Minimize Conjugate-Gradient

d680ec01-e4a4-481d-a757-e46bf7a9867c 72 3 12 decimal

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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 :

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 0.007 0.244 0.026 0.446 0.058 0.612 0.1 0.744 0.152 0.848 0.212 0.923 0.279 0.973 0.35 0.998 0.425 1 0.502 0.98 0.578 0.94 0.653 0.88 0.725 0.803 0.791 0.711 0.85 0.606 0.902 0.491 0.943 0.369 0.974 0.245 0.993 0.126 10212 mat :

independent parameters p 0.45 : xo 1 : 12 mat

Step.0 => compose/assign the model function

<= UFO model function x β f x x xo - (* β 1 el x β 3 el x * β 2 el + β 4 el x 2^* + (* + (* (p^:

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

Step.1 => size 'n' length of CG vector

size Gradient n 4 :

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

Step.2 => init of size 'n'... iter [user]

init 2 - 11 - 141 mat :

Observe . . .

Hover φ(x,β), view the CG Gradient vector .

This algo is not SSD, it is LS from the native

imbeded Cholesky solver. Not as robust as

MCD LM wrt initials. Robustness depends

upon the reflexivity of the model, ranging from

very robust to very difficult to initialize.

In this example, 'p' is user refine fit. Not being

included in the model it relaxes reflexivity.

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iter 5 :

algo solving for β β init iter Minimize :

assign/isolate ... β β : 6.678 - 10.19 12.851 - 7.736 41 mat

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

FIT U 01 0.001 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 31 line :

data FIT 2 1 sys

x β f x 01 int 0.785

About the UFO project . . .

A not clean data set originated from Russian Collab [Mathsoft].

His description induced an UFO ! ? . . .

1. The data set was cleaned via Bézier-Bernstein.

2. XY is a unary scaled version . . . that here:

3 . . . we fit for exporting a clean formula [pocket calculator].

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More statistical insight

Δfit absolute residuals ... XY only 3 Decimals δ i 1 XY rows range Δ i el Y i el X i el β f - 3 round eval : for X Y Δfit 13 mat X Y Δ augment stack X Δ augment 21 mat 21 line :

δ 1 el 11 mat 1 el X Y Δfit 000 0.007 0.244 0.006 - 0.026 0.446 0.004 0.058 0.612 0.001 0.1 0.744 0.001 - 0.152 0.848 0.001 - 0.212 0.923 0.001 - 0.279 0.973 0.001 0.35 0.998 0.002 0.425 1 0.002 0.502 0.98 0 0.578 0.94 0.002 - 0.653 0.88 0.003 - 0.725 0.803 0.002 - 0.791 0.711 0 0.85 0.606 0.003 0.902 0.491 0.006 0.943 0.369 0.002 0.974 0.245 0.004 - 0.993 0.126 0.008 - 100223 mat

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 0.0001

# 11 mat Explode ≡ # 11 mat 1 el Unest ≡ 21 sys

δ 2 el 0.005 BarSize

Pearson χ X Mean : μ Y Mean : n XY rows : 31 line

Pearson’s correlation coefficient corr 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 / / : 1.001

correlation between [X,Y data]

correlation between [X,fitted model]

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 / 21 mat 0.224818173 - 0.224632817 - 21 mat

.Example 2 . . .

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escape "logarithm 0 NOT defined" ε 1015 -^:

XY 00122 7.29 3 13.72 4 21.92 5 32.57 6 35.48 7 50.11 8 57.67 9 82.47 10701110512 111.4 13 124.7 14137152 mat :

<= model function x β f β 1 el x ε + (β 2 el^* :

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size Gradient n 2 :

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

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init 2221 mat :

iter 7 :

algo solving for β β init iter Minimize :

assign/isolate ... β β : 2.746 1.487 21 mat

X XY 1 col : Y XY 2 col : 12 mat data XY o 5 black plot : n XY rows : 31 sys

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 410.3

pilot 'iter'

FIT U 015 0.01 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 31 line :

data FIT 2 1 sys

x Φ 2.746 x 1.487^* :

The Procedure . . .

1. compose the model f(x,β)

2. assign 'n' the length of Gradient

3. introduce 'init' ... 'iter'

Unlike Mathcad, procedure is not self stopping

on some criteria. Works as fine in much simpler

steps ... Minimize(init,iter)

Convergence is lagerly succeptible to init guess.

Before the plunge . . .

1. You need a library of models.

2. From long expertise, on difficult models, it was

recognized a manual trial/error that would cross

twice the data set will generally converge to fit.

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The Gradient-vector of Genfit

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Vector Gradient ... Optimiz Symbolic x α f n φ ct 1 : i 1 n range PD ct el x α f (α ct el diff : stack f(x,β) & PD V 1 el x α f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

x α f α 1 el α 2 el x - α 3 el / exp * + α 4 el x - α 5 el / exp * + :

n last index of β ≡ x α φ x α f 5 φ : 21 line

x α φ α 2 el x α 3 el / - exp * α 4 el x α 5 el / - exp * + α 1 el + 1 x α 3 el / - exp x x α 3 el / - exp * α 2 el * α 3 el 2^/ x α 5 el / - exp x x α 5 el / - exp * α 4 el * α 5 el 2^/ 61 mat

Example 3: same model two families

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data 0.5 0000001305010185621328703654803497580658799483978879941005919990839610269810091849810377 mat :

j 2 :

XY data 1 col data j col augment :

Smath
unique
init 100 1.5 131 mat

MCD
unique
init 50 0.1 131 mat

Procedure

ε 1015 -^:

xo 0.5 :

<= model function x β f β 1 el 1 x ε + (xo - β 2 el / (β 3 el^- exp - (* :

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size Gradient n 3 :

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

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init 100 1.5 131 mat :

iter 7 :

algo solving for β β init iter Minimize :

assign/isolate ... β β : 95.824 2.803 2.935 31 mat

X XY 1 col : Y XY 2 col : 12 mat data XY o 5 black plot : n XY rows : 31 sys

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 23.2

FIT U 0.5 10 0.51 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 31 line :

data FIT x init f 3 1 sys

β 95.824 2.803 2.935 31 mat

β 95.824 2.803 2.935 31 mat 99.513 2.219 2.856 31 mat 101.047 1.044 0.534 31 mat 84.942 2.041 1.712 31 mat 98.197 1.63 1.506 31 mat 106.322 0.791 0.691 31 mat 16 mat ≡