Geoscience Reference
In-Depth Information
4. Rarefaction, Model Analysis and Fit to the Data
We have used a rarefaction technique to standardize the samples of different
sizes and compare them with the model predictions. The size of the samples
across all the temperature and salinity gradients ranged from [8
10
3
]to
10
8
]
individuals in the mysid community. For each dataset associated with each
particular environmental situation, we have calculated the empirical cumu-
lative distribution function using the empirical abundance values. Finally, we
sampled this empirical distribution function using a uniform distribution and
obtained the new abundance vector for a standardized sampling size of
J
M
¼
10
6
] individuals in the fish community and from [1.3
10
5
]to[3
[3.3
10
3
individuals. We then compared the mean after 100 replicates
of the standardized sampling size with the model predictions.
We have used a model to predict the empirical abundance value of interest
(N
o
, where N
o
is a positive number) for the fish and mysid communities.
Normally, we implicitly assume that our predictions will not reproduce our
observations exactly. Uncontrolled randomness from various unknown
sources will make observations deviate from the theoretical model predic-
tions (N). In order to consider this inevitable mismatch, we have used an
error model. We have considered a least-absolute values criterion, which is
known to be robust even when errors in the data are not normally distributed
(
Tarantola, 2006
).
We defined the following error function to model the probability of
observing N
o
absolute abundance of a species, given a model prediction, N:
J
F
¼
:
N
o
N
PN
o
j
ð
Þ
ð
Þ
N
exp
29
N
To satisfy
ð
1
PN
o
j
ð
N
Þ
dN
o
¼
1
;
ð
30
Þ
0
we obtain,
8
<
0
@
1
A
:
N
o
N
exp
N
o
>
N
N
1
0
@
1
A
:
PN
o
j
ð
N
Þ¼
ð
31
Þ
:
N 2
ð
e
1
Þ
N
N
o
exp
N
o
<
N
N
By assuming independent observational errors, it is straightforward to write
a likelihood for the species rank in abundance of the fish (F) and mysid (M)
communities as follows,
