# Estimation of Blooming Start with the Adaptation of the Unified Model for Three Apricot Cultivars (Prunus armeniaca L.) Based on Long-Term Observations in Hungary (1994–2020)

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Meteorological and Phenological Data

#### 2.2. The Unified Model

_{crit}and the total amount of chilling units C

_{tot}. The beginning of blooming is set off when F

_{tot}≥ F

_{crit}. Considering Equation (4), it is obvious that F

_{crit}and C

_{tot}are in a negative relationship [56]. This relationship expresses that a higher amount of chilling units means a lower amount of forcing units is needed to set the beginning of blooming off, which was proven experimentally [57,58,59,60,61].

- (1)
- According to Caffarra and Eccel [7], ${\mathit{b}}_{\mathit{c}}$ can be set to 0, so we described the c accumulation in the endodormancy with the following equation:$${{\mathit{C}}^{\prime}}_{*}\left(\mathit{T}\right)={{\displaystyle \sum}}^{\text{\hspace{1em}}}\frac{\mathbf{2}}{\mathbf{1}+{\mathit{e}}^{{\mathit{a}}_{\mathit{c}}{\left(\mathit{T}-{\mathit{c}}_{\mathit{c}}\right)}^{\mathbf{2}}}}.$$
- (2)
- It does not strongly constrain the model accuracy if we assume that the chilling unit accumulation ends at the beginning of ecodormancy (${\mathit{t}}_{\mathit{c}}={\mathit{t}}_{\mathbf{1}}$), [43]. We defined this day as the one when the string stage occurs [66,67]. The observed string stage data were available from the data base of the examined apricot cultivars in the time period 1994–2020. Assuming ${\mathit{t}}_{\mathit{c}}={\mathit{t}}_{\mathbf{1}}$, it follows that ${\mathit{C}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}}={\mathit{C}}_{\mathit{t}\mathit{o}\mathit{t}}$. In our study, ${\mathit{t}}_{\mathbf{2}}$ is the day of the beginning of blooming.

#### 2.3. Parameter Estimation with the Simulated Annealing Method

## 3. Results

- 1.
- First, we observe that for each apricot cultivar, the forcing parameters ${c}_{f}$ and $w$ are strongly related, although not linearly (see the upper panel in Figure 2). Therefore, these two parameters cannot be optimized independently. Using the empirical relationship between ${c}_{f}$ and $w$ obtained from the optimization process, we calculate the optimal values of ${c}_{f}$ for all fixed values of $w$;
- 2.
- Based on the resulted parameter vectors of all walks, we plot the histogram of the optimal parameter values of $w$ (see the lower panel in Figure 2). We see that the optimal parameter values of $w$ are dense mainly around three or four small to high values; more exactly, around one small, two medium, and one high value for ‘Rózsakajszi C.1406’, while around one small, one medium, and one high value for the other two cultivars. We immediately exclude the high values, because we obtained ${C}_{f}$ values in those cases between −10 °C and −30 °C, which are unlikely during the forcing period in Hungary [9,68];
- 3.
- The results of most walks are dense around the small value for each apricot cultivar, and we obtained the lowest RMSE values here, too. So, we fix the parameter value $w$ at the median of the preferred range of ‘small’ optimal parameters $w$: ${w}_{cb}$ = 15.6, ${w}_{gm}\text{}$ = 17.9, ${w}_{ro}$ = 19.4;
- 4.
- As a next step, we narrow the parameter space according to the biologically possible parameter values for Hungary (Table 2) [7,9,65,69,70]. In the original parameter space, we find several similarly good parameter vectors, that fit statistically very well to the observed blooming dates, but they are biologically impossible.

