Minimize Conjugate-Gradient

0c827f16-3007-47ff-9ef0-2710980708e1 32 4 5 decimal

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

Statistical fit ...

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 :

init iter Minimize XY f φ β LS 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 : β 1 el init : i 1 : i iter ≤ β i 1 + el XY f φ β i el LS : i i 1 + : 21 line while β β iter el : 41 line β 31 line :

Vector Gradient ... Optimiz Symbolic
seemingly a scalar kernel  not 
recognized in program loop 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 :

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 :

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 :

corr Pearson correlation 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 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 / / 31 line :

correlation between [X,Y data]

correlation ... [X,fitted model]

Absolute  residuals observer
fetches β from calculated δ j 1 XY rows range Δ j el Y j el X j el β f - eval : for X Δ augment 21 line :

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

The Fréchet (“Fr” for short) distribution is one of the important distributions in extreme value theory and it has applications ranging from accelerated life testing through floods, earthquakes, horse racing, rainfall, queues in supermarkets, wind speeds and sea waves.

XY 1 0.4897 2 1.3079 3 2.84 4 3.0364 5 1.7709 6 0.5491 7 0.4606 8 0.3345 9 0.2073 10 0.0012 11 0.0024 112 mat :

X XY 1 col : Y XY 2 col : 12 mat Data XY . 10 black plotG : 21 line

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

x β φ x β f 4 φ :

init 10573 - 14 mat transpose :

iter 3 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

SSD 0.472

corr 1.069

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 :

FIT range, populate U 011 0.05 range : i 1 U rows range FIT i el U i el β f eval : for U FIT augment 41 line :

Residuals 0.1 BarSize 0 2 1 sys

β 8.5418 40.7318 52.0514 48.7382 - 41 mat :

XY 0.1 BarSize FIT 2 1 sys

Cumulative discrete integration
wants β's isolated near the module Σ u 011 0.1 range : i 1 u rows range Cumul i el x β f x u 1 el u i el int eval : for u Cumul augment 31 line :

Σ Σ :

Σ FIT 2 1 sys

cumul ∫f(z) @ point Σ 111 row 11 10.9 12 mat

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 :

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 :

XY 040018002200034000460005470063300724008160091100115001240013400132 mat :

Select the support points

1. selct the support points
2. collect for al_nleqsolve list 13579111317 mat : xy XY list ListCollect : 21 sys

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

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

β 16537 40.1 - 44.0 44.13 25 - 27 - 40071 mat :

15800 40.1 - 44.0 44.125 25 - 27 - 40071 mat

16537 40.1 - 44.0 44.13 25 - 27 - 40071 mat

x β F β 1 el β 2 el x β 3 el - (/ (β 5 el^β 2 el x β 4 el - (/ (β 6 el^- exp * (* β 7 el + :

x β F ρaw support 3 1 sys

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 :

4 β F 6000.1

Absolute residuals

Residuals 0.1 BarSize 4 500 4 500 - 2 2 mat 2 1 sys

The support points X xy 1 col : Y xy 2 col : 12 mat

β f X 1 el β F Y 1 el - X 2 el β F Y 2 el - X 3 el β F Y 3 el - X 4 el β F Y 4 el - X 5 el β F Y 5 el - X 6 el β F Y 6 el - X 7 el β F Y 7 el - 71 mat :

Mathcad LM fails to fit better β f 0416 - 0434150 - 262771 mat

Project notes . . .

1. 1rst data set is some US rainfall.

2. 2nd data set from Smath Forum.

