Table 2.
Notation convention for curvature models in geometrical space. We include the biphasic and polynomial characterizations of the generalized allometric model of (3).GS stands for geometrical space, AS stands for arithmetical space, and CF means correction factor of retransformation bias of the regression error.
| Biphasic model | Eq. | Polynomial model | Eq. | |
|---|---|---|---|---|
| Basic form | w = βB(a)aαB(a) | (B.11) | w = βP(a)aαP(a) | (B.27) |
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| Regression equation | v = vB(a, u) + ϵ | (8), (D.4) | v = vP(a, u) + ϵ | (B.32), (D.8) |
|
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| Mean response GS | E B(v | u) = ∑12 ϑi(u(a))fi(u(a)) | (B.18) | E P(v | u) = ∑0mpku(a)k | (B.33) |
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| Back transformation AS | w = βB(a)aαB(a)eϵ | (B.21) | w = βP(a)aαP(a)eϵ | (B.34) |
|
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| Mean response AS |
E
Bδ(w | a) = βB(a)aαB(a)δ(ϵ) δ(ϵ) = E(eϵ) |
(B.22) |
E
P(w | a) = βP(a)aαP(a)δ(ϵ) δ(ϵ) = E(eϵ) |
(B.35) |
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| ||||
| CF Baskerville | E BB(w | a) = βB(a)aαB(a) δB(ϵ) | (B.22), (C.1) | EPB(w | a) = βP(a)aαP(a)δB(ϵ) | (B.35), (C.1) |
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| ||||
| CF Duan | E BD(w | a) = βB(a)aαB(a)δD(ϵ) | (B.22), (C.2) | EPD(w | a) = βP(a)aαP(a)δD(ϵ) | (B.35), (C.2) |
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| CF Zeng and Tang | E BZT(w | a) = βB(a)aαB(a)δZT(ϵ) | (B.22), (C.3) | EPZT(w | a) = βP(a)aαP(a)δZT(ϵ) | (B.35), (C.3) |
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| CF n-partial sum | E Bn(w | a) = βB(a)aαB(a)δn(ϵ) | (B.22), (C.4) | E Pn(w | a) = βP(a)aαP(a)δn(ϵ) | (B.35), (C.4) |