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Enhancing Short-Term Berry Yield Prediction for Small Growers Using a Novel Hybrid Machine Learning Model

Juan D. Borrero
1,* and
Juan-Diego Borrero-Domínguez
Department of Management and Marketing, University of Huelva, Pza. de la Merced s/n, 21002 Huelva, Spain
Agricultural Economics Research Group, University of Huelva, Pza. de la Merced s/n, 21002 Huelva, Spain
Author to whom correspondence should be addressed.
Horticulturae 2023, 9(5), 549;
Submission received: 4 April 2023 / Revised: 27 April 2023 / Accepted: 28 April 2023 / Published: 3 May 2023


This study presents a novel hybrid model that combines two different algorithms to increase the accuracy of short-term berry yield prediction using only previous yield data. The model integrates both autoregressive integrated moving average (ARIMA) with Kalman filter refinement and neural network techniques, specifically support vector regression (SVR), and nonlinear autoregressive (NAR) neural networks, to improve prediction accuracy by correcting the errors generated by the system. In order to enhance the prediction performance of the ARIMA model, an innovative method is introduced that reduces randomness and incorporates only observed variables and system errors into the state-space system. The results indicate that the proposed hybrid models exhibit greater accuracy in predicting weekly production, with a goodness-of-fit value above 0.95 and lower root mean square error (RMSE) and mean absolute error (MAE) values compared with non-hybrid models. The study highlights several implications, including the potential for small growers to use digital strategies that offer crop forecasts to increase sales and promote loyalty in relationships with large food retail chains. Additionally, accurate yield forecasting can help berry growers plan their production schedules and optimize resource use, leading to increased efficiency and profitability. The proposed model may serve as a valuable information source for European food retailers, enabling growers to form strategic alliances with their customers.

1. Introduction

Perishable agricultural products such as fruits and vegetables are highly time-sensitive and require swift and efficient delivery to maintain their freshness and quality. In addition, farmers are the weakest link in the modern food value chain [1], which is made up of complex sequences of operations with strong social, labor, economic, and environmental implications. Hence, the asymmetries in the food value chain have serious consequences for society in general and farmers in particular [2]. For this reason, the rise of digital technologies could help transform agriculture [3], and predictive models would be a viable option for farmers to improve sales channels with food retailers [4].
As a result of intense competition among large food supermarkets, more and more retailers now tend to demand on-time delivery schedules from their suppliers and are less tolerant of any deviations in their supply schedules. Retailers seek to maintain high-quality service without increasing prices, so maintaining profitability is always a challenge. In this context, the emergence of the omnichannel has revolutionized retail operations resulting in fundamental changes in the expectations of consumers and in the decision-making processes of retailers [5,6,7]. In this sense, researchers have recognized that omnichannel is the future of retail [8].
While demand forecasting is crucial in the supply chain, there has been little research on this topic in relation to omnichannel retailing [9]. To synchronize supply and demand in omnichannel retailing, a data-driven approach that utilizes k-means to identify sales patterns from historical data and uses neural networks to provide accurate forecasting through time series analysis has been proposed [10]. Other approaches include demand forecasting through exponential smoothing [11] and the application of different demand functions to replace the forecasting process [12]. To the best of our knowledge, no studies have addressed the issue of supply forecasting for the horticultural industry in omnichannel retailing, except for the research mentioned previously.
Horticultural products, in general, and the berry sector, in particular, involve seasonal production patterns and price instabilities that have been little studied in the literature but are important in commodity markets, not only operationally but also financially [13]. Due to the highly perishable nature of fresh berries and the globalized nature of the market, large-scale food retailers possess significant bargaining power in the global fresh berry industry through their ability to set prices for future weekly sales programs. For farmers, any deviation from the quantities specified in such programs can result in very significant economic losses when the quantity supplied is lower (penalties for non-compliance) or higher (lower sale prices) than the contractually agreed amount.
Because inaccurate production forecasts may lead to penalties as a result of contract overage, yield prediction systems have become fundamental tools for providing berry growers with better means of market planning [14].
In this study, we aimed to enhance the efficiency of fresh fruit and vegetable suppliers’ order-fulfillment processes while enabling retailers to offer prices of perishable produce to customers in advance [2]. With the emergence of advanced analytical techniques, including neural networks and machine learning (ML), there is a significant opportunity for horticultural farmers to improve yield prediction and generate greater revenue by determining the optimal harvest time and integrating this information into their customers’ e-commerce platforms [5,15].
Hence, accurate yield forecasting can be seen as a competitive factor for farmers due to large retailers typically demand consistent and reliable supplies of products to meet the needs of their customers. By producing accurate forecasts, they can enhance their reputation and increase their own bargaining power. This approach also has the potential to solve a key agronomic challenge, as yield prediction is important for many soft fruit growers because they are dealing with perishable, high-value, seasonally produced fruit and, in addition, harvesting depends on increasingly costly labor.
Our empirical study focuses on this market and proposes an improved method for short- and medium-term yield forecasts, building on the recent literature on time series predictive models. The motivation behind our approach is also to develop an accurate predictive algorithm that can efficiently exploit data from a single variable, such as yield, as other inputs, such as imagery, climate, or plant data, can be expensive and challenging for farmers to obtain.
In short, using only previous yield data as input, the main contribution of this work consists of the development of a new hybrid model for berry yield prediction based on statistical models, such as a time series analysis, in combination with ML techniques. A hybrid model typically consists of two or more models that are combined in a specific way to take advantage of their strengths and mitigate their weaknesses. In our study, the hybrid models for predicting yield in agriculture combine an ML algorithm, such as nonlinear autoregressive neural networks (NARs) and support vector regression (SVR), with a new Kalman filter approach. The statistical model identifies and accounts for trends and seasonality, while the ML algorithm helps to capture complex patterns on the time series of the error. To the best of our knowledge, this is the first approach to use this type of hybridization.
In addition, we compared different models, both non-hybrid and hybrid, using metrics such as root mean square error (RMSE), mean absolute error (MAE), and goodness-of-fit R2, to demonstrate the better performance of the hybrid models. To prove this, we analyzed three berry products separately and showed that the hybrid prediction models are valid for all these time series data.
Our results indicate that producers could take advantage of ML techniques to define digital marketing strategies that minimize their risks and improve their competitive position in the food value chain [16].
The paper is structured as follows: First, we review the different types of predictive models that exist in the literature, with emphasis on those recently used in agriculture. Second, we describe time series data and hybrid models. Third, we present the results of both hybrid and non-hybrid models. Finally, in the last two sections, we discuss the results and their implications.

