ListCollect ... Jacobian

8b3c5f17-e588-4bc3-89bc-2b08a9521bad 17 4 5 decimal

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

Initiatic visit to fitting

data list ListCollect 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 41 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 :

AlgLib Solver for a system of nonlinear equations(Uni) init Tol sol k 1 α f rows range u k el α k el : for α Jac u f u Jacobian : StepMax 0 : ε 1015 -^: 12 mat u init StepMax Tol α f α Jac al_nleqsolve : 41 line :

clear ln0 undefined ε 1012 -^:

0. support points solve al_nleqsolve

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UV ε 0.78 0.51 0.8 0.82 0.92 1.02 1.14 1.23 1.26 1.43 1.28 1.85 1.22 2.05 1.16 2.36 1.12 4.52 1.08 9.22 1.08 112 mat :

1. selct the support points
2. collect for al_nleqsolve list 136914 mat : new UV list ListCollect : 21 sys

raw UV o 5 black plot : support new . 14 blue plot : 21 line

support raw 2 1 sys

new 11012 -^* 0.78 0.82 0.92 1.43 1.28 2.36 1.12 42 mat

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

Automated al_nleqsolve ... NUMERIC/SYMBOLIC x α Φ α 2 el α 1 el α 2 el - 1 x α 3 el / (α 4 el^+ / + α 1 el α 2 el - 1 x α 3 el / (α 4 el^+ (2^/ x α 3 el / (α 4 el^* α 4 el * - : init 111114 mat transpose : α f ct 1 : j 1 x rows range Q ct el x j el α Φ y j el - : ct ct 1 + : 21 line for Q 31 line : 31 line

Setup similar Mathcad 1. model function 2. initials 3. implicit conditions 4. result

t0 1 time :

Tol 109 -^:

0.5 11114 mat transpose fails ≡

α init Tol sol : 0.78 1.063448 1.316802 4.850157 41 mat

1 time t0 - s 9

Original version 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 - 41 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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sanity @ support points α f 21010 -^* - 11010 -^* 31010 -^* - 31010 -^* 41 mat

α 0.78 1.063448 1.316802 4.850157 41 mat

x Clear 1

u 0 x ≤ (x 10 ≤ (* :

Fit " a la carte" from al_nleqsolve fitted exact @ support points ...

raw x α Φ u / support 3 1 sys 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 :

β Clear 1

1. Levenberg-Marquardt

solves global [this example] or from selection

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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 β 1 el β 2 el β 3 el β 4 el 41 mat :

Copy/paste model function in L-M  module 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 111114 mat transpose :

t0 1 time :

typical for this example tol 0.0001 :

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 :

recover @ 'tol' β β 4 round vectorize :

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 0.7773 1.0727 1.2781 4.8009 41 mat

1 time t0 - 22 s *

FIT U 012 0.1 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 31 line :

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.

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data x y o 5 black plotG : x c 10 x ≤ (x 12 ≤ (& cases : 21 sys

1.08 FIT data 3 1 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 :

x β f 1.08 - 0 ≡ x 0.5 5 solve 0.976 3.637 21 mat

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 :

2. Meritas Conjugate Gradient

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clear ln0 undefined ε 1015 -^:

XY ε 0.78 0.51 0.8 0.82 0.92 1.02 1.14 1.23 1.26 1.43 1.28 1.85 1.22 2.05 1.16 2.36 1.12 4.52 1.08 9.22 1.08 112 mat :

X XY 1 col : Y XY 2 col : 12 mat Data XY o 5 black plot : 21 line

3 appVersion 0.98.6179 ≡ Timing 5 sec ≡ 21 line

Model function ...
designed for => Art
secret application
Pharmacokinetics . 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 * - :

last index β # el ≡

<= Gradient Vector Function x β φ x β f 4 φ :

1. ~ 4 * faster than L-M 2. Different init [0.5 1 1 1]

