Search results for: gaussian copula CTGAN

Authors: Chandra Prayaga, Aaron Wade, Lakshmi Prayaga, Gopi Shankar Mallu

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This paper presents the use of machine learning algorithms to predict the success of students in an introductory physics course. Data having 140 rows pertaining to the performance of two batches of students was used. The lack of sufficient data to train robust machine learning models was compensated for by generating synthetic data similar to the real data. CTGAN and CTGAN with Gaussian Copula (Gaussian) were used to generate synthetic data, with the real data as input. To check the similarity between the real data and each synthetic dataset, pair plots were made. The synthetic data was used to train machine learning models using the PyCaret package. For the CTGAN data, the Ada Boost Classifier (ADA) was found to be the ML model with the best fit, whereas the CTGAN with Gaussian Copula yielded Logistic Regression (LR) as the best model. Both models were then tested for accuracy with the real data. ROC-AUC analysis was performed for all the ten classes of the target variable (Grades A, A-, B+, B, B-, C+, C, C-, D, F). The ADA model with CTGAN data showed a mean AUC score of 0.4377, but the LR model with the Gaussian data showed a mean AUC score of 0.6149. ROC-AUC plots were obtained for each Grade value separately. The LR model with Gaussian data showed consistently better AUC scores compared to the ADA model with CTGAN data, except in two cases of the Grade value, C- and A-.

Keywords: machine learning, student success, physics course, grades, synthetic data, CTGAN, gaussian copula CTGAN

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Authors: Lakshmi Prayaga, Chandra Prayaga. Aaron Wade, Gopi Shankar Mallu, Harsha Satya Pola

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Synthetic data generated by Generative Adversarial Networks and Autoencoders is becoming more common to combat the problem of insufficient data for research purposes. However, generating synthetic data is a tedious task requiring extensive mathematical and programming background. Open-source platforms such as the Synthetic Data Vault (SDV) and Mostly AI have offered a platform that is user-friendly and accessible to non-technical professionals to generate synthetic data to augment existing data for further analysis. The SDV also provides for additions to the generic GAN, such as the Gaussian copula. We present the results from two synthetic data sets (CTGAN data and CTGAN with Gaussian Copula) generated by the SDV and report the findings. The results indicate that the ROC and AUC curves for the data generated by adding the layer of Gaussian copula are much higher than the data generated by the CTGAN.

Keywords: synthetic data generation, generative adversarial networks, conditional tabular GAN, Gaussian copula

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Authors: Ishapathik Das, Sumen Sen, N. Rao Chaganty, Pooja Sengupta

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Dependent multivariate count data occur in several research studies. These data can be modeled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells, and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present a real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.

Keywords: copula, Gaussian copula, multivariate distributions, inflated distributios

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Authors: Giriraj Achari, Malay Bhattacharyya

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In this paper, the effectiveness of Copula Markov Switching Multifractal (MSM) models at forecasting Value-at-Risk of a two-stock portfolio is studied. The innovations are allowed to be drawn from distributions that can capture skewness and leptokurtosis, which are well documented empirical characteristics observed in financial returns. The candidate distributions considered for this purpose are Johnson-SU, Pearson Type-IV and α-Stable distributions. The two univariate marginal distributions are combined using the Student-t copula. The estimation of all parameters is performed by Maximum Likelihood Estimation. Finally, the models are compared in terms of accurate Value-at-Risk (VaR) forecasts using tests of unconditional coverage and independence. It is found that Copula-MSM-models with leptokurtic innovation distributions perform slightly better than Copula-MSM model with Normal innovations. Copula-MSM models, in general, produce better VaR forecasts as compared to traditional methods like Historical Simulation method, Variance-Covariance approach and Copula-Generalized Autoregressive Conditional Heteroscedasticity (Copula-GARCH) models.

Keywords: Copula, Markov Switching, multifractal, value-at-risk

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Authors: Hueiwang Anna Jeng, Norou Diawara, Nancy Welch, Cynthia Jackson, Rekha Singh, Kyle Curtis, Raul Gonzalez, David Jurgens, Sasanka Adikari

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Modeling effort is needed to predict the COVID-19 trends for developing management strategies and adaptation measures. The objective of this study was to assess whether SARS-CoV-2 viral load in wastewater could serve as a predictor for forecasting COVID-19 cases, hospitalization cases, and death cases using copula-based time series modeling. SARS-CoV-2 RNA load in raw wastewater in Chesapeake VA was measured using the RT-qPCR method. Gaussian copula time series marginal regression model, incorporating an autoregressive moving average model and the copula function, served as a forecasting model. COVID-19 cases were correlated with wastewater viral load, hospitalization cases, and death cases. The forecasted trend of COVID-19 cases closely paralleled one of the reported cases, with over 90% of the forecasted COVID-19 cases falling within the 99% confidence interval of the reported cases. Wastewater SARS-CoV-2 viral load could serve as a predictor for COVID-19 cases and hospitalization cases.

