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Appendix: Density Functions
Let
x
be the expression value of a gene probe and
j
be the component's in-
dex. The density function for each component is denoted as
p
(
x|j
). The density
function for each type of distribution used in our mixture models is as follows:
1. Normal distribution.
(
x − θ
A
j
)
2
2
θ
2
B
j
1
θ
B
j
√
2
π
exp
p
(
x|j
)=
−
2. Lognormal distribution.
−
− θ
A
j
)
2
2
θ
2
B
j
1
xθ
B
j
√
2
π
exp
(
ln
(
x
)
p
(
x|j
)=
3. Gamma distribution.
e
x/θ
B
j
θ
θ
A
j
B
j
p
(
x|j
)=
x
θ
A
j
−
1
G
(
θ
A
j
)
where
G
(
θ
A
)=2
0
e
−t
2
t
2
θ
A
−
1
dt
is the Gamma function.
θ
A
and
θ
B
correspond to the parameters in each distributions (i.e.
shape
,
scale
in gamma and
mean
,
standard deviation
in normal and lognormal distributions).
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