Data set

7f4b16d3-d644-4e76-a763-7239283963c5 109 15 12 decimal

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

Fourier DCT-II & IDCT-II

f dct2 DCT-II -> Discrete Cosine Transform n f length : g 1 el 1 n / sqrt f j 1 + el j 0 n 1 - sum * : k 1 n 1 - range g k 1 + el 2 n / sqrt f j 1 + el 2 j * 1 + (k * π * 2 n * / cos * (j 0 n 1 - sum * eval : for g n el 1 n / sqrt f j 1 + el 2 j * 1 + (n 1 - (* π * 2 n * / cos * (j 0 n 1 - sum * eval : g 61 line :

g idct2 IDCT-II Inverse Discrete Cosine IF k 0 ≡ 12 sqrt / 1 if : n g length : m g length 1 - : j n n m + range f j m - el 2 n / sqrt g k 1 + el IF (* 2 j * 1 + (k * π * 2 n * / cos * (k 0 n 1 - sum * eval : for f reverse 61 line :

Matlab DCT-II algo not same.

Data set

data_1 1 0.132 - 2 0.11 - 3 0.045 4 0.0045 5 0.138 - 6 0.467 7 0.5 8 0.78 9 1.342 10 0.993 11 1.105 12 1.119 13 1.18 14 1.143 15 0.655 16 1.008 17 0.771 18 1.208 19 0.938 20 0.783 21 0.734 22 0.763 23 1.044 24 0.754 25 0.797 252 mat :

data_2 26 1.086 27 0.426 28 0.563 29 0.814 30 0.665 31 0.561 - 32 0.429 - 33 0.187 34 0.153 - 35 0.422 - 36 0.127 - 37 0.018 - 38 0.247 - 39 0.422 - 40 0.738 - 41 0.711 - 42 0.863 - 43 0.769 - 44 0.485 - 45 0.79 - 46 0.737 - 47 0.708 - 48 0.965 - 49 0.899 - 50 0.947 - 252 mat :

data_3 51 0.871 - 52 1.138 - 53 1.191 - 54 0.046 - 55 1.016 - 56 1.275 - 57 0.617 - 58 0.908 - 59 1.09 - 60 0.972 - 61 0.594 - 62 0.5 - 63 0.346 - 64 0.191 - 65 0.177 - 66 0.362 - 67 0.547 - 68 0.455 69 0.417 70 0.129 - 71 0.352 72 0.906 73 0.386 74 0.047 - 75 0.214 252 mat :

data_4 76 0.309 77 0.615 78 0.846 79 0.698 80 0.819 81 1.113 82 1.042 83 1.114 84 0.653 85 0.667 86 0.657 87 0.846 88 0.547 89 0.053 90 1.244 91 1.046 92 0.793 93 0.383 94 0.335 95 0.272 96 0.537 97 0.248 98 0.67 99 0.226 - 100 0.208 252 mat :

XY data_1 data_2 data_3 data_4 stack :

X XY 1 col :

Y XY 2 col :

t0 1 time :

m XY rows : 100

Low 10 : High 0 : 13 mat

1. Set the window width

j 1 m range j Low ≤ (m High - j ≤ (¤ W j el 1 : 0 if for W 21 line

Project note: Smath smooths this data set in reasonable time. The window width 10 results in good smoothing, despite this abusive noisy data set

Y Y 1 m High - 11 submatrix :

X X 1 m High - 11 submatrix :

numeric Fx Y dct2 eval :

2. Enter the Fourier domain

3. Weight the dct2 ...... 4. Recover from idct2

U window weight Fx i 1 Fx rows range F i el W i el Fx i el * eval : for F 31 line :

Xf U idct2 eval :

dctII X Xf augment :

1 time t0 - 12 s *

up 2 : window height ≡

Stem Module

Spectrum stem plot for QuickPlot unicolor data Stem stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Pre-define Unit step x Φ 1 x sign + up / (:

Unit pulse:width/locate x p Φ 1 x p - sign + up / (:

α 1 - :

β 11 :

t f t α - Φ t β - Φ - :

L H N xd start/end @ L..H U 0 : dx H L - N 1 - / : i 1 N range U i el L dx - (dx i * + : for U 51 line :

L 0 : H 10 : N 11 : U L H N xd : 41 line

p i 1 U rows range vy i el U i el f : for U vy augment 21 line :

window p Stem :

p 01112131415161718191101112 mat

Fourier DCT-II window smoothing

XY dctII window 3 1 sys

Variable smoothing from Fourier "freaked" Budapest 39

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

XY data_1 data_2 data_3 data_4 stack :

data XY :

band 17 :

xy XY band 100 band - 1 + 12 submatrix :

xb xy 1 col : yb xy 2 col : N xy rows : 31 line

a b c spec . 10 black 13 mat : pts data a b c spec pts : 21 line

Prepare B39 Δx xb N el xb 1 el - (N / : ω 2 π N / * : u λ u xb 1 el - (ω Δx / * : 31 line

