Data, plot(data,char,size,clr)

ac895819-42d3-4d74-bc9c-7b2568a11f95 110 3 5 decimal

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

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

data 101002 0.020543 0.8175 0.1578 0.0247 3 0.039978 0.6727 0.246 0.0813 4 0.059347 0.5538 0.2984 0.1478 5 0.078943 0.4509 0.3312 0.2179 6 0.099077 0.3695 0.3362 0.2943 7 0.12023 0.3002 0.3306 0.3692 8 0.14139 0.2363 0.3304 0.4333 9 0.1643 0.1906 0.317 0.4924 10 0.18788 0.1533 0.3074 0.5393 11 0.21281 0.1194 0.2954 0.5852 12 0.23698 0.0931 0.2892 0.6177 13 0.26292 0.0728 0.2842 0.6430 14 0.29031 0.0562 0.28 0.6638 15 0.31787 0.0432 0.27 0.6868 16 0.34602 0.0343 0.2636 0.7021 17 0.37686 0.0254 0.2628 0.7118 18 0.40634 0.019 0.2622 0.7188 19 0.43691 0.0137 0.254 0.7323 20 0.47015 0.0098 0.2518 0.7384 21 0.50899 0.0067 0.2514 0.7419 22 0.54488 0.0051 0.2492 0.7458 23 0.59066 0.0031 0.2538 0.7432 24 0.64485 0.0015 0.255 0.7435 25 0.9178 0.00011 0.2564 0.7435 26 1.1252 0 0.2562 0.7438 27 1.1536 0 0.2506 0.7494 275 mat :

t data 2 col : A t data 3 col augment : B t data 4 col augment : C t data 5 col augment : 41 sys

A A o 5 blue plot : B B o 5 red plot : C C o 5 green plot : 31 sys

4 appVersion 0.98.6179.21440 ≡ iming 25 sec ≡ 21 mat

A B C 3 1 sys

The project . . .

A, B, C are Kinetics experimental data from Viktor.

An excellent fit was obtained from Mathcad Minerr ODEsolve.

In this document, we directly fit model functions to data.

Smath Genfit and Genfit-Observer conclude the project with

analytical Kinetics functions. Each model session is independent

and can be copied/pasted in new document.

]]> Genfit 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 :

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 :

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 :

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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Vector gradient ... Optimiz Symbolic 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 :

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 :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

XY 01 0.020543 0.8175 0.039978 0.6727 0.059347 0.5538 0.078943 0.4509 0.099077 0.3695 0.12023 0.3002 0.14139 0.2363 0.1643 0.1906 0.18788 0.1533 0.21281 0.1194 0.23698 0.0931 0.26292 0.0728 0.29031 0.0562 0.31787 0.0432 0.34602 0.0343 0.37686 0.0254 0.40634 0.019 0.43691 0.0137 0.47015 0.0098 0.50899 0.0067 0.54488 0.0051 0.59066 0.0031 0.64485 0.0015 0.9178 0.00011 1.1252 0 1.1536 0272 mat :

Exponential decay_2

<= model function x β f 0.01 β 1 el x - β 3 el / exp * + β 2 el x - β 4 el / exp * + :

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

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

X XY 1 col : Y XY 2 col : n XY rows : 31 line

range/populate L 0 : H 2 : N 100 : 13 mat U L H N xd : 21 line

1 0.025 0.125 0.5 41 mat 0.9999 0.007 - 0.0989 2.7952 - 41 mat 12 mat

β β 1 el 1 0.025 0.125 0.5 41 mat : iter 5 : i 1 : i iter ≤ β i 1 + el XY f φ β i el Minimize : i i 1 + : 21 line while β β iter el : 51 line :

β β : 0.999883 0.007032 - 0.098886 2.795226 - 41 mat

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 Error :

bar

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

data XY . 12 black plot :

data FIT 2 1 sys

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 0.00008

Genfit 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 :

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 :

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 :

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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Vector gradient ... Optimiz Symbolic 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 :

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 :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