Keywords: COVID-19, modeling, time series, copula function

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Authors: Alam Ali, Ashok Kumar Pathak

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Nowadays, copulas have attracted significant attention for modeling multivariate observations, and the foremost feature of copula functions is that they give us the liberty to study the univariate marginal distributions and their joint behavior separately. The copula parameter apprehends the intrinsic dependence among the marginal variables, and it can be estimated using parametric, semiparametric, or nonparametric techniques. This work aims to compare the coverage rates between an Elliptical and an Archimedean family of copulas via a fully parametric estimation technique.

Keywords: elliptical copula, archimedean copula, estimation, coverage rate

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Authors: Duhania O. Mahara, Santi W. Purnami, Aulia N. Fitria, Merissa N. Z. Wirontono, Revina Musfiroh, Shofi Andari, Sagiran Sagiran, Estiana Khoirunnisa, Wahyudi Widada

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For assessing interventions in numerous disease areas, the use of multiple time-to-event outcomes is common. An individual might experience two different events called bivariate time-to-event data, the events may be correlated because it come from the same subject and also influenced by individual characteristics. The bivariate time-to-event case can be applied by copula-based bivariate Cox survival model, using the Clayton and Frank copulas to analyze the dependence structure of each event and also the covariates effect. By applying this method to modeling the recurrent event infection of hemodialysis insertion on chronic kidney disease (CKD) patients, from the AIC and BIC values we find that the Clayton copula model was the best model with Kendall’s Tau is (τ=0,02).

Keywords: bivariate cox, bivariate event, copula function, survival copula

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Authors: Xiaoyu Shen, Fang Fang, Chujun Qiu

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Herewith we present a Fourier method for credit risk quantification and allocation in the factor-copula model framework. The key insight is that, compared to directly computing the cumulative distribution function of the portfolio loss via Monte Carlo simulation, it is, in fact, more efficient to calculate the transformation of the distribution function in the Fourier domain instead and inverting back to the real domain can be done in just one step and semi-analytically, thanks to the popular COS method (with some adjustments). We also show that the Euler risk allocation problem can be solved in the same way since it can be transformed into the problem of evaluating a conditional cumulative distribution function. Once the conditional or unconditional cumulative distribution function is known, one can easily calculate various risk metrics. The proposed method not only fills the niche in literature, to the best of our knowledge, of accurate numerical methods for risk allocation but may also serve as a much faster alternative to the Monte Carlo simulation method for risk quantification in general. It can cope with various factor-copula model choices, which we demonstrate via examples of a two-factor Gaussian copula and a two-factor Gaussian-t hybrid copula. The fast error convergence is proved mathematically and then verified by numerical experiments, in which Value-at-Risk, Expected Shortfall, and conditional Expected Shortfall are taken as examples of commonly used risk metrics. The calculation speed and accuracy are tested to be significantly superior to the MC simulation for real-sized portfolios. The computational complexity is, by design, primarily driven by the number of factors instead of the number of obligors, as in the case of Monte Carlo simulation. The limitation of this method lies in the "curse of dimension" that is intrinsic to multi-dimensional numerical integration, which, however, can be relaxed with the help of dimension reduction techniques and/or parallel computing, as we will demonstrate in a separate paper. The potential application of this method has a wide range: from credit derivatives pricing to economic capital calculation of the banking book, default risk charge and incremental risk charge computation of the trading book, and even to other risk types than credit risk.

Keywords: credit portfolio, risk allocation, factor copula model, the COS method, Fourier method

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Authors: Yo Fukutani, Shuji Moriguchi, Takuma Kotani, Terada Kenjiro

