WEBVTT
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So welcome back to the lecture on Automotive vision. Yeah,
Last monday We, started to talk about tracking objects
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in the three dimensional world and determining their Yah
movement. That means their velocity, the direction of movement,
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et cetera. And yeah, we started with a traditional regression
approach. And just to remind you of the last life that we
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discussed. That was actually this one. We assume that we have
a camera or another censor that is measuring the position
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of a vehicle that we observe over time. So in this case, we
have four different points in time for each point in time,
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we get a measurement where the vehicle is, of course, all these
measurements are suffering from imposition and noise. So
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they are not exactly the true position of the vehicle, but
just a measurement of it. When we said," Ok, if we assume
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that the vehicle is moving this constant velocity along this
road here, which is given by a certain coordinate system,
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then we can say that the position of the vehicle at a certain
point in time, T can be described by means of this linear
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law of movement, a certain initial position, plus the velocity
times, time. And of course, what we do is we measure the
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positions of the measurement itself is affected by a certain
amount of noise and impression that we might adhere. And
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then we were saying, okay, we do now the sensed position
on the left hand side for some points in time. We know
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the points in time tea at which we made the measurements and
what we do not know is the interposition x zero and the
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velocity v. So these are the two unknown variables and acts
of tea, the sensed position at a certain times, tea and the
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point in time tea is what we know. And now we want to well
determine the unknown parameters from our measurement and the
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way we can do so is that we say," Okay, for each measurement,
we can enter eggs of tea here on the left hand side, we
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can enter tea here on the right hand side, and we aim for
finding a zero and research that these equalities that we get
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for all the measurements are almost fulfilled if they have
more than two measurements, then it is very likely that we
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don't find any image of position in velocity with which all
the equations are met completely perfectly. Um, but um, we
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will. We aim for a solution that is good for, uh, for all
positive measurements. And this can be done by eliminating
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this, actually the difference. So this is the difference
of the left inside and the right inside of this equation,
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taking the square of it, and then summing up this era in this
equation. So to say that we get over all the measurements,
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the measurements that we have. So now we have a kind of arrow
measure that describes how good a certain choice of egg
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zero and V, um fits to their measurements. So the larger the
value is here of the sum then? well, the larger the sum is,
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the worries is the fit. That means and we do not fit
to the measurements. If the sum here is very small. If it is
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close to zero, then we know that the choice of and we
fits very good well to the measurements. That means that we
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found good choices of Ex, Zero and V. So how do we determine
the best, the best solution. While we take this term, we
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calculate the partial derivatives with suspect of the unknown
we zero those partial relatives and
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resolve the system of equation that we get. And this is shown
here. So once you do all the calculations, you will see
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that we end up with such a system of linear equations. And
as soon as the matrix on the left hand side. It has full
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rank, which typically is the case. As soon as we have at
least two measurements, then we can resolve the system of
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equations, and % uh, get the best values for egg Zero and
V. Well, if we like, we can also derive in a second step, a
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that means measures of uncertainty that tell us how
sure we can be about the values of that we have found
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now so we can even mhm extend our approach.
So after now, we were assuming the vehicle is moving this
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constant velocity. And however, in some situations, it is
obviously not true. So we assume the car is waiting in front
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of traffic lights, and now the traffic lights, which is to
green, and the car accelerates. Of course, in this, such a
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situation, definitely this assumption that the vehicle is
moving. Wisconsin velocity is not met. So in this case, we
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need another motion model that we apply. None to estimate the
movement of the vehicle. And for instance, we could assume
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in this case that the vehicle is moving with constant
acceleration. And in such a case, we will get a motion model like
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the one here on the slide that X of tier can be described
by an initial position ex zero, an velocity v zero, and an
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constant elect acceleration, a that means all the measurements
that we make, and we assume that all those are determined
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from such a linear for from such a relationship that we see
here. Um, yeah and um, that the movement of the vigor can be
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explained by these three parameters, external B, Zero and A.
Now we can proceed as in the case before we make a certain
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number of measurements where we measure the position of
the car at each point in time. Then, ah, we ah create an
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optimization problem, or we say," Okay, what we want is for
each measurement. We want that um the left hand side and the
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right inside of this equation fits as best as possible that
should hold for all measurements in parallel. That means
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what we do is recreate this difference of the left hand side
on the right hand side of this motion equation. Here we sum
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up the squares of those differences over all measurements.
And then we stayed or gave you one to find those parameters
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which minimize this term again, we calculate the
partial derivatives of this term, zero those and resolved
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the system of equations, respect to the unknown variables
A, B, Zero and . This also yields a system of linear
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equations. And if you like, just derive it at home. What are
the advantages of such an approach of regression approaches
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in general. Well so first of all, these approaches
are simple. Yeah, the calculations that you need to do are
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rather simple. It is easy to use. Ah, efficient calculations.
The only thing, the most expensive thing that you have to
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do computationally is you have to ah solve a system of
linear equations. But this a small system of linear
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equations but this still is not really time consuming @unoise@
so that works easily for all straight movements in one
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dimensional case. So if you only consider the position of a
vehicle along a road, for instance. It also works in a two
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dimensional case. If you want to estimate the two dimensional
movement of a vehicle saw a movement on the plane. So
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assume a parking lot or something large plan our area where
a vehicle is moving, and you want to estimate its velocity.
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And this can also be done with linear regression, and also
in three D, it would work. So if you have a spaceship or
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whatever, and you want to describe the movement of the
spaceship. It would also work. So my suggestion is give it a
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trial. So if you are faced with such a problem, try such a
regression approach, whether it works or not. And before you
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go to more complicated approaches to um to solve the task
@unoise@ shortcomings of these methods is, of course, as soon
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as we are not faced anymore with straight movements. And we
want to create motion models of not straight movements. So
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for instance, for a car that can turn so it can contrive
along a circle on a circle, our trajectory, then those
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approaches become nonlinear so that we get nonlinear
optimization problems. And so we have to use more complicated
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numerical solvers to solve this. It is still possible, but it
is computationally more demanding. So one example where we
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were using these kind of regression approaches is robot soccer.
That is a project that I did many years ago. Meanwhile,
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so more than ten years ago So we were operating
autonomous robots, these ones here with the pink or the violet,
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the violet number number, number signs @unoise@ they. Were
fully autonomous they. Had a camera on board with which they
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could detect their environment and recognize objects in their
environment. You see, the environment was color coded in a
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certain way. So there was an orange ball so that it is easy
to detect it. All the obstacles are black, and the ground
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was green, and there were white lines so. These things
are a color coded environment so that the perception is not
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too complicated, which was quite challenging, still challenging
at these days when the computers were not as powerful as
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to day. And the computer vision techniques as well were very
limited in what they were able to do, however. So the task
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was to let those robots play soccer autonomously. So
operated and no one at the sideline that was having a
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joystick to operate them, but they had to decide on their own
what to do and how to behave. And of course, for playing
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soccer, it is very important to know where the ball is. And
it is also very important to know in which way the ball is
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moving. Yeah, yeah, obviously. Now, if you want, for instance,
to implement a goalkeeper um and the goalkeeper only
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knows where the ball is at the point in time when the last
measurement was made, then the goalkeeper will not be very
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successful Yeah, because it doesn't know in which
direction to drive and where to expect the ball crossing
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the the goal line @unoise@ okay. So, we need the motion of
the ball. And also we need our ego motion. We, of course,
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need to know how fast we are moving in which direction we are
moving. So these were the two things that we were trying
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to estimate with regression. And of course, we would like
also to estimate the movement of other robots. But at that
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point in time, the recognition of the other robots was not
that stable and not that reliable that we could do it. Okay,
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so we are estimating the motion of the ball. Yeah, we did
that with the regression approach. Just a brief sketch of how
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we did it did it. So the first starting point of our
development was at the point in time where the other soccer
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teams could um not um do chip kicks so they could only kick
the ball flat on in a flat way, although the ball was not
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leaving the ground. So we could assume that the motion of the
ball is a two dimensional motion on the plane. So, and we
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also were assuming that at least for short periods of time.
We can assume that the ball booths with constant velocity.
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So if the ball is dribbled by another robot, or eh, if the
ball is was kicked, and now is moving afterwards. We are
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assuming that at least for the a couple of now same for
several hundreds of milliseconds. We can approximate the
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movement of the ball sufficiently well as a movement with
constant velocity. Of course, it was not perfectly true, but
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um, it was still okay. Um, for that purpose, what we did was
we um were estimating the velocity of the ball based on the
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last observations we took between three and fifteen observations.
The last three to fifteen observations, and we made
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@unoise@ one ah we we had a frame rate of the cameras
of thirty three milliseconds. That means % um. If we have
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just three observations, um the time between the first and
the last uh observation that we were using was roughly
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sixty-six to sixty-seven Millise seconds. So this was a
minimal time duration. And with fifteen observations we
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had a total tie observation time of roughly half a second.
