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/sci/ - Science & Math

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Anonymous Fri Sep 11 03:12:42 2020 No.12107832 [Reply] [Original]

I made a way to smooth functions. Here is a polynomial being smoothed:
https://www.desmos.com/calculator/cpvrudoiz7

What applications does this have?

>>	Anonymous Fri Sep 11 03:34:46 2020 No.12107857 Here's a crude smoothing of the step function https://www.wolframalpha.com/input/?i=0.5%28%2810-+x%29Heaviside%28-10%2B+x%29+%2B+%2810%2B+x%29*Heaviside%2810+%2B+x%29%29%2F10

>>	Anonymous Fri Sep 11 03:46:45 2020 No.12107870 >>12107832 sine wave being smoothed: https://www.desmos.com/calculator/il99dievkl

>>	Anonymous Fri Sep 11 03:51:48 2020 No.12107875 >>12107832 Define smooth

>>	Anonymous Fri Sep 11 03:52:25 2020 No.12107876 >>12107875 make things less jagged

>>	Anonymous Fri Sep 11 03:54:39 2020 No.12107881 >>12107876 Try in terms of differentiability. That would also provide light on your question.

>>	Anonymous Fri Sep 11 03:56:38 2020 No.12107883 >>12107881 My technique potentially works to make nondifferentiable functions like abs() continuous, haven't tried though

>>	Anonymous Fri Sep 11 03:59:06 2020 No.12107886 >>12107883 abs() is already continuous... it is also differentiable anywhere except its minima and maxima

>>	Anonymous Fri Sep 11 04:01:06 2020 No.12107889 >>12107886 I mean make it smooth

>>	Anonymous Fri Sep 11 04:02:54 2020 No.12107896 you might be mixing up continuity and differentiability.

>>	Anonymous Fri Sep 11 04:04:19 2020 No.12107900 >>12107889 Can you explain what smooth means? Like literally... saying less jagged does not really provide clarity. Either define smooth mathematically, or show us geometri ally.

>>	Anonymous Fri Sep 11 04:04:23 2020 No.12107901 >>12107876 Quantify "less jagged"

>>	Anonymous Fri Sep 11 04:06:21 2020 No.12107905 >>12107900 I'm talking about smoothing like filters like the savgol filter in scipy. The input is jagged and the output is smoother

>>	Anonymous Fri Sep 11 04:07:29 2020 No.12107907 >>12107900 I'm talking about smoothing like filters like the Savitzky-golay filter, but mine works on functions. The input is jagged and the output is smoother

>>	Anonymous Fri Sep 11 04:07:40 2020 No.12107908 >>12107905 Variable memory input XOR dynamic speed translator.

>>	Anonymous Fri Sep 11 04:08:21 2020 No.12107909 >>12107832 smooth = infinitely differentiable you're misusing standard terminology

>>	Anonymous Fri Sep 11 04:10:00 2020 No.12107912 >>12107909 redefining it. the death of language's ego

>>	Anonymous Fri Sep 11 04:11:47 2020 No.12107915 File: 27 KB, 640x480, images (40).jpg [View same] [iqdb] [saucenao] [google] >>12107875 no sharp

>>	Anonymous Fri Sep 11 04:13:33 2020 No.12107920 >>12107915 polynomial fluid dynamics and limits

>>	Anonymous Fri Sep 11 04:19:29 2020 No.12107932 smoothing a fifth degree polynomial https://www.desmos.com/calculator/nxvuwfk1ax

>>	Anonymous Fri Sep 11 04:30:05 2020 No.12107951 >>12107932 I've found another smoothing technique, check this out https://www.desmos.com/calculator/a0d3owy2fp

>>	Anonymous Fri Sep 11 04:34:20 2020 No.12107960 File: 109 KB, 1200x762, 1200px-WeierstrassFunction.svg[1].png [View same] [iqdb] [saucenao] [google] >>12107951 looks similar, it flips the function for delta>1 I tried to smooth this function but I couldn't take the integral

>>	Anonymous Fri Sep 11 04:36:22 2020 No.12107962 How does it work for data with stochastic nose, and is it computationally fast?

