Stochastic Signals Research Articles

Oscillations of Ultralight Dark Photon into Gravitational Waves

The discovery of gravitational waves (GWs) opens a new window for exploring the physics of the early universe. Identifying the source of GWs and their spectra at present turns out to be important tasks so as to assist the experimental detection of stochastic GW signal. In this paper, we investigate oscillations of the ultralight dark photon (ULDP) into GWs in the dark halo. Assuming dark matter is composed of the ULDP and there are primordial dark magnetic fields (PDMFs) arising from the axion inflation and/or the dark phase transition, then the ULDP can oscillate into the GW when it passes through an environment of PDMFs. We derive the local energy density of GWs in the galaxy cluster induced by the instaneous oscillation of ULDP in the PDMFs. These stochastic local GWs exhibit a pulse-like spectrum, with frequency depending on the mass of the ULDP, and can be detected in Pulsar Timing Arrays (PTAs) or future space-based interferometers. We also find that the low-frequency GW signal observed by the NANOGrav collaboration and other PTA experiments can be addressed by the oscillation of the ULDP in the PDMFs in the early universe.

Stochastic neuropeptide signals compete to calibrate the rate of satiation.

Neuropeptides have important roles in neural plasticity, spiking and behaviour1. Yet, many fundamental questions remain regarding their spatiotemporal transmission, integration and functions in the awake brain. Here we examined how MC4R-expressing neurons in the paraventricular nucleus of the hypothalamus (PVHMC4R) integrate neuropeptide signals to modulate feeding-related fast synaptic transmission and titrate the transition to satiety2-6. We show that hunger-promoting AgRP axons release the neuropeptide NPY to decrease the second messenger cAMP in PVHMC4R neurons, while satiety-promoting POMC axons release the neuropeptide αMSH to increase cAMP. Each release event is all-or-none, stochastic and can impact multiple neurons within an approximately 100-µm-diameter region. After release, NPY and αMSH peptides compete to control cAMP-the amplitude and persistence of NPY signalling is blunted by high αMSH in the fed state, while αMSH signalling is blunted by high NPY in the fasted state. Feeding resolves this competition by simultaneously elevating αMSH release and suppressing NPY release7,8, thereby sustaining elevated cAMP in PVHMC4R neurons throughout a meal. In turn, elevated cAMP facilitates potentiation of feeding-related excitatory inputs with each bite to gradually promote satiation across many minutes. Our findings highlight biochemical modes of peptide signal integration and information accumulation to guide behavioural state transitions.

Characterizing EEG signal dynamics in healthy, seizure-free, and seizure states using the chaos decision tree algorithm

Abstract Epilepsy is a multifaceted neurological condition marked by repetitive seizures that arise from irregular electrical activity in the brain. To understand this condition, a thorough examination of brain signals captured in different states is needed. In order to examine the dynamic behavior of brain signals in three different conditions: healthy, seizure-free, and seizure periods, this study uses the chaos decision tree algorithm. The findings show notable variations in these situations’ dynamics. Chaos is evident during seizure moments, showing extremely chaotic activity. The signals mostly exhibit stochastic behavior in the healthy condition, which is consistent with typical brain dynamics. It is noteworthy that an intermediate state exhibiting a blend of stochastic and chaotic signal dynamics is exhibited throughout the seizure-free time. Furthermore, the research shows that the frequency of chaotic signals rises with increasing proximity to the epileptogenic zone. These discoveries clarify the complex nature of epilepsy and offer insightful information about the dynamic properties of brain signals in various stages, aiding in improved understanding and potential diagnostic approaches.

A Study of Mixed Non-Motorized Traffic Flow Characteristics and Capacity Based on Multi-Source Video Data.