Yeah. So based on that, we were applying this linear
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regression approach that you would just sore assuming constant
velocity. We did that in a two dimensional manner. So we
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were estimating the velocity in ex direction and independently
in wide direction, um so that we get a velocity factor
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that describes in which way the ball is moving, of course.
Um, this assumption of constant velocity is sometimes
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violated, especially if the ball is kicked by a robot in
this moment when it is kicked. Of course, it is immediately
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changing its velocity in a very hard way. Or for instance,
if the ball collides with a robot or an obstacle, then it is
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moving. Direction is also also changes a lot. So at these
points in time. And of course, at that point in time, the
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assumption of concert velocity is violated. To deal with that,
we had this adaptive time for this adaptive observation
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length. That means we always observe whether a new measurement
fits to the motion model that we have so far. And if we
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observe that one or two times the deviation between the
observed position, and they observe the expected position using
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the estimated motion model and navigates very much. Then
we said," Okay, here it seems that something happened that
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violates our basic assumption. So in these cases, we shrink
again the number of observations to our minimum number of
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three, and otherwise we increase the observation time up to
the maximum observation time. Yeah, the more observations we
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have in general. If the assumption is not violated. Of course,
the better the estimate is the estimated movement is.
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However, if we observe that our basic assumption is violated.
We shrink the observation into all in order to um cope
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with these um changes @unoise@ okay. Then later on some teams,
some soccer teams started to kick the ball, not only in a
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flat way on the ground, on the soccer ground, but they started
to kick the ball in such a way. Now, doing some chip gigs
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in a um that caused a lot of problems for us, because we
didn't have a stereo vision system on board of our robots at
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that time. So we were not able to determine the three
dimensional position we would. However, we were able to detect
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that the ball is not on the ground by doing some plausibility
analysis @unoise@ um later on, we installed a stereo
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vision system, a binocular camera vision on one of the robots,
especially on the goalkeeper, and then the goalkeeper was
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able to determine the three dimensional position of the ball.
And then we were able to model also the vertical movement
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of the for % uh and in for and @unoise@ for that purpose we
were assuming that the movement in vertical direction is an
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accelerated movement. And we already knew the acceleration,
namely that is just ah % um ah Grammy team that accelerates
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the ball. That is the we know it. The only thing is that we
also had to consider that once the ball meets the ground, it
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is bouncing back and not continuing this motion. So again,
there is kind of model checking be done, whether or not the
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ball um bounces at the ground, and if it balances, we assuming
that it again moves upwards afterwards. So in total, what
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we did is a regression approach, plus some model checking
in order to cover situations in which the assumptions of
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constant or accelerated vertical motion are violated. So some
example here so. This is a test case Yeah that
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were recorded in our lap, and that is now played in a cycle
again and again. So we see the soccer ground, obviously.
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Then you see this Science Triangle um, where is it? mm, okay
% um. The Science Triangle % um @unoise@ here that. Is here
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that is actually our robot. Um. Here you see two circles, one
dark red circle, dark red, dashed circle. That is actually
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the position of the ball that was sensed with the camera at
that point in time. And then this a solid um red circle.
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That is um. The estimated position of the ball after having
done the regression. And then we see this line here that is
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indicating the movement direction and the velocity that was
estimated. So the longer the line, the larger the velocity
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that we estimated. So the motion actually starts here at this
point. The first two cycles that you observe, there will
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is only this observe for, because we do not have enough
measurements yet to estimate the velocity. And then this solid
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circle occurs and this indicates that we were able to
estimate the velocity of the ball. And if we look at the
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direction of this line, that it against the motion of the
ball. We see that it roughly fits to the real motion of the
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ball. Now, here, maybe this direction to the left, or the
the estimated motion is a little bit too much. The ah
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points to the right, but % um. Okay, the procession of this
Ah camera system was not that good. And so there were some
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mistakes there. Actually, the ball was rolling over the
field at that point in time. So that also means that the
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velocity was decreasing slowly over time. % um @unoise@
okay. So, that was the ball detection and the ball velocity
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estimation. Yeah, we also used that, as I said, to
estimate the motion of the robot. And for the Robert Motion
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that things were a little bit different. So we were not
assuming the straight movement of the robot. But move meant
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Wisconsin that means a movement on a circular tractory.
And the measurements that we had again were the position
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of the Robert, and also the orientation of the robot that
the Robert had. And based on that, we were deriving a
00:22:08.248 --> 00:22:17.270
technique M based on regression, with which we could efficiently
estimate also the movement of the robot @unoise@ okay.
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So, far this exclusion to robot soccer @unoise@ and so far
um the discussion of regression techniques for estimating
00:22:23.872 --> 00:22:28.382
@unoise@ @unoise@ @unoise@ @unoise@ @unoise@ @unoise@ @unoise@
@unoise@ @unoise@ @unoise@ @unoise@ @unoise@ @unoise@
00:22:28.382 --> 00:22:34.280
@unoise@ @unoise@ @unoise@ @unoise@ @unoise@
@unoise@ @unoise@ @unoise@ @unoise@ @unoise@ and now we want to
00:22:34.280 --> 00:22:40.873
go to a different technique, which is the technique of Beijing
filters including common filters and things like
00:22:40.873 --> 00:22:50.196
that. They have very much based on probability theory. And
since I guess that most of you do not have a lecture on
00:22:50.196 --> 00:23:00.048
probability theory so far, or maybe it is so long ago that
you don't remember the things I want to start with a very
00:23:00.048 --> 00:23:08.669
brief repetition of probability theory, so that we have the
basics again and can start based on these basic ideas of
00:23:08.669 --> 00:23:19.351
probability field. Okay, so it is actually I'M A said,
not mathematically completely correct this presentation,
00:23:19.351 --> 00:23:33.857
but it is a compromise in order to have it short and intuitive.
Okay, but let us start. So first of all, when we talk
00:23:33.857 --> 00:23:43.079
about probability theory, we have to talk about random events.
So things that might occur or might not occur randomly.
00:23:43.089 --> 00:23:52.917
And if we have such a random event, say, aim and this aid.
It might any event that you can imagine, and we might assume
00:23:52.917 --> 00:24:00.948
a probability for that. And the probability rough, intuitively
speaking, is something like a frequency with which we
00:24:00.948 --> 00:24:11.113
expect that this event occurs. So um, yeah. And what A might
be is up to the problem. A A might be in the context of
00:24:11.113 --> 00:24:19.380
autumn, maybe driving? how likely is it that a vehicle ahead
of an Eagle vehicle turns right at an intersection could be
00:24:19.380 --> 00:24:27.950
such an event, it could be the event it is raining tomorrow.
Whether or not it is raining tomorrow. It is up to random.
00:24:27.950 --> 00:24:37.278
We don't know in advance, but however, we can make a kind of
guess how likely it is that it is raining tomorrow. Okay,
00:24:37.278 --> 00:24:46.704
so that is the probability, then if we have several events,
those events might occur at the same time or not. Let us
00:24:46.704 --> 00:24:55.959
assume we have two events, A and B, and now we can define
something like a joint probability. Joint probability means,
00:24:55.959 --> 00:25:07.439
however, relative frequencies are to say with which we expect
that those two events occur at the same time @unoise@ so.
00:25:07.450 --> 00:25:15.903
For instance in, automate driving if we are facing that again.
And we ask whether a vehicle is turning right at the next
00:25:15.903 --> 00:25:24.362
intersection, that would be an event A and another event
would be that the vehicle ahead of us is activating the
00:25:24.362 --> 00:25:33.657
indicator lights, though, whether it is blinking, right or
not. These two events, these are two events, and we can
00:25:33.657 --> 00:25:46.799
somehow mhm make a guess how likely it is, how often we
expect that a vehicle ahead of us Ah turns right in an
00:25:46.799 --> 00:25:54.576
intersection and has its indicator lights activated. Yeah,
we see that already in this example, that there is a
00:25:54.576 --> 00:26:03.846
relationship between these two events, A and B. But of course,
from aid does not fully follows completely follows that B
00:26:03.846 --> 00:26:12.049
also holds, and vice versa. So they sometimes vehicles turn
right without activating the indicator lights. Sometimes
00:26:12.049 --> 00:26:22.819
they do. There is a stochastic relationship. Now there is some
probability that if a be also occurs, but it is not
00:26:22.819 --> 00:26:31.665
sure @unoise@ then the second the, third kind of probabilities
that we need to introduce are so called conditional
00:26:31.665 --> 00:26:42.373
probabilities, also referring to events A, and they say, more
or less, they model how often or is them how likely it is
00:26:42.373 --> 00:26:53.399
that event a occurs if, in those cases, and only in those
cases in which the event be occurred. And so if we consider
00:26:53.399 --> 00:27:01.645
again this example with the car that is approaching an
intersection might turn right, might have its indicator lights
00:27:01.645 --> 00:27:11.742
activated. Um A, this probability would mean, how often does
it happen that a car turns right at the intersection if it
00:27:11.742 --> 00:27:20.958
is having its indicator lights being activated. Mhm @unoise@
this should not be confused with the joint probability of
00:27:20.958 --> 00:27:31.900
A, N, B, the joint probability of A, N, B means for all cars
that are approaching an intersection. How? how probable?