2. Theoretical Background

Time series forecasting is a widely discussed problem, and we applied a novel approach in pursuit of a solution to this problem for perishable fruit and vegetable products.
Among the most commonly used time series predictive models, the auto-regressive integrated moving average (ARIMA) and seasonal auto-regressive moving average (SARIMA) use a linear regression based on the past values of a variable to be estimated, as well as errors previously made by the process. These models have been used for predictions in many areas [17,18,19,20,21,22,23,24,25,26], including the agri-food sector [4,27,28].
The Kalman filter has often been used to refine an ARIMA model by introducing a state-space system [29,30,31,32,33]. Despite the limited explicit references to the Kalman filter in the field of agri-food production [13], recent studies have utilized this technique for fitting predictive models within the horticultural sector [34].
Alongside linear models, several nonlinear algorithms and ML techniques have also been recently developed to predict time series data [15]. These include NAR, a generalization of the ARIMA model with more complex modeling, which has been applied for predicting the price of minerals [35], the price of certain vegetables [36], amounts of waste [37], water levels [38], COVID-19 case numbers [39], and other variables from different areas [40,41]. In addition, for some years now, numerous studies have adapted types of neural networks—especially SVR—to make time-series predictions in such areas as energy [42,43], finance [44,45,46], weather [47,48], water [49,50,51,52,53], COVID-19 [54], and agriculture [55,56,57,58].
Other recent studies have used hybrid prediction models, involving a combination of different methods, to predict future situations, including demand for flights in the aeronautical industry [59]; electrical energy consumption [60]; and in areas such as weather [61], finance [62], and agriculture [63,64,65,66]. In many such cases, the simplest hybrid algorithm has produced the best predictive result [67].
Currently, extensive research is being conducted in the agricultural sector to predict crop yields better using ML algorithms [68,69,70]. The information facilitated by remote sensing allows for the speedy collection of large amounts of data and offers the potential for extracting data from many variables quickly [71]. Most research using ML for yield forecasting has also involved the use of multispectral imagery data from satellites or unmanned aerial vehicles (UAVs) [71,72,73,74,75]. Some of these studies have sought to predict berry yields using ML-based predictive models [76,77,78].
However, the aforementioned studies on prediction in time series also reveal certain limitations in the field of perishable crop yield forecasting. Firstly, although ML has been identified as one of the most common techniques for yield prediction in agriculture [79], it is not always easy for yields to be predicted because ML algorithms require large amounts of data to provide reliable results [80], and this is not typically available from berry farms. For this reason, although big data analysis is becoming increasingly common in the agricultural sector [81], obtaining the amount of data needed for ML implementations remains a challenge. Secondly, the difficulty in developing yield forecasting models is related to the appropriate selection of independent variables [82]. Thirdly, the prediction models do not adequately capture trend changes in instabilities on the dependent variable. Consequently, this type of predictive system, which is based on the past information of the variable, exhibits a very characteristic deviation to the right, as has been demonstrated in a large body of recent research [83,84,85,86,87].

