# Using Bayesian growth models to predict grape yield

**Received :**20 December 2019;

**Accepted :**12 May 2020;

**Published :**9 July 2020

## Abstract

**Background and aims:** Seasonal differences in vine yield need to be managed to ensure appropriate fruit composition at harvest. Differences in yield are the result of changes in vine management (e.g., the number of nodes retained after harvest) and weather conditions (in particular, temperature) at key vine development stages. Early yield prediction enables growers to manage vines to achieve target yields and prepare the required infrastructure for the harvest.**Methods and results:** Bunch mass data was collected during the 2016/17, 2017/18 and 2018/19 seasons from a commercial vineyard on the Wairau Plains, Marlborough, New Zealand (41^{o}2’23”S; 173^{o}51’15”E). A Bayesian growth model, assuming a double sigmoidal curve, was used to predict the yield at the end of each season. The accuracy of the prediction was investigated using the Monte-Carlo simulation for yield prediction at various growth stages assuming different prior information.**Conclusion:** The results show that the model is sensitive to prior assumption and that having a non-informative prior may be more beneficial than having an informative prior based on one unusual year.

## Materials and methods

Data on grape bunch masses for the 2016/17, 2017/18, and 2018/19 seasons were collected from a commercial vineyard on the Wairau Plains (41^{o}2’23”S; 173^{o}51’15”E). For the 2016/17 and 2017/18 growing seasons, fifteen replicate plots were established in a single vineyard row. Each plot consisted of four vines planted 1.8 m apart. A shoot with two bunches was randomly selected from each plot on approximately a weekly basis, starting at flowering and continuing until shortly before the commercial harvest. In the 2018/19 the number of plots was increased to 30 and sampling continued as before. Bunches were individually bagged and taken to the laboratory for weighing. There were a total of one, five, and 63 missing data points for the seasons 2016/17, 2017/18, and 2018/19 respectively.

### 2.1 Bayesian Inference

Before describing the details of the model, we would like to quickly review the basics of Bayesian Inference. Consider the data y and the likelihood g(y$|\theta $) and the prior distribution for the parameter of interest g(θ), which describes our understanding of the probable parameter values before the data are observed. The prior distribution incorporates the prior information available to the researcher and may be based on the general understanding of the phenomenon or on prior experience and expert opinion. When no information is available, the prior distributions tend to be very wide and are often called *vague*. Because the choice of prior may be deemed somewhat subjective, a sensitivity analysis is often performed to determine to what extent it affects the modelling result. The posterior distribution for the parameter may be obtained from the Bayes Theorem as follows:

$g\left(\theta |y\right)=\frac{g\left(y|\theta \right)g\left(\theta \right)}{{\int}_{y}g\left(y|\theta \right)g\left(\theta \right)dy\propto g\left(y|\theta \right)g\left(\theta \right)}$

Because the above derivation can rarely be done analytically, computationally intensive Markov Chain Monte Carlo (MCMC) methods are used to produce samples from the posterior distributions, for which summary statistics (such as posterior mean and posterior 95 % credible intervals) can then be obtained.

If the data y can be divided into two batches, y_{1} and y_{2}, the equation above can be rewritten as

$g\left(\theta |{y}_{1},{y}_{2}\right)\propto g\left({y}_{1},{y}_{2}|\theta \right)g\left(\theta \right)\propto g\left({y}_{2}|\theta \right)g\left({y}_{1}|\theta \right)g\left(\theta \right)=g\left({y}_{2}|\theta \right)g\left(\theta |{y}_{1}\right).$

The posterior distribution of θ after observing the first batch y_{1} thus becomes the prior distribution for the experiment involving y_{2}, leading to sequential updating.

A posterior predictive distribution for a new (future) observation $\stackrel{~}{y}$ given the data y can then be obtained as

$g\left(\stackrel{~}{y}|y\right)={\int}_{y}g\left(\stackrel{~}{y}|\theta \right)g\left(\theta |y\right)d\theta .$

Again, numeric methods are customarily used to obtain a sample from such a distribution, which can then be summarised via sample statistics such as mean and quantiles. We now turn to the double-sigmoidal growth model.

