matter. All have the same influence. All the measurements
which are used in the linear regression, in a standard
01:14:25.745 --> 01:14:33.270
linear regression. Let us say a half the same influence on
the result in the common filter. It is not that like that.
01:14:33.279 --> 01:14:43.142
But m we could argue that somehow the influence decreases
over time. The last measurement has a strongest influence on
01:14:43.142 --> 01:14:51.636
the state estimation, while the oldest measurement has the
smallest influence. And roughly speaking, the influence
01:14:51.636 --> 01:15:02.241
decreases exponentially over time. You know, @unoise@ okay.
So, what else? okay, what do we need to know and to specify
01:15:02.241 --> 01:15:11.089
if we want to apply it, especially concerning the uncertainty,
the noise, these mattresses Q and R, we have to know them
01:15:11.089 --> 01:15:19.599
for implementing a common filter. So we have to know the
amount of measurement noise at the matrix. And this must
01:15:19.599 --> 01:15:28.352
be known. The big, a big advantage of linear regression that
we don't need to provide it, and we don't need to provide
01:15:28.352 --> 01:15:37.931
how certain the measurements are. We only need to know that
the uncertainty, this covance matrix, is the same for all
01:15:37.931 --> 01:15:47.571
measurements. But we do not need to specify it. Okay, so and
then the variances that we estimate that means what we have
01:15:47.571 --> 01:15:56.968
just seen. We have seen more or less in the comment filter.
The if we analyze them the the measures of
01:15:56.968 --> 01:16:05.632
uncertainty, they decrease us over time. So as a
rough, rough idea, while in a linear regression, if you do a
01:16:05.632 --> 01:16:13.421
repeat, a linear regression, and we add some measurements,
and then the variants that we can derive for a regression
01:16:13.421 --> 01:16:20.737
might also increase. So the regression does not have this
unintuitive behavior that if the measurements deviate very
01:16:20.737 --> 01:16:29.330
much from the % uh expected % um measurements that when still
the uncertainty decreases, that do not apply for linear
01:16:29.330 --> 01:16:36.736
regret. So in this way, the linear regression is more intuitive
concerning that, the boss possibility both techniques
01:16:36.736 --> 01:16:45.151
are possible. If you are faced with a problem. Try both. I
would say, try both some times. The common filter is easier
01:16:45.151 --> 01:16:53.208
or better to use. @unoise@ sometimes a regression approach is
easier, better to use @unoise@ so try both @unoise@ and do
01:16:53.208 --> 01:17:01.181
not change say just for because common feeler sounds that
crazy. We use it. No regression. Also, @unoise@ might have
01:17:01.181 --> 01:17:10.633
might be more beneficial in some situations, but sometimes
also the common filter is better than every question. Project
01:17:10.633 --> 01:17:22.300
@unoise@ okay. So, so now we have limited ourselves so far.
Linear systems, linear caution systems. The question is,
01:17:22.300 --> 01:17:30.286
what happens if the system is not linear anymore. And this
easily happens in practice. So it might happen that either
01:17:30.286 --> 01:17:37.619
the measurement depends in a nonlinear relationship from the
state or the state transition is nonlinear. That means we
01:17:37.619 --> 01:17:47.609
have to generalize things and say," Okay, now S T plus one is
not a linear function on N, S, T, but it is just an linear
01:17:47.609 --> 01:17:57.609
function. Say, if there is all that F and maybe nonlinear
function that morals, how as deep as one depends on his tea.
