Function Utilities

74051fee-70c0-495a-9ea9-8805eb047aee 44 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 :

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.

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

n 15

...n is total number of points in the data set

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x β g x β g

• Enter numerator and denominator polynomial orders

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p.n 3 : p.d 2 : 21 line

...p_n is the order of the numerator polynomial ...p_d is the order of the denominator polynomial

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k 1 p.n + p.d + : 6

...k is total order of rational fit

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"Normalized" rational fraction x β f β 1 el β ii 1 + el x ii^* ii 1 p.n sum + 1 β ii 1 + p.n + el x ii^* ii 1 p.d sum + / :

"Normalized"" rational fit equation x β f x β 3 el x * β 2 el + β 4 el x 2^* + (* β 1 el + 1 x β 6 el x * β 5 el + (* + /

x β φ x β f k φ :

• Enter k points from original data to be used for generating initial guess for β fit parameters

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FitPoints 1379121516 mat transpose :

...rows numbers for the k points

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Pts β.guess XY.g XY Pts 1 el row : ii 2 k range XY.g XY.g XY Pts ii el row stack : for X.g XY.g 1 col : Y.g XY.g 2 col : M.g k Ones : ii 1 p.n range M.g M.g X.g ii^vectorize augment : for ii 1 k p.n - 1 - (range M.g M.g X.g ii^Y.g * vectorize - augment : for M.g 1 -^Y.g * 81 line :

x β η x β g k φ :

β.orig b 00 stack :

β.orig b 0.09 0.001 stack :

β.orig FitPoints β.guess : 0.1913 19.6803 0.7466 0.0658 - 0.0485 0.0038 - 61 mat

β β 1 el β.orig : iter 9 : 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 :

β β :

Δβ' β k 2 + el : 4.71 103 -^*

N.iter β k 1 + el : 10

β β 1 k 11 submatrix : 0.1901 - 19.7287 0.7168 0.0653 - 0.0469 0.0038 - 61 mat

• Solve for roots of denominator of the rational to see the x values where the rational fit may "blow up"

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ii 1 p.d range P.den ii el β ii 1 + p.n + el : for 1 P.den stack 21 line polyroots 11.22 - 23.69 21 mat

ii 1 p.d range P.den ii el β.orig ii 1 + p.n + el : for 1 P.den stack 21 line polyroots 11.05 - 23.84 21 mat

1 0.09 0.001 13 mat transpose polyroots 12.98 - 77.02 - 21 mat

FIT.orig U 8.1 - 19.5 8.05 - range : i 1 U rows range FIT.orig i el U i el β.orig f eval : for U FIT.orig augment 31 line :

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 :

δ 8.0957 - 210.002 - 0.37 - 7.7963 - 195.802 - 0.28 7.4807 - 182.962 - 0.29 3.7187 - 78.476 - 0.53 - 1.9063 - 38.862 - 0.13 - 00 0.19 1.3739 26.87 0.3 5.2677 100 0.17 6.4886 122.37 0.06 8.3743 156.634 0.1 - 12.5517 231.9681 0.22 - 14.7427 271.442 0.1 - 17.4928 321.108 0.1 17.8462 327.502 0.11 19.456 356.66 0.06 - 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 :

data X Y . 12 black plotG :

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

data FIT FIT.orig 3 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 :

T.orig

MyBar Δelta 0.075 BarSize :

Mutual deviation between data set and rational fit.

0 MyBar 2 1 sys

More statistical insight

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

β SSD 0.852

β.orig SSD 2.054

RMS difference β SSD n / sqrt 0.238

RMS difference β.orig SSD n / sqrt 0.370

RMS difference b 00 stack SSD n / sqrt 2.8938

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 :

RMS difference b 0.09 0.001 stack SSD n / sqrt 195.8379

Fit.Y ii 1 n range Fit ii el X ii el β f : for Fit 21 line :

Y Fit.Y corr 0.99999921

X Y corr 0.996733

X Fit.Y corr 0.996733

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 / / : 0.99999921

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 0.996733 21 mat