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0.1.1 Gauss
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Rob Tillaart
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@ -6,6 +6,21 @@ The format is based on [Keep a Changelog](http://keepachangelog.com/)
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and this project adheres to [Semantic Versioning](http://semver.org/).
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## [0.1.1] - 2023-07-07
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- improve performance => reciprokeSD = 1.0/stddev
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- update readme.md
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- add performance section (UNO / ESP32)
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- elaborated future section
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- generated a more precise lookup table (8 decimals)
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- update unit tests
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- add default parameters to **bool begin(float mean = 0, float stddev = 1)**
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- allow negative **stddev** but return false if stddev <= 0.
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- add **float getMean()** convenience function.
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- add **float getStdDev()** convenience function.
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- clean up a bit
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## [0.1.0] - 2023-07-06
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- initial version
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@ -1,7 +1,7 @@
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//
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// FILE: Gauss.cpp
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// AUTHOR: Rob Tillaart
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// VERSION: 0.1.0
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// VERSION: 0.1.1
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// PURPOSE: Library for the Gauss probability math.
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// DATE: 2023-07-06
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@ -2,7 +2,7 @@
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//
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// FILE: Gauss.h
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// AUTHOR: Rob Tillaart
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// VERSION: 0.1.0
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// VERSION: 0.1.1
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// PURPOSE: Library for the Gauss probability math.
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// DATE: 2023-07-06
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@ -10,7 +10,7 @@
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#include "Arduino.h"
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#include "MultiMap.h"
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#define GAUSS_LIB_VERSION (F("0.1.0"))
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#define GAUSS_LIB_VERSION (F("0.1.1"))
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class Gauss
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@ -20,34 +20,48 @@ public:
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{
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_mean = 0;
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_stddev = 1;
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_reciprokeSD = 1;
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}
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bool begin(float mean, float stddev)
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// stddev should be positive.
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bool begin(float mean = 0, float stddev = 1)
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{
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_mean = mean;
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_stddev = abs(stddev);
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return true;
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_stddev = stddev; // should be positive
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_reciprokeSD = 1.0 / _stddev;
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return (stddev > 0);
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}
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float getMean()
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{
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return _mean;
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}
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float getStdDev()
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{
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return _stddev;
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}
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float P_smaller(float value)
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{
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if (_stddev == 0) return NAN;
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return _P_smaller((value - _mean) / _stddev);
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// normalize(value)
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return _P_smaller((value - _mean) * _reciprokeSD);
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}
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float P_larger(float value)
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{
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return 1.0 - P_smaller(value);
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// if (_stddev == 0) return NAN;
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// optimize math division?
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// return _P_larger((value - _mean) / _stddev);
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}
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float P_between(float p, float q)
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{
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if (_stddev == 0) return NAN;
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if (p >= q) return 0;
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return P_smaller(q) - P_smaller(p);
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}
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@ -56,8 +70,8 @@ public:
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float P_equal(float value)
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{
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if (_stddev == 0) return NAN;
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float n = (value - _mean)/_stddev;
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float c = 1.0 / (_stddev * sqrt(TWO_PI));
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float n = (value - _mean) * _reciprokeSD;
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float c = _reciprokeSD * (1.0 / sqrt(TWO_PI));
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return c * exp(-0.5 * n * n);
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}
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@ -70,7 +84,7 @@ public:
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float normalize(float value)
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{
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return (value - _mean)/_stddev;
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return (value - _mean) * _reciprokeSD;
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}
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@ -85,27 +99,40 @@ private:
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float _P_smaller(float x)
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{
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// TODO improve accuracy or reduce points.