01:17:57.630 --> 01:18:05.365
And let age be a function that maps the state onto the
observation now, of course, plus some editive gauge noise, which
01:18:05.365 --> 01:18:14.817
I is not mentioned here in this light, but which we still
assume, what can we do in such a case? observe that is not the
01:18:14.817 --> 01:18:23.590
what we want @unoise@ what can we do perhaps? So what
can we do? well, common filter doesn't work, because the
01:18:23.590 --> 01:18:31.211
common filter assumes that we have linear relationships,
cannot be used, but there are other techniques extensions. A
01:18:31.211 --> 01:18:38.079
one is a so called extended Carmen filter. That is the simpler
version. And then there is an common filter is also
01:18:38.079 --> 01:18:45.661
a kind of extension of the common filter for systems,
which is a little bit more difficult to to deal with. And a
01:18:45.661 --> 01:18:53.441
very general solution is a so called particle filter Ooh. We
will have a look at that later on. Okay, let us start with
01:18:53.441 --> 01:18:59.980
the extended comment filter. The basic idea of the extended
comment filter is very easy, and that is what you already
01:18:59.980 --> 01:19:07.147
know from from other lectures, also from this lecture. If
you deal with the nonlinear system and you want to use a
01:19:07.147 --> 01:19:14.058
linear technique on it. What do you do while you just linearize
the system locally, around the point of interest. That
01:19:14.058 --> 01:19:21.298
is actually the basic idea of the extended common filtering.
If you understood that you understood what the extended car
01:19:21.298 --> 01:19:31.244
and filter is it is just using a linear linearization of the
system around the present point of interest. So if you do
01:19:31.244 --> 01:19:38.668
that, we can ride the prediction surf and derive the prediction
and innovation stuff of the extent common viscer like
01:19:38.668 --> 01:19:49.734
that. The update of this um expected state Ah value, um U,
T platform predicted m can be done by just applying this
01:19:49.734 --> 01:19:58.656
mutual vector on to this day transition function. F, this
mute vector has the same shape as a state can be interpreted
01:19:58.656 --> 01:20:07.257
as a state factor so we can apply the state transition
matrix on it, and % Ah. So we consider all the nonlinearity,
01:20:07.257 --> 01:20:17.623
which is in F for the update this is not possible
we need something like a matrix that we can apply from the
01:20:17.623 --> 01:20:27.573
right and the left to the sigma and instead of
which matrix can be used here, while the matrix that we use
01:20:27.573 --> 01:20:37.646
is the of F of this function of fear. Em evaluated at
beauty at this point, beauty at the present best estimate of
01:20:37.646 --> 01:20:51.425
what the present state is. So what was the Jacquel of
a function F no one knows the Jacobine is the matrix that
01:20:51.425 --> 01:21:03.070
contains all the partial derivatives of the function never
heard of dodge the Yakovi mathematics is in matrix.
01:21:03.069 --> 01:21:11.724
Steve and health
01:21:11.724 --> 01:21:20.378
01:21:20.378 --> 01:21:30.094
ok so back to English the is the
the first order derivative of the function. F now So
01:21:30.094 --> 01:21:40.030
okay, that is a Jacobian. So we used the here. Instead
of dysfunction of this matrix, a T that we had in the
01:21:40.030 --> 01:21:50.063
Carmen Filter. So here the Jacobian is using playing the role
of this matrix now, if if is a linear function can
01:21:50.063 --> 01:21:59.737
be represented as eight times as plus you, and you calculate
the Jacobian, you get a so that makes sense. Okay, so that
01:21:59.737 --> 01:22:08.502
is the prediction step. The innovation step looks like that.
Actually, here again, where we can apply this predicted
01:22:08.502 --> 01:22:19.509
state to the measurement function age well directly, because
this looks like a state rector. So we can apply this
01:22:19.509 --> 01:22:27.544
nonlinear function. And here, so that and everywhere
else where we had this in the common filter innovation step,
01:22:27.544 --> 01:22:37.354
this matrix H, T. We also have a Matrix Hd here. But now this
Ht is a Jacobian of small of the function small age around
01:22:37.354 --> 01:22:47.103
this point for that point of view. So we take this measurement
function age, calculate its first order derivative at the
01:22:47.103 --> 01:22:56.586
point of interest, namely this predicted state. And you enter
it to the carbon filter innovation step formula. That is
01:22:56.586 --> 01:23:07.275
all. So that is it. So just a of the system model, and
that we use the linear rice model. That is the extended
01:23:07.275 --> 01:23:15.303
comment filter so. Remember back we didn't forget any
slide Ok now. The second version is a
01:23:15.303 --> 01:23:24.165
Carman filter the unsented Carmen filter is a little
bit different. So it also is using a kind of approximation,
01:23:24.165 --> 01:23:33.399
linear approximation or approximation of the nonlinear
function F, but it is a little bit different in how it works.