3. The fit of modified F(x,β) is done manually.

4. Mathcad 11 LM Minerr fails to compute β.

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Simpson fails unexpectedly ? Σ u 013 0.1 range : i 1 u rows range Cumul i el x β F x u 1 el u i el int eval : for u Cumul augment 31 line :

Σ Σ :

Σ x β F 2 1 sys

cumul ∫f(z) @ points Σ 131 row 132828212 mat

This 'Fr' that was found otherwise, will not be supported by the
numerical discrete integrator. Exponents are above max allowed.
It can be recast in more visible coefficients. The procedure is
applicable to specific project(s) otherwise difficult to manage. x Fr 143253 - x 43256.75 - (/ (25388.3 -^* 43253 - x 43256.75 - (/ (27354.7 -^- exp * :

m 40.21 :

x Φ m - x 44 - (/ (23 -^m - x 44 - (/ (25 -^- exp * :

x Δ x Fr x Φ - :

x Δ x Fr x Φ 2 1 sys Bar & Stem

Bar plot unicolor style, bar width user. data dt BarGraph 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 :

Spectrum stem plot for QuickPlot unicolor data Stem 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 :

Traditionally, Gumbel was the model "rainfall". It is now recognised that EV_II better represents "rainfall" from 154 years collection world wide. Here, we propose "Fréchet" Type II.

Fréchet: extreme value-II K α β x f K α β / * β x / (α 1 +^* β x / (α^- exp * :

US 5.123 4 5.36 6 5.597 14 5.834 9 6.071 7 6.308 24 6.545 10 6.782 4 7.019 9 7.256 5 7.494 11 7.731 7 7.968 4 8.205 2 8.442 1 8.679 2 8.916 3 9.153 0 9.39 1 9.627 1 9.864 2 10.101 1 10.338 0 10.575 2242 mat :

X US 1 col : Bar US 0.025 BarGraph : 21 line

XY i 1 X rows range Y i el K α β X i el f : for X Y augment 21 line :

K 31.664 : α 6.774 : β 6.212 : 31 line

plot K α β x f US Stem XY 31 sys :

plot Bar

α β x Fr α β / β x / (α 1 +^* β x / (α^- exp * :

Fr 0.5 1 x Fr 11 x Fr 21 x Fr 51 x Fr 41 sys :

Fréchet distribution

XY 5.123 4 5.36 6 5.597 14 5.834 9 6.071 7 6.308 24 6.545 10 6.782 4 7.019 9 7.256 5 7.494 11 7.731 7 7.968 4 8.205 2 8.442 1 8.679 2 8.916 3 9.153 0 9.39 1 9.627 1 9.864 2 10.101 1 10.338 0 10.575 2242 mat :

Fréchet: extreme value-II
Recognized model US rainfall. x β f β 1 el β 3 el (* β 2 el / β 2 el x / (β 3 el 1 +^* β 2 el x / (β 3 el^- exp * :

<= ConjugateGradient Vector Fnct x β φ x β f 3 φ :

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

'n' is of same legth as β

init 206513 mat transpose :

iter 7 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

β 31.665 6.2122 6.7726 31 mat

SSD 278.38

corr 0.767

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

FIT range, populate U 111 1.1 range : i 1 U rows range vy i el U i el β f eval : for U vy augment 41 line :

XY 0.025 BarSize FIT 2 1 sys

Absolute residuals from CG

Residuals 0.0375 BarSize 5 10 5 10 - 2 2 mat 2 1 sys percentile(v,p)..... qqplot(y1,y2)

v p percentile n v rows : pos1 p n * p + (0 - : position 0 pos1 0 ≤ pos1 0 pos1 < (n 1 - < n 1 - pos1 n 1 - ≥ cases : between position position Floor - : sorted v sort : percentile sorted position Floor el between sorted position Ceil el sorted position Floor el - (* + : percentile 71 line :

y1 y2 qqplot y1 & y2 are vectors λ1 y1 length : λ2 y2 length : pts λ1 λ1 λ2 ≤ λ2 cases : j 1 pts range p j el 100 j * pts / 0.3175 + 100 0.365 + / eval : qq j 1 el y1 p j el percentile : qq j 2 el y2 p j el percentile : 31 line for qq 61 line :