00:27:31.910 --> 00:27:42.306
how likely is it? how often do we observe that the car turns
right and has its indicator lives being activated, while
00:27:42.306 --> 00:27:50.419
the conditioner probability says only in those cases. For
those cars which have their indicator lights being activated?
00:27:50.430 --> 00:27:59.666
how probable? how often do we observe that such a car is
turning right. Yeah, so the two probabilities are related to
00:27:59.666 --> 00:28:08.119
each other. The joint probability and the traditional
probability, but they are not the same, and they cannot used um.
00:28:08.140 --> 00:28:18.719
They cannot be exchanged that easily. Okay, so these are
the three type of probabilities that we have to consider.
00:28:18.730 --> 00:28:28.619
So um, besides random events, there is something else
improbability theory that is called random variables. So often,
00:28:28.619 --> 00:28:38.868
improbability theory, we do not deal just with kind of crisp
events here, something like turns, right or not. Near can
00:28:38.868 --> 00:28:50.125
say either A or not, whether the event occurs or the event
does not occur but we are often faced with numbers with
00:28:50.125 --> 00:28:58.840
numbers that are somehow randomly chosen. Yeah, so. And these
numbers, which are somehow randomly chosen, are called
00:28:58.840 --> 00:29:08.469
random, very elbows. So there are two different cases that
we consider to be other, to distinguish the first one are
00:29:08.469 --> 00:29:17.396
discreet, um, Randomberry others, or random barriers, which
only ah might take on Integer numbers. So for instance, only
00:29:17.396 --> 00:29:27.751
positive Integer numbers or but, but just say indeches. And
the continuous variables are those which can take any real
00:29:27.751 --> 00:29:37.734
values that just real values. So for instance, if we say the
velocity of a car that we observe. That is, um, not just an
00:29:37.734 --> 00:29:46.499
Integer. Uh, the car cannot only drive one kilometer per
hour, and two and three and four and five, but it can also
00:29:46.499 --> 00:29:55.032
drive four point three or four point eight hours and
therefore this is a continuous render, very album. While, for
00:29:55.032 --> 00:30:04.461
instance, things that we can count. Yeah, um, or that can
only take on some Integer numbers, like, for instance, the
00:30:04.461 --> 00:30:11.713
question, how many bicycles are parked in front of this
lecture hall. That is an Integer. These are discrete random
00:30:11.713 --> 00:30:19.282
warriors. And the treatment of these discrete and continuous
renum bar apples is a little bit different, and therefore
00:30:19.282 --> 00:30:27.561
we have to make this distinction. So now these random barriers
are somehow related to random events. So what are typical
00:30:27.561 --> 00:30:35.849
random events which can be defined for
@unoise@ random very others. So for discreet , a typical
00:30:35.849 --> 00:30:44.396
event that that we might be interested in is something like
this one here. The event that this discrete random variable,
00:30:44.396 --> 00:30:55.210
why it takes on a certain now that why is equal to so
we write it here with this. Ah, with these Ah box brackets.
00:30:55.220 --> 00:31:04.769
And that means inside of these box brackets that are, to
say, a condition that describes the event or another
00:31:04.769 --> 00:31:14.715
event for air for discrete variables could be that the
why is the inside of a certain interval of numbers. Now,
00:31:14.715 --> 00:31:24.305
that is also a typical event. And another one would be that
we have a subset of the inteches and ask how and define that
00:31:24.305 --> 00:31:33.100
as an event that this variable why is takes on a value within
the subset of the indigenous for each of those events.
00:31:33.099 --> 00:31:40.579
Again, we can define probabilities, and we can say, what is
the probability of event of the event that why is equal to
00:31:40.579 --> 00:31:51.598
four or the other. That why is in the set of two, three, seven
and eleven for continuous variables, things are a little
00:31:51.598 --> 00:32:00.938
bit different. So for theoretical reasons for reasons
it is impossible to define an event that a continuous
00:32:00.938 --> 00:32:11.023
random variable takes on a certain value that A is exactly
equal to two point three or something like that. That is
00:32:11.023 --> 00:32:20.824
nothing that that is no random event. We can all for for
continues very hours. We can only define random events based on
00:32:20.824 --> 00:32:30.374
intervals. We can ask, we can define a random event,
for instance, whether X variable acts is within a certain
00:32:30.374 --> 00:32:38.923
interval. For instance, between seven point one and eight point
three. And we can also, of course, take union of several
00:32:38.923 --> 00:32:46.784
intervals and ask," How likely is it that the random variable
x is between four point five and eight point two, or
00:32:46.784 --> 00:32:54.952
between ten points for three and eleven point two. We can
have also unbounded intervals like the interval from
00:32:54.952 --> 00:33:04.704
fifteen on to infinity. That is also possible. But the important
thing is that for continuous, random variables, we can
00:33:04.704 --> 00:33:15.736
only define random events based on intervals and not on
individual numbers. Uh that comes from the theoretical
00:33:15.736 --> 00:33:28.206
aces of probability, the theory that we cannot. We can only
define a random events for continuous variables based on
00:33:28.206 --> 00:33:40.704
intervals. Okay, so now @unoise@ now once we defined
these random, these probabilities and these random events, we
00:33:40.704 --> 00:33:52.620
might introduce some rules to calculate these probabilities.
And the first rule is called a marginalization rule. It
00:33:52.620 --> 00:34:02.640
relates a simple probability for a single event with a
joint probability of two events, or one event and a
00:34:02.640 --> 00:34:13.559
Randomborough, and it is defined like that for a discrete
random barriers @unoise@ so we assume we have an event A. It
00:34:13.559 --> 00:34:23.062
doesn't matter what it is just an arbitrary event, a @unoise@
and we have a discrete random variable y % Ah. So now, ah,
00:34:23.062 --> 00:34:34.346
we might ask, in which way is this probability of A related
to the joint probability of A. And the event that why the
00:34:34.346 --> 00:34:46.912
y takes on a certain value and the solution or the this
relationship is shown here @unoise@ so. Once we know all
00:34:46.912 --> 00:34:58.069
these joined probabilities of the event A and the events that
the renewable capital. Why takes on a certain value, then
00:34:58.069 --> 00:35:08.386
we can derive the probability of A without considering why,
from these giant probabilities in this way, as it is shown
00:35:08.386 --> 00:35:17.185
on the slide. We consider all these joint probabilities of
Event A and all possible values of ran a variable. Why? and
00:35:17.185 --> 00:35:28.644
sum up over all these possibilities. And by doing that,
we get rid of this second random variable wire so we can
00:35:28.644 --> 00:35:39.064
eliminate the influence of a second random barrier, or a
certain random variable by summing up joint probabilities over
00:35:39.064 --> 00:35:50.189
all possible values at this random variable might take. And
by doing that now, we can get from joint probabilities to
00:35:50.189 --> 00:36:01.007
probabilities of a single event, or vice versa. Once we have
a probability for a single event, and we want to introduce
00:36:01.007 --> 00:36:11.994
a second render random variable y we can relate this um
probability of A to the joint probabilities of A and the random
00:36:11.994 --> 00:36:23.354
variable one. Okay, this rule is known as the marginalization
rule. And in this case, this probability on the left hand
00:36:23.354 --> 00:36:33.680
side, where the y has disappeared is called the
marginal distribution of this joint distribution? Ca yeah,
00:36:33.680 --> 00:36:42.619
marginal. The word " marginal " comes from the fact that if
we write down all these joint probabilities in a large table
00:36:42.619 --> 00:36:50.859
then we calculate the sums in each row, then we get
those marginal probabilities, how we write them at the margin
00:36:50.859 --> 00:37:00.460
of a table, and therefore they are called . Then the
second calculation rule is this one. Here it relates the
00:37:00.460 --> 00:37:09.657
conditional probability to the marginal and the joint probability.