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init 0.5 11114 mat transpose :

iter 7 :

t0 1 time :

algo solving for β β init iter Minimize :

assign/isolate .. β β 4 round vectorize :

1 time t0 - s 5

SSD 0.0015

corr 1.077

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 :

bar δ 0.0375 BarSize (:

bar

β 0.779 1.0725 1.2817 4.8248 41 mat

FIT U 012 0.025 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 31 line :

0.779 1.0725 1.2817 4.8248 41 mat

0.7773 1.0727 1.2781 4.8009 41 mat

Title 2 1.0625 Pharmacokinetics 14 blue 15 mat : datum 0 0.78 + 100 blue 15 mat : Δ δ Residuals 2 col augment : 31 sys

comparative abs.. residuals Δ 11015 -^* 0.001 0.001 0.51 0.001 0.001 0.82 0.021 - 0.021 - 1.02 0.023 0.023 1.23 0.002 - 0.002 - 1.43 0.014 - 0.014 - 1.85 0.014 0.014 2.05 0.005 - 0.005 - 2.36 0.005 - 0.005 - 4.52 0.005 0.005 9.22 0.007 0.007 113 mat

1.08 FIT Data Title datum 5 1 sys

Levenberg-Marquardt to fit a parametric function . . .

]]> Plot Utilities

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 :

L-M slightly refactored from ... Carlos 2018_09_20

x Clear 1

HEADER ITER β1 β2 β3 β4 Res1 μ dnor 18 mat :

t 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 :

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 * - (:

X x 1 el x 2 el x 3 el x 4 el 41 mat :

n X length : m t length : 21 line

3 appVersion 0.98.6179 ≡ Timing_LM ~ 10 s ≡ ConjugateGradient 5 s ≡ 31 line

t0 1 time :

Tol . . . Mathcad style

1. Start 0.01 . . . check residuals.

2. Adjust at will to minimize timing.

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Tol 0.0001 :

L-M Components

t X ia X 2 el X 1 el X 2 el - 1 t X 3 el / (X 4 el^+ / + X 1 el X 2 el - 1 t X 3 el / (X 4 el^+ (2^/ t X 3 el / (X 4 el^* X 4 el * - :

i 1 m range ΦΦ i el t i el X ia y i el - : for X Φ ΦΦ : 21 line

i 1 m range j 1 n range X j el X j el Tol + : jjh1 t i el X ia y i el - : X j el X j el 2 Tol * - : jjh2 t i el X ia y i el - : jjh0 i j el jjh1 jjh2 - 2 Tol * / : X j el X j el Tol + : 61 line for 11 line for

X derfhφ jjh0 :

initialise the solver x 0.5 11314 mat transpose :

L-M algorithm

h Tol abs : μ 0.01 : μ1 104.5069 : 31 sys

x x :

cont 0 : dnor 1 : update 1 : 31 line dnor Tol > update 1 ≡ f x Φ : J x 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 x s1 - (: f1 b1 Φ : res1 f1 norme (2^μ1 / : 51 line res1 res < x b1 : f f1 : 12 mat dnor s1 normi x normi / : cont cont 1 + : Sol cont 1 el cont : z 1 n range Sol cont z 1 + el x z el : 11 line for Sol cont n 2 + el res1 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

β x : 0.777336 1.072745 1.278127 4.800831 41 mat

recast for: SSD, corr X t : Y y : 12 mat

corr ... SSD

corr Pearson correlation χ X Mean : μ Y Mean : n X 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 :

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

FIT U 020 0.05 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 31 line :

β 0.77734 1.07274 1.27813 4.80083 41 mat

corr SSD 21 mat 1.079 0.001 21 mat

1 time t0 - 14 s *

t y augment FIT 2 1 sys

Absolute residuals Δ i 1 t rows range Δ i el y i el t i el β f - 3 round : for t Δ augment 21 line :

Absolute residuals

Δ 0.025 BarSize 1 1 sys

1 time t0 - min 0.4

digit data Table digit 9 :

Sol Sol digit round vectorize :

