Truncated Normal Distribution Research Articles

A truncated Gaussian distribution based multi-scale segment-wise fusion transformer model for multi-step commodity price forecasting

Accurately forecasting commodity price trends is crucial for producers, market participants, and related enterprises to make informed decisions regarding production planning and scheduling. However, achieving high accuracy in multi-step forecasting poses significant challenges due to the unique financial characteristics inherent in commodities. Thus, this paper proposes a novel truncated Gaussian distribution based multi-scale segment-wise fusion Transformer for multi-step commodity price forecasting. First, a multi-scale segment-wise fusion module, which capture the time dependencies from different time granularity, is designed to describe the time-varying trend characteristics of commodity prices. Second, considering the characteristics of price range fluctuation and truncation, a truncated Gaussian distribution is introduced to describe price uncertainty. Last, to evaluate the proposed method’s effectiveness, extensive experiments are conducted using real data on energy chemical product prices. The experimental results demonstrate that the proposed method accurately captures price change trends and effectively estimates price uncertainty. Compared to the widely adopted Autoformer, our approach achieves approximately 30% reductions in both root mean square error (RMSE) and mean absolute error (MAE) metrics. Additionally, it exhibits certain advantages over the current state-of-the-art (SOTA). In the 20-step and 60-step multi-step prediction tasks, the proposed method achieves RMSE values of 91.18 and 142.94, respectively, surpassing the current SOTA. The introduced research framework provides valuable insights for decision-makers engaged in analyzing and forecasting commodity markets. The code is available on https://github.com/dean-ob/TGD-MSSF.

A multi-objective two-stage stochastic programming approach for wind farms and demand side provider decisions using the weighted metric method

The rapid expansion of wind-based renewable energy sources highlights the critical need to enhance reserve capacity within power systems. This research aims to develop an effective model for harnessing the combined potential of flexible demand-side resources, acting as demand-side providers (DSPs), to address the inherent unpredictability of wind power. The DSP framework comprises three demand response aggregators: the capacity market program (CAP), energy storage systems (ESSs), and plug-in electric vehicle parking lots (PEVs PLs). To achieve this, a multi-objective two-stage stochastic programming model is introduced. It not only minimizes operational costs for system operators but also maximizes returns for demand-side resource owners supervised by DSPs in the energy and reserve market environment. The problem is tackled using a mixed-integer linear programming (MILP) approach employing a weighted metric methodology. Within this framework, the Weibull probability distribution function (WPDF) is utilized to model wind power uncertainty, while the truncated Gaussian distribution function (GDF) encapsulates the uncertainty regarding electric vehicle owner behavior. The results demonstrate that employing this method for simultaneous programming of energy and reserve not only reduces system operator costs but also enhances profits for resource owners.

Modeling bimodal capillary bundle two-phase flow in the shale matrix for predicting water saturation distribution

In gas field development, the distribution of water saturation is one of the most effective estimated two-phase flow law methods for the shale matrix. Water saturation has been widely studied; however, few studies exist to show that water saturation distribution is related to pore diameter distribution of shale matrix. Furthermore, based on experimental observations, imbibition and gas-driven water flow nuclear magnetic resonance experiments also show the phenomenon. Therefore, the objective of this work is to build a bimodal capillary bundle two-phase flow model to investigate the relation of water saturation with pore diameter distribution of shale matrix. Firstly, this work combines the bimodal pore size distribution with the truncated Gaussian distribution theory to present the shale matrix pore. Moreover, we develope a shale core imbibition model and a gas-driven water flow model to predict the distribution of the water saturation distribution during the two-phase flow, by using the gas-water two-phase seepage theory. Shale imbibition and gas-driven water flow experiments are carried out on shale samples from the underground Longmaxi Formation. The relative error of distribution curve of water saturation predicted by model and experiments has been calculated. At the initial stage of imbibition the shale matrix core is not completely dried model values do not match the measured values, but, the relative error is stable at 1–2% at the late stage of imbibition. The average relative error is 2.13% in gas-driven water flow process. As a result, the shale imbibition model and the displacement model can accurately characterize the two-phase (gas-water) flow process, providing a theoretical support to study the two-phase flow in the shale matrix.