And it says that the joint probability of two events,
00:37:09.657 --> 00:37:19.787
A and B, is equal to the conditional probability of A given
B times the marginal probability of B. And since the order
00:37:19.787 --> 00:37:28.158
of these events in the joint probability A and B doesn't
matter. It can be exchanged without any change. This is also
00:37:28.158 --> 00:37:37.544
equal to the conditional probability of P, given a times the
march in a probability of A. This is an important rule to
00:37:37.544 --> 00:37:45.341
get this relationship between joint probabilities and
conditional probabilities and describes in which they which way
00:37:45.341 --> 00:37:56.694
they are related to each other @unoise@ and. We can also
extend this rule if we are faced with three or more Ah, random
00:37:56.694 --> 00:38:04.361
events @unoise@ for. Instance if, we have three random
events then we can group those random events into two
00:38:04.361 --> 00:38:12.782
groups, and then apply this rule for those groups of of random
burial. So for instance, we could save the group. And the
00:38:12.782 --> 00:38:21.010
three random events A, B and C into one small group with only
event A and one larger group with events B and C, and then
00:38:21.010 --> 00:38:29.095
apply this rule. And then this means this is equal to
the probability of A given B and C. At times, the joint
00:38:29.095 --> 00:38:37.845
probability of @unoise@ we. Can also group it in a different
way. We can also say, okay, we group A, they events A, N, B
00:38:37.845 --> 00:38:46.755
in one group, and they men see in another group @unoise@ and
then we get the probability of A and be given sea times,
00:38:46.755 --> 00:38:54.179
the probability of C @unoise@ and like that @unoise@ we can
um yeah, calculate with those ah resources @unoise@ and.
00:38:54.190 --> 00:39:03.773
Finally the, third important rule that is actually derived
from from this rule for conditional probabilities is a
00:39:03.773 --> 00:39:15.879
base theory @unoise@ % um that is given here, and it is
actually that you can derive it directly from this row here by
00:39:15.879 --> 00:39:25.932
dividing this row by probability of B. Then we get this
theorem, it is as that the conditional probability of a given B
00:39:25.932 --> 00:39:34.887
is equal to the probability of B, given a times the
marginal probability of a over that march, improbability of be
00:39:34.887 --> 00:39:44.379
for which case is that interesting? so let us go back to our
example. We observe a car at an intersection. One event A
00:39:44.379 --> 00:39:55.145
is the car is moving right, turning right at the intersection,
the probability the event B is it is blinking, right? so
00:39:55.145 --> 00:40:09.992
its indicator lights are active @unoise@ so. Then this
probability means how likely is it that a car that is blinking,
00:40:09.992 --> 00:40:21.485
right? that has its indicator lights active is really turning
right. And this probability here means how likely is it
00:40:21.485 --> 00:40:32.074
that the carp links that wants to turn right. Yeah, so these
are two different probabilities, yeah, but they are related
00:40:32.074 --> 00:40:42.104
to each other, but they are not the same. They are related
by the relationship that is given by bay theory, and
00:40:42.104 --> 00:40:50.777
sometimes it is easy to determine one of those conditional
probabilities, but very difficult to determine the other kind
00:40:50.777 --> 00:41:01.440
of probability. So if maybe we might be able to stay that,
um? if a car wants to turn right. It is activating its
00:41:01.440 --> 00:41:11.144
indicator lights with a large probability, save if ninety
percent or so that might be easy. It is more difficult to
00:41:11.144 --> 00:41:22.557
conclude about if the car is blinking right how likely
is it really turning right? that is what we want to know
00:41:22.557 --> 00:41:32.538
when we are following this car and have to interact with it
and want to calculate its its future behave okay, so
00:41:32.538 --> 00:41:42.272
that Spacey room again, we can extend it to more than two
variables. And then again, we can group those variables and um
00:41:42.272 --> 00:41:51.711
deal with groups of or we can keep very elbows here in
this condition part without changing them. So in this case,
00:41:51.711 --> 00:42:00.486
for instance, we could keep see always in the condition part
of all the probabilities here, and only exchange the role
00:42:00.486 --> 00:42:10.514
of was also possible if you want to prove that you
just have to apply this definition here, or in the case of
00:42:10.514 --> 00:42:18.043
three variables, these definitions here @unoise@ and. Then
you can prove that this is also true @unoise@ okay. These,
00:42:18.043 --> 00:42:27.119
are the three major or the three really important rules for
dealing, calculating with probabilities. Once you know these
00:42:27.119 --> 00:42:37.429
rules, then you can actually do the whole probability theory
so. Now the next concept is the concept of sarcastic
00:42:37.429 --> 00:42:46.465
independence. What is that? well, if we observe these two
events, car is blinking, and car is turning on at an
00:42:46.465 --> 00:42:54.481
intersection. We easily see that there is some relationship
between these events. If we observe a car blinking, then it
00:42:54.481 --> 00:43:01.561
is actually very probable that it is also turning at the
intersection, though, there is a strong connection between
00:43:01.561 --> 00:43:11.956
these two events. However, if we consider two different events,
the car ahead of us is turning right at the intersection
00:43:11.956 --> 00:43:21.490
that event A and event B would be requested. The event that
it is raining tomorrow. We hardly see any relationship
00:43:21.490 --> 00:43:29.291
between these events. Yeah, if we know that it is raining
tomorrow. This doesn't help us at all to predict whether the
00:43:29.291 --> 00:43:38.349
car ahead of us will turn right or not at the intersection,
so they are not couple. They are independent of each other.
00:43:38.360 --> 00:43:46.964
And this idea of being independent, of not having any influence
on each other is called starcastic independence. And it
00:43:46.964 --> 00:43:57.583
is defined in such a way that two events A and B are um said
to be independent if the conditional probability of
00:43:57.583 --> 00:44:09.146
be given a is equal to the marginal probability of B. That
means, if we only consider cases in which a the event a
00:44:09.146 --> 00:44:22.532
occurs and we ask what happens with Event B? is it met
or not? then this knowledge of event A doesn't help us at
00:44:22.532 --> 00:44:34.969
all. If we ignore this knowledge, we cannot say more or
less than if we know whether A is too honored. So this is
00:44:34.969 --> 00:44:43.025
stochastic independence. And of course, many things are
assumed to be sarcastically independent. Now, what happens on
00:44:43.025 --> 00:44:53.312
the road here in Europe, of course, seems to be at least we
can assume that it is independent of what goes on on the
00:44:53.312 --> 00:45:04.428
roads, somewhere else on Earth. Now, for instance, or we might
also assume that @unoise@ then a decision of one driver
00:45:04.428 --> 00:45:16.855
and a decision of another driver that is not directly in the
vicinity of the first driver that those decisions and those
00:45:16.855 --> 00:45:26.787
behaviors are also independent of each other. So independence
is really a nice concept, because it simplifies all
00:45:26.787 --> 00:45:34.951
calculations. And that is the important thing. So actually,
the definition is, as I said, and joined the conditional
00:45:34.951 --> 00:45:43.493
probability of B given A is equal to be the probability
of B, then those events are said to be sarcastically
00:45:43.493 --> 00:45:51.589
independent. And this condition is equivalent to those two
conditions that are given here. Um you can use the rules for
00:45:51.589 --> 00:46:01.237
calculating those probabilities, and then you can easily show
that this is actually the same. So now let us go on
00:46:01.237 --> 00:46:11.021
for discrete, very animals. We saw that we can identify events,
probability random events where we can say, okay,
00:46:11.021 --> 00:46:21.234
we can be interested in the probability that this random
very of this discrete, when member ever takes on a certain
00:46:21.234 --> 00:46:30.181
Integer number and. If you like we can create a large
table with all the indigenous which this variable can take
00:46:30.181 --> 00:46:40.072
on and ask for each of these of these numbers of these Integer
numbers. How probable is it that this random variable
00:46:40.072 --> 00:46:50.599
takes on this value so we can represent the probabilities for
all individual explicitly explicitly, and by doing
00:46:50.599 --> 00:47:00.300
that, describe the whole probability distribution for
continuous random variables. As I said, this is not that easy.
00:47:00.309 --> 00:47:10.043
Because while all these individual numbers, the real numbers,
they are uncountable. So we cannot make such a table. It
00:47:10.043 --> 00:47:18.866
is impossible. And we have seen, or I've said that we can
only define probabilities for intervals. For that, the
00:47:18.866 --> 00:47:27.404
variable that the continuous variable is located inside of a
certain interval between two numbers. Now, the question is,
00:47:27.404 --> 00:47:35.467
how can we represent these probabilities efficiently? so there
are, of course, an uncountable amount of of intervals of
00:47:35.467 --> 00:47:44.738
the real axis. So we cannot make a table. And for each
interval, write down its probability. So how can we represent
00:47:44.738 --> 00:47:52.448
that efficiently and the solution is to introduce some, a
technique that is called probability density functions, or for
00:47:52.448 --> 00:48:02.800
short, P, D, F functions the probability density function.
