filtering – Smoothing a dataset

I have a dataset, named data as follows:

data = Transpose[{Table[i, {i, 1, 10, 0.1}], 
    N[Table[Log[i] + RandomReal[0.5], {i, 1, 10, 0.1}]]}];

I need to smooth the data using FFTFilters. I can do this in OrginLab software and the results look like this:

smooth = {{1, 0.55687}, {1.1, 0.57505}, {1.2, 0.59645}, {1.3, 0.62119}, {1.4, 0.64932},
  {1.5, 0.68081}, {1.6, 0.71555}, {1.7, 0.75331}, {1.8, 0.7938}, {1.9, 0.83666},
  {2, 0.88145}, {2.1, 0.92768}, {2.2, 0.97487}, {2.3, 1.02248}, {2.4, 1.07002}, 
  {2.5, 1.11701}, {2.6, 1.16303}, {2.7, 1.20772}, {2.8, 1.25078}, {2.9, 1.29202}, 
  {3, 1.3313}, {3.1, 1.36857}, {3.2, 1.40386}, {3.3, 1.43727}, {3.4, 1.46894},
  {3.5, 1.49905}, {3.6, 1.52781}, {3.7, 1.55543}, {3.8, 1.58213}, {3.9, 1.6081},
  {4, 1.6335}, {4.1, 1.65846}, {4.2, 1.68308}, {4.3, 1.70741}, {4.4, 1.73147}, 
  {4.5, 1.75526}, {4.6, 1.77875}, {4.7, 1.80188}, {4.8, 1.82461}, {4.9, 1.84689}, 
  {5, 1.86868}, {5.1, 1.88997}, {5.2, 1.91075}, {5.3, 1.93106}, {5.4, 1.95092}, 
  {5.5, 1.97041}, {5.6, 1.98959}, {5.7, 2.00855}, {5.8, 2.02737}, {5.9, 2.04612}, 
  {6, 2.06485}, {6.1, 2.08359}, {6.2, 2.10235}, {6.3, 2.12111}, {6.4, 2.13982}, 
  {6.5, 2.15839}, {6.6, 2.17672}, {6.7, 2.19468}, {6.8, 2.21216}, {6.9, 2.229},
  {7, 2.24507}, {7.1, 2.26026}, {7.2, 2.27446}, {7.3, 2.2876}, {7.4, 2.29962},
  {7.5, 2.31049}, {7.6, 2.32023}, {7.7, 2.32886}, {7.8, 2.33644}, {7.9, 2.34304}, 
  {8, 2.34873}, {8.1, 2.3536}, {8.2, 2.35774}, {8.3, 2.36123}, {8.4, 2.36416},
  {8.5, 2.36661}, {8.6, 2.36864}, {8.7, 2.37032}, {8.8, 2.37175}, {8.9, 2.37299}, 
  {9, 2.37417}, {9.1, 2.3754}, {9.2, 2.37684}, {9.3, 2.37869}, {9.4, 2.38116},
  {9.5, 2.38452}, {9.6, 2.38904}, {9.7, 2.39505}, {9.8, 2.40287}, {9.9, 2.41283}, 
  {10, 2.42525}};

ListPlot[{data, smooth}, Joined -> {False, True}]

data and smooth

Now, I tried to do it by Mathematica to get the result similar to that obtained by OriginLab, I did the following:

InverseFourier[LowpassFilter[Fourier[Transpose[data][[2]]], 4]];

But I couldn't smooth out the above dataset.
So help me get rid of it.
Thanks in advance

filtering – Smoothing a dataset

I have a dataset, named data as follows:
data = Transpose [{Table [i, {i, 1, 10, 0.1}],
N [array [Log [i] + RandomReal [0.5], {i, 1, 10, 0.1}]]}];
I need to smooth this data using FFTFilters. I can do this in the OrginLab software and the results are as follows:
{1 0.55687
1.1, 0.57505
1.2,0.59645
1.3, 0.62119
1.4, 0.64932
1.5, 0.68081
1.6, 0.71555
1.7, 0.75331
1.8, 0.7938
1.9, 0.83666
2, 0.88145
2.1, 0.92768
2.2, 0.97487
2.3, 1.02248
2.4, 1.07002
2.5, 1.11701
2.6, 1.16303
2.7, 1.20772
2.8, 1.25078
2.9, 1.29202
3, 1.3313
3.1, 1.36857
3.2, 1.40386
3.3, 1.43727
3.4, 1.46894
3.5, 1.49905
3.6, 1.52781
3.7, 1.55543
3.8, 1.58213
3.9, 1.6081
4, 1.6335
4.1, 1.65846
4.2, 1.68308
4.3, 1.70741
4.4, 1.73147
4.5, 1.75526
4.6, 1.77875
4.7, 1.80188
4.8, 1.82461
4.9, 1.84689
5, 1.86868
5.1, 1.88997
5.2, 1.91075
5.3, 1.93106
5.4, ​​1.95092
5.5, 1.97041
5.6, 1.98959
5.7, 2.00855
5.8, 2.02737
5.9, 2.04612
6, 2.06485
6.1, 2.08359
6.2, 2.10235
6.3, 2.12111
6.4, 2.13982
6.5, 2.15839
6.6, 2.17672
6.7, 2.19468
6.8, 2.21216
6.9, 2.229
7, 2.24507
7.1, 2.26026
7.2, 2.27446
7.3, 2.2876
7.4, 2.29962
7.5, 2.31049
7.6, 2.32023
7.7, 2.32886
7.8, 2.33644
7.9, 2.34304
8, 2.34873
8.1, 2.3536
8.2, 2.35774
8.3, 2.36123
8.4, 2.36416
8.5, 2.36661
8.6, 2.36864
8.7, 2.37032
8.8, 2.37175
8.9, 2.37299
9, 2.37417
9.1, 2.3754
9.2, 2.37684
9.3, 2.37869
9.4, 2.38116
9.5, 2.38452
9.6, 2.38904
9.7, 2.39505
9.8, 2.40287
9.9, 2.41283
10, 2.42525}

