Bertalanffy-Richards \begin{equation}y_{ij}=\beta_0{\small\{1-exp\left(-\beta_1x_{ij}\right)\small\}}^{\beta_2}+\varepsilon_{ij}\end{equation} |
\begin{equation}\beta_2=F\end{equation}
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\begin{equation}F_0=\frac{ln\left(\frac{y_{i1}}{\beta_0}\right)}{\ln(1-\exp{\left(-\beta_1x_{i1}\right)})}\end{equation}
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\begin{equation}y_{ij}=\beta_0\left(\frac{y_{i1}}{\beta_0}\right)^\frac{ln\left(1-\exp{\left(-\beta_1x_{ij}\right)}\right)}{ln\left(1-\exp{\left(-\beta_1x_{i1}\right)}\right)}+e_{ij}\end{equation}
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M1 |
Fixed-effects with no rainfall effect |
Lundqvist-Korf \begin{equation}y_{ij}=\beta_0exp\left(-\beta_1{x_{ij}}^{-\beta_2}\right)+\varepsilon_{ij}\end{equation} |
\begin{equation}beta_1=F\end{equation}
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\begin{equation}F_0=-ln\left(\frac{y_{i1}}{\beta_0}\right){x_{i1}}^{\beta_2}\end{equation}
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\begin{equation}y_{ij}=\beta_0\left(\frac{y_{i1}}{\beta_0}\right)^{\left(\frac{x_{i1}}{x_{ij}}\right)^{\beta_2}}+\varepsilon_{ij}\end{equation}
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M2 |
McDill-Amateis \begin{equation}y_{ij}=\frac{\beta_0}{1+\beta_2/{x_{ij}}^{\beta_1}}+\varepsilon_{ij}\end{equation} |
\begin{equation}\beta_2=F\end{equation}
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\begin{equation}F_0=\left(\frac{\beta_0}{y_{i1}}-1\right){x_{i1}}^{\beta_1}\end{equation}
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\begin{equation}y_{ij}=\frac{\beta_0}{1+\left(\frac{\beta_0}{y_{i1}}-1\right)\left(\frac{x_{i1}}{x_{ij}}\right)^{\beta_1}}+e_{ij}\end{equation}
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M3 |
Hossfeld I \begin{equation}y_{ij}=\frac{x_{ij}^2}{\beta_0+\beta_1{x_{ij}+\beta}_2x_{ij}^2}+\varepsilon_{ij}\end{equation} |
\begin{equation}beta_1=F\end{equation}
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\begin{equation}F_0=\frac{x_{i1}}{y_{i1}}-\frac{\beta_0}{x_{i1}}{-\beta}_2x_{i1}\end{equation}
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\begin{equation}y_{ij}=\frac{x_{ij}^2}{\beta_0+x_{ij}(\frac{x_{i1}}{y_{i1}}-\frac{\beta_0}{x_{i1}}{-\beta}_2\left(x_{ij}-x_{i1})\right)}+e_{ij}\end{equation}
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M4 |
Bertalanffy-Richards \begin{equation}y_{ij}=\beta_0\small\{{1-exp\left(-\beta_1x_{ij}\right)}\small\}^{\beta_2}+\varepsilon_{ij}\end{equation} |
\begin{equation}\beta_2=F\end{equation}
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\begin{equation}F_0=\frac{ln\left(\frac{y_{i1}}{\beta_0}\right)}{\ln(1-\exp{\left(-\beta_1x_{i1}\right)})}\end{equation}
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\begin{equation}y_{ij}=\beta_0^\ast\left(\frac{y_{i1}}{\beta_0}\right)^\frac{ln\left(1-exp{\left(-\beta_1^\ast x_{ij}\right)}\right)}{ln\left(1-exp{\left(-\beta_1^\ast x_{i1}\right)}\right)}+e_{ij}\end{equation}
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M5 |
Fixed-effects with rainfall effect |
Lundqvist-Korf \begin{equation}y_{ij}=\beta_0exp\left(-\beta_1{x_{ij}}^{-\beta_2}\right)+\varepsilon_{ij}\end{equation} |
\begin{equation}beta_1=F\end{equation}
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\begin{equation}F_0=-ln\left(\frac{y_{i1}}{\beta_0}\right){x_{i1}}^{\beta_2}\end{equation}
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\begin{equation}y_{ij}=\beta_0^\ast\left(\frac{y_{i1}}{\beta_0^\ast}\right)^{\left(\frac{x_{i1}}{x_{ij}}\right)^{\beta_2^\ast}}+e_{ij}\end{equation}
