ListCollect ... Jacobian

d440b766-3a5b-435c-ab63-e81a60e026da 13 15 12 decimal

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

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 :

sol(init,Tol) ... AlgLib Solver for a system of nonlinear equations
adapted from Viacheslav original 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 :

1. Point solve equalities al_nleqsolve

XY 0 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 135914 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 0 0.78 0.82 0.92 1.23 1.26 2.36 1.12 42 mat

x xy 1 col : y xy 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 0.5 11114 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 106 -^:

α init Tol sol : 0.78 1.06798 1.305237 5.080286 41 mat

1 time t0 - s 5

sanity @ support points α f 0.000000040 - 0.000000045 0.000000033 - 0.000000027 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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x Clear 1

u 0 x ≤ (x 10 ≤ (* :

Fit " a la carte" from al_nleqsolve

1.08 x α Φ u / raw support 4 1 sys

2. Refine global ... Minimize(init,iter)

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 :

Minimize(init,iter) ... adapted [Jean]  from Alyles contribution 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 :

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 3 sec ≡ 21 line

Model function ...
designed for => Art
confidential application 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. ~ 3 * faster than al_nleqsolve 2. Different init [0.5 1 1 4]

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init 0.5 11414 mat transpose :

iter 5 :

t0 1 time :

algo solving for β β init iter Minimize :

assign/isolate .. β β 4 round vectorize :

1 time t0 - s 3

SSD 0.0015

corr 1.0797

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 (:

δ 11015 -^* 0.002 0.51 0.002 0.82 0.023 - 1.02 0.02 1.23 0.003 - 1.43 0.014 - 1.85 0.014 2.05 0.006 - 2.36 0.005 - 4.52 0.005 9.22 0.007 112 mat

bar

FIT U 010 0.01 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 31 line :

Title 21 Pharmacokinetics 12 black 15 mat : datum 0 0.78 + 100 blue 15 mat : 21 sys

x β f 1.08 - 0 ≡ x 0.5 5 solve 0.976 3.635 21 mat

plot 1.08 FIT datum data Title 3.64 1.08 + 50 blue 15 mat 61 sys :

plot 0 x * 1.08 + FIT XY 3 1 sys