Fourier polynomial coefficients for B39 n 0 N range A n 1 + el 2 N 3 - / yb k 1 + el n k * ω * cos * k 0 N 1 - sum * eval : B n 1 + el 2 N 3 - / yb k 1 + el n k * ω * sin * k 0 N 1 - sum * eval : 21 line for

w 2 : length of B39 ≡

u B39 A 1 el 2 / A n 1 + el u λ n * cos * B n 1 + el u λ n * sin * + (n 1 w sum + 11 line :

fit x 25 - 150 24.75 - range : i 1 x length range v i el x i el B39 eval : 11 line for x v augment 31 line :

Fourier polynomial smoothing to periodic data set

pts fit 2 1 sys

The Smath Gaussian 'ksmmooth'

Data set

data_1 1 0.132 - 2 0.11 - 3 0.045 4 0.0045 5 0.138 - 6 0.467 7 0.5 8 0.78 9 1.342 10 0.993 11 1.105 12 1.119 13 1.18 14 1.143 15 0.655 16 1.008 17 0.771 18 1.208 19 0.938 20 0.783 21 0.734 22 0.763 23 1.044 24 0.754 25 0.797 252 mat :

data_2 26 1.086 27 0.426 28 0.563 29 0.814 30 0.665 31 0.561 - 32 0.429 - 33 0.187 34 0.153 - 35 0.422 - 36 0.127 - 37 0.018 - 38 0.247 - 39 0.422 - 40 0.738 - 41 0.711 - 42 0.863 - 43 0.769 - 44 0.485 - 45 0.79 - 46 0.737 - 47 0.708 - 48 0.965 - 49 0.899 - 50 0.947 - 252 mat :

data_3 51 0.871 - 52 1.138 - 53 1.191 - 54 0.046 - 55 1.016 - 56 1.275 - 57 0.617 - 58 0.908 - 59 1.09 - 60 0.972 - 61 0.594 - 62 0.5 - 63 0.346 - 64 0.191 - 65 0.177 - 66 0.362 - 67 0.547 - 68 0.455 69 0.417 70 0.129 - 71 0.352 72 0.906 73 0.386 74 0.047 - 75 0.214 252 mat :

data_4 76 0.309 77 0.615 78 0.846 79 0.698 80 0.819 81 1.113 82 1.042 83 1.114 84 0.653 85 0.667 86 0.657 87 0.846 88 0.547 89 0.053 90 1.244 91 1.046 92 0.793 93 0.383 94 0.335 95 0.272 96 0.537 97 0.248 98 0.67 99 0.226 - 100 0.208 252 mat :

XY data_1 data_2 data_3 data_4 stack :

X XY 1 col :

Y XY 2 col :

ksmooth # plotG

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

σ 0.3708158988058244 :

The Gaussian ksmooth

d K 12 π * sqrt σ * / d 2^2 σ 2^* / - exp * :

j X Y Ksmooth μ j 11 line : X i el X μ el - bw / K Y i el * (i 1 X rows sum X i el X μ el - bw / K (i 1 X rows sum / 21 line :

Gaussian Ksmooth Kernel

x K

W j 1 X rows range s j el j X Y Ksmooth eval : for s 21 line :

X XY 1 col : Y XY 2 col : Yo XY 2 col : 31 line

bw 15 : bandwidth ≡ XW X W augment : vx X : vy XW 2 col : 41 line

XW vx vy . 11 red plotG : data X Yo + 7 black plotG : 21 line

spline join/mesh the smoothed data set Cfit t 0100 0.5 range : i 1 t rows range v i el vx vy t i el cinterp eval : for t v augment 31 line :

collect plot plot Cfit data XW 31 sys :

plot

The Smath Fourier WINDOW 'Filter'

Data set

data_1 1 0.132 - 2 0.11 - 3 0.045 4 0.0045 5 0.138 - 6 0.467 7 0.5 8 0.78 9 1.342 10 0.993 11 1.105 12 1.119 13 1.18 14 1.143 15 0.655 16 1.008 17 0.771 18 1.208 19 0.938 20 0.783 21 0.734 22 0.763 23 1.044 24 0.754 25 0.797 252 mat :

data_2 26 1.086 27 0.426 28 0.563 29 0.814 30 0.665 31 0.561 - 32 0.429 - 33 0.187 34 0.153 - 35 0.422 - 36 0.127 - 37 0.018 - 38 0.247 - 39 0.422 - 40 0.738 - 41 0.711 - 42 0.863 - 43 0.769 - 44 0.485 - 45 0.79 - 46 0.737 - 47 0.708 - 48 0.965 - 49 0.899 - 50 0.947 - 252 mat :

data_3 51 0.871 - 52 1.138 - 53 1.191 - 54 0.046 - 55 1.016 - 56 1.275 - 57 0.617 - 58 0.908 - 59 1.09 - 60 0.972 - 61 0.594 - 62 0.5 - 63 0.346 - 64 0.191 - 65 0.177 - 66 0.362 - 67 0.547 - 68 0.455 69 0.417 70 0.129 - 71 0.352 72 0.906 73 0.386 74 0.047 - 75 0.214 252 mat :

data_4 76 0.309 77 0.615 78 0.846 79 0.698 80 0.819 81 1.113 82 1.042 83 1.114 84 0.653 85 0.667 86 0.657 87 0.846 88 0.547 89 0.053 90 1.244 91 1.046 92 0.793 93 0.383 94 0.335 95 0.272 96 0.537 97 0.248 98 0.67 99 0.226 - 100 0.208 252 mat :

