Mean angle is not a usual average. Means on circle - Intro to directional statistics (3B1B SoME1)

Подписаться 254

Просмотров 15 тыс.

50% 1

How to indicate the mean direction (or average direction) of the wind? How to calculate the mean position (or average position) on the circle?
[Timestamps below]
This video shows that such a simple thing like mean or average changes its meaning for the points belonging to the circle or when dealing with an angular variable. We present a generalised understanding of the mean based on the minimisation of the Fréchet function. This approach distinguishes between the intrinsic mean and the extrinsic mean. At the same time, it unifies the way of calculating the mean and median, not necessarily on the circle. It turns out that the circular median can also be intrinsic or extrinsic.
This is an introduction to circular statistics (for circular data) or, more generally, to directional statistics (for directional data). These are the simplest and most useful examples of statistics on topologically non-trivial manifolds (on manifold-valued data).
The lecture assumes the knowledge of mathematics at the level of a good high school graduate but includes a brief revision of the key issues.
(The video is submitted to the Summer of Math Exposition 1 carried out by 3blue1brown.)
By Karol Ławniczak
Timetable
00:00 Preliminary examples
01:27 Intro
01:50 Revision of the concept of mean
04:00 Revision and clarifications concerning directions, angles, arcs and positions on the circle
05:50 Failure of the usual mean
07:27 Circle as a figure on a plane vs as an autonomous space esp. labelling directions
08:34 Possible confusion with a usual mean over a circle
09:23 Extrinsic mean
11:08 New point of view on ordinary mean, Fréchet function
15:03 Intrinsic mean - optimisation approach
19:00 Intrinsic mean - analytic approach
21:50 Further visualizations and some properties
24:26 Median
29:54 Concluding comment
30:36 Neat physical example
32:06 Further reading
Links:
3blue1brown: Euler's formula with introductory group theory
• Euler's formula with i...
T. Hotz and S. Huckemann: Intrinsic Means on the Circle - Uniqueness, Locus and Asymptotics (2011)
arxiv.org/abs/1108.2141
A. Brun et al.: Intrinsic and Extrinsic Means on the Circle - A Maximum Likelihood Interpretation (2007)
ieeexplore.ieee.org/document/... (limited access)
Topic on StackExchange:
math.stackexchange.com/questi...

Наука

Опубликовано:

22 июл 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 89

@druid7456 2 года назад

PL: Miło mi poinformować, że napisy dostępne są w języku angielskim i polskim. ENG: I am pleased to inform you that the subtitles are available in English and Polish.

@cmyk8964 2 года назад

I once saw a survey analysis that said the “average color” of a few things were all green. I knew immediately that they averaged the color linearly in HSV or HSL color space without wrapping the hue around.

@druid7456 2 года назад

Excellent example.

@AlvaroALorite 2 года назад

But that's valid. The "wrapping" of the circle is artificial. In real life, the EM spectrum does not wrap as a circle.

@druid7456 2 года назад

@@AlvaroALorite The EM radiation spectrum does not really come in a circle. The thing is, we don't act like an EM spectrometer and we are interested in non-monochromatic light also. Or, to put it more precisely: we operate as an extremely low-resolution spectrometer: one that only distinguishes 3 (overlapping) ranges. Moreover, we not only want to express the colour of pure, monochromatic light, but we also need to describe mixed light. Fortunately, all that matters is the stimulation of these three receptors. These are independent nerve signals that span the 3D space of possible light sensations. It's RGB space. Along the main diagonal from (0,0,0)=B to (1,1,1)=W the total light intensity changes. If we normalize the "brightness", there will be two-dimensional sections with greys in the middle. If we introduce a polar coordinate system, the "saturation" changes along the radius and the "hue" - the kind/type of colour - changes along the perimeter or with angular coordinate. This is how the cylindrical and conical HSV and HSL spaces are defined (they differ in, roughly speaking, the way of "brightness" normalization). There, the "hue" parameter has an explicit circle topology. Averaging "hue" otherwise leads to bizarre effects such as red and pink averaging to green.