Characterization method of core pore structure based on truncated Gaussian and its application in shale cores

The core pore structure is mainly described by fractal theory. Conventional single fractal models cannot express the multi-peak distribution characteristics of coal rock, shale, carbonate rock, and other cores. A multifractal model has numerous dimensions and discontinuous functions, making the model parameter selection difficult. Based on the capillary bundle theory and the truncated Gaussian distribution model, a model for characterizing the multi-peak distribution structure of cores is developed in this study. The model's accuracy is verified using the tested NMR experimental data of shale bimodal distribution and the cited coal trimodal distribution. This study further analyzes the influence of tortuosity and tortuosity fractal dimensions on this model. The results show that tortuosity and tortuosity fractal dimensions are conducive to improving the accuracy of the multi-peak pore structure model. The structure model, which can characterize the multi-scale and multi-peak distribution characteristics of shale pore diameter, provides a new concept for describing the pore structure of porous media.

Rapid construction of Rayleigh wave dispersion curve based on deep learning

Introduction:The dispersion curve of the Rayleigh-wave phase velocity (VR) is widely utilized to determine site shear-wave velocity (Vs) structures from a distance of a few metres to hundreds of metres, even on a ten-kilometre crustal scale. However, the traditional theoretical-analytical methods for calculating VRs of a wide frequency range are time-consuming because numerous extensive matrix multiplications, transfer matrix iterations and the root searching of the secular dispersion equation are involved. It is very difficult to model site structures with many layers and apply them to a population-based inversion algorithm for which many populations of multilayers forward modelling and many generations of iterations are essential.Method:In this study, we propose a deep learning method for constructing the VR dispersion curve in a horizontally layered site with great efficiency. A deep neural network (DNN) based on the fully connected dense neural network is designed and trained to directly learn the relationships between Vs structures and dispersion curves. First, the training and validation sets are generated randomly according to a truncated Gaussian distribution, in which the mean and variance of the Vs models are statistically analysed from different regions’ empirical relationships between soil Vs and its depth. To be the supervised dataset, the corresponding VRs are calculated by the generalized reflection-transmission (R/T) coefficient method. Then, the Bayesian optimization (BO) is designed and trained to seek the optimal architecture of the deep neural network, such as the number of neurons and hidden layers and their combinations. Once the network is trained, the dispersion curve of VR can be constructed instantaneously without building and solving the secular equation.Results and Discussion:The results show that the DNN-BO achieves a coefficient of determination (R2) and MAE for the training and validation sets of 0.98 and 8.30 and 0.97 and 8.94, respectively, which suggests that the rapid method has satisfactory generalizability and stability. The DNN-BO method accelerates the dispersion curve calculation by at least 400 times, and there is almost no increase in computation expense with an increase in soil layers.

A convolutional Transformer-based truncated Gaussian density network with data denoising for wind speed forecasting

Wind speed forecasting plays an important role in the stable operation of wind energy power systems. However, accurate and reliable wind speed forecasting faces four challenges: how to reduce the data noise; how to find the optimal model inputs; how to describe the complex fluctuations in wind speed; and how to design a suitable loss function to tune the forecasting model. This study proposes a novel forecasting model to address the four challenges mentioned above. First, it uses a wavelet soft threshold denoising method to reduce noise in the original wind speed time series. Second, it uses the maximal information coefficient, which measures the linear and nonlinear relationships between historical wind speed data and forecasted targets, to determine the optimal model inputs. Third, a novel convolutional Transformer-based truncated Gaussian density network is designed to characterize the complex fluctuations in wind speed. The multi-scale information from different convolutional layers is weighted using the self-attention mechanism and then fed into the Transformer network to extract temporal information. The outputs are mapped into the forecasted targets with several fully connected layers. Fourth, considering the non-negativity of wind speed, the truncated Gaussian distribution, which shows a probability of zero when the wind speed is less than zero, is employed to model the uncertainty of wind speed forecasts. This leads to designing a truncated Gaussian distribution-based loss function to train the forecasting model. The forecasting results on three real-world datasets show that the proposed model not only provides accurate deterministic wind speed forecasts but also produces reliable probabilistic wind speed forecasts. The hypothesis testing also illustrates the effectiveness of the proposed model for deterministic and probabilistic wind speed forecasting.