That is a function um that is non negative for all value so
00:48:02.800 --> 00:48:13.408
that it can be Sierra. It can take on real positive numbers,
but never negative numbers. And we need that the integral
00:48:13.408 --> 00:48:22.503
for minus infinity to plus infinity of this probability density
function is equal to one that comes from the Exxims of
00:48:22.503 --> 00:48:31.990
probability theory. And with such a probably the density
function, we can represent the probability of intervals
00:48:31.990 --> 00:48:43.291
the intervals like that, the probability that the random
variable is located between two numbers, A and B, and by this
00:48:43.291 --> 00:48:52.500
definition. So if you use a probability density function, it
is said that the probability of acts being located between
00:48:52.500 --> 00:49:04.147
A and B is equal to the integral of this probability function,
ranging from A to B. So we use this representation to
00:49:04.147 --> 00:49:15.162
represent the probability distribution, we say probability
distribution um of such a continuous random variety now
00:49:15.162 --> 00:49:24.122
so. So P X, in this case, is the probability density function
for random variable capital acts evaluated a certain
00:49:24.122 --> 00:49:32.810
position. So the probability density function is not a
probability. So it is not that something that we can directly
00:49:32.810 --> 00:49:40.999
interpret as probability, and it can also be larger than one
different than probabilities, which cannot be larger than
00:49:40.999 --> 00:49:52.308
one. But this probably density function can become larger than
one. It can become very large um @unoise@ however. It, is
00:49:52.308 --> 00:50:00.858
somehow safe somehow proportional to the probability or
approximately proportionate to the probability that the random
00:50:00.858 --> 00:50:12.079
burial takes on a value in the vicinity of the special value
at which we evaluate the Ah probability density function.
00:50:12.090 --> 00:50:20.752
Now, @unoise@ okay. That, is a very important concept
so for those probability density functions, we can do
00:50:20.752 --> 00:50:28.223
actually the same calculations as we did for discrete
probabilities, for probabilities, of discrete by random marvels,
00:50:28.223 --> 00:50:36.573
but everywhere where we had probabilities so far. Now we have
to use probability density functions. And everywhere where
00:50:36.573 --> 00:50:44.686
we had summation, we have to replace it by integration.
That means this marginalization rule that we introduce for
00:50:44.686 --> 00:50:53.934
discreet random can be rewritten um for continuous random
variables in the way that is shown here on the slide. So
00:50:53.934 --> 00:51:05.902
we see that here on the right inside, we have a probability
density function for the joint, for the pair of continuous
00:51:05.902 --> 00:51:17.327
random barriers X, and why this is defined in actually the
same way as Ah, basic probability density functions just for
00:51:17.327 --> 00:51:27.119
a pair of numbers, and we can relate this probability, density,
function, ability, density, function only for variable
00:51:27.119 --> 00:51:37.534
capital acts by integrating out the influence of random
variable. Why? by taking the integral of the joint probability
00:51:37.534 --> 00:51:48.575
density function. And over the interval from my minus infinity
to plus infinity @unoise@ okay @unoise@ the next step
00:51:48.575 --> 00:51:57.023
the conditional probabilities. The same applies as
for discrete random barriers. We just replace the
00:51:57.023 --> 00:52:08.252
probability term by the probability density. Here are the
same, but the same rules actually apply. And for base rule um
00:52:08.252 --> 00:52:19.019
the same apply so as well. We exchange probabilities by
probability density functions, and still the theorem holds. So
00:52:19.019 --> 00:52:26.389
now these probability density functions, of course, all
functions which are not negative, and which for which the
00:52:26.389 --> 00:52:34.393
integral form meant minus infinity to infinity is equal to
one can be used as probability density functions. And of
00:52:34.393 --> 00:52:41.376
course, which one is suitable to represent the real
distribution of a certain random barrier. All depends on random
00:52:41.376 --> 00:52:51.134
variable and there yeah however, some
choices of probability density functions have become very
00:52:51.134 --> 00:53:01.911
successful and very powerful and very useful to represent
things in in natural life. And one of those, and maybe the
00:53:01.911 --> 00:53:10.886
most popular one is a so called or normal distribution
now that is a distribution that is in a certain way, very
00:53:10.886 --> 00:53:18.908
powerful, can be used in very many circumstances and has
some very nice properties. So let us introduce it. This
00:53:18.908 --> 00:53:28.563
density, this Gaussian distribution is the density function
is given here for the one dimensional case of a single
00:53:28.563 --> 00:53:39.693
random barrier? well, it is defined like that. So actually,
it is the exponential function of minus X qua. And
00:53:39.693 --> 00:53:48.059
then we have two parameters, View and Sigma, with which we
can control a little bit the shape of this function for the
00:53:48.059 --> 00:53:56.309
basic choice of mule being equal to zero and Zigma being
include one @unoise@ um. The plot of this function. The
00:53:56.309 --> 00:54:06.015
graph is given here. So @unoise@ symmetric around zero @unoise@
um being all positive @unoise@ um and tending to
00:54:06.015 --> 00:54:16.238
zero for very large and very small values x different trouble
etc . So with the parameter we can shift
00:54:16.238 --> 00:54:25.233
this curve a little bit to the left or the right so
Mu is actually the parameter that yield that defines the
00:54:25.233 --> 00:54:34.696
center of symmetry of this shape. So it means " eh " for the
basic choice. We equal to zero. The center of symmetry is
00:54:34.696 --> 00:54:43.112
at position zero. If we select new equal to, then the whole
curve is shifted by two to the right now, if we choose mute
00:54:43.112 --> 00:54:51.699
to be minus five, then the whole curve is shifted by minus
five to the left. And the roll of Sigma is to control
00:54:51.699 --> 00:55:00.506
actually, the width of this spell shaped curve. The larger
Sigma is, the wider this bell shaped curve becomes, the
00:55:00.506 --> 00:55:09.932
smaller Sigma is, the smaller and more narrow this above
shape curve becomes, and the higher the peak in the center
00:55:09.932 --> 00:55:21.521
becomes now. The more the larger Sigma is, the smaller the
peak is, the large the smaller Sigma is, the larger this
00:55:21.521 --> 00:55:29.095
maximum becomes. We can extend this er Gaussian distribution
also to the case of several random barriers, when we have
00:55:29.095 --> 00:55:35.820
several random barriers, and you want to describe the joint
density function for several renumberry albums. We can
00:55:35.820 --> 00:55:45.159
extend that. Let us assume that we ride those random
variables into the spectrum X here. And then we can define a
00:55:45.159 --> 00:55:53.085
probability density function like this one here. Now we see
that it is actually for the one dimension case, or if we
00:55:53.085 --> 00:56:02.440
assume that X is a one dimensional vector. This becomes the
same as this equation here. And yeah, looks like that again.
00:56:02.449 --> 00:56:10.650
We have this parameter mule, which controls the center of
symmetry, where it is so in the basic case, it is the zero
00:56:10.650 --> 00:56:19.069
vector than the scent of . Symmetry is zero @unoise@
but, we can shift it to any place @unoise@ um that we like.
00:56:19.079 --> 00:56:29.718
And ah, yeah, here this graph shows a plot of this function
in the two dimensional case. So with Ah, the center being at
00:56:29.718 --> 00:56:41.064
zero. And this metric's Sigma being the identity matrix. So
what? there is a role of this metric Sigma. So of course, it
00:56:41.064 --> 00:56:52.822
refers to the Small Sigma here. Somehow it controls how white
this curve is. No, whether it is very peaky or very wide,
00:56:52.822 --> 00:57:03.626
and it controls. And this or shape can can
change in different directions. So this this, this density
00:57:03.626 --> 00:57:13.316
function might be very peaky in some directions, and it might
be very wide in other directions at the same time, so we
00:57:13.316 --> 00:57:23.007
can control a little bit this kind of wide mass of this
function in different direction in this, in this space here in
00:57:23.007 --> 00:57:32.675
the space. And this is given with this so called metric
signal with matrix, Sigma has to be a matrix that is
00:57:32.675 --> 00:57:42.163
symmetric and that is positive, define it. That means all
iron values must be larger than zero. And all those for
00:57:42.163 --> 00:57:54.191
which these properties hold can be used as mattresses
and the entries in the diagonal of this magics, they control
00:57:54.191 --> 00:58:05.646
the whiteness, or to say in the different directions of the
coordinate system and the non diagonal elements somehow turn
00:58:05.646 --> 00:58:16.035
this curve a little bit. Ah, ah, we can say like that @unoise@
okay. So, and of course, caution distributions will be
00:58:16.035 --> 00:58:24.600
that once that we need all the lectures, and with which
we will calculate. So finally, let summarize our basic
00:58:24.600 --> 00:58:34.252
repetition of probability theory with a little bit of notation
stuff so. If we want to be fully correct in our
00:58:34.252 --> 00:58:43.546
notation. And we are faced with a random barrier will say
discreet random variable x. And we want to express the event
00:58:43.546 --> 00:58:52.404
that acts this random variable x takes on a certain value,
which is given here with small eggs, and then we would need
00:58:52.404 --> 00:59:01.740
to write it like that. P capital piece of the probability of
the event that the randombury of a couple acts takes on the
00:59:01.740 --> 00:59:10.247
value. Small axe. That is the full, correct, mathematically
correct notation. And for a probability density functions of
00:59:10.247 --> 00:59:18.676
capital. X is a continuous, random variable than we would
say. This is the probability to density function of the
00:59:18.676 --> 00:59:26.985
variable capital acts and be evaluated at position, which
is given by small acts. This is the full correct notation
00:59:26.985 --> 00:59:35.799
@unoise@ however. If, we use this notation we write formulas
that are very, very long, and therefore many method
00:59:35.799 --> 00:59:45.770
mathematicians prefer to have a short kind of writing and
shorter notation that looks like that. Sorry. It looks like
00:59:45.770 --> 00:59:58.449
that. P of capital X and small Pe of capital X instead of
this. And that notation, this might be confusing sometimes.