XFR model function from observer

XY 00 0.020543 0.0247 0.039978 0.0813 0.059347 0.1478 0.078943 0.2179 0.099077 0.2943 0.12023 0.3692 0.14139 0.4333 0.1643 0.4924 0.18788 0.5393 0.21281 0.5852 0.23698 0.6177 0.26292 0.643 0.29031 0.6638 0.31787 0.6868 0.34602 0.7021 0.37686 0.7118 0.40634 0.7188 0.43691 0.7323 0.47015 0.7384 0.50899 0.7419 0.54488 0.7458 0.59066 0.7432 0.64485 0.7435 0.9178 0.7435 1.1252 0.7438 1.1536 0.7494 272 mat :

'Kp' from the asymptote
'α' ± from trial/error Kp 0.75 - : α 10 : 21 sys

x β f Kp β 1 el - β 2 el + (β 2 el α x * β 3 el - (- β 2 el / exp * - β 1 el α x * β 3 el - (- β 1 el / exp * + β 1 el β 2 el - (/ * :

X XY 1 col : Y XY 2 col : n XY rows : 31 line

range/populate L 0 : H 2 : N 100 : 13 mat U L H N xd : 21 line

1.75 0.25 031 mat 1.040605 0.443615 0.022534 31 mat 12 mat

<= Gradient Vector Function x β φ x β f 3 φ :

β β 1 el 1.75 0.25 031 mat : iter 7 : i 1 : i iter ≤ β i 1 + el XY f φ β i el Minimize : i i 1 + : 21 line while β β iter el : 51 line :

β β : 1.040605 0.443615 0.022534 31 mat

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 Error :

Residuals observer for 'α'

bar

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

data XY . 12 black plot :

data FIT 2 1 sys

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 0.00026

Genfit 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 :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 :

Vector gradient ... Optimiz Symbolic 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 :

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 :

XY 00 0.020543 0.1578 0.039978 0.246 0.059347 0.2984 0.078943 0.3312 0.099077 0.3362 0.12023 0.3306 0.14139 0.3304 0.1643 0.317 0.18788 0.3074 0.21281 0.2954 0.23698 0.2892 0.26292 0.2842 0.29031 0.28 0.31787 0.27 0.34602 0.2636 0.37686 0.2628 0.40634 0.2622 0.43691 0.254 0.47015 0.2518 0.50899 0.2514 0.54488 0.2492 0.59066 0.2538 0.64485 0.255 0.9178 0.2564 1.1252 0.2562 1.1536 0.2506 272 mat :

Genfit-observer for reactive model(s)

from residuals observer
good guess proven ... fixed
good guess proven ... fixed
asymptote guess from data α 4 0.25 0.5 0.25 41 mat :

asymptote red 0.25 asymptote green-red 0.75-0.25 asymptote red 0.25

x β f α 4 el β 1 el 1 α 1 el x * (β 2 el - α 2 el / - exp - (* α 1 el x * (β 2 el - α 3 el / - exp * + :

X XY 1 col : Y XY 2 col : n XY rows : 31 line

range/populate L 0 : H 2 : N 200 : 13 mat U L H N xd : 21 line

0.2 0.1 21 mat 0.21256 0.157099 21 mat 12 mat

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

β β 1 el 0.2 0.1 21 mat : iter 5 : i 1 : i iter ≤ β i 1 + el XY f φ β i el Minimize : i i 1 + : 21 line while β β iter el : 51 line :

β β : 0.21256 0.157099 21 mat

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 Error :

Residuals observer [α1 = 4]

bar

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

data XY o 5 black plot :

data FIT 2 1 sys

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 0.00045

0 β f 0.005 -

sightly better fit @ x = 0 β 0.225 0.150113 21 mat :

0 β f 0.000001

This section is stand alone

Data, plot(data,char,size,clr)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

data 101002 0.020543 0.8175 0.1578 0.0247 3 0.039978 0.6727 0.246 0.0813 4 0.059347 0.5538 0.2984 0.1478 5 0.078943 0.4509 0.3312 0.2179 6 0.099077 0.3695 0.3362 0.2943 7 0.12023 0.3002 0.3306 0.3692 8 0.14139 0.2363 0.3304 0.4333 9 0.1643 0.1906 0.317 0.4924 10 0.18788 0.1533 0.3074 0.5393 11 0.21281 0.1194 0.2954 0.5852 12 0.23698 0.0931 0.2892 0.6177 13 0.26292 0.0728 0.2842 0.6430 14 0.29031 0.0562 0.28 0.6638 15 0.31787 0.0432 0.27 0.6868 16 0.34602 0.0343 0.2636 0.7021 17 0.37686 0.0254 0.2628 0.7118 18 0.40634 0.019 0.2622 0.7188 19 0.43691 0.0137 0.254 0.7323 20 0.47015 0.0098 0.2518 0.7384 21 0.50899 0.0067 0.2514 0.7419 22 0.54488 0.0051 0.2492 0.7458 23 0.59066 0.0031 0.2538 0.7432 24 0.64485 0.0015 0.255 0.7435 25 0.9178 0.00011 0.2564 0.7435 26 1.1252 0 0.2562 0.7438 27 1.1536 0 0.2506 0.7494 275 mat :

