Digital Signal Processing Reference
In-Depth Information
equilibrium. Accordingly, we propose the following modified ARE:
i
=
1
E[
2
]
(w
(
n
)
−
m
i
)
ARE(MMSOM, SOM)
i
=
2
]
.
(11.78)
i
=
1
E[
(w
Mi
(
n
)
−
m
i
)
From Equations 11.77 and 11.78, it is seen that the performance of SOM and
MMSOM with respect to the mean-squared estimation error is different for
small and large
n
. For large
n
, the dc-error dominates, and therefore the output
variance does not play any role. Therefore, all the conclusions drawn from the
analysis of bias are still valid: i.e., the MMSOM outperforms the linear SOM.
This is not the case for small
n
.Todemonstrate the role of output variance,
we evaluate Equation 11.78 for
n
1.
First, let us consider the influence functions of the SOM weights. They are
given by
=
IF
(
x
;
w
1
,F
)
=
x
[1
−
u
s
(
x
−
T
SOM
)
]
−
E[
x
|
x
∈ V
(
W
)
]
(11.79)
1
IF
(
x
;
w
2
,F
)
=
xu
s
(
x
−
T
SOM
)
−
E[
x
|
x
∈ V
(
W
)
]
,
(11.80)
2
where
u
s
(
x
)
is the unit-step function
1 f
x
≥
0
u
s
(
x
)
=
(11.81)
0
otherwise.
By applying Equation 11.75, we find
E[
x
2
2
i
V
(w
i
,F
)
=
|
x
∈ V
i
(
W
)
]
−
w
i
=
1
,
2
.
(11.82)
In the case of the Gaussian mixture model, for the first SOM weight and
T
=
T
SOM
,weobtain
E[
x
2
|
x
∈ V
(
W
)
]
1
1
2
+
erf
T
m
1
+
σ
2
+
(
1
−
m
1
=
1
−
)
F
(
T
)
σ
exp
2
1
2
+
erf
T
m
2
+
σ
T
2
−
−
m
2
√
2
1
2
−
m
1
×
(
m
1
+
T
)
−
σ
σ
exp
2
T
1
2
−
m
2
+
(
1
−
)(
m
2
+
T
)
−
.
(11.83)
σ
For the second SOM weight, we have
σ
]
.
(11.84)
1
E[
x
2
2
m
1
+
(
m
2
−
E[
x
2
|
x
∈ V
(
W
)
]
=
+
1
−
)
F
(
T
)
|
x
∈ V
(
W
)
2
1
1
−
F
(
T
)
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