A Practical Guide to Cancer Systems Biology

108 A Practical Guide to Cancer Systems Biology

l∈{ 1 , ...,L− 1 }. Hence, Eq. (9.5) can be written in the following matrix
form for target genei:

Xi=Φi·θi+Ei (9.6)

where

Xi=

⎡

⎢⎢

⎣

xi(2)

..

.

xi(L)

⎤

⎥⎥

⎦, Φi=

⎡

⎢⎢

⎣

φi(1)

..

.

φi(L−1)

⎤

⎥⎥

⎦,Ei=

⎡

⎢⎢

⎣

εi(1)

..

.

εi(L−1)

⎤

⎥⎥

⎦

In Eq. (9.6), we assume noisesεi(l) at different time points as independent
random variables of normal distribution with zero mean and unknown
variance σi^2 , i.e., the variance of εi is Σi = E{εiεTi} = σ^2 iI,whereI is
an identity matrix. Ifεiis assumed to be normally distributed withL− 1
elements, its probability density function is of the following form^6 :

p(εi)=

(

(2π)L−^1 det Σi

)− 1 / 2

exp

{

−

1

2

εTiΣ−i^1 εi

}

(9.7)

Considering Eqs. (9.6) and (9.7), the likelihood function can be expressed as

L(θi,σ^2 i)=p(θi,σ^2 i)

=(2πσ^2 i)−(L−1)/^2 exp

{

−

1

2 σ^2 i

(Xi−Φiθi)T(Xi−Φiθi)

}

(9.8)

Maximum likelihood estimation method aims at findingθiandσi^2 to maxi-
mize the likelihood function in Eq. (9.8). For the simplicity of computation,
it is practical to take the logarithm of the likelihood function, and we have
the following log-likelihood function:

lnL(θi,σ^2 i)=−

L− 1

2

ln(2πσ^2 i)−

1

2 σ^2 i

L∑− 1

l=1

(xi(l+1)−φi(l)·θi)^2 (9.9)

wherexi(l+1) andφi(l)arethel-th elements ofXiand Φi, respectively. Here,
the log-likelihood function is expected to have the maximum atθi=θˆiand
σ^2 i=ˆσ^2 i. The necessary conditions for determining the maximum likelihood