In French, the law of cosines is named Théorème d'Al-Kashi (Theorem of Al-Kashi), as al-Kashi was the first to provide an explicit statement of the law of cosines in a form suitable for triangulation. In one of his numerical approximations of pi, he correctly computed 2pi to 16 decimal places of accuracy. This was far more accurate than the estimates earlier given.
We aim in Al-Kashi project to provide a rich PHP package full of statistical functions useful for online business intelligent and data mining, possible applications may include an online log file analysis, Ad's and Campaign statistics, or survey/voting results on-fly analysis. It is published under GPL license; you can download it from PHPClasses.org website, and you can check the change log here.
Khaled Al-Sham'aa
Would you like to know more about statistical concepts and procedures implemented in this project? Please download this free electronic book assembled from Wikipedia articles to get detailed background information.
لمزيد من المعلومات عن هذا المشروع باللغة العربية إحيلكم إلى هذه التدوينات
Example Data
The data was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973-74 models). You can download example data file from here.
- Summary Statistics
- Statistical Graphics
- Correlation, Regression, and t-Test
- Distributions
- Chi-square test or Contingency tables (A/B testing)
- Diversity index
- Analysis of Variance (ANOVA)
- Cluster Analysis
- Time Series Analysis
- Matrix Functions
- Solve System of Linear Equations
- Example Data Description
$sep = "\t"; $nl = "\n";
$content = file_get_contents('data.txt');
$records = explode($nl, $content);
$header = explode($sep, trim(array_shift($records)));
$data = array_fill_keys($header, array());
foreach ($records as $id=>$record) {
$record = trim($record);
if ($record == '') continue;
$fields = explode($sep, $record);
$titles = $header;
foreach ($fields as $field) {
$title = array_shift($titles);
$data[$title][] = $field;
}
}
$x = $data['wt'];
$y = $data['mpg'];
require('kashi.php');
$kashi = new Kashi();
Summary Statistics:
 Mean (x) | 3.21725 |
| Mean (x, "geometric") | 3.0701885671208 |
| Mean (x, "harmonic") | 2.9182632148104 |
| Median (x) | 3.325 |
| Mode (x) | Array ( [0] => 3.44 ) |
| Variance (x) | 0.95737896774194 |
| SD (x) | 0.9784574429897 |
| %CV (x) | 30.412850819479 |
| Skewness (x) | 0.46591610679299 |
| Is it significant (i.e. test it against 0)? | bool(false) |
| Kurtosis (x) | 0.41659466963493 |
| Is it significant (i.e. test it against 0)? | bool(false) |
| Rank (x) | 9, 12, 7, 16, 18, 21, 23, 15, 13, 18, 18, 29, 25, 26, 30, 32, 31, 6, 2, 3, 8, 22, 17, 27, 28, 4, 5, 1, 14, 10, 23, 11 |
// $x is an array of values
echo 'Arithmetic Mean: ' . $kashi->mean($x) . '
';
echo 'Aeometric Mean: ' . $kashi->mean($x, "geometric") . '
';
echo 'Harmonic Mean: ' . $kashi->mean($x, "harmonic") . '
';
echo 'Mode: ' . print_r($kashi->mode($x)) . '
';
echo 'Median: ' . $kashi->median($x) . '
';
echo 'Variance: ' . $kashi->variance($x) . '
';
echo 'SD: ' . $kashi->sd($x) . '
';
echo '%CV: ' . $kashi->cv($x) . '
';
echo 'Skewness: ' . $kashi->skew($x) . '
';
echo 'Is it significant (i.e. test it against 0)? ';
var_dump($kashi->isSkew($x));
echo 'Kurtosis: ' . $kashi->kurt($x) . '
';
echo 'Is it significant (i.e. test it against 0)? ';
var_dump($kashi->isKurt($x));
echo 'Rank (x): ';
