Model for Agricultural Production in Colombia Using a Neuro-Fuzzy Inference System

Abstract

As mentioned by the Food and Agriculture Organization of the United Nations, agriculture has a primary role in food security. Given the advantageous conditions that Colombia has as a biodiverse country, creating and implementing sustainable and comprehensive agricultural systems is essential to generate agricultural decision-making tools. Therefore, this paper displays the design and deployment (training–validation) of a neuro-fuzzy model for the relevant agricultural production in Colombia. Four different configurations are proposed according to the data collected and the variables identified. The results show that a remarkable prediction of the models (configurations) is achieved by using training and validation data.

1. Introduction

According to [1], the diverse process of globalization integrating various markets and countries has been studied for its impact on climate change. The ecological modernization theory considers that environmental degradation is not necessarily the product of industrialization and economic growth [2]. However, it has been observed that economic growth, the use of fossil fuels, technological innovations, and globalization drive the emission of greenhouse gases [3], especially in low-income countries, which are the ones that suffer severely from the consequences [4]. Climate change is a global problem [5] that affects the agricultural sector, putting food security and ecosystem resilience at risk [6,7].

As a result of global warming and climate change, crops such as coffee [8], wheat [9], rice [10], cocoa [11], corn [12], cashews, and avocado [13] could disappear. Crop production is already suffering the consequences of climate change, and the degree of impact on a country depends on geographic location, adaptability, and Gross Domestic Product (GDP) [14]. For example, nearly half of the food in the UK is imported; thus, the food security and health of the inhabitants are susceptible to climate change in food-exporting countries [15]. In Colombia, the deterioration of ecosystems such as the Amazon contributes to climate change [16]. Forests lose carbon storage capacity; climatic phenomena become more prolonged or intense [17].

Colombia is located in a tropical zone, crossed by three mountain ranges, which provide advantageous conditions that position the country as one of the most biodiverse in the world, with regulating ecosystems such as wetlands [18], and with the third-largest number of plants [19]. Due to the variety of soils and temperatures [20], together with water availability, numerous crops such as coffee, corn, rice, sugar cane, and palm oil [21] grow easily, which are usually available all year round. Agriculture is one of the most representative sectors in Colombia. According to the 2019 National Agricultural Survey, the agricultural sector contributed 8.7% to the Gross Domestic Product, and the estimated total agricultural production for 2019 was 63 million tons [22]. Coffee is one of the star products, being the third-largest Colombian export product in 2022, with a total of USD 4.15 billion dollars, making Colombia the second-largest coffee exporter in the world [23].

Agriculture has a primary lead in food security, a concept that, regarding the Food and Agriculture Organization of the United Nations (FAO), is defined as follows: “at the individual, household, national and global levels, it is achieved when all persons, at all times, have economic and physical access to sufficient, safe and nutritious food to satisfy their dietary needs and preferences, in order to maintain a healthy and active life” [24]. The agricultural sector expands the productive and export base, generates employment, and provides material that serves as input for other sectors and, most importantly, ensures that food supply is adequate, safe, and affordable [25].

Colombian agriculture is characterized by extreme agroclimatic heterogeneity and often suffers from sparse, noisy data, posing significant challenges to accurately yield forecasting and decision-making. This study displays the advantages of the Adaptive Neuro-Fuzzy Inference System (ANFIS) in addressing these challenges. This approach can dynamically tune fuzzy membership functions to local microclimates—from the Caribbean coast to the Amazon rainforest and embedding linguistic rules that capture uncertainty (high rainfall, acidic soils, and intensive management). Its hybrid architecture robustly handles missing or biased records; for example, in an analysis of observations of agricultural employment, ANFIS generated clear decision rules for subsidy allocation and extension programs [26]. Regarding Stevia crop trials, ANFIS produces transparent, field-ready rules (e.g., if soil pH is medium and bulk density is high, then harvest is timely) that extension agents can readily interpret and adjust on-site [27]. Built on a lightweight five-layer Takagi–Sugeno structure with hybrid least squares (backpropagation) training, ANFIS trains rapidly on low-cost hardware and integrates new predictors without full retraining [28]. The attributes such as adaptability, interpretability, efficiency, and solid theoretical grounding make ANFIS an ideal tool for yield projection, investment guidance, and policy calibration across the various Colombian agricultural landscapes.

1.1. Related Works

This subsection covers different related works regarding data-driven prediction and decision-making. Among the main characteristics identified are traditional statistical methods, machine learning (ML) applications, hybrid statistical ML approaches, adaptive neuro-fuzzy inference systems, and agro-sustainability applications. Figure 1 shows the relationship of the identified characteristics of the related works, which are addressed in more detail below.

Technological development has made it possible to collect and store abundant data, from which information is gathered to suggest solutions and make decisions. Classification and prediction are two common actions that can be carried out with data [29,30,31]. In this order, traditional statistical methods have been used, such as the study by Parra et al. [19], which identifies cropping systems with greater flexibility in response to alterations in both climatic and non-climatic elements, or the work of Kim et al., which predicts the growth of garlic and onion bulbs [32], or Bai et al., who evaluate the impact of the foreign trade survey on the accuracy of foreign trade forecasting [33].

With the wide progress of artificial intelligence and the rise of machine learning, it is recommended to combine traditional statistical techniques with new tools that imitate human behavior and learn from the data they consume [34,35], from a metric that quantifies the error between the prediction obtained and the ideal response and feedback strategies that use the error to improve behavior in future events [36]. This approach also has the ability to analyze a large amount of data, and identify patterns, and it also has greater precision and captures the nonlinearity and sporadicity [37,38,39]. For example, González et al. use both statistical models (the linear regression model and Bayesian additive regression model) and machine learning approaches (neural networks, support vector machines, random forests, and the presence–absence approach) to predict possible coffee production scenarios [40].

Machine learning has demonstrated suitable predictive performance in different fields such as energy [41,42], environment [43,44], sustainability [45,46,47], medicine [48,49], economics [50,51], and agriculture [52]. These predictive models protrude for their capacity to produce inputs for well-informed policy and decision-making by analyzing, interpreting, and relating factors [50,53,54,55].

Using digital technology for sustainability is essential to find a balance between technological developments and sustainable practices, such as agroecology [56]. According to [57], rural sustainability, economic growth, and global food security depend on agricultural production. Numerous studies are under this approach, such as Xiong et al. [58], whose research offers various perspectives on food security and possible remedies for international organizations and governments, using machine learning methods such as random forest, K-nearest neighbors, artificial neural networks, and support vector machines. With a different approach, Florez et al. [59] developed a mobile application for plant recognition, using a family of convolutional neural networks to promote sustainable tourism.

The accurate and timely prediction of agricultural production behavior contributes to crop and harvest management, policy formulation, strategies, and decision-making in food security and sustainable development [60]. The neural network model can learn patterns of complex systems, an advantage for making predictions [61,62], as in the case of Kittichotsatsawat et al. [63], Abrougui et al. [64], Almady et al. [65], Thimmegowda et al. [66] and Satpath et al. [67], and Abraham [68], who study the production of coffee, citrus plants, potato, rice, and soybean, respectively. In addition to predicting crop yield or productivity, this machine learning method allows predicting soil quality with water deficit [69] or with specific nutrients [70], predicting outbreaks of fruit diseases [71], and even the composition of an agricultural product using nitrogen fertilization [72].

