<p> ksmooth # plotG </p>

e71d619e-f7cd-4b42-b522-0b660fdd0304 6 3 5 decimal

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

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 * :

Smath recognises the silent 'bw', something
like the Mathcad "global definition". 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 / :

Gaussian Ksmooth Kernel

x K

X Y bw 13 mat

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

W 11 mat

data 2.778 205 5.556 193 8.333 180 11.111 142 13.889 134 16.667 127 19.444 118 22.222 11025109 27.778 102 30.556 99 33.333 98 36.111 97 38.889 91 41.667 88 44.444 84 47.222 825082 52.778 79 55.556 75 58.333 74 61.111 70 63.889 68 66.667 67 69.444 63 72.222 627560 77.778 56 80.556 52 83.333 49 86.111 48 88.889 47 91.667 45 94.444 43 97.222 3510030362 mat :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : 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 augment 61 line :

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 deriv augment 101 line :

FLOW DATA Year 1 Year 2 Year 3 January 110180193 February 102118109 March 978899 April 847991 May 705682 June 625274 July 454768 August 673543 September 826030 October 1347548 November 2059849 December 14212763134 mat

data data Deriv data Σ

Prepare and code the Smath project

X data 1 col : Y data 2 col : 21 line

bw 10 : (bandwidth ≡ W W : (isolate W ≡ XW X W augment : 31 line

W 11 mat

isolate XW XW :

The working path: collect & return [X,Y]=>[bw]=>[Ksmooth(j,X,Y,bw)]=>[W]=>[XW]

XW X W o 7 red plotG : raw X Y + 7 black plotG : 21 line

===== Finish/collect the project =====

spline join/mesh the smoothed data set Cfit t 2.5 1003 range : i 1 t rows range v i el X W t i el cinterp eval : for t v augment 31 line :

collect plot plot Cfit raw XW 31 sys :

XY=>raw data, XW=>smoothed data, Cfit=> meshed spline export

plot

Absolute error applies to math functions. For data from collection we use the relative error ... here, totally negligible.

data Deriv XW Deriv 2 1 sys data Σ XW Σ 2 1 sys

J data Σ rows : 36

data Σ J row XW Σ J row 21 mat 100 8462.45 12 mat 100 8462.37 12 mat 21 mat

RelativeError 1 8462.45 8462.37 / - : 9106 -^* -

x f X W x cinterp eval :

h 0.1 :

1rst derivative  applied on interpolating functions x h D1 x 2 h * + f - x 2 h * - f + (- 4 h * / :

x c 1 2.5 x ≤ (x 100 ≤ (& cases :

x h D1 x c *