확률프론티어분석(SFA)을 이용한 국내 사립대학의 재정효율성 분석

Many private universities in Korea are facing the worst financial situation at present not only due to the decreasing school-age population and their structural problems but also the impact of COVID-19. This paper aims to estimate the university’s economic shut-down point and financial efficiency scores with 115 private universities in 2020. COLS, CMAD, and SFA(exponential and truncated-normal distribution models) methods are utilized to compare the deterministic and stochastic results. As the results of empirical estimations, we have found that first, 24.3% of the private universities are at or near the shut-down point, and their average financial inefficiency was 18.9% and 7.5% measured by two SF models. Second, a striking result shows that a 1% increase in financial inefficiency lowers the university’s academic and financial performance by 2.5 times(%). This illustrates that it is important not to fall behind other universities in financial efficiency. Third, the sum of elasticities of the financial variables shows 0.57 in both SF models, decreasing returns to scale(DRS). Thus, this paper provides an implication of the importance of managing the financial efficiency for policy makers of government and universities.

Design Under Uncertainties of the Thermal Ablation Treatment of Skin Cancer

Abstract This computational work deals with the optimal design of the thermal ablation treatment of skin cancer, by considering uncertainties in the model parameters. The tumor and other tissues were heated by a laser. Nanoparticles were used to improve the effects of the heating procedure and to promote thermal damage localized in the region containing the tumor. Treatment protocols examined in this work involved one single heating session with different prespecified durations, where the design variables were considered as the volume fraction of nanoparticles in the epidermis and tumor, as well as the time variation of the incident laser fluence rate. The optimal design problems were solved with the Markov Chain Monte Carlo method, by applying a modified version of the Metropolis-Hastings algorithm with sampling by blocks of parameters. The two parameter blocks were given by the properties of the tissues and by the design variables. The prior for the volume fraction of nanoparticles was given by a truncated Gaussian distribution, while a noninformative Gaussian Markov random field prior was used for the time variation of the laser fluence rate. The posterior distributions of the design variables were estimated by taking into account uncertainties in the model parameters and the desired statistical distribution of the thermal damage in the region of interest. The stochastic simulations resulted in optimal thermal damages with small uncertainties, which closely followed their desired statistical distribution functions.

Relaying Data With Joint Optimization of Energy and Delay in Cluster-Based UAV-Assisted VANETs

Vehicular networks are known for their dynamic topology, high mobility, and frequent disconnections. Unmanned aerial vehicles (UAVs) have been recently used as instant communication relays to bridge the communication gaps between terrestrial vehicles to improve connectivity in vehicular networks and overcome the aforementioned problems. Despite the existing work in the literature where each vehicle connects directly to UAVs, this work studies how clustering and different densities of vehicles affect delay and energy efficiency in a UAV-based vehicular network integrated with 5G technology. Consequently, this work addresses the problem of UAV enabling vehicular ad-hoc networks (VANETs) in a highway scenario, where UAVs serve as an effective complement to forward data packets between vehicles in the absence of sufficient fixed infrastructures in an emergency situation. The main objective of this work is to minimize the delay while maximizing the energy efficiency by minimizing the power consumption and maximizing the total data rate under realistic conditions, while nonorthogonal multiple access (NOMA) is also adopted as an alternative answer for the effective utilization of limited bandwidth. The free-flowing traffic follows a Poisson stochastic process where each vehicle is assigned a random speed selected from a truncated Gaussian distribution. To this end, a novel modification of fast global <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -means is adopted to partition vehicles, allowing communication between clusters by vehicle-to-vehicle links, while data packets between clusters are relayed through UAVs. By computing the convex approximation of the objective function and the constraints, the original problem with the mixed-integer, nonconvex, and nonlinear form is solved by the proposed iterative inner penalty function algorithm. Finally, extensive simulations are conducted to validate the superiority of the proposed method in terms of various metrics. The results indicate that the relaying task in the proposed UAV-assisted VANET based on 5G technology is perfectly suited to enhance the network connectivity.