Mixed non-motorized traffic is largely unaffected by motor vehicle congestion, offering high accessibility and convenience, and thus serving as a primary mode of "last-mile" transportation in urban areas. To advance stochastic capacity estimation methods and provide reliable assessments of non-motorized roadway capacity, this study proposes a stochastic capacity estimation model based on power spectral analysis. The model treats discrete traffic flow data as a time-series signal and employs a stochastic signal parameter model to fit stochastic traffic flow patterns. Initially, UAVs and video cameras are used to capture videos of mixed non-motorized traffic flow. The video data were processed with an image detection algorithm based on the YOLO convolutional neural network and a video tracking algorithm using the DeepSORT multi-target tracking model, extracting data on traffic flow, density, speed, and rider characteristics. Then, the autocorrelation and partial autocorrelation functions of the signal are employed to distinguish among four classical stochastic signal parameter models. The model parameters are optimized by minimizing the AIC information criterion to identify the model with optimal fit. The fitted parametric models are analyzed by transforming them from the time domain to the frequency domain, and the power spectrum estimation model is then calculated. The experimental results show that the stochastic capacity model yields a pure EV capacity of 2060-3297 bikes/(h·m) and a pure bicycle capacity of 1538-2460 bikes/(h·m). The density-flow model calculates a pure EV capacity of 2349-2897 bikes/(h·m) and a pure bicycle capacity of 1753-2173 bikes/(h·m). The minimal difference between these estimates validates the effectiveness of the proposed model. These findings hold practical significance in addressing urban road congestion.

Advanced Frequency Analysis of Signals with High-Frequency Resolution

In today’s era, it is important to analyze and utilize various signals in industrial or laboratory applications. Measured signals provide critical information about the controlled system, which can be contained precisely within a narrow frequency range. Many methods and algorithms exist to process such signals in both the time and frequency domains. In particular, signal processing in the frequency domain is primary in industrial practice because dominant components within a specific narrow frequency band are sought. The discrete Fourier transformation (DFT) algorithm is the tool used in practice to find these frequency components. The DFT algorithm provides the full frequency spectrum with a higher number of calculation steps, and its spectrum frequency resolution is low. Therefore, research has focused on finding a method to achieve high-frequency spectrum resolution. An important factor in selecting the technique was that such an algorithm should be implementable on a microprocessor-based system under harsh industrial conditions. Research results showed that the DFT ZOOM method meets these requirements. The frequency zoom has many advantages but requires some modification. It is implemented in high-performance analyzers, but a thorough and detailed description of the respective algorithm is lacking in technical articles and literature. This article mathematically and theoretically describes the modified frequency zoom algorithm in detail. The steps of the frequency zoom, from creating an analytical signal through frequency shifting and decimation to the frequency analysis of the signal, are realized. The algorithm allows for the analysis of a signal with high-frequency resolution in a limited frequency band. A significant modification of DFT ZOOM is that of using the Hilbert transform to create an analytic signal. This resolves the aliasing issue caused by the overlap between fundamental and sideband spectra. Results from processing deterministic and stochastic signals using the modified DFT ZOOM are presented. The presented experimental results contribute to a more detailed frequency analysis of the signal. As part of this scientific research, the issues of frequency zoom were thoroughly addressed, solving the partial problems of this algorithm, both in theory and in the context of signal theory.

Mechanics of reproductive differentiation in the land plants: a paradigm shift?

This article addresses the physical mechanics of gametogenesis in vascular plants. The earliest events that initiate reproductive differentiation in the land plants are not well understood. How are the few cells that initiate reproductive differentiation specified and how is that information translated into action at the cellular level? In this article I propose a physical mechanism that resolves the problem of spatial targeting without invoking dependence on diffusible morphogens or other external signals. I suggest that the initiation of archesporial differentiation can instead be attributed to the confluence of organ geometry, surficial topography, and the physical mechanics of sporangial growth, resulting in the spontaneous emergence of an isotropic singularity that locates and precipitates archesporial differentiation. In discussing the logic of single-cell target selection and the limits of stochastic molecular signaling I propose that the sporangium would be better understood as a pressurized stress-mechanical lens that focuses turgor-generated growth forces on a central location, generating a physical singularity that locates and specifies the cell or cells that become the archesporium and initiates their transition from somatic proliferation to reproductive differentiation.

A Stochastic Dynamic Operator Framework That Improves the Precision of Analysis and Prediction Relative to the Classical Spike-Triggered Average Method, Extending the Toolkit.