echo implode(', ', $kashi->rank($x)) . '
';
TopStatistical Graphics:
 Boxplot | Array
(
[min] => 1.513
[q1] => 2.62
[median] => 3.325
[q3] => 3.73
[max] => 5.282
[outliers] => Array
(
[0] => 5.345
[1] => 5.424
)
)
|
 Histogram | Array
(
[1.513-2.002] => 4
[2.002-2.491] => 4
[2.491-2.98] => 4
[2.98-3.469] => 9
[3.469-3.957] => 7
[3.957-4.446] => 1
[4.446-4.935] => 0
[4.935-5.424] => 3
)
|
 Normal Q-Q Plot | x = -0.62609901275838, -0.36012989155586, -0.83051087731871, -0.039176085543034, 0.11776987461046, 0.36012989155586, 0.53340970683585, -0.11776987461046, -0.27769043950814, 0.19709908415753, 0.27769043950814, 1.2298587580185, 0.72451438304624, 0.83051087731871, 1.417797139161, 2.1538746917937, 1.6759397215193, -0.94678175657479, -1.6759397215193, -1.417797139161, -0.72451438304624, 0.44509652516901, 0.039176085543034, 0.94678175657479, 1.0775155681381, -1.2298587580185, -1.0775155681381, -2.1538746917937, -0.19709908415753, -0.53340970683585, 0.62609901275838, -0.44509652516901 y = 2.62, 2.875, 2.32, 3.215, 3.44, 3.46, 3.57, 3.19, 3.15, 3.44, 3.44, 4.07, 3.73, 3.78, 5.25, 5.424, 5.345, 2.2, 1.615, 1.835, 2.465, 3.52, 3.435, 3.84, 3.845, 1.935, 2.14, 1.513, 3.17, 2.77, 3.57, 2.78 |
 Ternary Plot | x = 0.729, 0.722, 0.734, 0.706, 0.695, 0.675, 0.659, 0.723, 0.701, 0.692, 0.679, 0.663, 0.676, 0.654, 0.577, 0.574, 0.625, 0.779, 0.785, 0.788, 0.716, 0.667, 0.664, 0.645, 0.691, 0.763, 0.766, 0.796, 0.689, 0.723, 0.672, 0.718 y = 0.356, 0.36, 0.369, 0.382, 0.376, 0.419, 0.407, 0.364, 0.406, 0.387, 0.408, 0.398, 0.395, 0.422, 0.463, 0.459, 0.403, 0.312, 0.317, 0.31, 0.394, 0.407, 0.417, 0.41, 0.368, 0.34, 0.323, 0.3, 0.375, 0.354, 0.381, 0.377 |
echo 'Boxplot:
';
print_r($kashi->boxplot($x));
echo '
';
echo 'Histogram:
';
print_r($kashi->hist($x, 8));
echo '
';
echo 'Normal Q-Q Plot:
';
$qq = $kashi->qqnorm($x);
echo 'x = ' . implode(', ', $qq['x']) . '
';
echo 'y = ' . implode(', ', $qq['y']) . '
';
echo 'Ternary Plot:
';
$xy = $kashi->ternary($data['wt'], $data['mpg'], $data['qsec']);
echo 'x = ' . implode(', ', $xy['x']) . '
';
echo 'y = ' . implode(', ', $xy['y']) . '
';
TopCorrelation, Regression, and t-Test:
| Covariance (x, y) | -5.1166846774194 |
| Correlation (x, y) | -0.86765937651723 |
| Significant of Correlation | 1.2939593840855E-10 |
| Path Analysis | Array
(
[1] => -0.70763801614376
[2] => -0.20274707094052
[3] => 0.15145821845688
)
|
| Regression (y = a + b*x) | Array
(
[intercept] => 37.285126167342
[slope] => -5.3444715727227
[r-square] => 0.75283279365826
[adj-r-square] => 0.74459388678021
[intercept-se] => 1.8776273372559
[intercept-2.5%] => 33.450499570026
[intercept-97.5%] => 41.119752764658
[slope-se] => 0.55910104509932
[slope-2.5%] => -6.486308238383
[slope-97.5%] => -4.2026349070623
[F-statistic] => 91.375325003762
[p-value] => 1.2939604943085E-10
)
|
| Multiple Regression (y = a + b1*x1 + b2*x2) | Array
(
[intercept] => 37.227270116447
[b1] => -3.8778307424046
[b2] => -0.031772946982161
[r-square] => 0.82678545188279
[adj-r-square] => 0.81483962097816
[intercept-se] => 0
[intercept-2.5%] => 37.227270116447
[intercept-97.5%] => 37.227270116447
[b1-se] => 0
[b1-2.5%] => -3.8778307424046
[b1-97.5%] => -3.8778307424046
[b2-se] => 0
[b2-2.5%] => -0.031772946982161
[b2-97.5%] => -0.031772946982161
[F-statistic] => 69.211213391777
[p-value] => 9.1090543852236E-12
)
|
| t-Test unpaired | -15.632569384303 |
