<content> Utilities </content>

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

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 :

collect all the counted values in sorted 'N' N histogram value table count k 1 : j k table rows range table j el value ≡ index k el j : k k 1 + : 21 line index k el 0 : if for index rows 31 line : i 1 N max range x i el i : count i el i N count : 21 line for x count augment 31 line :

Error 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 :

Central Limit Theorem [CLT] demo

'h' is very closely normaly distributed [CLT] n 500 : h 10 n Random * 20 n Random * + 30 n Random * + 40 n Random * + 50 n Random * + : N h Ceil vectorize sort : H N histogram : 41 line

In this demo, λ is a "fake scale" factor to see PDFnormal, tobe optimized  "Genfit" λ H 2 col sum : μ H rows 2 / : 21 sys

N Min N Max h Mean H Mean 41 mat 19134 74.552 36.112 41 mat

λ μ 21 mat 6336721 mat

plot H Stem λ x μ μ 3 / pdf.Normal * μ 20 μ 5 - 22 mat 31 sys :

H rows 134

repeat ▲ calculation

plot

resize QuickPlot ...

H λ x μ μ 3 / pdf.Normal * 2 1 sys

jmG: 20230923

<content> Utilities </content>

n Low High Srnd rnd n Random : rndS High Low - (rnd rnd Min - rnd Max rnd Min - 1012 -^+ / (* Low + : rndS 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 :

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 :

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

1. The Central Limit Theorem

In probability theory, the central limit theorem (CLT) establishes that,
in most situations, when independent random variables are added,
their properly normalized sum tends toward a normal distribution
(informally a "bell curve") even if the original variables themselves
are not normally distributed ...[Wikipedia]

n 200 :

bias 1 - :

Experiment: 5, 10 ... Ω 5 :

Unform ...  -1 < rnd < 1 n rnd μ n 11 line : Ω μ Random * 21 line :

idx 1 n range :

N1 n rnd Floor : N2 n rnd Ceil : N N1 N2 + bias + : 31 line

μ N Mean : σ N StdDev : 21 mat 4.12 021 mat

N N :

N Min N Mean N Max 31 mat 010325 / 831 mat

½noise => scaled noise [-1, 1] ½noise idx n 1 - 1 Srnd augment :

½noise 0.001 BarSize

If the histogram turns red,
F9 for a new attempt.
Red H "index undefined"
missing value(s) sorted 'N'

N N sort :

Number of bins = max(N) + 1 N max 8

count specific values in sorted vectorcollect counted values in user 'N' value vector count N histogram 21 sys

H N histogram :

h N Max Ceil N Min Floor - H rows / : λ n h * : 21 mat 0.89 177.78 21 mat

H 0.125 BarSize x 0.25 * 1 0.5 pdf.Normal 40 * 2 1 sys

H 0811822233343753061871581992 mat

http://www.embedded.com/design/configurable-systems/
4211153/DSP-Tricks--Generating-normally-distributed-random-data-

In the development of many signal processing algorithms where it is necessary to generate
normally distributed random data, a problem that is sometimes encountered is how actually
to generate random data samples whose distribution follows that normal (Gaussian) curve.
There’s a straightforward way to solve this problem using any software package that can
generate uniformly distributed random data, as most of them do.

As it turns out, there’s a principle in advanced probability theory, known as the
Central Limit Theorem, that says when random data from an arbitrary distribution is
summed over M samples, the probability distribution of the sum begins to approach a
normal distribution as M increases.

In other words, if we generate a set of N random samples that are uniformly distributed
between zero and one, we can begin adding other sets of N samples to the first set.
As we continue summing additional sets, the distribution of the N-element set of sums
becomes more and more normal.

2. Thermal noise generator

<content> Stem Construction Utility </content>

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 :

scale n Low High Srnd rnd n Random : rndS High Low - (rnd rnd Min - rnd Max rnd Min - 1012 -^+ / (* Low + : rndS 31 line :

N var ThermalNoise n 1 N range c n el 2 var * 10 Random 1 el ln * - sqrt 1002 π * Srnd 1 el cos * : for c 21 line :

N 128 : var 0.5 : 12 mat

ThN N var ThermalNoise :

ThN ThN eval :

idx 1 ThN rows range :

ThN Mean 0.17

ThN idx ThN augment :

ThN Stem var var - 4 1 sys

2 var * dummy Random 1 el ln * - sqrt dummy 02 π * Srnd 1 el cos * dummy 10 typical, reasonable ≡

Of the dummy length vector, we take the first value [index 1]
This value is random as well as any other otherwise indexed.

<content> Stem Construction Utility </content>

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 :

3. Logistic noise generator

L 1 : S 0.75 : size 200 : 13 mat

size L S rlogis λ 10 : xRnd 2 λ / size 1 λ Random.N * : size L S rlogis L 1 xRnd / ln vectorize S * - : size L S rlogis 41 line :

rlogis size L S rlogis :

rlogis Min rlogis Mean rlogis Max 31 mat 0.207 - 0.926 1.52 31 mat

idx 1 rlogis rows range : rlogis idx rlogis augment : 21 line

rlogis Stem rlogis 2 col Mean 2 1 sys