plot(data,char,size,clr) ..... Strip paired 0's

4d267959-c54f-4374-b14e-50d5552976a3 77 6 12 decimal

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

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Strip paired 0's on length

data Strip2 u data 1 col : v data 2 col : k 0 : i 1 data rows range u i el v i el * 0 ≠ R k k 1 + : 1 el u i el : R k 2 el v i el : 21 line End of strip if 11 line for 21 line R 41 line :

data n Decimate collect every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for 21 line u v augment 41 line :

Carlos Pile-Redux [project in collaboration]

D 18 in * :

s 3.5 D * :

3 appVersion 0.98.6179 ≡

Rows Cols η 1 D s / atan 90 deg * / Cols 1 - (Rows * Rows 1 - (Cols * + Rows Cols * / (* - :

r 110 1.1 range :

i 1 r length range A1 i 1 el r i el : A1 i 2 el A1 i 1 el z η 0.95 ≡ z solve : 21 line for

i 4 r length range A4 i 1 el r i el : A4 i 2 el A4 i 1 el z η 0.8 ≡ z solve : 21 line for

i 1 r length range A2 i 1 el r i el : A2 i 2 el A2 i 1 el z η 0.9 ≡ z solve : 21 line for

i 10 r length range A5 i 1 el r i el : A5 i 2 el A5 i 1 el z η 0.75 ≡ z solve : 21 line for

i 1 r length range A3 i 1 el r i el : A3 i 2 el A3 i 1 el z η 0.8524 ≡ z solve : 21 line for

i 32 r length range A6 i 1 el r i el : A6 i 2 el A6 i 1 el z η 0.7 ≡ z solve : 21 line for

strip the unwanted 0's A4 A4 Strip2 : A5 A5 Strip2 : A6 A6 Strip2 : 31 sys

Target sols

A1 1 row A2 1 row A3 1 row 31 mat 1 1.393 12 mat 1 2.296 12 mat 1 5.991 12 mat 31 mat

A4 1 row A5 1 row A6 1 row 31 mat 1.3 9.812 12 mat 1.9 15.97 12 mat 4.1 15.919 12 mat 31 mat

X 1 1.3 1.9 31 mat :

Y 5.991 9.812 15.97 31 mat :

a X Y Intercept : b X Y Slope : 21 sys 4.767 - 10.97 21 sys

N.rows 3 :

N.cols 3 :

Decimate

a1 sub1 A1 110 range 12 range el : sub2 A1 1191 range 12 range el : sub3 sub2 5 Decimate : sub1 sub3 stack 41 line :

a2 sub1 A2 110 range 12 range el : sub2 A2 1191 range 12 range el : sub3 sub2 5 Decimate : sub1 sub3 stack 41 line :

a3 sub1 A3 110 range 12 range el : sub2 A3 1191 range 12 range el : sub3 sub2 5 Decimate : sub1 sub3 stack 41 line :

a1 a1 o 5 black plot :

a2 a2 o 5 black plot :

a3 a3 o 5 black plot :

a4 sub1 A4 115 range 12 range el : sub2 A4 1688 range 12 range el : sub3 sub2 5 Decimate : sub1 sub3 stack 41 line :

a5 sub1 A5 121 range 12 range el : sub2 A5 2282 range 12 range el : sub3 sub2 5 Decimate : sub1 sub3 stack 41 line :

a6 sub1 A6 119 range 12 range el : sub2 A6 2060 range 12 range el : sub3 sub2 3 Decimate : sub1 sub3 stack 41 line :

a4 a4 o 5 black plot :

a5 a5 o 5 black plot :

a6 a6 o 5 black plot :

a1 a2 a3 a4 a5 a6 A1 A2 A3 A4 A5 A6 1 20 1 5 - 2 2 mat N.rows N.cols + 200 blue 1 5 mat a b x * + 15.97 16 1 sys

N.rows N.cols N.rows N.cols η 31 mat 33 0.764 31 mat

A1 rows A2 rows A3 rows 31 mat A4 rows A5 rows A6 rows 31 mat 12 mat 91919131 mat 88826031 mat 12 mat