- 5.
- Then, we searched for the global optimum of the parameter space for each apricot cultivar. We define the global optimum parameter vector as the parameter vector with the lowest root-mean-square error (RMSE) among the grid of values of Table 4. It is seen that, in many cases (${a}_{c}^{cb}$, ${c}_{c}^{cb}$, ${b}_{f}^{cb}$, ${a}_{c}^{gm}$, ${c}_{c}^{gm}$, ${b}_{f}^{gm}$, ${a}_{c}^{ro}$, ${c}_{c}^{ro}$, ${b}_{f}^{ro}$, and ${c}_{f}^{ro}$), the global optimal parameter values do not fall in the local optimum bins. This is most surprising for the parameter ${c}_{f}^{ro}$, where more than 70% of the walk limits fall in the local optimal bin, but the global optimum parameter value does not;
- 6.
- Using the global optimum parameter vector, we estimated the blooming date for each apricot cultivar with an average error less than 2.5 days (RMSE < 2.5). For comparison, if we take the mean blooming data calculated over all the years as a constant [64], the average error of the estimation (i.e., the error of the base model) is as high as 9.7–10.6 days, depending on cultivars;
- 7.
- Finally, based on the daily average temperature and the global optimal parameter vectors, we calculated the critical amount of chilling and forcing units, and determined the chilling and forcing process for each cultivar in the period 1994–2020 (Figure 1 and Figure 3). We provide the parameter values that are optimized with the simulated annealing method, applying the unified model and the observed string stage and blooming data of years 1994–2000 (Table 4). The temperature that is optimal for the plant for chilling unit accumulation (${c}_{c}$) is 1.50 °C for ‘Ceglédi bíborkajszi’, 2.13 °C for ‘Gönci magyar kajszi’, and 2.42 °C for ‘Rózsakajszi C.1406’ in the period 1994–2020. According to our calculations, the most chilling units (${C}_{crit}=$ 29.8 units) are necessary for ‘Gönci magyar kajszi’, and the least chilling units (${C}_{crit}=$ 12.7 units) are required by ‘Ceglédi bíborkajszi’ for breaking the endodormancy (Table 4). The inflection point of the forcing unit accumulation (i.e., ${c}_{f}$) is between 8.30 and 9.04 °C, depending on the cultivars. This curve has no maximum point, but the forcing unit accumulation is close to the maximum (1 unit) at 12–15 °C (more than 0.9 units) that could be considered as ‘optimal temperature’ for the plant in their preparation for blooming. The average accumulated forcing units for the blooming are between 14.0 and 16.4 units for each cultivar in the period 1994–2020 (Table 4). Surprisingly, the absolute value of parameter $k$ of our results is larger than is reported in the publications of other researchers (i.e., in between −10
^{−4}and −10^{−8}) [8,9,43,64]. This may lead to a conclusion that, in the case of Hungarian apricots, the chilling unit accumulation has a relatively larger effect on forcing unit accumulation.

## 4. Discussion and Conclusions

^{−7}or more), a simplification can be applied with $k=0$ and $w={F}_{crit}$. But, in view of our research, this simplification is not justified, because the absolute value of parameter $k$ is larger (between −0.0072 and −0.0086) than it is in publications of other researchers (−10

^{−8}–−10

^{−4}) [8,9,43,64]. It can mean that, in the case of Hungarian apricots, the chilling unit accumulation has a larger effect on forcing unit accumulation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The schematic diagram of the chilling (upper panel: (

**a**)) and forcing (lower panel: (

**b**)) unit accumulation depending on the daily mean temperature for ‘Ceglédi bíborkajszi’, ‘Gönci magyar kajszi’, and ‘Rózsakajszi C.1406’.

**Figure 2.**The forcing parameter values of ${c}_{f}$ and $w$ (upper panel: (

**a**)) and the histogram of the optimal forcing parameter values of $w$ (lower panel: (

**b**)) for ‘Ceglédi bíborkajszi’, based on 10,000 random walk optimization processes. (We obtained similar diagrams for ‘Gönci magyar kajszi’ and ‘Rózsakajszi C.1406’).

**Figure 3.**The annual chilling (blue curves) and forcing unit (orange curves) accumulation in the time period September 1994–April 2020 (highlighted: September 2014–April 2015 as a ‘pattern period’: this period has the lowest RMSE compared to the chilling units calculated from the mean daily temperatures of the twenty-six studied periods, i.e., the results of the year that can be considered as the most similar one to the ‘average year’) for ‘Ceglédi bíborkajszi’ (

**a**), ‘Gönci magyar kajszi’ (

**b**), and ‘Rózsakajszi C.1406’ (

**c**).

**Table 1.**The boundaries of the parameter space and the step length

^{1}used for the first estimation.

Parameter | Minimum Value | Maximum Value | Step Length |
---|---|---|---|

${a}_{c}$ | 0 | 10 | 0.01 |

${c}_{c}$ | −50 | 50 | 0.10 |

${c}_{c}$ | −10 | 0 | 0.01 |

${b}_{f}$ | −30 | 30 | 0.10 |

${c}_{f}$ | 0 | 200 | 0.10 |

$w$ | 2 | 9 | 0.01 |

^{1}These are not fixed step lengths: the $\sigma $ parameter of the Gaussian distribution was used to select the step length.

Parameter | Minimum Value | Maximum Value | Step Length |
---|---|---|---|

${a}_{c}$ | 0.2 | 1.0 | 0.001 |

${c}_{c}$ | 1 | 5 | 0.005 |

${b}_{f}$ | −0.9 | −0.1 | 0.001 |

${c}_{f}$ | 6 | 14 | 0.010 |

${k}_{exp}$ | 2 | 6 | 0.005 |

**Table 3.**Characterization of the local optimum parameter vectors of the unified model: the minimum and maximum values of the bins with significantly higher number of optimized limit values (optimum bins), the number of the limits in them, the median and the standard deviation, together with the maximum, the median, and standard deviation of the RMSE. The values belonging to the cultivars ‘Ceglédi bíborkajszi’, ‘Gönci magyar kajszi’, and ‘Rózsakajszi C.1406’ are denoted by the superscripts ‘cb’, ‘gm’, and ‘ro’ respectively.