Neural networks can improve prediction accuracy using historical data and current variables. In this regard, the structure of neural networks has been modified with feedback to process data sequences [73], or to automate the learning of complex data relationships [74], or with convolutional layers to process data with special structures [75]. Mainly, the neural network design depends on the data to be worked with and the objective to obtain. The learning capacity of artificial neural networks can be leveraged by combining it with the reasoning capacity of fuzzy logic, thus providing improved forecasting capabilities compared to traditional models [76]. This combination is called Adaptive Neuro-Fuzzy Inference System (ANFIS).

The model resulting from the combination of fuzzy logic and neural networks has been used for predictive approach in the agricultural sector. For example, Dutta et al. faced a lack of sufficient and accurate collective data to forecast agricultural product inflation and capture the virtue of fuzzy systems and neural networks to rely on time-series analysis with restricted data [77]. Combined with optimization methods, Abd Elfattah uses an ANFIS system to predict coal prices [78]. Factors that influence agricultural productivity, such as soil quality [79], oil spills on land [80], and floods [81], are also predicted with neuro-fuzzy systems. Garg presents a time-series model (based on fuzzy logic) to predict rice crop yield, which facilitates government entities in proper crop resource planning and management [82]. On the other hand, Gunasundari in [83] employs a bee swarm intelligence-inspired algorithm to classify and predict crop yield. As can be seen in these investigations using neuro-fuzzy systems, the interpretability of the prediction is favored, i.e., the results can be conceptually associated with human language. However, it is necessary to explore the advantages that models built from neuro-fuzzy systems can provide in their predictive, and interpretative capacity.

Forecasting Approach

Forecasting consists of determining weather behavior in the future relevant for prediction models. Existing approaches that combine environmental forecasting with fuzzy logic can inspire future extensions of the proposed model. For example, in [84], regarding a pump storage operation, a proposal using energy price as a key input to fuzzy waste water level control is shown. The authors propose a pump control strategy for storage tanks considering energy cost fluctuations. This work demonstrates the importance of energy demand forecasting in balancing supply and demand as it considers a system that incorporates fluctuations in energy prices, liquid level, inflow rate, and storm forecasts as input variables. According to [85], since weather directly impacts human life and is significant in sectors such as agriculture, transportation, energy, and natural disaster management, it is a relevant scientific, social, and economic concern. The authors in [85] analyzed the use of fuzzy logic for weather condition analysis, focused on handling imprecise meteorological data through fuzzy sets. In [86], a combination of fuzzy logic and other multi-criteria decision-making techniques to identify and evaluate the primary climate transition hazards are employed. This method enables the handling of the inherent imprecision of these hazards and the improvement of their description by using language phrases. This analysis emphasizes how crucial it is to have a strong framework to predict and alleviate the effects of climate change. Finally, an approach to evaluating the financial risk to crops in areas with extremely variable weather is presented in [87]. In a low-income economy, authors employ data-driven strategies that utilize Singular Spectrum Analysis (SSA) and Principal Component Analysis (PCA). The risk measure was initially developed for Colombian coffee plantations productive from 2010 to 2019. The research is focused on comprehending how index insurance technologies will affect the strategic economic crops of the nation in the future.

1.2. Approach and Paper Organization

The link among knowledge, technological tools, and the advantageous conditions that Colombia has as a biodiverse country is essential for the creation and implementation of sustainable and comprehensive agricultural systems. In this sense, and in order to generate input for decision-making in the agricultural sector, a neuro-fuzzy model is proposed to predict the behavior of production as a pivotal element.

This paper presents the design and deployment (training–validation) of a neuro-fuzzy model for the production of relevant agricultural products in Colombia. The performance and features are observed regarding the applicability of the fuzzy models (interpretability level).

The document is organized as follows: Section 2 describes the employed techniques and Section 3 the methodology utilized; Section 4 displays the dataset utilized. Meanwhile, Section 5 describes the model design and the implementation process. Then, Section 6 contains the comparison results; Section 7 displays an alternative for selecting training and testing data, and, in Section 8, the interpretability of neuro-fuzzy systems obtained is addressed. Finally, in Section 9 and Section 10, the discussion and conclusions are given.

2. Neuro-Fuzzy Systems and Clustering

The Adaptive Network Fuzzy Inference System (ANFIS) merges the tuning capacity of neural networks and the qualitative approach provided by fuzzy logic, resulting in a technique for learning membership function parameters [88].

ANFIS is a multilayer feedforward network where every node serves a specific purpose both in the input signals and in the set of parameters that belong to that node. The operation can change from neuron to neuron depending on the function (input–output) required by the network. In Figure 2, the connections (arrows) in the network indicate the direction flow of signals [28]. The architecture of an ANFIS network consists of the following layers:

• Layer 1: The function parameters are fitted in every node in this layer. The membership degree value provided by the membership functions’ input is each node’s output. At this stage, $A_i$ and $B_i$ correspond to fuzzy sets.
• Layer 2: Each node located this layer is not adaptive-type. Outputs are the result of multiplying the signals entering the respective node.
• Layer 3: In this layer, each node is fixed (not adaptive-type). The normalized firing strength of the i-th rule in each node is normalized.
• Layer 4: All nodes in this layer are matched to an output-defined function. These nodes have a function defined as $O_{4,i}={\overline{w}}_i{f}_i$, where $f_i$ is the output function of the respective rule i.
• Layer 5: In this layer, the node is fixed and generates the system output by summing all signals coming from the preceding nodes [89].

Fuzzy C-Means

Fuzzy C-Means is an unsupervised clustering algorithm that treats a data item as belonging to multiple clusters based on membership. This algorithm allows the clustering of data in a multidimensional space. By using fuzzy partitioning, data items can be assigned to more than one cluster according to the membership degree matrix, which contains values ranging from 0 to 1. At each iteration, the cluster centers and membership values are updated, and the algorithm finds the cluster center that minimizes the dissimilarity function.

For each datum in each cluster is given a membership value according to how distant it is from the cluster center. The datum with the highest membership value within a cluster is located near that cluster center. All of the data items’ membership values within a cluster must add up to one.

One advantage of the algorithm is that each data item can have a membership value to more than two clusters; therefore, tolerance measures can be established based on the clustering accuracy required for each problem.

To perform the classification using the Fuzzy C-Mean algorithm, the object function J given in Equation (1) is minimized at each iteration.

$$J=\sum _{i=1}^N\sum _{j=1}^C{\left({\mu }_{ij}\right)}^m{\parallel x_i-c_j\parallel }^2$$

(1)

where C is the quantity of clusters, N is the total data, ${\mu }_{ij}$ is the membership degree of data item $x_i$ in cluster j, the parameter m is the fuzzification coefficient, and $c_j$ is the center vector for cluster j. The metric $\parallel x_i-c_j\parallel$ computes the proximity of the datum to the center of the cluster. The algorithm modifies the center of the vector for every cluster at each iteration. The membership degree for each datum is calculated as follows:

$${\mu }_{ij}={\displaystyle \frac{1}{{\displaystyle \sum _{k=1}^C}{\left(\frac{\parallel x_i-c_j\parallel }{\parallel x_i-c_k\parallel }\right)}^{\frac{2}{m-1}}}}$$

(2)

The fuzzification coefficient m allows us to establish the clustering tolerance where $1<m<\infty$; therefore, m controls the level of overlapping between clusters [90]. Finally, the center of the respective cluster $c_j$ is calculated as follows:

$$c_j={\displaystyle \frac{{\displaystyle \sum _{i=1}^N}{\left({\mu }_{ij}\right)}^m{x}_i}{{\displaystyle \sum _{i=1}^N}{\left({\mu }_{ij}\right)}^m}}$$

(3)

3. Methodology

The research carried out has a quantitative approach since it uses data from the agricultural sector and employs metrics to determine the capacity of the developed predictive models and their interpretability. It also has an experimental part in which different configurations for neural networks and neuro-fuzzy systems are evaluated. Figure 3 presents a summary of the phases carried out.

