Tree Inference: Response Time and Other Measures in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters
Abstract
:1. Introduction
1.1. Selective Influence
1.2. Vertex Arrangements
2. Results
2.1. Representation
= rh [p(i, j)t(i, j) − p(i, n)t(i, n)] − ri [ p(h, j)t(h, j) − p(h, n)t(h, n)]
= −rh{[1 − p(i, j)]tw(i, j) − [1 − p(i, n)]tw(i, n)]}
+ ri{[1 − p(h, j)]tw(h, j) − [1 − p(h, n)]tw(h, n)]}.
= rh[pA(i)pD − pA(i)pD] = 0.
= ri[p(h, j) − k] − ri[p(h, n) − k]
= ri[p(h, j) − p(h, n)].
= pA(i)pD + pB(i)pF(j)
= pA(i)pD[μ*Δ(i) + 0] + pB(i)pF(j)[μ*B(i) + μ*F(j)].
2.2. Uniqueness of Parameters
2.2.1. Numerical Example
2.2.2. Admissible Transformations
= pA(i)pD [tA(i) + tD] + pB(i)pF(j) [tB(i) + tF(j)] − pA(i)pD [tA(i) + tD]
− pB(i)pF(j′) [tB(i) + tF(j′)]
= pB(i)pF(j) [tB(i) + tF(j)] − pB(i)pF(j′) [tB(i) + tF(j′)].
= p*B(i)p*F(j) [t*B(i) + t*F(j)] − p*B(i)p*F(j′) [t*B(i) + t*F(j′)].
= p*B(i)p*F(j) [t*B(i) + t*F(j)] − p*B(i)p*F(j′) [t*B(i) + t*F(j′)].
= cpB (i)[pF(j)/c + (c − 1) pD/c][t*B(i) + t*F(j)]
− cpB (i)[pF(j′)/c + (c − 1) pD/c][t*B(i) + t*F (j′)]
= pF(j)t*B(i) + pF(j)t*F(j) + (c − 1) pD[t*F(j) − t*F(j′)]
− pF(j′)t*B(i) − pF(j′)t*F(j′)
= pF(j)t*F(j) + (c − 1)pD[t*F(j) − t*F(j′)] − pF(j′)t*F(j′)
− pF(j)tF(j) + pF(j′)tF(j′)
p(i, j)t(i, j) = p*A(i)p*D [t*A(i) + t*D] + p*B(i)p*F(j) [t*B(i) + t*F(j)].
2.2.3. Remarks on Nonnegative Measure Values
2.3. Degrees of Freedom
3. Discussion
Author Contributions
Funding
Conflicts of Interest
Appendix A
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Old | New 1 | |
---|---|---|
pA(i) | 0.50 | 0.20 |
pB(i) | 0.50 | 0.80 |
pD | 0.40 | 0.40 |
pF(j) | 0.16 | 0.25 |
tA(i) | 4.50 | 7.50 |
tD | 4.00 | 7.00 |
tB(i) | 2.00 | 3.00 |
tF(j) | 5.00 | 2.50 |
Standard Binary Tree for Ordered Processes 1 |
---|
p*B(i) = cpB(i) p*D = pD p*F(j) = pF(j)/c + (c − 1) pD/c If p*B(i) ≠ 0, t*B(i) = tB(i) + f If p*F(j) ≠ 0, |
If p*D ≠ 0, t*D = tD + e If p*A(i) ≠ 0, |
If p*C ≠ 0, t*C = tC + k If p*E(J) ≠ 0, |
0 < c ≤ 1/max{pB(i)} |
pD − min{pF(j)} ≤ pD* c |
max{pF(j)} − pD ≤ (1 − pD)* c |
} |
} |
} |
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Schweickert, R.; Zheng, X. Tree Inference: Response Time and Other Measures in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters. Mathematics 2022, 10, 267. https://doi.org/10.3390/math10020267
Schweickert R, Zheng X. Tree Inference: Response Time and Other Measures in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters. Mathematics. 2022; 10(2):267. https://doi.org/10.3390/math10020267
Chicago/Turabian StyleSchweickert, Richard, and Xiaofang Zheng. 2022. "Tree Inference: Response Time and Other Measures in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters" Mathematics 10, no. 2: 267. https://doi.org/10.3390/math10020267
APA StyleSchweickert, R., & Zheng, X. (2022). Tree Inference: Response Time and Other Measures in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters. Mathematics, 10(2), 267. https://doi.org/10.3390/math10020267