where the posterior on
W
is proportional to the product of the
Gaussian distribution of weight vectors, which correspond to
each target element as follows:
p W AGB
p AGB p
w
i
i
i
n
i
i
i
( | S A
|W S) W|A ( |
, , )
(
,
, )
∝
(
)
∝
=
∏
1
µ Σ
(11)
where
μ
i
=σ
σ
i
-2
∑
i
Φ
T
AGB
i
and
∑
i
=(
σ
i
-2
Φ
T
Φ
Φ+
A
)
-1
are the model
parameter (
w
i
) mean and the covariance of distribution,
respectively. The optimum values for the hyperparameters,
A
opt
=
diag
[
α
1
opt
,α
α
2
opt
,…,α
α
P
opt
], are then used to obtain the posterior
mean and covariance, as follows (Thayananthan
et al.
, 2006):
i
opt
i
opt
T
opt
A
∑
=
(
)
+
(
)
−
−
σ
Φ Φ
2
1
(12)
µ σ
Σ
Φ
i
opt
i
opt
i
T
i
AGB
=
(
)
−
2
opt
(13)
W
opt
=
diag
[
μ
1
opt
,α
μ
2
opt
,…,α
μ
m
opt
]
(14)
The coef cient of determination (
R
2
), the root mean square
error (
RMSE
), the mean absolute error (
MAE
), and the mean er-
ror (
ME
) are calculated as the accuracy statistics. The predict-
ed
AGB
is underestimated when
ME
is negative and overesti-
mated when
ME
is positive (Englhart
et al.
, 2012):
RMSE
f y
n
i
n
=
−
=
∑
(
)
2
1
(15)
MAE
f y
n
i
i
i
n
=
=
=
∑
1
(16)
ME
=
f
i
– y
i
(17)
where
f
is predicted
AGB
,
y
is measured
AGB
, and
n
is number
of predicted
AGB
Results
The models were implemented using Matlab 2013, and the
results were averaged from the results of ten iterations. The
estimation of
AGB
from
SAR
data using the
MLR
model resulted
in Equation 18:
AGB
=64.4+47.76×〖10
3
exp(
σ
0
HH
)+4.21×10
7
exp(
σ
0
HV
) (18)
where
σ
0
is the average of multi-temporal backscattering coef-
ficients.
In the
MLPNN
model, the actual multitemporal backscatter
values of the input layer were converted to the corresponding
output values with the assumption that the number of hidden
layers is two. In addition, the
SVR
model was used to estimate
the
AGB
with a kernel function parameter, g, equal to 1.5 and
a regularization parameter, C, equal to 50. The error tolerance
was set to 0.001, and the number of support vectors was 88.
The
MVRVR
model was modified using the Gaussian kernel
function with a scale parameter,, equal to 1.85, where the
number of relevance vectors was approximately 12 percent of
training data. In this model, non-zero columns of the opti-
mized weight matrix correspond to the training data.
The results of validation are shown in Table 2, where it can
be seen that the
MVRVR
model has both the highest
R
2
value
(
R
2
= 0.9) and the lowest error measures (
RMSE
= 32.05,
MAE
= 24.20, and
ME
= 32.80). In the
MVRVR
model, the training
data were pruned to a few relevance vectors, in contrast to the
approach used in other models. The overfitting problem can
be solved by reducing the impact of outliers (Tipping, 2001)
and the
MVRVR
model achieved better results using only 12
percent of training data (after reducing the impact of outliers)
compared to other models. Also, the use of fewer observations
can leads to reduction of both the time and cost involved in
collecting ground data. In addition, the results of the
MVRVR
shows the lowest overestimation in the biomass range be-
tween 0 and 200 Mg/ha and the lowest underestimation in the
biomass range between 0 and 400 Mg/ha.
The
SVR
model showed reduced errors in relation to
MAE
measurements than the
MLPNN
, due to the proper estimations
made before the saturation point (224.75 Mg/ha), while its
ME
was greater than that of other models because of its lower
saturation point. Despite the lower
ME
and higher saturation
point, the
MLPNN
model showed random behavior in compari-
son with the
SVR
and
MLR
models. Utilizing the
MLR
model re-
sulted in the lowest
R
2
(
R
2
= 0.47) value, and the highest error
measures (
RMSE
= 75.35,
MAE
= 68.61 and
ME
= -78.11), which
led to an underestimation of
AGB
because of the negative
ME
.
Figure 2 shows a comparison of the different estimation
approaches used in
AGB
estimation, and it can be observed
that the highest saturation point were recorded for the
MVRVR
model (297.81 Mg/ha). Because the selection of relevance vec-
tors in the
MVRVR
model is aimed at minimizing the whole ap-
proximation errors, this model followed the trend of ground
data for the whole range of biomass, unlike the
SVR
model. In
addition, the over- and under-estimation of validation data are
evident in
MLPNN
and
MLR
, respectively.
Figure 3 shows scatterplots of the observed versus predict-
ed values of
AGB
. The scatterplots for
MVRVR
show that the re-
sults are scattered over the whole biomass range. In addition,
as the
MVRVR
tries to minimize the whole error before satura-
tion point, the scatters tend to lie near the line one to one. It
is possible to see that the real biomass value before 200 [Mg/
ha] is extremely overestimated using
MLR
, but after saturation
points it is underestimated using all models.
Plate 2 shows a biomass map of the study area produced
by the
MVRVR
model. All four models have the ability to es-
timate biomass when it has a value of under 200 Mg/ha. The
biomass range between 250 Mg/ha and 300 Mg/ha emphasizes
the importance of increasing the saturation point in biomass
estimation. This is the area in which the
MVRVR
model is bet-
ter than the other three models; in these cases, it is the only
model that can accurately estimate biomass.
T
able
2. R
esults
O
f
V
alidation
F
or
F
our
E
stimation
M
odels
A
t
A 95 P
ercent
C
onfidence
I
nterval
. T
he
C
oefficient
O
f
D
etermination
(R
2
), R
oot
M
ean
S
quare
E
rror
(RMSE), M
ean
A
bsolute
E
rror
(MAE), A
nd
M
ean
E
rror
(ME) A
re
D
epicted
Methods
R
2
RMSE
(Mg/ha)
MAE
(Mg/ha)
ME
(Mg/ha)
AGB<200
AGB>200
Total
MLR
0.47 ± 0.08
75.35 ± 15.5
68.61 ± 14.8
590.71
-668.82
-78.11 ± 16.0
MLPNN
0.59 ± 0.13
65.99 ± 18.2
52.69 ± 15.3
520.82
-387.18
133.64 ± 22.4
SVR
0.62 ± 0.14
63.58 ± 16.7
40.57 ± 11.2
351.99
-633.83
-281.84 ± 32.3
MVRVR
0.90 ± 0.06
32.05 ± 8.5
24.20 ± 8.2
151.93
-184.73
-32.80 ± 8.1
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
January 2016
45
