* Example: $kpca = new KernelPCA(KernelPCA::KERNEL_RBF, null, 2, 15.0);
* will initialize the algorithm with an RBF kernel having the gamma parameter as 15,0.
* This transformation will return the same number of rows with only 2 columns.
*
* @param float $totalVariance Total variance to be preserved if numFeatures is not given
* @param int $numFeatures Number of columns to be returned
* @param float $gamma Gamma parameter is used with RBF and Sigmoid kernels
*
* @throws InvalidArgumentException
*/
public function __construct(int $kernel = self::KERNEL_RBF, ?float $totalVariance = null, ?int $numFeatures = null, ?float $gamma = null)
{
if (!in_array($kernel, [self::KERNEL_RBF, self::KERNEL_SIGMOID, self::KERNEL_LAPLACIAN, self::KERNEL_LINEAR], true)) {
throw new InvalidArgumentException('KernelPCA can be initialized with the following kernels only: Linear, RBF, Sigmoid and Laplacian');
}
parent::__construct($totalVariance, $numFeatures);
$this->kernel = $kernel;
$this->gamma = $gamma;
}
/**
* Takes a data and returns a lower dimensional version
* of this data while preserving $totalVariance or $numFeatures.
* $data is an n-by-m matrix and returned array is
* n-by-k matrix where k <= m
*/
public function fit(array $data): array
{
$numRows = count($data);
$this->data = $data;
if ($this->gamma === null) {
$this->gamma = 1.0 / $numRows;
}
$matrix = $this->calculateKernelMatrix($this->data, $numRows);
$matrix = $this->centerMatrix($matrix, $numRows);
$this->eigenDecomposition($matrix);
$this->fit = true;
return Matrix::transposeArray($this->eigVectors);
}
/**
* Transforms the given sample to a lower dimensional vector by using
* the variables obtained during the last run of fit.
*
* @throws InvalidArgumentException
* @throws InvalidOperationException
*/
public function transform(array $sample): array
{
if (!$this->fit) {
throw new InvalidOperationException('KernelPCA has not been fitted with respect to original dataset, please run KernelPCA::fit() first');
}
if (is_array($sample[0])) {
throw new InvalidArgumentException('KernelPCA::transform() accepts only one-dimensional arrays');
}
$pairs = $this->getDistancePairs($sample);
return $this->projectSample($pairs);
}
/**
* Calculates similarity matrix by use of selected kernel function
* An n-by-m matrix is given and an n-by-n matrix is returned
*/
protected function calculateKernelMatrix(array $data, int $numRows): array
{
$kernelFunc = $this->getKernel();
$matrix = [];
for ($i = 0; $i < $numRows; ++$i) {
for ($k = 0; $k < $numRows; ++$k) {
if ($i <= $k) {
$matrix[$i][$k] = $kernelFunc($data[$i], $data[$k]);
} else {
$matrix[$i][$k] = $matrix[$k][$i];
}
}
}
return $matrix;
}
/**
* Kernel matrix is centered in its original space by using the following
* conversion:
*
* K′ = K − N.K − K.N + N.K.N where N is n-by-n matrix filled with 1/n
*/
protected function centerMatrix(array $matrix, int $n): array
{
$N = array_fill(0, $n, array_fill(0, $n, 1.0 / $n));
$N = new Matrix($N, false);
$K = new Matrix($matrix, false);
// K.N (This term is repeated so we cache it once)
$K_N = $K->multiply($N);
// N.K
$N_K = $N->multiply($K);
// N.K.N
$N_K_N = $N->multiply($K_N);
return $K->subtract($N_K)
->subtract($K_N)
->add($N_K_N)
->toArray();
}
/**
* Returns the callable kernel function
*
* @throws \Exception
*/
protected function getKernel(): Closure
{
switch ($this->kernel) {
case self::KERNEL_LINEAR:
// k(x,y) = xT.y
return function ($x, $y) {
return Matrix::dot($x, $y)[0];
};
case self::KERNEL_RBF:
// k(x,y)=exp(-γ.|x-y|) where |..| is Euclidean distance
$dist = new Euclidean();
return function ($x, $y) use ($dist): float {
return exp(-$this->gamma * $dist->sqDistance($x, $y));
};
case self::KERNEL_SIGMOID:
// k(x,y)=tanh(γ.xT.y+c0) where c0=1
return function ($x, $y): float {
$res = Matrix::dot($x, $y)[0] + 1.0;
return tanh((float) $this->gamma * $res);
};
case self::KERNEL_LAPLACIAN:
// k(x,y)=exp(-γ.|x-y|) where |..| is Manhattan distance
$dist = new Manhattan();
return function ($x, $y) use ($dist): float {
return exp(-$this->gamma * $dist->distance($x, $y));
};
default:
// Not reached
throw new InvalidArgumentException(sprintf('KernelPCA initialized with invalid kernel: %d', $this->kernel));
}
}
protected function getDistancePairs(array $sample): array
{
$kernel = $this->getKernel();
$pairs = [];
foreach ($this->data as $row) {
$pairs[] = $kernel($row, $sample);
}
return $pairs;
}
protected function projectSample(array $pairs): array
{
// Normalize eigenvectors by eig = eigVectors / eigValues
$func = function ($eigVal, $eigVect) {
$m = new Matrix($eigVect, false);
$a = $m->divideByScalar($eigVal)->toArray();
return $a[0];
};
$eig = array_map($func, $this->eigValues, $this->eigVectors);
// return k.dot(eig)
return Matrix::dot($pairs, $eig);
}
}