N

c0b6ec20-9ed3-4495-badd-452f9fcaf386 50 15 12 decimal

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

N

N 15214334555667138109610911151213137147151016617218121913201021122252312241125112617277282299305313329331634113513367379389397409417428431344114544611479486497501851115255313548551256165785813591660561862763664765116646796856917708711372137313741275137617747887914808811182118312841285486487188689159011911092293129413958961597139889910100141011210271036104121051510681074108111091211031111311281131311416115101161111712118101198120161215122161236124161251112614127212891290130113141327133713491351213681371113891391014017141414213143414417145121461614712148101491415031514152121532154315511156815715158111591016091618162916310164116551661716717168316911170111711017281734174121751717681771217811179101801218110182131831118491850186161872188318911190319117192121935194919511196919761983199720052002 mat :

Utilities

M Min Max Scale Max Min - (M M min - M max M min - 106 -^+ / (* Min + :

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 :

Stem plot for QuickPlot unicolor data Stem stem 0 : 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 : 81 line :

size min max vx Hist Paul W. MCD 11 => Smath vx last vx rows : numbins max min - size / : hist numbins el 0 : i 1 numbins range bin_mid i el i 1 + (size * size 2 / - : j 1 vx last range i size * vx j el ≤ (i 1 + (size * < hist i el hist i el 1 + : N/A if for 21 line for bin_mid hist vx last / augment 61 line :

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 :

μ N 2 col Mean : 9.29

'N' is randomNormal generated, histogram is therefore PDFnormal. Cumulative weighted classification can be used to identify the PDF that best fit the data set. Is our data set PDFnormal ?.. YES.

rn N 0 μ Scale :

bin 20 : size 1 bin / : min 0 : max 1 : 41 line

vy rn 2 col : H size min max vy Hist : 21 line

μ vy Mean : σ vy StdDev : 21 mat 0.4315 0.1956 21 mat

3 appVersion 0.98.6179 ≡

σ μ sqrt 2 / : 0.328

Cumulative weighted clssification

H 0.0125 BarSize x 2 μ * σ pdf.Normal 1 * 0.825 * 2 μ * 1.5 2 μ * 0.5 - 2 2 mat 3 1 sys

Normal random from source creator

N Stem 0 2 1 sys

=== The histogram companion ... integer data set ===============

counts specific values in a sorted vector value table count k 1 : j k table rows range table j el value ≡ index k el j : k k 1 + : 21 line No match if for index rows 31 line :

N N 2 col sort :

N max 18

collect all the counted values in sorted 'N' N bin histogram bin N max ≤ i 0 bin range x i 1 + el i : count i 1 + el i N count : 21 line for x count augment 21 line over bin if 11 line :

H N 18 histogram :

bin N max ≤

Brute force alternative: for small values data set scale and round integers.

μ N Mean : σ N StdDev : σ½ 0.5 σ * : 31 mat 9.29 4.21 2.11 31 mat

low N min : upp N max : 21 mat 01821 mat

n N length : h upp low - H rows / : 21 mat 200 0.9474 21 mat

s 0.25 :

To best Engineer collected data: 1. Measuring equipment should be 10 times more accurate than the measured. 2. Make sure the collected data haven't been freaked by else, i.e: raw from collection. Data set not raw from source => recycle.

Sigma μ σ - s - μ σ - s μ σ + s μ σ + s - μ σ - s - 52 mat :

Stem & bar plots stem H Stem : bar H 0.1 BarSize : 21 line

H 02142639411512697138179161014112212201317145155169178181192 mat

vx 0 n 1 range :

vy vx μ σ pdf.Normal n * h * vectorize :

xy vx vy augment :

xy xy . 11 black plot :

Stem style histogram fitted PDFnormal... μ, σ

x μ σ pdf.Normal n * h * stem μ 40 μ 10 - 2 2 mat xy Sigma 7 1 sys bar x μ σ pdf.Normal n * h * μ 40 μ 10 - 2 2 mat 3 1 sys

μ σ ±

Exercise 1: Mathcad style . . . hist uniform

]]> hist(intervals,values) ... [reduced version from Davide Carpi]

intervals values hist binBounds intervals sort : binMin binBounds 1 el : binCount intervals length 1 - 1 matrix : sortedValues values sort : lastIndex 1 : j 2 intervals length range binMax binBounds j el : k lastIndex values length range v sortedValues k el : lastIndex k : v binMin < value below the range continue 21 line v binMax ≥ value belong to next bins break 21 line found a value for this bin binCount j 1 - el binCount j 1 - el 1 + : continue 31 line if if 31 line for binMin binMax : 31 line for binCount 71 line eval :

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 :

Stem plot for QuickPlot unicolor ... redefined/isolate 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 :

Uniform random data 1000100 Random.N :

Intervals to be populated by 'h' intervals 02040608010016 mat transpose :

h intervals data hist :

h transpose 221725112415 mat

j 1 h rows range : frequency j h augment : stem frequency Stem : 31 sys

frequency 0.5 BarSize stem 2 1 sys

data Min data Max data rows 31 mat 210010031 mat

h sum 99

Exercise 2: Mathcad style . . . hist normal

]]> hist(intervals,values) Utilities

intervals values hist binBounds intervals sort : binMin binBounds 1 el : binCount intervals length 1 - 1 matrix : sortedValues values sort : lastIndex 1 : j 2 intervals length range binMax binBounds j el : k lastIndex values length range v sortedValues k el : lastIndex k : v binMin < value below the range continue 21 line v binMax ≥ value belong to next bins break 21 line found a value for this bin binCount j 1 - el binCount j 1 - el 1 + : continue 31 line if if 31 line for binMin binMax : 31 line for binCount 71 line eval :