Experimental data set t data 2 col : A t data 3 col augment : B t data 4 col augment : C t data 5 col augment : 41 sys

A A o 5 black plot : B B o 5 black plot : C C . 10 black plot : 31 sys

t Ω k D k 1 el Ω 1 el * - k 1 el Ω 1 el * k 2 el Ω 2 el * - k 3 el Ω 3 el * + k 2 el Ω 2 el * k 3 el Ω 3 el * - 31 mat :

S 10031 mat 0 1.25 50 t Ω 1015531 mat D Rkadapt :

t S 1 col : Xs t S 2 col augment : Ys t S 3 col augment : Zs t S 4 col augment : 41 sys

Xs Ys Zs A B C 6 1 sys

k 1 el α ≡

k1 is independent and quickly found [10] From k2=k3=1 ... an inductive aspect ratio 3/1 is sugestive. Then attempt the same principle as for k1. By successive trials [6,2 ... 9,3 ... 12,4 ... 15,5] the superb match is found.

k 11131 mat ≡

k 101131 mat ≡

k 103131 mat ≡

k 1015531 mat ≡

ratio k2/k3 Kp α 2 el / - 3 ≡

Numerical setup  ODEsolve k 1 el Ω 1 el * - k 1 el Ω 1 el * k 2 el Ω 2 el * - k 3 el Ω 3 el * + k 2 el Ω 2 el * k 3 el Ω 3 el * - 31 mat

Data, plot(data,char,size,clr)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

data 101002 0.020543 0.8175 0.1578 0.0247 3 0.039978 0.6727 0.246 0.0813 4 0.059347 0.5538 0.2984 0.1478 5 0.078943 0.4509 0.3312 0.2179 6 0.099077 0.3695 0.3362 0.2943 7 0.12023 0.3002 0.3306 0.3692 8 0.14139 0.2363 0.3304 0.4333 9 0.1643 0.1906 0.317 0.4924 10 0.18788 0.1533 0.3074 0.5393 11 0.21281 0.1194 0.2954 0.5852 12 0.23698 0.0931 0.2892 0.6177 13 0.26292 0.0728 0.2842 0.6430 14 0.29031 0.0562 0.28 0.6638 15 0.31787 0.0432 0.27 0.6868 16 0.34602 0.0343 0.2636 0.7021 17 0.37686 0.0254 0.2628 0.7118 18 0.40634 0.019 0.2622 0.7188 19 0.43691 0.0137 0.254 0.7323 20 0.47015 0.0098 0.2518 0.7384 21 0.50899 0.0067 0.2514 0.7419 22 0.54488 0.0051 0.2492 0.7458 23 0.59066 0.0031 0.2538 0.7432 24 0.64485 0.0015 0.255 0.7435 25 0.9178 0.00011 0.2564 0.7435 26 1.1252 0 0.2562 0.7438 27 1.1536 0 0.2506 0.7494 275 mat :

Experimental data t data 2 col : A t data 3 col augment : B t data 4 col augment : C t data 5 col augment : 41 sys

This section is stand alone

A A o 5 black plot : B B o 5 black plot : C C o 5 black plot : 31 sys

β 1 0.007 - 0.0989 2.7952 - 41 mat :

0.999883 0.007032 - 0.098886 2.795226 - 41 mat

<= model function x β f1 0.01 β 1 el x - β 3 el / exp * + β 2 el x - β 4 el / exp * + :

'Kp' from the asymptote
'α' ± from trial/error Kp 0.75 - : α 10 : 21 sys

x b f2 Kp b 1 el - b 2 el + (b 2 el α x * b 3 el - (- b 2 el / exp * - b 1 el α x * b 3 el - (- b 1 el / exp * + b 1 el b 2 el - (/ * :

