Input data preparation.

Each data file was converted to the CSV format and pre-processed using the DataFrames.jl package [31].

We used two data pre-processing strategies depending on the machine learning method being used.

For models that can take time series data as input (for example, recurrent neural networks), we used the past pollen concentration data for an appropriate interval (1-20 days, depending on the experimental variant) preceding the predicted day by n days. For example, for time window 14, to predict 4 days in the future, we used data from the interval between 18 and 4 days in the past. We used the same approach for meteorological data, including the weather forecast simulation, so we included meteorological data for the predicted day and n days earlier. In the experiments, the weather forecast simulation used historical meteorological data rather than an external forecast model. Specifically, for time window w and prediction horizon m, we treated the recorded daily meteorological values on the target day and the preceding m + w days as forecast inputs. This design isolates the performance of the pollen-forecasting algorithms from uncertainties in real-time weather predictions.

We used rolling windows and moving averages for models that prefer tabular data (for example, decision trees). For each feature, additional columns were prepared that contained exponential moving averages (EMAs) [32] and their derivatives. The exponential moving average is given by Eq (1) where t is the size of the window, α is a smoothing factor, and x_t is a feature value at point t.

(1)

Let i be an index of a data point, and f be a feature name, then the following features were constructed:

Similarly, given a time window w, to predict n days in the future, we used data from the interval between n + w and n days in the past. For meteorological data, we also included preprocessed meteorological data for the predicted day and n days before.

Target variable and measures.

In this work, the main goal of predictive models is to forecast pollen concentrations 1, 4, and 7 days into the future using supervised learning. Since the target variable is numeric, this is a regression problem. To assess the quality of the regression, the mean absolute error (MAE) described by Eq (2) and root mean square error (RMSE) described by Eq (3) metrics were used.

(2)

(3)

In these equations, y, , , and N denote a reference target value, a predicted target value, the mean of the target variable, and the number of observations, respectively.

However, from a clinical point of view, it is more important to classify pollen concentration into a specific category, allowing appropriate preventive measures to be taken. Three selected categories of daily pollen concentrations that cause allergic rhinitis symptoms were modified according to personal observation of the symptoms in Krakow patients [14]. So, from a technical point of view, this is a classification problem in which the target variable is categorical, with three possible classes for each taxon. The categories, along with the frequency of occurrence for each class, are as follows:

1. Betula
   1. (a) low: 1 – 10 , 76.0%,
   2. (b) medium: 11 – 75 , 12.8%,
   3. (c) high: >75 , 11.2%,
2. Poaceae
   1. (a) low: 1 – 10 , 68.3%,
   2. (b) medium: 11 – 50 , 22.6%,
   3. (c) high: >50 , 9.1%.

To assess the quality of classification for each class, we used the accuracy defined by Eq (4) where TP, TN, FP, FN denote a true positive value, a true negative value, a false positive value, and a false negative value, respectively.

(4)

Multi-associative graph network.

Multi-Associative Graph Networks (MAGN) [33] is a novel graph-based approach to representing and processing large-scale training data along with key relationships between data elements. Unlike conventional feedforward neural networks, where backpropagation is the primary training mechanism, MAGN uses a recursive, feedback-oriented graph structure inspired by how the human brain stores and retrieves information. This design makes incorporating new data on the fly easier and adapting existing models without a complete retraining phase.

The features are represented as sensory fields composed of sensory neurons using a dedicated structure called ASA-graphs [48]. This data structure vertically relates feature values, aggregates duplicates, and is seamlessly combined with other MAGN neurons. The data neurons in the graph represent objects from the database, whereas the connections capture existing and newly discovered relationships. A tuning algorithm further refines the graph by learning which neurons (and their associated objects) should be prioritized, improving the network capacity for more accurate classification and more substantial relational dependencies. Overall, MAGN aims to offer a flexible, brain-inspired architecture that can quickly handle new information, preserve and exploit relational structures, and enable more efficient computational intelligence processes than standard feed-forward systems.

