Description:FIELD OF INVENTION
The invention pertains to an IoT module integrating AI-driven mathematical models for curve fitting of sensory data. This technology enables real-time analysis and interpretation of sensor readings, enhancing data accuracy and facilitating informed decision-making in various applications such as environmental monitoring, industrial automation, and healthcare.
BACKGROUND OF INVENTION
In modern industries and applications such as environmental monitoring, healthcare, and industrial automation, the integration of Internet of Things (IoT) devices has become ubiquitous for real-time data collection. However, interpreting and making sense of the vast amounts of sensory data generated by these devices presents a significant challenge. Curve fitting, a fundamental mathematical technique, plays a crucial role in extracting meaningful insights from such data by identifying underlying patterns or trends. Traditional curve fitting methods often struggle with the complexities and nonlinearities inherent in sensory data. To address this challenge, the invention proposes an IoT module equipped with an AI-based mathematical model specifically designed for curve fitting sensory data. By leveraging artificial intelligence techniques such as machine learning, this module can adaptively learn and model complex relationships within the data, improving accuracy and reliability in curve fitting tasks. The AI-based mathematical model embedded within the IoT module enables automated and intelligent processing of sensory data in real-time. By employing sophisticated algorithms, the module can effectively handle nonlinearities, noise, and uncertainties present in the data, thereby providing more accurate and robust curve fitting results. This innovation has the potential to revolutionize various fields reliant on IoT devices by enabling enhanced data analysis capabilities, leading to improved decision-making, predictive maintenance, and overall operational efficiency.
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SUMMARY
The invention introduces an IoT module featuring an AI-driven mathematical model tailored for curve fitting sensory data. In contemporary scenarios like environmental monitoring and industrial automation, IoT devices are pivotal for gathering extensive sensory data. However, accurately interpreting this data poses challenges due to its complexity and nonlinearity. Traditional curve fitting methods often struggle with these complexities. This IoT module overcomes these challenges by integrating advanced artificial intelligence techniques, specifically crafted for analyzing sensory data and extracting meaningful insights through curve fitting. The embedded AI-based mathematical model utilizes machine learning algorithms to dynamically learn and model complex relationships within the data. This empowers the module to effectively handle nonlinearities, noise, and uncertainties inherent in sensory data, yielding more precise and reliable curve fitting outcomes. By offering automated and intelligent processing of sensory data in real-time, the IoT module enhances data analysis capabilities, enabling improved decision-making, predictive maintenance, and operational efficiency across various domains reliant on IoT technology. This innovation holds promise for transforming the interpretation and utilization of sensory data, thereby facilitating more efficient deployment of IoT devices across diverse applications.
LITERATURE SURVEY
Evolutionary computation, heuristic, and stochastic search techniques offer global optimization solutions for large-scale problems, distinguishing them from conventional gradient-based methods prone to local minima traps. Evolutionary computation algorithms like genetic algorithms (GA) resolve optimization problems through population evolutions such as reproduction, mutation, and recombination. Researchers initially proposed GA for curve fitting to minimize least-squared errors, using binary coding schemes to produce near-optimal solutions. Later, they extended their studies to consider noisy data and outliers, achieving satisfactory results. Real-valued genetic algorithms (RVGA) encode candidate solutions as vectors of real-valued coefficients, addressing premature convergence issues with a sequential evolution mechanism. Recent studies show the effectiveness of real-coded GA in tasks like spline fitting.
Particle Swarm Optimization (PSO) is another widely used optimization method for parameter optimization in curve fitting tasks. It demonstrates superior accuracy and computational efficiency compared to other algorithms like GA. Researchers have applied PSO to various tasks, such as fitting potential energy functions and analyzing optical spectral data, achieving robust results even with noisy datasets.
Bayesian probabilistic models have also shown promise in automatic and accurate curve fitting, particularly in handling uncertainties in measured or observed data points. Bayesian inference, incorporating probabilistic modeling, accurately determines knot configuration parameters for piecewise polynomial fitting. The reversible jump Markov Chain Monte Carlo (MCMC) method calculates posterior distributions efficiently, producing accurate fittings for various benchmark curves, although it may require considerable computation time due to repeated sampling. Recent advancements, like Bayesian regression splines, aim to improve computational efficiency while maintaining accuracy in curve fitting tasks.
DETAILED DESCRIPTION OF INVENTION
Curve fitting involves identifying a parameterized function that best aligns with a given dataset, essentially serving as a function approximation task where minimizing an error measure is essential. This representation of data in a parameterized function holds significance in data analysis, visualization, and computer graphics, providing valuable insights for modeling and prediction techniques. Scientists and engineers often find it beneficial to fit measured or observed data to an empirical relationship to gain insight into related scientific phenomena. The resulting empirical formula facilitates tasks such as interpolation, extrapolation, differentiation, and locating maximum or minimum points on the curve without requiring extensive mathematical treatment based on underlying theories. Additionally, curve fitting is commonly employed to estimate parameters in various modeling studies. For instance, in climate science, parameters in models like the notch-delay solar model are estimated using curve fitting methods.
Traditionally, fitting a dataset to a curve involves selecting a parameterized function or a set of basis functions based on data distribution. Then, a numerical method such as the Levenberg-Marquardt (LM) algorithm is utilized to minimize the sum of squared residuals of data points and determine fitting coefficients. However, the LM algorithm often encounters challenges such as slow convergence, getting stuck in parameter space, and sensitivity to initial parameter guesses, especially for large problems with numerous parameters. Consequently, recent studies on parameter optimization problems have shifted from classical numerical methods to heuristic and stochastic search-based techniques like genetic algorithms (GA), particle swarm optimization (PSO), and Bayesian approaches.
Neural Network Approaches to Curve Fitting
In curve fitting problems, the objective is to find a parametric function 𝑓(𝑥;𝑎1,𝑎2,…𝑎𝑘) that best matches a given dataset of N points {𝑥𝑖,𝑦𝑖}𝑖=1𝑁 in terms of minimizing a specific error measure, typically the sum of squared residuals. Mathematically, this can be formulated as:

where the functional form of the regression function 𝑓(𝑥) is unknown, coefficients 𝑎1,𝑎2,…𝑎𝑘, are to be determined algorithmically, and 𝜖𝑖 represents the random error of the data points.
This problem can be approached as a function approximation task, where the complex function 𝑓(𝑥) is approximated by combining a linear combination of a series of simple parameterized basis functions ℎ(𝑥;𝑎). This can be expressed as:

where m is the number of basis functions used, βs are coefficients, and each basis function has the same analytical form but different parameter values.
Neural network techniques for curve fitting leverage the function approximation theorem of neural networks, which states that any continuous target function 𝑓(𝑥) can be approximated by a single-layer feedforward network (SLFN) with a sufficiently large number of hidden neurons. This implies that given an arbitrarily small positive number 𝜖>0, an SLFN with enough hidden nodes can always find a set of free parameters 𝑤w such that:

Here, 𝐹(𝑥,𝑤) represents a function realized by an SLFN, serving as an approximation of the target function 𝑓(𝑥).
In the context of SLFN approximation, if logistic activation functions are chosen for the hidden layer and linear functions for the output layer, the function to be fitted can be represented approximately as a linear combination of logistic functions. The weights and network parameters can be determined using the back-propagation (BP) training algorithm. However, BP has limitations, such as the tendency to become stuck in local minima.
Alternatively, radial basis function (RBF) neural networks offer a compelling approach for function approximation due to their universal approximation capabilities, simple architecture, and ease of training. Originating from the early work of Broomhead and Lowe (1988), RBF networks have been proven to possess universal approximation capabilities. Huang, Zhu, and Siew (2006, 2012) extended this concept and introduced extreme learning machines (ELM), which share similarities with SLFNs and RBF networks. In RBF networks and ELM, the function to be fitted is modeled approximately as the output of the network:

where 𝑀 is the number of neurons in the hidden layer, 𝑤𝑖 is the output weight, ci is the center vector for the 𝑖th neuron, ϕ(r) is an RBF that depends solely on the distance from the center, and ∣∣⋅∣∣ denotes the Euclidean distance.
In curve fitting, our objective is to determine the optimal parameters of a network to minimize the sum of least-squared errors. We adopt the Extreme Learning Machine (ELM) paradigm to train the network, where individual neuron parameters are randomly assigned while network weight parameters are determined analytically. Recognizing that both conventional Radial Basis Function (RBF) networks and ELMs often employ Gaussian functions as basis functions due to their favorable analytical properties, we propose enhancing these models by adding a linear neuron in the hidden layer. This addition helps adjust the localized behavior of Gaussian functions, particularly at boundary regions, where they may fail to model lower and upper boundary regions accurately.



Applied to each data point in the dataset using the adjusted approximation scheme, we obtain a linear system of equations. These equations can be compactly represented in matrix form, where the weight vector is determined using the Moore-Penrose pseudo-inverse, calculated via the Single Value Decomposition (SVD) method.
In summary, our approach utilizes ELM to approximate the curve to be fitted. Neuron parameters are randomly assigned, and output weights are computed analytically. The number of hidden layer neurons is determined using Particle Swarm Optimization (PSO).
PSO is an optimization algorithm that maintains a number of candidate solutions, or particles, in the search space simultaneously. Initially, random solutions are assigned, and the algorithm iteratively evaluates the fitness of each solution according to the objective function. Each particle is characterized by its position, velocity, and fitness, and the movement of particles through the fitness landscape constitutes the optimization process to find the minimum or maximum of the objective function.
In our application, we define the number of hidden layer nodes as the particle position. The objective function is the sum of squared residuals. The PSO algorithm updates particle positions and velocities iteratively based on personal and global best solutions, acceleration constants, and random numbers. At the end of all iterations, the global best solution is determined from all particles.