batch data

batch 345.3 382.1 415.1 426.1 445.1 443.7 454.1 489.9 462.3 503.9 470.4 515.6 472.4 528.2 489.7 531 507.6 535.9 512.1 544 519.9 546.7 529.6 547.1 533.7 562.6 534.8 571.9 537.5 578.1 538.2 586.8 539.8 590.7 540.3 592.1 542.7 596 543.3 597.4 545 600.4 548.5 603 549.2 608.1 549.3 609.8 549.5 611.6 551.6 612 551.9 612 552.3 614.6 553.8 617.8 554.7 618.6 554.8 619.1 555.6 619.9 555.7 624.5 555.7 625 555.9 625 557.6 625558 625.9 563 628.8 563.8 630.3 565.3 632.5 565.7 633.4 566.1 635 566.9 636.7 567.5 637.4 569.2 638.8 569.2 639.1 569.5 640.2 569.6 640.9 569.7 642.5 569.9 642.6 571 643.4 571 644.2 571.6 650.8 571.8 655 571.9 656.8 574.5 657.7 574.9 658.1 575.1 658.8 575.4 658.8 576.7 659.5 577.2 659.6 577.4 660.2 577.4 663 577.8 663.6 579.2 663.9 579.6 664.8 581.1 666.8 581.6 667 581.7 667.8 581.9 668.7 582.7 670.4 583 670.9 583.2 671.7 583.2 674 583.8 674.5 584 674.8 584.2 676.2 584.4 677.9 584.7 678.2 584.9 680.3 585.5 681.3 585.8 681.7 586 682.1 586.2 682.5 587.2 682.6 587.5 683.4 587.5 683.8 587.8 684.2 588.2 684.6 588.5 684.9 589.3 685.4 589.7 686.4 590.9 687.5 592.1 687.7 593688 593.5 688.4 593.8 689 594.5 689.6 594.7 690 595.4 690.3 595.7 690.5 595.9 690.8 596.8 690.9 599 691.1 599.4 691.6 599.8 691.9 599.9 693601 693.4 602.8 693.8 603.4 694.6 604 695.1 604.5 695.3 605.3 695.4 605.4 695.7 605.5 695.8 605.8 696.4 606.3 697.1 606.5 697.4 607 697.7 608 698.2 608.6 698.9 608.7 699.4 609.4 700.4 610.5 700.8 610.9 701 611.1 701.5 611.4 702.2 611.8 703.4 611.9 704.1 612706 612.2 707.3 612.4 708.1 612.7 708.2 612.9 708.2 614 708.2 615.4 708.4 616.1 708.7 616.6 709.2 617 709.7 617.3 710 617.4 710.5 617.4 711.4 618.1 712.2 619.2 712.3 619.7 713 620.1 714.2 620.5 714.9 620.9 715.1 621.2 715.3 621.6 715.4 621.8 715.8 621.8 716.5 621.8 716.9 622.1 717.1 622.8 717.7 623.2 718.8 623.3 719.5 623.8 720 624.2 720.3 624.4 720.7 624.8 721.1 625.5 721.8 626.8 722.8 627.6 723.3 627.8 723.4 628 723.6 628.2 723.9 629.7 724.9 631.3 725.6 631.8 726 632.1 726.8 632.5 727.9 633.1 728.6 634729 634.9 729.2 636.1 729.4 637730 638.4 730.5 639.8 730.8 640.3 731.3 640.8 731.9 642.5 733.3 643.9 734.1 644.3 735.3 644.9 735.9 645.3 736.4 645.7 737 646.8 737.8 647.7 738.2 648.6 738.3 650.5 738.8 651.7 740 652.3 740.7 653.8 741.3 655.4 743.2 656.7 744.5 659.1 744.8 661.5 744.8 662.4 745.3 664.3 746.2 665.6 747.2 669.8 749.5 671.9 750.5 674 750.6 676.1 751.4 676.7 751.8 678.8 752.2 681.7 752.4 683.7 753.1 684.4 755.2 685.6 756.6 688.7 757.2 689.9 758.1 695.1 759.3 697.9 759.7 698.3 759.8 698.3 760700 760.5 700.7 761.2 702.1 761.4 703.7 762.8 706.5 763.7 707.5 766 710.6 766.6 711.6 767.8 713.4 769.2 715.4 769.4 719.9 772.6 722.3 774.7 724.1 775.1 725.5 776.1 728.8 776.4 744.9 777.3 748 778.2 755.9 785.3 760.9 789.8 775.9 795.1 807.8 797.4 820.5 813.6 821.7 815.8 2402 mat :