Fig 2 illustrates how MAGN encodes numerical and categorical information into a structured, multi-layer graph that supports learning through association rather than gradient-based optimization. Each sensory field (A and B) corresponds to a specific input feature, and the sensory neurons within them (A.1, A.2, A.3; B.1, B.2, B.3) represent distinct observed feature values aggregated across the dataset. The duplicate counter beneath each sensory neuron reflects the frequency of that value, allowing MAGN to prioritize more statistically relevant information. Object neurons (O.1, O.2, O.4) serve as integrative units connecting the individual feature values that co-occur within a training instance. Defining connections (solid arrows) establish the composition of an object in the feature space, whereas similarity connections (dotted lines) capture the statistical co-occurrence of feature values across many objects. This relational structure enables MAGN to retrieve patterns by following associative paths rather than performing iterative optimization, which explains its strong performance across accuracy, MAE, execution time, and memory metrics. The interactions between sensory neurons, object neurons, and associations enable MAGN to store training data, discover new relationships, and generalize over previously seen data.

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Fig 2. MAGN structure.

Schematic structure of the Multi-Associative Graph Network (MAGN). Sensory fields (rectangular nodes A and B) contain sensory neurons representing unique feature values (green circles), each with an associated duplicate counter indicating how many times the value appeared in the training data. Object neurons (blue circles) represent individual data samples and are linked to the sensory neurons through defining connections, which encode the feature composition of each object. Similarity connections between sensory neurons capture statistical associations between feature values across the dataset, while duplicate counters on object neurons reflect the frequency of feature values and their patterns. Together, these components form a hierarchical associative graph that enables incremental learning, relational reasoning, and efficient retrieval.

https://doi.org/10.1371/journal.pone.0332093.g002

This work used the lazy regressor and classifier based on neural activation propagation in MAGN for predictions. It utilized similarity connections to fuzzy inputs represented by the sensory neurons. With fuzzy activation, the signal propagates to data neurons by defining connections. Then, an algorithm called similarity voting estimates the target value using different techniques depending on the type of target variable (classification for categorical variables and regression for numerical variables).

In addition, due to its efficient structure, MAGN was used to calculate mutual information [49] to analyze the relationship between input and target features. Mutual information measures how much knowledge of one random variable reduces the uncertainty about another. Mathematically, mutual information is defined by Eq (5) [50]. Here, P_(X,Y) is the joint probability distribution for the pair of random variables X and Y, both defined in the same probability space. At the same time, P_X and P_Y are the marginal distributions for X and Y, respectively.

(5)

MAGN was also used to extract frequent patterns and association rules to characterize the pollen season. A frequent pattern refers to a combination of feature values (captured by the sensory neurons) that appears at least as often as the specified minimum support threshold [51]. A support of pattern A measures how frequently a pattern appears in a dataset to be considered significant and is defined by Eq (6).

(6)

An association rule is an if-then statement of the form , where A and B are frequent patterns [52]. It indicates that whenever pattern A appears in an object, pattern B is also likely to appear. Frequent patterns and association rules are highly effective in data mining to uncover hidden relationships within datasets. For a given rule , the confidence measures the strength of the implication by calculating the conditional probability that pattern B occurs given that pattern A occurs [53]. Confidence is a measure of certainty of a rule and is defined as in Eq (7) where is the conditional probability (the probability of an event occurring, given that another event is already known to have occurred).

(7)

Lift measures the strength of association between the feature values in a rule compared to what would be expected if they were independent [54]. Lift is defined by Eq (8).

(8)

K-Nearest neighbors.