• Data collection: Data collected by the National Administrative Department of Statistics in the 2019 National Agricultural Survey were used. These data provide information on the agricultural production of various products in the five natural Colombian regions from 2012 to 2019.
  To carry out the training process, all input and output variables are normalized to have values between 0 and 1, and then for the simulation, they are scaled to their real values. Data imputation is also performed.
• Model development: The model is implemented using neuro-fuzzy systems. The output corresponds to the production of the respective agricultural product in tons. The neuro-fuzzy models are developed using $80\%$ of the data for training. Different input–output configurations are proposed.
• Validation and evaluation: In this stage, $20\%$ of the data are used for validation. Tables are built to show the mean squared error (MSE) results for training and validation data, where the minimum, maximum, average, and standard deviation (STD) values are presented. To determine the best configuration of the neuro-fuzzy system, an experimental design is performed considering different configurations.
• Analysis of results: The best model can be determined by considering the best values obtained from the MSE in validation. Additionally, to obtain better interpretability, the Sugeno-type system obtained is converted to Mamdani, with linear functions and output constants. According to the output membership functions resulting from the systems, their capacity to fit the data and interpret it is analyzed.

4. Dataset

The dataset employed is obtained from DANE (https://www.dane.gov.co/index.php/estadisticas-por-tema/agropecuario/encuesta-nacional-agropecuaria-ena (accessed on 5 August 2024)), which provides a dataset with 215 usable entries acquired from 2012 to 2019 and records the production of traditional agricultural products (in tons) as displayed in Figure 4.

The model is implemented using neuro-fuzzy systems. The output corresponds to the production of the respective agricultural product in tons. The considered inputs of the system are as follows:

• Type of product: banana, cocoa, coffee, sugar cane, orange, and plantain.
• Natural region of Colombia: Andean, Caribbean, Pacific, Orinoco, and Amazon.
• Year of production (optional): from 2012 to 2019.
• Planted area: in hectares.
• Productive area: in hectares.
• Previous production value (optional): in tons.

For model training and validation, the data for inputs and outputs are normalized in the range $[0,1]$, which avoids disproportionate values and shapes of the fuzzy sets used in the model.

5. Models Description and Implementation

For the model development, different configurations of the previous variables can be used; therefore, different alternatives are proposed. Figure 5 displays the models considered according to the input–output configurations.

Figure 5a shows the configuration of model $M_1$, where a total of 14 inputs are employed. The description of the respective inputs and outputs of the model is as follows:

• $X_1,\dots X_6$: entries associated with each product encoded in binary.
• $W_1,\dots W_5$: entries associated with each region encoded in binary.
• U: input associated with the year in which the production measurement is made.
• $Z_1$: planted area (input).
• $Z_2$: productive area (input).
• Y: output corresponding to production.

The schematic representation of model $M_2$ is observed in Figure 5b; similar to the $M_1$ model, 14 inputs are used; however, instead of using U (the year of production), T is used corresponding to the previous output value. The description of the input variables and the output of this model is as follows:

• $X_1,\dots X_6$: entries associated with each product encoded in binary.
• $W_1,\dots W_5$: entries associated with each region encoded in binary.
• T: the previous value of the output.
• $Z_1$: planted area (input).
• $Z_2$: productive area (input).
• Y: output corresponding to production.

Figure 5c displays the configuration of model $M_3$, which has a smaller number of inputs, five in total. In this way, the aim is to have a more compact model. The description of the respective inputs and outputs of the model is as follows:

• X: input associated with each product coded according to its production level (from lowest to highest).
• W: input associated with each region coded according to its production level (from lowest to highest).
• U: input associated with the year in which the production measurement is made.
• $Z_1$: planted area (input).
• $Z_2$: productive area (input).
• Y: output corresponding to production.

The representation of model $M_4$ is presented in Figure 5d; in the same way, to obtain a compact model, the $M_4$ model employs five inputs. In contrast to the $M_3$ model that utilizes U (year of production), model $M_4$ employs T, which is the previous value of the output. The description of the input variables and the output of this model is as follows:

• X: input associated with each product coded according to its production level (from lowest to highest).
• W: entries associated with each region coded according to its production level (from lowest to highest).
• T: previous value of the output.
• $Z_1$: planted area (input).
• $Z_2$: productive area (input).
• Y: output corresponding to production.

The ordinary codification for product and geographic region is given considering the amount of production (tons); it first appears the product and regions with low production and in the end products and regions with the maximum production. The binary codification for products is displayed in

Table 1. Binary codification for products.

Product	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$
Banano	1	0	0	0	0	0
Cocoa	0	1	0	0	0	0
Coffee	0	0	1	0	0	0
Sugar cane	0	0	0	1	0	0
Orange	0	0	0	0	1	0
Plantain	0	0	0	0	0	1

; meanwhile,

Table 2. Binary codification for geographic region.

Region	${\mathit{W}}_1$	${\mathit{W}}_2$	${\mathit{W}}_3$	${\mathit{W}}_4$	${\mathit{W}}_5$
Andean	1	0	0	0	0
Caribbean	0	1	0	0	0
Pacific	0	0	1	0	0
Orinoco	0	0	0	1	0
Amazon	0	0	0	0	1

shows the binary codification for geographic region.

5.1. Implementation Process

This section shows the results of implementing neuro-fuzzy systems for each model considered. For neuro-fuzzy models, an initial assignment based on Fuzzy C-Means (FCM) is used, which performs a random assignment of the clusters’ initial centers. Considering the random initialization of the model parameters, each configuration is trained 20 times.

For implementation, 80% of the data are used in training process and 20% for validation. In this way, the respective tables show the MSE results for training and validation data, where the minimum, maximum, average, and standard deviation values are presented.

To carry out the training process, all input–output variables are normalized to have values between 0 and 1, and then they are scaled to their real values for the simulation.

Linear and constant output functions are employed for neuro-fuzzy systems with the Sugeno model. For these systems, the configurations are as follows:

• Output membership functions: linear and constant.
• Number of clusters: 2, 3, 4, and 5.
• Fuzzy partition exponent: 1.1, 2, 3, and 4.

5.1.1. Implementation for Model $M_1$

Regarding the training stage, the statistical values as minimum, maximum, mean, and STD obtained for each configuration is displayed in

Table 3. MSE results for model $M_1$ using training data.

