ListCollect

95044f1e-0048-4956-91aa-79e3fcc2f80f 7 15 12 decimal

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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 :

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 :

XY 0.132 0.1 0.322 0.258 0.511 0.543 0.701 0.506 0.891 0.606 1.081 0.622 1.27 0.569 1.46 0.453 1.65 0.438 1.839 0.316 2.029 0.29 2.219 0.195 122 mat :

1. selct the support points
2. collect for al_nleqsolve list 61112 mat : xy XY list ListCollect : 21 sys

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

support raw 2 1 sys

xy 1.081 0.622 2.029 0.29 22 mat

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

vector of guesses ... 
from robust2crucifying ! X0 0.1 112 mat transpose :

t0 1 time :

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 : x α Φ α 1 el α 2 el * x α 2 el 1 -^* α 1 el x α 2 el^* - exp * : α f x 1 el α Φ y 1 el - x 2 el α Φ y 2 el - 21 mat : 31 line

Notes . . . Smath 6179

1. solves system [n equalities = n parameters.

2. The order of equalities is immaterial.

3. Guess wisely the best support equalities !

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α sol : 0.42628 2.20381 21 mat

setup f(α) wrt model/support

1 time t0 - s 1

α f 0.000000000 0.000000000 21 mat

x c 10 x ≤ (x 4 ≤ (& cases :

Next . . . the LM Carlos companion solver

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x 0.132 0.322 0.511 0.701 0.891 1.081 1.27 1.46 1.65 1.839 2.029 2.219 121 mat :

y 0.1 0.258 0.543 0.506 0.606 0.622 0.569 0.453 0.438 0.316 0.29 0.195 121 mat :

β β 1 el β 2 el 21 mat :

m x length :

n β length :

Weibull model function
yet inactive in project x β f β 1 el β 2 el * x β 2 el 1 -^* β 1 el x β 2 el^* - exp * :

Vector of initial guesses β α :

1. from al_nleqsolve 2. β from LM Minimze(β)

LEVENBERG MARQUARDT METHOD

4. Use an equation to minimize inside a solve block.

5. Use the Levenberg-Marquardt method to minimize this problem.

prepare β fx i 1 m range f i el β 1 el β 2 el * x i el (β 2 el 1 -^* β 1 el x i el (β 2 el^* - exp * y i el - eval : for f 21 line : u derfh i 1 m range j 1 n range β j el β j el h + : jjh1 β fx i el : β j el β j el 2 h * - : jjh2 β fx i el : jjh0 i j el jjh1 jjh2 - 2 h * / : β j el β j el h + : 61 line for 11 line for jjh0 21 line : Eps 1.110 1016 -^* : eta1 Eps sqrt : eta2 eta1 : h eta1 : 14 mat J β derfh : f β fx : 12 mat A J transpose J * : g J transpose f * : ng g normi : F f transpose f * 2 / : 14 mat mu eta1 A Diag max * : nu 2 : stop 0 : 13 mat 61 sys :

derfh(u) Jacobian matrix for finite differences

β Minimize prepare stop ¬ ng eta2 ≤ stop 1 : p A mu n identity * + (1 -^g - (* : break try np β normi : nx eta2 β normi + : np eta2 nx * ≤ stop 2 : stop ¬ xnew β p + : fn xnew fx : Jn xnew derfh : Fn fn transpose fn * 2 / : dL p transpose mu p * g - (* 2 / : dF F Fn - : dL 1 el 0 > (dF 1 el 0 > (& β xnew : F Fn : J Jn : f fn : A J transpose f * : g J transpose f * : ng g normi : mu mu 13 / 12 dF 1 el dL 1 el / * 1 - (3^- Max * : nu 2 : 91 line mu mu nu * : nu 2 nu * : 21 line if 71 line if if 41 line if 11 line while β 21 line 21 line :

The parameters for best fit are the calculated values:

copy/paste in initials
to refit as desired ... β β Minimize : 0.4987 1.9637 21 mat

i.e: in 'α' above

6. Calculate the sum of squares implicitly minimized by this method.

t x :

x β f β 1 el β 2 el * x β 2 el 1 -^* β 1 el x β 2 el^* - exp * :

Relative residuals Residuals i 1 t rows range Δ i el 1 y i el t i el β f / - eval : for t Δ augment 21 line :

SSD = "Sum Square Differences" SSD y i el t i el β f - (2^(i 1 m sum : 0.019368

RMS difference RMSD SSD m / sqrt : 0.0402

Relative residuals

Residuals RMSD BarSize 1 1 sys

FIT range, populate U 04 0.01 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 41 line :

7. Plot the best Weibull fit versus the x-y data.

raw FIT support 3 1 sys

1 time t0 - s 9.6

β fx i 1 m range f i el β 1 el β 2 el * x i el (β 2 el 1 -^* β 1 el x i el (β 2 el^* - exp * y i el - eval : for f 21 line :

β derfh i 1 m range j 1 n range β j el β j el h + : jjh1 β fx i el : β j el β j el 2 h * - : jjh2 β fx i el : jjh0 i j el jjh1 jjh2 - 2 h * / : β j el β j el h + : 61 line for 11 line for jjh0 21 line :

β derfh 0.27 0.21 - 0.59 0.18 - 0.78 0.03 - 0.82 0.13 0.71 0.26 0.5 0.32 0.23 0.31 0.05 - 0.24 0.28 - 0.14 0.44 - 0.04 0.53 - 0.05 - 0.54 - 0.12 - 122 mat

Jacob β derfh :

Jacob