Plot Utilities

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Plot 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 :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 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 :

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].

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 :

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 :

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 :

Construction

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 :

's' slant
'λ shift
'n' curvature s 0.05 : λ 1 : n 1 : 31 sys

x n f s x * 1 x λ - (exp + ln n^(* :

L 3 : H 13 : N 10 : 13 mat X L H N xd : XY X X 3 - (n f vectorize augment : data XY o 5 black plot : 41 sys

x 1 f x 1 - ( n f x 2 - ( n f x 3 - ( n f x 4 - ( n f x 5 - ( n f data 7 1 sys Prepare NOT used

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 : 31 line for data r3 r4 r5 augment 21 line :

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 :

illustrate ListCollect

Select the support points not used in this example

1. selct the support points
2. collect for al_nleqsolve list 135791116 mat : xy XY list ListCollect : 21 sys

raw XY o 5 black plot : support xy . 14 red plot : 21 line

raw support 2 1 sys

Example_1:fitting a data set XY to a model function

Part 1

Construction should have no interest, as we are going to fit a function to a data set, no matter its origin, most generally experimental data. al_nleqsolve is robust wrt guesses, though not foolproof. The found β's are used to initialize Genfit Part 2

About the model . . .

It models real parabolic phenomenons

of shifted origin in either/both X_Y

directions. In fact, it takes into account

5 physical constituants.

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vector of guesses X0 0.001 110115 mat transpose :

data set & constrains XY 304 0.03466 5 0.13133 6 0.31904 7 0.60972 8 1.00454 9 1.50201 10 2.10087 11 2.80036 12 3.60015 13 4.50006 112 mat :

AlgLib Solver for a system of nonlinear equations(Uni) sol Levenberg-Marquardt non-linear solver k 1 β f rows range u k el β k el : for β Jac u f u Jacobian : StepMax 0 : ε 1014 -^: 12 mat u X0 StepMax ε β f β Jac al_nleqsolve : 51 line :

A generalized adjustable parabolic function x β Φ β 1 el x * 1 x β 5 el - β 3 el / exp + ln (β 2 el^(* β 4 el + :

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

equalities 30.14 red 50.14 red 70.14 red 100.14 red 130.14 red 55 mat :

x c 13 x ≤ (x 13 ≤ (& cases :

The system of equalities β f 7 β Φ 0.60972 - 13 β Φ 4.50006 - 3 β Φ 0.0 - 10 β Φ 2.10087 - 5 β Φ 0.13133 - 51 mat :

β sol : 0.028705 1.147516 0.972812 0.013686 - 4.459577 51 mat

Notes . . . Smath 6179

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

2. The order of equalities is immaterial.

3. From not knowing more we get an exact fit at the

support points, i.e: at equalities.

Observe => by how much β's are so ≠ than

those that constructed the data set. Immaterial for

this demo example. This property reflexivity may

blind/fool the solver, left for manual fit.

]]> x 3 - ( n f x c * x β Φ data 3 1 sys

Exact fit from exact data set

Absolute residuals Residuals i 1 XY rows range Δ i el Y i el X i el β Φ - eval : for X Δ augment 21 line :

Absolute residuals ... 5 red points reputed "exact" from al_nleqsolve .

Residuals 0.1 BarSize equalities 2 1 sys

mutual deviation
@ the equalities,
within the decimals
of the constrains. β f 5.3 106 -^* 5.6 107 -^* 2.6 106 -^* 2.4 106 -^* - 6.1 106 -^* - 51 mat

Genfit 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 :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 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 :

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].

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 :

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 :

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 :

Part 2

β refine/refit

Aren't we happy al_nleqsolve ? . . . Answer is NO, why ?

Unlike al_nleqsolve that tends to fit exactly at the equalities points,

ConjugateGradient is an SSD algorithm. It then fits more glogally and

more precisely. That's what we are looking for from a fitting session

given some noisy experimental data.

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XY 304 0.03466 5 0.13133 6 0.31904 7 0.60972 8 1.00454 9 1.50201 10 2.10087 11 2.80036 12 3.60015 13 4.50006 112 mat :

Model function x β f β 1 el x * 1 x β 5 el - β 3 el / exp + ln (β 2 el^(* β 4 el + :

<= ConjugateGradient Vector Fnct x β φ x β f 5 φ :

nleqsolve 0.028705 1.147516 0.972812 0.013686 - 4.459577 51 mat

'n' = length(β)

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

β β 1 el 0.025 1.5 1 0.01 - 515 mat transpose : iter 7 : 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 :

Absolute residuals Residuals i 1 XY rows range Δ i el Y i el X i el β f - eval : for X Δ augment 21 line :

ConjugGradient β 0.03017 1.14152 1.0003 0.01572 - 4.49076 51 mat

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 9106 -^*

FIT range, populate U 015 0.025 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 41 line :

RMS difference SSD n / sqrt 0.0009

data FIT 2 1 sys

Absolute residuals from CG

Residuals 0.1 BarSize

About fitting data sets . . .