Here we test the stochastic dynamic operator (SDO) as a new framework for describing physiological signal dynamics relative to spiking or stimulus events. The SDO is a natural extension of existing spike-triggered average (STA) or stimulus-triggered average techniques currently used in neural analysis. It extends the classic STA to cover state-dependent and probabilistic responses where STA may fail. In simulated data, SDO methods were more sensitive and specific than the STA for identifying state-dependent relationships. We have tested SDO analysis for interactions between electrophysiological recordings of spinal interneurons, single motor units, and aggregate muscle electromyograms (EMG) of major muscles in the spinal frog hindlimb. When predicting target signal behavior relative to spiking events, the SDO framework outperformed or matched classical spike-triggered averaging methods. SDO analysis permits more complicated spike-signal relationships to be captured, analyzed, and interpreted visually and intuitively. SDO methods can be applied at different scales of interest where spike-triggered averaging methods are currently employed, and beyond, from single neurons to gross motor behaviors. SDOs may be readily generated and analyzed using the provided SDO Analysis Toolkit We anticipate this method will be broadly useful for describing dynamical signal behavior and uncovering state-dependent relationships of stochastic signals relative to discrete event times.

The evolution of two‐stage production and order equilibrium in a random yield supply chain with demand information updating

AbstractThe mismatch between production, order, and demand seriously affects supply chain performance. However, most research focus on the mismatch between the retailer's order and customer's demand, which ignores the influence of the supplier's random yield on supply chain members' decision‐making. This paper investigates a two‐stage optimization problem within a two‐echelon supply chain, featuring a supplier with random yield and a retailer updating demand information in real‐time. Faced with a long production lead time, the retailer can either place advance orders at the production season's onset (first‐stage advance order) or opt for instant orders at the beginning of the sales season (second‐stage instant order). To ensure timely order fulfillment, the supplier initially employs a cost‐effective regular production mode with random yield during the production season. If yields are insufficient during sales, a pricier emergency production mode with guaranteed output becomes available. Utilizing a dynamic programming approach, we formulate the two‐stage optimization problem to derive optimal production and order decisions. Our analysis uncovers how realized random yield and stochastic market signals influence emergency production and instant order quantities in the second stage. We compare expected profits in scenarios with perfect and imperfect market signals, probing the members' preferences regarding order strategies. An intriguing finding emerges: as instant wholesale prices rising, the supplier's preferred order strategy diverges from the retailer. By strategic adjustments to the instant wholesale price, we demonstrate the potential for unanimous agreement on preferred order strategies among supply chain members — a quality enhancing the chain's flexibility and performance. Moreover, we extend the model to hybrid order strategies and identify conditions for unanimous preference among the three strategies. To bolster our theoretical findings, we provide numerical examples, lending practical support to our study.

Timely ICU Outcome Prediction Utilizing Stochastic Signal Analysis and Machine Learning Techniques with Readily Available Vital Sign Data.

The ICU is a specialized hospital department that offers critical care to patients at high risk. The massive burden of ICU-requiring care requires accurate and timely ICU outcome predictions for alleviating the economic and healthcare burdens imposed by critical care needs. Existing research faces challenges such as feature extraction difficulties, low accuracy, and resource-intensive features. Some studies have explored deep learning models that utilize raw clinical inputs. However, these models are considered non-interpretable black boxes, which prevents their wide application. The objective of the study is to develop a new method using stochastic signal analysis and machine learning techniques to effectively extract features with strong predictive power from ICU patients' real-time time series of vital signs for accurate and timely ICU outcome prediction. The results show the proposed method extracted meaningful features and outperforms baseline methods, including APACHE IV (AUC = 0.750), deep learning-based models (AUC = 0.732, 0.712, 0.698, 0.722), and statistical feature classification methods (AUC = 0.765) by a large margin (AUC = 0.869). The proposed method has clinical, management, and administrative implications since it enables healthcare professionals to identify deviations from prognostications timely and accurately and, therefore, to conduct proper interventions.

Identifying chaotic dynamics in noisy time series through multimodal deep neural networks

Abstract Chaos detection is the problem of identifying whether a series of measurements is being sampled from an underlying set of chaotic dynamics. The unavoidable presence of measurement noise significantly affects the performance of chaos detectors, as discerning chaotic dynamics from stochastic signals becomes more challenging. This paper presents a computationally efficient multi-modal deep neural network tailored for chaos detection by combining information coming from the analysis of time series, recurrence plots and spectrograms. The proposed approach is the first one suitable for multi-class classification of chaotic systems while being robust with respect to measurement noise, and is validated on a dataset of 15 different chaotic and non-chaotic dynamics subject to white, pink or brown coloured noise.

