Linear Discrimant Analysis (LDA) (#82)

* Linear Discrimant Analysis (LDA)

* LDA test file

* Matrix inverse via LUDecomposition

* LUDecomposition inverse() and det() applied

* Readme update for LDA

README.md

@@ -88,8 +88,9 @@ Example scripts are available in a separate repository [php-ai/php-ml-examples](
     * [Token Count Vectorizer](http://php-ml.readthedocs.io/en/latest/machine-learning/feature-extraction/token-count-vectorizer/)
     * [Tf-idf Transformer](http://php-ml.readthedocs.io/en/latest/machine-learning/feature-extraction/tf-idf-transformer/)
 * Dimensionality Reduction
-    * PCA
+    * PCA (Principal Component Analysis)
     * Kernel PCA
+    * LDA (Linear Discriminant Analysis)
 * Datasets
     * [Array](http://php-ml.readthedocs.io/en/latest/machine-learning/datasets/array-dataset/)
     * [CSV](http://php-ml.readthedocs.io/en/latest/machine-learning/datasets/csv-dataset/)

src/Phpml/DimensionReduction/EigenTransformerBase.php

@@ -0,0 +1,98 @@
+<?php declare(strict_types=1);
+
+namespace Phpml\DimensionReduction;
+
+use Phpml\Math\LinearAlgebra\EigenvalueDecomposition;
+use Phpml\Math\Matrix;
+
+/**
+ * Class to compute eigen pairs (values & vectors) of a given matrix
+ * with the consideration of numFeatures or totalVariance to be preserved
+ *
+ * @author hp
+ */
+abstract class EigenTransformerBase
+{
+    /**
+     * Total variance to be conserved after the reduction
+     *
+     * @var float
+     */
+    public $totalVariance = 0.9;
+
+    /**
+     * Number of features to be preserved after the reduction
+     *
+     * @var int
+     */
+    public $numFeatures = null;
+
+    /**
+     * Top eigenvectors of the matrix
+     *
+     * @var array
+     */
+    protected $eigVectors = [];
+
+    /**
+     * Top eigenValues of the matrix
+     *
+     * @var type
+     */
+    protected $eigValues = [];
+
+    /**
+     * Calculates eigenValues and eigenVectors of the given matrix. Returns
+     * top eigenVectors along with the largest eigenValues. The total explained variance
+     * of these eigenVectors will be no less than desired $totalVariance value
+     *
+     * @param array $matrix
+     */
+    protected function eigenDecomposition(array $matrix)
+    {
+        $eig = new EigenvalueDecomposition($matrix);
+        $eigVals = $eig->getRealEigenvalues();
+        $eigVects= $eig->getEigenvectors();
+
+        $totalEigVal = array_sum($eigVals);
+        // Sort eigenvalues in descending order
+        arsort($eigVals);
+
+        $explainedVar = 0.0;
+        $vectors = [];
+        $values = [];
+        foreach ($eigVals as $i => $eigVal) {
+            $explainedVar += $eigVal / $totalEigVal;
+            $vectors[] = $eigVects[$i];
+            $values[] = $eigVal;
+
+            if ($this->numFeatures !== null) {
+                if (count($vectors) == $this->numFeatures) {
+                    break;
+                }
+            } else {
+                if ($explainedVar >= $this->totalVariance) {
+                    break;
+                }
+            }
+        }
+
+        $this->eigValues = $values;
+        $this->eigVectors = $vectors;
+    }
+
+    /**
+     * Returns the reduced data
+     *
+     * @param array $data
+     *
+     * @return array
+     */
+    protected function reduce(array $data)
+    {
+        $m1 = new Matrix($data);
+        $m2 = new Matrix($this->eigVectors);
+
+        return $m1->multiply($m2->transpose())->toArray();
+    }
+}

src/Phpml/DimensionReduction/KernelPCA.php

@@ -86,7 +86,7 @@ public function fit(array $data)
         $matrix = $this->calculateKernelMatrix($this->data, $numRows);
         $matrix = $this->centerMatrix($matrix, $numRows);
 
-        list($this->eigValues, $this->eigVectors) = $this->eigenDecomposition($matrix, $numRows);
+        $this->eigenDecomposition($matrix, $numRows);
 
         $this->fit = true;
 