Fitting session to a selected data set

1. The data set is clean, thus experiment al_nleqsolve.

2. Refine Genfit as required for noisy data sets.

]]> data sets [a1, a2, a3, a4, a5, a6]

a1 1 1.393171 1.1 1.236559 1.2 1.130642 1.3 1.054234 1.4 0.996511 1.5 0.951366 1.6 0.915091 1.7 0.885307 1.8 0.860414 1.9 0.839299 2 0.821162 2.4 0.768569 2.8 0.734946 3.2 0.711599 3.6 0.69444 4 0.681298 4.4 0.670909 4.8 0.662491 5.2 0.655532 5.6 0.649682 6 0.644695 6.4 0.640395 6.8 0.636648 7.2 0.633353 7.6 0.630434 8 0.62783 8.4 0.625493 8.8 0.623383 9.2 0.621469 9.6 0.619724 10 0.618128 312 mat :

a2 1 2.295822 1.1 1.899397 1.2 1.660466 1.3 1.500729 1.4 1.386409 1.5 1.300547 1.6 1.233694 1.7 1.180166 1.8 1.13634 1.9 1.099798 2 1.068863 2.4 0.981444 2.8 0.927273 3.2 0.890414 3.6 0.86371 4 0.843473 4.4 0.827608 4.8 0.814836 5.2 0.804333 5.6 0.795543 6 0.78808 6.4 0.781663 6.8 0.776087 7.2 0.771197 7.6 0.766874 8 0.763025 8.4 0.759575 8.8 0.756465 9.2 0.753649 9.6 0.751085 10 0.748742 312 mat :

a3 1 5.991366 1.1 3.878736 1.2 2.99784 1.3 2.514609 1.4 2.209353 1.5 1.99904 1.6 1.845336 1.7 1.728096 1.8 1.635721 1.9 1.561058 2 1.49946 2.4 1.332907 2.8 1.234928 3.2 1.170403 3.6 1.124696 4 1.090623 4.4 1.064244 4.8 1.043217 5.2 1.026063 5.6 1.011802 6 0.99976 6.4 0.989456 6.8 0.980538 7.2 0.972746 7.6 0.965878 8 0.959779 8.4 0.954327 8.8 0.949424 9.2 0.944991 9.6 0.940964 10 0.937289 312 mat :

a4 1.3 9.811927 1.4 6.375036 1.5 4.890435 1.6 4.062607 1.7 3.534666 1.8 3.168649 1.9 2.899969 2 2.694352 2.1 2.531924 2.2 2.400403 2.3 2.29167 2.4 2.200341 2.5 2.122484 2.6 2.055382 2.7 1.996896 2.8 1.945511 3.2 1.790038 3.6 1.685317 4 1.60994 4.4 1.553101 4.8 1.508743 5.2 1.473148 5.6 1.44394 6 1.419525 6.4 1.398802 6.8 1.381047 7.2 1.36564 7.6 1.352146 8 1.34023 8.4 1.32963 8.8 1.320139 9.2 1.311592 9.6 1.303855 10 1.296818 342 mat :

a5 1.9 15.969699 2 11.244246 2.1 8.869659 2.2 7.441089 2.3 6.487113 2.4 5.804918 2.5 5.292843 2.6 4.894308 2.7 4.575319 2.8 4.314222 2.9 4.096568 3 3.912347 3.1 3.754406 3.2 3.617495 3.3 3.497677 3.4 3.391938 3.5 3.297935 3.6 3.213816 3.7 3.138101 3.8 3.069589 3.9 3.007301 4 2.950424 4.4 2.765015 4.8 2.627423 5.2 2.521263 5.6 2.436867 6 2.368166 6.4 2.311153 6.8 2.263081 7.2 2.221998 7.6 2.186483 8 2.155477 8.4 2.128172 8.8 2.103943 9.2 2.082298 9.6 2.062844 10 2.045264 372 mat :

a6 4.1 15.918798 4.2 14.571734 4.3 13.483792 4.4 12.586766 4.5 11.834455 4.6 11.194454 4.7 10.64336 4.8 10.163849 4.9 9.742826 5 9.370205 5.1 9.038092 5.2 8.740223 5.3 8.47156 5.4 8.22801 5.5 8.006208 5.6 7.803366 5.7 7.617152 5.8 7.445603 5.9 7.287051 6 7.140073 6.2 6.876116 6.4 6.645788 6.6 6.443046 6.8 6.263214 7 6.102618 7.2 5.958326 7.4 5.827977 7.6 5.709643 7.8 5.601735 8 5.502934 8.2 5.412133 8.4 5.328399 8.6 5.250938 8.8 5.179069 9 5.112209 9.2 5.049852 9.4 4.991557 9.6 4.936941 9.8 4.885665 10 4.837432 402 mat :

select data "a" XY a4 :

renders best a4

ListCollect

data list ListCollect collects index list select list : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Minimize Conjugate-Gradient

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

init iter Minimize XY f φ β LS X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx-> raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line : β 1 el init : i 1 : i iter ≤ β i 1 + el XY f φ β i el LS : i i 1 + : 21 line while β β iter el : 41 line β 31 line :