Stochastic Optimization and Uncertainty Quantification of Natrium-based Nuclear-Renewable Energy Systems for Flexible Power Applications in Deregulated Markets

Rapid integration of variable renewable energy sources (VRES) has made modeling and stochastic optimization of hybrid energy systems crucial for studying their long-term performance and viability. However, most studies have focused on just historical data, which may be unreliable for capturing short-term fluctuations, rare events, and long-term patterns of energy demand, price, and the variability of renewable energy sources. For this study, optimal synthetic time series models were developed using Wasserstein distance. The models were validated by comparing the key statistical measures against those of the historical data. They were then used to optimize the integrated Natrium-style advanced energy systems and their long-term (30 years) economics. The stochastic model performs bi-level optimization to find the optimal sizes for the balance of plant and thermal energy storage, while also optimizing energy dispatch to achieve the maximum net present value.In studies of two deregulated markets (California ISO and the Electric Reliability Council of Texas), the integrated Natrium-style system performed better in CAISO than in ERCOT, given higher and more consistent electricity prices during peak-demand periods. The potentially enlarged cost associated with the variable operation and maintenance of the TES system also plays a significant role in driving the system sizing, thus its impacts on the system are investigated in detail through comparison against a baseline case. The study also finds that the bi-level optimization results based on stochastic gradient descent closely match the grid search results. The uncertainty quantification of the stochastic signals provides further NPV-related insights and probability distributions for the case studies. The normal standard error of the mean of NPV for the case with and without TES VOM for CAISO were found to be 7.73M ± 1.09M USD and 104.99M ± 1.25M USD, respectively based on a 95% confidence. Given the relatively small NPV variance based on 150 samples, the analysis affords the most robust possible prediction of the techno-economic performance of the integrated Natrium-style energy systems.

Spectral Variance in a Stochastic Gravitational-wave Background from a Binary Population

A population of compact object binaries emitting gravitational waves that are not individually resolvable will form a stochastic gravitational-wave signal. While the expected spectrum over population realizations is well known from Phinney, its higher-order moments have not been fully studied before or computed in the case of arbitrary binary evolution. We calculate analytic scaling relationships as a function of gravitational-wave frequency for the statistical variance, skewness, and kurtosis of a stochastic gravitational-wave signal over population realizations due to finite source effects. If the time derivative of the binary orbital frequency can be expressed as a power law in frequency, we find that these moment quantities also take the form of power-law relationships. We also develop a numerical population synthesis framework against which we compare our analytic results, finding excellent agreement. These new scaling relationships provide physical context to understanding spectral fluctuations in a gravitational-wave background signal and may provide additional information that can aid in explaining the origin of the nanohertz-frequency signal observed by pulsar timing array campaigns.

Damping wings in the Lyman-α forest: A model-independent measurement of the neutral fraction at 5.4 < z < 6.1

Context. Recent observations have positioned the end point of the Epoch of Reionization (EoR) at a redshift of z ∼ 5.3. However, observations of the Lyman-α forest have not yet been able to discern whether reionization occurred slowly and late, with substantial neutral hydrogen persisting at a redshift of ∼6, or rapidly and earlier, with the apparent late end driven by the fluctuating ultraviolet background. Gunn-Peterson (GP) absorption troughs are solid indicators that reionization is not complete until z = 5.3, but whether they contain significantly neutral gas has not yet been proven. Aims. We aim to answer this question by directly measuring, for the first time, the neutral hydrogen fraction (xHI) at the end of the EoR (5 ≲ z ≲ 6) in high-redshift quasar spectra. Methods. For high neutral fractions, xHI ≳ 0.1, GP troughs exhibit damping wing (DW) absorption extending over 1000 km s−1 beyond the troughs. While conclusively detected in Lyman-α emission lines of quasars at z ≥ 7, DWs are challenging to observe in the general Lyman-α forest due to absorption complexities and small-scale stochastic transmission features. Results. We report the first successful identification of the stochastic DW signal adjacent to GP troughs at redshifts of z = 5.6 through careful stacking of the dark gaps in the Lyman-α forest (S/N = 6.3). We use the signal to present a measurement of the corresponding global xHI = 0.19 ± 0.07 (−0.16+0.11) at 1σ (2σ) at z = 5.6 and a limit of xHI < 0.44 at z = 5.9. Conclusions. The detection of this signal demonstrates the existence of substantially neutral islands near the conclusion of the EoR, unequivocally signaling a late-and-slow reionization scenario.

