Function Utilities

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This Smath document will help you complement the "Stat Treasury". 1. maple indefinite integral wiil solve for the cumulative CDF. 2. maple solve will solve for the quantile(s) ICDF. Most PDF from the treasury are already "normalised", i.e: the PDF are scaled to provide CDF from range(0..1)

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

α β x Fr_p x int maple α β x / ln * exp - exp

y α β x / ln * exp - exp ≡ x solve maple β y ln - ln α / exp /

α β x Fr_c α β x / ln * exp - exp :

α β x Fr_q β x ln - ln α / exp / :

PDF 0.5 1 x Fr_p 11 x Fr_p 21 x Fr_p 51 x Fr_p 41 sys :

CDF 0.5 1 x Fr_c 11 x Fr_c 21 x Fr_c 51 x Fr_c 41 sys :

QDF 0.5 1 x Fr_q 11 x Fr_q 21 x Fr_q 51 x Fr_q 41 sys :

Fréchet PDF "probability distribution"

PDF

Fréchet QDF "quantile"

QDF

Fréchet CDF "cumulative distribution"

CDF

Filling a statistical region [PDF]

Prepare the project

Construct the project

x f 51 x Fr_p :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

sx 0.75 :

N 1000 :

xmin 0.75 :

xmax 1.5 :

ymin 0 :

ymax 2 :

space_x xmax xmin - (:

PDF x 1012 -^2 0.005 range : i 1 x rows range y i el x i el f eval : 11 line for x y augment 31 line :

space_y ymax ymin - (:

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x f ≤ (x xmax ≤ (* :

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 7 #808080 13 mat :

fill Pts a b c spec pts :

Domain

Xmin xmin 1 xmin 0.25 - 22 mat :

Xmax xmax 1 xmax 0.25 - 22 mat :

Filling a range of a PDF

PDF Xmin Xmax fill 4 1 sys x f x xmin ≥ ( x xmax ≤ ( * / Xmin Xmax 3 1 sys

a xmin : b xmax : 21 line

x f x a b int 0.861827897942007

51 b Fr_c 51 a Fr_c - 0.861827896719184

f a b Int x f x a b int 32 / b a - (* w G w 01 int * ≡ memo ≡ w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 41 line :

f a b Int 0.861827894670677

Weibull distribution

α β t Sw t α / (β^- exp :

The survival function: the probability that a unit survives for at least time 't'.

The cumulative failure function: the probability that a unit failss for at time T<t.

α β t CDFw 1 α β t Sw - :

α β t PDFw t α / (β^β t / (* t α / (β^- exp * :

The probability density function of failure @ time 't'.

1 0.5 x Sw 1 1 x Sw 1 2 x Sw 3 1 sys

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

x α β L α β * x i el β 1 - (^* α x i el β^* - exp * i 0 n 1 - product : (Vraisemblance ≡

n α β * ln * β 1 - (x i el ln i 0 n 1 - product * + (α x i el β^i 0 n 1 - product * - log vraisemblance ≡

x 1.304 1.088 0.335 0.493 1.946 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 :

n x length : 20

x x sort :

i 1 n range :

X i x augment :

α 0.725 : β 1.444 : 21 line

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

Ω i 1 n range w i el x i el α β W eval : for x w augment 21 line :