Physical layer security for indoor GSM VLC

We consider the physical layer security (PLS) of the generalized spatial modulation (GSM) enabled indoor multiple-input single-output (MISO) visible light communication (VLC) downlink, wherein an optical artificial noise (OAN)-assisted scheme is proposed to enhance the secrecy of the considered system. Due to the transmitter (Alice) sending confidential messages and OAN with the power and amplitude constraints, a truncated Gaussian distribution is exploited. With a finite discrete distribution of the input signals, the secrecy performance is analyzed in the OAN-assisted GSM VLC system. Specifically, the average mutual information (AMI) is first proposed, then the lower bound and approximation of AMI, and the achievable secrecy rate (ASR) are evaluated as well. In addition, considering the proposed power allocation problem of the considered OAN-assisted GSM VLC system, the particle swarm optimization (PSO) algorithm is explored to attain the global optimal power allocation coefficient, which is capable of maximizing the ASR. The numerical and simulation results validate our theoretical results of the OAN-assisted secrecy performance of the considered indoor GSM VLC system.

Modelling the growth of atmospheric nitrous oxide using a global hierarchical inversion

Abstract. Nitrous oxide is a potent greenhouse gas (GHG) and ozone-depleting substance, whose atmospheric abundance has risen throughout the contemporary record. In this work, we carry out the first global hierarchical Bayesian inversion to solve for nitrous oxide emissions, which includes prior emissions with truncated Gaussian distributions and Gaussian model errors, in order to examine the drivers of the atmospheric surface growth rate. We show that both emissions and climatic variability are key drivers of variations in the surface nitrous oxide growth rate between 2011 and 2020. We derive increasing global nitrous oxide emissions, which are mainly driven by emissions between 0 and 30∘ N, with the highest emissions recorded in 2020. Our mean global total emissions for 2011–2020 of 17.2 (16.7–17.7 at the 95 % credible intervals) Tg N yr−1, comprising of 12.0 (11.2–12.8) Tg N yr−1 from land and 5.2 (4.5–5.9) Tg N yr−1 from ocean, agrees well with previous studies, but we find that emissions are poorly constrained for some regions of the world, particularly for the oceans. The prior emissions used in this and other previous work exhibit a seasonal cycle in the extra-tropical Northern Hemisphere that is out of phase with the posterior solution, and there is a substantial zonal redistribution of emissions from the prior to the posterior. Correctly characterizing the uncertainties in the system, for example in the prior emission fields, is crucial for deriving posterior fluxes that are consistent with observations. In this hierarchical inversion, the model-measurement discrepancy and the prior flux uncertainty are informed by the data, rather than solely through “expert judgement”. We show cases where this framework provides different plausible adjustments to the prior fluxes compared to inversions using widely adopted, fixed uncertainty constraints.

Design and Performance Analysis of Intra-Vehicle VLC System With Random Receiver Orientation

Visible light communication is a promising technology for communication in an intra-vehicle scenario which can support high throughput without causing any electromagnetic interference. In contrast to RF link, the VLC link is not isotropic, i.e., it highly depends on the line of sight links (LOS). Therefore, it is important to understand the impact of receiver orientation and user mobility on the performance of VLC system. In the first part of the work, the effect of receiver orientation on the LOS channel is explored. The receiver orientation is modeled as a truncated Gaussian distribution. Based on the statistics of receiver orientation, channel and received signal to noise ratio (SNR) statistics are obtained for both the single input single output (SISO) and multiple input single output (MISO) systems. For the SISO system, closed-form expression for the bit error rate (BER) and outage probability are obtained. For the MISO VLC system, the BER and outage probability are evaluated numerically. The impact of Field of View (FOV) which corresponds to the aperture on the receiver from the transmitter on the performance metrics such as outage probability and BER are analyzed. To capture the impact of mobility on the performance of the system, random way-point mobility model is considered. Closed-form expression for the PDF of the LOS channel gain and BER is derived under the condition of angular variations are more dominant than the overall radial-gain spread for a coherence-period. The derived results help to explore the joint impact of random receiver orientation and mobility on the performance.