Fitting data sets is an art [F.B. Hildebrand ] . . . that is more than true.

It takes an arsenal of maths, immagination and at least the two tools [LM & CG].

1. If data sets aren't raw from source, they have been freaked, so you will fit freak !

2. Some model function(s) are so reflexive that the fit can only be manual [trial/error].

3. DRT [Data Reduction Technique(s)] is often necessary ... be immaginative/patient.

4. Most importantly, you need an immense library of model functions.

5. Often, CG works well from tentative initials. It depends mostly upon the direction

of the evolution of CG. Reflexive/reactive model(s) renders the fit painful.

This Smath document illustrates a successful combined [CG, LM] fitting session.

CG => Conjugate Gradient ... LM => Levenberg-Marquardt.

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Example_2: fitting the reduced type 'T' thermocouple

Utilities

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 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 :

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 6.25629 - 268.935 - 6.22919 - 259.34 - 6.19773 - 252.87 - 6.17138 - 248.595 - 6.15358 - 246.048 - 5.87302 - 218.789 - 5.75328 - 210.002 - 5.53559 - 195.802 - 5.31472 - 182.962 - 2.7407 - 78.476 - 1.43494 - 38.862 - 1.0679 26.87 4.2773 100 5.3414 122.37 7.0364 156.634 11.0133 231.9681 13.2188 271.442 16.0953 321.108 16.4733 327.502 18.2179 356.66 202 mat :

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

The data set is Type 'T' T/C [Thermocouple] from NBS Monograph 125. A damned good fit ! Former NBS is now NIST. Check Pearson stat.

Treduced X Y X / vectorize augment :

jmG Mathcad DAEP [2003] β 25.8149913045858 12.7668104944864 - 1.25727409574702 3.04185012250532 0.611123673599964 0.040154623061311 0.0006438455716873 0.46562548019054 - 0.030977484107051 0.119835340769071 0.027040212533505 0.002100716189884 0.0000488329299122 131 mat :

Treduced o 4 black plot

β β 13 round vectorize :

fit to the reduced data set.
Normalized Padé rational 6/6.
RoundOFF wisely ... rational
are sensitive to true decimals x β Tr β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + β 5 el x 4^* + β 6 el x 5^* + β 7 el x 6^* + 1 β 8 el x * + β 9 el x 2^* + β 10 el x 3^* + β 11 el x 4^* + β 12 el x 5^* + β 13 el x 6^* + / :

reverse the fit to the original data set x β f x x β Tr * :

Absolute residuals δ i 1 T rows range Δ i el Y i el X i el β f - 3 round eval : for X Y Δ augment 21 line :

FIT U 6.26 - 18.25 6.2 - range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 31 line :

data T . 12 black plot :

β 15 D 0 0.001 0.003 - 0.001 0.003 0.001 - 0.005 - 0.001 0.004 0.004 - 0.002 0 0.003 - 0 0.004 0.004 0.004 - 0.003 0.002 0.007 - 201 mat

δ 6.2563 - 268.935 - 0.001 6.2292 - 259.34 - 0.001 6.1977 - 252.87 - 0.003 - 6.1714 - 248.595 - 0.001 6.1536 - 246.048 - 0.003 5.873 - 218.789 - 0.001 - 5.7533 - 210.002 - 0.005 - 5.5356 - 195.802 - 0.001 5.3147 - 182.962 - 0.004 2.7407 - 78.476 - 0.004 - 1.4349 - 38.862 - 0.002 1.0679 26.87 0 4.2773 100 0.003 - 5.3414 122.37 0 7.0364 156.634 0.004 11.0133 231.9681 0.004 13.2188 271.442 0.004 - 16.0953 321.108 0.003 16.4733 327.502 0.002 18.2179 356.66 0.007 - 203 mat

NBP NormalBoilingPoint ≡ TP TriplePoint ≡ SP SublimationPoint ≡ BP BoilingPoint ≡ FP FreezingPoint ≡ 51 line

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

data FIT 2 1 sys

Δelta X δ 3 col augment : stem Δelta Error : bar Δelta 0.075 BarSize : datum X 0 Y * . 12 black plotG : 41 line

0 stem datum 3 1 sys

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 n sum : 0.000208

χ X Mean : μ Y Mean : n T rows : 31 line

χ μ n 31 mat 1.754059 14.306345 - 2031 mat

More statistical insight

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 / / : 0.9999999

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 0.986337977 0.98633812 21 mat

Example_3: fitting the reduced Kiby2

Prepare

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 : 31 line for data r3 r4 r5 augment 21 line :

Part 1: Search β's from robust al_nleqsolve

Reduced Kirby2 [Y/X] XY 9.65 0.00085 24 0.00538 62.63 0.0515 104.46 0.1376 134.79 0.1918 157.76 0.2222 176.25 0.2401 191.48 0.251 207.58 0.2591 229.92 0.2657 254.22 0.2684 278.83 0.2675 302.02 0.2645 324.42 0.2602 347.02 0.2548 371.3 0.2483 162 mat :