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M6 |
McDill-Amateis \begin{equation}y_{ij}=\frac{\beta_0}{1+\beta_2/{x_{ij}}^{\beta_1}}+\varepsilon_{ij}\end{equation} |
\begin{equation}\beta_2=F\end{equation}
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\begin{equation}F_0=\left(\frac{\beta_0}{y_{i1}}-1\right){x_{i1}}^{\beta_1}\end{equation}
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\begin{equation}y_{ij}=\frac{\beta_0^\ast}{1+\left(\frac{\beta_0^\ast}{y_{i1}}-1\right)\left(\frac{x_{i1}}{x_{ij}}\right)^{\beta_1^\ast}}+e_{ij}\end{equation}
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M7 |
Hossfeld I \begin{equation}y_{ij}=\frac{x_{ij}^2}{\beta_0+\beta_1{x_{ij}+\beta}_2x_{ij}^2}+\varepsilon_{ij}\end{equation} |
\begin{equation}beta_1=F\end{equation}
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\begin{equation}F_0=\frac{x_{i1}}{y_{i1}}-\frac{\beta_0}{x_{i1}}{-\beta}_2x_{i1}\end{equation}
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\begin{equation}y_{ij}=\frac{x_{ij}^2}{\beta_0^\ast+x_{ij}(\frac{x_{i1}}{y_{i1}}-\frac{\beta_0^\ast}{x_{i1}}-\beta_1^\ast\left(x_{ij}-x_{i1})\right)}+e_{ij}\end{equation}
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M8 |
Bertalanffy-Richards \begin{equation}y_{ij}=\beta_0\small\{{1-exp\left(-\beta_1x_{ij}\right)}\small\}^{\beta_2}+\varepsilon_{ij}\end{equation} |
\begin{equation}\beta_2=F\end{equation}
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\begin{equation}F_0=\frac{ln\left(\frac{y_{i1}}{\beta_0}\right)}{\ln(1-\exp{\left(-\beta_1x_{i1}\right)})}\end{equation}
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\begin{equation}y_{ij}=\beta_0^u\left(\frac{y_{i1}}{\beta_0^u}\right)^\frac{ln\left(1-\exp{\left(-\beta_1^ux_{ij}\right)}\right)}{ln\left(1-\exp{\left(-\beta_1^ux_{i1}\right)}\right)}+e_{ij}\end{equation}
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M9 |
Mixed-effects with rainfall effect |
Lundqvist-Korf \begin{equation}y_{ij}=\beta_0exp\left(-\beta_1{x_{ij}}^{-\beta_2}\right)+\varepsilon_{ij}\end{equation} |
\begin{equation}beta_1=F\end{equation}
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\begin{equation}F_0=-ln\left(\frac{y_{i1}}{\beta_0}\right){x_{i1}}^{\beta_2}\end{equation}
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\begin{equation}y_{ij}=\beta_{0i}^u\left(\frac{y_{i1}}{\beta_{0i}^u}\right)^{\left(\frac{x_{i1}}{x_{ij}}\right)^{\beta_2^u}}+e_{ij}\end{equation}
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M10 |
McDill-Amateis \begin{equation}y_{ij}=\frac{\beta_0}{1+\beta_2/{x_{ij}}^{\beta_1}}+\varepsilon_{ij}\end{equation} |
\begin{equation}\beta_2=F\end{equation}
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\begin{equation}F_0=\left(\frac{\beta_0}{y_{i1}}-1\right){x_{i1}}^{\beta_1}\end{equation}
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\begin{equation}y_{ij}=\frac{\beta_0^u}{1+\left(\frac{\beta_0^u}{y_{i1}}-1\right)\left(\frac{x_{i1}}{x_{ij}}\right)^{\beta_1^u}}+e_{ij}\end{equation}
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M11 |
Hossfeld I \begin{equation}y_{ij}=\frac{x_{ij}^2}{\beta_0+\beta_1{x_{ij}+\beta}_2x_{ij}^2}+\varepsilon_{ij}\end{equation} |
\begin{equation}beta_1=F\end{equation}
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\begin{equation}F_0=\frac{x_{i1}}{y_{i1}}-\frac{\beta_0}{x_{i1}}{-\beta}_2x_{i1}\end{equation}
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\begin{equation}y_{ij}=\frac{x_{ij}^2}{\beta_0^u+x_{ij}(\frac{x_{i1}}{y_{i1}}-\frac{\beta_0^u}{x_{i1}}-\beta_1^u\left(x_{ij}-x_{i1})\right)}+e_{ij}\end{equation}
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M12 |