In this section, we will consider a traditional linear interpolation between the recorded data and show that the correlation between the features does not change much proving that unlike extrapolation, the interpolation is a stable process. In the next sections, we will demonstrate 2 methods PM GenAI and CAIHA that are able to make accurate predictions beyond the recorded data. a 73-year-old man complained about hypertension. The blood test was taken. The history of the bmp panel is given below ( Table 2 ):

Time is measured in months. Figure 1 shows the correlation between features (heatmap). For example, potassium highly correlates with chloride values while CO₂ correlates with sodium.

Figure 2 shows the interpolated data for Chloride (a) and CO₂ (b).

Figure 3 shows the heatmap (correlation between interpolated features). It demonstrates slightly different correlation values compared to original data correlation. This shows that correlation is a stable process.

Augmented data can be calculated in the form of Gaussian posterior probabilities:

$p\left(x\right)={\displaystyle {\sum }_{k=1}^N{\pi }_k\left(x_k,\mu ,{\Sigma }_k\right)}$ (1)

which is the multivariate normal distribution of the data set x with mean μ and covariance matrix Σ_k. K is the number of Gaussian components and ${\pi }_k$ is a coefficient for k-Gaussian distribution:

${\displaystyle {\sum }_{k=1}^N{\pi }_k}=1$

The covariance matrix is calculated from:

${\Sigma }_k=\frac{1}{N_k}{\displaystyle {\sum }_{i=1}^N{\gamma }_{ik}\left(x_i-{\mu }_k\right){\left(x_i-{\mu }_k\right)}^{\text{T}}}$ (2)

In this formula,

$N_k=\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\gamma }_{ik}$

Figure 4(a) and Figure 4(b) shows the scattered plots of augmented data. Figure 5(a) and Figure 5(b) depicts Gaussian and Gibbs data augmentation. Figure 6 depicts original + interpolated data and augmented data (red)

Let’s build now the confidence ellipse. Given a 2D Gaussian distribution of variables X = original chloride and Y = augmented data. The ellipse is defined by:

${\chi }^2={\left(x-\mu \right)}^{\text{T}}{\Sigma }^{-1}\left(x-\mu \right)$ (3)

Here ${\chi }^2=5.99$ chi-squared critical value with 2 degrees of freedom.

Σ is the 2 × 2 covariance matrix between X and Y.

$\mu =\left({\mu }_X,{\mu }_Y\right)$ is the mean vector of X and Y.

Figure 5(a) and Figure 5(b) illustrates Gaussian and Gibbs augmentation of the recorded data.

Figure 6(a) depicts the confidence ellipse of augmented and original + interpolated data (BUN vs. Chloride) and Figure 6(b) Chloride vs. creatinine. Figure 7 shows the confidence ellipse of augmented and original + interpolated data (BUN vs. creatinine.

Let us build now the confidence ellipse.

A confidence ellipse is a graphical representation of the uncertainty (or confidence region) around an estimated mean vector in two dimensions. It is often used in statistics, regression analysis, and multivariate analysis to show where the “true” values are likely to fall with a certain probability (e.g., 95%).

Key Ideas:

Shape: The ellipse is centered at the sample mean of two features. Axes: The major and minor axes of the ellipse correspond to the eigenvectors of the covariance matrix of the data, and their lengths depend on the eigenvalues (which reflect variance along each direction).

Confidence Level: A 95% confidence ellipse means that if you were to repeat the sampling many times, the true population mean would lie within the ellipse about 95% of the time. Figure 7 shows the 95% confidence ellipse of creatinine and BUN.

The characteristics of the confidence ellipse are given by:

LSTM (Long Short-Term Memory) is a refined version of Recurrent Neural Networks (RNNs), designed to better handle long-term dependencies by controlling information flow through special structures called gates. The core of LSTM consists of four interacting components:

1) Forget Gate

Controls what portion of the previous cell state $c_{t-1}$ to retain ( $\sigma$ is a sigmoid function).

$f_t=\sigma \left(W_f{x}_t+U_f{h}_{t-1}+b_f\right)$ (4)

2) Input Gate

Controls what new information to write into the cell.

$i_t=\sigma \left(W_i{x}_t+U_i{h}_{t-1}+b_i\right)$ (5)

3) Cell Candidate (New Content)

Proposes new content to be added to the cell state.

${\tilde{c}}_t=\tanh \left(W_c{x}_t+U_c{h}_{t-1}+b_c\right)$ (6)

4) Cell State Update combines old state and new candidate using the gates:

$c_t=f_t\odot c_{t-1}+i_t\odot {\tilde{c}}_t$ (7)

5) Output Gate decides how much of the cell state to expose:

$o_t=\sigma \left(W_o{x}_t+U_o{h}_{t-1}+b_o\right)$ (8)

LSTM prediction is highly inaccurate and unstable for small data sets. This is why the sparce data were interpolated (Time index 40 corresponds to month 24). Interpolation significantly stabilizes the LSTM algorithm ( Figure 8 and Figure 9 ).

PM GenAI (Principal Model Generative Algorithm) was introduced by Philip de Melo [7] [8] .

