Utilities: C.G, plotG, Stem

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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 :

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 :

Minimize model "ConjugateGradient" wrt parameters [β's]

This Smath document will minimize the model for the best set of the parameters. The success will depend upon the wise choice of the initial β's. This algo is not as robust as the Levenberg-Marquardt, the final fit is iterative but will finally fit<=>LM. At each run, you can observe the goodness of fit on the graph. The core algorithm Minimize(XY,f,φ,β) operates indirectly on the SSD(β). In other words: Minimize(XY,f,φ,β) is implicit of the objective criterium SSD(β). Many models are pretty intuitive, others are very reactive [reflexive] wrt the parameters. If not wisely initialised, may crash Smath. Good idea to start the fitting session on a scratch pad work sheet. Hover φ(x,β), for ConjugateGradient. This open code model "Minimize" is identical in fit to Mathcad/Mathsoft, OriginLab.

XY data set

XY 2 - 0 1.875 - 0 1.75 - 0.003 1.625 - 0.009 1.5 - 0.021 1.375 - 0.041 1.25 - 0.07 1.125 - 0.112 1 - 0.167 0.875 - 0.236 0.75 - 0.315 0.625 - 0.398 0.5 - 0.479 0.375 - 0.552 0.25 - 0.612 0.125 - 0.652 0 0.667 0.125 0.652 0.25 0.612 0.375 0.552 0.5 0.479 0.625 0.398 0.75 0.315 0.875 0.236 1 0.167 1.125 0.112 1.25 0.07 1.375 0.041 1.5 0.021 1.625 0.009 1.75 0.003 1.875 020332 mat :

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

Model function

in φ(x,β,f,n) -> set 'n' to last index β # el ≡

x β φ x β f 5 φ :

Conjugate Gradient

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

Initial 'β' β 0.6667 10 - 1 6.667 051 mat ≡

β 0.677966 9.908587 - 0.000058 - 7.422175 0.009526 - 51 mat :

re-inject to iterate manually

Genfit coeff's of fit β XY f φ β Minimize :

FIT range, populate U 2 - 2 1.99 - range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 41 line :

β 0.677968 9.89608 - 0.000001 7.41538 0.009539 - 51 mat

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

data X Y . 12 black plotG :

data FIT 2 1 sys

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

β SSD 0.000372

RMS difference β SSD n / sqrt 0.003357

region s - h - s - h s h s h - s - h - 52 mat :

s 1 : h 0.05 : 21 line

Residuals Error Residuals region x β f x diff 4 1 sys

Note: Functions that have absolute reference, like functions from mathematical libraries, are appreciated in term of "absolute residuals". For model functions, the brute appreciation of goodness of fit is given in term of ... ----------- relative residuals ----------- OBSERVE: rows(XY)=33 ... it is great mistake attempting to fit an over-populated data set. There is no rule to best conserve population.

Copy/paste this section for next model fit

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 :

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 :