Github user tgaloppo commented on a diff in the pull request:
https://github.com/apache/spark/pull/3871#discussion_r22504722
--- Diff:
mllib/src/main/scala/org/apache/spark/mllib/stat/impl/MultivariateGaussian.scala
---
@@ -17,23 +17,84 @@
package org.apache.spark.mllib.stat.impl
-import breeze.linalg.{DenseVector => DBV, DenseMatrix => DBM, Transpose,
det, pinv}
+import breeze.linalg.{DenseVector => DBV, DenseMatrix => DBM, diag, max,
eigSym}
-/**
- * Utility class to implement the density function for multivariate
Gaussian distribution.
- * Breeze provides this functionality, but it requires the Apache
Commons Math library,
- * so this class is here so-as to not introduce a new dependency in
Spark.
- */
+import org.apache.spark.mllib.util.MLUtils
+
+/**
+ * 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
+ */
private[mllib] class MultivariateGaussian(
val mu: DBV[Double],
val sigma: DBM[Double]) extends Serializable {
- private val sigmaInv2 = pinv(sigma) * -0.5
- private val U = math.pow(2.0 * math.Pi, -mu.length / 2.0) *
math.pow(det(sigma), -0.5)
-
+
+ /**
+ * Compute distribution dependent constants:
+ * rootSigmaInv = D^(-1/2) * U, where sigma = U * D * U.t
--- End diff --
U D^-1 * U.t => U D^(-1/2) D^(-1/2) U.t => (U D^(-1/2)) (D^(-1/2) U.t)
...both are U and D are symmetric, so...
(U D^(-1/2)) = (U.t (D^(-1/2)).t) => (D^(-1/2) U).t
and
(D^(-1/2) U.t) = (D^(-1/2) U)
thus
U D^-1 U.t => (D^(-1/2) U).t (D^(-1/2) U)
... bringing in the delta we get
delta.t (D^(-1/2) U).t (D^(-1/2) U) delta
=> ((D^(-1/2) U) delta).t (D^(-1/2) U) delta = norm(D^(-1/2) U delta)^2
as indicated by @mengxr
(phew, hope I did that OK! :) )
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