Pearson correlation

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Pearson correlation

p# corr Pearson correlation X XY 1 col : Y XY 2 col : 12 mat χ X Mean : μ Y Mean : n XY rows : 13 mat 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 p# f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el p# f eval μ - (2^(i 1 n sum * sqrt / / 41 line :

===== Thurber data set =====

ListCollect

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 :

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 :

XY Thurber

XY 2.981 - 84.248 2.921 - 87.264 2.912 - 87.185 2.84 - 89.076 2.797 - 89.608 2.702 - 89.868 2.699 - 90.101 2.633 - 92.405 2.481 - 95.854 2.363 - 100.696 2.322 - 101.06 1.501 - 401.672 1.46 - 390.724 1.274 - 567.534 1.212 - 635.316 1.1 - 7330.54 1.046 - 759.087 0.915 - 894.206 0.714 - 990.785 0.566 - 1090 0.545 - 1080 0.4 - 1120 0.309 - 1180 0.109 - 1260 0.103 - 1270 0.01 1290 0.119 1330 0.377 1350 0.79 1420 0.963 1430 1.006 1420 1.115 1440 1.572 1460 1.841 1470 2.047 1450 2.2 1460362 mat :

1. selct the support points
2. collect for al_nleqsolve list 113243414 mat : xy XY list ListCollect : 21 sys

raw XY o 5 red plot : support xy . 16 black plot : 21 line

support raw 2 1 sys

xy 2.981 - 84.248 1.46 - 390.724 0.109 - 1260 1.841 147042 mat

x xy 1 col : y xy 2 col : 21 sys

vector of guesses ... 
from robust2crucifying ! X0 8014001014 mat transpose :

t0 1 time :

AlgLib Solver implicit equalities [Courtesy  Viacheslav] sol k 1 μ f rows range u k el μ k el : for μ Jac u f u Jacobian : StepMax 0 : ε 1012 -^: 12 mat u X0 StepMax ε μ f μ Jac al_nleqsolve : 41 line :

SGompertz . . .

is extremely reflexive, declared unfittable

in Smath [L-M, CG]. Possible attempt

OriginLab, Mathcad 11. As offered, and

fitted by al_nleqsolve, at this point, you

may want to round/refactor the coefficients.

The fillet region is a bit sparse !

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SGompertz Thurber model fnct x m Φ m 1 el m 2 el m 3 el x m 4 el - (* - exp - exp * + :

Implicit equalities = 0 μ f x 1 el μ Φ y 1 el - x 2 el μ Φ y 2 el - x 3 el μ Φ y 3 el - x 4 el μ Φ y 4 el - 41 mat :

μ sol : 84.25 1396.05 1.61 1.2 - 41 mat

Notes . . . Smath 6179

1. solves a system of [n equalities = n parameters].

2. The order of equalities is immaterial.

3. Guess wisely the best support equalities !

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1 time t0 - s 1

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

μ f 0.000000000000 0.000000000002 - 0.000000000002 - 0.000000000001 41 mat

x c 14 - x ≤ (x 3 ≤ (& cases :

native al_nleqsolve from

113243414 mat

μ 84.25 1396 1.61 1.2 - 41 mat 87.5 1375 1.6 1.25 - 41 mat 901375 1.6 1.25 - 41 mat 801400 1.5 1.25 - 41 mat 901370 1.6 1.25 - 41 mat 15 mat :

observe fillet region j 2 :

data x μ 1 j el Φ x c * support 3 1 sys

x μ f x μ 1 j el Φ :

μ corr 0.4512