y1 batch 1 col : y2 batch 2 col : 12 mat

sanity MCD DAEP y1 0.25 percentile y2 0.25 percentile y1 0.75 percentile y2 0.75 percentile 41 mat 576.8 659.5 640.7 731.8 41 mat

x y1 Min y1 Max range :

x x x augment :

SYMBOLIC qq y1 y2 qqplot :

qq rows qq cols 21 mat 240221 mat

Ignore Smath complaining qq is @ kernel level ...

i 1 qq rows range qqX i 1 el qq i 1 el : for

i 1 qq rows range qqY i 1 el qq i 2 el : for

NUMERIC qqplotXY qqX qqY augment :

x qqplotXY 2 1 sys

qqplotXY 398.6 415.7 438 439.5 452479 460.4 500.6 468.5 512.9 471.9 525.2 485.7 530.3 503.4 534.8 511.1 542.1 518.1 546.1 527.4 547 532.8 559 534.5 569.8 536.9 576.7 538 584.8 539.4 589.8 540.2 591.8 172 mat

Quantiles and qqplot . . .

of these two batches shows that they have not come from the same distribution

although the values for very small and very large quantiles are similar.

The 1/1 reference line is plotted to emphasize how the data vary. The large

deviation between the two quantiles suggests very different processing conditions.

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t f 2 t 300 / exp cos - :

stdevwn 0.2 :

stdevout 1 :

j whitenoise j Random 0 stdevwn ICDF.Normal eval :

j outliers j Random 0 stdevout ICDF.Normal eval :

rn 1000 :

j 1 rn range :

white j rn whitenoise 2 col augment : out j rn outliers 2 col augment : 21 line

F j f vectorize :

white white :

out out :

i 40 rn 80 range U i el 0 : U i 19 - el 1 : U i 18 - el 1 : U i 17 - el 1 : U i 16 - el 1 : U i 15 - el 1 : U i 14 - el 1 : U i 13 - el 1 : U i 12 - el 1 : U i 11 - el 1 : U i 10 - el 1 : U i 9 - el 1 : U i 8 - el 1 : U i 7 - el 1 : U i 6 - el 1 : U i 5 - el 1 : U i 4 - el 1 : U i 3 - el 1 : U i 2 - el 1 : U i 1 - el 1 : 201 line for

U U :

Create bursts of outliers i 1 rn range Ω i el U i el out 2 col i el * eval : for

white j Ω augment 2 1 sys

U rows 1000

C F white 2 col + Ω + :

chirp j C augment :

chirp j F augment 2 1 sys medsmooth

Y w medsmooth n Y rows : w must be odd m w 1 - 2 / : i 1 n 1 - (range mvgavg i el Y j el j 12 i * sum 2 i * 2 + / i m < Y j el j 2 i * n - n sum 2 n i - (* / i n m - > Y j el j i m - 1 + i m + sum w / cases eval : for mvgavg 51 line :

Part_1: presmooth outliers

Project remark . . .

1. medsmooth is independent of outliers magnitude.

