. 2022 Jul 8;9(4):045501. doi: 10.1117/1.JMI.9.4.045501

Table 7.

Alternative parameterization for RMH model parameterization.

(a) Alternative RMH parameters expressed in terms of RMH model parameters.

V

is defined by

V = 1 + b^{- 2} + 2 (σ_{R}^{2} + σ_{T R}^{2})

. Note that

x_{6} = x_{5} + σ_{T C (-)}^{2}

x_{7} = x_{5} + σ_{R C (-)}^{2}

, and

1 - x_{4} = [1 + b^{- 2}] / V

Alternative RMH model parameters

RMH model parameters

b

b

x_{1}

(δ_{1}) / \sqrt{V}

x_{2}

(δ_{2}) / \sqrt{V}

x_{3}

2 σ_{R}^{2} / V

x_{4}

[2 (σ_{R}^{2} + σ_{T R}^{2})] / V

x_{5}

σ_{C (-)}^{2}

x_{6}

σ_{T C (-)}^{2} + σ_{C (-)}^{2}

x_{7}

σ_{R C (-)}^{2} + σ_{C (-)}^{2}

(b) Constraints on RMH alternative parameters. These follow from the equations in part (a), nonnegativity of the RMH variance components and constraints Eqs. (6) and (8).

0 \leq x_{1}; 0 \leq x_{2}; 0 \leq x_{i} \leq 1, i = 3, \dots, 7; x_{4} \geq x_{3}; x_{6} \geq x_{5}; x_{7} \geq x_{5}; x_{6} + x_{7} - x_{5} \leq 1

(23)

(c) RMH parameters expressed in terms of alternative RMH parameters. Note that in terms of the alternative parameterization,

V = (1 + 1 / b^{2}) / (1 - x_{4})

RMH parameter

Alternative RMH parameter

b

b

δ_{1}

x_{1} \sqrt{V}

δ_{2}

x_{2} \sqrt{V}

σ_{R}^{2}

. 5 x_{3} V

σ_{T R}^{2}

. 5 V (x_{4} - x_{3})

σ_{C (-)}^{2}

x_{5}

σ_{T C (-)}^{2}

x_{6} - x_{5}

σ_{R C (-)}^{2}

x_{7} - x_{5}

σ_{ε (-)}^{2}

1 - (x_{6} + x_{7} - x_{5})