Function Utilities

191d3314-331e-4b9a-b3fe-34005a306916 15 6 12 decimal

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

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 8.0957 - 210.002 - 7.7963 - 195.802 - 7.4807 - 182.962 - 3.7187 - 78.476 - 1.9063 - 38.862 - 00 1.3739 26.87 5.2677 100 6.4886 122.37 8.3743 156.634 12.5517 231.9681 14.7427 271.442 17.4928 321.108 17.8462 327.502 19.456 356.66 152 mat :

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

num order
denom order p.n 4 : p.d 3 : 21 sys

...k is total order of rational fit k 1 p.n + p.d + : 8

The rational fit procedure :

a slight recast from Jeff rev.3. In some ways,

it works same as Mathcad GenfitMatrix, except for

the vector of support points that must equal 'k' .

1. βinit is a rational fit through the support points.

2. Next step iterates Minimize ConjugateGradient

over all data points, kind of average fit.

3. Higher order rational haven't been tested for

robustness . . . RemToDo.

4. High order poly on large X variate need large

decimals. For large X, the procedure is scaled.

]]>

"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^* + 1 β 6 el x * + β 7 el x 2^* + β 8 el x 3^* + /

recast the system x β φ x β f k φ :

support points ≡ k support 134579121518 mat :

iter 3 :

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 :

β β 1 el βinit : iterate ConjugateGradient i 1 : Δβ'.max 1 : i iter ≤ (Δβ'.max 106 -^> (& β i 1 + el XY f φ β i el Minimize : i i 1 + : n 1 k range Δβ' n el β i el n el β i 1 - el n el - β i 1 - el n el / 1 try abs : for Δβ'.max Δβ' max : 41 line while β β i el i Δβ'.max stack : 61 line :

1. iterates
2. Minimize N.iter β k 1 + el : Δβ' β k 2 + el : 21 mat 4 0.0056 21 mat

b β 1 k 11 submatrix :

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

β ct 1 : j 1 b rows range u ct el b j el 8 round : ct ct 1 + : 21 line for u 31 line :

CWD C:\SmathFile CurrentDirectory :

β CWD βrational wfile 1

β CWD βrational rfile :

βinit 0.0023 19.8516 2.9582 0.1117 0.0014 0.1611 0.0066 0.0001 81 mat

β 0.00807823 19.84766385 3.08381444 0.11462235 0.00082808 0.16745272 0.00681467 0.00002951 81 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^* + 1 β 6 el x * + β 7 el x 2^* + β 8 el x 3^* + / :

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 8.0957 - 210.002 - 0 7.7963 - 195.802 - 0.01 7.4807 - 182.962 - 0.01 - 3.7187 - 78.476 - 0.01 1.9063 - 38.862 - 0.01 - 00 0.01 - 1.3739 26.87 0 5.2677 100 0.01 6.4886 122.37 0.01 8.3743 156.634 0.01 - 12.5517 231.9681 0.02 - 14.7427 271.442 0 17.4928 321.108 0.02 17.8462 327.502 0.01 19.456 356.66 0.02 - 153 mat

FIT u 8.1 - 19.5 8.05 - 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 fraction to Type 'J' T/C, NBS Monograph 125

data FIT support 3 1 sys

Support points [init] xy 8.0957 - 210.002 - 7.4807 - 182.962 - 3.7187 - 78.476 - 1.9063 - 38.862 - 1.3739 26.87 6.4886 122.37 14.7427 271.442 19.456 356.66 82 mat

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 4.97308 80.56334 1531 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 / / : 0.99999988

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 0.99673 0.99673 31 mat

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

χ μ n 31 mat 4.97308 80.56334 1531 mat