<p> linfit modules </p>

f9308b91-5ac3-4882-b300-bfa4531f0ab9 21 6 9 decimal

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

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

Applicable parts are: Optimiz "numeric" XY f φ β βlinfit 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 Gradient Matrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for Cholesky solver [LS] 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 :

linfit technique ...

"linfit" is a one time use function. All β coefficients are set to '1'. On the unique first pass, it produces a linear combination solution of the "model vector function" wrt the XY data set. Used in two situations: 1. "linfitCheby" to combined a vector of ChebyShev orthogonal polynomials [other than ChebyShev can be attempted]. This will be much superior to the ordinary simple polynomial "Polyfit". 2. In other instances, the data set may be so weird that nothing will do from model function. In that case [example below], you will have to invent the vector of "model functions" made of whatever will best pass a function through the data set. The two examples reproduce Mathcad 11 [1/1].

Example 1: A vector of inventions

XY 0 0.43 0.2 0.22 0.4 0.8 0.6 0.1 0.8 11262 mat :

X XY 1 col : Y XY 2 col : 21 line

A vector of inventions x β f β 1 el 1 x + / β 2 el x * + β 3 el x 3^* + :

β 11131 mat :

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

x β φ x β f 3 φ :

β XY f φ β βlinfit :

β 0.4859 0.4813 - 2.1846 31 mat

data X Y . 14 red plotG :

FIT x 0.25 - 1.25 0.245 - range : i 1 x rows range fit i el x i el β f eval : for x fit augment 31 line :

linfit to a weird data set

FIT data 2 1 sys

Example 2: A sum of reciprocal monomials

XY 75274 - 100160 - 125104 - 150 71.5 - 200 35.2 - 250 16.2 - 300 4.2 - 4009500 16.9 600 21.3 70024112 mat :

X XY 1 col : Y XY 2 col : 21 line

β 111141 mat :

Sum of reciprocal monomials x β f β 1 el β 2 el x 1 -^* + β 3 el x 2 -^* + β 4 el x 3 -^* + :

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

x β φ x β f 4 φ :

β XY f φ β βlinfit :

β 42.1478 11189.2499 - 803760.955 - 10127030.7033 - 41 mat

x G β n el 1 x / (n^* (n 14 sum :

data X Y . 14 red plotG :

FIT x 7075071 range : i 1 x rows range fit i el x i el β f eval : for x fit augment 31 line :

x G x 300350 solve 326.2

linfit to Broida data set [sum of linear scaled reciprocal monomials]

FIT data 326.2 30 326.2 30 - 2 2 mat 3 1 sys