3. Material and Methods

3.1. Description of the Data

More than 50 percent of all Spanish farms are small farms with less than 5 hectares [88]. Such farms face the need to concentrate supply in order to become more competitive. This is especially the case for Spanish berry growers. In this study, we focused specifically on the Huelva area of Spain because this region is the main producer and exporter of fresh berries in Europe [89] and because Huelva berry growers are usually grouped into cooperatives for the distribution of their products [90].
During the summer, berry cooperatives carry out preliminary deals with their clients (retailers and wholesalers) that are formalized in purchase commitments. When the harvest season starts, these agreements are put into effect by means of weekly sales programs in which prices and daily quantities to be supplied by the cooperative are established. Any non-compliance with agreed conditions results in penalties for producers. If the quantity supplied is less than the agreed amount, a financial penalty is applied. If the quantity supplied exceeds the agreed amount, the product is sold to customers at a much lower price.
To demonstrate that our model was fully functional, we fitted it using a berry yield dataset. This dataset contained data for three berry fruits from farms of Huelva (Spain), corresponding to three periods or seasons.
We worked with a daily berry yield dataset containing data for strawberries, raspberries, and blueberries; these data were obtained from 328 small berry growers from four representative agri-food cooperatives. A longer time series was not possible because of the difficulty in obtaining reliable yield data.
In order to ensure that seasonal time periods were equal, September 1st was chosen as the start date, and July 15th of the following year was chosen as the end date for each agricultural season. In total, we obtained a time series involving approximately 1000 observations of each berry fruit, giving a total of 2835 records in the dataset, which we considered sufficient to run our algorithm.
For the training and testing of our model, we used the first two seasons (2017–2018 and 2018–2019) as the training dataset and the last season (2019–2020) as the testing dataset.

