XY raw data [100]

a5baa0f5-5ceb-4f1c-938e-1e29616fdfb2 21 4 5 decimal

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

Smoothing by convolving ksmooth

XY raw data [100]

XY 0.5 - 0.105 0.48 - 0.115 0.46 - 0.126 0.44 - 0.138 0.42 - 0.151 0.4 - 0.165 0.38 - 0.181 0.36 - 0.198 0.34 - 0.217 0.32 - 0.238 0.3 - 0.261 0.28 - 0.286 0.26 - 0.313 0.24 - 0.343 0.22 - 0.375 0.2 - 0.411 0.18 - 0.45 0.16 - 0.493 0.14 - 0.539 0.12 - 0.59 0.1 - 0.645 0.08 - 0.705 0.06 - 0.77 0.04 - 0.841 0.02 - 0.918 01 0.02 1.09 0.04 1.186 0.06 1.288 0.08 1.398 0.1 1.515 0.12 1.638 0.14 1.767 0.16 1.902 0.18 2.043 0.2 2.186 0.22 2.333 0.24 2.48 0.26 2.627 0.28 2.773 0.3 2.914 0.32 3.05 0.34 3.18 0.36 3.326 0.38 3.415 0.4 3.52 0.42 3.62 0.44 3.73 0.46 3.78 0.48 3.848 0.5 3.903 0.52 3.971 0.54 4.023 0.56 4.072 0.58 4.117 0.6 4.16 0.62 4.178 0.64 4.208 0.66 4.235 0.68 4.259 0.7 4.281 0.72 4.301 0.74 4.319 0.76 4.338 0.78 4.351 0.8 4.362 0.82 4.377 0.84 4.387 0.86 4.4 0.88 4.407 0.9 4.419 0.92 4.424 0.94 4.436 0.96 4.444 0.98 4.451 1 4.457 1.02 4.463 1.04 4.468 1.06 4.475 1.08 4.48 1.1 4.484 1.12 4.489 1.14 4.493 1.16 4.497 1.18 4.501 1.2 4.505 1.22 4.508 1.24 4.511 1.26 4.513 1.28 4.518 1.3 4.52 1.32 4.523 1.34 4.525 1.36 4.527 1.38 4.531 1.4 4.533 1.42 4.535 1.44 4.537 1.46 4.539 1.48 4.54 1002 mat :

ksmooth kernel

σ 0.3708158988058244 :

The Gaussian ksmooth

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

Smath recognises the silent 'bw', something
like the Mathcad "global definition". W j X Y Ksmooth X i el X j el - bw / K Y i el * (i 1 X rows sum X i el X j el - bw / K (i 1 X rows sum / : j 1 X rows range s j el j X Y Ksmooth eval : for s 31 line :

Gaussian Ksmooth Kernel

x K

X Y bw 13 mat

from below

Discrete derivative by convolution ... better than finite differences. data scale convol vx data 1 col : vy data 2 col : 12 mat f 1 - 0113 mat transpose : Dy vy f al_convr1d eval : Dy Dy 2 Dy rows 1 - 11 submatrix : Deriv vx scale Dy * augment : Deriv Deriv 2 Deriv rows 1 - 12 submatrix : 61 line :

scale XY 2 col Max XY 1 - convol 2 col Max / (- : 15.4422 -

NOT smooth for sub-use ►

XY XY scale convol

X XY 1 col : Y XY 2 col : 12 mat bw 0.125 : (bandwidth ≡ XW X W augment : 31 line

Explain Project:

We are given a data set to quality control. Though it loks good,

noisy and unfit for publishing. The Gaussian ksmooth reveals an

excellent candidate. The discrete derivative is a noise amplifier.

XW is the smoothed derivative.

You can't ksmooth this data set XY directly . . . not visible.

Differently of derivative noise amplifier, cumulative integration

is a noise absorber . . . even excessive noise.

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Sanity smoothed derivative ... XW ► smoothed deriv. Disable blue to inspect quality of red... adjust 'bw'.

XY scale convol XW scale convol 2 1 sys

Y 0 :

smoothed XW scale convol eval :

XY 2 col Max 4.54

data Σ Finite differences Cumulative ∫ vx data 1 col : vy data 2 col : 12 mat Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y XY 12 el + augment 51 line :

Cleaned from FD Σ smoothed Σ :

∫ Σ Σ rows row : 1.46 2.8281 12 mat

scale XY 2 col Max ∫ 12 el / : 1.6053

vx Σ 1 col : vy Σ 2 col scale * : 12 mat x Σ vx vy x ainterp : x λ 1 0.5 - x ≤ (x 1.48 ≤ (& cases : 31 sys

interpolate in-situ u 1.25 :

track 22 0.75 2 u u Σ 32 mat :

The original noisy data set has been cleaned via the convolving adjustable ksmooth. At this point, we can export a more reliable data set.

x Σ x λ * track 2 1 sys

Export vx vy augment :

∫

Export XY 2 1 sys

u u Σ 12 mat 1.25 4.516 12 mat

Sanity check the Project:

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data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 Y i 1 + el Y i el - X i 1 + el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X 1 deriv * augment 101 line :

Export Deriv