@karius85 Год назад

@@druid7456 Yes, but I would specify that the averaging of light spectra depends on the application. If you're asking a person what the "average" of red and violet is, green seems nonsensical, and most people would feel an answer of pink/magenta is more intuitive. However, the average of the EM frequencies of these colours correspond to green. Even if this seems counterintuitive from a human perspective of perceived colour, it can be extremely misleading in other contexts, for instance in the field of astrophotography. Somewhat analogously, the concept of musical pitch has a perceived circular quality to it, where multiplying the frequency of a sine wave by 2 yields a harmonic relation of an octave. However, because the audible spectrum of most humans is greater than a single octave, in this case, human perception would more align with an "average" frequency (possibly in base 12 log2 or something). I guess the point is that context matters, and that colour can have either a physical definition, as well as a human cognitive/aesthetic definition, however none of the two are more correct in an a priori sense.

@marcelo55869 2 года назад

Did you record it during a storm? Wearing headphones I can clearly hear the rain in the background. Very relaxing vibe to it...

@donnybeegoode 2 года назад

Exquisite, compelling and very comprehensible. Chapeau bas for the amount of work you’ve put into making this video, it was a pleasure to watch and learn some new stuff. Please, keep your channel alive!

@druid7456 2 года назад

I am glad that you liked the material and you perceive it as valuable. Thank you for your words of appreciation. It means a lot to the creator.

@Wagon_Lord 2 года назад

I especially love at the end where you said you dislike such questions as "what are the applications of this". I agree wholly, sometimes it's fun to just think on a problem. Thank you very much for this!

@ositchukwu7502 2 года назад

Świetny film, pozdrawiam z Polski

@vitorguilhermecoutinhodeba3253 2 года назад

It's really good having this kind of content here on RU-vid and nicely illustrated!

@druid7456 2 года назад

I am glad you like it and I hope it will be useful.

@jackozeehakkjuz 2 года назад

this is fantastic! I'm still finding videos from the SoME1 and this one is delightful! may be more advanced than what I'd consider for the "general public" but absolutely enjoyable. Thanks for your work!

@druid7456 2 года назад

Thanks for the appreciation. It may not be very easy, but it definitely should be more widely known, as it is a perfectly natural extension of statistics in topologically trivial space, without which many things simply cannot be done properly.

@nifets 2 года назад

Really well made video! Learned a lot!

@dsagman 2 года назад

truly impressive. thank you for all the hard work to share this.

@druid7456 2 года назад

Thanks :) There was a lot of work indeed.

@druid7456 2 года назад

RU-vid is still processing HD video. Since yesterday afternoon. Apparently, they are not keeping up with the volume of videos uploaded. Please be patient. As soon as the service is ready, the video will be broadcast.

@druid7456 2 года назад

Ok, HD is available.

@KirkWaiblinger 2 года назад

I have been looking for a solution to taking the mean on a circle for years. Thanks for making this video. Even now it seems Wikipedia has nothing to say about these intrinsic means - maybe you want to write the article?

@druid7456 2 года назад

It was the same with me. Maybe not for years, but for a long time I was looking for an answer to this seemingly fundamental question. Finally, I worked out the problem myself. And only then did I find works introducing these concepts. There is a mess on Wikipedia on this topic. And on some related too. Maybe it really needs to be written properly.

@Czeckie 2 года назад

this was really inspiring! Seems like the problem of means on submanifolds could have some connections to topology.

@MrObear 2 месяца назад

Great video!

@alexanderbrady5486 2 года назад

Great video. One issue you didn't bring up is that for some distributions multiple points can minimize the L2 distance. In those cases it is impossible to say what the "true" mean should be, but it is an issue that can trip up your algorithm if you aren't prepared to handle it. It can also cause havoc with your statistics, as a stable distribution of points around the circle can produce a wild distribution of means if that distribution of points happens to be right on the edge between two mean candidates.

@druid7456 2 года назад

Indeed, I was not concerned with the issue of global uniqueness (unambiguity). This is also an interesting issue, but I had to limit the threads somewhere. I limited myself to the milder requirement of local uniqueness and I said something about it when discussing the median. The global ambiguity (non-uniqueness) appears in the physical example presented at the end. Could you please explain what exactly you have in the last sentence? It sounds intriguing.

