Special Topics - The Kalman Filter (21 of 55) Finding the Covariance Matrix, Numerical Ex. 1

[@ma888u]

I guess this guy will be one day one of the most famous profs in the world... I think any teacher can learn a lot from him about how to teach difficult stuff. How does he make it that it feels like being in elementary school while he is teaching something so complex like the kalman filter... :DDDDD

[@cheatmagnet]

Basically all professors have different views on how to teach a certain subject. Most of the time, the students are not the central element of their carreer , so they treat them as such. Having a clear, elaborate and succesfull youtube series doesn't mean this professor would be the same in a lecture with all the time constraints that it brings.

[@ryanhooper5680]

Professor, this example made me realize that I am not completely clear about the practical meaning of variance. A couple videos back in this series (video 19 of 55) you explained that plus/minus 1 standard deviation around the mean covers about two-thirds of observed values, while plus/minus the variance around the mean covers almost 100% of the values we might observe, such that variance is a broader range than standard deviation. But doesn't this example illustrate that this is only true for values of standard deviation greater than 1? Here where the standard deviation was less than 1, variance is actually smaller than the standard deviation and does not cover the range of observed values. But then again, a simple change in units of the measurements before calculating variance (e.g. say if the numbers in your example were in meters, we could transform those measurements to millimeters by multiplying by 1000) can recover the kind of relationship you mentioned. So I'm fairly confused. Now that I think about it this is something that has always mystified me about variance every time I have run across the topic, and has been a barrier to my conceiving of variance as a very practical measure. But as I've been learning recently, this whole variance-covariance matrix is foundational for many eminently practical analytical tools. I'm really enjoying the lecture series so far.

[@MichelvanBiezen]

+Ryan Hooper Yes, I see your point. I am going to look into this a little more and get back to you on that.

[@RyanBahneman]

+Ryan Hooper The matrix in the example contains the square of the variances. The standard deviation is the square root of those numbers, which bring the units back into check. For example: The height measurements have a sigma^2 value of 0.0006 but a sigma value of approximately 0.0245, so most of the measurements should be within 0.6 +/- 2*sigma = 0.6 +/- 0.049, which is the case.

[@vincenzodelzoppo9125]

+Ryan Hooper I see Your point. In fact there is a mistake about that. I found these lessons very useful for understanding the concept behind the Kalman filter (I think the professor is really clear) but i found some little mistakes about statistics. You can understand it by simply using dimensional measurements. Let's say our measurements are distances so that you have a set of 2cm,4cm,5cm,7cm,7cm . When You calculate the standard deviation it is about 1.9cm and it is possible to compare the measurements with the standard deviation so you can say they are in or out of the interval. But when you caclulate the variance has the dimension of an area of 3.6cm^2. There is no way to compare distance with areas so it has no meaning to compare measurements with the variance (that has the dimensions of the square of the measurements).

[@ryanhooper5680]

+Vincenzo Del Zoppo Excellent point, thank you. Perhaps it is best to continue to think of it how it's always been explained to me previously, that if you want an interval to capture nearly all observed measurements, you take around +/-3 standard deviations around the mean. Variance is variance.

[@vincenzodelzoppo9125]

+Ryan Hooper +- 3 sigma means 99.7% of measurments should be in. ... Variance has the dimension of the square of the measurement. So even if it is a pure number has no meaning to compare it to measurements.

[@RESC_Eng]

I hope you have long life to learn more and more from you,, Thank you so much my professor.

[@MichelvanBiezen]

You are very welcome

[@markjurik5548]

I think the symbolic representation of the covariance between X and Y is (sigma-xy)^2, rather than (sigma-x)*(sigma-y), which is the product of two standard deviations.

[@NickAlexeev]

I noticed that too. I think that symbolic representation of the covariance between X and Y should be sigma_xy. Without the square. edit: Michel gets to that notation by part 25.

[@chaluvadi]

In this case the variance doesn't cover entire data set. Length has a mean of 4.1 and variance is 0.02, which makes bounds as [4.08 4.12] which clearly doesn't cover entire data set.

[@levk4321]

I thought that generally the off-diagonal elements of the covariance matrix cannot be factorized like in the presentation, apart from some very special cases. In fact, for independent variables the off-diagonal elements will be zeros (because the sums will run over all posible pairs drawn from the corresponding distributions) and not be just the products of the corresponding standard deviations. ?

[@codyheiner3636]

Yes, these videos are good for a rough idea of what Kalman Filters are and what purpose they serve, but the math details have a lot of errors, especially regarding variance and covariance. Edit: it seems he does the computations correctly in the next video, although his notation is confusingly wrong. (Covariance is NOT simply the product of variances of the two variables, that obviously does not represent the relationship between the two variances)

[@levk4321]

@@codyheiner3636 somehow those errors do not affect the final result much, besides they are so obvious that are fun excercise to correct them as y go over the videos :)

[@Randomsimus]

I appreciate the spoon feedings xD Like i know it all, but I don't mind being spoon fed once a while

[@spyhunter0066]

Not sure the formulas on the bottom left and in the cover matrix on the top middle with sums and x,y,z. Why did you change the notation from L,W,H to x,y,z? For that formula to be correct, x_i should equal to L_average - L_i . However, you defined x_i as an individual measurement not the deviation from the average. It's clear what you meant in the next videos, but the sigma notation usage is better and clearer without any confusion.

[@ashumohanty566]

long live sir thank you

[@jobaidajarin356]

Thank you sir

[@MichelvanBiezen]

Glad to be of help.

[@jobaidajarin356]

@@MichelvanBiezen Sir, It's been a great help. Salute.