| Test of null hypothesis that mean of x = mean of y (assumed equal variances) | 2.2204460492503E-16 |
| Test of null hypothesis that mean of x = mean of y (assumed unequal variances) | 0 |
| t-Test paired | -13.847209446072 |
| Test of null hypothesis that mean of x-y = 0 Probability is | 8.1046280797636E-15 |
echo 'Covariance: ' . $kashi->cov($x, $y) . '
';
echo 'Correlation: ' . $kashi->cor($x, $y) . '
';
$r = $kashi->cor($x, $y);
$n = count($x);
echo 'Significant of Correlation: ' . $kashi->corTest($r, $n) . '
';
echo 'Path Analysis: ' . print_r($kashi->path($y, array(1=>$x, $data['hp'], $data['qsec'])), true) . '
';
echo 'Regression: ' . print_r($kashi->lm($y, $x), true) . '
';
echo 'Multiple Regression: ' . print_r($kashi->lm($data['mpg'], $data['wt'], $data['hp'])), true) . '
';
echo 't-Test unpaired: ' . $kashi->tTest($x, $y, false) . '
';
echo 'Test (assumed equal variances): ' . $kashi->tDist($kashi->tTest($x, $y, false), $kashi->tTestDf($x, $y, true, false)) . '
';
echo 'Test (assumed unequal variances): ' . $kashi->tDist($kashi->tTest($x, $y, false), $kashi->tTestDf($x, $y, false, false)) . '
';
echo 't-Test paired: ' . $kashi->tTest($x, $y, true) . '
';
echo 'Test: ' . $kashi->tDist($kashi->tTest($x, $y, true), $kashi->tTestDf($x, $y, false, true)) . '
';
TopDistributions:
| Normal distribution (x=0.5, mean=0, sd=1) | 0.3520653267643 |
| Probability for the Student t-distribution (t=3, n=10) one-tailed | 0.01334365502257 |
| Probability for the Student t-distribution (t=3, n=10) two-tailed | 0.0066718275112848 |
| Probability for F distribution (f=2, df1=12, df2=15) | 0.10268840717083 |
| Inverse of the standard normal cumulative distribution, with a probability of (p=0.95) | 1.6448536251337 |
| t-value of the Student's t-distribution for the probability $p and $n degrees of freedom (p=0.05, n=29) | 2.0452296438589 |
| Standardize (x) (mean=0 & variance=1) | -0.61039956748153, -0.34978526910097, -0.91700462439985, -0.002299537926887, 0.22765425476185, 0.24809459188973, 0.36051644609311, -0.027849959336746, -0.068730633592521, 0.22765425476185, 0.22765425476185, 0.8715248742903, 0.52403914311621, 0.57513998593593, 2.0775047648356, 2.2553356978483, 2.1745963661931, -1.0396466471672, -1.6375265081579, -1.4126827997511, -0.76881218022266, 0.3094156032734, 0.22254417047987, 0.63646099731959, 0.64157108160156, -1.3104811141117, -1.1009676585508, -1.7417722275101, -0.048290296464633, -0.45709703902238, 0.36051644609311, -0.44687687045844 |
echo 'Normal distribution (x=0.5, mean=0, sd=1): ' . $kashi->norm(0.5, 0, 1) . '
';
echo 'Probability for the Student t-distribution (t=3, n=10) one-tailed: ';
echo $kashi->tDist(3, 10, 1) . '
';
echo 'Probability for the Student t-distribution (t=3, n=10) two-tailed: ';
echo $kashi->tDist(3, 10, 2) . '
';
echo 'F probability distribution (f=2, df1=12, df2=15): ' . $kashi->fDist(2, 12, 15) . '
';
echo 'Inverse of the standard normal cumulative distribution (p=0.95): ';
echo $kashi->inverseNormCDF(0.95) . '
';
echo 't-value of the Student\'s t-distribution (p=0.05, n=29): ';
echo $kashi->inverseTCDF(0.05, 29) . '
';
echo 'Standardize (x) (i.e. mean=0 & variance=1): ';
echo implode(', ', $kashi->standardize($x)) . '
';
TopChi-square test or Contingency tables (A/B testing):
|
Calculate the probability that number of cylinders distribution in automatic and manual transmission cars is same |