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0080970067	0.0006520287	0.0005858619	0.0006163937	0.0249175713	0.0052764293	0.0052706751	0.0052861664
Min	0.0014464026	0.0003501009	0.0003490070	0.0003447610	0.0117108493	0.0051697209	0.0051806223	0.0052047847
Mean	0.0067106225	0.0004471362	0.0005047289	0.0004856341	0.0189376298	0.0052271251	0.0052329461	0.0052357157
STD	0.0016604761	0.0001058163	0.0000907092	0.0001115018	0.0036777510	0.0000277730	0.0000244748	0.0000259180
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0073218711	0.0002474675	0.0002564712	0.0002637826	0.0217646402	0.0007907762	0.0007628882	0.0008177421
Min	0.0023902338	0.0001435086	0.0001356283	0.0001550074	0.0123038452	0.0005510538	0.0005491428	0.0005835823
Mean	0.0056626572	0.0001842301	0.0001735653	0.0001830093	0.0164622460	0.0006806250	0.0006540321	0.0007090027
STD	0.0015862618	0.0000262825	0.0000322725	0.0000241794	0.0042100060	0.0000700964	0.0000631287	0.0000615479
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0045380144	0.0001090544	0.0001342405	0.0001131843	0.0211840993	0.0006555216	0.0006108346	0.0010487058
Min	0.0016559719	0.0000486169	0.0000498980	0.0000480308	0.0120151509	0.0003943222	0.0004207806	0.0004225335
Mean	0.0038691020	0.0000785151	0.0000855817	0.0000845127	0.0130940059	0.0005117056	0.0005137023	0.0005594526
STD	0.0007621738	0.0000165233	0.0000161395	0.0000155993	0.0024534777	0.0000624163	0.0000500255	0.0001286913
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0060855859	0.0000606032	0.0000611575	0.0000671140	0.0231051217	0.0010458541	0.0005400185	0.0005874693
Min	0.0009424633	0.0000362497	0.0000366652	0.0000369707	0.0027984844	0.0004273985	0.0003833400	0.0003728754
Mean	0.0036256780	0.0000495981	0.0000449943	0.0000463380	0.0123707484	0.0005140109	0.0004550287	0.0004679564
STD	0.0012175436	0.0000074309	0.0000058180	0.0000076688	0.0041170621	0.0001340749	0.0000406167	0.0000628095

. Meanwhile,

Table 4. MSE for model $M_1$ using validation dada.

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0177270665	0.0017306865	0.0016536181	0.0016825147	0.0380765329	0.0033676067	0.0033351392	0.0033845704
Min	0.0049653296	0.0007409856	0.0007724999	0.0009364512	0.0262334413	0.0032905059	0.0032860476	0.0032862555
Mean	0.0147951478	0.0013472609	0.0014447894	0.0013908057	0.0323204918	0.0033077720	0.0033054794	0.0033150417
STD	0.0029340639	0.0002370994	0.0002461751	0.0002274188	0.0038952120	0.0000159508	0.0000134508	0.0000277304
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0161630609	0.0015018339	0.0016231431	0.0031428354	0.0386211841	0.0013500815	0.0014401097	0.0011474682
Min	0.0049583293	0.0004787222	0.0003209476	0.0003192737	0.0262220038	0.0007167421	0.0007071209	0.0007056395
Mean	0.0129764764	0.0008683176	0.0011153713	0.0009696354	0.0304980798	0.0008774251	0.0009008648	0.0008391093
STD	0.0029232616	0.0002887291	0.0003697260	0.0006032062	0.0049499088	0.0001941584	0.0001999813	0.0001262611
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0115235927	0.0048586801	0.0026209845	0.0131033631	0.0373192962	0.0011724928	0.0011374321	0.0020260329
Min	0.0030184090	0.0003339608	0.0003691388	0.0002626520	0.0259555461	0.0007780004	0.0006801332	0.0007437698
Mean	0.0094294556	0.0017391407	0.0010667581	0.0021194622	0.0275721857	0.0009383672	0.0009151844	0.0010106556
STD	0.0025344718	0.0014325040	0.0005853278	0.0027979129	0.0032272685	0.0001131231	0.0001031128	0.0002552771
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0146152499	0.0239827729	0.0023409806	0.0092491020	0.0372450722	0.0019471766	0.0010586234	0.0017299126
Min	0.0027474770	0.0003747091	0.0003172813	0.0004326232	0.0072052374	0.0005863290	0.0005309827	0.0005327444
Mean	0.0094949746	0.0020764764	0.0009556653	0.0015936895	0.0257768722	0.0009473666	0.0008034998	0.0008209868
STD	0.0029479844	0.0051699035	0.0005913655	0.0020833794	0.0062185545	0.0003490905	0.0001212417	0.0002468106

encloses the values from using validation data. For the $M_1$ model, the best values obtained are as follows:

• Linear:

  –

  MSE training: $0.3625\times {10}^{-4}$.

  –

  MSE validation: $2.6265\times {10}^{-4}$.

• Constant:

  –

  MSE training: $3.7288\times {10}^{-4}$.

  –

  MSE validation: $5.3098\times {10}^{-4}$.

5.1.2. Implementation for Model $M_2$

For the $M_2$ configuration, the MSE training results can be seen in

Table 5. MSE results for model $M_2$ using training data.

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0043934754	0.0006233736	0.0006295172	0.0005384344	0.0293803543	0.0055045847	0.0054772233	0.0057506063
Min	0.0029855557	0.0003555075	0.0003483190	0.0003494133	0.0115164188	0.0049961235	0.0050100447	0.0050115597
Mean	0.0041232171	0.0004068735	0.0004425067	0.0004043567	0.0236613301	0.0051179760	0.0050968506	0.0051524248
STD	0.0003276122	0.0000666921	0.0000907333	0.0000507982	0.0052578472	0.0001238407	0.0001037868	0.0001743111
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0041614789	0.0004674037	0.0004165398	0.0004781249	0.0255641611	0.0034971303	0.0035295927	0.0035163343
Min	0.0027758755	0.0001358777	0.0001239747	0.0001173833	0.0163570851	0.0007398282	0.0007050922	0.0007424269
Mean	0.0032761198	0.0002261646	0.0002160883	0.0002296571	0.0203340710	0.0024660962	0.0028801868	0.0025841186
STD	0.0006050975	0.0001130203	0.0000984471	0.0001305877	0.0036724829	0.0010968779	0.0010478754	0.0011448758
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0028167201	0.0001171199	0.0001098330	0.0002845319	0.0239702270	0.0035729387	0.0036808833	0.0033936204
Min	0.0010043421	0.0000352087	0.0000490594	0.0000498544	0.0144242214	0.0005373898	0.0005598528	0.0005331879
Mean	0.0026057086	0.0000726908	0.0000656849	0.0000800069	0.0178406682	0.0012369186	0.0011307390	0.0011542260
STD	0.0004307405	0.0000218625	0.0000158173	0.0000502282	0.0016170392	0.0008345779	0.0007968789	0.0007561475
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0028387853	0.0000411117	0.0000468695	0.0000436516	0.0176777479	0.0033342812	0.0032632847	0.0031787204
Min	0.0005556958	0.0000146977	0.0000192886	0.0000153475	0.0081891746	0.0003173075	0.0003215709	0.0005053861
Mean	0.0023273360	0.0000304944	0.0000287421	0.0000297959	0.0162558365	0.0008894551	0.0009721553	0.0007936620
STD	0.0007763069	0.0000065186	0.0000075674	0.0000071628	0.0023087981	0.0007353002	0.0007013533	0.0006100003

where the statistical summary (min, max, mean, and STD) is detailed.

Table 6. MSE results for model $M_2$ using validation data.