2. LinfitCheby, LSQ piece wise are the sole smoothers.

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

must be odd w 15 :

n X rows :

m w 1 - 2 / : 7

kind of balance  ... accept ! scale 112 m * / + :

mvgavg Y w medsmooth : X X 1 mvgavg rows 11 submatrix : Y mvgavg 1 mvgavg rows 11 submatrix : 31 line

scale 1.0714

chirp X scale Y * augment 2 1 sys

========================================================

ε 1015 -^:

Mathcad x 0 stdev ICDF.Normal x 0 stdev qnorm ≡

x α f x 0 α ICDF.Normal :

x ε 1 0.01 range :

data x x 0.5 f vectorize augment :

x y axes x y / - :

data 1 10 1 10 - 2 2 mat axes 3 1 sys

Project Note . . .

1. Spline_2xV6 freeware, takes *.dat from file.

2. I was unable to finish converting Mathcad 11

LSQ ORIGIN[0] => ORIGIN[1] ... => Smath

3. LinfitCheby . . . RemToDo

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LSQ [Robert] Mathcad 11 ORIGIN:=0

Ucertainty on vector of target values

4 decimals x Γ x Gamma :

vy 25.527 25.51 25.512 25.511 25.511 25.508 25.487 25.501 81 mat :

vy incertitude n vy length : u vy StdDev n sqrt / n 1 - 2 / Γ n 1 - sqrt * n 2 / Γ 2 sqrt * / * : 21 line :

vy Mean 25.508375

MCD 0.00418 vy incertitude 0.00413

U vy incertitude ± : 0.0041 0.0041 - 21 sys

vy Mean U + 25.5125 25.50425 21 sys

i 1 vy rows range :

i vy 25.5125 - augment i vy 25.50424 - augment 2 1 sys

Trending . . . raw data

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t0 1 time :

ksmooth # plotG

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 :

σ 0.3708158988058244 :

The Gaussian ksmooth

d K 12 π * sqrt σ * / d 2^2 σ 2^* / - exp * :

Smath recognises the silent 'bw', something
like the Mathcad "global definition". j X Y Ksmooth X i el X j el - bw / K Y i el * (i 1 X rows sum X i el X j el - bw / K (i 1 X rows sum / :

Gaussian Ksmooth Kernel

x K

X Y bw 13 mat

W j 1 X rows range s j el j X Y Ksmooth eval : for s 21 line :

W 11 mat

raw samples XY 02116121212209932120421135209862067720718207092056102083112090122072132064142081152074162054172056182058192061202 mat :

Prepare and code the Smath project

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

bw 5 : (bandwidth ≡ W W : (isolate W ≡ XW X W augment : 31 line

W 11 mat

The working path: collect & return [X,Y]=>[bw]=>[Ksmooth(j,X,Y,bw)]=>[W]=>[XW]

XW XW o 7 red plot : raw XY + 7 black plot : 21 line

===== Finish/collect the project =====

u Decay 2053.7374 70.0025 u - 8.0493 / exp * + :

spline join/mesh the smoothed data set Cfit t 019 0.1 range : i 1 t rows range v i el X W t i el cinterp eval : for t v augment 31 line :

collect plot Cfit raw x Decay XW XY 51 sys

x Decay x 019 int 39531

XY=>raw data, XW=>smoothed data, Cfit=> meshed spline export

Cfit raw x Decay XW XY 5 1 sys

1 time t0 - 3 s *

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 00102 0.143 3 0.727 4 0.889 5 0.875 6 0.714 7182 mat :

β 7 2.5 21 mat :

Transmute Sigmoid

x β f β 1 el 1 x / (β 2 el^* - exp :

plot XY o 5 black plot :

U 2.5 :

track 10 0.25 5 0.25 U U β f 32 mat :

User U U β f + 20 blue 15 mat :

track x β f plot User 41 sys

track x β f plot User 4 1 sys

Rotavirus . . . Pathogen, Microorganism

Collected from Steven Finch [Mathsoft].

Solved otherwise [jmG], Transmute Sigmoid

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U U β f 12 mat 2.5 0.49 12 mat

1000 β f 1

Statistical composite error ±

where ε(1,2,,,) are in Engineering units ε ε1 2^ε2 2^+ εn 2^+ + sqrt ≡