${\mathit{a}}_{\mathit{c}}^{\mathit{c}\mathit{b}}$ | ${\mathit{c}}_{\mathit{c}}^{\mathit{c}\mathit{b}}$ | ${\mathit{b}}_{\mathit{f}}^{\mathit{c}\mathit{b}}$ | ${\mathit{c}}_{\mathit{f}}^{\mathit{c}\mathit{b}}$ | ${\mathit{k}}_{\mathit{e}\mathit{x}\mathit{p}}^{\mathit{c}\mathit{b}}$ | ${\mathit{a}}_{\mathit{c}}^{\mathit{g}\mathit{m}}$ | ${\mathit{c}}_{\mathit{c}}^{\mathit{g}\mathit{m}}$ | ${\mathit{b}}_{\mathit{f}}^{\mathit{g}\mathit{m}}$ | ${\mathit{c}}_{\mathit{f}}^{\mathit{g}\mathit{m}}$ | ${\mathit{k}}_{\mathit{e}\mathit{x}\mathit{p}}^{\mathit{g}\mathit{m}}$ | ${\mathit{a}}_{\mathit{c}}^{\mathit{r}\mathit{o}}$ | ${\mathit{c}}_{\mathit{c}}^{\mathit{r}\mathit{o}}$ | ${\mathit{b}}_{\mathit{f}}^{\mathit{r}\mathit{o}}$ | ${\mathit{c}}_{\mathit{f}}^{\mathit{r}\mathit{o}}$ | ${\mathit{k}}_{\mathit{e}\mathit{x}\mathit{p}}^{\mathit{r}\mathit{o}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Minimum | 0.568 | 1.400 | −0.436 | 7.600 | 2.080 | 0.248 | 3.160 | −0.420 | 7.280 | 2.040 | 0.312 | 2.640 | −0.556 | 7.360 | 2.040 |

Maximum | 0.576 | 1.440 | −0.428 | 9.280 | 2.480 | 0.256 | 3.200 | −0.380 | 9.040 | 2.400 | 0.320 | 2.720 | −0.484 | 8.560 | 2.600 |

No. of items | 128 | 129 | 156 | 7480 | 1592 | 143 | 138 | 735 | 7950 | 1680 | 132 | 270 | 1318 | 7180 | 2210 |

Median | 0.572 | 1.420 | −0.432 | 8.210 | 2.260 | 0.252 | 3.180 | −0.400 | 7.880 | 2.220 | 0.316 | 2.680 | −0.520 | 7.710 | 2.310 |

St. deviation | 0.002 | 0.010 | 0.002 | 0.420 | 0.110 | 0.002 | 0.010 | 0.011 | 0.430 | 0.100 | 0.002 | 0.020 | 0.021 | 0.310 | 0.160 |

Maximum RMSE | 4.40 | 4.17 | 3.40 | 4.60 | 4.39 | 5.04 | 4.66 | 4.58 | 7.18 | 4.68 | 4.78 | 5.63 | 4.62 | 5.78 | 5.05 |

Median RMSE | 2.92 | 2.94 | 2.84 | 2.86 | 2.66 | 2.82 | 2.81 | 2.79 | 2.81 | 2.49 | 2.47 | 2.38 | 2.41 | 2.41 | 2.15 |

St. dev. RMSE | 0.33 | 0.33 | 0.18 | 0.21 | 0.31 | 0.49 | 0.43 | 0.27 | 0.25 | 0.33 | 0.57 | 0.63 | 0.31 | 0.31 | 0.50 |

**Table 4.**The values of the global optimum parameter vectors of the unified model and the corresponding RMSE values for ‘Ceglédi bíborkajszi’, ‘Gönci magyar kajszi’, and ‘Rózsakajszi C.1406’.