@alexanderbrady5486 2 года назад

@@druid7456 I have had data where the angles were something like 50% +X axis, 50% -X axis (with some tight distribution around those two directions). Using any circular mean method (linear average, complex average, minimization of distance) on a single data set will give you a "mean" of +Y or -Y. But if you do that across multiple datasets you will see the mean flip randomly between +Y and -Y. Thus, a "stable" distribution of data ends up with a calculated mean whose variance is very high. Not too deep, really it just goes to show that the "mean" isn't always meaningful for circular data (as you mention in the video)

@druid7456 2 года назад

@@alexanderbrady5486 Yes, you are right. These sudden jumps in the mean are unpleasant. And, in fact, they apply to both the extrinsic and intrinsic mean (though in a different manner and to a different extent). But that's the nature of the circle. When the centre of mass passes through the centre of the circle (continuously), the extrinsic mean jumps to the antipodes. The intrinsic mean behaves differently here. We only have such jumps in the case of discrete data. For continuous distributions, consisting of two symmetrical and almost equal peaks at mutual antipodes, when one of them loses its advantage to the other, the mean ceases to be unique and follows two paths (continuously!) to the new dominant peak. You can see something like this in the final simulation (31:38).

@anatoly-k Год назад

exhaustive explanation! could you make the same explanation for compositional statistics?

@computationaltrinitarianism 2 года назад

Fantastic introduction.

@druid7456 2 года назад

Thanks for the appreciation.

@sajjadakbar6649 2 года назад

Very well explained, preciesly.

@druid7456 2 года назад

Thank you very much.

@yupbank 2 года назад

i am working on ML application with directional target! run into tons of familiar conclusions. on the median, there are some weird cases when even number of points are evenly spaced in the circle, then effectively every point in the circle is median.

@druid7456 2 года назад

It's nice to meet a colleague! I also work with ML. Two issues belonging to two separate fields of my research interests led me to this topic. Firstly, it was quantum mechanics on topological non-trivial manifolds (not related to ML), and secondly, it was machine learning applied to the problems of aquatic ecology, where I also dealt with a target that did not belong to the Euclidean domain.

@AmitKumar-xw5gp 2 года назад

very good video..good use of manim..

@druid7456 2 года назад

Manim is a great tool, but this video has been made without Manim. :) Try to guess what the graphics and animations were generated in.

@AmitKumar-xw5gp 2 года назад

@@druid7456 There are a lot of softwares out there, please let us know how did you make those animations..

@druid7456 2 года назад

@@AmitKumar-xw5gp Wolfram Mathematica :)

@LeetMath 2 года назад

for continuous distribution: try to describe the distribution as a series of sums of sinusoidal moments, average direction is peak of the 1st moment

@druid7456 2 года назад

Indeed, the notion of trigonometric moments (or Fourier coefficients) provides valuable insight into the problem of the mean on the circle (the extrinsic one). If I understand you correctly, you are talking about the first trigonometric moments argument. For n = 1, the definition of the trigonometric moments comes down to the formula for the centre of mass on the complex plane of some mass distribution on the circle. (By the way, these moments are nothing else than the Fourier coefficients (in the complex version).) The projection of this centre of mass on the circle is simply its argument. However, the extrinsic mean is neither the only nor the best mean on the circle. In particular, it does not fully deserve the term "mean direction" (which statement I justify in the video). I remain an advocate of the intrinsic mean. ;)

@LeetMath 2 года назад

i think maybe its better to think of it in terms of trying to summarize a distribution rather than extend the idea of an average or representative value. in f: R->R+ of probability distributions on R, you have the standardized moments, which i’m not sure if they can be used to correspond to weights for a set of basis functions to reconstruct any distribution. in the circle space you have something different. specifically, there can be nonzero zeroth moment, which maybe corresponds to variance. maybe you have a distribution of something like ‘directions of roads in a city’ which will likely have bipole and quadripole moments. i’m not sure what something like a gaussian convolved / wrapped into a circle looks like in terms of trigonometric moments, you want something that reproduces the mean / variance of that kind of shape if you have it.

