0aa49c80-4b1e-4e4f-add1-00c72aa23728 24 6 9 decimal

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

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 :

Randomize a function

To regenerate new set of randomized function, click in any active gray region => press F9. "level" = max level of random excursion...± level.

Uniform ...  -1 < rnd < 1 n level rnd level n Random * level 2 / - 11 line :

x f span level populate randomize x 0 span 1 populate / range : i 1 x rows range y i el x i el f : for x y x rows level rnd + augment x y augment 21 mat 31 line :

XY 1 el noisy ≡ XY 2 el clean ≡ 21 mat

XY x sin 4 π * 0.25 20 randomize :

a b c spec . 8 black 13 mat : data XY 1 el a b c spec pts : 21 line

2 periods of noisy sine

data XY 2 el 2 1 sys

Collect for B39 data XY 1 el : X data 1 col : Y data 2 col : N data rows : 41 line

over the 4π, @ population 20 randomize(f(x),n,level,populate) produces 20*4*π = 252 data pairs. 20*4*π=251.327412

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

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

fit x 1 - 15 0.9 - range : i 1 x length range v i el x i el B39 eval : 11 line for x v augment 31 line :

Most interesting: from much higher 'w'length [try w=19] generate high frequecy random Δ

a b c spec . 9 black 13 mat : pts data a b c spec pts : 21 line

Fourier polynomial smoothing to periodic data set. Observe the effectiveness of B39 ... get new randomized => observe the Δ. A measuring chain must be at least 10 times more accurate than the measured, implying noise level of 0.1 or less... so: decrease noise level,though it does not significantly improve B39 recovery ... pretty robust in this example.

pts fit x sin 3 1 sys

Absolute deviation ... observe:
click on B39 => press F9 Δ i 1 X rows range true i el X i el sin eval : B39 i el X i el B39 eval : 21 line for X true B39 - (augment 21 line :

χ X Mean : μ Y Mean : n X rows : 31 line

χ μ n 31 mat 6.275 0.004897 - 25231 mat

x f x B39 :

Pearson's correlation coefficient corr X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el f eval μ - (2^(i 1 n sum * sqrt / 21 mat : 0.392564 - 0.39258 - 21 mat

correlation between [X,Y]

C corr 1 el corr 2 el / : 1

correlation between [X,fitted model]