3.2. Description of the Hybrid Model

In recent years, ML algorithms have achieved remarkable success in various areas, including agriculture, as described in the theoretical background above. However, the superiority of ML over other simpler algorithms has not been so apparent in the field of forecasting [67], where data availability is often limited, and regressors are not available [91].
Hybrid prediction models strategically combine several algorithms to exploit the particular advantages of different methods, resulting in more accurate estimations compared to other models [92,93]. In this study, we built on previous research [61,94,95,96,97,98] to propose two hybrid forecasting models. The first model uses a novel approach of the Kalman filter [34] combined with a multilayer perceptron (MLP) neural network that has one hidden layer as its base architecture. The second model replaces NAR with SVR.
Thus, our two hybrid models calculated the prediction of the time series with the Alternative Kalman Filter (AKF) and corrected the error-making predictions about the errors that the predictive system will commit, either through a NAR or an SVR process.
Hence, because hybrid models applied to time series forecasting tend to achieve better performance than pure statistical or pure ML methods [67], by considering Xt as the time series that we intended to estimate and at ~ N (0, σ2) as the white noise process associated with the predictive system, the hybrid model can be expressed as in Equation (1):
x ^ t + 1 = L ^ t + 1 + a ^ t + 1
where L ^ t + 1 is the linear prediction of the AKF on the time series, and a ^ t + 1 is the estimated error deduced by a NAR or SVR process.
The ARIMA process is one of the most widely used models for time series forecasting. It is a regressive process that depends on past values of the variable to be estimated and on any errors made by the process itself. Our algorithm automatically chooses the optimal ARIMA(p, d, q) parameters that minimize both the Akaike information criterion (AIC) and Bayesian information criterion (BIC).
The introduction of an ARIMA model into a state-space system, if it is carried out using the classical method [99,100], causes problems when the standard deviation of the white noise associated with the system is too large because the algorithm requires the introduction of a random value that can distort the result. In order to eliminate this potential problem, in this study, we followed a new method, namely, AKF [34], which eliminates the introduction of a random value generated by the machine and which can be defined by the equation of state (2), as follows:
X t X t 1 X t m = ϕ 1 ϕ m 1 ϕ m 1 0 0 0 1 0 X t 1 X t 2 X t m 1 + θ 1 θ m 0 0 0 0 a t 1 a t 2 a t m 1
where Xt is a stationary zero-mean time series; at is a white noise process associated with the ARIMA(p, d, q) process; m = max{p, q}, where p and q are the delays calculated according to Box–Jenkins methodology; and φi, θj ∈ R are the parameters associated with the ARIMA(p, d, q) process, such that φi = 0 if i > p and θj = 0 if j > q.
The observation Equation (3) can now be expressed as follows:
z t = 1 0 X t X t 1 X t m , w h e r e   υ t = 0
Our predictive system, therefore, applies the Kalman filter [101] to Equations (2) and (3) and thereby solves the above-mentioned convergence problems.
A NAR process is a type of MLP neural network consisting of a nonlinear regression of the variable to be estimated with respect to its own past that can be defined by the expression Xt = f (Xt−1, …, Xtp) + at, where Xt is the time series, p ∈ N is the number of delays used by the system, at ~ N (0, σ2) is the white noise process of the system, and f is a nonlinear function, generally unknown.
We used the NAR process with a single hidden layer [102], as in Equation (4), as follows:
X ^ t = ψ 2 j = 1 M ω j o ψ 1 i = 1 p ω i j + b j + b 0
where ψ1 and ψ2 are the activation functions in the hidden layer and in the output layers, respectively; wij and wj0 are the weights from input i to hidden unit j and from hidden unit j to output o, respectively; and bj and bo are the biases in the corresponding units (Figure 1).
The number of inputs p and the number of units in the hidden layer M were obtained using a growth technique, first in the input layer and then in the hidden layer, adding units step by step and calculating the RMSE on the training set. Finally, p and M were determined when a minimum RMSE value was found.
For the activation functions, we tested the sigmoidal function and the hyperbolic tangent in the hidden layer, choosing the one that obtained the lowest RMSE in the training set, and the identity function in the output layer, as has been previously described for regression problems [37,61,62,103,104].
Finally, as a learning algorithm, we used the backpropagation algorithm, which can correct the weights minimizing the error made in the training set. Therefore, if we consider Wl the weight matrix of layer l = 1, 2, the backpropagation algorithm calculates the increments as follows:
Δ W l = η E W l
where η ∈ R is the learning rate, and E is the error made by the system in each step of the training. For the learning rate, we used the value in the interval [0.01, 0.1] that minimizes the RMSE in the training set, tested in increments of 0.01.
SVRs are a type of ML derived from the adaptation of SVM used for nonlinear regressions. For calculation purposes, we used Equation (5) as follows:
y ^ i = j = 1 N α j α j * k x i , x j + b
where k is the kernel function, and (αj − αj*) ≠ 0 values are support vectors.
We made error corrections by the linear system and computed a prediction on the time series of the error, which is a type of white noise variable, using the Gaussian function k x i , x j = e γ x i x j 2 as the kernel function. Lastly, we chose the number of inputs in a similar way to the method used in the NAR model by increasing the number of lags one by one and observing the RMSE in the training set until we found a minimum through package e1071 of the R programming language to determine the value of the parameters needed for this model.