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0025004015	0.0012171711	0.0012486271	0.0012544732	0.0200408172	0.0080520530	0.0079636284	0.0082941530
Min	0.0017151553	0.0003584566	0.0003429069	0.0003413725	0.0044835418	0.0076621132	0.0076718162	0.0076801546
Mean	0.0019799632	0.0005580612	0.0006626626	0.0005890328	0.0146996705	0.0077323982	0.0077181862	0.0077600574
STD	0.0002396672	0.0002021701	0.0002678870	0.0002564924	0.0050679074	0.0000931165	0.0000656214	0.0001422642
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0025334958	0.0057841969	0.0086395508	0.0235916360	0.0173879882	0.0055505935	0.0055594446	0.0055526054
Min	0.0017628329	0.0003211319	0.0003893655	0.0002791330	0.0075944727	0.0009849162	0.0009187042	0.0009509613
Mean	0.0023136221	0.0013952595	0.0016740056	0.0023678043	0.0102814095	0.0038696104	0.0046698654	0.0040424918
STD	0.0002383873	0.0012532080	0.0017825421	0.0050587283	0.0036520839	0.0019060835	0.0016200637	0.0019767727
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0025996079	0.0072061428	0.0093240427	0.1591769209	0.0177997723	0.0057965730	0.0058095432	0.0055260254
Min	0.0012774951	0.0008266469	0.0004873639	0.0006863235	0.0023396645	0.0008652328	0.0009536177	0.0009072779
Mean	0.0023362433	0.0019367101	0.0023185716	0.0098046723	0.0078798024	0.0021384581	0.0019968072	0.0020126783
STD	0.0002543260	0.0014902849	0.0020416854	0.0351940641	0.0026174278	0.0015932382	0.0013030162	0.0011910029
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0066337975	0.0069880224	0.0757956205	0.0031532754	0.0168650985	0.0061098327	0.0066425154	0.0062429654
Min	0.0012631418	0.0008839887	0.0006815937	0.0012493217	0.0022755568	0.0008785804	0.0006645863	0.0005455133
Mean	0.0025081702	0.0024120951	0.0063911199	0.0020131347	0.0069720418	0.0016237977	0.0022132220	0.0015372202
STD	0.0010402884	0.0016886826	0.0167336134	0.0004807928	0.0031121560	0.0015011146	0.0020694847	0.0012245736

was obtained by employing validation data, the best values obtained are as follows:

• Linear:

  –

  MSE training: $0.1469\times {10}^{-4}$.

  –

  MSE validation: $2.7913\times {10}^{-4}$.

• Constant:

  –

  MSE training: $3.1731\times {10}^{-4}$.

  –

  MSE validation: $5.4551\times {10}^{-4}$.

5.1.3. Implementation for Model $M_3$

Regarding the training stage,

Table 7. MSE results for model $M_3$ using training data.

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0069982325	0.0044072995	0.0029144193	0.0043935860	0.0119370878	0.0109344334	0.0078518059	0.0078569119
Min	0.0053127815	0.0025352697	0.0023432721	0.0023304691	0.0113277092	0.0079464922	0.0076566094	0.0077534165
Mean	0.0064576709	0.0036425895	0.0024385145	0.0025172939	0.0116541721	0.0092527565	0.0077944349	0.0078061320
STD	0.0006455609	0.0005778870	0.0001791452	0.0004676873	0.0002782727	0.0013988212	0.0000536557	0.0000208386
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0064857536	0.0023260011	0.0017355063	0.0017512598	0.0123169849	0.0079584507	0.0048800610	0.0046525032
Min	0.0064855693	0.0011245956	0.0008549682	0.0009070289	0.0123119506	0.0064035964	0.0043256984	0.0043299794
Mean	0.0064856187	0.0017165399	0.0012871068	0.0015373040	0.0123143437	0.0073239501	0.0044097660	0.0044146060
STD	0.0000000583	0.0002326932	0.0003444591	0.0002368792	0.0000023760	0.0002998893	0.0001240684	0.0000902811
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0064857149	0.0018103145	0.0011540461	0.0011394766	0.0123188079	0.0050374271	0.0039618373	0.0040610837
Min	0.0064643914	0.0012150584	0.0005462101	0.0004670573	0.0123140631	0.0045186805	0.0023164727	0.0023652725
Mean	0.0064844557	0.0015622306	0.0008899317	0.0008583213	0.0123147456	0.0048044510	0.0032953912	0.0029579023
STD	0.0000047237	0.0001604705	0.0001624248	0.0001883688	0.0000010522	0.0001379490	0.0004582761	0.0005270630
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0064846653	0.0014671957	0.0007941515	0.0007146169	0.0123109728	0.0045526289	0.0027694723	0.0027418833
Min	0.0057198122	0.0012111682	0.0002903829	0.0002635031	0.0082344246	0.0044703654	0.0017080855	0.0017304822
Mean	0.0063699165	0.0013111511	0.0004796575	0.0004897928	0.0116994137	0.0045110743	0.0022679496	0.0022310550
STD	0.0002801882	0.0000622567	0.0001240400	0.0001149768	0.0014933809	0.0000225322	0.0002620396	0.0002977945

displays the max, min, mean, and STD metrics obtained for the considered configurations. Meanwhile, the results in

Table 8. MSE results for model $M_3$ using validation data.

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0172352278	0.0122765967	0.0064946191	0.0121738573	0.0224340945	0.0291427487	0.0120644621	0.0121034889
Min	0.0152803532	0.0063540796	0.0056154445	0.0055803549	0.0212473722	0.0124733872	0.0098370246	0.0110405045
Mean	0.0163954571	0.0088372379	0.0057546236	0.0060467222	0.0219563922	0.0200806291	0.0114545983	0.0116125450
STD	0.0008384501	0.0018361548	0.0002853881	0.0014635442	0.0004787592	0.0077178175	0.0006250579	0.0002143604
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0133220201	0.0072065295	0.0049744339	0.0052839471	0.0267205716	0.0165710026	0.0104229922	0.0094742808
Min	0.0133170225	0.0025137405	0.0015224211	0.0018813195	0.0267113535	0.0124020612	0.0088845156	0.0087871646
Mean	0.0133192786	0.0055544990	0.0032927033	0.0043509156	0.0267169661	0.0130355495	0.0091337105	0.0091131271
STD	0.0000015980	0.0009132302	0.0013481999	0.0009846865	0.0000034036	0.0010974823	0.0003228145	0.0001548588
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0133465021	0.0548908212	0.0055334731	0.0056975141	0.0267270703	0.0127582696	0.0088304911	0.0081842984
Min	0.0133253104	0.0053407206	0.0013708250	0.0010200057	0.0267112382	0.0111728886	0.0052701950	0.0052908074
Mean	0.0133416334	0.0111383126	0.0032425136	0.0030137443	0.0267139164	0.0123813723	0.0069063848	0.0062151911
STD	0.0000039935	0.0119201163	0.0011019821	0.0013588233	0.0000037412	0.0004618051	0.0010659602	0.0009436170
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0133356916	0.1606264866	0.0058242326	0.0062292837	0.0267069815	0.0126451056	0.0064600390	0.0064658082
Min	0.0117594177	0.0045636519	0.0008688279	0.0006732161	0.0159652613	0.0125385554	0.0043919896	0.0040299573
Mean	0.0130993767	0.0371743021	0.0027645668	0.0029558510	0.0250956281	0.0125947296	0.0051014085	0.0051233568
STD	0.0005765763	0.0503018352	0.0014379277	0.0016549429	0.0039345795	0.0000275380	0.0006025698	0.0007006443

were gathered through validation data. The best values obtained for model $M_3$ are as follows:

• Linear:

  –

  MSE training: $2.6350\times {10}^{-4}$.

  –

  MSE validation: $6.7322\times {10}^{-4}$.

• Constant:

  –

  MSE training: $17.0000\times {10}^{-4}$.

  –

  MSE validation: $40.0000\times {10}^{-4}$.

5.1.4. Implementation for Model $M_4$

Considering the implementation for model $M_4$,

Table 9. MSE results for model $M_4$ using training data.