@LeetMath 2 года назад

this video is pretty good, it made me think in new ways about the concept of measures of averageness in general, in terms of being derived from variance minimization

@druid7456 2 года назад

@@LeetMath Yes, my assumption was to find the mean value that is as close as possible (in a reasonable sense) to all the elements of the set. This condition is met by the intrinsic mean (as well as the int. median, albeit slightly differently). The picture with the moments is attractive. It should be noted, however, that there are several similar representations on R: with moments, central moments, cumulants, something else for sure. There may be more options for the circle too. Perhaps one of them has even more desirable properties. As for the analogy of ordinary trigonometric moments to moments in R, it should be noted that the exponentiation does something different with x and different with e^(i*x). Unless there is a special interpretation in R, in S it means scaling an angle variable (or indicating the winding number). This is because the parameter measuring the position (and then the natural distances) in the space of the circle ended up in the exponent. From a group point of view (Lie groups), the analogy is very imperfect. Thanks for those comments about the moments. I think that a lot of interesting things can be found here. I will certainly work on it more.

@tomekwas8018 2 года назад

Very good material! It seems to me that in this way one can generalize mean and median concept to an arbitrary metric space (including the weird ones). Do you have, by chance, any references to formulation of something like that? As mentioned in other comments, the Wikipedia is pretty bad in these topics, and google scholar turned out not very helpful too.

@druid7456 2 года назад

Yes, this approach allows you to define a meaningful mean, median and other descriptive statistics for variables belonging to arbitrary manifolds. Some interesting phenomena arise in doing so. Does it work for completely arbitrary metric spaces? I don't know, but if you've researched this question, I'd love to know your results. I can only recommend the work already cited: 1) T. Hotz and S. Huckemann: Intrinsic Means on the Circle - Uniqueness, Locus and Asymptotics (2011) (available on ArXiV) 2) A. Brun et al.: Intrinsic and Extrinsic Means on the Circle - A Maximum Likelihood Interpretation (2007) (I don't have a link to the free version, but you can always find it on SciHub) Maybe I'll take up this topic myself with more insight.

@echolambda 2 года назад

I have a question: it seems that the choice of N candidates in the intrinsic mean do not depend on the weights (p_j) and only the "true mean" depends on them. Is that correct?

@druid7456 2 года назад

No, it doesn't work this way. Let's assume the weights are normalized (for simplicity). The number of candidates depends on the weights. If the weights are irrational, it's a dead-end, and you have to use the optimization formula (or approximate the weights with rational numbers, but this is a sensitive matter). If they are rational, it is best to expand their sequence to a common denominator. Take the numerators as the new weights and the denominator as the normalization factor (inverse). Then this common denominator will indicate the number of (evenly spaced) candidates. There will always be a "normal" weighted mean among them. And this is the place the positions of the other candidates can be measured from. It's like counting individual data points with multiplicities. Try the formula 20:05 with weights introduced.

@TrainerInukuma 2 года назад

Why don't you just see every angle as a vector? You could then just add them all together and calculate the angle of the result vector to get an average angle. This also covers the corner cases with oposing angles.

@druid7456 2 года назад

But I propose just such a solution in approach #1 (9:30). It does not matter whether the position on the plane is parameterized by one complex number or by a two-dimensional real vector. When you add these vectors, you get a vector that points to a point inside or outside the circle. If you divide the sum by the number of vectors added (like in every mean), the result is always inside the circle. In order for the thus "mean" to belong to the space of the circle, these points must be projected onto the circle along the radius. In the case of averaging the antipodal points, this mean (as a projection) will not be defined. I explain the properties and disadvantages of this approach (10:46 and further).

@rv706 2 года назад

A question that I may have missed from the video: what's the extrinsic circular average of two antipodal points?

@druid7456 2 года назад

The centre of mass is right in the middle of the disk and its projection on the circle is not defined. You can read it from formulas (10:08 and 10:43) or see it on the animated chart (22:07).