src/Phpml/DimensionReduction/LDA.php

@@ -0,0 +1,247 @@
+<?php
+
+declare(strict_types=1);
+
+namespace Phpml\DimensionReduction;
+
+use Phpml\Math\Statistic\Mean;
+use Phpml\Math\Matrix;
+
+class LDA extends EigenTransformerBase
+{
+    /**
+     * @var bool
+     */
+    public $fit = false;
+
+    /**
+     * @var array
+     */
+    public $labels;
+
+    /**
+     * @var array
+     */
+    public $means;
+
+    /**
+     * @var array
+     */
+    public $counts;
+
+    /**
+     * @var float
+     */
+    public $overallMean;
+
+    /**
+     * Linear Discriminant Analysis (LDA) is used to reduce the dimensionality
+     * of the data. Unlike Principal Component Analysis (PCA), it is a supervised
+     * technique that requires the class labels in order to fit the data to a
+     * lower dimensional space. <br><br>
+     * The algorithm can be initialized by speciyfing
+     * either with the totalVariance(a value between 0.1 and 0.99)
+     * or numFeatures (number of features in the dataset) to be preserved.
+     *
+     * @param float|null $totalVariance Total explained variance to be preserved
+     * @param int|null $numFeatures Number of features to be preserved
+     *
+     * @throws \Exception
+     */
+    public function __construct($totalVariance = null, $numFeatures = null)
+    {
+        if ($totalVariance !== null && ($totalVariance < 0.1 || $totalVariance > 0.99)) {
+            throw new \Exception("Total variance can be a value between 0.1 and 0.99");
+        }
+        if ($numFeatures !== null && $numFeatures <= 0) {
+            throw new \Exception("Number of features to be preserved should be greater than 0");
+        }
+        if ($totalVariance !== null && $numFeatures !== null) {
+            throw new \Exception("Either totalVariance or numFeatures should be specified in order to run the algorithm");
+        }
+
+        if ($numFeatures !== null) {
+            $this->numFeatures = $numFeatures;
+        }
+        if ($totalVariance !== null) {
+            $this->totalVariance = $totalVariance;
+        }
+    }
+
+    /**
+     * Trains the algorithm to transform the given data to a lower dimensional space.
+     *
+     * @param array $data
+     * @param array $classes
+     *
+     * @return array
+     */
+    public function fit(array $data, array $classes) : array
+    {
+        $this->labels = $this->getLabels($classes);
+        $this->means  = $this->calculateMeans($data, $classes);
+
+        $sW = $this->calculateClassVar($data, $classes);
+        $sB = $this->calculateClassCov();
+
+        $S = $sW->inverse()->multiply($sB);
+        $this->eigenDecomposition($S->toArray());
+
+        $this->fit = true;
+
+        return $this->reduce($data);
+    }
+
+    /**
+     * Returns unique labels in the dataset
+     *
+     * @param array $classes
+     *
+     * @return array
+     */
+    protected function getLabels(array $classes): array
+    {
+        $counts = array_count_values($classes);
+
+        return array_keys($counts);
+    }
+
+
+    /**
+     * Calculates mean of each column for each class and returns
+     * n by m matrix where n is number of labels and m is number of columns
+     *
+     * @param type $data
+     * @param type $classes
+     *
+     * @return array
+     */
+    protected function calculateMeans($data, $classes) : array
+    {
+        $means = [];
+        $counts= [];
+        $overallMean = array_fill(0, count($data[0]), 0.0);
+
+        foreach ($data as $index => $row) {
+            $label = array_search($classes[$index], $this->labels);
+
+            foreach ($row as $col => $val) {
+                if (! isset($means[$label][$col])) {
+                    $means[$label][$col] = 0.0;
+                }
+                $means[$label][$col] += $val;
+                $overallMean[$col] += $val;
+            }
+
+            if (! isset($counts[$label])) {
+                $counts[$label] = 0;
+            }
+            $counts[$label]++;
+        }
+
+        foreach ($means as $index => $row) {
+            foreach ($row as $col => $sum) {
+                $means[$index][$col] = $sum / $counts[$index];
+            }
+        }
+
+        // Calculate overall mean of the dataset for each column
+        $numElements = array_sum($counts);
+        $map = function ($el) use ($numElements) {
+            return $el / $numElements;
+        };
+        $this->overallMean = array_map($map, $overallMean);
+        $this->counts = $counts;
+
+        return $means;
+    }
+
+
+    /**
+     * Returns in-class scatter matrix for each class, which
+     * is a n by m matrix where n is number of classes and
+     * m is number of columns
+     *
+     * @param array $data
+     * @param array $classes
+     *
+     * @return Matrix
+     */
+    protected function calculateClassVar($data, $classes)
+    {
+        // s is an n (number of classes) by m (number of column) matrix
+        $s = array_fill(0, count($data[0]), array_fill(0, count($data[0]), 0));
+        $sW = new Matrix($s, false);
+
+        foreach ($data as $index => $row) {
+            $label = array_search($classes[$index], $this->labels);
+            $means = $this->means[$label];
+
+            $row = $this->calculateVar($row, $means);
+
+            $sW = $sW->add($row);
+        }
+
+        return $sW;
+    }
+
+    /**
+     * Returns between-class scatter matrix for each class, which
+     * is an n by m matrix where n is number of classes and
+     * m is number of columns
+     *
+     * @return Matrix
+     */
+    protected function calculateClassCov()
+    {
+        // s is an n (number of classes) by m (number of column) matrix
+        $s = array_fill(0, count($this->overallMean), array_fill(0, count($this->overallMean), 0));
+        $sB = new Matrix($s, false);
+
+        foreach ($this->means as $index => $classMeans) {
+            $row = $this->calculateVar($classMeans, $this->overallMean);
+            $N = $this->counts[$index];
+            $sB = $sB->add($row->multiplyByScalar($N));
+        }
+
+        return $sB;
+    }
+
+    /**
+     * Returns the result of the calculation (x - m)T.(x - m)
+     *
+     * @param array $row
+     * @param array $means
+     *
+     * @return Matrix
+     */
+    protected function calculateVar(array $row, array $means)
+    {
+        $x = new Matrix($row, false);
+        $m = new Matrix($means, false);
+        $diff = $x->subtract($m);
+
+        return $diff->transpose()->multiply($diff);
+    }
+
+    /**
+     * Transforms the given sample to a lower dimensional vector by using
+     * the eigenVectors obtained in the last run of <code>fit</code>.
+     *
+     * @param array $sample
+     *
+     * @return array
+     */
+    public function transform(array $sample)
+    {
+        if (!$this->fit) {
+            throw new \Exception("LDA has not been fitted with respect to original dataset, please run LDA::fit() first");
+        }
+
+        if (! is_array($sample[0])) {
+            $sample = [$sample];
+        }
+
+        return $this->reduce($sample);
+    }
+}