The real database consists of a set of 10 sequences of 75 or 100 data points (for each of the 4 sensors) acquired using the WSN located at Universidad del Quindío. The timing of the measurements is random (minutes, hours, or days). The original file, named Datos.csv, was processed to obtain the 10 sequences in .mat format, with the following names: datosprueba1.mat, datosprueba2.mat, datosprueba10.mat. Each file contains variables D, D2, D3, and D4, which correspond to temperature (°C), relative humidity (%), atmospheric pressure (Bar), and soil humidity (%), respectively. Figure 1 shows the time series for sequence 1, where data align with standard values for temperature, relative humidity, atmospheric pressure, and soil humidity, considering that the altitude of the city of Armenia is approximately 1550 meters above sea level.

Figure 1

A wireless sensor network (WSN) is defined as a self-configurable network composed of a small number of sensor nodes, also known as motes, which are spatially distributed and interconnected. Recently, a new method has been developed to enhance data performance, achieving a reliability range sufficient for continuous monitoring in systems [15]. Additionally, WSNs serve as an important communication bridge between the virtual and the real physical world. Their wide range of applications makes them a significant contributor to fields such as industry, science, transportation, security, infrastructure, the military, and even commercial applications [16], [17].

1) Gaussian Process (GP). A Gaussian Process is a collection of random variables where a finite number of them follow a joint Gaussian distribution [18]. A GP is completely defined by its mean function m(x), and its covariance function k(x,x'), such that f(x) ~ GP (m(x),k(x,x')) . In supervised learning, the radial basis kernel (RBF) is commonly employed as the covariance function, and is given by:

Where θ and σ² are the hyper-parameters of scale and variance, respectively. Consider a noisy data, indexed by an independent variable (may be time) and its related values {(xi, yi)}n i=1,..,n, where the yi values follow the standard regression model yi = fi + ε, with fi = f(xi) and a Gaussian noise ε ∼ N(0,σ²n). The joint distribution of the traini ng values y, and the validation values, y*, is given by:

Being Ky,y*, Ky,y and Ky*,y* the covariance matrix between the training-validation, training-training, and validation-validation points, respectively. Here, the conditional distribution for the validation points is given by y*|X,y,X* ~ N (y*, cov(y*)), where:

2) Multi-Output Gaussian Processes (MOGP). The general method for Multiple Output Gaussian Processes (MOGP) describes D output tasks { f d x } d = 1 D , x∈ R p , through convolution integrals of latent functions { u q i x } q = 1 ,   i = 1 Q , R q , with soft kernels { G d , q i x - z } D = 1 ,   q = 1 ,   i = 1 D ,   Q , R q

Assuming that the latent functions u_q ⁱ(x) are independent Gaussian processes with covariance function k_q(x,x'), the outputs f_d(x) conform to a joint Gaussian process with covariance function k_dd,(x,x') with d,d' = 1,...,D, given by:

Which is known as convoluted multiple output covariance or CMOC. If it is assumed that G j , q i (x − z) = a j , q i δ(x − z), being δ(x) the Dirac delta function, there is a particular case for the covariance function known as the linear model of coregionalization (LMC). The Covariance k j , j ´ (x,x′) simplifies to:

Where b j , j ´ q = ∑ i = 1 R q a j , q i a j ´ , q i . In this case, RBF kernels (see Equation (1)) are used to build the LMC. Once the LMC is developed, the classical inference method based on maximum likelihood is used to find the posterior distribution over the outputs f j J (x). For a detailed explanation of Multiple Output Gaussian Processes [19]. In the context of a WSN, each output f j (x) corresponds to the data delivered by each sensor.

1) Support Vector Machine. The Support Vector Machine for Regression (SVR) is a variant of the Support Vector Machine (SVM) used for regression problems rather than classification [12].

2) Random Forest (RF). It is a supervised learning algorithm used for both classification and regression tasks in machine learning. Instead of relying on a single tree to make decisions, RF combines the predictions from multiple trees to improve accuracy. Its main feature is randomization during tree construction and decision making. In training, a subset of the features is randomly selected, and variations are also introduced into the training data.

The details of each algorithm, as well as the evaluation and partitioning of the database for training and validation, are described as follows:

Training partition: The dataset was split into 80% for training and 20% for testing across four atmospheric variables: temperature, relative humidity, atmospheric pressure, and soil humidity.

MOGP: The linear model of co-regionalization was used to compute the posterior distribution, along with a radial basis function (RBF) kernel for the covariance matrix calculation. Training was set to 150 iterations with 4 latent functions.

GP: An RBF kernel and maximum likelihood estimation were applied to compute the posterior distribution. Training was set to 50 iterations.

SVR: An RBF kernel was used with parameters C=20 and Gamma=0.5, determined via grid search.

RF: An ensemble of 100 decision trees was employed.

The Root Mean Square Error (RMSE) was calculated on the validation sets for all methods.

Figure 2 shows data modeling with MOGP for Frame 1. The red line corresponds to the real data of the WSN, the blue line to the regression with MOGP, and the green points are the data necessary for reconstruction, that is, it indicates the dimensionality reduction having a given RMSE.