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// NORM.DIST(mean, stddev, x, true)
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float __gauss[] = {
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0.5000, 0.5398, 0.5793, 0.6179, 0.6554, 0.6915, 0.7257, 0.7580,
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0.7881, 0.8159, 0.8413, 0.8643, 0.8849, 0.9032, 0.9192, 0.9332,
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0.9452, 0.9554, 0.9641, 0.9713, 0.9772, 0.9821, 0.9861, 0.9893,
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0.9918, 0.9938, 0.9953, 0.9965, 0.9974, 0.9981, 0.9987, 1.0000
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0.50000000, 0.53982784, 0.57925971, 0.61791142,
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0.65542174, 0.69146246, 0.72574688, 0.75803635,
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0.78814460, 0.81593987, 0.84134475, 0.86433394,
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0.88493033, 0.90319952, 0.91924334, 0.93319280,
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0.94520071, 0.95543454, 0.96406968, 0.97128344,
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0.97724987, 0.98213558, 0.98609655, 0.98927589,
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0.99180246, 0.99379033, 0.99533881, 0.99653303,
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0.99744487, 0.99813419, 0.99865010, 0.99996833,
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0.99999971, 1.00000000
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};
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// 0..60000 uint16_t = 68 bytes less
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float __z[] = {
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0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,
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0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5,
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1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3,
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2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 10.0
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2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 4,0,
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5.0, 6.0
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};
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if (x < 0) return 1.0 - multiMap<float>(-x, __z, __gauss, 32);
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return multiMap<float>(x, __z, __gauss, 32);
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// a dedicated MultiMap could exploit the fact that
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// the __z[] array is largely equidistant.
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// that could remove the __z[] array (almost) completely.
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if (x < 0) return 1.0 - multiMap<float>(-x, __z, __gauss, 34);
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return multiMap<float>(x, __z, __gauss, 34);
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}
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float _mean = 0;
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float _stddev = 1;
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float _stddev = 1; // not needed as _reciprokeSD holds same info?
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float _reciprokeSD = 1;
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};
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// -- END OF FILE --
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@ -16,7 +16,7 @@ Library for the Gauss probability math.
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Gauss is an experimental Arduino library to approximate the probability that a value is
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smaller or larger than a given value.
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These under the premises of a Gaussian distribution with parameters **mean** and **stddev**
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(a.k.a. average / mu and standard deviation / sigma).
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(a.k.a. average / mu / µ and standard deviation / sigma / σ).
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If these parameters are not given, 0 and 1 are used by default (normalized Gaussian distribution).
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The values are approximated with **MultiMap()** using a 32 points interpolated lookup.
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@ -28,12 +28,14 @@ Return values are given as floats, if one needs percentages, just multiply by 10
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#### Accuracy
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The lookup table used has 32 points with 4 significant digits.
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Do not expect a higher accuracy / precision.
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The lookup table has 34 points with 8 decimals.
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This matches the precision of float data type.
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Do not expect a very high accuracy / precision as interpolation is linear.
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For many applications this accuracy is sufficient.
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(links to a table with more significant digits is welcome).
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Values of the table are calculated with ```NORM.DIST(mean, stddev, x, true)```.
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Note: 0.1.0 was 32 points 4 decimals. Need to investigate reduction of points.
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#### Applications
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@ -42,9 +44,20 @@ For many applications this accuracy is sufficient.
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- compare population data with individual
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#### Character
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| parameter | name | ALT-code | char |
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|:-----------:|:------:|:----------:|:-----:|
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| mean | mu | ALT-230 | µ |
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| stddev | sigma | ALT-229 | σ |
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- https://altcodesguru.com/greek-alt-codes.html
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#### Related
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- https://en.wikipedia.org/wiki/Normal_distribution
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- https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_probability/bs704_probability9.html
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- https://github.com/RobTillaart/Multimap
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- https://github.com/RobTillaart/Statistic (more stat links there).
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@ -59,19 +72,32 @@ For many applications this accuracy is sufficient.
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#### Base
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- **Gauss()** constructor. Uses mean = 0 and stddev = 1 by default.
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- **bool begin(float mean, float stddev)** set the mean and stddev.
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Returns true on success. If needed stddev is made positive.
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- **bool begin(float mean = 0, float stddev = 1)** set the mean and stddev.
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Returns true if stddev > 0 which should be so.
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Returns false if stddev <= 0, which could be a user choice.
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Note that if ```stddev == 0```, probabilities cannot be calculated
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as the distribution is not Gaussian.
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The default values (0,1) gives the normalized Gaussian distribution.
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**begin()** can be called at any time to change the mean or stddev.
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- **float getMean()** returns current mean.
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- **float getStddev()** returns current stddev.
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#### Probability
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Probability functions return NAN if stddev == 0.
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- **float P_smaller(float f)** returns probability **P(x < f)**.
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Multiply by 100.0 to get the value as a percentage.
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A.k.a. **CDF()** Cumulative Distribution Function.
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- **float P_larger(float f)** returns probability **P(x > f)**.
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Multiply by 100.0 to get the value as a percentage.
