Markov chain

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(Markoff), Andrei, Russian mathematician, 1865-1922.
Markov chain - number of steps or events in sequence.
Markov chaining - a theory used in psychiatry.
Markov process - a process such that the conditional probability distribution for the state at any future instant, given the present state, is unaffected by any additional knowledge of the past history of the system.
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At this moment, the time step for weighted Kappa is decided according to the steady state nature of the transition probability matrix. If transition probability matrix approaches their steady state at s time point, then, we calculate weighted Kappa from 1 to s steps accordingly.
We normalize the transition probability matrix obtained after abstraction to meet the requirements of the Markov chain model.
The probability can be obtained by multiplying the matrix P by itself, resulting transition probability matrix [P.sup.2].
The system's future state will be determined by investigating state transition probability matrix, and gray change interval of relative prediction value in the future moments also will be determined; it can be expressed as [[[cross product].sub.1i], [[cross product].sub.2i]].
(i) [T.sub.i] = ([Q.sup.(I).sub.i], [[GAMMA].sup.(I)]), with 1 [less than or equal to] i [less than or equal to] 5, characterized by a transition probability matrix [Q.sup.(I).sub.i] derived from [Q.sup.(I)] by multiplying the terms of the two pseudodiagonals by a coefficient [[alpha].sub.i] [member of] {[10.sup.4], [10.sup.2], [10.sup.0], [10.sup.-2], [10.sup.-4], [10.sup.-6]}.
Consequently, the general form of the transition probability matrix defined for a deteriorating element is
The model parameters mainly consist of two parts: weights of driving factors and land use transition probability matrix. Driving factors here mainly refer to static factors influencing land use change.
Therefore, the transition probability matrix P can be obtained by the foregoing expected-value method.
Transition probability matrix of the textural types in the 0-100 cm subinterval in the study area Upper layer Lower layer Sand Loamy Sandy Silt Loam sand loam loam Sand 0 0.30 0.25 0.00 0.05 Loamy sand 0.56 0 0.44 0.00 0.00 Sandy loam 0.15 0.31 0 0.00 0.38 Silt loam 0.00 0.00 1.00 0 0.00 Loam 0.09 0.09 0.00 0.00 0 Clay loam 0.04 0.07 0.07 0.04 0.11 Silty clay loam 0.00 0.10 0.00 0.20 0.00 Silty clay 0.00 0.00 0.13 0.00 0.13 Clay 0.00 0.00 0.00 0.00 0.00 Upper layer Clay Silty clay Silty Clay loam loam clay Sand 0.00 0.10 0.20 0.10 Loamy sand 0.00 0.00 0.00 0.00 Sandy loam 0.15 0.00 0.00 0.00 Silt loam 0.00 0.00 0.00 0.00 Loam 0.64 0.09 0.09 0.00 Clay loam 0 0.48 0.15 0.04 Silty clay loam 0.30 0 0.40 0.00 Silty clay 0.13 0.50 0 0.13 Clay 0.00 0.00 1.00 0 Table 3.
Then, we note by P the transition probability matrix (the element of row i column/represents the probability P^).
{[r.sub.t], t [greater than or equal to] 0} is a homogeneous, finite-state Markovian process with right continuous trajectories taking values in a finite set S = {1,2, 3, ..., N}, with the mode transition probability matrix being

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