Modeling residual-vehicle effects on uncertainty estimation of the connected vehicle penetration rate

In the transition to full deployment of connected vehicles (CVs), the CV penetration rate plays a key role in bridging the gap between partial and complete traffic information. Several innovative methods have been proposed to estimate the CV penetration rate using only CV data. However, these methods, as point estimators, may lead to biased estimations or suboptimal solutions when applied directly in modeling or system optimization. To avoid these problems, the uncertainty and variability in the CV penetration rate must be considered. Recently, a probabilistic penetration rate (PPR) model was developed for estimating such uncertainties. The key model input is a constrained queue length distribution composed exclusively of queues formed by red signals in undersaturation conditions with no residual vehicles. However, in real-world scenarios, due to random arrivals, residual vehicles are commonly carried over from one cycle to another in temporary overflow cycles in undersaturation conditions, which seriously restricts the applicability of the PPR model. To address this limitation, this paper proposes a Markov-constrained queue length (MCQL) model that can model the complex effects of residual vehicles on the CV penetration rate uncertainty. A constrained queue with residual vehicles is decomposed into four vehicle groups: observable constrained residual vehicles, unobservable constrained residual vehicles, unconstrained residual vehicles, and new arrivals. Although the first vehicle group is observable in the former cycle, the focus of this work is to model the residual vehicles from the second and third vehicle groups in combination with the new arrivals. The MCQL model includes four sub-models, namely, the residual-vehicle model, convolutional constrained queue model, constrained residual queue model, and observable residual queue model, to isolate and derive the distribution of the constrained vehicle set formed by the three latter vehicle groups. This distribution is then substituted into the PPR model to estimate the uncertainty. Comprehensive VISSIM simulations and applications to real-world datasets demonstrate that the proposed MCQL model can accurately model the residual-vehicle effect and estimate the uncertainty. Thus, the applicability of the PPR model is truly extended to real-world settings, regardless of the presence of residual vehicles. A simple stochastic CV-based adaptive signal control example illustrates the potential of the proposed model in real-world applications.

Probing flavor violation and baryogenesis via primordial gravitational waves

We show that observations of primordial gravitational waves of inflationary origin can shed light into the scale of flavor violation in a flavon model which also explains the mass hierarchy of fermions. The energy density stored in oscillations of the flavon field around the minimum of its potential redshifts as matter and is expected to dominate over radiation in the early universe. At the same time, the evolution of primordial gravitational waves acts as bookkeeping to understand the expansion history of the universe. Importantly, the gravitational wave spectrum is different if there is an early flavon dominated era compared to radiation domination expected from a standard cosmological model and this spectrum gets damped by the entropy released in flavon decays, determined by the mass of the flavon field mS and new scale of flavor violation ΛFV. We derive analytical expressions of the frequency above which the spectrum is damped, as-well-as the amount of damping, in terms of mS and ΛFV. We show that the damping of the gravitational wave spectrum would be detectable at BBO, DECIGO, U-DECIGO, μ−ARES, LISA, CE and ET detectors for ΛFV = 105−10 GeV and mS = OTeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left(\ extrm{TeV}\\right) $$\\end{document}. Furthermore, the flavon decays can source the baryon asymmetry of the universe. We identify the mS – ΛFV parameter space where the observed baryon asymmetry η ~ 10−10 is produced and can be tested by gravitational wave detectors like LISA and ET. We also discuss our results in the context of the recently measured stochastic gravitational background signals by NANOGrav.

Gaussian bistable cascade double feedback stochastic resonance weak signal enhancement detection

Stochastic resonance (SR), as a nonlinear weak signal enhancement detection method, has found extensive applications in fields like fault diagnosis and biological information processing. However, further research has revealed that single stochastic resonance systems often fail to meet the requirements of practical engineering applications, leading to a growing interest in multi-system collaborative stochastic resonance. This article first constructs a cascaded SR system based on the Gaussian bistable model, and introduces the idea of feedback into the cascaded system. A Gaussian bistable cascaded dual feedback SR system is proposed to improve the output signal enhancement performance of the system. This system effectively utilizes the memory characteristics inherent in the output feedback and enables multi-level particle transitions and energy transfers in noisy input signals, thereby reducing noise and enhancing the extraction of weak signal features. Furthermore, the BSO optimization algorithm was utilized to optimize the system parameters of the model, resulting in an adaptive Gaussian bistable cascaded double feedback stochastic resonance (GBCDFSR) system. This system is applied to the processing of noisy periodic signals across various noise types. Simulation experiments and engineering applications show that the proposed model has stronger signal detection capabilities.

