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 :

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

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 :

x ft 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 * - :

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 0.5 11141 mat :

Tol 1016 -^:

t0 1 time :

x y β Tol βs LMSA 0.7773 1.073 1.278 4.801 41 mat

0.77717 1.07277 1.27792 4.79477 41 mat

0.779 1.07253 1.28167 4.82484 41 mat

1 time t0 - s 7.493

t βs ft βs 2 el βs 1 el βs 2 el - 1 t βs 3 el / (βs 4 el^+ / + βs 1 el βs 2 el - 1 t βs 3 el / (βs 4 el^+ (2^/ t βs 3 el / (βs 4 el^* βs 4 el * - :

t x :

m x rows :

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 : 0.0015

RMS difference SSD m / sqrt 0.0116

RMSD SSD m / sqrt :

Relative residuals

Residuals RMSD BarSize 1 1 sys

data x y . 12 blue plotG :

FIT range, populate U 010 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

x β Clear 1.000

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 0.0000000001 :

t0 1 time :

x y β Tol βs LMSA 0.5022 2.0004 21 mat

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

1 time t0 - s 1.109

t x :

m x rows :

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 :

data FIT 2 1 sys

FIT range, populate U 03 0.025 range : i 1 U rows range vy i el U i el βs ft eval : for U vy augment 41 line :

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 :

XY D2 :

Point solve equalities al_nleqsolve

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

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 00 0.079 3.782 0.29 0.77 0.545 0.068 1.154 052 mat

vector of guesses ... 
from robust2crucifying ! β 1012 -^105 - 0.1 5 - 15 mat transpose :

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

Tol 1016 -^:

Jean Solution 1.1077 142.40751 1.01751 - 0.07235 13.58616 - 51 mat

x y β Tol βs LMSA 2.1283 142.4114 2.0372 - 0.0723 26.376 - 51 mat

t0 1 time :

1 time t0 - s 0

α βs :

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 2.12827 142.41144 2.03717 - 0.07235 26.37602 - 51 mat

Notes . . . Smath 7219

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 f 0000051 mat

The support points X XY 1 col : Y XY 2 col : 12 mat data XY o 5 black plot : 21 sys

x c 10 x ≤ (x 1.25 ≤ (& cases :

Fit " a la carte" from LMA

data x α Φ x c * support 3 1 sys

Support points xy 00 0.079 3.782 0.29 0.77 0.545 0.068 1.154 052 mat

XY D3 :

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

D3 rows 27

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

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 - (* - :

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

x 0.000 0.07900 0.2900 0.5450 1.154 51 mat

y 10.86 0.7060 0.2260 - 0.03100 - 0.000 51 mat

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

vector of guesses ... 
from robust2crucifying ! β 01 0.1 0.075 0.1 15 mat transpose :

Tol 1016 -^:

x y β Tol βs LMSA 0.00024 1.03264 0.1063 0.06231 0.12544 51 mat

t0 1 time :

Jean Solution 0.00071 1.0028 0.10647 0.06114 0.12774 51 mat

1 time t0 - s 0

β βs :

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 0.00024 1.03264 0.1063 0.06231 0.12544 51 mat

The support points X XY 1 col : Y XY 2 col : 12 mat data XY o 5 black plot : 21 sys

x c 10 x ≤ (x 1.25 ≤ (& cases :

Fit " a la carte" from al_nleqsolve

data x β Φ x c * support 3 1 sys

β f 5.588 1012 -^* 6.377 1013 -^* 5.117 1011 -^* 1.627 1010 -^* - 1.112 1010 -^* 51 mat

Jean 2.932 106 -^* 2.063 106 -^* 0.0001973 0.0006440 - 0.0004336 51 mat

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

vector of guesses ... 
from robust2crucifying ! β 8014001014 mat transpose :

Tol 1016 -^:

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

SGompertz . . .

is extremely reflexive, declared unfittable

in Smath [L-M, CG]. Possible attempt

OriginLab, Mathcad 11. As offered, and

fitted by al_nleqsolve, at this point, you

may want to round/refactor the coefficients.

The fillet region is a bit sparse !

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t0 1 time :

x y β Tol βs LMSA 84.24796852 1396.05301219 1.61215217 1.20180484 - 41 mat

84.25 1396.05 1.61 1.2 - 41 mat

1 time t0 - s 1

SGompertz Thurber model fnct x m Φ m 1 el m 2 el m 3 el x m 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 βs : 84.25 1396.05 1.61 1.2 - 41 mat

Notes . . . Smath 7251

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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The support points X XY 1 col : Y XY 2 col : 12 mat data XY o 5 black plot : 21 sys

μ f 0.000000000051 0 0.00000000001 - 0.000000000031 - 41 mat

0.000000000000 0.000000000003 - 0.000000000000 0.000000000002 41 mat

x c 14 - x ≤ (x 4 ≤ (& cases :

μ 84.25 1396 1.61 1.2 - 41 mat 11 mat :

j 1 :

data x μ 1 j el Φ x c * support 3 1 sys

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