Ω Ω : vx Ω 1 col : vy Ω 2 col : W vx vy . 12 black plotG : 41 line

x α β W vectorize supports x as a defined vector ... x:= ≡

W x α β W 2 1 sys

Fit a PDF data set

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

XY 0.013 12 0.038 57 0.063 96 0.088 91 0.113 83 0.138 97 0.163 111 0.188 114 0.213 109 0.238 142 0.263 111 0.288 139 0.313 103 0.338 119 0.363 127 0.388 118 0.413 156 0.438 137 0.463 133 0.488 125 0.513 101 0.538 105 0.563 116 0.588 110 0.613 94 0.638 104 0.663 90 0.688 87 0.713 84 0.738 73 0.763 87 0.788 82 0.813 68 0.838 70 0.863 65 0.888 66 0.913 51 0.938 49 0.963 48 0.988 44 1.013 41 1.038 39 1.063 38 1.088 37 1.113 32 1.138 35 1.163 30 1.188 29 1.213 23 1.238 30 1.263 21 1.288 23 1.313 26 1.338 23 1.363 24 1.388 19 1.413 14 1.438 17 1.463 13 1.488 22 1.513 12 1.538 14 1.563 10 1.588 18 1.613 14 1.638 13 1.663 10 1.688 14 1.713 8 1.738 8 1.763 4 1.788 8 1.813 6 1.838 9 1.863 10 1.888 9 1.913 3 1.938 2 1.963 8 1.988 4 2.013 4 2.038 10 2.063 4 2.088 3 2.113 1 2.138 5 2.163 2 2.188 5 2.213 3 2.238 4 2.263 7 2.288 9 2.313 1 2.338 1 2.363 5 2.388 6 2.413 3 2.438 5 2.463 3 2.488 2 2.513 1 2.538 1 2.563 2 2.588 1 2.613 4 2.638 4 2.663 1 2.688 3 2.713 4 2.738 4 2.763 4 2.788 1 2.813 4 2.838 0 2.863 2 2.888 4 2.913 2 2.938 0 2.963 0 2.988 0 3.013 1 3.038 2 3.063 1 3.088 2 3.113 0 3.138 0 3.163 1 3.188 0 3.213 0 3.238 0 3.263 1 3.288 0 3.313 1 3.338 0 3.363 1 3.388 3 3.413 1 3.438 0 3.463 1 3.488 0 3.513 0 3.538 1 3.563 1 3.588 3 3.613 0 3.638 0 3.663 1 3.688 0 3.713 0 3.738 0 3.763 0 3.788 0 3.813 1 3.838 1 3.863 0 3.888 1 3.913 0 3.938 2 3.963 1 3.988 2 4.013 0 4.038 0 4.063 1 4.088 1 4.113 0 4.138 0 4.163 1 4.188 0 4.213 0 4.238 1 4.263 0 4.288 0 4.313 0 4.338 0 4.363 1 4.388 0 4.413 0 4.438 1 4.463 0 4.488 0 4.513 1 4.538 0 4.563 0 4.588 0 4.613 1 4.638 0 4.663 0 4.688 1 4.713 0 4.738 0 4.763 0 4.788 0 4.813 0 4.838 0 4.863 0 4.888 0 4.913 1 4.938 0 4.963 1 4.988 12002 mat :

The user PDF model x f scale x xo - (k 1 -^x xo - μ / (- exp * k Gamma μ (k^* / * :

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

scale 115 : k 2.0 : μ 0.313 : xo 0.0 - : 41 line

scale 114.208 : k 2.091 : μ 0.311 : xo 0.024 - : 41 line

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

FIT U 05 0.025 range : i 1 U rows range FIT i el U i el f eval : for U FIT augment 31 line :

δ 0.013 12 2.6 - 0.038 57 17.5 0.063 96 35.5 0.088 9113 0.113 83 9.4 - 0.138 97 7.2 - 63 mat

data X Y . 10 black plotG :

The Smath "Genfit Minimize" does not perform to best fit this user model. You can attempt manually or take it from done in Mathcad. The original data set is immense, the size of the "bin" is user choice. It influences the choice of the model. A PDF model may be a "statistical ly", deeper analysis is required and there is no rule, fit to preference.

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

0 f 0

SSD 7030

data FIT μ 200 μ 0 2 2 mat 3 1 sys

Δelta X δ 3 col augment : stem Δelta Error : 21 line

0 stem μ 50 μ 50 - 2 2 mat 3 1 sys

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

χ μ n 31 mat 2.5005 23.3 20031 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 f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el f eval μ - (2^(i 1 n sum * sqrt / / : 70.025437 -

correlation between [X,Y]

correlation between [X,fitted model]

A wise Preference

You would not be wrong thinking the Gamma(k)*(μ)^k model is a bit not digestible. This next model is "too beautiful not to be true" ["..." Paul Dirac].

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

The user PDF model x a m f m x * a x * - exp * :

a 203 / : m 2467 : p 0.48 : 31 line

fit U 05 0.025 range : i 1 U rows range fit i el p U i el * a m f : for U fit augment 31 line :

px 320 / 1 p / * : 0.3125

data fit px 200 px 0 2 2 mat 3 1 sys

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

SSD 7162.2

The "local void ratio"

data m p * x * a p * x * - exp * px 200 px 0 2 2 mat 3 1 sys

The project originates from the Mathsoft Collaboratory: "local void ratio" porosity or void fraction. The original data was reduced Ydata/Xdata The Gamma model was first explored.

Reduce Ydata Xdata / ≡

bin 200 ≡

H bin Reduce histogram ≡

H our XY data set ≡

m p * x * a p * x * - exp *

m p * a p * - 21 mat 1184.16 3.2 - 21 mat

The final "Void Ratio" 
original data set. "Genfit" 
supports the 'peak' model. x β f β 1 el x * β 2 el x * exp * :

β 1184.2 3.2 - 21 mat :