linfit modules

f9308b91-5ac3-4882-b300-bfa4531f0ab9 48 3 12 decimal

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linfit modules

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 :

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 :

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 4: An effective recognition test

XY 0.1 10 0.2 5 0.3 3.333 0.4 2.5 0.5 2 0.6 1.667 0.7 1.429 0.8 1.25 0.9 1.111 112 0.5 3 0.333 4 0.25 5 0.2 142 mat :

X XY 1 col : Y XY 2 col : 12 mat data XY . 12 black plot : 21 line

Guess by inspection x β f β 1 el β 2 el x 1 -^* + :

β 0121 mat :

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

x β φ x β f 2 φ :

Does NOT retro-length φ(x,β,f,2) β XY f φ β βlinfit :

β 0.00000239 0.99999838 21 mat

data x β f 2 1 sys

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 : 12 mat data XY . 12 black plot : n XY rows : 31 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 :

0.4859 0.4813 - 2.1846 31 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 XY . 14 red plot :

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 :

x Φ 11 x + / x x 3^13 mat transpose :

Cholesky companion solver j 1 x Φ length range i 1 X length range M i j el X i el Φ j el : for for Q M transpose M * : d M transpose Y * : b Q 1 -^eval d * : 41 line

linfit to a weird data set

FIT data 2 1 sys

b 0.485946 0.481346 - 2.184594 31 mat

x fit b x Φ * :

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 : 12 mat data XY . 12 black plot : n XY rows : 31 line

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 ≡

β 111114 mat transpose :

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 11 line :

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

data XY . 14 red plot :

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

Example 3: An effective combination

Data XY

XY 1 4.18 2 4.67 3 5.3 4 5.37 5 5.45 6 5.74 7 5.65 8 5.84 9 6.36 10 6.38 11 6.31 12 6.46 13 6.48 14 6.59 15 6.74 16 6.58 17 6.85 18 6.53 19 6.68 20 7.01 21 6.82 22 6.65 23 6.95 24 7.12 25 7.14 26 6.91 27 6.94 28 7.23 29 6.9 30 7.39 31 6.94 312 mat :

X XY 1 col : Y XY 2 col : 12 mat data XY . 12 black plot : n XY rows : 31 line

Associate primary functions x β f β 1 el β 2 el x ln * + β 3 el x sqrt * + :

β 11131 mat :

4.1834 0.9982 0.0805 - 31 mat

x β φ x β f 3 φ :

β XY f φ β βlinfit :

β 4.1834 0.9982 0.0805 - 31 mat

FIT x 0.001 50 0.1 range : i 1 x rows range fit i el x i el β f eval : for x fit augment 31 line :

data FIT 2 1 sys

Example 4: An effective recognition test

XY 0.1 10 0.2 5 0.3 3.333 0.4 2.5 0.5 2 0.6 1.667 0.7 1.429 0.8 1.25 0.9 1.111 112 0.5 3 0.333 4 0.25 5 0.2 142 mat :

X XY 1 col : Y XY 2 col : 12 mat data XY . 12 black plot : 21 line

Guess by inspection x β f β 1 el β 2 el x 1 -^* + :

β 0121 mat :

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

x β φ x β f 2 φ :

Does NOT retro-length φ(x,β,f,2) β XY f φ β βlinfit :

β 0121 mat

data x β f 2 1 sys