Plot Utilities

4a5daaf5-ffab-4c1f-b38c-a7b2d22990b4 24 15 12 decimal

&[DATE] &[TIME] - &[FILENAME]

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 :

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 :

Levenberg Marquardt's second algorithm . . . Carlos 20200115

x y β Tol βs LMSA Eps Tol : eta1 Eps sqrt : eta2 eta1 : h eta1 : stop 0 : 15 mat m Clear 1 : m x rows : n β rows : 12 mat β fx i 1 m range ff i el x i el ft y i el eval - : for ff 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 : A n n Zeros : g n Zeros : J m n Zero : 13 mat Fn 1 Zero : dL 1 Zero : dF 1 Zero : F 1 Zero : 14 mat f β fx : J β derfh : 12 mat A J transpose J * : g J transpose f * : ng g normi : 13 mat F f transpose f * 2 / : mu eta1 A Diag max * : nu 2 : 13 mat stop ¬ ng eta2 ≤ stop 1 : p A mu n identity * + invert 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 11 line if if 41 line if 11 line while βs β : 121 line :

Example_1 ... global fit Weibull data set

x β Clear 1

Weibull model function x ft β 1 el β 2 el * x β 2 el 1 -^* β 1 el x β 2 el^* - exp * :

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 0.8 121 mat :

Tol 106 -^:

t0 1 time :

x y β Tol βs LMSA 0.5022 2.0002 21 mat

==========================

1 time t0 - s 55

FUNCTION t βs ft βs 1 el βs 2 el * t βs 2 el 1 -^* βs 1 el t βs 2 el^* - exp * :

m x rows :

t x :

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

SSD = "Sum Square Differences" SSD y i el t i el βs ft - (2^(i 1 m sum : 0.0186

RMS difference SSD m / sqrt 0.0394

RMSD SSD m / sqrt :

Relative residuals

Residuals RMSD BarSize 1 1 sys

data x y . 12 blue plotG :

FIT range, populate U 04 0.025 range : i 1 U rows range vy i el U i el βs ft eval : for U vy augment 41 line :

data FIT 2 1 sys D2 ... D3

0.38 -

D2 00 0.021 2.22 0.04 3.375 0.059 3.789 0.079 3.782 0.099 3.538 0.12 3.164 0.141 2.754 0.164 2.322 0.188 1.919 0.213 1.552 0.237 1.253 0.263 0.991 0.29 0.77 0.318 0.595 0.346 0.456 0.377 0.341 0.406 0.257 0.437 0.192 0.47 0.14 0.509 0.096 0.545 0.068 0.591 0.044 0.645 0.026 0.918 0.002 1.125 0 1.154 0272 mat :

D3 0 10.861 0.021 6.056 0.04 3.327 0.059 1.687 0.079 0.706 0.099 0.133 0.12 0.187 - 0.141 0.338 - 0.164 0.395 - 0.188 0.394 - 0.213 0.362 - 0.237 0.32 - 0.263 0.272 - 0.29 0.226 - 0.318 0.185 - 0.346 0.149 - 0.377 0.118 - 0.406 0.093 - 0.437 0.073 - 0.47 0.056 - 0.509 0.041 - 0.545 0.031 - 0.591 0.022 - 0.645 0.014 - 0.918 0.002 - 1.125 0 1.154 0272 mat :

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 :

Example_2 ... selected fit Pharmacokinetics data set

x β Clear 1

model to D2(x) x ft x β 1 el β 2 el x β 4 el / - exp * + β 3 el x β 5 el / - exp * + (* :

XY D2 :

D2 rows 27

1. selct the support points
2. collect for al_nleqsolve list 1514222715 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 00 0.079 3.782 0.29 0.77 0.545 0.068 1.154 052 mat

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

vector of guesses ... β 1012 -^105 - 0.1 5 - 15 mat transpose :

Tol 106 -^:

t0 1 time :

α x y β Tol βs LMSA : 0.807 142.4039 0.7175 - 0.0724 9.8189 - 51 mat

1 time t0 - min 1.9

model to D2(x) x α Φ x α 1 el α 2 el x α 4 el / - exp * + α 3 el x α 5 el / - exp * + (* :

