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Chapter 8: Probability Density Estimation 303

min MD x ({ 2 i ( n + 1) )} > t C , (8.45)
i

is a threshold to create a new term. The rule in Equation 8.45 states
where t C
that if the smallest squared Mahalanobis distance is greater than the thresh-
old, then we create a new term. In the univariate case, if t C = 1 is used, then
a new term is created if a new observation is more than one standard devia-
tion away from the mean of each term. For t C = 4 , a new term would be cre-
ated for an observation that is at least two standard deviations away from the
existing terms. For multivariate data, we would like to keep the same term
based on
creation rate as in the 1-D case. Solka [1995] provides thresholds t C
the squared Mahalanobis distance for the univariate, bivariate, and trivariate
cases. These are shown in Table 8.3.

T A B L E E E8 8 .3
A
T
AB
T
TA
BL
8.3
.3
.3
8
B
L
LE
Recommended Thresholds for Adaptive Mixtures
Dimensionality Create Threshold
1 1
2 2.34
3 3.54
When we create a new term, we initialize the parameters using
Equations 8.46 through 8.48. We denote the current number of terms in the
model by N.
ˆ n +( 1) ( n + 1)
µ N + 1 = x , (8.46)
ˆ ( n + 1) 1
p N + 1 = ------------ , (8.47)
n + 1
ˆ n +( 1) ()
ˆ
Σ N + 1 = ℑΣ i , (8.48)
ˆ
() is a weighted average using the posterior probabilities. In prac-
where ℑΣ i
tice, some other estimate or initial covariance can be used for the new term.
To ensure that the mixing coefficients sum to one when a new term is added,
ˆ ( n + ) must be rescaled using
1
the p i
ˆ n ()
1
ˆ ( n + ) ------------; , ,
np i
p i = i = 1 … N .
n + 1

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