Several experiments were conducted to test the new method and evaluate its performance using datasets falling into three distinct categories:
Generated data from benchmark functions: This category includes datasets generated from benchmark functions widely used to verify newly developed non-linear regression algorithms (Denison, Mallick, and Smith, 1998). The benchmark functions used are:
𝑓1(𝑥)=(4𝑥−2)+2𝑒−16(4𝑥−2)2,𝑥∈[0,1]
𝑓2(𝑥)=sin (2(4𝑥−2))+2𝑒−16(4𝑥−2)2,𝑥∈[0,1]
Simulated data (200 points) were generated from these functions with zero-mean normal noise (σ = 0.3) added for training and testing purposes.
Synthesized spectral data: This category comprises data from synthetic spectroscopic data, essential for detecting the underlying structure of matter and analyzing spectral curves. Spectral data usually appear in the form of Gaussian, Lorentzian, and Voigt profiles with added noise. Three spectral datasets were considered, each representing one of the line shapes. These datasets were synthesized from true spectral line shape functions with zero-mean normal noise added. The datasets include:
• Gaussian2 dataset: a double-peak Gaussian with an exponential baseline (250 data points).
• Lorentzian dataset: a distorted Lorentzian function (200 data points).
• Pseudo-Voigt dataset: an approximate combination of Gaussian and Lorentzian (200 data points).
• Real-world measured data: This category involves real-world data obtained from measurements in various domains, including Raman and Infrared spectroscopy.
In Figures 1 and 2, true functions described in Equations (21) and (22) are displayed, and fitting results from different methods are compared. Notably, our method with M = 32 hidden nodes produces highly accurate fitting results closely overlapping the true functions throughout the domain. Comparative methods such as hierarchical genetic algorithm (HGA) and k-mean RBF fitting show slight deviations or fluctuations from the true functions in certain regions.

Figure 1. Comparative Analysis of Fitting Results for Benchmark Function 1

Figure 2. Comparative Analysis of Fitting Results for Benchmark Function 2
Table 1 presents the mean-squared errors (MSE) values from different methods, indicating that our method outperforms others in terms of accuracy (Denison, Mallick, and Smith, 1998; Trejo-Caballero and Torres-Huitzil, 2015).
Table 1. Mean-Squared Error Comparison for Benchmark Functions

The examples above illustrate the effectiveness of our adapted and optimized ELM in accurately fitting various datasets to smooth curves represented by intrinsic regression functions. The universal approximation theorems proposed by Hornik and Park and Sandberg underscore the importance of a sufficiently large number of neurons in the hidden layer for approximating a target function with arbitrary precision. However, in practical applications, achieving this concept of sufficient neurons doesn't necessitate an excessively large number. Instead, no more than 40 neurons are generally adequate for nonlinear transformations from input to output space.
Despite the finite number of neurons, each hidden layer neuron is characterized by different internal parameter values, randomly assigned according to a probabilistic distribution. This diversity contributes to local generalization, with each neuron providing effective approximations in its vicinity. The collective contribution of all neurons constructs a global approximation, with individual neuron contributions fine-tuned through output weights during the learning process. Lastly, we demonstrate the fitting results for real-world measured data—stopping power curve fitting from high-energy physics experiments. Stopping power, crucial in ion beam analysis (IBA) and heavy ion radiation therapy, is experimentally measured in high-energy physics facilities. Accurate stopping power curves are essential for the precise analysis of elemental composition and depth distribution in IBA, as well as for incident ion beam dosimetry in heavy ion radiation therapy.
Our results show closer agreement with measured points and better reflect the data distribution trends. For instance, in the case of oxygen projectiles incident on a carbon target, Paul's fitting values are higher than actual data points at lower energy regions, and lower than the data at higher energy positions. Conversely, Paul's results are slightly lower at mid-energy positions and slightly higher at higher energy regions for oxygen projectiles incident on silicon. Similar trends are observed for nitrogen projectiles incident on carbon, aluminum, and silicon targets, with our results showing closer alignment with the actual data distribution.
DETAILED DESCRIPTION OF DIAGRAM
Figure 1. Comparative Analysis of Fitting Results for Benchmark Function 1
Figure 2. Comparative Analysis of Fitting Results for Benchmark Function 2 , Claims:1. IoT module with AI based mathematical model for curve fitting for sensory data claims that a neural network-based method for curve fitting, presenting an intelligent alternative to traditional numerical analysis.
2. The proposed technique combines the strengths of the ELM algorithm and PSO optimization to improve accuracy and efficiency.
3. Optimizing the number of hidden layer nodes using PSO is crucial for achieving robust generalization in the model.
4. An additional linear neuron is incorporated into the ELM architecture to address localized behavior issues in basis functions.
5. The method is evaluated using various datasets, including benchmark problems and spectroscopic line shape data.
6. Promising results are obtained, demonstrating the effectiveness of the approach in challenging fitting tasks.
7. Future research will focus on optimizing all parameters of the network configuration to reduce network size.
8. Additionally, exploration of the method's applicability in multidimensional and sparse data fitting will be pursued.
9. The study contributes to advancing curve fitting methodologies by leveraging neural network and optimization techniques.
10. Overall, the proposed approach shows potential for enhancing accuracy and efficiency in curve fitting applications.