- Dynamic Modeling 109
estimatesθˆiand ˆσi^2 must conform to the following two equations:
∂lnL(θi,σi^2 )
∂θi∣∣
∣∣
θi=θˆi=0(9.10)
∂lnL(θi,σ^2 i)
∂σi^2∣
∣∣
∣
σ^2 i=ˆσ^2 i=0After some computational deduction, the estimated parametersθˆiand ˆσ^2 i
are given as
θˆi=(ΦTiΦi)−^1 ΦTiXi (9.11)ˆσ^2 i=1
L− 1L∑− 1l=1(xi(l+1)−φi(l)·θˆi)^2=1
L− 1
(Xi−Φiθˆi)T(Xi−Φiθˆi) (9.12)In this manner, given the transcriptomics data, the parameters in the discrete
dynamic model can be identified based on Eq. (9.11) Simply speaking,
regardless of all the mathematic deduction, we can use the transcriptomics
data to build the matrices ΦiandXiin (9.6) according to Eq. (9.5). Then,
with the formula in (9.11), simple matrix manipulation can be employed to
have the estimated parameter vectorθˆi, from which all the model parameters
can be identified.
3.4. Model selection
In the discrete dynamic model describing the temporal expression of gene,
the parameter aij denotes the regulatory ability of the j-th regulatory
gene for the target gene i, which implies the regulatory relationships
between the genes. Since the objective is to identify the mechanism of
regulatory relationships of the selected genes of interest, the significance
of the regulatory abilities should be determined for the identification of
significant regulations in the gene regulatory network. Therefore, Akaike
information criterion (AIC)6,8and Student’st-test^3 are then employed for
model order selection and to determine the significance of the regulatory
relationships. AIC, which includes both estimated residual error and model
complexity in one statistics, quantifies the relative goodness of fit of a model.
For a discrete dynamic model withNiparameters (or regulatory genes) to