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As the distribution is continuous **P_larger(f) == 1 - P_smaller(f)**.
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- **float P_between(float f, float g)** returns probability **P(f < x < g)**.
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Multiply by 100.0 to get the value as a percentage.
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- **float P_equal(float f)** returns probability **P(x == f)**.
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This is the bell curve formula.
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This uses the bell curve formula.
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#### Other
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@ -82,43 +108,74 @@ E.g if mean == 50 and stddev == 14, then 71 ==> +1.5 sigma.
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- **float bellCurve(float f)** returns probability **P(x == f)**.
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## Performance
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Indicative numbers for 1000 calls, timing in micros.
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Arduino UNO, 16 MHz, IDE 1.8.19
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| function | 0.1.0 | 0.1.1 | notes |
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|:--------------|:--------:|:--------:|:--------|
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| P_smaller | 375396 | 365964 |
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| P_larger | 384368 | 375032 |
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| P_between | 265624 | 269176 |
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| normalize | 44172 | 23024 |
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| bellCurve | 255728 | 205460 |
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| approx.bell | 764028 | 719184 | see examples
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ESP32, 240 MHz, IDE 1.8.19
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| function | 0.1.0 | 0.1.1 | notes |
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|:--------------|:--------:|:--------:|:--------|
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| P_smaller | - | 4046 |
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| P_larger | - | 4043 |
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| P_between | - | 3023 |
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| normalize | - | 592 |
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| bellCurve | - | 13522 |
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| approx.bell | - | 7300 |
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## Future
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#### Must
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- documentation
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- mu + sigma character
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- unit tests
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#### Should
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- optimize performance
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- remove division by stddev
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- optimize accuracy
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- revisit lookup of MultiMap
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- (-10 .. 0) might be more accurate (significant digits)?
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- double instead of floats? (good table?)
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- make use of equidistant \_\_z\[] table
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#### Could
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- **void setMean(float f)**
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- **float getMean()**
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- **void setStddev(float f)**
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- **float getStddev()**
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- default values for **begin(0,1)**
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- add examples
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- e.g. temperature (DS18B20 or DHT22)
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- e.g. loadcell (HX711)
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- does the stddev needs to be positive,
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- what happens if negative values are allowed?
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- equality test Gauss objects
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- move code to .cpp file? (rather small lib).
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- embed MultiMap hardcoded instead of library dependency
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- **bellCurve()** => **Z()**?
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- add unit tests
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- remove **\_stddev** as **\_reciprokeSD** holds same information.
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- reverse normalization
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- G(100,25) which value has stddev 0.735?
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- **VAL(probability = 0.75)** ==> 134 whatever
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- Returns the value of the distribution for which the **CDF()** is at least probability.
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- Inverse of **P_smaller()**
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- **float P_outside(float f, float g)** returns probability **P(x < f) + P(g < x)**.
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- assuming no overlap. Use **P_outside() = 1 - P_between()**
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#### Won't (unless requested)
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- equality test Gauss objects
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- does the stddev needs to be positive? Yes.
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- what happens if negative values are allowed? P curve is reversed.
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- move code to .cpp file? (rather small lib).
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- **void setMean(float f)** can be done with begin()
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- **void setStddev(float f)** can be done with begin()
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@ -20,7 +20,7 @@ void setup(void)
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Serial.print("GAUSS_LIB_VERSION: ");
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Serial.println(GAUSS_LIB_VERSION);
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Serial.println();
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Serial.println("Timing in micros");
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Serial.println("Timing in micros (1000 calls)");
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Serial.println();
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test_1();
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@ -0,0 +1,16 @@
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Arduino UNO
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IDE 1.8.19
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Gauss_performance.ino
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GAUSS_LIB_VERSION: 0.1.1
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Timing in micros (1000 calls)
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P_smaller: 365964
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P_larger: 375032
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P_between: 269176
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normalize: 23024
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bellCurve: 205460
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approx.bell: 719184
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done...
