Country # of positive 7-day stretches Training Testing Thailand 95 12 Malaysia 78 8 Philippines 85 9 Indonesia 83 7 Cambodia 88 7 Table 3:

State transition matrix (25 cells).

Let A denote the [(L + 1).sup.K] x [(L + 1).sup.K]

state transition matrix of MC.

However, the control update depends on the computation of the

state transition matrix. Both the eigenvalue-eigenvector method in AMPC and the Taylor/Pade series expansion method in general MPC for computing the matrix exponential of a large-scale system are time consuming.

[State.sub.1] [State.sub.2] [State.sub.3] 0.250 0.750 0.000 [State.sub.4] [State.sub.5] 0.000 0.000 TABLE 2: The

state transition matrix of DDoS_HMM.

Let C be the n-cell 60/102 NBCA whose

state transition matrix T is as the following:

By constantly checking the

state transition matrix, we can find out whether the system will reach a steady state.

If there is a positive integer n such that all the entries in the

state transition matrix [P.sup.n] are positive, the Markov chain is said to be regular.

The

state transition matrix will be populated by calculating the probabilities for each state's likelihood to transition to any other state in the model.

Since [PSI](t, [t.sub.0]) ([t.sub.0] [greater than or equal to] 0) is the

state transition matrix of (4), [mathematical expression not reproducible].

Lemma 6 gives the

state transition matrix of AMC after n transitions.

where the

state transition matrix [mathematical expression not reproducible] with [mathematical expression not reproducible] ([T.sub.S] is time interval between adjacent positioning points).

After determining the second layer hidden states of the training set TD, the next step is compute the other three basic parameters of HMM in the condition of the second layer hidden states, including the initial probability matrix [pi],

state transition matrix A and confusion matrix B.