@rv706 2 года назад

@@druid7456: Ok as one would expect it's not well defined. Thanks

@nicrule4424 2 года назад

I think the section at 10:45 exemplifies my biggest problem with your video. Everything you say during this section makes sense to me. However, the math on the right contains 13 symbols (ER, er, D, C, x, y, z, c, r ,e, N, R, Φ), not 1 of them is on the drawing, and you give no explanation. This implies 1 of 2 things. Either it isn’t important and can be ignored, or the viewer is meant to already understand this information. If 1, remove it, if 2, show what all these values are on each blue point and explain what the equations are doing. I have a degree in mechanical engineering, so I can state with some confidence, you are using notation only someone with a bachelor’s in mathematics would know. This puts a hard limit on who is capable of following your video, restricting it mostly to people who already understand most of what you are trying to teach. You start with a question “how do you find a useful average of 2d directions?” This isn’t exactly a master’s thesis; students just need to understand algebra and polar/complex coordinates. With that starting point: you need to show the standard linear approach doesn’t work, show how complex coordinates mitigate the disproportionate gap between points bridging the linear -> cyclical mapping seam, and show an example of the standard 8th grade averaging equation working in these coordinates. That's it. Instead, you made a video a licensed engineer couldn’t follow. You are under no obligation to target the same audience as 3B1B, but this subject did not need to be this complicated. Some random notes: For much of your presentation, your font size is really small and there is a lot of empty space. At 480p, some of the subscript can get blurred out by the compression. Also, don’t add borders, and just try to use the space you have available. The chapter transition animations don’t add anything to the video and waste 5 seconds each. Starting at 2:30, it would be helpful if you highlight or point to each noun as you say it, (number of events, weight, unity). This would be nice throughout the video; this was just the first I noticed. At 4:00, you say something about simply/multiply connected spaces with no explanation as to what those are. If this is important, it needs an explanation, if it isn’t it shouldn’t be there at all. At 4:21 you give a quick definition for azimuth, then never use the word again. The graphic at 6:40 took me a while to understand. So the yellow numbers at the bottom map to the two possible values of the lowest BLUE dot. The two yellow dots map to the RED values. Dots are frequently covering up angle numbers, and the bottom half of the circle’s angle labels are unreadable. During this section, you talk about reading the circle as a line segment, but the visuals for this are at 11:12 for some reason. I would have replaced averaging “the usual way” with Linearly vs Cyclically. At 7:20, your graphic from 5:30 would make it more clear what this means. 7:30 I don’t think this section needs to exist as it seems to only be there to set up for projecting a point from inside the circle onto the circle, which is pretty easy to understand on its own. I don’t understand the section starting at 8:40. You jump from 2d to 3d, discrete points to continuums, and bring in integrals all at the same time. This section doesn’t have a stated goal I could recognize or a connection to previous sections beyond “circle.” I admit, this section is on the edge of my knowledge in this subject, but your video isn’t helping me learn more. You have clicks in your audio occasionally (14:47). Assuming you recorded audio and video separately, you can just cut these out. It sounds like you are recording outside. I can still understand you, but some RU-vidr’s record in a coat closet to reduce echo if you wanted to try that. I won’t go into detail about the rest of the video because I kept getting lost after about halfway through, and I think the first two paragraphs cover most of my issues. Starting at 22:35, you have a lot of very nice animations, but no accompanying equations. If you show the 3 graphed points, with their values filled in in the equations, you could show the entire algebraic process from input to output at the same time. 3B1B does this sometimes where he has 1 animation which pretty much summarizes the entire video. Your video is well presented; I just think you should make things more understandable for viewers who haven’t taken calculus 4.

@druid7456 2 года назад

Oh, I think most inconveniencies you mentioned follow from my intention to make a two-level explanation. Some viewers may stick to intuitive understanding and its visual support; and some can benefit from the formal side. This is also why I omitted the explanation of these symbols, which are customarily used in a given context and should be clear to the person using the mathematical apparatus. Finally, those symbols, that are explained, are usually explained only once, which actually requires the viewer to focus. This film aimed not only to answer the question about the correct way of averaging points in space with a circle topology but also to discuss the mathematical problems that arise on this occasion. Hence the slightly digressive style that can give you the impression of an unnecessary complication. In any case, thank you for such a comprehensive discussion and valuable comments. Some of them convince me (including these concerning video production, but not only these) and some do not. However, I will definitely take these things into account in the future.