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@ -17,9 +17,9 @@ void setup(void)
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Serial.println();
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test_1();
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test_2();
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test_3();
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test_4();
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// test_2();
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// test_3();
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// test_4();
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Serial.println("\ndone...");
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}
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@ -6,6 +6,8 @@ Gauss KEYWORD1
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# Methods and Functions (KEYWORD2)
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begin KEYWORD2
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getMean KEYWORD2
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getStdDev KEYWORD2
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P_smaller KEYWORD2
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P_larger KEYWORD2
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@ -23,7 +23,7 @@
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"version": "^0.1.7"
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}
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],
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"version": "0.1.0",
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"version": "0.1.1",
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"license": "MIT",
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"frameworks": "arduino",
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"platforms": "*",
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@ -1,9 +1,9 @@
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name=Gauss
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version=0.1.0
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version=0.1.1
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author=Rob Tillaart <rob.tillaart@gmail.com>
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maintainer=Rob Tillaart <rob.tillaart@gmail.com>
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sentence=Library for the Gauss probability math.
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paragraph=
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paragraph=normal distribution.
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category=Data Processing
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url=https://github.com/RobTillaart/Gauss
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architectures=*
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@ -41,9 +41,13 @@ unittest(test_constructor)
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{
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Gauss G;
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G.begin(0, 1);
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assertEqualFloat(0.0, G.getMean(), 0.0001);
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assertEqualFloat(1.0, G.getStdDev(), 0.0001);
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assertEqualFloat(0.5, G.P_smaller(0), 0.0001);
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assertEqualFloat(0.5, G.P_smaller(0), 0.001);
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G.begin(10, 3);
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assertEqualFloat(10.0, G.getMean(), 0.0001);
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assertEqualFloat(3.0, G.getStdDev(), 0.0001);
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}
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@ -53,13 +57,17 @@ unittest(test_P_smaller)
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G.begin(0, 1);
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assertEqualFloat(0.0013, G.P_smaller(-3.0), 0.001);
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assertEqualFloat(0.0228, G.P_smaller(-2.0), 0.001);
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assertEqualFloat(0.1587, G.P_smaller(-1.0), 0.001);
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assertEqualFloat(0.5000, G.P_smaller(0.0), 0.001);
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assertEqualFloat(0.8413, G.P_smaller(1.0), 0.001);
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assertEqualFloat(0.9772, G.P_smaller(2.0), 0.001);
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assertEqualFloat(0.9987, G.P_smaller(3.0), 0.001);
