Prepare

c587bd48-9e90-4ef3-8dab-a1e8b6158235 61 4 5 decimal

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

Prepare

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 : 31 line for data r3 r4 r5 augment 21 line :

Part 1: Search β's from robust al_nleqsolve

Reduced Kirby2 [Y/X] XY 9.65 0.00085 24 0.00538 62.63 0.0515 104.46 0.1376 134.79 0.1918 157.76 0.2222 176.25 0.2401 191.48 0.251 207.58 0.2591 229.92 0.2657 254.22 0.2684 278.83 0.2675 302.02 0.2645 324.42 0.2602 347.02 0.2548 371.3 0.2483 162 mat :

1. selct the support points
2. collect for al_nleqsolve list 159121515 mat : xy XY list ListCollect : 21 sys

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

pre-graph of the support points

raw support 2 1 sys

μ 3 :

optimum for this fit

Declare the model function x β Φ β 1 el x β 2 el^* β 3 el x β 4 el - (1 μ -^* - exp * x β 4 el - (μ -^* β 5 el + ≡

The support points x xy 1 col : y xy 2 col : 12 mat

The equalities system.
Expand/reduce to match
length of support points x 1 el β Φ y 1 el - x 2 el β Φ y 2 el - x 3 el β Φ y 3 el - x 4 el β Φ y 4 el - x 5 el β Φ y 5 el - 51 mat

vector of guesses X0 3000 1.5 50000100 - 015 mat transpose :

AlgLib Solver for a system of nonlinear equations(Uni) sol k 1 β f rows range u k el β k el : for β Jac u f u Jacobian : StepMax 0 : ε 106 -^: 12 mat u X0 StepMax ε β f β Jac al_nleqsolve : 41 line :

The support points xy 9.65 0.0008 134.79 0.1918 207.58 0.2591 278.83 0.2675 347.02 0.2548 52 mat

Model function x β Φ β 1 el x β 2 el^* β 3 el x β 4 el - (1 μ -^* - exp * x β 4 el - (μ -^* β 5 el + :

The system of equalities
is constructed manually β f x 1 el β Φ y 1 el - x 2 el β Φ y 2 el - x 3 el β Φ y 3 el - x 4 el β Φ y 4 el - x 5 el β Φ y 5 el - 51 mat :

β sol : 2065.0903 1.6168 50207.6598 82.5509 - 0.0001 - 51 mat

raw x β Φ 2 1 sys

Mutual conformity
@ support points. β f 0.0007 - 0.0009 0.0005 0.0001 0.0009 - 51 mat

Part 2: Refine β's from SSD Conjugate Gradient

Genfit Utilities

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

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 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 :

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 :

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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Vector gradient ... Optimiz Symbolic 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 :

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 :

NIST Kirby2 data Kirby 9.65 0.0082 24 0.129 62.63 3.223 104.46 14.377 134.79 25.853 157.76 35.06 176.25 42.31 191.48 48.06 207.58 53.78 229.92 61.1 254.22 68.23 278.83 74.6 302.02 79.88 324.42 84.4 347.02 88.43 371.3 92.2 162 mat :

x β f β 1 el x β 2 el^* β 3 el x β 4 el - (1 μ -^* - exp * x β 4 el - (μ -^* β 5 el + :

vector gradient function x β φ x β f 5 φ :

β β 1 el 2000 1.6 5000090 - 015 mat transpose : iter 5 : i 1 : i iter ≤ β i 1 + el XY f φ β i el Minimize : i i 1 + : 21 line while β β iter el : 51 line :

β β :

β 2371.3622 1.5984 52309.9886 86.1521 - 0.001 51 mat

al_nleqsolve 2065.0903 1.6168 50207.6598 82.5509 - 0.0001 - 51 mat

Genfit [CG] 2371.3622 1.5984 52309.9886 86.1521 - 0.001 51 mat

round export. β 2371 1.5984 52310 86.15 - 0.001 51 mat :

Part 3: Recover Kirby2 from the "reduced"

Kirby2 x β f x * ≡

Kirbyfit x 04001 range : i 1 x rows range vy i el x i el β f x i el * eval : for x vy augment 31 line :

X Kirby 1 col : Y Kirby 2 col : n Kirby rows : 31 line

Kirbydata Kirby . 12 black plot : Title 16025 Transmute OVNI fit to Kirby2 12 black 15 mat : 21 line

j 1 XY rows range Δ j el Y j el X j el β f X j el * - eval : for δ X 100 Δ * augment : bar δ 1.5 BarSize (: SSD Y i el X i el β f X i el * - (2^(i 1 n sum : 41 line

SSD 0.0304

RMS SSD n / sqrt 0.0436

bar Kirbyfit x β f x * Kirbydata Title 5 1 sys

Part 4: Explain the project

Model f(x,β) => Courtesy Jean Giraud. ------------------------------------ ...This fit results from 'data reduction Y/X' from which the model is developped and reconstructed Kir(x)... Kirby2(x)*x. The reduced Y/X looks weird to fit, OVNI model will digest. Observe this model 'pins' the fit to datum 0 ! Reference: Kirby, R., NIST (1979). Scanning electron microscope line width standards.

jmGiraud: 2018-08-09

Explain/Proceed . . .

Part 1: This fitting session takes advantage of the robust al_nleqsolve, robust wrt to initials.

As it is fast, you can tentativly try up to you start getting some kind of fit, then refine further.

Once a fit is obtained, plug the β's in the next Genfit Minimize. al_nleqsolve passes through

support points. Next ConjugateGradient is an SSD method, thus refit/refine the model function.

To ease the fitting session, copy/paste Part 1 in new work sheet ... once done the pre-fit,

transport the β's to proceed with Part 2.

The list is the user selected support points, highlighted in the pre-graph.

There are no rule(s) about finding initials, purely trial/error X0. From observation, the very

pre-fit that passes through the data set, is likely to produce a fit.

Finally => best round the parameters.

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