Now, I tried to do it in mathematics to get the result similar to that obtained by OriginLab, I did the following operations:
InverseFourier [LowpassFilter [Fourier [Transpose [data] [[2]], 4]];
But I couldn't smmothen the above dataset.
So please stop by to get rid of it.
Thanks in advance

pde – Application of the heat equation smoothing proof (Evans) to the wave equation smoothing proof

Theorem 8 from the book of Evans Chapter 2.3 Heat equation:

Assume $ u in C ^ 2_1 (U_T) $ solves the heat equation $ U_T $. so $ u in C ^ infty (U_T) $.

Here, the heat equation is $ u_t – Delta u = f $ in $ Bbb R ^ n times (0, T) $ with $ u = 0 $ sure $ Bbb R ^ n times {t = 0} $ or $ u in C ^ 2_1 ( Bbb R ^ n times (0, T)) bigcap C ( Bbb times (0, T)) $.

Now, I would like to prove an analog of Theorem 8 for the wave equation:

Assume $ u (x; t) $ is a solution to the wave equation $ u_ {tt} – Delta u = f (x; t) $. Let $ f (x; t) in C ^ infty_ {x, t} ( bar V_T) $ and $ u (x; t) = C ^ {2,1} _ {x, t} ( bar V_t) $. (It looks like the equal sign should be an "in" sign, but I'm not sure.) Prove that $ u (x; t) in C ^ infty (V_T) $.

I believe that, to prove this analog, I have to adapt the proof of Theorem 8 that Evans made to him. I don't know if I should change the use of evidence of cylindrical means by spherical means. I understand Evans' proof for the regularity of the heat equation, but I'm completely lost on how to prove the above regularity for the wave equation. What would be the proof?

reference request – Smoothing of $ f sqrt {1 + g ^ 2} $ and $ fg sqrt {1 + g ^ 2} $ for the functions $ f $ and $ g $ such as $ f, fg, fg ^ $ 2 are smooth

Assume that $ f $ and $ g $ are functions of $ mathbb R $ at $ mathbb R $ so that the functions $ f, fg, fg ^ 2 $ are smooth, that is to say $ C ^ infty ( mathbb R) $. Does it necessarily follow that
the functions $ f sqrt {1 + g ^ 2} $ and $ fg sqrt {1 + g ^ 2} $ are smooth?

Of course, the problem here is that the function $ g $ must not be smooth, or even continuous, at function zeros $ f $.

We can also note that the continuity of the
the functions $ f sqrt {1 + g ^ 2} $ and $ fg sqrt {1 + g ^ 2} $ (to zeros of $ f $ and therefore everywhere) easily results from inequalities $ | f sqrt {1 + g ^ 2} | le | f | + | fg | $ and $ | fg sqrt {1 + g ^ 2} | le | fg | + | fg ^ 2 | $.

quaternion – Smoothing from Slerp to Slerp

p0 ------- p1 ------- p2

I am trying to successfully slip between 2 points (p0-p1; p1-p2). But the transition between two slerped angles (p1_in-p1_out) doesn't match, and that sounds shocking

Video of his appearance:

the green squares are points, the position and the angle between each are interpolated.

The problem is most visible at the second 6.00

I have tried hermitite splines, although this reduces the effect, it is still visible.

I tried to convert Catmull-Rom to use it with quaternions by simply adding the 4th index to the vector functions. This works perfectly, except on certain occasions, where some are interpolated from the wrong (long) "path", which switches the view between these points. I tried to line up, didn't help.

The other SO questions related to slerp do not ask this question, and I could not get usable answers from some questions on 4-point slerps.

Is this a common problem, do I supervise something?

Thank you

machine learning – What type of bigram probability smoothing is this?

