Weibull MLE N/A

34612184-37d5-4fe7-bdc9-2688f8b51f20 21 4 5 decimal

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

MCD 0.25 16.207 21 mat

Martin 0.331 30.71 21 mat

Jean 0.3055 18.17 21 mat

MLE Maximum Likelyhood Estimate

low high N logpts linear-log populate step high low / 10 log N / : i 1 N range x i el low 10 step i *^* : for x 41 line :

XY 10 0.000000000001 1000 0.75 10000 0.85 100000 0.88 1000000 0.91 100000000162 mat :

x y Axes x y / - :

X XY 1 col : Y XY 2 col : 12 mat

x log x log10 :

xo 1 : α 0.351 : β 30.71 : 13 mat xo α β x f 11 x xo - β / + (α -^- : u 11010^100 logpts : data X log vectorize Y augment : Ω u log vectorize xo α β u f vectorize augment : 51 sys

1010^11010^*

The 'x' range 1010^log 10

vx Ω 1 col : vy Ω 2 col : 12 mat x MLE vx vy x linterp : 21 sys

data 111012 -^* 3 0.75 4 0.85 5 0.88 6 0.91 8162 mat

day 2.5 :

day MLE 0.573

MLE day day MLE augment :

MLE @ 'day' MLE 2.5 0.5726 12 mat

Given MLE solve 'x day'

Ω data Axes MLE 4 1 sys

x f x MLE 0.75 - :

a 0 : b 10 : ε 0.00001 : 13 mat xn b a + (2 / : 21 line

Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn xn f n 31 mat 3.1946 01231 mat

slow timing U x MLE 0.75 - 0 ≡ x 020 solve : 3.6167

Remark ... 1. Martin coefficients look better fit. However, we look for best fit upper region. 2. Thus, Mathcad coeffs are prefered from two equivalent Conjugate Gradient algorithms Maximize and Minerr.

SSD = "Sum Square Differences" SSD vy i el xo α β vx i el f - (2^(i 1 XY rows sum :

SSD Martin 0.0036 SSD 0.0036

MCD 0.25 16.207 21 mat

Weibull MLE N/A Mathcad Maximize Attempted MLE ..... FREAK !

α β Clear 1

α β x f 11 x β / + (α -^- :

α β x f ln β x + β / (α^1 - ln α β x + β / ln * -

β x + β / (α^1 - ln α β x + β / ln * - α diff β x + β / ln 1 - β x + β / (α^+ /

β x + β / (α^1 - ln α β x + β / ln * - β diff x α * β β x + (* 1 - β x + β / (α^+ (* / -

X 13456861 mat :

length _X n 6 :

system 1 β X i el (i 1 n sum + β / ln 1 - β X i el (i 1 n sum + β / (α^+ / * 0 ≡ 1 X i el (i 1 n sum * α * β β X i el (i 1 n sum + (* 1 - β X i el (i 1 n sum + β / (α^+ (* / - 0 ≡ 21 mat :

init α 0.1 ≡ β 1 ≡ 21 mat :

fucken freak !

α β sol system init 0.001 FindRoot :

α β 21 mat α β sol : 3.0549 1.8169 21 mat

α 110 / * β 101 / * 21 mat 0.3055 18.1686 21 mat

sanity β X i el (i 1 n sum + β / ln 1 - β X i el (i 1 n sum + β / (α^+ / X i el (i 1 n sum α * β β X i el (i 1 n sum + (* 1 - β X i el (i 1 n sum + β / (α^+ (* / - 21 mat : 0.00059531 0.00033933 - 21 mat

Working with Weibull

α β Clear 1

plot(data,char,size,clr)

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 :

Weibull PDF model x α β W α β * x (β 1 -^* α x β^* - exp * :

X 1.304 1.088 0.335 0.493 1.945 1.123 1.274 0.476 2.03 1.402 3.645 1.28 0.326 0.153 0.961 0.3 1.236 2.007 0.264 0.964 201 mat :

length _X n 20 :

log vraisemblance α β x L n α β * ln * β 1 - (X i el ln i 1 n sum * + (α X i el β^i 1 n sum * - :

partial derivatives and
1rst order conditions. α β x L α diff n α / X i el (β^i 1 n sum - ≡ (0 ≡ α β x L β diff n β / X i el ln i 1 n sum + α X i el (β^(X i el ln * i 1 n sum * - ≡ (0 ≡ 21 sys

system n α / X i el (β^i 1 n sum - 0 ≡ n β / X i el ln i 1 n sum + α X i el (β^(X i el ln * i 1 n sum * - 0 ≡ 21 mat :

init α 1 ≡ β 1 ≡ 21 mat :

α β sol system init 0.001 FindRoot :

α β 21 mat α β sol : 0.724588 1.444003 21 mat

sanity n α / X i el (β^i 1 n sum - n β / X i el ln i 1 n sum + α X i el (β^(X i el ln * i 1 n sum * - 21 mat : 0.000755 0.000369 21 mat

α β 21 mat 0.724588 1.444003 21 mat :

Weibull  log likely
probability PDF x α β W α β * x (β 1 -^* α x β^* - exp * eval :

i 1 X rows range Y i el X i el α β W eval : for XY X Y augment : data XY o 5 black plot : 31 line

data x α β W 2 1 sys

Weibull log vraisemblance . . . [ loglikelyhood]

Two sets of conditions are required for solving the two

parameters α, β. Nicely enough, more savant Mathematicians

concluded the L(α,β,x) and the two solving conditions wrt the

data vector X. Smath spits out solution with great elegance !

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