Figure 2

Tables 1 and 2 show RMSE values for all frames and dimensionality reduction percentages, the average value and the corresponding standard deviation.

Table 1

	S1:Temperature	S2:Humidity	S3:ATM	S4:Soil humidity
Frame 1	0.043	1.040	1.25	0.018
Frame 2	1.486	2.606	0.35	0.132
Frame 3	0.781	1.831	0.75	0.129
Frame 4	0.788	2.275		0.183
Frame 5	0.255	3.807		0.155
Frame 6	0.691	3.616		0.197
Frame 7	0.453	3.324	0.8	0.109
Frame 8	0.690	1.841	0.4	0.093
Frame 9	0.673	3.415		0.139
Frame 10	0.374	3.337		0.077
Average	0.623±0.390	2.709±0.931	(1.09±0.52)	0.123±0.052

Table 2

	S1:Temperature	S2:Humidity	S3:ATM	S4:Soil humidity
Frame 1	78.7	73.3	54.7	84.0
Frame 2	57.3	64.0	53.3	52.0
Frame 3	53.3	66.7	58.7	53.3
Frame 4	56.0	65.0	56.0	50.0
Frame 5	53.0	54.0	55.0	51.0
Frame 6	52.0	56.0	56.0	52.0
Frame 7	56.0	66.0	58.0	51.0
Frame 8	53.0	64.0	57.0	51.0
Frame 9	64.0	63.4	56.0	49.5
Frame 10	56.0	60.0	58.0	52.0
Average	57.9±8.1	63.2±5.5	56.3±1.7	54.6±10.4

Figure 3 shows the data modeling with GPs for Frame 1. The color coding is the same as described for MOGP.

Figure 3

Figure 4 shows data modeling with independent SVRs for Frame 1. The color coding is the same as in the previous cases; however, there are no green points, since SVR does not allow for redundant data reduction.

Figure 4

Figure 5 shows the data modeling with Random Forest for Frame 1. The color coding is the same for SVR.

Figure 5

1) Analysis in terms of RMSE. Table 3 provides comparative results of the Root Mean Square Error (RMSE) for different machine learning methods (MOGP, GP, SVR, RF) applied to data from four sensors (S1, S2, S3, S4), each related to different environmental magnitudes (temperature, humidity, atmospheric pressure and soil humidity).

Table 3

	S1:Temperature	S2:Humidity	S3:ATM	S4:Soil Humidity
MOGP	0.623±0.390	2.709±0.931	(1.09±0.52)	0.123±0.052
GP	0.808±0.324	2.879±1.772	(1.16±0.53)	0.132±0.027
SVR	1.311±0.996	5.115±3.494	(2.01±1.08)	0.232±0.085
RF	0.786±0.281	3.574±1.301	(1.55±0.41)	0.179±0.054

According to the results described in tables and figures, the following can be established:

MOGP (Multi-Output Gaussian Process): This method exhibits the lowest RMSE across all sensors: S1 (temperature), S2 (humidity), and S4 (soil humidity). This indicates that MOGP is the most accurate approach for predicting these magnitudes compared to other methods. For S3 (atmospheric pressure), MOGP also achieves the lowest RMSE, although the difference with some other methods is less pronounced. The accuracy of MOGP is particularly high, as it shows the lowest RMSE in all four scenarios. This suggests that MOGP is robust and well-suited for adapting to different environmental magnitudes. Additionally, MOGP demonstrates low variability, which indicates greater robustness and reduced sensitivity to changes in input data or model configuration.

GP (Gaussian Process): This method performs well for S1 (temperature) and S4 (soil humidity), though not as effective as MOGP. Overall, it ranks second in terms of accuracy. Simple Gaussian Processes remain a strong alternative for regression problems; however, in this context, as many independent GPs are required as there are sensors in the WSN. In contrast, MOGP requires only a single model. Whether single-output or multi-output, Gaussian Processes are particularly efficient for modeling relationships in data with potentially nonlinear patterns.

SVR (Support Vector Regression): This method tends to show intermediate performance across all scenarios. It is neither the most accurate nor the least, which may suggest that it is a more balanced model and less prone to overfitting. SVR follows the temporal trends of the WSN measurements; however, it struggles to capture abrupt changes. It is observed that SVR exhibits higher standard deviations compared to other methods, thus reflecting greater variability in its results. This could be attributed to its sensitivity to hyperparameter configuration.

RF (Random Forest): Similar to GP, it has a fairly consistent performance in all scenarios but fails to outperform MOGP in terms of accuracy. RF is an interesting model, simple and easy to implement. It is certainly a good learning machine for modeling WSN data, with acceptable errors.

2) Analysis of Redundant Data Reduction (Dimensionality). Results in Table 4 provide relevant information about the redundant data reduction capability using GP and MOGP in a WSN.

Table 4

	S1:Temperature	S2:Humidity	S3:ATM	S4:Soil Humidity
MOGP	57.9±8.1	63.2±5.5	56.3±1.7	54.6±10.4
GP	55.6±7.1	67.7±5.0	56.6±2.5	49.7±1.0

In summary, results support the effectiveness of Gaussian process-based models (MOGP and GP) for dimensionality reduction in wireless sensor network environments by using uncertainty information effectively.