### 2.2 Double sigmoidal growth model

Let Yi describe the bunch mass observed at time xi , i = 1...n, where n is the total number of observations. In order to guarantee a non-negative response and control for heteroscedasticity apparent from Figures 1a, 1b, and 1c, it is common to assume that the logarithm of mass has a Gaussian distribution:

*log(Y*_{i}*)** **~** **N(μ*_{i}*,** **τ)**(1)*

where τ is the *precision* or inverse variance, and the mean expected log-mass μ_{i} can be modelled via the double logistic curve as follows:

* **μ*_{i}* **=** **f(x*_{i}*,** **α*_{0}*,** **α*_{1}*,** **β*_{0}*,** **β*_{1}*,** **γ*_{0}*,** **γ*_{1}*)** **=*** **$\frac{{\alpha}_{0}}{1+{e}^{-{\gamma}_{0}\left({t}_{i}-{\beta}_{0}\right)}}$

*+***$\frac{{\alpha}_{1}}{1+{e}^{-{\gamma}_{1}\left({t}_{i}-{\beta}_{1}\right)}}$**

*+***${\epsilon}_{i}$**

**(2)**

where α_{0} and α_{1} describe the first and second asymptotes respectively, β0 and β1 are the inflection points, and γ_{0} and γ_{1} are the slope parameters. Figure 1 illustrates the role of these parameters further. In order to analyse the model within the Bayesian framework, we need to provide prior distributions for these parameters. It should be noted that we expect both slopes γ0 and γ1 to be positive. We also do not expect the second asymptote to be less than the first; i.e., α_{1} ≥ α_{0}. Note that when α_{1} = α_{0}, the second term in Equation 2 disappears and the double sigmoidal curve collapses to a single sigmoidal curve. In order to reflect these restrictions, the following priors are assumed:

α_{0} ~ N(μ_{α0}, τ_{α0}) (3)

α_{1} ~ TN(μ_{α1}, τ_{α1}, α_{0}) (4)

β_{0} ~ N(μ_{β0}, τ_{β0}) (5)

β_{1} ~ TN(μ_{β1}, τ_{1}, β_{0}) (6)

γ_{0} ~ TN(μ_{γ0}, τ_{γ0}, 0) (7)

γ_{1} ~ TN(μ_{γ1}, τ_{γ1}, 0) (8)

τ ~ Gamma(a_{τ}, b_{τ}) _{}_{(9)}

Where TN(μ, τ, L) denote a normal distribution with mean μ and precision τ left-truncated at L. In order to have *a priori *independent parameters, the model can be reparametrized in terms of Δα = α_{1} – α_{0} and Δβ = β_{1} - β_{0}, and the priors in Eq. (4) and Eq. (6) recast as follows:

Δα ~ N(μ_{Δα}, τ_{Δα}, 0) (10)

Δβ ~ N(μ_{Δβ}, τ_{Δβ}, 0) (11)

The analysis consisted of two parts. In the first part, for each growing season, the model was fitted to the grape bunch data available after the first, second, and up to the final (fifteenth) observation time, assuming independent residuals over time, due to the destructive measuring procedure. This was done in order to investigate the trade-off between an earlier prediction and prediction accuracy, as well as to look at the growth curves fitted after one season. In the second part, four different sets of priors were taken into account when analysing the bunch mass data for the 2018/19 growing season in order to examine the effects of incorporating historical data into the modelling process. The weakly informative priors were based on the general expectations of the shape of the grape growth curve and had high variance, and thus small precision. The more informative prior distributions were obtained by using the means and variances of the posterior sample’s parametric approximation of the posterior distributions from the analysis of the 2017/18 bunches, the 2016/17 bunches, and by combining the two. This was done by taking the informed priors from the 2017/18 season’s bunches and informing these on the 2016/17 grape bunches. The three sets of priors are listed in Table 1.

