Modeling and Multivariate Methods - SAS

476 Clustering Data Chapter 18
Normal Mixtures
Complete Tours is the number of times to restart the estimation process. This helps to guard against the
process finding a local solution.
Initial Guesses is the number of random starts within each tour. Random starting values for the
parameters are used for each new start.
Max Iterations is the maximum number of iterations during the convergence stage. The convergence
stage starts after all tours are complete. It begins at the optimal result out of all the starts and tours, and
from there converges to a final solution.
Platform Options
For details on the red-triangle options for Normal Mixtures and Robust Normal Mixtures, see “K-Means
Platform Options” on page 472.
Details of the Estimation Process
Normal Mixtures uses the EM algorithm to do fitting because it is more stable than the Newton-Raphson
algorithm. Additionally we're using a Bayesian regularized version of the EM algorithm, which allows us to
smoothly handle cases where the covariance matrix is singular. Since the estimates are heavily dependent on
initial guesses, the platform will go through a number of tours, each with randomly selected points as initial
centers.
Doing multiple tours makes the estimation process somewhat expensive, so considerable patience is required
for large problems. Controls allow you to specify the tour and iteration limits.
Additional Details for Robust Normal Mixtures
Because Normal Mixtures is sensitive to outliers, JMP offers an outlier robust alternative called Robust
Normal Mixtures. This uses a robust method of estimating the normal parameters. JMP computes the
estimates via maximum likelihood with respect to a mixture of Huberized normal distributions (a class of
modified normal distributions that was tailor-made to be more outlier resistant than the normal
distribution).
The Huberized Gaussian distribution has pdf Φ k
( x)
.
Φ k
( x)
=
---------------------------
exp( – ρ( x)
)
c k
ρ( x)
=
x 2
---- if x ≤ k
2
k 2
kx
– ---- if x > k
2
c k
=
exp( – k 2 ⁄ 2)
2πΦk [ ( ) – Φ( – k)
] + 2----------------------------
k