01:23:33.409 --> 01:23:43.945
So assume we have assumed in this case, we have a two
dimensional state space this is x and this is me for
01:23:43.945 --> 01:23:52.373
our motion model case, and then % um a Gaugeian distribution
is described by, of course, the expectation value
01:23:52.373 --> 01:24:03.342
@unoise@ that would be this point here and by the covance
matrix. And if we ask for such a Gaussian density which
01:24:03.342 --> 01:24:15.683
is, if you say okay, um, for which values does this ocean
density function yields the same value, then we end up with
01:24:15.683 --> 01:24:25.724
ellipses. Ah, so an ellipse. This kind of lips is somehow
um modeling this caution distribution. Yeah, the center
01:24:25.724 --> 01:24:35.434
position is Ah expectation value, and the covance matrix is
some holidays represented by the shape. And the farther the
01:24:35.434 --> 01:24:43.568
larger the diameter is of this, of this ellipse. The
larger is the uncertainty in this direction. So here, the
01:24:43.568 --> 01:24:51.369
uncertainty in this direction is smaller than the uncertainty
in this direction. Yeah, so that is actually how this can
01:24:51.369 --> 01:24:59.102
be interpreted. So now, what we do in an unsended Karl filter
to, for instance, implement the prediction step is that we
01:24:59.102 --> 01:25:07.076
create some points, which are called Sigma points. One of the
Sigma points is this point, the center. And then there are
01:25:07.076 --> 01:25:15.822
some other points which are spread around the center, following
somehow the shape of this ellipse. This is done in a
01:25:15.822 --> 01:25:24.781
sematical, systematical way, so not in a random way. We do not
randomly sample points, signal points, but we analyze the
01:25:24.781 --> 01:25:33.186
matrix and based on the matrix regenerate
these points. And by construction these points are reap
01:25:33.186 --> 01:25:41.872
implicitly, can be used to represent this Gaugeian
distribution. So we based, if we are given these points, if we
01:25:41.872 --> 01:25:49.804
calculate their expectation value, we end up with this point.
And if we calculate their matrix, we end up with the
01:25:49.804 --> 01:25:56.931
matrix, which is given by this bloomer lips. That means
representing such a caution distribution can be done with
01:25:56.931 --> 01:26:04.621
these so called signal points. Its signal points, however,
can be also interpreted as a state vector. So if it is state
01:26:04.621 --> 01:26:11.729
vector, we can apply the state transition function effort
to it. Yeah, and map it. Um and look at the results. So in
01:26:11.729 --> 01:26:19.419
this case, if we apply F or age, doesn't matter on to that.
Maybe this point is map there here. And this one there.
01:26:19.430 --> 01:26:27.237
There is one here, this one here, this one here. Now we again
have a set of segment points. And now what we can do is
01:26:27.237 --> 01:26:34.018
based on these signal points, we can add equal variance
matrix, again, based on it. So we can estimate a gulf and
01:26:34.018 --> 01:26:42.920
distribution based on. So what we did now is that we have
solved this problem of nonlinear function by first generating
01:26:42.920 --> 01:26:50.447
these signal points as a kind of representative points
that describe this Gaugeian distribution, then apply this
01:26:50.447 --> 01:26:58.962
nonlinear function to each of these signal points. And based
on the Sigma points, which we get as result, we re estimate
01:26:58.962 --> 01:27:06.473
a gulf and distribution. And by doing that, we can overcome
these nonlinear, these problems of non so that is a
01:27:06.473 --> 01:27:14.339
basic idea of the unsented common filters. Of course, the
details are a little bit more technical, but for reasons of
01:27:14.339 --> 01:27:22.582
time, I've omitted those technical details here in the lecture.
If you are interested, just read the original papers, or
01:27:22.582 --> 01:27:31.529
read the chapter about the karma filter techniques in the
book on probabilistic robotics. There it is explained in a
01:27:31.529 --> 01:27:42.460
nice way, and you find the details. Yeah, so that is
the matter so okay so far, I think now time is up for today.
01:27:42.470 --> 01:27:54.399
So let us continue next time. Then with an example that
shows how these can be used in practice @unoise@ Um. Yeah.