A Metropolis-Hasting sampler was written in R (Team (2015)). In each case, 10^{5} iterations were run after a 2 × 10^{4} burn-in and the convergence was visually assessed.

**Table 1. Prior distributions used for modelling the growth of grape bunches in the 2018/19 season. The vague prior is based on general expectations, while the parameters of the 2017, 2018, and 2017 + 2018 priors are based on parametric approximations of the posterior distributions resulting from combining vague prior with the 2017 and 2018 seasons, and a combination of the two years of bunch mass data.**

Coefficient |
Vague Prior |
2017 Prior |
2018 Prior |
2017+2018 Prior |
---|---|---|---|---|

α |
N(4.09,0.11) |
N(4.40,32.61) |
N(4.58, 155.24) |
N(4.31, 100.77) |

Δα |
TN(0.69,4,0) |
TN(0.76,13.28,0) |
TN(0.74,121.99,0) |
TN(2.01,27.62,0) |

β |
N(200,0.02) |
N(186.89,0.93) |
N(171.26,2.97) |
N(181.25,1.75) |

Δβ |
TN(49,0.11,0) |
TN(58.48,0.05,0) |
TN(53.38,0.28,0) |
TN(58.61,0.05,0) |

γ |
TN(0.3,44.44,0) |
TN(0.09,28591.51,0) |
TN(0.20,104.75,0) |
TN(0.07,94918.84,0) |

γ |
TN(0.3,44.44,0) |
TN(0.12,182.66,0) |
TN(0.27,90.17,0) |
TN(5.19*10 |

τ |
Gamma(4,1) |
Gamma(208.72,31.55) |
Gamma(207.62,22.22) |
Gamma(202.81,26.61) |

**Figure 1. The parameterisation of the double sigmoidal growth curve, and the three main phases of phenological growth exhibited by grapes during the growing season.**

### 2.3 Simulation studies

In order to examine the effect of prior assumptions on yield prediction, as well as to investigate the value of information in the context of yield prediction, 100 data sets were simulated based on the parameters drawn from the posterior distribution, produced after fitting the model to the 2018/19 season’s bunch mass data. The simulated data consisted of the mass of 15 grape bunches being measured at 15 time points throughout the growing season. The model described in Equations (1)-(11) was then fitted with the four different priors described in Table 2. The estimated posterior average bunch mass at day 273 (the harvest day for the year 2019) was then evaluated after each of the 14 measurement time points. In addition, the mean absolute error (MAE) was defined as

$MAE=\Vert E\left(Y\vee x=273\right)-{\overline{y}}_{273}^{sim}\Vert $

12)

Where

$E\left(Y|x=273,{y}^{sim}\right)=E\left\{\frac{{\alpha}_{0}}{1+{e}^{-{\gamma}_{0}\left(273-{\beta}_{0}\right)}}+\frac{{\alpha}_{1}}{1+{e}^{-{\gamma}_{1}\left(273-{\beta}_{1}\right)}}\right\}$

Y^{sim} is the simulated data set and ${\overline{y}}_{273}^{}$ is the simulated average bunch mass on day 273. Similarly, the mean percentage error (MPE) was evaluated as

MPE = log$\frac{E\left(Y\vee x=273\right)}{{\overline{y}}_{273}^{}}$ (13)

## Results

Individual bunch masses with the running (geometric) mean are shown in Figure 2. Interesting differences between the seasons are apparent. A clear Phase II is apparent in the 2017/18 season between days 25 and 50 (15th January to 8th February 2018), which was not observed in 2016/17. This reflects the lower average temperature over the flowering period in 2016/17 when compared to 2017/18, resulting in a longer flowering duration.