I hope this is not off topic but I need to understand this example. Given the corpus 12 1 13 12 15 234 2526 and smoothing factor of k=1. The example does the following:

Consider OOV words (excluding vocabulary) and assign them a value zero times, after that k=1 is added to the time of appearance of each word, to avoid zero probabilities. Thus, the result of smoothing the probability of bigrams will be:

$ P (1 | 12) = (1 + k) / (2 + 2 + 6 * k) = $ 0.2
$ P (15 | 12) = (1 + k) / (2 + 2 + 6 * k) = $ 0.2
$ P (13 | 1) = (1 + k) / (2 + 6 * k) = $ 0.25
$ P (12 | 13) = (1 + k) / (2 + 6 * k) = $ 0.25
$ P (234 | 15) = (1 + k) / (2 + 6 * k) = $ 0.25
$ P (2526 | 234) = (1 + k) / (2 + 6 * k) = $ 0.25

My question is, what kind of smoothing is this? shouldn't it be like that? $ P (1 | 12) = (1 + k) / (2 + 6 * k) = $ 0.25
In addition, it also says "If OOV words appear, you must use smoothing to return a value; P $ (234 | 12) = 1 / ((2/7) * 6 + 6) = $ 0.1296"

PS: I take this example in a small section of the translated version of this Chinese web page, it just explains a code implementation.

How the term dispersive in PDE plays the role of smoothing

In the KdV equation,
$$ u_t + u_ {xxx} -6uu_x = 0, $$
$ uu_x $ is the nonlinear term that causes the explosion, and $ u_ {xxx} $ is the dispersive term, I wonder how the dispersive term smoothes the solution, otherwise the solitary wave solution will not exist.

pde – Instant smoothing effect on the sphere

the instant smoothing effect of the equation of heat is the property that, if $ f in L ^ infty ( mathbb R ^ d) $, then the solution for
$$ begin {cases} partial_t u = Delta u, & t> 0 \ u (0, x) = f (x), & x in mathbb R ^ d, end {cases} $$
is such that $ u (t, cdot) in C ^ infty $ for everyone $ t> 0 $.


Question. Assume that $ omega in L ^ infty ( mathbb R ^ d; mathbb S ^ {d-1}) $; in particular, it is defined up to a zero measurement set. Are there $ omega (t, x) $ such as

  1. $ omega (t, x) in mathbb S ^ {d-1} $ for everyone $ t> 0, x in mathbb R ^ d $;
  2. $ omega (t, cdot) in C ^ infty ( mathbb R ^ d; mathbb S ^ {d-1}) $;
  3. $ displaystyle lim_ {t downarrow 0} int _ { mathbb R ^ d} lvert omega (t, x) – omega (x) rvert ^ 2 , dx = 0 $ ?

This question appeared in the context of this answer, step 2. There, I tried to give a solution by solving
$$ begin {cases}
partial_t eta = Delta eta, & t> 0, \
eta (0, x) = omega (x), & x in mathbb R ^ d,
end {cases}
$$

who produces $ eta colon (0, infty) times mathbb R ^ d to mathbb R ^ d $then taking
$$
omega (t, x): = frac { eta (t, x)} { lvert eta (t, x) rvert}. $$

However, I am not too sure that it is correct. Although it is true that $ eta $ is smooth (see for example Evans "Partial Differential Equations", 2nd Edition, Theorem 8 page 59), I'm not sure that $ eta (t, x) $ 0 for everyone $ t> 0 $.

preferences – Application-specific setting for "Use font smoothing when available"?

Historically, this setting (System Preferences -> General) controls
sub-pixel font anti-aliasing.

Since Mojave depreciates the rendering of subpixels, he controls
The anti-aliasing in grayscale is applied. That is, the text looks bolder
when checked.

Is it possible to modify this parameter by application (defaults write stuff), especially for the menu bar?

Flash smoothing. (Flash Smoothing) [on hold]

Hi guys, I know that sounds like a pretty stupid question, but I wondered how I could literally dim my flash. I take pictures at night in a prayer group, and it's usually quite dark and people are moving around a lot, I need a very fast shutter speed, but for that, j & rsquo; I need a lot of light, that is to say that I have to use the flash, but the flash may scatter people closed eyes praying. I have therefore looked for broadcasters, there is even q is an octabox for camera flash, but I would love to hear from you. Is it possible for me to make this flash weaker so as not to burst with joy in the eyes? At home, I took a bunch of paper and I put it in front of the flash kkk. Are there any other professional equipment that can offer me this? thank you for

Google Translate:

Hi people, I know it seems like it's more important, but I wanted to know how to literally get some light in the flash light, on the pictures of a group of night prayers, and in general, it's enough Moving, bright enough, need a fast shutter speed, be sure to be precise in this light, whether it's necessary to pass through the flash, by flash or by flash, or disperse the people with whom you are dating, praying, then looking for q is an octabox for camera flash, would like to know more about you, is there any way to say that the flash is no longer difficult to do so much against people? In the house of Picked up, a sheet of paper mounted in front of the flash kkk, is there a more professional equipment that I offer? thanks for understanding