Implicit equalities = 0 α f x 1 el α Φ y 1 el - x 2 el α Φ y 2 el - x 3 el α Φ y 3 el - x 4 el α Φ y 4 el - x 5 el α Φ y 5 el - 51 mat :

α f 0 0.000002674 - 0.000003665 0.000002599 - 0.000000408 51 mat

The support points data XY o 5 black plot : x c 10 x ≤ (x 1.25 ≤ (& cases : 21 sys

Fit " a la carte" from LMA

data x α Φ x c * support 3 1 sys

xy 00 0.079 3.782 0.29 0.77 0.545 0.068 1.154 052 mat

Example_3 ... selected fit Pharmacokinetics data set

XY D3 :

===== D3 data set =====

D3 rows 27

x β Clear 1

model to D3(x) x ft β 1 el β 2 el x - β 3 el + β 5 el / exp x - β 3 el + (β 4 el β 5 el + β 4 el β 5 el * / * exp - (* - :

1. selct the support points
2. collect for LMA list 1514222715 mat : xy XY list ListCollect : 21 sys

raw XY o 5 black plot : support xy . 15 blue plot : 21 line

support raw 2 1 sys

xy 0 10.861 0.079 0.706 0.29 0.226 - 0.545 0.031 - 1.154 052 mat

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

vector of guesses ... β 01 0.1 0.075 0.1 15 mat transpose :

Tol 106 -^:

t0 1 time :

β x y β Tol βs LMSA : 0.00024 1.0326 0.1063 0.0623 0.12545 51 mat

1 time t0 - min 2.4

model to D3(x) x β Φ β 1 el β 2 el x - β 3 el + β 5 el / exp x - β 3 el + (β 4 el β 5 el + β 4 el β 5 el * / * exp - (* - :

Implicit equalities = 0 β f x 1 el β Φ y 1 el - x 2 el β Φ y 2 el - x 3 el β Φ y 3 el - x 4 el β Φ y 4 el - x 5 el β Φ y 5 el - 51 mat :

β f 0.000002493 0.000000045 0.000000226 0.000000686 - 0.000000470 51 mat

data XY o 5 black plot : x c 10 x ≤ (x 41.25 ≤ (& cases : 21 sys

Fit " a la carte" from listcollect(XY,list)

data x β Φ x c * support 3 1 sys

xy 0 10.861 0.079 0.706 0.29 0.226 - 0.545 0.031 - 1.154 052 mat

Example_4 ... selected fit NIST Thurber data set

XY Thurber

XY 2.981 - 84.248 2.921 - 87.264 2.912 - 87.185 2.84 - 89.076 2.797 - 89.608 2.702 - 89.868 2.699 - 90.101 2.633 - 92.405 2.481 - 95.854 2.363 - 100.696 2.322 - 101.06 1.501 - 401.672 1.46 - 390.724 1.274 - 567.534 1.212 - 635.316 1.1 - 7330.54 1.046 - 759.087 0.915 - 894.206 0.714 - 990.785 0.566 - 1090 0.545 - 1080 0.4 - 1120 0.309 - 1180 0.109 - 1260 0.103 - 1270 0.01 1290 0.119 1330 0.377 1350 0.79 1420 0.963 1430 1.006 1420 1.115 1440 1.572 1460 1.841 1470 2.047 1450 2.2 1460362 mat :

x β Clear 1

SGompertz Thurber model fnct x ft β 1 el β 2 el β 3 el x β 4 el - (* - exp - exp * + :

1. selct the support points
2. collect for LMA list 113243414 mat : xy XY list ListCollect : 21 sys

raw XY o 5 red plot : support xy . 16 black plot : 21 line

support raw 2 1 sys

xy 2.981 - 84.248 1.46 - 390.724 0.109 - 1260 1.841 147042 mat

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

vector of guesses ... β 8014001014 mat transpose :

Tol 1012 -^:

t0 1 time :

μ x y β Tol βs LMSA : 84.247968764855 1396.05301175173 1.612152171278 1.201804844806 - 41 mat

1 time t0 - s 19

SGompertz Thurber model fnct x μ Φ μ 1 el μ 2 el μ 3 el x μ 4 el - (* - exp - exp * + :