**Figure 2. Individual observations and average bunch masses (g) (solid black line) of grape bunches over the 2016/17, 2017/18 and 2018/19 growing seasons. **

The black dotted lines in each plot represent the posterior estimated mean growth curve for each season, with the surrounding grey ribbons representing the 10 % and 90 % estimation bounds. 2a. 2016/17 Growing season, 2b. 2017/18 Growing season, 2c. 2018/19 Growing season

The posterior estimated mean growth curves for the grape bunches in the three seasons examined are shown in Figure 2. While all the models show a reasonable fit, there are a couple of aspects worth noting. A clear phase II and double sigmoid curve is apparent in the 2017/18 season between days 25 and 50 (15th January to 8th February 2018) and to a lesser extent in 2018/19. This was not observed in 2016/17, for which the resulting growth appears to be a simple sigmoid curve. Some of these differences may be explained by the seasonal differences in average temperature during flowering, and in rainfall from flowering to véraison. The average daily temperature, calculated using the model described in Trought (2005), in 2016/17 was 17.9 ^{o}C, while those in 2017/18 and 2018/19 were 18.8 and 18.6 ^{o}C respectively. Lower temperatures result in a longer flowering duration and greater asynchronous berry development. The rainfall from flowering to véraison was significantly higher in 2017/18 (135 mm) than in 2016/17 and 2018/19 (49 and 41 mm respectively).

Rainfall during fruit ripening resulted in the onset of *Botrytis cinerea* leading to a sharp deterioration of bunch mass in 2017/18 (Figure 2b). The data for the last observational time point were therefore not included in the model, because the sharp decrease in bunch mass would have affected the harvest mass estimates produced from the Bayesian model for that particular year; here, the standard double sigmoidal growth curve struggles to accommodate the sharp rise in the middle of the season.

The posterior estimates for the average bunch mass were also obtained after adding the data as it became available consecutively to the model along with the various sets of priors. The resulting mean absolute errors are shown in Figure 3. The value of information is evident here, along with the importance of having informed priors, particularly at the starting of the growing season. This is seen where the prediction errors for the model with vague priors are quite poor and erratic, although they do improve markedly once five or more data points are added (23^{rd} January 2019).

**Figure 3. The posterior mean absolute error for the models with the vague prior, along with the 2017, 2018, and 2017+18 priors, fitted to the grape bunch masses from the 2018/19 season after different amount of observations have been made available.**

Figure 4 shows the posterior mean curves and associated 95 % credible bounds, which were estimated after fitting the model to the entire 2018/19 bunch mass data with the vague, the 2017, the 2018, and the 2017 + 2018 priors respectively. The results show that there is some apparent sensitivity to the choice of priors at different points in the growing season. For example, for the results from the incorporation of two years of priors, the estimation bounds are somewhat larger than those from the other three sets of priors. However, this switches to having lower estimations for the bunch masses at the end of the growing season.

**Figure 4. Posterior mean bunch mass curves with bounds for the double sigmoidal model fitted to the data for the 2018/19 growing season with the vague priors, along with priors informing using the 2016/17 data, the 2017/18 data, and combining them together in different orders (2016/17 to 2017/18 and vice versa). The 5 % and 95 % estimation bounds are included for each set of priors.**

Figure 5 compares the posterior estimates for the final bunch masses of the 2018/19 bunch mass data, using each of the three informed sets of priors (2017, 2018, and 2017 + 2018). When estimates are made using two years of data, they become more consistent earlier in the growing season, after about half of the data has been included in the Bayesian model. However, the means of these estimates appear to be lower than the actual final yield, as well as the estimates made from the other two models (2017, 2018).

**Figure 5. The predicted final bunch masses for the 2018/19 growing season, compared over the 3 sets of informed priors. The 5 % and 95 % estimation bounds are included for each set of priors. **

The black horizontal line represents the observed average final yield for the 2018/19 growing season.