Parameter | ‘Ceglédi bíborkajszi’ | ‘Gönci magyar kajszi’ | ‘Rózsakajszi C.1406’ |
---|---|---|---|

${a}_{c}$ | 0.949 | 0.216 | 0.608 |

${c}_{c}$ | 1.50 | 2.13 | 2.42 |

${b}_{f}$ | −0.626 | −0.443 | −0.365 |

${c}_{f}$ | 8.30 | 9.04 | 8.84 |

$w$ | 15.60 | 17.90 | 19.40 |

${k}_{exp}\left(k\right)$ | 2.14 (−0.0072) | 2.08 (−0.0083) | 2.07 (−0.0086) |

${t}_{c}={t}_{1}$ | 14th of January | 22nd of January | 30th of January |

${t}_{2}$ | 27th of March | 29th of March | 1st of April |

${C}_{crit}={C}_{tot}$ | 12.73 | 29.78 | 19.69 |

${F}_{crit}$ | 14.24 | 13.99 | 16.41 |

${F}_{tot}$ | 14.57 | 14.31 | 16.82 |

RMSE | 2.37 | 2.10 | 1.49 |

**Table 5.**The mean, standard deviation (StDev), range, lower and upper 95% confidence limits (LCI, UCI) of the observed and estimated blooming start (BM), and the parameters of the unified model ${F}_{crit}$, ${C}_{crit}={C}_{tot}$, and ${F}_{tot}$, together with their slope in the measured 26 years (1994–2020) with their corresponding significance levels ($p$)of the varieties ‘Ceglédi bíborkajszi’, ‘Gönci magyar kajszi’, and ‘Rózsakajszi C.1406’.

Observed BM | Estimated BM | ${\mathit{F}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}}\text{}$ | ${\mathit{C}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}}={\mathit{C}}_{\mathit{t}\mathit{o}\mathit{t}}\text{}$ | ${\mathit{F}}_{\mathit{t}\mathit{o}\mathit{t}}\text{}$ | ||
---|---|---|---|---|---|---|

Ceglédi bíborkajszi | Mean | 207.9 | 208.4 | 14.2 | 12.7 | 14.6 |

StDev | 10.8 | 12.1 | 0.4 | 3.8 | 0.5 | |

Range | 48.0 | 51.0 | 1.5 | 14.8 | 1.7 | |

LCI | 203.8 | 203.7 | 14.1 | 11.3 | 14.4 | |

UCI | 212.1 | 213.1 | 14.4 | 14.2 | 14.7 | |

Slope | −0.224 | −0.219 | 0.004 | −0.041 | −0.001 | |

p | 0.438 | 0.500 | 0.696 | 0.686 | 0.933 | |

Gönci magyar kajszi | Mean | 210.3 | 210.2 | 14.0 | 29.8 | 14.3 |

StDev | 10.6 | 11.2 | 0.8 | 6.5 | 0.8 | |

Range | 46.0 | 47.0 | 2.6 | 22.3 | 2.8 | |

LCI | 206.2 | 205.9 | 13.7 | 27.3 | 14.0 | |

UCI | 214.3 | 214.5 | 14.3 | 32.3 | 14.6 | |

Slope | −0.283 | −0.213 | 0.001 | −0.005 | −0.003 | |

p | 0.316 | 0.479 | 0.978 | 0.977 | 0.881 | |

Rózsakajszi C.1406 | Mean | 212.9 | 212.8 | 16.4 | 19.7 | 16.8 |

StDev | 10.0 | 10.2 | 0.7 | 4.7 | 0.7 | |

Range | 41.0 | 40.0 | 2.6 | 18.9 | 3.2 | |

LCI | 209.0 | 208.9 | 16.2 | 17.9 | 16.5 | |

UCI | 216.7 | 216.7 | 16.7 | 21.5 | 17.1 | |

Slope | −0.305 | −0.257 | −0.001 | 0.012 | 0.003 | |

p | 0.252 | 0.344 | 0.933 | 0.925 | 0.880 |

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**MDPI and ACS Style**

Mesterházy, I.; Raffai, P.; Szalay, L.; Bozó, L.; Ladányi, M.
Estimation of Blooming Start with the Adaptation of the Unified Model for Three Apricot Cultivars (*Prunus armeniaca* L.) Based on Long-Term Observations in Hungary (1994–2020). *Diversity* **2022**, *14*, 560.
https://doi.org/10.3390/d14070560

**AMA Style**

Mesterházy I, Raffai P, Szalay L, Bozó L, Ladányi M.
Estimation of Blooming Start with the Adaptation of the Unified Model for Three Apricot Cultivars (*Prunus armeniaca* L.) Based on Long-Term Observations in Hungary (1994–2020). *Diversity*. 2022; 14(7):560.
https://doi.org/10.3390/d14070560

**Chicago/Turabian Style**

Mesterházy, Ildikó, Péter Raffai, László Szalay, László Bozó, and Márta Ladányi.
2022. "Estimation of Blooming Start with the Adaptation of the Unified Model for Three Apricot Cultivars (*Prunus armeniaca* L.) Based on Long-Term Observations in Hungary (1994–2020)" *Diversity* 14, no. 7: 560.
https://doi.org/10.3390/d14070560