>>	Anonymous Fri Sep 11 04:38:07 2020 No.12107964 >>12107912 based

>>	Anonymous Fri Sep 11 04:43:40 2020 No.12107971 >>12107962 It can smooth any polynomial. If you fit a polynomial to the data then it can smooth it. It's fast for polynomials, it can't compute complicated functions though, because it involves integration

>>	Anonymous Fri Sep 11 04:58:28 2020 No.12107995 >>12107971 Sorry for asking such a n00b question but how does one make a polynomial that isn't smooth to begin with?

>>	Anonymous Fri Sep 11 05:02:11 2020 No.12107998 >>12107995 For example if you make a lagrange polynomial it will be extreme because it will overfit the data, but a smoothing method will keep the values in range

>>	Anonymous Fri Sep 11 05:04:43 2020 No.12108002 >>12107998 Then why the fuck would I bother making a lagrange polynomial in the first fucking place?

>>	Anonymous Fri Sep 11 05:07:53 2020 No.12108007 >>12108002 That's not an application, he just asked how a polynomial could not be smooth

>>	Anonymous Fri Sep 11 05:25:59 2020 No.12108028 File: 12 KB, 378x248, index.png [View same] [iqdb] [saucenao] [google] Smoothed abs(x)

>>	Anonymous Fri Sep 11 05:27:28 2020 No.12108031 File: 12 KB, 378x248, index.png [View same] [iqdb] [saucenao] [google] abs(x)

>>	Anonymous Fri Sep 11 05:30:09 2020 No.12108035 File: 9 KB, 215x234, 1585491076230.jpg [View same] [iqdb] [saucenao] [google] >>12108031 That's a cool hat bruh.

>>	Anonymous Fri Sep 11 05:36:43 2020 No.12108042 File: 23 KB, 380x248, a.png [View same] [iqdb] [saucenao] [google] >>12107960 Smoothed it

>>	Anonymous Fri Sep 11 05:40:24 2020 No.12108049 >>12108042 Looks like an awesome snake to me if it was only made out of its spine and the patterns on the skin, but without the actual skin.

>>	Anonymous Fri Sep 11 05:52:57 2020 No.12108066 File: 20 KB, 380x248, index.png [View same] [iqdb] [saucenao] [google]

>>	Anonymous Fri Sep 11 06:00:05 2020 No.12108075 File: 21 KB, 380x248, index.png [View same] [iqdb] [saucenao] [google]

>>	Anonymous Fri Sep 11 06:01:52 2020 No.12108078 If you want to try it, take the integral of f(x) from x-delta to x+delta several times. It works best if you do it infinitely many times.

>>	Anonymous Fri Sep 11 06:07:23 2020 No.12108081 >>12108078 I agree, everything is better when you do it infinitely many times. I remember all those things I did infinite many times. Ah, those were the categorical imperatives of my exploration.

>>	Anonymous Fri Sep 11 07:01:08 2020 No.12108135 File: 25 KB, 386x248, a.png [View same] [iqdb] [saucenao] [google]

>>	Anonymous Fri Sep 11 07:37:11 2020 No.12108189 File: 26 KB, 386x248, a.png [View same] [iqdb] [saucenao] [google]

>>	Anonymous Fri Sep 11 09:23:03 2020 No.12108320 >>12107832 does it work for x(x+1)/2

>>	Anonymous Fri Sep 11 16:53:56 2020 No.12109447 >>12108320 Yep. Quadratic

>>	Anonymous Fri Sep 11 17:38:33 2020 No.12109538 >>12109447 >>12108320 tested it, it doesn't work

>>	Anonymous Fri Sep 11 17:42:34 2020 No.12109546 it works, but x(x+1)/2 is smooth so it doesn't change https://www.desmos.com/calculator/pzud5aw3jo

>>	Anonymous Fri Sep 11 17:45:29 2020 No.12109548 it works by interpolating nearby values, so a parabola doesn't change by much

>>	Anonymous Fri Sep 11 20:09:39 2020 No.12109881 >>12107832 Signal denoising kek

>>	Anonymous Fri Sep 11 20:18:36 2020 No.12109909 Study low pass filters. You can probably tune one to smooth however you want.

Anonymous Fri Sep 11 20:23:53 2020 No.12109919

>>12109909
samefagging to say: I would come up with a metric for smoothness because your idea of "smooth" is still not well defined. Some people have asked if you mean removing discontinuities or places where it's not continuously differentiable, and you don't mean that. Try figuring out what you mean by smooth and testing different methods.