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0048506596	0.0015469246	0.0015480940	0.0017401581	0.0178432591	0.0111986106	0.0097820331	0.0088869809
Min	0.0048506304	0.0012754479	0.0012626486	0.0012602458	0.0178432500	0.0096069131	0.0086511995	0.0081913207
Mean	0.0048506367	0.0014055594	0.0013655549	0.0015285514	0.0178432548	0.0099709277	0.0093596313	0.0084057658
STD	0.0000000085	0.0001149538	0.0001067231	0.0001859197	0.0000000018	0.0003142496	0.0003490482	0.0003096864
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0048484363	0.0016329469	0.0016935848	0.0016138973	0.0178522968	0.0061628552	0.0056893464	0.0079089000
Min	0.0048484322	0.0015103215	0.0016644175	0.0008647555	0.0178513826	0.0061492122	0.0056151903	0.0048539310
Mean	0.0048484346	0.0015829170	0.0016753089	0.0010192659	0.0178518055	0.0061552988	0.0056652303	0.0077344605
STD	0.0000000018	0.0000444477	0.0000086412	0.0001466348	0.0000002461	0.0000041701	0.0000141565	0.0006782212
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0048482991	0.0014018601	0.0014409998	0.0013582577	0.0287252744	0.0057405322	0.0108752987	0.0105797261
Min	0.0033456302	0.0012224327	0.0012299191	0.0007079524	0.0050223410	0.0056009590	0.0054390467	0.0015664727
Mean	0.0039960672	0.0012871517	0.0013648569	0.0012148570	0.0105779337	0.0056358986	0.0098777599	0.0059346292
STD	0.0007045022	0.0000622807	0.0000914213	0.0002153173	0.0073881520	0.0000258046	0.0018572749	0.0026029491
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0047792054	0.0011756434	0.0012008834	0.0012184889	0.0287250489	0.0057751863	0.0106043722	0.0105243163
Min	0.0033373937	0.0011137324	0.0010482460	0.0003951683	0.0050283807	0.0057261509	0.0104706630	0.0017787307
Mean	0.0039215831	0.0011469064	0.0011778816	0.0008540607	0.0062237791	0.0057414341	0.0105654177	0.0047988146
STD	0.0004966080	0.0000197239	0.0000387720	0.0002731092	0.0052962689	0.0000146702	0.0000265286	0.0023025050

contains the results (max, min, average, and STD) after the training process for each configuration of the fuzzy system. After the training procedure,

Table 10. MSE results for model $M_4$ using validation data.

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0013853527	0.0027097016	0.0027802999	0.0024513282	0.0076208483	0.0101675704	0.0101137009	0.0107280161
Min	0.0013825454	0.0010521765	0.0009316701	0.0008875321	0.0076208373	0.0092721406	0.0092977764	0.0089434370
Mean	0.0013851873	0.0019912551	0.0020739087	0.0014539695	0.0076208389	0.0099682814	0.0096148623	0.0101331653
STD	0.0000006228	0.0006970058	0.0005801907	0.0005032252	0.0000000029	0.0002441021	0.0001789323	0.0007498761
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0013959173	0.0040678350	0.0024629558	0.0418754454	0.0076244810	0.0026626105	0.0035996567	0.0108567520
Min	0.0013957871	0.0010439709	0.0022345675	0.0009070154	0.0076234150	0.0026339686	0.0032497170	0.0056295300
Mean	0.0013958398	0.0021272849	0.0022622197	0.0051085485	0.0076237915	0.0026445806	0.0034718168	0.0104920583
STD	0.0000000631	0.0011529093	0.0000484240	0.0091267681	0.0000004332	0.0000100515	0.0000613838	0.0011464662
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0013973232	0.0736593288	0.0073906263	0.0037214665	0.0182143549	0.0065044786	0.0042540944	0.0086863591
Min	0.0007338815	0.0008559171	0.0008669978	0.0008956907	0.0013061990	0.0058693040	0.0024263960	0.0014432050
Mean	0.0011426088	0.0057090444	0.0020340595	0.0017610129	0.0042577531	0.0062652735	0.0039959552	0.0058096363
STD	0.0002132448	0.0160362096	0.0017869834	0.0007081161	0.0044012690	0.0001750238	0.0005820412	0.0021182686
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0013638019	0.0134376327	0.0039151203	0.0039882157	0.0182299648	0.0067477803	0.0044040427	0.0075041045
Min	0.0009420390	0.0008632560	0.0008559350	0.0006603341	0.0013047184	0.0037314160	0.0041232760	0.0016793748
Mean	0.0011764402	0.0018769806	0.0029698930	0.0015041857	0.0022570614	0.0054866117	0.0042231588	0.0046657596
STD	0.0001270135	0.0027839459	0.0011450323	0.0009118754	0.0037632159	0.0006430364	0.0000479420	0.0020469260

was obtained through validation data. The best obtained values are as follows:

• Linear:

  –

  MSE training: $3.9517\times {10}^{-4}$.

  –

  MSE validation: $6.6033\times {10}^{-4}$.

• Constant:

  –

  MSE training: $16.0000\times {10}^{-4}$.

  –

  MSE validation: $13.0000\times {10}^{-4}$.

6. Comparison Results

Regarding the four proposed models, the results comparison is shown in this section. The simulation outcomes for the best configuration found for each model utilizing training and validation data are shown in Figure 6. In general, a superior fit is observed with fuzzy systems using linear functions in the output.

Table 11. MSE results taking the best configuration determined from each model.

	Linear		Constant
Model	Training	Validation	Training	Validation
$M_1$	$0.3625\times {10}^{-4}$	$2.6265\times {10}^{-4}$	$3.7288\times {10}^{-4}$	$5.3098\times {10}^{-4}$
$M_2$	$0.1469\times {10}^{-4}$	$2.7913\times {10}^{-4}$	$3.1731\times {10}^{-4}$	$5.4551\times {10}^{-4}$
$M_3$	$2.6350\times {10}^{-4}$	$6.7322\times {10}^{-4}$	$17.0000\times {10}^{-4}$	$40.0000\times {10}^{-4}$
$M_4$	$3.9517\times {10}^{-4}$	$6.6033\times {10}^{-4}$	$16.0000\times {10}^{-4}$	$13.0000\times {10}^{-4}$

shows the MSE value obtained for each model. Regarding the results with validation data, the best model with linear and constant neuro-fuzzy systems is $M_1$.

7. Alternative for Training and Testing Data Selection

Some limitations identified for model training and validation are the small amount of data and unavailable data segments, which is why random sampling was used to select the training and test data. However, for a more forecasting approach, data from previous years should be used to predict the next. For this reason, an experimental test is carried out to demonstrate this approach, where data from the 2012–2017 period are used for training and data from 2018–2019 for testing. Given the limitations, this test is carried out with the best configuration found in the previous section (model $M_1$). For subsequent work where a complete set of historical data can be obtained, the models can be adjusted using this approach.

Using this selection for training and validation data, the statistical values as minimum, maximum, mean, and STD obtained for each configuration are displayed in

Table 12. MSE results for model $M_1$ using training data (2012–2017).