### 3.1 Simulation Studies

For each of the 100 simulated datasets, the posterior mean bunch masses at harvest (273^{rd} day) and the associated 95 % credible interval were evaluated *a priori* and after each of the 14 observation time points. The results are shown in Figures 6, 7, 8 and 9. The informative priors naturally have narrower credible intervals to begin with. As the season progresses and more observations are collected, the intervals become narrower and the predictions for the average bunch mass at harvest become individually dependent on the dataset and closer to the black line, indicating a perfect guess. It is worth noting that the prior based on the 2017/18 growing seasons, which was very different from the 2016/17 one, is overly conservative and is not swayed by the data, resulting in relatively poor predictions, even at the end of the season. The model based on priors informed from both of these years of data suffers even more from this, as the increased precision on the asymptote parameters (α_{0} and Δα) keeps the means of the posterior estimates more fixed and with tighter bounds. The model with priors informed using two seasons of bunch mass data tended to overestimate the bunch masses with less data, and underestimate with all the data.

**Figure 6. Posterior mean masses at harvest (273rd day) and the associated 95 % CIs for the 100 simulated datasets modelled using the vague prior a priori and after the 1st (18th December), 6th (31st January) and 15th (31st March) observation time points respectively. The red line indicates equivalence.**

**Figure 7. Posterior mean masses at harvest (273rd day) and the associated 95 % CIs for the 100 simulated datasets modelled using the 2017 prior a priori and after the 1st, 6th and 15th observation time points respectively. **

The red line indicates equivalence.

**Figure 8. Posterior mean masses at harvest (273rd day) and the associated 95 % CIs for the 100 simulated datasets modelled using the 2018 prior a priori and after the 1st, 6th and 15th observation time points respectively. **

The red line indicates equivalence.

**Figure 9. Posterior mean masses at harvest (273rd day) and the associated 95 % CIs for the 100 simulated datasets modelled using the 2017 + 2018 prior a priori and after the 1st, 6th and 15th observation time points respectively.**

The red line indicates equivalence.

The resulting mean absolute error, averaged over the 100 datasets, is shown in Figure 10. Here, the vague prior only starts to perform just as well as the other priors once approximately 14 of the 15 days of data have been included, which is not suitable given the desire to produce a model which can produce reliable estimates early in the growing season. However, the results shown in Figures 10 and 11 suggest a high consistency throughout the growing season when using the other three priors.

**Figure 10. Mean absolute error for the bunch mass at harvest averaged over the 100 simulated datasets for the vague, 2017, 2018, and 2017 + 2018 priors respectively.**

**Figure 11. Mean percentage error for the mass bunch at harvest averaged over the 100 simulated datasets for the vague, 2017, 2018, and 2017 + 2018 priors respectively.**

## Discussion

Bayesian methods are capable of systematically incorporating prior knowledge. This feature is especially relevant to viticulture and to grape growth modelling, since there is substantial, often vineyard-specific, expert knowledge available. The ability of the Bayesian framework to seamlessly update model estimates as new data comes in is especially useful given the dynamic nature of the phenomenon being modelled. Starting out with a yield estimate based on historical data, and perhaps a general weather forecast for the coming season, and revising that estimate as new information becomes available is a goal well-worth achieving.

The model examined in this study considers the bunch mass to be a function of time only. This is clearly an oversimplification, since plant growth in general, and grape growth in particular, is known to be affected by temperature, usually expressed as growing degree days (Coombe, 1986). Wang and Engel (1998) introduced a function describing the relationship between the daily temperature and the daily growth date and applied it to wheat growth, whereas, for example, Parker *et al.* (2011) and Parker (2012) extended these ideas to model phenological stages of grape development. Therefore, our model may be improved by replacing days with the growth rate expressed as a function of growing degree days. Climate variables other than temperature, such as the amount of solar radiation, may also influence grape growth and development (Dokoozlian and Kliewer, 1996; Bergqvist *et al.*, 2001; Fernandes de Oliveira and Nieddu, 2016). Additional variations may arise due to characteristics of the land, such as soil and topography (Trought and Bramley, 2011; Bramley *et al.*, 2011), as well as management practices. When available, this information can be incorporated into the framework by hierarchically modelling the growth curves parameters. Thus, for example, the inflection points β_{0} and β_{1} can be construed as functions of climate and spatial covariates.

