Genfit Utilities

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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 :

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 :

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 : Data X Y . 10 black plotG : 31 line

10573 - 41 mat

8.5418 40.7318 52.0514 48.7382 - 41 mat

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 :

x β f number param in f(x,β φ

β β 1 el 10573 - 14 mat transpose : iter 3 : 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 8.5418 40.7318 52.0514 48.7382 - 41 mat

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 Collab NDTMA.

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