Plot Utilities

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

Levenberg-Marquardt style 1

4 appVersion 0.98.6179.21440 ≡

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 :

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 :

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 20200122

x y β Tol βs LMSA Eps Tol : eta1 Eps sqrt : eta2 eta1 : h eta1 : stop 0 : 15 mat m Clear 1 : β fx i 1 m range ff i el x i el β ft y i el eval - : for ff 21 line : β m n h derfh i 1 m range j 1 n range β j el β j el h + : jjh1 x i el β ft y i el - : β j el β j el 2 h * - : jjh2 x i el β ft y i el - : jjh0 i j el jjh1 jjh2 - 2 h * / Re : β j el β j el h + : 61 line for 11 line for jjh0 21 line : m x rows : n β rows : 12 mat A n n Zeros : g n Zeros : J m n Zeros : 13 mat Fn 1 Zeros : dL 1 Zeros : dF 1 Zeros : F 1 Zeros : 14 mat f β fx : J β m n h 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 ¬ cont cont 1 + : 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 : i 11 range stop ¬ xnew β p + : fn xnew fx : Jn β m n h derfh : brake 11 line try 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 for if 41 line if 21 line while βs β : 121 line :

m Clear 1 :

x 1234567891011111 mat :

y 0.4897 1.3079 2.84 3.0364 1.7709 0.5491 0.4606 0.3345 0.2073 0.0012 0.0024 111 mat :

x β ft β 1 el β 3 el x β 4 el - / (β 2 el 1 +^* β 3 el x β 4 el - / (β 2 el^- exp * :

UV x y augment :

XY x y augment : data XY . 10 black plot : 21 line

1. selct the support points
2. collect for al_nleqsolve list 12345678919 mat : new UV list ListCollect : 21 sys

raw UV o 5 black plot : support new . 14 blue plot : x1 new 1 col : y1 new 2 col : 12 mat 31 line

x1 12345678991 mat

y1 0.4897 1.3079 2.84 3.0364 1.7709 0.5491 0.4606 0.3345 0.2073 91 mat

Style 1... rejects too low Tol

β 857 3.75 - 14 mat transpose :

Tolerance Tol 0.00005 :

t0 1 time :

SS 6179 OK 9 data β x1 y1 β Tol βs LMSA : 8.3294 14.947 19.6175 16.2284 - 41 mat

1 time t0 - min 2.1

857 3.75 - 14 mat

recover @ 'tol' β β 4 round vectorize :

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

x α F α 1 el α 3 el x α 4 el - / (α 2 el 1 +^* α 3 el x α 4 el - / (α 2 el^- exp * :

Fréchet & many more 
are reflexive ... β 8.3294 14.947 19.6175 16.2284 - 41 mat

conjugated
gradient α 8.5418 40.7318 52.0514 48.7382 - 41 mat :

x c 10 x ≤ (x 12 ≤ (& cases :

x α F x c * x β f x c * data 3 1 sys

857 3.75 - 14 mat transpose

new 1 0.4897 2 1.3079 3 2.84 4 3.0364 5 1.7709 6 0.5491 7 0.4606 8 0.3345 9 0.2073 92 mat

0.0001

Levenberg-Marquardt style 2

x y β Clear 1

t0 1 time :

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

n X length :

m t length :

Function adjusted to the data i 1 m range f i el X 2 el X 1 el X 2 el - 1 t i el X 3 el / (X 4 el^+ / + X 1 el X 2 el - 1 t i el X 3 el / (X 4 el^+ (2^/ t i el X 3 el / (X 4 el^* X 4 el * - y i el eval - : for

Jacobian matrix Function i 1 m range jj i 1 el f i el β 1 el diff : jj i 2 el f i el β 2 el diff : jj i 3 el f i el β 3 el diff : jj i 4 el f i el β 4 el diff : 41 line for

can be grouped but 
NOT collected {X derf jj : X fx f : tol 104 -^β normi * : 31 line

cont 0 : dnor 1 : update 1 : 13 mat μ 0.01 : μ1 100 : 12 mat 21 mat

normi("matrix") - Returns the infinity norm of the matrix

eval symbolic to see the descending gradient X derf

Difficult to initialize
depends upon reflevxivity
of the model,eg: SGompertz
supports NO fit method !
except: al_nleqsolve. β 0.5 0.75 1441 mat :

Adjust/observe residuals plot tol 0.0004

Carlos L-M algorithm

dnor tol > update 1 ≡ f β fx : J β derf : jtj J transpose J * : res f norme (2^: 41 line continue if a jtj μ n identity * + : s a 1 -^J transpose f * (* : b β s - : f1 b fx : res1 f1 norme (2^μ1 / : res1 res < β b : f f1 : dnor s normi β normi / : cont cont 1 + : Sol cont 1 el cont : Sol cont 2 el β 1 el : Sol cont 3 el β 2 el : Sol cont 4 el β 3 el : Sol cont 5 el β 4 el : Sol cont 6 el res1 μ1 * : Sol cont 7 el μ : Sol cont 8 el dnor : μ μ 10 / : update 1 : 141 line μ μ 10 * : update 0 : 21 line if 71 line while

β 0.777334934017 1.072745027895 1.278126168617 4.800807543461 41 mat

1 time t0 - 14.1 s *

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 : 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 :

XY 1012 -^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 :

ε 10^-12 to clear
Smath ln0 undefined. 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 XY 1 col : Y XY 2 col : 12 mat data XY . 12 black plot : n XY rows : 31 line

Absolute  residuals δ Δ 0 : j 1 XY rows range Δ j el Y j el X j el β f - 3 round eval : for X Δ augment 31 line :

FIT U 010 0.025 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 31 line :

bar δ 0.0375 BarSize (:

β 0.77733 1.07275 1.27813 4.80081 41 mat

bar

data XY o 5 black plot :

Title 21 Pharmacokinetics 12 black 15 mat :

collect 1.08 FIT 0.78 data Title 51 sys :

collect

header iter β1 β2 β3 β4 res μ dnorm 18 mat :

Table header Sol stack :

Table iter β1 β2 β3 β4 res μ dnorm 1 0.789361 1.074749 1.35451 3.890834 1.18207 0.01 0.091114 2 0.773371 1.070441 1.243124 4.630308 0.018636 0.001 0.159703 3 0.778593 1.072707 1.28057 4.810854 0.002883 0.0001 0.037529 4 0.777268 1.072755 1.278058 4.798605 0.001501 1105 -^* 0.002553 5 0.777342 1.072745 1.278138 4.800981 0.001492 1106 -^* 0.000495 6 0.777335 1.072745 1.278126 4.800808 0.001492 1107 -^* 3.615337 105 -^* 78 mat

Levenberg-Marquardt . . .

Fits data sets to model function(s). Both styles need initial β

Example 1 does not like the last two points, distracted from the project.

It complains on too low TOL About equivalent fit Cholesky Conjugate Gradient.

Example 2 enables tabulation of the project. About equivalent Cholesky C G.

Both methods [Cholessky/Levenberg-Marquardt] are difficult to initialize.

This difficulty results from the model function. Some model are midly reflexive,

others are very reflexive. More reflexive they are , more difficult to initialize.

I know one model function that can't be fitted by no methods I have attempted.

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