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/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.spark.mllib.stat.distribution
import breeze.linalg.{diag, eigSym, max, DenseMatrix => DBM, DenseVector => DBV, Vector => BV}
import org.apache.spark.annotation.{DeveloperApi, Since}
import org.apache.spark.mllib.linalg.{Matrices, Matrix, Vector, Vectors}
import org.apache.spark.mllib.util.MLUtils
/**
* :: DeveloperApi ::
* This class provides basic functionality for a Multivariate Gaussian (Normal) Distribution. In
* the event that the covariance matrix is singular, the density will be computed in a
* reduced dimensional subspace under which the distribution is supported.
* (see [[http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case]])
*
* @param mu The mean vector of the distribution
* @param sigma The covariance matrix of the distribution
*/
@Since("1.3.0")
@DeveloperApi
class MultivariateGaussian @Since("1.3.0") (
@Since("1.3.0") val mu: Vector,
@Since("1.3.0") val sigma: Matrix) extends Serializable {
require(sigma.numCols == sigma.numRows, "Covariance matrix must be square")
require(mu.size == sigma.numCols, "Mean vector length must match covariance matrix size")
private val breezeMu = mu.toBreeze.toDenseVector
/**
* private[mllib] constructor
*
* @param mu The mean vector of the distribution
* @param sigma The covariance matrix of the distribution
*/
private[mllib] def this(mu: DBV[Double], sigma: DBM[Double]) = {
this(Vectors.fromBreeze(mu), Matrices.fromBreeze(sigma))
}
/**
* Compute distribution dependent constants:
* rootSigmaInv = D^(-1/2)^ * U.t, where sigma = U * D * U.t
* u = log((2*pi)^(-k/2)^ * det(sigma)^(-1/2)^)
*/
private val (rootSigmaInv: DBM[Double], u: Double) = calculateCovarianceConstants
/**
* Returns density of this multivariate Gaussian at given point, x
*/
@Since("1.3.0")
def pdf(x: Vector): Double = {
pdf(x.toBreeze)
}
/**
* Returns the log-density of this multivariate Gaussian at given point, x
*/
@Since("1.3.0")
def logpdf(x: Vector): Double = {
logpdf(x.toBreeze)
}
/** Returns density of this multivariate Gaussian at given point, x */
private[mllib] def pdf(x: BV[Double]): Double = {
math.exp(logpdf(x))
}
/** Returns the log-density of this multivariate Gaussian at given point, x */
private[mllib] def logpdf(x: BV[Double]): Double = {
val delta = x - breezeMu
val v = rootSigmaInv * delta
u + v.t * v * -0.5
}
/**
* Calculate distribution dependent components used for the density function:
* pdf(x) = (2*pi)^(-k/2)^ * det(sigma)^(-1/2)^ * exp((-1/2) * (x-mu).t * inv(sigma) * (x-mu))
* where k is length of the mean vector.
*
* We here compute distribution-fixed parts
* log((2*pi)^(-k/2)^ * det(sigma)^(-1/2)^)
* and
* D^(-1/2)^ * U, where sigma = U * D * U.t
*
* Both the determinant and the inverse can be computed from the singular value decomposition
* of sigma. Noting that covariance matrices are always symmetric and positive semi-definite,
* we can use the eigendecomposition. We also do not compute the inverse directly; noting
* that
*
* sigma = U * D * U.t
* inv(Sigma) = U * inv(D) * U.t
* = (D^{-1/2}^ * U.t).t * (D^{-1/2}^ * U.t)
*
* and thus
*
* -0.5 * (x-mu).t * inv(Sigma) * (x-mu) = -0.5 * norm(D^{-1/2}^ * U.t * (x-mu))^2^
*
* To guard against singular covariance matrices, this method computes both the
* pseudo-determinant and the pseudo-inverse (Moore-Penrose). Singular values are considered
* to be non-zero only if they exceed a tolerance based on machine precision, matrix size, and
* relation to the maximum singular value (same tolerance used by, e.g., Octave).
*/
private def calculateCovarianceConstants: (DBM[Double], Double) = {
val eigSym.EigSym(d, u) = eigSym(sigma.toBreeze.toDenseMatrix) // sigma = u * diag(d) * u.t
// For numerical stability, values are considered to be non-zero only if they exceed tol.
// This prevents any inverted value from exceeding (eps * n * max(d))^-1
val tol = MLUtils.EPSILON * max(d) * d.length
try {
// log(pseudo-determinant) is sum of the logs of all non-zero singular values
val logPseudoDetSigma = d.activeValuesIterator.filter(_ > tol).map(math.log).sum
// calculate the root-pseudo-inverse of the diagonal matrix of singular values
// by inverting the square root of all non-zero values
val pinvS = diag(new DBV(d.map(v => if (v > tol) math.sqrt(1.0 / v) else 0.0).toArray))
(pinvS * u.t, -0.5 * (mu.size * math.log(2.0 * math.Pi) + logPseudoDetSigma))
} catch {
case uex: UnsupportedOperationException =>
throw new IllegalArgumentException("Covariance matrix has no non-zero singular values")
}
}
}
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