As a way of emphasising this, we took into consideration climate information for the region in which the data was collected. Meteorological data was sourced from the National Institute of Water and Atmospheric Research at the Marlborough Research station 1.1 km south west of the trial site (49^{o}21’51’S; 173^{o} 47’ 56”E). Figure 12 demonstrates the temperature anomalies for the three growing seasons relative to the last 10 years. The accumulated temperature experienced in the 2016/17 growing season was lower than in the other seasons. Above-average temperatures were noted in the 2017/18 season over the flowering period (as indicated by the slope of the deviation from the long-term mean), while the other two seasons were close to the long-term average over flowering. Further heatwaves were experienced in both the 2017/18 and 2018/19 seasons at certain times in the period between flowering and véraison (Salinger *et al.*, 2019; Salinger *et al.*, 2020). The dates of flowering and véraison were estimated using the Grapevine Flowering Véraison (GFV) (Parker *et al.*, 2011). This appeared to have more of a direct impact on the yield for the 2017/18 seasons, shown by the increase in the average bunch mass measurements. Figure 13 shows the accumulated rainfall for the three growing seasons. It can be seen that the 2017/18 growing season experienced higher amounts of rainfall during the course of the growing season. The 2018/19 season is typified by extended periods of no rainfall in the second half of the season. This may explain the very limited amount of growth in the bunch masses seen after the period of véraison in this particular case.

**Figure 12. Temperature anomalies for the 2016/17, 2017/18 and 2018/19 growing seasons, relative to the average daily temperatures over the last 10 years.**

**Figure 13. Temperature anomalies for the 2016/17, 2017/18 and 2018/19 growing seasons, relative to the average daily temperatures over the last 10 years.**

It can be seen in Figure 2a that the 6-parameter double sigmoidal curve was not a good fit for the grape growth observed in the 2016/17 season. Archontoulis and Miguez (2015) have reviewed a wide variety of nonlinear regression models used in agricultural research and they have made comparisons between various specifications of sigmoid functions. Of particular interest to us are the Richards model (Richards,1959), the Gompertz model (Gompertz, 1825) and the Weibull curve (Weibull, 1951), all of which we intend to compare with our current implementation of the double sigmoidal model. It is also worth noting another more recent model developed by Yin *et al.* (2003), which involves incorporating the maximum growth rate of the fruit to help calculate fruit mass.

The current model assumed conditionally independent priors and was estimated via the standard Metropolis-Hasting sampler (Gelman *et al.*, 2014). Taking joint prior specifications into account will increase the flexibility of the model, allowing a wider range of expert opinion formulations to be adapted, and may also improve the efficiency of the algorithm.

Finally, applying the resulting range of parametrisations to a wider spatio-temporal set of data will improve our understanding of the modelling framework, and ultimately produce a better tool for early grape yield prediction.

## Conclusion

In this study, we illustrated the use of a Bayesian framework for fitting a standard double sigmoidal growth curve to the Sauvignon blanc grape bunch mass data collected over the 2016/17 2017/18 and 2018/19 growing seasons in Marlborough, New Zealand. We also performed a simulation study to investigate both the sensitivity of the model to prior assumptions and the value of information. The latter refers to the role of additional consecutive observations throughout the season in improving the accuracy of the estimation of the grape bunch mass at harvest.

The results from this analysis show that the model is sensitive to prior assumptions made for the parameters of the double sigmoidal model. From an early yield prediction perspective, the incorporation of non-informative (vague) priors to the model resulted in poor results, only becoming similar to the results seen from models using more informed priors once half of the growing season data was incorporated into the Bayesian model. This led to the conclusion that some information about the parameterisation of the double sigmoidal model is influential in producing useful results.

## Acknowledgements

We would like to thank the Marlborough Research Centre Trust for the use of their vineyard.

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