@user-uy8yt7ku4w 2 года назад

Really good video, except for the sound.

@jacobolus 2 года назад

What happens if you use stereographic distance (tangent of half of the angle measure) instead of angle measure or chordal distance?

@druid7456 2 года назад

The absolute value of this half-angle tangent may be used as a metric. (The tangent itself may also be negative.) It is not an intrinsic metric for the circle. For antipodal points, it runs to infinity. If we have a cluster on one side of the circle and even a single point on the antipodes, the Frechet function cannot be minimized in this cluster because it escapes to infinity there. Therefore, this method will not indicate a cluster as the location of the mean. This is not what we expect from the mean.

@druid7456 2 года назад

I saw you posted a link to an article on related issues on the sphere. Now I can't find it. Perhaps youtube deleted it (for some reason). What is this article? Could you give me a hint on how to find it?

@jacobolus 2 года назад

@@druid7456 I linked to Buss/Fillmore (2001) "Spherical Averages and Applications..." But also see Dorst (2021) "Optimal Combination of Orientation Measurements ..."

@druid7456 2 года назад

@@jacobolus Thanks. I'll read it.

@larrs73 5 дней назад

More circular statistics

@d.o.584 2 года назад

Your "extrinsic" mean is not much of an extrinsic. It is the direction which minimizes sum(1-cos(phi_i-phi_0)), which is a good (square) metric for a circle. The video quality is superb. Some lettering maybe small on some "slides", but it is a nitpick.

@druid7456 2 года назад

Thanks for quoting the cosine metric formula. And for words of appreciation. Yes, the cosine metric is the valid metric on the circle. It is equivalent to the aforementioned chordal metric. And for the purpose of measuring (a kind of) distance on the circle with this metric, it is not necessary to immerse the circle in the plane and measure the distance through the interior of the disk. However, it is NOT an intrinsic metric of the circle. The metric space consisting of a circle with cosine (or chordal) metric is not length (or path) metric space. The only intrinsic metric of the circle is the metric induced from the Euclidean metric on the plane. It is this one that applies to the definition of the intrinsic mean.

@rv706 2 года назад

At the end of the day, it's not clear why topology would have anything to do with the intrinsic mean at all. From what I seem to have understood from your video, the only structure (on the circle) that you are really using is the metric: argmin {Σf(d(xi, x))pj} seems defined for every metric space (given a choice of positive monotone function f on the positive reals), no matter what. [edit: provided the min is unique] Am I mistaken?

@MagicGonads 2 года назад

The topology is referring to the modular behaviour on the circle, which eliminates metrics where the minimisation does not produce a well defined function under the modular identification

@rv706 2 года назад

@@MagicGonads: what do you mean by "eliminates metrics"?

@MagicGonads 2 года назад

@@rv706 makes them not suitable as the intrinsic

@Tom-pc7lb 2 года назад

I think the movie “A Clockwork Orange” was about this.

@rv706 2 года назад

At first I thought: "the circle is a divisible abelian Lie group; so just take the sum of the n points of the group and then divide by n". But the "n-th root" in a divisible group need not be unique!

@rv706 2 года назад

Ah, now I've reached 19:30 and you're saying exactly (something equivalent to) this

@MadsterV 2 года назад

NASA has a method called Singular Value Decomposition which allows you to average rotations represented as quaternions, that would work too

@druid7456 2 года назад

Singular value decomposition is a fairly general matrix operation. Since you speak of quaternions, you probably mean 3D rotation (related to the sphere), not plane rotation (to circle). Of course, quaternions can be represented by matrices (complex 2x2 or real 4x4). However, I do not know what method of averaging the rotations you can mean (or maybe orientations on the sphere - because they are not the same, and it is impossible to map them 1 to 1, although it was possible in the case of a circle). Say something more or cite the source, and we will see what kind of mean it is.