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0053292600	0.0004060109	0.0004438863	0.0004180377	0.0196978942	0.0050409284	0.0050386693	0.0050158100
Min	0.0027803512	0.0001827136	0.0001819501	0.0001812179	0.0081160434	0.0031746558	0.0029127259	0.0034310978
Mean	0.0046439670	0.0002644968	0.0002316052	0.0002539799	0.0156070691	0.0041595898	0.0041126637	0.0043089835
STD	0.0009531646	0.0000784435	0.0000745914	0.0000930047	0.0032238989	0.0005918965	0.0006206530	0.0004791798
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0050750971	0.0001243341	0.0001405120	0.0001257583	0.0175915709	0.0005825397	0.0005492823	0.0005491278
Min	0.0009980269	0.0000712486	0.0000810241	0.0000769846	0.0090454796	0.0004188570	0.0004352400	0.0004177740
Mean	0.0032113183	0.0000989969	0.0001078496	0.0000973204	0.0119598296	0.0004916083	0.0004760829	0.0004715419
STD	0.0011408935	0.0000156722	0.0000172312	0.0000161272	0.0029840736	0.0000491722	0.0000301938	0.0000337984
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0027485577	0.0000549521	0.0000693590	0.0000597283	0.0153248692	0.0004350121	0.0005236697	0.0005168433
Min	0.0004909324	0.0000268831	0.0000274818	0.0000263048	0.0085658775	0.0002851397	0.0002751776	0.0003024105
Mean	0.0023312164	0.0000388382	0.0000399444	0.0000403465	0.0104176357	0.0003411169	0.0003656765	0.0003709913
STD	0.0005935942	0.0000079775	0.0000103665	0.0000102101	0.0013559253	0.0000479596	0.0000741969	0.0000568863
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0025455464	0.0000231807	0.0000421453	0.0000366212	0.0100999430	0.0008487727	0.0003951713	0.0005972200
Min	0.0004977681	0.0000129153	0.0000057140	0.0000063398	0.0051396564	0.0002645714	0.0002067919	0.0002349152
Mean	0.0021095840	0.0000178495	0.0000183418	0.0000167561	0.0095058617	0.0003558566	0.0002984997	0.0003407384
STD	0.0007972064	0.0000029969	0.0000074028	0.0000066778	0.0013287222	0.0001256467	0.0000504845	0.0000877377

. Meanwhile,

Table 13. MSE for model $M_1$ using validation dada (2018–2019).

Clusters	2
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0235563819	0.0112156916	0.0105737760	0.0117430602	0.0455201380	0.0229539038	0.0230566278	0.0232422471
Min	0.0173422891	0.0025090957	0.0021403051	0.0026365829	0.0287684057	0.0047471863	0.0047836978	0.0046726179
Mean	0.0218417286	0.0055234911	0.0059155567	0.0062953060	0.0386389990	0.0193798864	0.0201932601	0.0199095693
STD	0.0023511721	0.0027038297	0.0024340078	0.0029942847	0.0051706466	0.0052305545	0.0040612598	0.0053811967
Clusters	3
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0226924509	0.0134878641	0.0253683213	0.0630020225	0.0427741650	0.0031073942	0.0031819043	0.0030090038
Min	0.0067113956	0.0028004050	0.0036740169	0.0030820502	0.0291867968	0.0017104703	0.0018080674	0.0018142282
Mean	0.0176036047	0.0080245692	0.0082541929	0.0148593111	0.0324997501	0.0023298785	0.0022610204	0.0022041344
STD	0.0036603990	0.0031563687	0.0052181174	0.0159469333	0.0047415560	0.0004391231	0.0003605095	0.0002996196
Clusters	4
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.0170129544	0.1405747492	0.0646945647	0.1686702544	0.0421768214	0.0049480763	0.0033480070	0.0033080639
Min	0.0026643098	0.0027932122	0.0027656525	0.0038577444	0.0244981629	0.0019479774	0.0018024609	0.0015143868
Mean	0.0142102810	0.0212833587	0.0126934967	0.0253835209	0.0298068348	0.0028120517	0.0026689420	0.0025681594
STD	0.0044322575	0.0333101615	0.0152146240	0.0413667737	0.0031255302	0.0006155931	0.0004353978	0.0004861269
Clusters	5
Output Functions	Linear				Constant
Exponent	1.1	2.0	3.0	4.0	1.1	2.0	3.0	4.0
Max	0.2618335799	0.0987098440	0.1313852727	0.0752032269	0.0368387540	0.0061134887	0.0108268125	0.0063217016
Min	0.0022546367	0.0033117612	0.0023646510	0.0040732959	0.0223592712	0.0021569927	0.0022836379	0.0017891166
Mean	0.0261613324	0.0168706148	0.0191141660	0.0223257217	0.0289779644	0.0031098090	0.0041498095	0.0033573824
STD	0.0556222357	0.0207570155	0.0275837322	0.0208547204	0.0026656601	0.0009840858	0.0026467951	0.0011534380

contains the values obtained using validation data. In this way, for the $M_1$ model the best values obtained are as follows:

• Linear:

  –

  MSE training: $0.05714\times {10}^{-4}$.

  –

  MSE validation: $21.0000\times {10}^{-4}$.

• Constant:

  –

  MSE training: $2.0679\times {10}^{-4}$.

  –

  MSE validation: $15.0000\times {10}^{-4}$.

Figure 7 displays the simulation results taking the best system obtained in the validation process. This figure shows the spaces associated with the lack of available data for training, which lowers the performance of the system to make predictions.

Table 14. MSE results taking best configurations for $M_1$.

	Linear		Constant
Data Selection	Training	Validation	Training	Validation
Random sampling	$0.3625\times {10}^{-4}$	$2.6265\times {10}^{-4}$	$3.7288\times {10}^{-4}$	$5.3098\times {10}^{-4}$
Segmented data	$0.0571\times {10}^{-4}$	$21.0000\times {10}^{-4}$	$2.0679\times {10}^{-4}$	$15.0000\times {10}^{-4}$

shows the MSE value obtained for model $M_1$ using random sampling data and segmented data (regarding years). As can be seen in this table, the strategy of segmenting data for training and validation according the period (2012–2017 for training and 2018–2019 for testing) presents a better MSE in the training process, but, in validation, the MSE worsens, which is why it is used random sampling for training and validation due to the low amount of data and incomplete data available in this application.

8. Interpretability of Neuro-Fuzzy Models

In order to obtain the best interpretability of neuro-fuzzy systems, the Sugeno-type system is converted into Mamdani-type using the Fuzzy Logic Designer tool of MATLAB (R2023a) [91]. When the Sugeno system is transformed into a Mamdani system, the obtained Mamdani system is composed in the output with triangular membership functions where the center is located on the value of the constant term of the output functions of the Sugeno system. The width of the triangular fuzzy set depends on the coefficients of the linear functions of the Sugeno system. By performing the conversion, the following Mamdani-type fuzzy systems are obtained:

• FIS-L: system attained by converting the Sugeno system with linear functions at the output.
• FIS-C: system determined from the conversion of the Sugeno system with constant functions at the output.

Considering the system achieved from the conversion of the Sugeno system with linear functions at the output, Figure 8a shows the structure of the Mamdani-type fuzzy system; Figure 8b presents the output fuzzy sets, and, finally, Figure 9 contains the set of rules of the Mamdani-type fuzzy system FIS-L.

On the other hand, having the system obtained from the conversion of the Sugeno system with constant functions at the output (FIS-C), Figure 10a shows the structure of the Mamdani-type fuzzy system. Additionally, Figure 10b contains the output fuzzy sets, and, finally, Figure 11 presents the set of rules of the Mamdani-type fuzzy system FIS-C. As seen, the Mamdani FIS-C fuzzy system allows for a better interpretation of the rules given the shape of the fuzzy sets used at the output. For example, Figure 10b shows that, although the fuzzy sets cover the entire output range, greater activation at lower output values are present.