Cancer mutationscape: revealing the link between modular restructuring and intervention efficacy among mutations

There is increasing evidence that biological systems are modular in both structure and function. Complex biological signaling networks such as gene regulatory networks (GRNs) are proving to be composed of subcategories that are interconnected and hierarchically ranked. These networks contain highly dynamic processes that ultimately dictate cellular function over time, as well as influence phenotypic fate transitions. In this work, we use a stochastic multicellular signaling network of pancreatic cancer (PC) to show that the variance in topological rankings of the most phenotypically influential modules implies a strong relationship between structure and function. We further show that induction of mutations alters the modular structure, which analogously influences the aggression and controllability of the disease in silico. We finally present evidence that the impact and location of mutations with respect to PC modular structure directly corresponds to the efficacy of single agent treatments in silico, because topologically deep mutations require deep targets for control.

Finite-Time Fault-Tolerant Control of Nonlinear Spacecrafts with Randomized Actuator Fault: Fuzzy Model Approach

The primary objective of this paper is to address the challenge of designing finite-time fault-tolerant control mechanisms for nonlinear flexible spacecraft systems, which are particularly vulnerable to randomized actuator faults. Diverging from traditional methodologies, our research harnesses the capabilities of the Takagi–Sugeno (T–S) fuzzy framework. A unique feature of our model is the representation of actuator failures as stochastic signals following a Markov process, thereby offering a robust solution for addressing timeliness concerns. In this paper, we introduce a generalized reciprocally convex inequality that includes adjustable parameters, broadening the scope of previous results by accommodating them as special cases. Through the amalgamation of this enhanced inequality and flexible independent parameters, we propose an innovative controller design strategy. This approach establishes a stability standard that guarantees mean-square H∞ performance. In order to validate the efficacy of the suggested strategy, we present a numerical illustration involving a nonlinear spacecraft system, showcasing the practical advantages and feasibility of our proposed technique.

Efektifitas Penggunaan Analisis Stochastic Oscillator Dan Moving Average Convergence-Divergence Pada Perdagangan Saham-Saham Perbankan di Bursa Efek Indonesia

The objective of the research is to find out how effective stochastic oscillator and moving average convergence-divergence (MACD) technical analysis is in yielding capital gain and abnormal return, viewed from descriptive analysis, analysis of effects, and disparity test analysis by using some theories and previous research results. This research uses daily stock closing price data from the issuers in banking su-sector of the research period of 2015-2019. The research employs descriptive analytic approach, regression analysis and disparity test, supported by IBM SPSS version 25 software. The result of the descriptive analysis shows that buy and sell stochastic oscillator and MACD signals are effective in yielding capital gain. The result of regression analysis shows that gain signal yielded by indicator of Stochastic Oscillator and MACD has significant effects on capital gain in which the probability value (Asymp. Sig) of Stochastic Oscillator on capital gain is 0.000, and the probability value (Asymp.Sig) of MACD on capital gain is 0.000. The result of the disparity test analysis shows that there is no difference in abnormal return in the pre and post appearance of gain signal in Stochastic Oscillator and in MACD

Constraints on primordial curvature power spectrum with kination era: Insights from NANOGrav 15-year data set

The stochastic signal detected by the NANOGrav, PPTA, EPTA, and CPTA collaborations can be attributed to gravitational waves induced by the primordial curvature perturbations generated during inflation. These scalar-induced gravitational waves provide valuable insights into the small-scale inflation and reheating epochs. In this paper, we assume an equation of state of w=1 during reheating and adopt a log-normal form for the primordial curvature power spectrum to elucidate the observed stochastic signal. The inflation and reheating scenarios are rigorously constrained utilizing Bayesian methods applied to the NANOGrav 15-year data set. The analysis yields constraints on the reheating temperature, indicating Trh≳0.01Gev, a result consistent with constraints derived from Big Bang nucleosynthesis. Furthermore, the NANOGrav 15-year data set necessitates the primordial curvature power spectrum's amplitude and width to satisfy A∼0.01 and Δ≲0.1, respectively. Due to the change in the equation of state w, there exists a turning point in the energy density spectrum of scalar-induced gravitational waves. This suggests that if more data about the scalar-induced gravitational waves is observed, it could potentially provide constraints on the time when the reheating epoch transitions to radiation domination.