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)) . '
';
    

Statistical 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, 0.62609901275838, -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, 0.53340970683585, -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']) . '
';

Correlation, 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)) . '
';

Distributions:

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)) . '
';

Chi-square test or Contingency tables (A/B testing):

Calculate the probability that number of cylinders distribution in automatic and manual transmission cars is same

0.012646605046107

    $table['Automatic'] = array('4 Cylinders' => 3, '6 Cylinders' => 4, '8 Cylinders' => 12);
    $table['Manual']    = array('4 Cylinders' => 8, '6 Cylinders' => 3, '8 Cylinders' => 2);

    $results     = $kashi->chiTest($table);
    $probability = $kashi->chiDist($result['chi'], $result['df']);
    echo 'Chi-square test probability: ' . $probability . '
';

Diversity index:

Shannon index for number of forward gears	1.0130227035447
Simpson index for number of cylinders	0.357421875

    $gear = array('3' => 15, '4' => 12, '5' => 5);
    $cyl  = array('4' => 11, '6' => 7, '8' => 14);

    echo 'Shannon index for gear: ' . $kashi->diversity($gear) . '
';
    echo 'Simpson index for cyl: ' . $kashi->diversity($cyl, 'simpson') . '
';

Analysis of Variance (ANOVA):

Analysis of variance procedure (ANOVA) Typical ANOVA example output (mpg ~ cyl): ANOVA table Variate: mpg Source of variation d.f. s.s. m.s. v.r. F pr. cyl 2 824.78 412.39 39.70 <.001 Residual 29 301.26 10.39 Total 31 1126.05 Tables of means Grand mean 20.09 cyl 4 6 8 26.66 19.74 15.10 rep. 11 7 14 Standard errors of means e.s.e. 1.218 min.rep 0.861 max.rep Standard errors of differences of means s.e.d. 1.723X min.rep 1.218X max.rep Least significant differences of means (5% level) l.s.d. 3.524X min.rep 2.492X max.rep Stratum standard errors and coefficients of variation d.f. s.e. cv% 29 3.223 16.0

Array ( [TDF] => 2 [EDF] => 29 [TotDF] => 31 [SST] => 824.7845900974 [SSE] => 301.2625974026 [SSTot] => 1126.0471875 [MST] => 412.3922950487 [MSE] => 10.388365427676 [VRT] => 39.697515255869 [F] => 4.9789191744003E-9 [Mean] => 20.090625 [Means] => Array ( [4] => 26.6636364 [6] => 19.7428571 [8] => 15.1000000 ) [Reps] => Array ( [4] => 11 [6] => 7 [8] => 14 ) [SE] => Array ( [min] => 1.2182168131961 [max] => 0.86140936956643 ) [SED] => Array ( [min] => 1.7228187391329 [max] => 1.2182168131961 ) [LSD] => Array ( [min] => 3.5235599562701 [max] => 2.491533138996 ) [CV] => 16.042799717154 )

require('kashi_anova.php');

// $obj = new KashiANOVA($dbname, $dbuser, $dbpass, $dbhost);
$obj = new KashiANOVA('test', 'root', '', 'localhost');

$str = file_get_contents('anova_data.txt');
$obj->loadString($str); 

// mpg ~ cyl
$result = $obj->anova('cyl', 'mpg');
print_r($result);

Cluster Analysis:

K-Means Clustering

Array ( [Chrysler Imperial] => 0 [Pontiac Firebird] => 0 [AMC Javelin] => 0 [Camaro Z28] => 0 [Lincoln Continental] => 0 [Cadillac Fleetwood] => 0 [Merc 450SLC] => 0 [Merc 450SL] => 0 [Merc 450SE] => 0 [Dodge Challenger] => 0 [Duster 360] => 0 [Ford Pantera L] => 0 [Hornet Sportabout] => 0 [Maserati Bora] => 0 [Toyota Corona] => 1 [Porsche 914-2] => 1 [Lotus Europa] => 1 [Ferrari Dino] => 1 [Fiat X1-9] => 1 [Mazda RX4] => 1 [Toyota Corolla] => 1 [Honda Civic] => 1 [Fiat 128] => 1 [Mazda RX4 Wag] => 1 [Merc 280C] => 1 [Merc 280] => 1 [Merc 230] => 1 [Merc 240D] => 1 [Valiant] => 1 [Hornet 4 Drive] => 1 [Datsun 710] => 1 [Volvo 142E] => 1 )