4. Results

Supply agreements are key contributors to berry farm profitability; however, profit levels for berries ultimately depend not on growers but on retailers. In addition, time series of prices and yields are characterized by temporal variability (Figure 2).
Specifically, we can observe in the cross-correlation functions (Figure 3) that, in all three cases, both time series present a large negative correlation, especially in lag 0 when the value of the linear correlation is greater than −0.7. This adds further value to a predictive system.
In this section, we analyzed the results obtained for the weekly predictions of three berry crops during the 2019–2020 season and compare the results obtained using (1) the classical Kalman filter-based model and (2) the AKF-based model with those obtained using (3) the hybrid AKF-SVR model and (4) the hybrid AKF-NAR model. The selection of parameters for each model is detailed in Section 3.
We compared the results of the four models using RMSE, MAE, and goodness-of-fit R2 criteria to minimize errors while not penalizing a higher number of parameters, as required in nonlinear processes.
For the weekly strawberry season forecast, our algorithm automatically chose the optimal ARIMA parameters that minimized the AIC and BIC criteria, resulting in the ARIMA(3, 1, 1) model. For the AKF-SVR model, we used a Gaussian kernel function with γ = 0.1, C = 1, and ϵ = 0.1, and eight inputs; for the AKF-NAR model, a hyperbolic tangent activation function in the hidden layer, with η = 0.01, p = 4, and M = 1.
Table 1 summarizes the results for the 2019–2020 season. These indicate that AKF improved upon the classical Kalman filter, and neural network correction further improved performance. Thus, the AKF-SVR and AKF-NAR models were found to be the most accurate.
Similarly, for raspberry, we used an ARIMA(1, 1, 1) model and a Gaussian kernel function with γ = 0.071, C = 1, ϵ = 0.1, and 14 inputs for the AKF-SVR model. For the AKF-NAR model, we used a hyperbolic tangent function as the activation function in the hidden layer, with η = 0.1, p = 16, and M = 1. Again, the hybrid AKF-SVR was the best predictive model, followed closely by the hybrid AKF-NAR (see Table 2).
For blueberry, we used an ARIMA(6, 1, 1) model and a Gaussian kernel function with γ = 0.1, C = 1, and ϵ = 0.1, and 10 inputs, for the AKF-SVR model. For the AKF-NAR model, we used a sigmoid function as the activation function in the hidden layer, with η = 0.1, p = 10, and M = 1. Table 3 shows that the AKF-SVR model was the most accurate in predicting blueberry season values.
Finally, it is worth noting that our two hybrid models exhibit a goodness-of-fit greater than 0.95 across all three data series. These results demonstrate how the predicted yields from the hybrid models align better with the actual yield values compared to the non-hybrid models. To visualize this improvement, Figure 4, Figure 5 and Figure 6 display a comparison of the four predictive models for each berry fruit in terms of the observed real variable, which is the weekly berry yield measured in kilograms.