Regarding the interpretability of the fuzzy system (for input variables), as an example, the case shown in Figure 9 and Figure 11 can be considered where it is observed that the inputs $W_1$ to $W_4$ have a greater influence on the inference and, therefore, on the output calculation. This depends on the approach of the rules. For example, it could be taken to the interpretation of the form: if the region is Andean and the crop is banana, the production will be high. The choice of model depends on the the preference of the user, and the accuracy of the prediction provides the necessary information to observe the model performance. On the other hand, the interpretative capacity of the models can make these technological tools closer to the users in the sense of handling a linguistic structure similar to the normal language. In contrast, the variable $W_5$ has little influence on the activation of the rules; this variable represents the Amazon region. On the other hand, the membership function for variable U in all the rules exhibits significant activation.

In addition, most of the membership functions observed in Figure 9 and Figure 11 cover the entire input range, indicating that each input has a continuous impact on rule activation. In the case of FIS-C, rules 2 and 4, having a wider output range, may have a greater influence on the overall inference.

Table 15. MSE results for Mamdani fuzzy systems.

System	Training	Validation
FIS-L	$0.9580\times {10}^{-4}$	$2.6265\times {10}^{-4}$
FIS-C	$4.2950\times {10}^{-4}$	$5.3098\times {10}^{-4}$

presents the MSE results for the Mamdani fuzzy systems. These results show that the best performance of the MSE is achieved with the FIS-L fuzzy system.

Figure 12a shows the simulation outcomes using training and validation data for the FIS-L system, while in Figure 12b, they are presented for the Mamdani FIS-C system. As can be seen, FIS-L presents an exceeding data fit; however, the FIS-C system allows better interpretability of the rules, where concepts such as high, medium-low, very low, and close to zero output can be associated (which can be used in an authentic context).

In summary of all results obtained in testing four versions of the neuro-fuzzy predictor ($M_1$–$M_4$), it is found that the simplest input layout ($M_1$) gives the most reliable results and that using linear output functions instead of fixed value outputs slashes forecast errors. Adding more clusters (mini groups of similar field conditions) can nudge accuracy even higher; however, each extra cluster also adds more “if this then that” rules and makes the system harder to interpret. For the model selection, the sweet spot is $M_1$ with a moderate number of clusters allowing sharp yield predictions from the fuzzy model (Sugeno) while still keeping a manageable rule set and can be transformed in the Mamdani version to obtain clear, plain language guidelines for agronomists and farmers.

9. Discussion

As seen, the fuzzy system with linear functions at the output allows for greater adaptability, which is why a better MSE value is obtained with this system. However, when greater interpretability is sought, the fuzzy system with constant functions at the output allows the association of output sets with linguistic labels according to their location.

Proposing several models and different configurations for each model made, it is possible to choose the best predictive performance. Ordering the data in ascending order proved not to provide an advantage for the accuracy of the prediction. The choice between fuzzy systems with linear and constant output functions depends on the trade-off between adaptability and interpretability. The FIS-L system achieves better adaptability and a lower MSE but at the cost of reduced interpretability due to less-defined fuzzy sets. In contrast, the FIS-C system enhances interpretability by associating output sets with linguistic labels and providing more structured membership functions. Therefore, selecting the appropriate model should consider factors such as interpretability, computational efficiency, and ease of implementation, ensuring an optimal balance between adaptability and clarity.

Neuro-fuzzy systems offer both accurate forecasts and human-readable rules for Colombian agriculture by balancing Sugeno-type models’ precision with Mamdani-type models’ interpretability. While the Sugeno approach minimizes error through adaptive linear functions, Mamdani translates those outputs into straightforward “if this then that” rules. Choosing the right model involves trading off raw accuracy for interpretability and considering hardware constraints, user expertise, and ease of updating rule sets to ensure both reliable predictions and actionable decision support on the farm.

It is also observed that the FIS-L system presents a smaller set of rules compared to the FIS-C system, which would allow its generalization; however, the fuzzy sets obtained in the FIS-L output make interpretability more complex. The membership functions of the FIS-C system, observed in Figure 10b, are more well defined than those displayed in Figure 8b, which may indicate a more structured and possibly more detailed distribution of the output data, allowing for better interpretation. For choosing the suitable model, aspects including interpretability, computational requirements, and ease of implementation might be taken into account. A balance must be sought between system adaptability (better MSE) and interpretability (rules and sets).

Regarding the reduced amount of data (215 records) and the use of 80% of the data for training models and validating them with 20% the risk of overfitting is possible; however, evaluating and choosing the models considering the interpretability feature allows one to avoid this risk since the labels of the fuzzy sets obtained must have an associated output concept. In this way, a model with better interpretability is chosen (usable) instead of a model with better MSE. This case is displayed in Section 8 where the FIS-L model has a better MSE value; however, the model FIS-C is useful from the interpretability feature. Comparison with other traditional prediction methods is an aspect to consider in order to determine the most suitable model usable in a real context; for this, it is necessary to set guidelines to obtain a fair comparison. In this way, aspects such as different performance metrics, complexity, interpretability, ease of use, and scalability must be regarded for comparison. For these models and techniques, an evaluation can be performed in a future work since this paper is oriented to display a proposal of codifications to implement the model for the production of relevant agricultural products in Colombia.

In order to obtain a more accurate forecasting model, climate information is necessary to be included for each agricultural product; however, these specialized data are not available since Colombia has wide variety of thermal zones making it difficult to obtain a uniform and usable dataset. The proposal displayed in this paper addresses the problem in a general way so that when specialized climate data become available, they can be used for the training models.

In addition, the use of a single model to predict the production of different crops may be more effective by training separate models for each product (e.g., cocoa and sugar cane). Nevertheless, this work wanted to test a coding approach that considered several products to identify what level of performance the model can achieve and thus be able to improve it for future developments.

Data are a relevant issue to train the models; in this work, different aspects were addressed. The first is the low amount of data (215 records), and the second is the missing data imputed (in three cases). In addition, data for region and product are not always available. It should be also noted that the collection of this type of data depends on the government institution in charge, which is subject to change depending on the policies of the current government (affecting data quality). Even with the data limitation, the models were trained and evaluated obtaining suitable results, especially in terms of interpretability.

10. Conclusions

This research can be regarded as an investigation of the connections among the attributes specified for agricultural production in Colombia and the strengths and weaknesses of each model, as well as the correlation between the input and output variables.

The suggested models could be clearly qualified thanks to the methodology and metrics taken into consideration. Different encodings were proposed for the input data, these being models $M_1$, $M_2$, $M_3$, and $M_4$. Considering the results, it was determined that model $M_1$ is the most convenient encoding for the fuzzy system. Comparing the results obtained across the four models, it is observed that arranging the inputs in ascending order has no contribution to improving prediction accuracy.

When analyzing the MSE values over 20 training processes (runs) with different configurations, a trend is observed in which the MSE decreases as the number of clusters increases. However, when more clusters are employed, the number of rules increases, which also increases the complexity of the fuzzy system and decreases its interpretability.

It was observed that the FIS-L system is more adaptable and provides a better MSE value; however, when converting the Sugeno system to a Mamdani, the FIS-C system achieves better interpretability. With the FIS-C system, linguistic labels can be associated with membership functions, considering their location in the output discourse universe. This feature allows for an exceeding interpretability rule.

Future works may consider developing a software tool to establish rules that allow human decision-making. Additionally, the method of building the fuzzy logic system can be investigated, considering the preliminary knowledge of a group of experts.