XY data_1 data_2 data_3 data_4 stack :

XY XY 110012 submatrix :

X XY 1 col :

Y XY 2 col :

Smath/User redefinition v n cfft v n al_fftc1d :

m Y rows :

v n icfft v n al_fftc1dinv (:

r X Y Slope eval : 0.001165 -

dd Y X r * - : D dd m cfft : w 3 : window size ≡ 31 line

Low w : High w : 13 mat

dw mwabs dw mat2sys abs sys2mat ≡

Create the Fourier window j 1 m range j Low ≤ (m High - j ≤ (¤ DW j el D j el : 11 line 0 if for DW 21 line

dw DW m icfft eval X r * + :

vy dw mwabs :

vy1 vy 13311 submatrix : vy2 vy 346811 submatrix : vy3 vy 6910011 submatrix : 31 line

vy vy1 vy2 - vy3 stack :

Though I don't understand the segmentation, denoising results. Segments depend upon 'w'.

Rate smoothing

Filter X vy augment :

________ function ........ points ===================== 1. B39 ______ 2. linfitCheby ______ 3. DCT-II ...... 4, 5 ~ = 4. ksmooth ...... 5. WINDOW ......

Filter XY 2 1 sys

................NEXT................ Fourier DCT-II [Data linfitCheby].sm Fourier DCT-II [Data Genfit Minimize].sm

Conclude the project: deceptively simple

initial refined 1 0.09 0.06 - 31 mat 0.942 0.094 0.06 - 31 mat 22 mat

The five proposals above are not futile as they may apply otherwise in closely similar projects. However, none of the choices were wise. By inspection, the data set has sine format, it may not be of amplitude '1', it may have some phase shift. We just need to guess wisely for the frequency. Now, we can construct a model function from these assumptions and guess wisely from trial/error. Once the guesses are in target, we get a final fit in one shot "Genfit Minimize". We conserve β below

β 1 0.094248 031 mat :

x β f β 1 el β 2 el x * β 3 el + sin * :

FIT range, populate U 10 - 120 9.5 - range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 41 line :

... that's not all:

Methods based on the SSE [SumSquareError] are affected by the noise distribution. Thus, in this example, you may prefer to survive with a NON-Minimzed fit/smooth. The "Quizer" didn't give the original.

data X Y . 12 black plotG :

1 0.094248 031 mat

data FIT x B39 XY 4 1 sys

the data set spans over 3π [likely]

3 π * 100 / 0.094248

Conclusion

No method will fit/smooth this data set. Noise is non sense, much too big and not random, just crappy invention. Easy to prove ... visit Fourier DCT-II [Reconciliation B39]

Stem Construction Utility

Core component of the stem style P v replicate k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 31 line :

Stem construction vy stem N vy length : i 1 N range q i el 3 : q N el 2 : 21 line for 11 line i 1 N range v i el i : for i 1 N range S 3 i * el vy i el : for u q v replicate : V u S augment : V V 3 V rows 12 submatrix : 71 line :

NoiseRatio i 1 X rows range y i el X i el sin eval : r i el Y i el y i el / : 21 line for X r augment 21 line :

Collect the project N 100 : vy NoiseRatio 2 col : SNR vy stem : 31 line

snr SNR 55 - 22502250 - 22 mat 41 sys :

snr

XY XY 148612 submatrix :

X XY 1 col :

Y XY 2 col :

Smath/User redefinition v n cfft v n al_fftc1d :

m Y rows : 73

v n icfft v n al_fftc1dinv (:

r X Y Slope eval : 0.002989 -

dd Y X r * - : D dd m cfft : w 2 : window size ≡ 31 line

Low w 1 + : High w 1 - : 13 mat

dw mwabs dw mat2sys abs sys2mat ≡

Create the Fourier window j 1 m range j Low ≤ (m High - j ≤ (¤ DW j el D j el : 11 line 0 if for DW 21 line

dw DW m icfft eval X r * + :

sn dw mwabs :

sn1 sn 12011 submatrix : sn2 sn 215511 submatrix : sn3 sn 567311 submatrix : 31 line

sn sn1 sn2 - sn3 stack :

Filter X sn augment :

data Filter XY 3 1 sys