Table Sol :

Table digit res1 HEADER Δ datum Title iter Fit.Y corr δ xd Error Minimize Residuals c FIT f z Sol SSD res update dnor cont μ1 μ h jjh2 jjh1 j i ia tol m n BarSize plotG u Tol t0 list ε sol plot ListCollect lastError Table digit res1 t HEADER Δ iter φ Y X Fit.Y corr δ xd Error Minimize Residuals c FIT f z Sol SSD f1 b1 s1 res update dnor cont μ1 μ h jjh2 jjh1 j Φ ΦΦ i ia tol m n β BarSize plotG u α Tol t0 init y x ε sol plot ListCollect lastError digit res1 iter f z SSD res update dnor cont μ1 μ h jjh2 jjh1 j i ia tol m n u Tol t0 ε

LM components ITER evolution
Table
1	0.780615261	1.085046575	1.244797232	3.371776441	0.151897526	0.01	0.110261296
2	0.782374072	1.06920314	1.297555075	4.639383094	0.021517126	0.001	0.273227416
3	0.777193471	1.072743477	1.276955153	4.793088981	0.00310503	0.0001	0.032068232
4	0.777375064	1.072740551	1.278173509	4.801943712	0.001494009	0.00001	0.001843989
5	0.77733217	1.072745165	1.278121414	4.800734758	0.001492354	0.000001	0.000251827
6	0.777335788	1.072744963	1.278127443	4.800830971	0.001492346	0.0000001	0.000020041
7	0.777482226	1.072734906	1.278389538	4.804237387	0.001492461	0.00000001	0.010816169
8	0.77732302	1.072745947	1.278108667	4.800476156	0.001492346	0.000000001	0.000783512
9	0.777336547	1.072744859	1.278128152	4.8008539	0.001492346	0	0.000078683
#
ITER	β1	β2	β3	β4	Res1	μ	dnor
#
#
#

Table HEADER Sol stack :

Table digit res1 HEADER Δ datum Title iter Fit.Y corr δ xd Error Minimize Residuals c FIT f z SSD res update dnor cont μ1 μ h jjh2 jjh1 j i ia tol m n BarSize plotG u Tol t0 list ε sol plot ListCollect lastError Table digit res1 t HEADER Δ iter φ Y X Fit.Y corr δ xd Error Minimize Residuals c FIT f z Sol SSD f1 b1 s1 res update dnor cont μ1 μ h jjh2 jjh1 j Φ ΦΦ i ia tol m n β BarSize plotG u α Tol t0 init y x ε sol plot ListCollect lastError digit res1 iter f z SSD res update dnor cont μ1 μ h jjh2 jjh1 j i ia tol m n u Tol t0 ε

Table
ITER	β1	β2	β3	β4	Res1	μ	dnor
1	0.780615261	1.085046575	1.244797232	3.371776441	0.151897526	0.01	0.110261296
2	0.782374072	1.06920314	1.297555075	4.639383094	0.021517126	0.001	0.273227416
3	0.777193471	1.072743477	1.276955153	4.793088981	0.00310503	0.0001	0.032068232
4	0.777375064	1.072740551	1.278173509	4.801943712	0.001494009	0.00001	0.001843989
5	0.77733217	1.072745165	1.278121414	4.800734758	0.001492354	0.000001	0.000251827
6	0.777335788	1.072744963	1.278127443	4.800830971	0.001492346	0.0000001	0.000020041
7	0.777482226	1.072734906	1.278389538	4.804237387	0.001492461	0.00000001	0.010816169
8	0.77732302	1.072745947	1.278108667	4.800476156	0.001492346	0.000000001	0.000783512
9	0.777336547	1.072744859	1.278128152	4.8008539	0.001492346	0	0.000078683
#
ITER	β1	β2	β3	β4	Res1	μ	dnor
#
#
#

in Table settings: 1. check caption ... 2. type Table%T in that "caption space" 2. ... by default Tble2 ... set Table