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https://issues.apache.org/jira/browse/SPARK-18581?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=15695942#comment-15695942
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Hao Ren edited comment on SPARK-18581 at 11/25/16 2:09 PM:
-----------------------------------------------------------
After reading the code comments, I find it takes into consideration the
degenerate case of multivariate normal distribution:
https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case
I agree that the covariance matrix need not be invertible.
However, the pdf of gaussian should always be smaller than 1, shouldn't it ?
According to wikipedia, I can't see any reason the pdf could be larger than 1.
Let's focus on the MultivariateGaussian.calculateCovarianceConstants() function:
The problem I faced is that my covariance matrix gives an eigenvalue vector 'd'
as the following:
DenseVector(2.7681862718766402E-17, 9.204832153027098E-5, 8.995053289618483E-4,
0.0030052504431952055, 0.006867041289040775, 0.030351586260721354,
0.03499956314691966, 0.04128248388411499, 0.055530636656481766,
0.09840067120993062, 0.13259027660865316, 0.16729084354080376,
0.18807175387781094, 0.19009666915093745, 0.19065188805766764,
0.19116928711151343, 0.19218984168511, 0.22044130291811304,
0.23164643534046853, 0.32957890755845165, 0.4557354551695869,
0.639320905646873, 0.8327082373125074, 1.7966679300383896, 2.5790389754725234)
Meanwhile, the non-zero tolerance = 1.8514678433708895E-13
thus,
{code}
val logPseudoDetSigma = d.activeValuesIterator.filter(_ > tol).map(math.log).sum
{code}
logPseudoDetSigma = -58.40781006437829
-0.5 * (mu.size * math.log(2.0 * math.Pi) + logPseudoDetSigma) =
6.230441702072326 = u (same variable name in the code)
Knowing that
{code}
private[mllib] def logpdf(x: BV[Double]): Double = {
val delta = x - breezeMu
val v = rootSigmaInv * delta
u + v.t * v * -0.5 // u is used here
}
{code}
If [ v.t * v * -0.5 ] is a small negative number for a given x, then the logpdf
will be about 6 => pdf = exp(6) = 403.4287934927351
In the gaussian mixture model case, some of the gaussian distributions could
have a 'u' value much bigger, which results in a pdf = 2E10
was (Author: invkrh):
After reading the code comments, I find it takes into consideration the
degenerate case of multivariate normal distribution:
https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case
I agree that the covariance matrix need not be invertible.
However, the pdf of gaussian should always be smaller than 1, shouldn't it ?
Let's focus on the MultivariateGaussian.calculateCovarianceConstants() function:
The problem I faced is that my covariance matrix gives an eigenvalue vector 'd'
as the following:
DenseVector(2.7681862718766402E-17, 9.204832153027098E-5, 8.995053289618483E-4,
0.0030052504431952055, 0.006867041289040775, 0.030351586260721354,
0.03499956314691966, 0.04128248388411499, 0.055530636656481766,
0.09840067120993062, 0.13259027660865316, 0.16729084354080376,
0.18807175387781094, 0.19009666915093745, 0.19065188805766764,
0.19116928711151343, 0.19218984168511, 0.22044130291811304,
0.23164643534046853, 0.32957890755845165, 0.4557354551695869,
0.639320905646873, 0.8327082373125074, 1.7966679300383896, 2.5790389754725234)
Meanwhile, the non-zero tolerance = 1.8514678433708895E-13
thus,
{code}
val logPseudoDetSigma = d.activeValuesIterator.filter(_ > tol).map(math.log).sum
{code}
logPseudoDetSigma = -58.40781006437829
-0.5 * (mu.size * math.log(2.0 * math.Pi) + logPseudoDetSigma) =
6.230441702072326 = u (same variable name in the code)
Knowing that
{code}
private[mllib] def logpdf(x: BV[Double]): Double = {
val delta = x - breezeMu
val v = rootSigmaInv * delta
u + v.t * v * -0.5 // u is used here
}
{code}
If [ v.t * v * -0.5 ] is a small negative number for a given x, then the logpdf
will be about 6 => pdf = exp(6) = 403.4287934927351
In the gaussian mixture model case, some of the gaussian distributions could
have a 'u' value much bigger, which results in a pdf = 2E10
> MultivariateGaussian does not check if covariance matrix is invertible
> ----------------------------------------------------------------------
>
> Key: SPARK-18581
> URL: https://issues.apache.org/jira/browse/SPARK-18581
> Project: Spark
> Issue Type: Bug
> Components: MLlib
> Affects Versions: 1.6.2, 2.0.2
> Reporter: Hao Ren
>
> When training GaussianMixtureModel, I found some probability much larger than
> 1. That leads me to that fact that, the value returned by
> MultivariateGaussian.pdf can be 10^5, etc.
> After reviewing the code, I found that problem lies in the computation of
> determinant of the covariance matrix.
> The computation is simplified by using pseudo-determinant of a positive
> defined matrix.
> In my case, I have a feature = 0 for all data point.
> As a result, covariance matrix is not invertible <=> det(covariance matrix) =
> 0 => pseudo-determinant will be very close to zero,
> Thus, log(pseudo-determinant) will be a large negative number which finally
> make logpdf very biger, pdf will be even bigger > 1.
> As said in comments of MultivariateGaussian.scala,
> """
> Singular values are considered to be non-zero only if they exceed a tolerance
> based on machine precision.
> """
> But if a singular value is considered to be zero, means the covariance matrix
> is non invertible which is a contradiction to the assumption that it should
> be invertible.
> So we should check if there a single value is smaller than the tolerance
> before computing the pseudo determinant
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