Security analysis and defense strategy of distributed filtering under false data injection attacks

This paper investigates the distributed state estimation for multi-sensor networks under false data injection attacks. The well-known χ2 detector is first considered for detecting the authenticity of the transmitted data. A necessary and sufficient condition for the insecurity of the distributed estimation system is derived under which the hostile attacks can bypass the false data detector and degrade the estimation performance. Moreover, an algorithm for generating false data is provided to keep the attack stealthy. In order to overcome the detection vulnerability, a new protection strategy is proposed to ensure that the distributed estimator is secure under false data injection attacks. It is worth emphasizing that the strategy adopts a stochastic rule instead of a fixed threshold to detect suspicious data, which effectively avoids the occurrence of the truncated Gaussian distribution. A simulation example of moving vehicle is presented to demonstrate the effectiveness of the developed approaches.

Combination treatment optimization using a pan-cancer pathway model.

The design of efficient combination therapies is a difficult key challenge in the treatment of complex diseases such as cancers. The large heterogeneity of cancers and the large number of available drugs renders exhaustive in vivo or even in vitro investigation of possible treatments impractical. In recent years, sophisticated mechanistic, ordinary differential equation-based pathways models that can predict treatment responses at a molecular level have been developed. However, surprisingly little effort has been put into leveraging these models to find novel therapies. In this paper we use for the first time, to our knowledge, a large-scale state-of-the-art pan-cancer signaling pathway model to identify candidates for novel combination therapies to treat individual cancer cell lines from various tissues (e.g., minimizing proliferation while keeping dosage low to avoid adverse side effects) and populations of heterogeneous cancer cell lines (e.g., minimizing the maximum or average proliferation across the cell lines while keeping dosage low). We also show how our method can be used to optimize the drug combinations used in sequential treatment plans—that is, optimized sequences of potentially different drug combinations—providing additional benefits. In order to solve the treatment optimization problems, we combine the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm with a significantly more scalable sampling scheme for truncated Gaussian distributions, based on a Hamiltonian Monte-Carlo method. These optimization techniques are independent of the signaling pathway model, and can thus be adapted to find treatment candidates for other complex diseases than cancers as well, as long as a suitable predictive model is available.

Propagation characteristic of adoption thresholds heterogeneity in double-layer networks with edge weight distribution

People normally show different relation intensity and adoption behaviors, which can affect information spreading on social networks. To describe the two phenomena, we consider edge weight distribution and adoption thresholds heterogeneity on the networks separately, where the adoption thresholds of nodes obey truncated gaussian distribution. Therefore, we propose a double-layer network model with edge weight distribution to explore the impact of heterogeneous adoption thresholds of nodes on information spreading. And we also develop an edge-based compartmental theory to analyze the information spreading mechanism. Through numerical simulations and theoretical analysis, we find that increasing the values of mean can suppress information spreading, and the impact of the values of standard deviation on information spreading depends on the values of mean. Specifically, for a small value of mean, the size of information spreading will decrease with the increase of the value of standard deviation. Meanwhile, when the values of standard deviation are fixed, the size of information spreading will increase continuously with the increase of transmission probability. For a large value of mean, the size of information spreading will increase first and then decrease with the increase of the value of standard deviation. In addition, this paper shows that the theoretical method agrees with the numerical simulations in the experimental results.

Loss function search for person re-identification

In recent years, person re-identification, which learns discriminative features for the specific person retrieval problem across non-overlapping cameras, has attracted extensive attention. One of the main challenges in person re-identification with deep neural networks is the design of the loss function, which plays a vital role in improving the discrimination of the learned features. However, most existing models utilize the hand-designed loss functions, which are usually sub-optimal and time-consuming. The search spaces of the two existing AutoML-based methods are either too complicated or too simple to include various forms of loss functions. In order to solve the irrationality of the above search spaces, in this paper, we propose a method of AutoML for loss function search named LFS-ReID for person ReID in the framework of the margin-based softmax loss function. Specifically, we first analyze the margin-based softmax loss function and conclude four key properties. Then we carefully design a sampling distribution based on the non-independent truncated Gaussian distributions to sample the loss function, which conforms to the above four properties. Finally, a method based on reinforcement learning is adopted to optimize the sampling distribution dynamically. Experimental results demonstrate that the proposed method achieves state-of-the-art performance on four commonly used datasets.