1.040605 0.443615 0.022534 31 mat

b 1.040605 0.443615 0.022534 31 mat :

from residuals observer
good guess proven ... fixed
good guess proven ... fixed
asymptote guess from data α 4 0.25 0.5 0.25 41 mat :

ρ 0.225 0.150113 21 mat :

x c 10 x ≤ cases :

x ρ f3 α 4 el ρ 1 el 1 α 1 el x * (ρ 2 el - α 2 el / - exp - (* α 1 el x * (ρ 2 el - α 3 el / - exp * + :

x β f1 x c * x b f2 x c * x ρ f3 x c * A B C 6 1 sys 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

The two interesting Kinetics derivatives

x D2 x b f2 x diff : x D3 x ρ f3 x diff : 21 sys

Locate min/max [Golden ratio]

x D2 x c * x D3 x c * 2 1 sys

max min 21 mat 0.068 0.175 21 mat ≡

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

Max/Min Golden ratio

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

x f x D2 : max f 0 0.125 y1 y2 ≥ 1012 -^Extrema 1 el : 21 sys

x f x D3 : min f 0.125 0.25 y1 y2 ≤ 1012 -^Extrema 1 el : 21 sys

max/min Derivatives max min 21 mat 0.068 0.175 21 mat

x f x ρ f3 : max f 0 0.125 y1 y2 ≥ 1012 -^Extrema 1 el : 21 sys

max of green 'C' max 0.106

Experiment a kinetics system

Setup numerical ODE fill-in  => [IC, k] t Ω k D k 1 el Ω 1 el * - k 1 el Ω 1 el * k 2 el Ω 2 el * - k 3 el Ω 3 el * + k 2 el Ω 2 el * k 3 el Ω 3 el * - 31 mat :

t X t diff k 1 el t X * - ≡ t Y t diff k 1 el t X * k 2 el t Y * - k 3 el t Z * + ≡ t Z t diff k 2 el t Y * k 3 el t Z * - ≡ 31 mat

Smath executable ... fill-in  [IC, k] sol 10031 mat 01199 t Ω 105531 mat D Rkadapt :

t sol 1 col : X t sol 2 col augment : Y t sol 3 col augment : Z t sol 4 col augment : 41 sys

X Y Z 3 1 sys

Similar models => Kinetics Derivatives

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 :

data 0 0.91 1 0.693 2 0.471 5 0.084 11017 0.017 23 0.055 29 0.124 35 0.079 41 0.111 47 0.095 53 0.121 59 0.092 132 mat :

possible model to D3(x) ... RemToDo for future. x β f 1 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 - (* - :

data data o 5 black plot :

β 0.894 0.497 4.84 7.231 7.227 51 mat :

Φarmaco x 060 0.25 range : i 1 x rows range f i el x i el β f eval : for x f augment 31 line :

Data Medico/Pharmacokinetics from unabstracted source

data Φarmaco 2 1 sys

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 :

Y

Reduced data Y/X

XYR 13.125 154.614 11.780114 34.026 246.218 7.236172 55.446 252.327 4.55086 76.021 242.368 3.188172 92.273 191.45 2.074821 105.182 152.065 1.445732 117.288 142.612 1.215913 127.59 121.356 0.95114 137.523 117.018 0.850898 147.304 115.21 0.782124 103 mat :

Reduced fit suggestive of D2(x) x μ fD2 μ 1 el μ 2 el x μ 4 el / - exp * + μ 3 el x μ 5 el / - exp * + :

Recovered fit x q x x μ fD2 * :

μ 0.02024 14.848 4.376 47.1 6.158 51 mat :

x c 10 x ≤ cases :

Reduced data set to fit XR XYR 1 col XYR 3 col augment : XR XR . 10 black plot : 21 sys

Original data set for recovery XY XYR 1 col XYR 2 col augment : XY XY . 10 black plot : 21 sys

x user 80 :

x q x user - 0 ≡ x 150175 solve 162.4

Reduce x 1020011 range : i 1 x rows range R i el x i el μ fD2 eval : for x R augment 31 line :

Fit x 02501 range : i 1 x rows range fit i el x i el q eval : for x fit augment 31 line :

The fitted reduced data set

XR Reduce 2 1 sys

Fit of the original data set

XY Fit 2 1 sys

The scalar model fit

x q x c * 162.4 80 + 30 red 1 5 mat 2 1 sys