@MadsterV 2 года назад

@@druid7456 I didn't post the URL to the paper because the comment might get deleted, but it's an easy search. It would be a problem to have rotations not constrained to a sphere, but perhaps there's an equivalent representation for 2D rotations?

@MadsterV 2 года назад

@@druid7456 Nevermind, it doesn't make sense in 2D, you could end up aligned with the wrong axis

@druid7456 2 года назад

The 3D rotation can be reduced to 2D by fixing the axis, so the method that works in 3D must also work in 2D. In addition, the statistics on a sphere (immersed in 3D) is even more interesting than on a circle (in 2D). I would gladly see what the method you mentioned is. I know yt deletes links. You can always provide bibliographic information (author, title, year) or DOI number.

@MadsterV 2 года назад

@@druid7456 oh nice I did comment the link (and sadly some thoughts too) but I can see it got deleted. If you search for this it should be among the first hits-> nasa Singular Value Decomposition quaternion average EDIT: I remembered the deleted thoughts: what happens if the angles are in exact opposite sides of the circle? is the mean still defined?

@lucassaito2842 2 года назад

I mean this in the best way possible, but I have some criticisms. Before though, I'm leaving this comment because I liked the video a lot (and I also feel like many more could be made much better with just a few tweaks), so please don't take them as hate. I just want to watch cool maths stuff. First , the video looks very good and technical, but it feels too much like a lecture without the lecture benefits (like doing exercises for instance). It also hard to keep up at times, especially since it's not really clear when something on the screen is relevant later on in the video (and I personally feel like a video on a more 'out there' topic as this one should feel more self contained, unless it is part of some sort of series of videos). If I'm allowed to be a bit more picky, audio is super super super important on RU-vid. Any reasonable mic would have done the job, and they aren't that expensive for the quality they bring (like the 40$ ones are very much ok for a long time).

@druid7456 2 года назад

Thanks, any kind advice is appreciated. I think I know what you mean about lecture / non-lecture. As I answered one viewer: "I think most inconveniencies you mentioned follow from my intention to make a two-level explanation. Some viewers may stick to intuitive understanding and its visual support; and some can benefit from the formal side. This is also why I omitted the explanation of these symbols, which are customarily used in a given context and should be clear to the person using the mathematical apparatus. Finally, those symbols, that are explained, are usually explained only once, which actually requires the viewer to focus. This film aimed not only to answer the question about the correct way of averaging points in space with a circle topology but also to discuss the mathematical problems that arise on this occasion. Hence the slightly digressive style that can give you the impression of an unnecessary complication. " The pace resulted from the guidelines of the competition for which I was preparing this video. I wanted to convey a lot of content, including necessary repetitions and side issues, without intimidating viewers with the overall length of the video. In the beginning, I encourage viewers to pause the playback whenever necessary. As for the strange sound of the audio track, well, I can hear it too. I didn't have time to correct it as the video was submitted to a contest with an inexorable deadline. However, it seems to me that everything is understandable. Certainly, in the next video, I will try to make the sound better. Meanwhile, it remains to enjoy the relaxing sound of rain in the background. ;)

@lucassaito2842 2 года назад

@@druid7456 Yes, for sure it makes sense. I enjoyed the video and wish you the best from now on if you wish to upload more like this one!

@rv706 2 года назад

11:39 - What is pj? You don't say it and it's not obvious from the context or standard common conventions.

@druid7456 2 года назад

The weight. Normalized one. ("p" indicates probability.) Introduced in preliminary revision (3:26) and commented just past the moment you mentioned (11:57).

@rv706 2 года назад

@@druid7456: I see, thank you. So, this does not cover just the average "geometric" position of n points but more generally the "expected position" of a distribution on the circle.

@rv706 2 года назад

"Both type of means do not coincide" * the two types of means do not coincide

@rv706 2 года назад

RU-vid keeps deleting my comment (probably because I tried to write a link to a pronounciation site). Well, so, what I wrote was: 1) Please write the formulas in bigger fonts. 2) The word "determine" is not pronounced "deter mine", it's pronounced "determyn". 3) The Fréchet guy was French, so it's pronounced more or less "freshé" (the ending T is not heard, and the "ch" is like an English "sh").