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<title>Examples with matrix class</title>
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<?php
include_once("/Users/pabsta/Desktop/matrix/matrix.php");
?>
<ol>
<li>The simplest example is to solve a system of linear equations : <br>
<table>
<tr><td>x_1</td><td>+</td><td>3 x_2</td><td>+</td><td>5 x_3</td><td>=</td><td>1</td></tr>
<tr><td>3x_1</td><td>+</td><td>5 x_2</td><td>+</td><td>11 x_3</td><td>=</td><td>2</td></tr>
<tr><td>5x_1</td><td>+</td><td>11 x_2</td><td>+</td><td>9 x_3</td><td>=</td><td>3</td></tr>
</table><br>
Which translates into the following matricial system : <br>
<table>
<tr><td>
<table>
<tr align="center"><td>1</td><td>3</td><td>5</td><td></tr>
<tr align="center"><td>3</td><td>5</td><td>11</td><td></tr>
<tr align="center"><td>5</td><td>11</td><td>9</td><td></tr>
</table>
</td>
<td>
<table><tr><td>x_1</td></tr><tr><td>x_2</td></tr><tr><td>x_3</td></tr></table>
</td>
<td>=</td>
<td><table><tr><td>1</td></tr><tr><td>2</td></tr><tr><td>3</td></tr></table></td>
</tr>
</table>
Or :
<table>
<tr><td>A</td><td>x</td><td>=</td><td>b</td></tr>
</table><br>
The solution is obviously to inverse the A matrix :
<table>
<tr><td>x</td><td>=</td><td>A^(-1)b</td></tr>
</table><br>
We can do this using this class :<br>
<code>
$A = new matrix(array(array(1,3,5), array(3,5,11), array(5,11,9)));<br>
$b = new matrix(array(array(1), array(2), array(3)));<br>
$A_inv = $A->inv();<br>
$sol = $A_inv->times($b);<br>
$sol->print_matrix();<br>
</code><br>
Which yields the following solution :
<?php
$A = new matrix(array(array(1,3,5), array(3,5,11), array(5,11,9)));
$b = new matrix(array(array(1), array(2), array(3)));
$A_inv = $A->inv();
$sol = $A_inv->times($b);
$sol->print_matrix();
?><br>
<li>We can also perform a regression : <br>
Let X be the following matrix of data : <br>
<table>
<tr align="center"><td>1</td><td>0.21</td><td>10</td></tr>
<tr align="center"><td>1</td><td>0.35</td><td>11</td></tr>
<tr align="center"><td>1</td><td>0.45</td><td>12</td></tr>
<tr align="center"><td>1</td><td>0.52</td><td>13</td></tr>
<tr align="center"><td>1</td><td>0.63</td><td>14</td></tr>
<tr align="center"><td>1</td><td>0.79</td><td>15</td></tr>
</table><br>
(Note that the matrix X is almost colinear)<br>
<table>
<tr align="center"><td>103.20</td></tr>
<tr align="center"><td>113.30</td></tr>
<tr align="center"><td>123.40</td></tr>
<tr align="center"><td>133.50</td></tr>
<tr align="center"><td>143.60</td></tr>
<tr align="center"><td>153.70</td></tr>
</table><br>
The model is then y = b_1 * 1 + b_2 * x_2 + b_3*x_3 + e, or y = Xb + e and the minimised norm of e is given by : b = (X*X)^(-1)*X'y. This can be done with the following code : <br>
<code>
$arr = array(array(1,0.21, 10), array(1,0.35, 11), array(1,0.45, 12), array(1,0.52, 13), array(1,0.63, 14), array(1,0.79, 15));<br>
$X = new matrix($arr);<br>
$arr = array(array(103.20), array(113.30), array(123.40), array(133.50), array(143.60), array(153.70));<br>
$y = new matrix($arr);<br>
$X_p = $X->prime();<br>
$XX = $X_p->times($X);<br>
$XX_inv = $XX->inv();<br>
$X_py = $X_p->times($y);<br>
$b = $XX_inv->times($X_py); <br>
$b->print_matrix();<br>
</code>
<?php
$arr = array(array(1,0.21, 10), array(1,0.35, 11), array(1,0.45, 12), array(1,0.52, 13), array(1,0.63, 14), array(1,0.79, 15));
$X = new matrix($arr);
$arr = array(array(103.20), array(113.30), array(123.40), array(133.50), array(143.60), array(153.70));
$y = new matrix($arr);
$X_p = $X->prime();
$XX = $X_p->times($X);
$XX_inv = $XX->inv();
$X_py = $X_p->times($y);
$b = $XX_inv->times($X_py);
$b->print_matrix();
?><br>
The predicted values of y and e, the variance of the error terms and the covariance matrix can then be found with : <br>
<code>
$y_hat = $X->times($b);<br>
$e_hat = $y->minus($y_hat);<br>
$e_hat_p = $e_hat->prime();<br>
$sigma2 = $e_hat_p->times($e_hat);<br>
$sigma2 = $sigma2->s_times(1/($X->get_num_rows() - $X->get_num_columns()));<br>
$cov = $XX_inv->s_times($sigma2->get_value(1,1));<br>
$cov->print_matrix();<br>
</code><br>
<?php
$y_hat = $X->times($b);
$e_hat = $y->minus($y_hat);
$e_hat_p = $e_hat->prime();
$sigma2 = $e_hat_p->times($e_hat);
$sigma2 = $sigma2->s_times(1/($X->get_num_rows() - $X->get_num_columns()));
$cov = $XX_inv->s_times($sigma2->get_value(1,1));
$cov->print_matrix();
?><br>
</li>
<li>We can also perform some fixed point algorithms. Let X be the following Markov transition matrix :<br>
<table>
<tr align="center"><td>0.6</td><td>0.4</td><td>0.0</td></tr>
<tr align="center"><td>0.2</td><td>0.1</td><td>0.7</td></tr>
<tr align="center"><td>0.1</td><td>0.3</td><td>0.6</td></tr>
</table><br>
One can then find the limiting probability distribution by iterating until convergence : <br>
<code>
$arr = array(array(0.6, 0.4, 0), array(0.2, 0.1, 0.7), array(0.1, 0.3, 0.6));<br>
$X = new matrix($arr);<br>
$power = $X;<br>
$tolerance = 0.000001;<br>
$check = 10;<br>
$iter = 0;<br><br>
while($check > $tolerance)<br>
{<br>
$last_max = $max;<br>
$power = $power->times($X);<br><br>
$max = $power->mat_max();<br>
$max = $max->prime();<br>
$max = $max->mat_max();<br>
$max = $max->get_value(1,1);<br>
$check = abs($max - $last_max);<br><br>
$iter++;<br><br>
}<br>
$power->print_matrix();
echo "Number of iterations : $iter";<br>
</code><br><br>
<?php
$arr = array(array(0.6, 0.4, 0), array(0.2, 0.1, 0.7), array(0.1, 0.3, 0.6));
$X = new matrix($arr);
$power = $X;
$tolerance = 0.000001;
$check = 10;
$iter = 0;
while($check > $tolerance)
{
$last_max = $max;
$power = $power->times($X);
$max = $power->mat_max();
$max = $max->prime();
$max = $max->mat_max();
$max = $max->get_value(1,1);
$check = abs($max - $last_max);
$iter++;
}
$power->print_matrix();
echo "Number of iterations : $iter<br><br>";
?>
</li>
</ol>
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