1. selct the support points
2. collect for al_nleqsolve list 159121515 mat : xy XY list ListCollect : 21 sys

raw XY o 5 black plot : support xy . 14 red plot : 21 line

pre-graph of the support points

raw support 2 1 sys

μ 3 :

optimum for this fit

Declare the model function x α Φ α 1 el x α 2 el^* α 3 el x α 4 el - (1 μ -^* - exp * x α 4 el - (μ -^* α 5 el + ≡

The support points x xy 1 col : y xy 2 col : 12 mat

vector of guesses X0 3000 1.5 50000100 - 015 mat transpose :

AlgLib Solver for a system of nonlinear equations(Uni) sol k 1 α f rows range u k el α k el : for α Jac u f u Jacobian : StepMax 0 : ε 1014 -^: 12 mat u X0 StepMax ε α f α Jac al_nleqsolve : 41 line :

The support points xy 9.65 0.0008 134.79 0.1918 207.58 0.2591 278.83 0.2675 347.02 0.2548 52 mat

Model function x α Φ α 1 el x α 2 el^* α 3 el x α 4 el - (1 μ -^* - exp * x α 4 el - (μ -^* α 5 el + :

The system of equalities
is constructed manually.
Expand/reduce to match
length of support points α f x 1 el α Φ y 1 el - x 2 el α Φ y 2 el - x 3 el α Φ y 3 el - x 4 el α Φ y 4 el - x 5 el α Φ y 5 el - 51 mat :

α sol : 2065.0903 1.6168 50207.6598 82.5509 - 0.0001 - 51 mat

raw x α Φ 2 1 sys

Mutual conformity
@ support points [SP]
within accuracy of SP α f 0.0007 - 0.0009 0.0005 0.0001 0.0009 - 51 mat

Part 2: Refine β's from LS Conjugate Gradient

Genfit 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 :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 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 :

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 :

NIST Kirby2 data Kirby 9.65 0.0082 24 0.129 62.63 3.223 104.46 14.377 134.79 25.853 157.76 35.06 176.25 42.31 191.48 48.06 207.58 53.78 229.92 61.1 254.22 68.23 278.83 74.6 302.02 79.88 324.42 84.4 347.02 88.43 371.3 92.2 162 mat :

x β f β 1 el x β 2 el^* β 3 el x β 4 el - (1 μ -^* - exp * x β 4 el - (μ -^* β 5 el + :

vector gradient function x β φ x β f 5 φ :

β β 1 el 2000 1.6 5000090 - 015 mat transpose : iter 5 : 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 :

β 2371.3622 1.5984 52309.9886 86.1521 - 0.001 51 mat

al_nleqsolve 2065.0903 1.6168 50207.6598 82.5509 - 0.0001 - 51 mat

Genfit [CG] 2371.3622 1.5984 52309.9886 86.1521 - 0.001 51 mat

round export. β 2371 1.5984 52310 86.15 - 0.001 51 mat :

Part 3: Recover Kirby2 from the "reduced"

Kirby2 x β f x * ≡

Kirbyfit x 04001 range : i 1 x rows range vy i el x i el β f x i el * eval : for x vy augment 31 line :

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

Kirbydata Kirby . 12 black plot : Title 16025 Transmute OVNI fit to Kirby2 12 black 15 mat : 21 line

j 1 XY rows range Δ j el Y j el X j el β f X j el * - eval : for δ X 100 Δ * augment : bar δ 1.5 BarSize (: SSD Y i el X i el β f X i el * - (2^(i 1 n sum : 41 line

SSD 0.0304

RMS SSD n / sqrt 0.0436

bar Kirbyfit x β f x * Kirbydata Title 5 1 sys

Part 4: Explain the project

Model f(x,β) => Courtesy Jean Giraud. ------------------------------------ ...This fit results from 'data reduction Y/X' from which the model is developped and reconstructed Kir(x)... Kirby2(x)*x. The reduced Y/X looks weird to fit, OVNI model will digest. Observe this model 'pins' the fit to datum 0 ! Reference: Kirby, R., NIST (1979). Scanning electron microscope line width standards.

jmGiraud: 2018-08-09

Explain/Proceed . . .

Part 1: This fitting session takes advantage of the robust al_nleqsolve, robust wrt to initials.

As it is fast, you can tentativly try up to you start getting some kind of fit, then refine further.

Once a fit is obtained, plug the β's in the next Genfit Minimize. al_nleqsolve passes through

support points. Next ConjugateGradient is an LS method, thus refit/refine the model function.

To ease the fitting session, copy/paste Part 1 in new work sheet ... once done the pre-fit,

transport the β's to proceed with Part 2.

The list is the user selected support points, highlighted in the pre-graph.

There are no rule(s) about finding initials, purely trial/error X0. From observation, the very

pre-fit that passes through the data set, is likely to produce a fit.

Finally => best round the parameters.

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