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  • 00:04

    In this video, we’ll discuss the working principles of the Kalman filter algorithm.

  • 00:09

    Let’s start with an example.

  • 00:11

    While you’re desperately staring at your bills, an ad in a magazine catches your eye.

  • 00:18

    You can earn $1 million by joining a competition where you design a self-driving car, which uses a GPS sensor to measure its position.

  • 00:28

    Your car is supposed to drive 1 km on 100 different terrains.

  • 00:34

    Each time, it must stop as close as possible to the finish line.

  • 00:40

    At the end of the competition, the average final position is computed for each team,

  • 00:46

    and the owner of the car with the smallest error variance and an average final position closest to 1 km gets the big prize.

  • 00:56

    Here’s an example.

  • 00:58

    Let these points represent the final position and the red ones the average final position for different teams.

  • 01:06

    Based on these results, team 1 would lose due to the biased average final position although it has small variance.

  • 01:15

    Team 2 would lose as well.

  • 01:18

    Its average final position is on the finish line but it has high variance.

  • 01:24

    The winner would be team 3 since it has the smallest variance and its average final position is on the finish line.

  • 01:32

    If you want to want to be a millionaire, you don’t want to rely purely on GPS readings, since they can be noisy.

  • 01:39

    In order to meet the required criteria to win the competition, you can estimate the car’s position using a Kalman filter.

  • 01:49

    Let’s look at this system to understand how the Kalman filter works.

  • 01:54

    The input to the car is the throttle.

  • 01:56

    The output that we’re interested in is the car’s position.

  • 02:01

    For such a system, we would have multiple states.

  • 02:04

    But here to give you intuition, we’ll assume an overly simplistic system where the input to the car is the velocity.

  • 02:13

    This system will have a single state: the car’s position.

  • 02:17

    And we’re measuring this state, so matrix C is equal to 1.

  • 02:22

    It’s important to know y as accurately as possible, since we want the car to finish as close as possible to the finish line.

  • 02:31

    But the GPS readings will be noisy.

  • 02:33

    We’ll show this measurement noise with v, which is a random variable.

  • 02:39

    Similarly, there’s process noise, which is also random and can represent the effects of wind or changes in the car’s velocity.

  • 02:49

    Although these random variables don’t follow a pattern, using probability theory we can tell something about their average properties.

  • 02:58

    V, for example, is assumed to be drawn from a Gaussian distribution with zero mean and covariance R.

  • 03:07

    This means if we measured the position of the car - let’s say a hundred times at the same location -

  • 03:14

    the noise in these readings would take on values with most of them located near the zero mean and fewer located further away from it.

  • 03:24

    And this results in the Gaussian distribution, which is described by the covariance R.

  • 03:31

    Since we have a single output system, the covariance R is scalar and is equal to the variance of the measurement noise.

  • 03:40

    Similarly, the process noise is also random and assumes a Gaussian distribution with covariance Q.

  • 03:48

    Now we know that the measurement is noisy, and therefore what we measure doesn’t quite reflect the true position of the car.

  • 03:55

    If we know the car model, we can run the input through it to estimate the position.

  • 04:01

    But this estimate also won’t be perfect. Because now we’re estimating x, which is uncertain due to the process noise.

  • 04:10

    This is where the Kalman filter comes into play; it combines these two pieces of information to come up with the best estimate of the car’s

  • 04:18

    position in the presence of process and measurement noise.

  • 04:23

    We’ll discuss the working principle of the Kalman filter visually with the help of probability density functions.

  • 04:31

    At the initial time step k-1, the actual car position can be anywhere around the estimate x^hat_(k-1) ,

  • 04:42

    and this uncertainty is described by this probability density function.

  • 04:47

    What this plot also tells us is that the car is going to be most likely around the mean of this distribution.

  • 04:56

    At the next time step, the uncertainty in the estimate has increased, which is shown with the larger variance.

  • 05:06

    This is because between time step k-1 and k, the car might have run over a pothole, or maybe the wheels may have slipped a little bit.

  • 05:17

    Therefore, it may have traveled a different distance than what we’ve predicted by the model.

  • 05:24

    As we discussed before, another source of information on car’s position comes from the measurement.

  • 05:31

    Here, the variance represents the uncertainty in the noisy measurement.

  • 05:37

    Again, the true position can be anywhere around the mean.

  • 05:43

    Now that we have the prediction and measurement, the question is what is the best estimate of the car’s position?

  • 05:51

    It turns out that the optimal way to estimate the car’s position is by combining these two pieces of information.

  • 05:59

    And this is done by multiplying these two probability functions together.

  • 06:05

    The resulting product is also a Gaussian function.

  • 06:10

    This estimate has a smaller variance than either of the previous estimates.

  • 06:15

    And the mean of this probability density function gives us the optimal estimate of the car’s position.

  • 06:23

    This is the basic idea behind Kalman filters.

  • 06:25

    But to win the competition, you need to be able to implement the algorithm. We’re going to discuss this in the next video.

