Plot Utilities

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Plot 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 :

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 :

t0 1 time :

1. Define two vectors.

x 0.132 0.322 0.511 0.701 0.891 1.082 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 :

n β length :

m x length :

2. Define a fitting function (Weibull density with unknown parameters).

FUNCTION x ft β 1 el β 2 el * x β 2 el 1 -^* β 1 el x β 2 el^* - exp * :

Jacobian matrix for finite differences B 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 :

3. Define initial guess values for the two parameters.

Initial Assumption Values Vector β 0.8 112 mat transpose :

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

Function adjusted to the data β fx i 1 m range f i el x i el ft y i el eval - : for f 21 line :

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

LEVENBERG MARQUARDT METHOD

Eps 1.110 1016 -^* : eta1 Eps sqrt : eta2 eta1 : h eta1 : tol Eps sqrt β normi * : dnor 1 : 16 mat

f β fx :

J β derfh :

A J transpose J * :

g J transpose f * :

ng g normi :

F f transpose f * 2 / :

mu eta1 A Diag max * :

nu 2 :

stop 0 :

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 : if 41 line if 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 J * : 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 11 line if 21 line while

The parameters for best fit are the calculated values:

SOLUTION β 0.502 2.00 21 mat

6. Calculate the sum of squares implicitly minimized by this method (By JEAN).

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

t x :

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 : 0.0186

Relative residuals

Residuals 0.0394 BarSize 1 1 sys

RMS difference SSD m / sqrt 0.0394

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

data x y . 12 blue plotG :

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 :

data FIT 2 1 sys

1 time t0 - s 2.172