Stem plot for QuickPlot unicolor ... redefined/isolate 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 :

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 :

Cols 40 :

@ 1000 => 1/1 PDFnormal length 100 :

length of 'M': user M length Cols Random :

rn j 1 M cols range S j 1 el M j col : for S sum 21 line eval :

Generate randon normal

i 1 M rows range : rn i rn Floor augment : 21 sys

'rnd => vector of user scaled normal random

rn Stem 1 1 sys

upper rn 2 col Max Ceil : lower rn 2 col Min Floor : 21 sys 261621 sys

n M length : μ rn 2 col Mean : σ rn 2 col StdDev : 31 mat 4000 19.36 1.899 31 mat

Intervals to be populated by 'h'
bin wisely wrt Min/Max ... intervals 01517192123253018 mat transpose :

data rn 2 col :

h intervals data hist :

h transpose 053139195117 mat

h sum 100

j 1 h rows range : frequency j h augment : stem frequency Stem : center intervals rows 2 / : 41 sys

ε upper lower - h rows / : 1.429

x f x center σ 2 / pdf.Normal length * 1 * :

bin frequency random-normal

frequency 0.5 BarSize stem x f 3 1 sys

copy Exercise 2

paste in blank sheet

experiment . . .

]]>

Exercise 3: revisit Big Foot

]]> N

N 15214334555667138109610911151213137147151016617218121913201021122252312241125112617277282299305313329331634113513367379389397409417428431344114544611479486497501851115255313548551256165785813591660561862763664765116646796856917708711372137313741275137617747887914808811182118312841285486487188689159011911092293129413958961597139889910100141011210271036104121051510681074108111091211031111311281131311416115101161111712118101198120161215122161236124161251112614127212891290130113141327133713491351213681371113891391014017141414213143414417145121461614712148101491415031514152121532154315511156815715158111591016091618162916310164116551661716717168316911170111711017281734174121751717681771217811179101801218110182131831118491850186161872188318911190319117192121935194919511196919761983199720052002 mat :

Utilities

M Min Max Scale Max Min - (M M min - M max M min - 106 -^+ / (* Min + :

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 :

Stem plot for QuickPlot unicolor data Stem stem 0 : 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 : 81 line :

size min max vx Hist Paul W. MCD 11 => Smath vx last vx rows : numbins max min - size / : hist numbins el 0 : i 1 numbins range bin_mid i el i 1 + (size * size 2 / - : j 1 vx last range i size * vx j el ≤ (i 1 + (size * < hist i el hist i el 1 + : N/A if for 21 line for bin_mid hist vx last / augment 61 line :

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 :

μ N 2 col Mean : 9.29

N rows 200

rn N 0 μ Scale :

bin 20 : size 1 bin / : min 0 : max 1 : 41 line

vy rn 2 col : H size min max vy Hist : 21 line

μ vy Mean : σ vy StdDev : 21 mat 0.4315 0.1956 21 mat

σ μ sqrt 2 / : 0.328

pts H . 10 black plot :

x fit x 2 μ * σ pdf.Normal 1 * 0.825 * :

Cumulative weighted clssifiation

H Stem x fit 2 μ * 1.5 2 μ * 0.5 - 2 2 mat pts 4 1 sys

Cumulative Weighted Classification ... is a technique to best foot the PDF from trial/error. This technique was used in Mathsoft Collaboratory to finalize the "Void ratio" project that lasted > 1 month !NO thanks Originator! No merit in this demo as we knew in advance 'N' was random-normal. Credit to Paul W. [McGill University]. Commonly, the "Big-Foot" fitting session will result to some of the familiar PDF. Often, un-listed model will fit better.

Exercise 4: two well fitted histograms

]]> 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 stem 0 : 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 : 81 line :

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

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 :

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

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

K 31.664 : α 6.774 : β 6.212 : 31 line

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

plot K α β x fr US Stem XY 31 sys :

plot Utilities

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 :

Error stem plot for QuickPlot unicolor data Error stem 0 : 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 : 81 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 :

Otherwise Pearson independent of f(x,ParameterName) x y corr n x length : xy x y * vectorize : x2 x x * vectorize : y2 y y * vectorize : n xy sum (* x sum (y sum (* - n x2 sum (* x sum (2^- (n y2 sum (* y sum (2^- (* (sqrt / 51 line :

Fit.Y ii 1 n range Fit ii el X ii el β f : for Fit 21 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 :

Void ratio PDF model x β f β 1 el x * β 2 el x * exp * :

X XY 1 col : Y XY 2 col : 12 mat data XY . 8 black plot : n XY rows : 31 line

n 200

β 1146.0242 3.1887 - 21 mat :

Histogram is fitted from Genfit collection

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

px 0.3125 :

Absolute residuals δ i 1 XY rows range Δ i el Y i el X i el β f - 0 round eval : for X Y Δ augment 21 line :

corr 1.0041

x c 10 x ≤ (x 10 ≤ (& cases :

data x β f x c * px 200 px 25 - 2 2 mat 3 1 sys

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

0 stem px 50 px 50 - 2 2 mat 3 1 sys