Function Utilities

4445e17f-f093-4ce4-b915-82c263cfff4f 14 6 12 decimal

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Rational Fraction Auto Procedure

Function 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 : 13 mat 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 :

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 :

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 :

XY 0 2113.3 1 2111.7 2 2109.2 3 2105.2 4 2099.4 5 2092.1 6 2084.7 7 2078.6 8 2075.1 9 2073.9 10 2074.1 11 2074.4 12207413 2072.5 14 2070.1 15 2067.2 16 2064.4 17206218 2060.5 19 2059.6 202 mat :

This Procedure . . . is courtesy from Jeff rev. 3

For this data set wrt rational best num/den order the ConjugateGradient is not needed.

Thus, the βinit becomes β , rounded/transited for Export f(x,β)

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

num order
denom order p.n 5 : p.d 5 : 21 sys

...k is total order of rational fit k 1 p.n + p.d + : 11

"Normalized" rational fraction have
leading "1" in denom.  Reputed stable
numerically. Most if not all computer
approximations are  "Normalized".
Generally, Num_order is Denom_order+1.
Some fit were found not "Normalized. x β f β 1 el β j 1 + el x j^* j 1 p.n sum + 1 β j 1 + p.n + el x j^* j 1 p.d sum + / maple :

x β f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + β 5 el x 4^* + β 6 el x 5^* + 1 β 7 el x * + β 8 el x 2^* + β 9 el x 3^* + β 10 el x 4^* + β 11 el x 5^* + /

support points ≡ k support 13579111315171920111 mat :

βinit ct 1 : j support X.g ct el XY j 1 el : Y.g ct el XY j 2 el : 12 mat ct ct 1 + : 21 line for M.g k Ones : i 1 p.n range M.g M.g X.g i^vectorize augment : for i 1 p.d range M.g M.g X.g i^Y.g * vectorize - augment : for M.g 1 -^Y.g * 61 line :

1. selected  support pts
2. collect for βinit list support : xy XY list ListCollect : 21 sys

Support points xy 0 2113.3 2 2109.2 4 2099.4 6 2084.7 8 2075.1 10 2074.1 12207414 2070.1 16 2064.4 18 2060.5 19 2059.6 112 mat

b βinit :

β ct 1 : j 1 b rows range u ct el b j el 12 round : ct ct 1 + : 21 line for u 31 line :

CWD C:\SmathFile CurrentDirectory :

β CWD βrational wfile 1

β CWD βrational rfile :

β 2113.3 214.400207455929 174.43781522443 - 29.968120725868 2.04458520289 - 0.054526755395 0.10206536214 0.082414875377 - 0.014211693449 0.000974159604 - 0.000026130212 111 mat

x β f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + β 5 el x 4^* + β 6 el x 5^* + 1 β 7 el x * + β 8 el x 2^* + β 9 el x 3^* + β 10 el x 4^* + β 11 el x 5^* + / :

Absolute residuals [mutual conformity] δ i 1 XY rows range Δ i el Y i el X i el β f - 2 round eval : for X Y Δ augment 21 line :

δ 0 2113.3 01 2111.7 0.03 - 2 2109.2 03 2105.2 0.02 4 2099.4 05 2092.1 0.08 - 6 2084.7 07 2078.6 0.04 - 8 2075.1 09 2073.9 0.03 - 10 2074.1 011 2074.4 0.01 - 122074013 2072.5 0.02 - 14 2070.1 015 2067.2 0.01 - 16 2064.4 0172062 0.1 - 18 2060.5 019 2059.6 0203 mat

FIT u 020 0.1 range : i 1 u rows range FIT i el u i el β f eval : for u FIT augment 31 line :

Δelta X δ 3 col augment : stem Δelta Error : data X Y . 12 black plotG : support xy + 20 blue plot : 41 line

Rational fit weekly Corn Thermies pre-smoothed "ksmooth"

data FIT support 3 1 sys 0 stem 2 1 sys BarSize Unicolor QuickPlot or any color X_Y

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 :

SSD = "Sum Square Differences" parameter SSD Y i el X i el parameter f - (2^(i 1 XY rows sum 11 line :

Pearson’s correlation coefficient 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 :

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

χ μ n 31 mat 9.5 2081.1 2031 mat

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 / / : 0.99993725

correlation between [X,Y data]

correlation between [X,fitted model]

Fit.Y i 1 n range Fit i el X i el β f : for Fit 21 line :

More statistical insight

Y Fit.Y corr X Y corr X Fit.Y corr 31 mat 1.00000 0.95152 - 0.95158 - 31 mat

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

χ μ n 31 mat 9.5 2081.1 2031 mat