Vector Gradient ... Optimiz Symbolic
seemingly a scalar kernel  not 
recognized in program loop x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

Error stem plot for QuickPlot unicolor data Error 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 :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

corr Pearson correlation χ X Mean : μ Y Mean : n XY rows : 13 mat 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 β f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el β f eval μ - (2^(i 1 n sum * sqrt / / 31 line :

correlation between [X,Y data]

correlation ... [X,fitted model]

Absolute  residuals observer
fetches β from calculated δ j 1 XY rows range Δ j el Y j el X j el β f - eval : for X Δ augment 21 line :

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 XY rows sum 11 line :

Otherwise Pearson independent of f(x,ParameterName) x y corr n x length : xy x y * vectorize : x2 x x * vectorize : y2 y y * vectorize : n xy sum (* x sum (y sum (* - n x2 sum (* x sum (2^- (n y2 sum (* y sum (2^- (* (sqrt / 51 line :

Fit.Y ii 1 n range Fit ii el X ii el β f : for Fit 21 line :

1. selct the support points
2. collect for al_nleqsolve
3. recover [X, Y] list 2102513 mat : xy XY list ListCollect : X XY 1 col : Y XY 2 col : 12 mat 31 sys

raw XY o 5 black plot : support xy . 14 blue plot : 21 line

Observe . . .

1. Distribute wisely this clean data set.

2. By inspection, [1,10,25] will generate

division by zero wrt the model !

]]>

xy 1.4 6.375 2.2 2.4 6.4 1.399 32 mat

x xy 1 col : y xy 2 col : 21 sys

support X 1 el 15 X 1 el 5 - 2 2 mat raw 3 1 sys

vector of guesses ... 
from robust2crucifying ! X0 0 0.5 413 mat transpose :

AlgLib Solver for a system of nonlinear equations(Uni) sol k 1 α f rows range u k el α k el : for α Jac u f u Jacobian : StepMax 0 : ε 1012 -^: 12 mat u X0 StepMax ε α f α Jac al_nleqsolve : 41 line :

Homographic model to XY x α Φ 1 x X 1 el - (- α 1 el α 2 el x X 1 el - (* + / α 3 el + :

Implicit equalities = 0 α f x 1 el α Φ y 1 el - x 2 el α Φ y 2 el - x 3 el α Φ y 3 el - 31 mat :

fitted parameters α sol : 0.13298 0.87429 2.29169 31 mat

Notes . . . Smath 6179

1. Solves a system of [n equalities = n parameters.

2. The order of equalities is immaterial.

3. Guess wisely the best support equalities !

]]>

sanity fit ...
@ support points α f 0021015 -^* - 31 mat

X XY 1 col : Y XY 2 col : 12 mat data XY o 5 black plot : n XY rows : 31 line

Fitting session to a selected data set

1. From initial al_nleqsolve parameters

2. Refine Genfit as required for noisy data sets.

]]>

parametric model fit x β f 1 x X 1 el - (- β 1 el β 2 el x X 1 el - (* + / β 3 el + :

<= ConjugateGradient Vector Fnct x β φ x β f 3 φ :

fetch initial β from al_nleqsolve init α transpose :

iter 3 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

β transpose 0.132974 0.874327 2.291673 31 mat

SSD 0.00000001

corr 0.999998

FIT range, populate U X 1 el 12 X 1 el 0.01 + (range : i 1 U rows range vy i el U i el β f eval : for U vy augment 41 line :

Absolute residuals Residuals i 1 XY rows range Δ i el Y i el X i el β f - eval : for X 200 Δ * augment 21 line :

user a6 < 4.1 user 4.25 : λ user β f : 21 sys

Residuals 0.0125 BarSize FIT X 1 el 15 X 1 el 2 - 2 2 mat data 5 11 Pile-Redux factor 'f' 14 black 1 5 mat 0 8 'f' 14 green 1 5 mat 0 8 5 8 5 8 user λ 4 2 mat 7 1 sys

user λ 12 mat 4.25 1.573 12 mat

Declare the model fit => exact

Here, the 15 D residuals are compared to 6 D data set.

Exact model to experimental data is very rare.

]]>

data β 1 el β 2 el β 3 el 31 mat 21 mat a1 a2 a3 a4 a5 a6 1.748223 4.183793 0.821162 31 mat 0.815023 2.686169 1.068863 31 mat 0.222623 1.556437 1.499460 31 mat 0.132974 0.874327 2.291673 31 mat 0.084224 0.416912 4.096568 31 mat 0.145334 0.173076 9.038092 31 mat 26 mat ≡