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 :

1. Define two vectors.

4 appVersion 0.98.6179.21440 ≡

t0 1 time :

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 :

LM code from Carlos ... compacted algo style ... Conjugate Gradient delivers a better fit ... Conjugate Gradient auto-iter runs faster

β β 1 el β 2 el 21 mat :

m x length :

n β length :

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

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 * :

3. Define initial guess values for the two parameters.

Vector of initial guesses β 0.1 121 mat :

[1 min] 0.1 121 mat

al_leqsolve 
 [1.5 min] 0.42628 2.20381 21 mat

[3 min] 0.8 121 mat

Iterate the solver ... in collapsed.

0.8 121 mat

0.325 1.607 21 mat

0.501 1.842 21 mat

0.502 2.004 21 mat

Mathcad LM 0.502 221 mat

LM algo style in collapsed

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 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 if if 41 line if 11 line while β 21 line 21 line :

A:=J^T*f is wrong, it must to be A:=J^T*J.

The parameters for best fit are the calculated values:

copy/paste in initials β β Minimize : 0.5022 2.0003 21 mat

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 : 0.018606

RMS difference RMSD SSD m / sqrt : 0.0394

Relative residuals

Residuals RMSD BarSize 1 1 sys

data x y . 12 black 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 :

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

data FIT 2 1 sys

1 time t0 - min 1

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 0.26 0.2 - 0.58 0.18 - 0.78 0.04 - 0.83 0.13 0.72 0.26 0.5 0.32 0.21 0.31 0.07 - 0.24 0.31 - 0.13 0.47 - 0.03 0.55 - 0.07 - 0.55 - 0.13 - 122 mat

Jacob β derfh :

Jacob

Iterative Conjugate Gradient . . .

]]> Minimize Conjugate-Gradient

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 :

init iter Minimize XY f φ β LS 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 : β 1 el init : i 1 : i iter ≤ β i 1 + el XY f φ β i el LS : i i 1 + : 21 line while β β iter el : 41 line β 31 line :

Vector Gradient ... Optimiz Symbolic
seemingly a scalar kernel  not 
recognized in program loop 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 :

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 :

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 :

corr Pearson correlation χ X Mean : μ Y Mean : n XY rows : 13 mat X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el β f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el β f eval μ - (2^(i 1 n sum * sqrt / / 31 line :

correlation between [X,Y data]

correlation ... [X,fitted model]

Absolute  residuals observer
fetches β from calculated δ j 1 XY rows range Δ j el Y j el X j 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 XY rows sum :

Otherwise Pearson independent of f(x,ParameterName) x y corr n x length : xy x y * vectorize : x2 x x * vectorize : y2 y y * vectorize : n xy sum (* x sum (y sum (* - n x2 sum (* x sum (2^- (n y2 sum (* y sum (2^- (* (sqrt / 51 line :

Fit.Y ii 1 n range Fit ii el X ii el β f : for Fit 21 line :

XY 0.132 0.1 0.322 0.258 0.511 0.543 0.701 0.506 0.891 0.606 1.082 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 :

<= Weibull model function x β f β 1 el β 2 el * x β 2 el 1 -^* β 1 el x β 2 el^* - exp * :

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

t0 1 time :

<= Gradient Vector Function x β φ x β f 2 φ :

X XY 1 col : (simply an index] ≡ Y XY 2 col : (time to failure in days ≡ n XY rows : data XY o 5 black plot : 41 line

L 0 : H 5 : N 199 : 13 mat U L H N xd : 21 line

hyper robust wrt init init 0.1 112 mat transpose :

iter 4 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

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

SSD 0.019

β 0.5017 2.003 21 mat

β 0.5017 2.003 21 mat

corr 0.6797

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

bar δ 0.025 BarSize :

barDev bar 1 col 1 bar 2 col * augment :

Weibull fit to Carlos data set

barDev FIT data 3 1 sys

δ 0.132 0.031 - 0.322 0.048 - 0.511 0.093 0.701 0.044 - 0.891 0.005 1.082 0.018 1.27 0.001 1.46 0.051 - 1.65 0.015 1.839 0.022 - 2.029 0.032 2.219 0.007 122 mat

1 time t0 - 2.3 s *

al_nleqsolve fixed through points . . .

]]> 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 :

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 :

AlgLib Solver for a system of nonlinear equations(Uni) sol original proposal Viacheslav 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 : 51 line :

Weibull model function x α Φ α 1 el α 2 el * x α 2 el 1 -^* α 1 el x α 2 el^* - exp * :

Implicit equalities = 0 α f x 1 el α Φ y 1 el - x 2 el α Φ y 2 el - 21 mat :

Notes . . . Smath 6179

1. solves a system of [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

1 time t0 - s 0.9

α f 0.000000000 0.000000000 21 mat

x c 10 x ≤ (x 4 ≤ (& cases :

Fit " a la carte" from al_nleqsolve

raw x α Φ x c * support 3 1 sys

Support points xy 1.081 0.622 2.029 0.29 22 mat

raw x α Φ x c * support 31 sys

Notes . . . @ this point

1. plug al_nleqsolve in LM Carlos above.

2. otherwise: initialize LM . . . from

. . . al_nleqsolve . . . any selected support.

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