00:59:58.460 --> 01:00:07.984
This might sometimes be really confusing if you are confused
by that. Rewrite everything in this basic notation. Yeah,
01:00:07.984 --> 01:00:17.299
yeah. However, you will easily see that the terms that we
get become very large, and therefore I will also use the
01:00:17.299 --> 01:00:26.204
simplified notation if it is not too confusing. But if you
face such a term and you think I don't understand it. I am
01:00:26.204 --> 01:00:34.129
too much confused. First, rewrite it through the long form
to understand it. In another thing is this kind of notation.
01:00:34.139 --> 01:00:43.679
You might write Capital X tilder. And then we get this
Endscript and of view. And this is a temptation to write
01:00:43.679 --> 01:00:51.210
that we assume that this random, very Albert Capital X is
distributed according to a Gaussian distribution. That means
01:00:51.210 --> 01:01:00.588
we say that with this notation, we stayed that the probably
the density function of this running variable. X is a
01:01:00.588 --> 01:01:08.889
probability density function with these parameters, view and
signal, which are given me. Yeah, so this is the notation.
01:01:08.900 --> 01:01:19.280
So this says we assume that the Randombury our X is distributed
according to a Gaugeian distribution. That means that we
01:01:19.280 --> 01:01:31.050
have to use this Gaussian probability density function to
describe the probabilities of this . Okay, so far, the
01:01:31.050 --> 01:01:42.011
repetition of the very brief and repetition of the basics of
probabilities. You now let us use that to develop a model
01:01:42.011 --> 01:01:52.280
which is very useful to do this. Kind of, yeah, tracking of
objects, estimating motion of objects over time, integrating
01:01:52.280 --> 01:02:03.075
measurement in an incremental way. So the main idea that we
need for this is that we want to understand the world, or
01:02:03.075 --> 01:02:14.680
the part of the world in which we want to model as a system,
and the system is assumed to have a state. And we assume
01:02:14.680 --> 01:02:24.555
that this state contains all information that describe in which
the system behaves. Now, the just knowing the state of a
01:02:24.555 --> 01:02:36.833
system should be sufficient to understand or to be able to
describe how the system behaves no @unoise@ so. And this
01:02:36.833 --> 01:02:45.838
means that we can describe the changes of the system of a
time, by means of describing the changes of its state over
01:02:45.838 --> 01:02:55.375
time. So if we know the state of the system. We know everything.
So that is the basic thing idea behind it. So let us go
01:02:55.375 --> 01:03:03.651
back to our son. An example, a car that is moving Wisconsin
velocity. If you want to describe this car that is moving
01:03:03.651 --> 01:03:11.428
Wisconsin velocity. It is fully sufficient to know its
position and its velocity. Now, at a certain point in time. If
01:03:11.428 --> 01:03:18.304
you know its position and velocity. Then these two pieces
of information are completely sufficient to describe the
01:03:18.304 --> 01:03:27.201
behavior of this car to make predictions? where will it be in
ten seconds? if I know where it is at the moment. And if I
01:03:27.201 --> 01:03:34.875
know how fast it is. I can easily predict where it will be
maybe up to some randomness that still might occur. But
01:03:34.875 --> 01:03:44.237
@unoise@ in general I,'m able to do this prediction. I'm also
able to describe where has this vehicle been ten seconds
01:03:44.237 --> 01:03:52.113
ago, something like that. So the behavior, at least the
relevant behavior of this vehicle is fully described by these
01:03:52.113 --> 01:03:59.369
state barriers. If the vehicle is moving Wisconsin velocity.
If the vehicle is not moving Wisconsin velocity. If this
01:03:59.369 --> 01:04:06.380
assumption is not fulfilled, then, of course, knowing just
the position of the velocities definitely not sufficient to
01:04:06.380 --> 01:04:15.863
describe the behavior of the vehicle, then we are not able
to predict where it will be in ten seconds. If we don't know
01:04:15.863 --> 01:04:25.983
in which way it is accelerating. So that means then maybe we
would need in such a case, to add the acceleration of the
01:04:25.983 --> 01:04:34.487
vehicle as an additional barrier to this state factor, to
the state information. And if the car is not moving on a
01:04:34.487 --> 01:04:42.219
straight road, but maybe on a curved road. Maybe we also
need to know the steering angle of the vehicle, or the draw
01:04:42.219 --> 01:04:50.952
rate of the vehicle in order to describe in which way it is
turning now. So it depends on the system that we observe on
01:04:50.952 --> 01:04:59.032
the assumptions that we can make in which way it behaves,
and which in pieces of information must be added to this
01:04:59.032 --> 01:05:07.539
state. Information are the system, but we assume that it is
possible to define it, it be assumed that it is possible to
01:05:07.539 --> 01:05:15.309
select a certain number of pieces of information, a finite
number of information of pieces of information, to add it to
01:05:15.309 --> 01:05:22.820
a state factor, or to assemble off pieces of information.
And then with this knowledge, we can fully describe the
01:05:22.820 --> 01:05:31.180
behavior of the system @unoise@ okay so are based on
that, then we can maybe create a transition model and say,
01:05:31.180 --> 01:05:39.519
okay, based on these pieces of information, we can predict
where will the vehicle be in a certain Ah! number of seconds.
01:05:39.530 --> 01:05:53.544
How will it be if we go back to that later @unoise@ okay.
We, also assume that we can observe the system. And for
01:05:53.544 --> 01:06:03.066
instance, the card that is moving. We observe it with a
sensor. So we also need to describe in a certain obstruct way
01:06:03.066 --> 01:06:12.820
how this measurement works. So in this case, we assume that
we can make a measurement, we can measure something. And
01:06:12.820 --> 01:06:25.521
this measurement depends on the state of the system. And we
assume that there is a function that explains in which which
01:06:25.521 --> 01:06:37.012
measurement we can achieve if the system is in a certain
state. So we assume that the measurement, which is now written
01:06:37.012 --> 01:06:46.770
here throughout the slides with a letter said that it is created
by mapping the state of the system at the present point
01:06:46.770 --> 01:06:56.699
in time that is now denoted with S for state on to the
measurement. And maybe we have some measurement noise. That is
01:06:56.699 --> 01:07:04.614
the second part here that is written. That means there is
just some random influence that is affecting the measurement
01:07:04.614 --> 01:07:15.166
as we assume this. First of all, there is a clear, well
defined relationship that explains how the measurement is
01:07:15.166 --> 01:07:25.743
related to the present state of the object of the system,
and some additional noise, some additional random influence
01:07:25.743 --> 01:07:37.433
that occurs that is somehow disturbing the measurement so that
we don't get the real measurement value said, but that we
01:07:37.433 --> 01:07:48.909
get % um % uh, an randomly disturbed measurement. So
for a car that we observe with the camera, we might observe,
01:07:48.909 --> 01:07:58.766
for instance, the position of the of the car. And then we
say," Okay, said of tea, the measurement that we get is equal
01:07:58.766 --> 01:08:07.107
to eggs of tea, are one of the state variables, plus some
measurement noise, or which we could in this case, even
01:08:07.107 --> 01:08:15.068
rewrite into this matrix form that it is something like a
row vector one, zero times this column vector that contains
01:08:15.068 --> 01:08:24.523
all the state, very others okay important for us is
what we can observe, of course, is set of tea. Mhm said," Of
01:08:24.523 --> 01:08:33.588
tea is what we can observe. That is what we can measure. That
is what we get. And of course, what we want to determine
01:08:33.588 --> 01:08:42.279
is the state of the system here, which contains the forces
velocity of the car. This is what we for. We aim to estimate
01:08:42.279 --> 01:08:52.389
what we want to get. We don't assume that we can observe
that directly. We don't assume that we can observe all the
01:08:52.389 --> 01:09:00.660
state variables mhm. Maybe that is hidden so maybe
we don't have a center that can sense the velocity of the
01:09:00.660 --> 01:09:08.082
vehicle but of goes based on our measurements. We
want to draw conclusions about the state of the system and
01:09:08.082 --> 01:09:15.663
estimate the current state of the system. So for doing that,
let us introduce some sarcastic model, a so called hidden
01:09:15.663 --> 01:09:23.659
Markov model. What is a hidden mark of model a hidden Markov
model is a time discreet sarcastic state transition system.