5. Discussion

Berry fruits are highly valued, but they pose a significant agronomic challenge for growers due to their seasonality and perishability. The cost of “just-in-time” staffing for harvesting is a major concern for growers. Additionally, the time series of berry prices are negatively correlated with the production time series, as noted by some authors [105,106] and confirmed by our study (see Figure 3). If the yield forecast is known, farmers can infer higher incomes for the farm when the variables are correlated.
Therefore, accurate yield forecasting is crucial for these crops, especially since sale prices are agreed upon in advance and are highly time-sensitive. This ability can significantly improve logistical planning, scheduling of harvest tasks, financial planning, and transportation of fruit [72]. With the proliferation of digital technologies and big data, most research is using data from different sources [76,77,78], while only a few studies have focused on short-term agricultural yield forecasting based solely on past yield data. In this study, we addressed this issue by developing a new hybrid predictive system that combines time series with ML techniques.
Our new hybrid predictive system introduces the ARIMA model into a state-space system by AFK adjusting the error through NAR or SVR models. The ARIMA model captures the trend and autocorrelation in the data, while the ML algorithm captures complex patterns and relationships in the error.
Our findings indicate that our hybrid forecasting models outperform non-hybrid models and hybrid models that use more inputs as variables than the previous yield.
Thus, our hybrid systems outperform non-hybrid models by achieving higher goodness-of-fit values and lower values for RMSE and MAE, as indicated in Table 1, Table 2 and Table 3. Furthermore, when comparing our results with those of similar studies that employed different prediction models over time series, such as ARIMA, KF, SVR, or NAR, we arrive at three interesting conclusions.
Firstly, our algorithm produces better performance than other methods [22,38,64,82,107,108,109,110,111], as confirmed by the values of R2 reported in the previous section. In particular, for similar datasets, our study improves the results of the Kalman filter, and the alternative Kalman models presented in [34].
Secondly, predictions based on simple models such as KF or AKF tend to deviate to the right with respect to observed variables, as these predictive models are not able to quickly capture trend changes [32,37,42,86,87,112,113]. Our hybrid models smooth out the trend changes, as seen in Figure 4, Figure 5 and Figure 6.
Thirdly, our forecasting model uses only past performance as an input, and it is a type of time-series model that relies on historical data to predict future results and provides, in all cases, a goodness-of-fit higher than 0.95. These results allow us to conclude that our model is better than those presented in [68,69,70,71,72,73,74,75,76,77,78] as it is simpler to model, requires fewer data, and are easier and cheaper to obtain.
Therefore, our results show that statistical model prediction with only previous yield data could be a first step toward berry prediction models that help growers plan their harvest and marketing operations [114]. In summary, our hybrid predictive models can be useful for predicting yield in agriculture because they combine the strengths of multiple models to provide more accurate and reliable predictions using single-variable past data. By leveraging the strengths of different models, they improve the accuracy of yield predictions and become more flexible and adaptable to changing conditions.
However, our hybrid models still have some limitations, particularly in accurately predicting maximums, minimums, and turning points. To address this issue, we propose that new time series approaches used in other sectors should also be explored by researchers, including adding exogenous variables to the ARIMA models [115,116,117] or directly applying neural networks to the time series under study [118,119,120,121,122,123,124,125]. Such an approach could be adopted in studies that use time series ML processes other than those presented here [126,127,128,129,130].
In the future, researchers could explore the combination of climate data from satellite images, public weather stations, and on-farm dataloggers, as well as soil and plant genotype data collected by UAVs. These actions could help farmers better prepare to maximize their income, especially in a world of low profitability due to high production costs and low market prices.
Moreover, when building a predictive model at the micro level to determine the weekly yield of berries for a berry farm, it is important to take into account the potential impact of macroeconomic shocks and global uncertainty (i.e., COVID-19 or the Russia–Ukraine war) on the model’s predictions. To improve predictions, strategies such as considering external factors such as weather conditions, consumer demand, and supply chain disruptions; testing the model under different sets of assumptions regarding macroeconomic conditions; and using ensemble methods such as combining multiple models could be employed. By taking these steps, the hybrid model can provide more accurate predictions and help the berry farm make better-informed decisions.

6. Conclusions

The commercial production of berries is a significant agricultural activity in Spain and Europe, and accurate forecasting of weekly yield can bring economic benefits to farmers. In this study, we proposed a hybrid model that combines the ARIMA model with neural networks and other machine learning techniques to increase the accuracy of short-term berry yield prediction. The results demonstrate that the proposed hybrid models have a goodness-of-fit value above 0.95 and lower RMSE and MAE values compared with simpler models, making it a useful tool for predicting future production not only of berries but also of other agricultural products.
This study has important implications for the horticultural sector. Firstly, small growers can use digital strategies to offer crop forecasts and increase sales by promoting loyalty in their relationships with large food retail chains. This can be achieved by providing retailers with accurate information about berry availability. Secondly, accurate yield forecasting can help growers plan their production schedules and optimize resource utilization, leading to increased efficiency and profitability. Thirdly, the hybrid model developed in this study can serve as a valuable information source for European food retailers with online stores and e-platforms, allowing them to set offers in advance and form strategic alliances with their suppliers. Overall, this study presents a useful hybrid time-series forecasting method that can improve yield prediction accuracy, thereby benefiting both growers and retailers in the horticulture industry.
The study highlights the potential benefits of using advanced time series forecasting methods in horticultural production. We suggest that public institutions offer large, reliable, and anonymized time series data of agricultural prices, consumer demand, and yields to practitioners and researchers. Facilitating the development of more accurate and useful predictive models for agricultural time series data could enhance the authority of the horticultural industry in handling future economic crises.
Moreover, the results of this research can inform policy decisions related to the use of these methods and their impact on food supply chains. For example, if the hybrid model proposed in this study is widely adopted, it could lead to increased efficiency in the production and distribution of berries and other agricultural products. Furthermore, the suggestion to implement this algorithm on the agricultural price time series has implications for policy related to data sharing and accessibility in the horticulture industry.
The research presented in this paper also contributes to the literature on time series forecasting in agriculture by proposing a novel hybrid model that combines two different algorithms to increase the accuracy of short-term berry yield prediction. The article also introduces an innovative method to enhance the prediction performance of the ARIMA model. These contributions could inspire further research into hybrid models and methods for improving time series forecasting accuracy in agriculture and other industries.
Finally, the paper has theoretical implications for the development of hybrid time series forecasting models. This approach can potentially be applied to other fields beyond agriculture, where accurate short-term forecasting is critical, such as finance or energy markets. Furthermore, the introduction of an innovative method to eliminate randomness and incorporate only observed variables and system errors into the state-space system could have implications for the development of other statistical models.