The k-Nearest Neighbors (k-NN) algorithm [34] is a non-parametric classification and regression method that assigns target values to predicted data based on the values of the target variable of the k-closest training samples in the n-dimensional feature space. It calculates the distance (commonly using the Euclidean distance) between the query point and all training points to determine the nearest neighbors. k-NN is a lazy algorithm, meaning that it does not build a model but instead makes predictions at runtime based on the raw training data. The choice of k is crucial: a small k may lead to overfitting, while a large k may oversmooth the decision boundaries [55]. Despite its simplicity, k-NN performs well in low-dimensional spaces but struggles with high-dimensional data due to the curse of dimensionality [56] while all dimensions influence results in the same way, even if some of them are irrelevant. There is no attention mechanism that could prioritize more essential data features.

The k-NN algorithm has also been widely analyzed in the context of practical data pre-processing requirements. In many applications, the performance of k-NN depends strongly on feature scaling, because distance metrics such as Euclidean or Manhattan distance are sensitive to differences in feature ranges. Standardization or normalization is therefore typically applied to ensure that no single variable disproportionately influences the computed distances [57]. Additionally, various distance metrics can be used to better adapt k-NN to specific data distributions or to reduce the impact of correlated features. These enhancements enable the algorithm to be more robust in heterogeneous datasets, such as meteorological time series combined with biological variables.

Linear regression.

Linear regression is a fundamental machine learning method used to model the relationship between a dependent variable y and n independent variables x_n by fitting a linear equation Eq (9) [37]. The parameters (β coefficients) are usually estimated by minimizing the sum of squared residuals (residual is the difference between the predicted value and observed value, e.g., ). Despite its simplicity and interpretability, linear regression assumes linearity, independence, homoscedasticity (an assumption of equal or similar variances in different groups), and normality of residuals, making it sensitive to outliers and multicollinearity [58].

(9)

In practical applications, several extensions of classical linear regression are used to mitigate its limitations and improve predictive performance. Techniques such as regularization, including Ridge (L2) and Lasso (L1) regression [59], add penalty terms to the loss function to reduce overfitting and address multicollinearity by shrinking or eliminating coefficients. These modifications help stabilize parameter estimates and improve generalization, particularly when predictors are highly correlated or when the number of features is large relative to the number of observations. Regularized linear models have been widely used in environmental and atmospheric sciences due to their robustness and ability to handle noisy real-world data [15].

Tree-based methods.

A decision tree is a supervised machine learning algorithm structured as a tree where each internal node represents a decision based on a feature, each branch represents a subset of feature values, and each leaf node represents a predicted class or value [39]. The tree is constructed using algorithms such as ID3 or CART, which recursively split data based on criteria such as information gain, Gini impurity, or variance reduction [60]. Decision trees are interpretable and can handle both numerical and categorical data. However, they are prone to overfitting, especially with deep trees, which can be mitigated using pruning or more advanced models such as gradient boosting and random forests.

Random Forest is an ensemble learning method that builds multiple decision trees and combines their results to improve predictive accuracy and reduce overfitting [41]. It works by training each tree on a random subset of the data and selecting a random subset of features at each split to enhance diversity. Predictions are made by voting among trees. The model is highly robust to overfitting and works well with large datasets, but it is less interpretable than simpler models. Random Forest is widely used in various applications due to its versatility, scalability, and ability to effectively handle missing data and imbalanced datasets [61].

XGBoost (Extreme Gradient Boosting) [42] is a machine learning model based on gradient boosting. This technique assumes that the next model minimizes the overall prediction error when combined with previous models. It builds an ensemble of decision trees sequentially, where each new tree corrects the errors of the previous ones using gradient-based optimization. XGBoost incorporates regularization techniques (such as L1 and L2 penalties) to prevent overfitting, making it more robust than traditional gradient boosting methods. Due to its high predictive accuracy and execution speed, XGBoost has become a dominant choice in machine learning competitions and real-world applications, including medical diagnosis [62].

Deep neural networks.

Deep neural networks (DNNs) are multilayer artificial neural networks that consist of multiple hidden layers between the input and output layers. Using techniques such as backpropagation and activation functions, DNNs have achieved state-of-the-art performance in various fields, including time series prediction [44].