Hierarchical Clustering

32 15 14 0.034867528963888 33 12 11 0.046511652279906 34 1 0 0.048063902847295 35 10 9 0.048146270217687 36 33 13 0.048374485470338 37 24 4 0.06456633193609 38 19 17 0.067898627038737 39 22 21 0.092305891561629 40 39 37 0.11301195978463 41 32 16 0.11529825256692 42 31 2 0.1155541020107 43 5 3 0.11717892926293 44 40 36 0.11995870908923 45 23 6 0.12445889917409 46 38 25 0.12703468709516 47 46 42 0.19819935352147 48 8 7 0.20845446781686 49 48 20 0.22553907135502 50 45 44 0.23476357897562 51 47 18 0.24068916220486 52 50 41 0.25528946686225 53 34 29 0.26595333894602 54 51 27 0.27674027068183 55 54 26 0.28056404941297 56 49 43 0.28521660028422 57 56 35 0.30779338554525 58 30 28 0.35715746216011 59 55 53 0.37801491177356 60 59 57 0.42234403985919 61 60 52 0.52592878486916 62 61 58 0.49319668374021

require('kashi_cluster.php');
$obj = new KashiCluster();
$obj->dataLoad($data);

$result = $obj->kMean(2);
print_r($result);

// Heretical tree output has no header, and consists of four columns. For each row, the first column is the 
// identifier of the node, the second and third columns are child nodes identifier, and the fourth column used 
// to determine the height of the node when rendering a tree.
$tree = $obj->hClust();
echo "
$tree
";

Time Series Analysis:

Moving Average

2.894, 3.062, 3.201, 3.375, 3.362, 3.362, 3.358, 3.458, 3.566, 3.692, 4.054, 4.4508, 4.7058, 4.3998, 3.9668, 3.2838, 2.692, 2.327, 2.574, 3.019, 3.421, 3.315, 3.039, 2.6546, 2.5206, 2.3056, 2.6326, 2.7606

    echo 'Moving Average for x: ' . implode(', ', $kashi->movingAvg($x, 5)) . '
';

Matrix Functions:

     | 1   2 |         | 5   7 | 
 A = | 3   4 |  ,  B = | 6   8 |

A + B	| 6 9 | | 9 12 |
B - A	| 4 5 | | 3 4 |
A * 2	| 2 4 | | 6 8 |
A * B	| 17 23 | | 39 53 |
Transpose of B, t(B)	| 5 6 | | 7 8 |
Determinant of A, |A|	-2
Cofactor of A	| 4 -3 | | -2 1 |
Adjoint of A	| 4 -2 | | -3 1 |
Inverse of A	| -2 1 | | 1.5 -0.5 |

    $A[1][1] = 1;
    $A[1][2] = 2;
    $A[2][1] = 3;
    $A[2][2] = 4;

    $B[1][1] = 5;
    $B[1][2] = 7;
    $B[2][1] = 6;
    $B[2][2] = 8;

    echo 'A + B = ', print_r($kashi->mAddition($A, $B), true), '
';

    echo 'B - A = ', print_r($kashi->mSubtraction($B, $A), true), '
';

    echo 'A * 2 = ', print_r($kashi->mMultiplication($A, 2), true), '
';

    echo 'A * B = ', print_r($kashi->mMultiplication($A, $B), true), '
';

    echo 'Transpose of B, t(B) = ', print_r($kashi->mTranspose($B), true), '
';

    echo 'Determinat of A, |A| = ', print_r($kashi->mDeterminant($A), true), '
';

    echo 'Cofactor of A = ', print_r($kashi->mCofactor($A), true), '
';

    echo 'Adjoint of A = ', print_r($kashi->mAdjoint($A), true), '
';

    echo 'Inverse of A = ', print_r($kashi->mInverse($A), true), '
';

Solve System of Linear Equations:

System of Linear Equations 2*X2 + X3 = 9 X1 + 4*X2 - X3 = 23 -X1 + X2 + X3 = 2

Array ( [1] => 2 [2] => 5 [3] => -1 )

	$X[1] = array(1=>0, 2, 1);
	$X[2] = array(1=>1, 4, -1);
	$X[3] = array(1=>-1, 1, 1);

	$Y = array(1=>9, 23, 2);
	
	print_r($kashi->solve($X, $Y));

To-do list:

Example Data Description (Motor Trend Car Road Tests):

You can download example data file from here.