I'd look at the fourier transform before and after smoothing and see if that shows the difference you're noticing in the high frequency end.

>>	Anonymous Fri Sep 11 20:36:02 2020 No.12109963 >>12107951 lol, genius. i simplified your discovery further. can we split the nobel prize? https://www.desmos.com/calculator/8rq4idll8m

>>	Anonymous Fri Sep 11 20:39:21 2020 No.12109977 Why dont you just do it better, like me https://www.desmos.com/calculator/10wkzfwy8h

>>	Anonymous Fri Sep 11 20:41:44 2020 No.12109985 What the fuck is this thread

>>	Anonymous Fri Sep 11 20:43:14 2020 No.12109989 >>12109985 Methods of smoothing

>>	Anonymous Fri Sep 11 20:43:25 2020 No.12109990 >>12109985 integropolynomial interpolation

>>	Anonymous Fri Sep 11 20:47:49 2020 No.12110008 is this smooth? https://www.desmos.com/calculator/mnugjqauzc

>>	Anonymous Fri Sep 11 20:55:04 2020 No.12110029 i call this art "smooth criminal" https://www.desmos.com/calculator/j0ixwyhihw

>>	Anonymous Fri Sep 11 21:21:51 2020 No.12110092 I submitted it to vixra. Here's the pdf if you want to read: https://drive.google.com/file/d/1VzzvxT5XDhBIAuPjbsy_c9ItKjbVvJ6R/view?usp=sharing

>>	Anonymous Sat Sep 12 01:23:55 2020 No.12110520 >>12107998 What the fuck are you even talking about? A Lagrange polynomial is smooth and can't be any smoother. Does your technique make it infinitely+1 differentiable?

Anonymous Sat Sep 12 10:03:15 2020 No.12111383
File: 14 KB, 598x227, 1590857525204.png [View same] [iqdb] [saucenao] [google]

>>12107832
That has been done before and is a standard trick in analysis. You usually just do the same trick as with kernel smoothers from statistics but instead of a weighted sum you have a weighted integral.
One aplication is often when you want to show that some subset of smooth functions is dense in some Banachspace.
For example to proof that the set of smooth functions with compact support is dense in L^p you usually use an approximation of the L^p function by smoothing it with such a kernel.

Anonymous Sat Sep 12 10:15:20 2020 No.12111413
File: 63 KB, 1000x885, 1481839809871.png [View same] [iqdb] [saucenao] [google]

>>12110092
>>12111383
to add on this:
It looks like what you are doing is exactly what i described where your kernel function k(x,y) is 1 when |x-y|<delta and 0 otherwise. you seem to have also failed to mention, that you devide the result by 2delta or else the resulting function should go to 0 as delta goes to zero.

>>	Anonymous Sat Sep 12 10:20:54 2020 No.12111429 >reinventing the wheel >presenting it as a breakthrough >redefining words >paper on vixra We have reached peak /sci/

>>	Anonymous Sat Sep 12 10:25:51 2020 No.12111447 >>12107886 Sorry if I'm being retarded, but abs(x) is not continuous, because its derivative is not continuous. It goes from -1 to 0 to 1 literally instantly.

>>	Anonymous Sat Sep 12 10:26:32 2020 No.12111453 >>12111413 Yeah you divide it by 2*delta every time you integrate, I forgot to mention that. A google search didn't show the method. At least my paper will show up when someone searches

>>	Anonymous Sat Sep 12 10:27:27 2020 No.12111456 >>12111447 \|x\| is continuous, what are you talking about

>>	Anonymous Sat Sep 12 10:28:19 2020 No.12111457 >>12111447 That's sign(x) not abs(x)

>>	Anonymous Sat Sep 12 10:31:31 2020 No.12111470 >>12111447 it is continuous but not continuously differentiable

>>	Anonymous Sat Sep 12 10:35:28 2020 No.12111479 >>12110520 Sharpness is caused by high curvature. Lagrange polynomials tend to have high curvature

>>	Anonymous Sat Sep 12 10:40:10 2020 No.12111488 >inefficient approximation to to the Gauß filter Revolutionary.

>>	Anonymous Sat Sep 12 10:46:10 2020 No.12111499 >>12110520 OP doesn't what smoothness means. ignore it

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