Author Contributions

Conceptualization, J.D.B.; methodology, J.D.B.; validation, J.D.B. and J.-D.B.-D.; formal analysis, J.D.B.; investigation, J.D.B.; resources, J.D.B.; data curation, J.D.B. and J.-D.B.-D.; writing—original draft preparation, J.D.B. and J.-D.B.-D.; writing—review and editing, J.D.B. All authors have read and agreed to the published version of the manuscript.


This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Restrictions apply to the availability of these data. Data were obtained from a third party. The data are not publicly available due to privacy concerns.


The authors acknowledge the support provided by the companies that released the data used for the analysis.

Conflicts of Interest

The authors declare no conflict of interest.


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Figure 1. NAR neural network.
Figure 1. NAR neural network.
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Figure 2. Relationship between price and production for time series data, 2018–2019: Strawberry (above), raspberry (middle), and blueberry (bottom).
Figure 2. Relationship between price and production for time series data, 2018–2019: Strawberry (above), raspberry (middle), and blueberry (bottom).
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Figure 3. Cross-correlation functions and lagged regressions, 2018–2019: Strawberry (above), raspberry (middle), and blueberry (bottom).
Figure 3. Cross-correlation functions and lagged regressions, 2018–2019: Strawberry (above), raspberry (middle), and blueberry (bottom).
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Figure 4. Strawberry forecasting graphs, 2019–2020.
Figure 4. Strawberry forecasting graphs, 2019–2020.
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Figure 5. Raspberry forecasting graphs, 2019–2020.
Figure 5. Raspberry forecasting graphs, 2019–2020.
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Figure 6. Blueberry forecasting graphs, 2019–2020.
Figure 6. Blueberry forecasting graphs, 2019–2020.
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Table 1. Results for strawberry time series, 2019–2020.
Table 1. Results for strawberry time series, 2019–2020.
Classical Kalman filter (KF)0.89952,906.0090,847.01
Alternative Kalman filter (AKF)0.95336,901.7462,269.74
Hybrid AKF-SVR0.95435,147.7161,339.73
Hybrid AKF-NAR0.95435,640.5361,311.99
Table 2. Results for raspberry time series, 2019–2020.
Table 2. Results for raspberry time series, 2019–2020.
Classical Kalman filter (KF)0.93812,576.6725,323.00
Alternative Kalman filter (AKF)0.94711,037.7723,419.50
Hybrid AKF-SVR0.9806772.1414,248.64
Hybrid AKF-NAR0.9609324.9520,425.92
Table 3. Results for blueberry time series, 2019–2020.
Table 3. Results for blueberry time series, 2019–2020.
Classical Kalman filter (KF)0.78327,351.5051,876.18
Alternative Kalman filter (AKF)0.88022,108.3338,562.61
Hybrid AKF-SVR0.97310,677.0618,150.67
Hybrid AKF-NAR0.95810,768.4722,706.09
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Borrero, J.D.; Borrero-Domínguez, J.-D. Enhancing Short-Term Berry Yield Prediction for Small Growers Using a Novel Hybrid Machine Learning Model. Horticulturae 2023, 9, 549.

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Borrero JD, Borrero-Domínguez J-D. Enhancing Short-Term Berry Yield Prediction for Small Growers Using a Novel Hybrid Machine Learning Model. Horticulturae. 2023; 9(5):549.

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Borrero, Juan D., and Juan-Diego Borrero-Domínguez. 2023. "Enhancing Short-Term Berry Yield Prediction for Small Growers Using a Novel Hybrid Machine Learning Model" Horticulturae 9, no. 5: 549.

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