01:09:23.670 --> 01:09:31.412
Its observation in its successes, they depend entirely on
its present state and do not depend on previous dates or
01:09:31.412 --> 01:09:37.744
observations. Okay, let us go through the definition step by
step, though. It is a state transitioning system. What does
01:09:37.744 --> 01:09:46.118
it mean? now we assume the system has a certain state at a
certain point in time, and then it makes a transition to
01:09:46.118 --> 01:09:54.280
another state for the next point in time. The definition says
also that it is a time discreet state transition system.
01:09:54.289 --> 01:10:03.816
That means we are not considering time as a continuum, but
we are considering time. Ah, only up to a certain discrete
01:10:03.816 --> 01:10:13.314
sequence of points in time now. So we assume, for instance,
that we consider the state of the system only for Ah points
01:10:13.314 --> 01:10:22.113
in time, which are spaced by, say, one second were interested
in the state of the system. Now, one sick second later,
01:10:22.113 --> 01:10:30.108
two seconds later, three seconds later and so on, but not
we are not considering what happens between second zero and
01:10:30.108 --> 01:10:39.029
second one. We only considered these Integer positions
in time. So that is time discreet, then stochastic means.
01:10:39.039 --> 01:10:47.869
Well, there is some randomness in now. So this day transition
the transition from one state to another. State is not
01:10:47.869 --> 01:10:53.906
fully deterministic. It is not fully explained by a transition
function. But there is also some randomness that is
01:10:53.906 --> 01:11:00.750
affecting this, this state transition. That means, if we
know the state at the present point in time, we cannot
01:11:00.750 --> 01:11:07.724
completely determine what will be the success or state, but
we can only determine that up to a probability distribution,
01:11:07.724 --> 01:11:14.920
because there is some randomness that is affecting this day
transition. Okay, its observation and its successor state
01:11:14.920 --> 01:11:25.110
depend entirely on its present state and do not depend on
previous dates or observations. So what does it say? okay, so
01:11:25.110 --> 01:11:34.329
its successor state depends entirely on its present state
and not on pre on its previous date observations. You know,
01:11:34.329 --> 01:11:45.940
this means once we know the present state, we know everything
that we need to predict what is going on in future to
01:11:45.940 --> 01:11:58.045
explain the behavior of the system. In future, we can forget
about the past. We do not need to know how, in which way we
01:11:58.045 --> 01:12:07.797
entered a certain state, the system ended a certain day. But
knowing the state is sufficient to describe its behavior
01:12:07.797 --> 01:12:20.317
now. So think of % um. You want to drive through a place
where you are driving on the road, you arrive at a certain
01:12:20.317 --> 01:12:30.833
place. Mhm not yet at your final destination, and you
ask yourself," How do I get from this place to my final
01:12:30.833 --> 01:12:41.118
destination, then it is definitely, it doesn't matter at all
how you manage to come to this place. It only depends on
01:12:41.118 --> 01:12:50.653
where you are to plan your future path to your goal
and it doesn't matter whether you went there in a straight
01:12:50.653 --> 01:12:58.945
manner or not, straight manner that is completely irrelevant
for describing how you can arrive at the at your final
01:12:58.945 --> 01:13:08.106
destination. So that is meant with it depends entirely
on its present state and not on the past, and also the
01:13:08.106 --> 01:13:17.684
observation that is related to the state only depends on
the present state and not on the past. That means when we
01:13:17.684 --> 01:13:26.637
measure the position of a car with a camera. We assume that
this measurement is completely independent of the fact from
01:13:26.637 --> 01:13:38.158
where the car came. Yeah, yeah, or whether it was exhilarating
or accelerating in past, that is completely irrelevant
01:13:38.158 --> 01:13:46.833
for this measurement. Yeah, okay. So that is an assumption
that we have to make. And if these assumptions hold, then we
01:13:46.833 --> 01:13:56.935
are faced with something that is called a hidden Markov model.
In probability theory, this definition is summarized in
01:13:56.935 --> 01:14:05.991
these two equations here there. These two equations actually
describe this independence assumption. They state that
01:14:05.991 --> 01:14:14.048
while the probability of a certain successful state, knowing
the present state, knowing the present observation, knowing
01:14:14.048 --> 01:14:24.948
the previous day, the previous observation and so on, up to
the very first state can be simply fight, uh, and is equal
01:14:24.948 --> 01:14:34.155
to just the probability of the success estate. Given the
present state. That means, if we want to predict what is going
01:14:34.155 --> 01:14:42.458
on in the future. If we want to say," okay, what in which way
will the will the system behave in future? only with what
01:14:42.458 --> 01:14:51.280
we need to know is the present state. And we can forget all
the past states. We can forget all the past observations.
01:14:51.289 --> 01:15:00.209
They don't have any influence on the future behavior @unoise@
as well for the observation, the probability to make a
01:15:00.209 --> 01:15:08.683
certain observation at a certain Ah, given knowing the
present state of the system, it is completely sufficient as
01:15:08.683 --> 01:15:17.150
annoying. The present state is completely sufficient. If we
know previous observations a previous day. This doesn't help
01:15:17.150 --> 01:15:26.783
us to explain better the present observation. Um, as long
as we know the present state. So again, we can forget about
01:15:26.783 --> 01:15:36.007
all the past, because the past doesn't have influence on the
future. If we know the present state of the system. So the
01:15:36.007 --> 01:15:43.586
present state contains all the relevant information that
is necessary to describe the behavior of the system and to
01:15:43.586 --> 01:15:52.216
describe the measurement process @unoise@ so. Now if we
assume that we can model a system as a hid Markov model. Then
01:15:52.216 --> 01:16:02.383
what we want to get, or what we usually aim for is that we
want to get an idea of what is the present state of the
01:16:02.383 --> 01:16:09.588
system. So we assume we are observing the system several
points in time. We make several measurements where, from point
01:16:09.588 --> 01:16:17.841
in time, one to point in time t in this case. So having
observed the whole sequence of observations, said one to set
01:16:17.841 --> 01:16:26.476
tea. We want to know, well, what is the present state
of the system. So we want to calculate a probability or
01:16:26.476 --> 01:16:34.210
probability distribution, and of the present state that we
want to know ball. We made some observations of the position
01:16:34.210 --> 01:16:42.429
of the car. What is it present? velocity? what is this present
state? for instance, a variant of this question is given
01:16:42.429 --> 01:16:50.617
him, namely, we made the sequence of observations, and
we are not interested in what is the present state of the
01:16:50.617 --> 01:16:59.451
system. But we want to know what is the future state of the
system. Now for the next point in time, if we observed up to
01:16:59.451 --> 01:17:07.024
now what is what is happening up to now. We want to know, well,
what do we have to expect in future from the system. How
01:17:07.024 --> 01:17:16.222
will it behave? now, what will be the next state @unoise@
okay. So, let us derive that. Okay, for that purpose, I
01:17:16.222 --> 01:17:24.898
present to you these formulas, and we will go through all
these formulas in detail. Let us start. Where do we start
01:17:24.898 --> 01:17:37.593
here? @unoise@ hey now, let us derive them step by step. Okay,
let us start here. So the probability of S, T, given the
01:17:37.593 --> 01:17:46.705
sequence of observations up to that point in time. What
is it? we apply first the Beijing Rule base rule. And we
01:17:46.705 --> 01:17:57.308
exchange the role of S, T and Sad team. Now this is done
here. So just S, T and said," Ti are exchanged."" yes." and
01:17:57.308 --> 01:18:07.633
said one up to Sat. T minus one is preserved in the condition
part with base rules. This means this is equal to this
01:18:07.633 --> 01:18:17.464
equation, to this Turner. Well, okay. And now we simplify
things. We look at the denominator, and we see that the
01:18:17.464 --> 01:18:26.444
denominator only contains observations. The observations from
pointed time one up to the present point in time tea. We
01:18:26.444 --> 01:18:35.140
assume that we know those. We have observed them so they,
they don't change. They don't vary. They are fixed. And that
01:18:35.140 --> 01:18:44.127
means that this denominator is just a constant. It is independent
of S, T. It is just a constant. And that means we can
01:18:44.127 --> 01:18:52.975
say this term here is proportional to its . Furthermore,
we assume that the system is a hit mark of moral. And
01:18:52.975 --> 01:19:02.753
since it is a hit mark of model. We know that if we know
the present state, we know everything that is necessary to
01:19:02.753 --> 01:19:10.880
describe the probability distribution over the observations
and all the past observations do not help us to make a
01:19:10.880 --> 01:19:19.435
better prediction of the observation, or to make it better
to know more about the present salvation that is actually
01:19:19.435 --> 01:19:27.857
this independent assumptions that comes from the hidden mark
of one. That means we can simplify this term and leave away
01:19:27.857 --> 01:19:35.649
those old observations. They are not helpful. We don't need
them, because we assume that we are in a hidden mark of
01:19:35.649 --> 01:19:44.902
modern. So this means this simplifies and becomes this term
yet. Now, the second factor here is preserved. It is just
01:19:44.902 --> 01:19:53.164
the same. So that is the first equation. The second equation
starts here. So that is about this future state that we
01:19:53.164 --> 01:20:02.037
about which we want to to say something. Once we are given
the sequence of observations up to point in time tea. For
01:20:02.037 --> 01:20:10.165
that purpose, we use a marginalization rule that we have
introduced and we add another additional random variable
01:20:10.165 --> 01:20:19.576
to this probability namely, the random variable s
T, which we need is a kind of linking element to later on
01:20:19.576 --> 01:20:28.491
simplify these equations. So we add this additional random
variable as T here. And we know if we added at that point
01:20:28.491 --> 01:20:37.992
due to the marginalization rule. We have to sum up over
all possible values that this random variable as T might
01:20:37.992 --> 01:20:46.914
take. So that is actually the marginalization rule. Now we
use the rule of the definition of conditional probabilities,
01:20:46.914 --> 01:20:55.151
and that so actually, what we have here is a joint
probability being that is conditioned on some other variables,
01:20:55.151 --> 01:21:02.894
some other, very others. So now we use the definition of
conditional distributions to change this joint probability into
01:21:02.894 --> 01:21:11.347
a conditional probability Ah, by pushing. So to say this
Sd variable into the condition part. Now that we get this
01:21:11.347 --> 01:21:18.351
condition probability, and we need the correction factor.
