Biomedical Engineering Reference
In-Depth Information
5.1 Functional Enviromics Algorithm
Among the whole set of elementary modes, the subset that is tightly linked to the
envirome can be effectively determined by regression analysis of flux data,
R
¼
b
fg
, against envirome data, X
¼
c
i
;
j
, satisfying the following criteria:
(a) Maximize the captured variance of envirome data X
¼
c
i
;
j
and of flux data
R
¼
b
fg
.
(b) Maximize the correlation between elementary mode weighting factors and
envirome variables.
(c) Minimize the number of elementary modes required to capture a given vari-
ance of R
¼
b
fg
and X
¼
c
i
;
j
, i.e., minimize redundancy.
These criteria can be fulfilled by maximizing the covariance between envirome
data, X
¼
c
i
;
j
, and respective measured flux data, R
¼
b
fg
, according to the
formula
Maximize
I
cov X
;
R
ð
Þ
ð
13
Þ
R
¼
K
EM
T
K
¼
X
I
T
s
:
t
:
with
EM
¼
e
fg
a
q
K
matrix
of
K
elementary
cellular
functions,
e
i
dim e
ðÞ¼
q
½
, K
¼
k
fg
a M
K matrix of weight vectors k
i
of elementary
modes dim k
ðÞ¼½
, and I
¼
I
i
;
j
a K
N matrix of intensity parameters,
which are the degrees in Eq. (
13
). Several methods can be used to solve Eq. (
6
).
One efficient method consists in one-by-one decomposition of elementary modes
according to Eqs. (
14
-
16
)
X
¼
T
W
T
þ
EF
X
ð
14
Þ
R
¼
K
EM
T
þ
EF
R
ð
15
Þ
K
¼
T
B
T
þ
EF
K
ð
16
Þ
with EF
i
residuals matrices that are minimized, W a matrix of loading coefficients,
and B a matrix of regression coefficients. Finally, the intensity matrix I is given by
I
¼
B
W
T
ð
17
Þ
The result of this procedure is the discrimination of a minimal set of elementary
modes that are tightly linked with medium composition. The information can
finally be organized into an N 9 K data array, called a functional enviromics map:
Functional enviromics map
¼
I
T
¼
I
j
;
i
; j
¼
1
;
...
;
N; i
¼
1
;
...
;
K
ð
18
Þ
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