Implicit equalities = 0 μ f x 1 el μ Φ y 1 el - x 2 el μ Φ y 2 el - x 3 el μ Φ y 3 el - x 4 el μ Φ y 4 el - 41 mat :

μ f 0.000000242172 0.000000005597 - 0.000000021279 0.000000173642 - 41 mat

data XY o 5 black plot : x c 14 - x ≤ (x 4 ≤ (& cases : 21 sys

data x μ Φ x c * support 3 1 sys

xy 2.981 - 84.248 1.46 - 390.724 0.109 - 1260 1.841 147042 mat

Else fit Companion

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 :

Weibull reference x k λ W k λ / x λ / (k λ -^* x λ / (k^- exp * :

t0 1 time :

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 :

<= 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 ≡

<= 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 5 :

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 :

corr 0.6678

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

β 0.5022 2.0003 21 mat

β 0.5022 2.0003 21 mat

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 0.049 - 0.511 0.093 0.701 0.044 - 0.891 0.005 1.082 0.018 1.27 0.001 1.46 0.05 - 1.65 0.016 1.839 0.022 - 2.029 0.032 2.219 0.007 122 mat

1 time t0 - 3 s *

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

SSD 0.019

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 :

x 1012 -^0.51 0.82 1.02 1.23 1.43 1.85 2.05 2.36 4.52 9.22 111 mat :

y 0.78 0.8 0.92 1.14 1.26 1.28 1.22 1.16 1.12 1.08 1.08 111 mat :

β Clear 1

Piecewise L_M Carlos Original

β β 1 el β 2 el β 3 el β 4 el 41 mat :

Model function to fit the given data set β 2 el β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ / + β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ (2^/ x β 3 el / (β 4 el^* β 4 el * -

n β length : m x length : 21 line

initialise the solver init 0.5 11114 mat transpose :

t0 1 time :

typical for this example tol 0.00000001 :

L-M Components => algo solves 'β'

x β ia β 2 el β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ / + β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ (2^/ x β 3 el / (β 4 el^* β 4 el * - : ................................... i 1 m range ΦΦ i el x i el β ia y i el - : for β Φ ΦΦ : 21 line ................................... i 1 m range j 1 n range β j el β j el h + : jjh1 x i el β ia y i el - : β j el β j el 2 h * - : jjh2 x i el β ia y i el - : jjh0 i j el jjh1 jjh2 - 2 h * / : β j el β j el h + : 61 line for 11 line for ................................. β derfhφ jjh0 : 71 line

h tol : μ 0.01 : μ1 100 : 31 sys

cont 0 : dnor 1 : update 1 : 31 line dnor tol > update 1 ≡ f init Φ : J init derfhφ : jtj J transpose J * : res f norme (2^: 41 line if 11 line a jtj μ n identity * + : s1 a 1 -^J transpose f * (* : b1 init s1 - (: f1 b1 Φ : SSD f1 norme (2^μ1 / : 51 line SSD res < init b1 : f f1 : 12 mat dnor s1 normi init normi / : cont cont 1 + : Sol cont 1 el cont : z 1 n range Sol cont z 1 + el init z el : 11 line for Sol cont n 2 + el SSD μ1 * : Sol cont n 3 + el μ : Sol cont n 4 + el dnor : μ μ 10 / : update 1 : 101 line μ μ 10 * : update 0 : 21 line if 31 line while 21 line

assign 'β' from solved 'init' β init :

model function to fit wrt data set x β f β 2 el β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ / + β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ (2^/ x β 3 el / (β 4 el^* β 4 el * - (:

β 0.777335 1.072745 1.278127 4.800823 41 mat

1 time t0 - 29 s *

Levenberg-Marquardt . . .

1. Model [Jean] designed for Art MCD 11.

2. LM slight recast from Carlos 2018_09_20

3. This model is robust as experimented.

4. Not all initials can be same. Attempt to

cross the model twice through the data set.

]]>

FIT range, populate U 012 0.1 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 41 line :

data x y o 5 black plotG : datum 0 0.78 + 100 blue 15 mat : 21 sys

Absolute residuals Residuals i 1 x rows range Δ i el y i el x i el β f - eval : for x Δ augment 21 line :

1.08 FIT data datum 4 1 sys