Convolution is a mathematical operation used in signal processing and deep learning, where a filter (kernel) slides over input data to extract features by computing element-wise multiplications and summing the results [44]. In deep learning, 1-dimensional convolutional layers help capture temporal dependencies in sequences. The convolution operation is efficient because it enables weight sharing and local connectivity, significantly reducing the number of parameters compared to fully connected layers [63].

Long Short-Term Memory (LSTM) is a type of recurrent neural network (RNN) designed to overcome the vanishing gradient problem, allowing it to learn long-term dependencies in sequential data [46]. LSTMs use input, forget, and output gates to regulate the flow of information, ensuring that relevant data is retained while irrelevant information is discarded. The cell state acts as a memory unit, allowing LSTMs to store and update information across long sequences, making them well-suited for time series forecasting. Despite their effectiveness, LSTMs can be computationally expensive, which reduces their practical application.

Similarly to LSTMs, Gated Recurrent Units (GRUs) are a type of RNN designed to capture long-term dependencies in sequential data while addressing the problem of vanishing gradients [47]. Unlike LSTMs, GRUs use only two gates: the reset and update gates, making them computationally more efficient. The update gate determines how much past information should be carried forward, while the reset gate controls how much new information should be incorporated. GRUs have been widely applied in time series forecasting, often performing comparably to LSTMs with fewer parameters [64], effectively reducing computational complexity without performance sacrifice.

Fig 3 shows the architecture of the deep neural network used in the experiments.

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Fig 3. Deep neural network architecture.

The input features, such as pollen concentration or meteorological data, are in the form of time series. The convolutional layers, followed by the LSTM/GRU layers, and the dense layers were used sequentially. The output is a time series predicting the pollen concentration in the following days.

https://doi.org/10.1371/journal.pone.0332093.g003

Experiment 1.

The goal of Experiment 1 was to systematically compare the forecasting accuracy of nine machine learning algorithms listed in Table 3 across three forecast horizons (1, 4, and 7 days ahead).

For each forecast feature vectors were prepared as detailed in the Input Data Preparation section.

We evaluated each model on two parallel tasks:

• Regression: predicting the continuous pollen concentration, assessed via Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE).
• Classification: predicting discrete pollen concentration categories, assessed by overall accuracy as described in the Target Variable and Measures subsection.

To prevent temporal leakage, we grouped records by calendar year and randomly assigned 70% of years to the training set and the remaining 30% to the independent test set, ensuring no overlap in years between partitions.

All algorithms were trained using their default hyperparameter settings to maintain fair comparability. Performance metrics were computed exclusively on the held-out test set for each horizon.

This design allowed us to identify which methods most effectively capture both the numeric trends and categorical thresholds of birch and grass pollen seasons under varying lead times.

Experiment 2.

Building on the predictive performance established in Experiment 1, Experiment 2 aimed to interpret and characterize the relationships between the input and target variables. It comprised three components:

• Feature importance analysis: feature importance learned by the top-performing model to rank each predictor’s contribution to model accuracy.
• Mutual information assessment: normalized mutual information between each input variable and the target (pollen concentration or category) to quantify their statistical dependence without assuming linear relationships.
• Association rule mining: association rules mined by MAGN uncover patterns of meteorological and pollen variables that precede high or low pollen days.

These analyses collectively reveal which factors and combinations of them most strongly influence pollen forecasts and offer an interpretable characterization of pollen season dynamics.

Software and hardware.

All experiments were implemented in the Julia programming language v1.11.3 [66]. The packages used are listed in the appropriate subsections above. The source code is publicly available under the MIT license [67].

The experiments were carried out on a dedicated workstation with one AMD Ryzen Threadripper Pro 5965WX CPU, 128 GB RAM, and 2 x GPU NVidia GeForce 3090. To ensure stable thermal conditions during testing, the CPU and GPUs are cooled with a custom water cooling system, and the workstation is placed in a server cabinet with a dedicated connection for mechanical ventilation.