Function Utilities

74051fee-70c0-495a-9ea9-8805eb047aee 21 4 5 decimal

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

Data 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 :

Initialise the num.

Here, we start by initialising the numerator of the rational. From the allure of the data set, a 3rd order polynomial is the right choice. This part is stricly stand alone. Next, we will guess about the denom of the rational fraction. This part will be manual trial/error, helped from some reasonable iterative suite.

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

x φ 1 x x 2^x 3^14 mat transpose :

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

b 0.8437 21.1566 0.4085 - 0.0138 41 mat

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 :

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 :

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 :

"Normalised" rational fraction x β f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + 1 β 5 el x * + β 6 el x 2^* + / :

x β φ x β f 6 φ :

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

The data set is Type 'J' T/C [Thermocouple] from NBS Monograph 125. A damned good fit ! Former NBS is now NIST. Observe 'iter=9' vs residuals ... check Pearson correlation.

Absolute residuals 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 :

β β 1 el 0.8437 21.1566 0.4085 - 0.0138 0.09 0.001 61 mat : iter 9 : i 1 : i iter ≤ β i 1 + el XY f φ β i el Minimize : i i 1 + : 21 line while β β iter el : 51 line :

β β :

δ 8.0957 - 210.002 - 0.03 7.7963 - 195.802 - 0.13 7.4807 - 182.962 - 0.2 - 3.7187 - 78.476 - 0.24 1.9063 - 38.862 - 0.24 - 00 0.23 1.3739 26.87 0.38 5.2677 100 0.19 6.4886 122.37 0.04 8.3743 156.634 0.22 - 12.5517 231.9681 0.54 - 14.7427 271.442 0.43 - 17.4928 321.108 0.01 17.8462 327.502 0.1 19.456 356.66 0.54 153 mat

β 0.22518 - 19.605525 6.915739 0.435004 0.361548 0.024788 61 mat

FIT U 8.1 - 19.5 8 - range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 31 line :

data X Y . 12 black plotG :

Rational fraction to Type 'J' T/C, NBS Monograph 125

data FIT 2 1 sys

Δelta X δ 3 col augment : stem Δelta Error : 21 line

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 :

MyBar Δelta 0.075 BarSize :

Mutual deviation between data set and rational fit.

0 MyBar 2 1 sys

More statistical insight

β 0.22518 - 19.605525 6.915739 0.435004 0.361548 0.024788 61 mat :

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

RMS difference β SSD n / sqrt 0.28

β SSD 1.2

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 / / : 0.999997

correlation between [X,Y data]

correlation between [X,fitted model]

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 / 21 mat 0.996733 0.996735 21 mat