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assertEqualFloat(0.0000, G.P_smaller(-6.0), 0.0001);
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assertEqualFloat(0.0001, G.P_smaller(-4.0), 0.0001);
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assertEqualFloat(0.0013, G.P_smaller(-3.0), 0.0001);
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assertEqualFloat(0.0228, G.P_smaller(-2.0), 0.0001);
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assertEqualFloat(0.1587, G.P_smaller(-1.0), 0.0001);
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assertEqualFloat(0.5000, G.P_smaller(0.0), 0.0001);
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assertEqualFloat(0.8413, G.P_smaller(1.0), 0.0001);
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assertEqualFloat(0.9772, G.P_smaller(2.0), 0.0001);
|
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assertEqualFloat(0.9987, G.P_smaller(3.0), 0.0001);
|
||||
assertEqualFloat(0.9999, G.P_smaller(4.0), 0.0001);
|
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assertEqualFloat(1.0000, G.P_smaller(6.0), 0.0001);
|
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}
|
||||
|
||||
|
||||
@ -69,13 +77,13 @@ unittest(test_P_larger)
|
||||
|
||||
G.begin(0, 1);
|
||||
|
||||
assertEqualFloat(0.9987, G.P_larger(-3.0), 0.001);
|
||||
assertEqualFloat(0.9772, G.P_larger(-2.0), 0.001);
|
||||
assertEqualFloat(0.8413, G.P_larger(-1.0), 0.001);
|
||||
assertEqualFloat(0.5000, G.P_larger(0.0), 0.001);
|
||||
assertEqualFloat(0.1587, G.P_larger(1.0), 0.001);
|
||||
assertEqualFloat(0.0228, G.P_larger(2.0), 0.001);
|
||||
assertEqualFloat(0.0013, G.P_larger(3.0), 0.001);
|
||||
assertEqualFloat(0.9987, G.P_larger(-3.0), 0.0001);
|
||||
assertEqualFloat(0.9772, G.P_larger(-2.0), 0.0001);
|
||||
assertEqualFloat(0.8413, G.P_larger(-1.0), 0.0001);
|
||||
assertEqualFloat(0.5000, G.P_larger(0.0), 0.0001);
|
||||
assertEqualFloat(0.1587, G.P_larger(1.0), 0.0001);
|
||||
assertEqualFloat(0.0228, G.P_larger(2.0), 0.0001);
|
||||
assertEqualFloat(0.0013, G.P_larger(3.0), 0.0001);
|
||||
}
|
||||
|
||||
|
||||
@ -85,13 +93,13 @@ unittest(test_P_between)
|
||||
|
||||
G.begin(0, 1);
|
||||
|
||||
assertEqualFloat(0.4987, G.P_between(-3.0, 0.0), 0.001);
|
||||
assertEqualFloat(0.4772, G.P_between(-2.0, 0.0), 0.001);
|
||||
assertEqualFloat(0.3413, G.P_between(-1.0, 0.0), 0.001);
|
||||
assertEqualFloat(0.0000, G.P_between(0.0, 0.0), 0.001);
|
||||
assertEqualFloat(0.3413, G.P_between(0.0, 1.0), 0.001);
|
||||
assertEqualFloat(0.4772, G.P_between(0.0, 2.0), 0.001);
|
||||
assertEqualFloat(0.4987, G.P_between(0.0, 3.0), 0.001);
|
||||
assertEqualFloat(0.4987, G.P_between(-3.0, 0.0), 0.0001);
|
||||
assertEqualFloat(0.4772, G.P_between(-2.0, 0.0), 0.0001);
|
||||
assertEqualFloat(0.3413, G.P_between(-1.0, 0.0), 0.0001);
|
||||
assertEqualFloat(0.0000, G.P_between(0.0, 0.0), 0.0001);
|
||||
assertEqualFloat(0.3413, G.P_between(0.0, 1.0), 0.0001);
|
||||
assertEqualFloat(0.4772, G.P_between(0.0, 2.0), 0.0001);
|
||||
assertEqualFloat(0.4987, G.P_between(0.0, 3.0), 0.0001);
|
||||
}
|
||||
|
||||
|
||||
@ -101,13 +109,29 @@ unittest(test_P_equal)
|
||||
|
||||
G.begin(0, 1);
|
||||
|
||||
assertEqualFloat(0.004432, G.P_equal(-3.0), 0.001);
|
||||
assertEqualFloat(0.053991, G.P_equal(-2.0), 0.001);
|
||||
assertEqualFloat(0.241971, G.P_equal(-1.0), 0.001);
|
||||
assertEqualFloat(0.398942, G.P_equal(0.0), 0.001);
|
||||
assertEqualFloat(0.241971, G.P_equal(1.0), 0.001);
|
||||
assertEqualFloat(0.053991, G.P_equal(2.0), 0.001);
|
||||
assertEqualFloat(0.004432, G.P_equal(3.0), 0.001);
|
||||
assertEqualFloat(0.004432, G.P_equal(-3.0), 0.0001);
|
||||
assertEqualFloat(0.053991, G.P_equal(-2.0), 0.0001);
|
||||
assertEqualFloat(0.241971, G.P_equal(-1.0), 0.0001);
|
||||
assertEqualFloat(0.398942, G.P_equal(0.0), 0.0001);
|
||||
assertEqualFloat(0.241971, G.P_equal(1.0), 0.0001);
|
||||
assertEqualFloat(0.053991, G.P_equal(2.0), 0.0001);
|
||||
assertEqualFloat(0.004432, G.P_equal(3.0), 0.0001);
|
||||
}
|
||||
|
||||
|
||||
unittest(test_normailze)
|
||||
{
|
||||
Gauss G;
|
||||
|
||||
G.begin(100, 25);
|
||||
|
||||
assertEqualFloat(-3.0, G.normalize(25), 0.0001);
|
||||
assertEqualFloat(-2.0, G.normalize(50), 0.0001);
|
||||
assertEqualFloat(-1.0, G.normalize(75), 0.0001);
|
||||
assertEqualFloat(0.0, G.normalize(100), 0.0001);
|
||||
assertEqualFloat(1.0, G.normalize(125), 0.0001);
|
||||
assertEqualFloat(2.0, G.normalize(150), 0.0001);
|
||||
assertEqualFloat(3.0, G.normalize(175), 0.0001);
|
||||
}
|
||||
|
||||
|
||||
@ -115,3 +139,4 @@ unittest_main()
|
||||
|
||||
|
||||
// -- END OF FILE --
|
||||
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user