All

The example sentences of GAUSSIAN in videos (15 in total of 34)

similarly adverb , the determiner process noun, singular or mass noise noun, singular or mass is verb, 3rd person singular present also adverb random adjective and coordinating conjunction assumes verb, 3rd person singular present a determiner gaussian proper noun, singular distribution noun, singular or mass with preposition or subordinating conjunction covariance noun, singular or mass q proper noun, singular .
they personal pronoun began verb, past tense by preposition or subordinating conjunction transforming verb, gerund or present participle the determiner liouville proper noun, singular field noun, singular or mass into preposition or subordinating conjunction a determiner far adverb milder noun, singular or mass object noun, singular or mass called verb, past participle a determiner gaussian proper noun, singular
make verb, base form another determiner duplicate noun, singular or mass of preposition or subordinating conjunction the determiner layer noun, singular or mass and coordinating conjunction double adjective - click noun, singular or mass the determiner gaussian proper noun, singular blur proper noun, singular smart adjective filter noun, singular or mass to to edit verb, base form its possessive pronoun
the determiner statistical adjective term noun, singular or mass for preposition or subordinating conjunction it personal pronoun is verb, 3rd person singular present gaussian proper noun, singular distribution noun, singular or mass , but coordinating conjunction many adjective people noun, plural call verb, non-3rd person singular present it personal pronoun the determiner bell proper noun, singular curve proper noun, singular as preposition or subordinating conjunction
and coordinating conjunction with preposition or subordinating conjunction the determiner duplicated verb, past participle layer noun, singular or mass selected verb, past participle , go verb, base form to to filter verb, base form > noun, singular or mass blur proper noun, singular > noun, singular or mass and coordinating conjunction choose verb, base form the determiner " gaussian proper noun, singular blur proper noun, singular " filter noun, singular or mass .
after preposition or subordinating conjunction you personal pronoun satisfied verb, past tense all determiner your possessive pronoun cravings verb, 3rd person singular present it personal pronoun is verb, 3rd person singular present now adverb time noun, singular or mass for preposition or subordinating conjunction the determiner gaussian proper noun, singular nightlife noun, singular or mass start noun, singular or mass the determiner gaussian proper noun, singular nightlife noun, singular or mass at preposition or subordinating conjunction the determiner black adjective dog noun, singular or mass an determiner
then adverb you personal pronoun could modal also adverb go verb, base form into preposition or subordinating conjunction filter proper noun, singular , blur proper noun, singular , gaussian proper noun, singular blur proper noun, singular , to to blur verb, base form this determiner layer noun, singular or mass because preposition or subordinating conjunction
blur noun, singular or mass and coordinating conjunction then adverb i personal pronoun 'm verb, non-3rd person singular present going verb, gerund or present participle to to select verb, base form gaussian noun, singular or mass blur noun, singular or mass so preposition or subordinating conjunction i personal pronoun 'm verb, non-3rd person singular present going verb, gerund or present participle to to go verb, base form three cardinal number times verb, 3rd person singular present the determiner
and coordinating conjunction i personal pronoun 'm verb, non-3rd person singular present going verb, gerund or present participle to to choose verb, base form blur noun, singular or mass and coordinating conjunction sharp adjective and coordinating conjunction and coordinating conjunction go verb, base form for preposition or subordinating conjunction a determiner gaussian noun, singular or mass blur proper noun, singular right adverb here adverb
is verb, 3rd person singular present operating verb, gerund or present participle even adverb close verb, base form to to the determiner perfect adjective gaussian proper noun, singular beam noun, singular or mass , so adverb if preposition or subordinating conjunction it personal pronoun 's verb, 3rd person singular present say verb, non-3rd person singular present ten cardinal number times noun, plural
let verb, base form s proper noun, singular say verb, non-3rd person singular present our possessive pronoun gaussian proper noun, singular blur noun, singular or mass i personal pronoun created verb, past tense a determiner custom noun, singular or mass keyboard noun, singular or mass shortcut noun, singular or mass for preposition or subordinating conjunction my possessive pronoun gaussian proper noun, singular
actual adjective filter noun, singular or mass name noun, singular or mass there existential there to to bring verb, base form up preposition or subordinating conjunction the determiner gaussian proper noun, singular blur proper noun, singular dialogue noun, singular or mass box noun, singular or mass , but coordinating conjunction i personal pronoun 'm verb, non-3rd person singular present quite adverb happy adjective with preposition or subordinating conjunction
i personal pronoun 'm verb, non-3rd person singular present going verb, gerund or present participle to to search verb, base form for preposition or subordinating conjunction blur proper noun, singular in preposition or subordinating conjunction the determiner effects proper noun, singular library proper noun, singular right adverb here adverb , the determiner gaussian proper noun, singular blur noun, singular or mass .
how wh-adverb can modal we personal pronoun convert verb, non-3rd person singular present a determiner gaussian proper noun, singular to to something noun, singular or mass that determiner kind noun, singular or mass of preposition or subordinating conjunction approximates verb, 3rd person singular present this determiner circular adjective shape noun, singular or mass ?
do verb, non-3rd person singular present something noun, singular or mass like preposition or subordinating conjunction go noun, singular or mass filter noun, singular or mass , blur noun, singular or mass , and coordinating conjunction apply verb, base form a determiner gaussian noun, singular or mass blur noun, singular or mass and coordinating conjunction say verb, non-3rd person singular present look verb, base form , i personal pronoun

Use "gaussian" in a sentence | "gaussian" example sentences

How to use "gaussian" in a sentence?

  • [On the Gaussian curve, remarked to Poincaré:] Experimentalists think that it is a mathematical theorem while the mathematicians believe it to be an experimental fact.
    -Gabriel Lippmann-
  • One of the laws of nature," Gordon said, "is that half the people have got to be below average.""For a Gaussian distribution, yeah," Cooper said. "Sad, though.
    -Gregory Benford-

Definition and meaning of GAUSSIAN

What does "gaussian mean?"

/ˈɡousēən ˌdistrəˌbyo͞oSH(ə)n/

noun
undefined.
other
Of or relating to Karl Gauss or his mathematical theories of magnetics or electricity or astronomy or probability.