There may be the probability of S of T, this marginal, so to
01:21:18.351 --> 01:21:26.743
say, of S and T. However, since we kept all these ah
of old observations here in the condition part. We also have
01:21:26.743 --> 01:21:36.786
to have it here @unoise@ so. Now we again use the knowledge
that we are facing hidden Markov model. And in a hidden mark
01:21:36.786 --> 01:21:46.840
of model, the state transition from known state S T to his
future state S T plus one is independent as we defined. It is
01:21:46.840 --> 01:22:00.439
independent from all the past observations. That means we
can leave those observations away and simplify this to this
01:22:00.439 --> 01:22:07.765
simpler probability. Yeah, so no magic insight of these
calculations, just applying the basic calculation rules for
01:22:07.765 --> 01:22:16.545
probabilities, and using the fact that we are facing a hidden
mark of model that we are assuming that we are facing a
01:22:16.545 --> 01:22:24.180
hidden Markov model in which the present observation only
depends on the present state and not on past dates and
01:22:24.180 --> 01:22:31.814
observations, and in which the present, the transition from
the present state to the future state only depends on the
01:22:31.814 --> 01:22:40.671
present state and not on past states and observations. Okay,
now we can have a look at what we see. So this is actually
01:22:40.671 --> 01:22:47.953
just the state transition probability that describes in which
way the states are changing over time. This term here that
01:22:47.953 --> 01:22:56.572
we need here is actually something that we get as result
from this calculation here. So we, that is actually the same
01:22:56.572 --> 01:23:19.684
term that we can see here. I am furthermore up. So that is
not what we want. Mhm Mhm. Furthermore, what we see is that
01:23:19.684 --> 01:23:28.603
this term here is just the simple probability of an observation.
It just says," How likely is a certain observation if
01:23:28.603 --> 01:23:40.016
we are in a certain state. And we can also see that this
probability is something that is calculated here, but for the
01:23:40.016 --> 01:23:49.445
previous point in time. So you see, this has the same
structure as this term, and the structure is the same, the
01:23:49.445 --> 01:23:59.448
probability of the state at a certain point in time. Given all
the observations up to that point in time. And here it is
01:23:59.448 --> 01:24:10.466
the same kind of probability, but just for one point in
time later. Well, that means we get some the current
01:24:10.466 --> 01:24:21.734
relationship here. Now, if we want to calculate the output.
If we want to calculate this term here. We need these day
01:24:21.734 --> 01:24:32.790
transition probabilities, and we need the result of the
calculation of the upper equation here. So we need to say, if we
01:24:32.790 --> 01:24:43.337
interpret that as a kind of calculation. Where are we get
as a result, the left hand side. And we need that result as
01:24:43.337 --> 01:24:53.907
input to the equation here and if we want to evaluate
this term, what we need is the probability of a certain
01:24:53.907 --> 01:25:04.017
observation and the result of a calculation which we did one
step before by evaluating the equation at the bottom of the
01:25:04.017 --> 01:25:13.397
slide so we get a kind of recurrent cursive
whether to say recursive calculation over time. And this
01:25:13.397 --> 01:25:25.084
interpreted as an algorithm looks like that. So we have
two steps. One is called the prediction step. And this
01:25:25.084 --> 01:25:35.954
prediction stuff actually implements Mhm. This lower
equation here. So it takes us input these two things, and it
01:25:35.954 --> 01:26:00.210
calculates the thing. Yeah, yeah, what does there is some
sign this one here. Well, we sung over all possible values
01:26:00.210 --> 01:26:12.264
that Este can take @unoise@ okay. This, is for the discrete
state for the discrete case. Yeah, let us assume at this
01:26:12.264 --> 01:26:21.136
point that we are dealing with discreet random barriers
@unoise@ the. Same applies for continuous then we replace
01:26:21.136 --> 01:26:31.480
the sum by the integral and the probabilities by the probability
density functions. Yeah, okay. But let us first
01:26:31.480 --> 01:26:40.335
assume that we have discreet Runbury albums now. So we sum
up over all possible values of a discrete random, very
01:26:40.335 --> 01:26:48.626
honesty @unoise@ okay. So, that is a robot of this prediction
step. It calculates the probability distribution over the
01:26:48.626 --> 01:27:00.015
future states. So we assume that we observed. We made the
measurements up to point in time tea. And we want to ask,
01:27:00.015 --> 01:27:08.830
well, what is the expected state at the next point in time.
That is a prediction step, therefore called prediction step.
01:27:08.840 --> 01:27:18.113
This, the other step is called innovation or correction step.
That is actually the step that is shown here at the upper
01:27:18.113 --> 01:27:26.710
equation here that implements this equation, and it is
calculating. So to say, well, we observed all measurements up to
01:27:26.710 --> 01:27:38.215
now. And now we want to know for this point in time, what
is the present state of the system @unoise@ and what we do
01:27:38.215 --> 01:27:47.767
what is so to say the difference we know also the present
observation in this prediction step. We so to say the
01:27:47.767 --> 01:27:55.576
knowledge, the the measurements that we made do not cover the
point in time for which we know the state of the system,
01:27:55.576 --> 01:28:04.366
but they end up one point in time before for the innovation
step. We assume that we made all the observations up to that
01:28:04.366 --> 01:28:13.539
point in time. So that also means that we integrate. So to
say the last observation in our reason. So, and we execute
01:28:13.539 --> 01:28:24.939
these two steps, one after the other. Yeah, every time we
make a new observation, we make one innovation step. Then we
01:28:24.939 --> 01:28:33.912
get these % um. Are these these kind of probabilities. And
then we make a prediction step to get to guess what will be
01:28:33.912 --> 01:28:42.446
the future state of the system one point in time later.
Then we wait for a new measurement. Once we get the new
01:28:42.446 --> 01:28:50.410
measurement, we apply this new measurement in the innovation
step. And we go from this side to this side. Of course,
01:28:50.410 --> 01:28:59.270
what we also need is some . So at the beginning, we need
to start at some point. And while it depends a little bit
01:28:59.270 --> 01:29:07.945
on the application, whether we want to start here on the right
inside, or whether we want to start at the left hand side
01:29:07.945 --> 01:29:15.537
for some applications, it might be that we know really well
the initial conditions of the system that we really know
01:29:15.537 --> 01:29:27.044
very well. What is the initial state? then we start here on
the left hand side. If we don't know very well. What is the
01:29:27.044 --> 01:29:39.089
mhm? um, yeah. Sometimes, in other cases, we might start here
on the right inside. That is up to the application
01:29:39.089 --> 01:29:47.267
so that means the whole process looks like that. So we start
with an energy guess, a with an initial gas for the first
01:29:47.267 --> 01:29:53.673
state, an initial probability distribution for the first
state, then we do an innovation step to integrate the first
01:29:53.673 --> 01:30:00.079
measurement, then we make a prediction step to guess
the probability distribution for the second point in time,
01:30:00.079 --> 01:30:07.159
knowing just the first measurement, then we integrate the
second measurement in an innovation step and so on and so on.
01:30:07.170 --> 01:30:14.865
And as I said, we might also start with a prediction step,
if we like, if that helps more than starting with an
01:30:14.865 --> 01:30:23.747
innovation. St okay, that is it for today. And this is both
possible. That is it for today. Ah, yeah. We continue next
01:30:23.747 --> 01:30:25.129
week.