370c454c-2813-436f-addc-d91706d8d852 54 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Filling a statistical region [PDF]

t0 1 time :

Prepare the project

Construct the project

xo x f 1150 x xo - (2^* + / :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

sx 0.25 :

xo 0.5 :

N 1000 :

xmin sx :

xmax 1 sx - :

ymin 0 :

ymax 1 :

space_x xmax xmin - (:

PDF x 01 0.0025 range : i 1 x rows range y i el xo x i el f eval : 11 line for x y augment 31 line :

space_y ymax ymin - (:

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y xo x f ≤ (x xmax ≤ (* :

jmGiraud: 2015/12/31

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 6 #808080 13 mat :

fill Pts a b c spec pts :

Domain

Xmin sx 1 sx 0.25 - 22 mat :

Xmax xmax 1 xmax 0.25 - 22 mat :

Xo xo 1 xo 022 mat :

fill 9.514124 1013^* 1.6 1014^* / 2.007961 1011^* 21012^* / . 6 #808080 4.01846 1013^* 81013^* / 1.320522 1014^* 2.5 1014^* / . 6 #808080 6.136739 1013^* 11014^* / 3.910666 1013^* 51014^* / . 6 #808080 6.197768 1013^* 11014^* / 1.298811 1014^* 2.5 1014^* / . 6 #808080 3.095331 1013^* 81013^* / 4.91874 1013^* 51014^* / . 6 #808080 3.832813 1013^* 81013^* / 4.264847 1014^* 11015^* / . 6 #808080 7.585859 1012^* 21013^* / 7.420502 1013^* 11015^* / . 6 #808080 1.933257 1013^* 51013^* / 3.701556 1014^* 11015^* / . 6 #808080 85 mat

Explain things::

These definitions [*] output nested elements. To get the value, Un-nest by indexing [1]

Filling a range of a PDF

PDF fill Xmin Xmax Xo 5 1 sys

1 Random space_x * xmin + 0.456159 11 mat

[*]

1 Random space_y * ymin + 0.078667 11 mat

[*]

1 Random space_x * xmin + 1 el 0.523002

1 Random space_y * ymin + 1 el 0.242885

In Mathcad, rnd(,) outputs a number, that can be fractional. In Smath, we make it fractional by multiplying [in this case]. In Smath, Random(,) creates a vector. We want a number, therefore Random(1)to make things simpler.

1 time t0 - 20.7 s *

Construct/simulate an experimental data set

Noisy data sets are often used to test filter modules. So, here is a typical construction. We just take a smaller magnitude of the noise. It represents a good resolution of the measuring chain that must be at least 10 times more accurate [resolution] than the sampled data. Noise from measurement is due to digital "quantization" whether "quantizaton" is from visual or digital.

Superpose noise on a function data n rnd μ n 11 line : 2 μ Random * 1 - 21 line : x 01 0.01 range : i 1 x rows range y i el xo x i el f eval : rn x rows rnd : 21 line for x y 0.1 rn * + (augment 41 line :

Create an experimental noisy data set

data PDF 2 1 sys

To generate a new noisy experiment, F9

About denoising

"denoising" is a big word doing little/bad or useles. You may doubt finding the function. Just denoising is in some way useless as we want a function. Hint:: from deeper knowledge, a Fourier polynomial would be the best candidate, near exact.

Silent observation

The "Pts" routine acts as a "random compressor". A "random copressor" ? YES, a random compressor ! It fills N random pairs in the critical region. Depending upon the size of the critical region, at constant 'N density', the points will look clear could or dark cloud. Very impressive when applied to partially filled circular shape. Convince yourself: set sx:= 0.375