Outlier Detection in Wellness Data using Probabilistic Mapped Mean-Shift Algorithms

In this paper, the Probabilistic Mapped Mean-Shift Algorithm is proposed to detect anomalous data in public datasets and local hospital children’s wellness clinic databases. The proposed framework consists of two main parts. First, the Probabilistic Mapping step consists of k-NN instance acquisition, data distribution calculation, and data point reposition. Truncated Gaussian Distribution (TGD) was used for controlling the boundary of the mapped points. Second, the Outlier Detection step consists of outlier score calculation and outlier selection. Experimental results show that the proposed algorithm outperformed the existing algorithms with real-world benchmark datasets and a Children’s Wellness Clinic dataset (CWD). Outlier detection accuracy obtained from the proposed algorithm based on Wellness, Stamps, Arrhythmia, Pima, and Parkinson datasets was 93%, 94%, 80%, 75%, and 72%, respectively.

Free-Space Optical Communication Using Non-Mode-Selective Photonic Lantern-Based Coherent Receiver

A free-space optical communication system using non-mode-selective photonic lantern (PL) based coherent receiver is studied. Based on the simulation of photon distribution, the power distribution at the single-mode fiber end of the PL is quantitatively described as a truncated Gaussian distribution over a simplex. The signal-to-noise ratios (SNRs), bit-error rate (BER), and outage probability of PL based receiver using selection combining (SC), equal-gain combining (EGC), and maximal-ratio combining (MRC) over Gamma-Gamma turbulence channels are analyzed and compared with the single-mode fiber (SMF) receiver and multimode fiber (MMF) receiver. We demonstrate that EGC is the most suitable combining for PL based receiver because the PL power distribution has limited influence on the BER and outage probability when EGC is used and can greatly affect the BER and outage probability when SC is used. Numerical results show that PL based receiver with EGC requires 6 dB less SNR than the SMF receiver and 5.5 dB less SNR than the MMF receiver at BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−6</sup> in moderate turbulence.

Performance Analysis of Long Short-Term Memory-Based Markovian Spectrum Prediction

Dynamic Spectrum Access (DSA) solutions equipped with spectrum prediction can enable proactive spectrum management and tackle the increasing demand for radio frequency (RF) bandwidth. Among various prediction techniques, Long Short-Term Memory (LSTM) is a deep learning model that has demonstrated high performance in forecasting spectrum characteristics. Although well-performing, the theoretical characterization of LSTM prediction performance has not been well developed in the literature. Therefore, in this article, we examine an LSTM based temporal spectrum prediction model and characterize its prediction performance through theoretical analysis. To this end, we analyze the LSTM prediction outputs over simulated Markov-model-based spectrum data and spectrum measurements data. Our results suggest that the predicted scores of the LSTM based system model can be described using mixtures of truncated Gaussian distributions. We also estimate the performance metrics using the mixture model and compare the results with the observed prediction performance over simulated and measured datasets.

Wandering walk of chimera states in a continuous medium

Abstract The coexistence of coherent and incoherent domains in discrete coupled oscillators, chimera state, has been attracted the attention of the scientific community. Here we investigate the macroscopic dynamics of the continuous counterpart of this phenomenon. Based on a prototype model of pattern formation, we study a family of localized states. These localized solutions can be characterized by their sizes, and positions, and Yorke-Kaplan dimension. Chimera states in continuous media correspond to chaotic localized states. As a function of parameters and their size, the position of these chimera states can be bounded or unbounded. This allows us to classify these solutions as wandering or confined walk. The wandering walk is characterized by a chaotic motion with a truncated Gaussian distribution in its displacement as well as memory effects.