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 :

About modeling ...

... often, two or more models will fit as well. This rational model is short, but not "normalised". 11 samples from depopulated to preserve the arc region. The flat/vertical regions were stripped of superflous samples, just enough points. In Computer Approximation of Functions, error is given as absolute |ε| wrt their range of base approximation which is sgnificant in that case. Differently, in modeling sampled data |ε| has no meaning. Instead, we use the "relative error" ... ... ε:= 1-Y/fit(X)

XY 0.5 83.67 1 74.7 2 70.26 5 67.79 10 66.76 20 65.94 50 64.88 100 64.13 200 63.65 500 62.58 1000 62.3 112 mat :

Experiment Smath "Genfit Algorithm"

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

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

1100 0.5 0.01 41 mat 8.76 66.167 0.527 0.009 41 mat 12 mat

Gradient vector fnct x β φ x β f 4 φ :

β β 1 el 1100 0.5 0.01 41 mat : 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 :

Relative residuals Δ i 1 XY rows range Δ i el 1 Y i el X i 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 n sum : 0.137

L 0.5 : H 100 : N 199 : 13 mat

Relative residuals

Δ Error

U L H N xd :

FIT i 1 U rows range vy i el U i el β f eval : for U vy augment 21 line :

data X Y . 12 black plotG :

data FIT 2 1 sys

Δ 0.5 0.0009 - 1 0.0016 2 0.002 5 0.0018 - 10 0.0019 - 20 0.0004 - 50 0.0018 100 0.0014 200 0.0031 - 500 0.0013 1000 0.0001 - 112 mat

Error plot Construction

data Δ :

data ERR 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 stem 31 line :

λ data ERR 1 el data ERR 2 el data ERR 3 el data ERR 4 el data ERR 5 el data ERR 6 el data ERR 7 el data ERR 8 el data ERR 9 el data ERR 10 el data ERR 11 el 111 sys :

datum 25 - 0250022 mat :

λ datum 2 1 sys

Construct the J_Fraction

Δ 0.5 0.001 - 1 0.002 2 0.002 5 0.002 - 10 0.002 - 20050 0.002 100 0.001 200 0.003 - 500 0.001 10000112 mat

Define Symbolic ... Optimiz = NONE x collect x β f confrac x convert evalf maple :

Optimiz NONE x collect 123672969920000000 / 25431291215000000 x 2308385232000000 / + 46689663200000 x 12641321362500000 / + (* / - (* / +

61.8361 508.852379 x 115.466131 + 233.53475 x 2.022539 + / - / +