The method consists of the following steps:

PM GenAI generates new data by sampling from the posterior distribution of the model weights. This allows it to create diverse outputs based on the learned distributions, rather than relying on a fixed, deterministic model.

By modeling uncertainty in the parameters, PM GenAI produces a distribution of outputs instead of a single prediction. This is particularly useful for tasks that benefit from multiple plausible outcomes, such as data generation or decision-making under uncertainty.

Design the neural network architecture and specify prior distributions for the model parameters. Use techniques such as variational inference or Markov Chain Monte Carlo (MCMC) to approximate the posterior distribution of the weights.

Once the posterior is approximated (typically via backpropagation), sample weights from the distribution and perform feedforward passes through the network to generate predictions or synthetic data points that reflect the learned uncertainty. PM GenAI generates data that resembles real data, which is especially valuable in scenarios with limited labeled examples. By incorporating uncertainty, PM GenAI enables models to be more robust to outliers and noise, ultimately improving generalization.

In summary, using Bayesian inference in deep learning for data generation enables the incorporation of uncertainty into models, generate realistic synthetic data, and make more reliable and accurate predictions as expressed in Figures 10-12 .

To analyze the structure of the original data and predict (such as sharp variations), we use the CWT method (Continuous Wavelet Transform). The Continuous Wavelet Transform transforms a 1D signal into a 2D representation that shows how the frequency content of the data variations changes over time, using wavelets—localized oscillatory functions.

Larger scales capture low frequencies (slow trends) while smaller scales correspond to high frequencies (sharp changes). Figure 13 and Figure 14 show CWT of original and predicted values of BUN and Glucose.

We will follow now [9] and consider a new algorithm that uses stochastic characteristics of the data to predict values such as vitals, etc. beyond the time of observations. We will consider real data (irregular, sparse, incomplete). [10] considered predicting disease progress with imprecise lab test results. [11] discussed time series prediction with Long Short-Term Memory (LSTM). [12] introduce fuzzy logics in biomedical prediction of time series. [13] demonstrated an advanced machine learning algorithm in diabetes dynamics. [14] used machine learning applied to blood glucose forecasting.

We generate augmented data in effort to identify the most likely predicted value for BUN from 20 to 30. We have chosen the center at 25 ( Figure 15 ). Now we can explore other possibilities. Table 3 shows statistical data of other possibilities with different centers. Figure 16 shows KDE curves and BUN = 22 was chosen in accordance with Table 3 .

To decide which BUN value to choose, you want the value that best balances:

From this Table, it is seen that the next BUN value will be 22 which very well approximates the PM GenAI prediction ( Figure 10 ).

BUN values 22, 24, 20, and 26 all have the same KDE value (0.168) and identical histogram count (166). However, BUN = 22 was still chosen because of its higher (less negative) log-probability, which leads to the highest weighted score. BUN = 22 served as the initial condition for AI time-series prediction.

The method is called “Self-Adaptive Informed Healthcare Algorithm” because it concurrently uses synthetic data and stochastic property of the recorded data. The method gives a number of solutions, but the choice is based on the maximum of log probability.

This paper tackles a critical challenge in healthcare analytics: the prediction of Basic Metabolic Panel (BMP) values from sparse and irregular clinical data, which is a common issue in real-world Electronic Health Records (EHRs).

Accurate forecasting of BMP components—such as blood urea nitrogen, creatinine, and electrolytes—is vital for monitoring metabolic function, especially in patients with chronic or acute conditions. However, the sparsity and irregular sampling of clinical lab data hinder the effectiveness of traditional time-series models.

To address this, the study introduces a multi-component methodology consisting of 1) data interpolation, 2) Gaussian Mixture Model (GMM)-based data augmentation, and 3) the development of two novel predictive algorithms: PM GenAI (Predictive Modeling with Generative AI) and SAIHA (Self-Adaptive Informed Healthcare Algorithm).

The interpolation step ensures a more continuous representation of patient data, while the GMM augmentation enriches the dataset with plausible synthetic samples that reflect the underlying statistical distribution of the observed values. This addresses the data sparsity problem and improves the robustness of downstream prediction models.

The core contribution lies in the design and application of PM GenAI and SAIHA. These algorithms are specifically tailored to leverage both real and synthetic data for enhanced predictive performance. PM GenAI integrates generative modeling with feature-aware prediction, while SAIHA introduces a self-adaptive mechanism that dynamically adjusts to the patient’s evolving clinical state.

While this study presents a novel and promising methodology for predicting Basic Metabolic Panel (BMP) values using sparse and irregular clinical data, several limitations must be acknowledged:

The methodology was evaluated on a single patient case study. While this allowed for a detailed analysis, it limits the generalizability of the findings. The variability across patient populations, disease profiles, and care pathways may significantly affect the model’s performance in broader clinical settings.

The PM GenAI and SAIHA algorithms, while powerful, introduce an additional layer of model complexity that may hinder interpretability for clinicians. Clinical deployment requires transparency and explainability, which remains a challenge for generative and adaptive models.

The proposed methodology has not yet been validated on independent datasets or across multiple healthcare institutions. External validation is necessary to ensure robustness, reproducibility, and applicability in diverse clinical environments.