Genfit Utilities

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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 :

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].

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 :

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 :

Construction

's' slant
'λ shift
'n' curvature s 0.05 : λ 1 : n 1 : 31 sys

x n f s x * 1 x λ - (exp + ln n^(* :

L 3 : H 13 : N 10 : 13 mat X L H N xd : XY X X 3 - (n f vectorize augment : data XY o 5 black plot : 41 sys

x 1 f x 1 - ( n f x 2 - ( n f x 3 - ( n f x 4 - ( n f x 5 - ( n f data 7 1 sys

Part 1

Construction should have no interest, as we are going to fit a function to a data set, no matter its origin, most generally experimental data. al_nleqsolve is robust wrt guesses, though not foolproof. The found β's are used to initialize Genfit Part 2

About the model . . .

It models real parabolic phenomenons

of shifted origin in either/both X_Y

directions. In fact, it takes into account

5 physical constituants.

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vector of guesses X0 0.001 110115 mat transpose :

data set XY 304 0.03466 5 0.13133 6 0.31904 7 0.60972 8 1.00454 9 1.50201 10 2.10087 11 2.80036 12 3.60015 13 4.50006 112 mat :

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 :

A generalized adjustable parabolic function x β Φ β 1 el x * 1 x β 5 el - β 3 el / exp + ln (β 2 el^(* β 4 el + :

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

x c 13 x ≤ (x 13 ≤ (& cases :

The system of equalities β f 7 β Φ 0.60972 - 13 β Φ 4.50006 - 3 β Φ 0.0 - 10 β Φ 2.10087 - 5 β Φ 0.13133 - 51 mat :

β sol : 0.028705 1.147516 0.972812 0.013686 - 4.459577 51 mat

Notes . . . Smath 6179

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

2. The order of equalities is immaterial.

3. From not knowing more we get an exact fit.

Observe => by how much β's are so ≠ than

those that constructed the data set. Immaterial for

this demo example. This property reflexivity may

blind/fool the solver, left for manual fit.

]]> x 3 - ( n f x c * x β Φ data 3 1 sys

Exact fit from exact data set

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

Absolute residuals

Residuals 0.1 BarSize

mutual deviation
@ the equalities. β f 5.3 106 -^* 5.6 107 -^* 2.6 106 -^* 2.4 106 -^* - 6.1 106 -^* - 51 mat

Part 2

β refine/refit

Aren't we happy al_nleqsolve ? . . . Answer is NO, why ?

Unlike al_nleqsolve that tends to fit exactly at the equalities points,

ConjugateGradient is an SSD algorithm. It then fits more glogally and

more precisely. That's what we are looking for from a fitting session

given some noisy experimental data.

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Model function x β f β 1 el x * 1 x β 5 el - β 3 el / exp + ln (β 2 el^(* β 4 el + :

XY 304 0.03466 5 0.13133 6 0.31904 7 0.60972 8 1.00454 9 1.50201 10 2.10087 11 2.80036 12 3.60015 13 4.50006 112 mat :

<= ConjugateGradient Vector Fnct x β φ x β f 5 φ :

nleqsolve 0.028705 1.147516 0.972812 0.013686 - 4.459577 51 mat

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

'n' is of same legth as β

data X Y . 12 black plotG :

β β 1 el 0.025 1.5 1 0.01 - 515 mat transpose : iter 7 : 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 :

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

ConjugGradient β 0.03017 1.14152 1.0003 0.01572 - 4.49076 51 mat

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 9106 -^*

FIT range, populate U 015 0.1 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 41 line :

RMS difference SSD n / sqrt 0.0009

data FIT 2 1 sys

Absolute residuals from CG

Residuals 0.1 BarSize