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If risk management of the assets owned by companies, risk assessment of real estate portfolio, and risk identification of the entire region are to be implemented, it is necessary to consider simultaneous damage to multiple buildings. In this research, the Sagami Trough earthquake tsunami that could have a significant effect on the Japanese capital region is focused on, and a method is proposed for simultaneous damage assessment using copulas that can take into consideration the correlation of tsunami depths and building damage between two sites. First, the tsunami inundation depths at two sites were simulated by using a nonlinear long-wave equation. The tsunamis were simulated by varying the slip amount (five cases) and the depths (five cases) for each of 10 sources of the Sagami Trough. For each source, the frequency distributions of the tsunami inundation depth were evaluated by using the response surface method. Then, Monte-Carlo simulation was conducted, and frequency distributions of tsunami inundation depth were evaluated at the target sites for all sources of the Sagami Trough. These are marginal distributions. Kendall’s tau for the tsunami inundation simulation at two sites was 0.83. Based on this value, the Gaussian copula, t-copula, Clayton copula, and Gumbel copula (n = 10,000) were generated. Then, the simultaneous distributions of the damage rate were evaluated using the marginal distributions and the copulas. For the correlation of the tsunami inundation depth at the two sites, the expected value hardly changed compared with the case of no correlation, but the damage rate of the ninety-ninth percentile value was approximately 2%, and the maximum value was approximately 6% when using the Gumbel copula.

Keywords: copulas, Monte-Carlo simulation, probabilistic risk assessment, tsunamis

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Authors: Alam Ali, Ashok Kumar Pathak

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Path analysis is a statistical technique used to evaluate the strength of the direct and indirect effects of variables. One or more structural regression equations are used to estimate a series of parameters in order to find the better fit of data. Sometimes, exogenous variables do not show a significant strength of their direct and indirect effect when the assumption of classical regression (ordinary least squares (OLS)) are violated by the nature of the data. The main motive of this article is to investigate the efficacy of the copula-based regression approach over the classical regression approach and calculate the direct and indirect effects of variables when data violates the OLS assumption and variables are linked through an elliptical copula. We perform this study using a well-organized numerical scheme. Finally, a real data application is also presented to demonstrate the performance of the superiority of the copula approach.

Keywords: path analysis, copula-based regression models, direct and indirect effects, k-fold cross validation technique

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Authors: Yibin Qiu, Yubo Ouyang, Shihan Li, Guorui Zhang, Qi Li, Weirong Chen

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This paper aims at exploring the impacts of high dimensional dependencies of wind speed among wind farms on probabilistic optimal power flow. To obtain the reactive power optimization faster and more accurately, a mixture vine Copula structure model combining the K-means clustering, C vine copula and D vine copula is proposed in this paper, through which a more accurate correlation model can be obtained. Moreover, a Modified Backtracking Search Algorithm (MBSA), the three-point estimate method is applied to probabilistic optimal power flow. The validity of the mixture vine copula structure model and the MBSA are respectively tested in IEEE30 node system with measured data of 3 adjacent wind farms in a certain area, and the results indicate effectiveness of these methods.

Keywords: mixture vine copula structure model, three-point estimate method, the probability integral transform, modified backtracking search algorithm, reactive power optimization

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Authors: Adrian O'Hagan, Robert McLoughlin

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Considerable focus in the world of insurance risk quantification is placed on modeling loss values from lines of business (LOBs) that possess upper tail dependence. Copulas such as the Joe, Gumbel and Student-t copula may be used for this purpose. The copula structure imparts a desired level of tail dependence on the joint distribution of claims from the different LOBs. Alternatively, practitioners may possess historical or simulated data that already exhibit upper tail dependence, through the impact of catastrophe events such as hurricanes or earthquakes. In these circumstances, it is not desirable to induce additional upper tail dependence when modeling the joint distribution of the loss values from the individual LOBs. Instead, it is of interest to accurately assess the degree of tail dependence already present in the data. The empirical copula and its associated upper tail dependence coefficient are presented in this paper as robust, efficient means of achieving this goal.

Keywords: empirical copula, extreme events, insurance loss reserving, upper tail dependence coefficient

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Authors: Islem Kedidi, Rihab Bedoui Bensalem, Faysal Manssouri

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In traditional models of insurance data, the number and size of claims are assumed to be independent. Relaxing the independence assumption, this article explores the Vine copula to model dependence structure between multivariate frequency and average severity of insurance claim. To illustrate this approach, we use the Wisconsin local government property insurance fund which offers several insurance protections for motor vehicles, property and contractor’s equipment claims. Results show that the C-vine copula can better characterize the multivariate dependence structure between frequency and severity. Furthermore, we find significant dependencies especially between frequency and average severity among different coverage types.

Keywords: dependency modeling, government insurance, insurance claims, vine copula

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Authors: Serge Provost, Yishan Zang

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Copulas are currently utilized in finance, reliability theory, machine learning, signal processing, geodesy, hydrology and biostatistics, among several other fields of scientific investigation. It follows from Sklar's theorem that the joint distribution function of a multidimensional random vector can be expressed in terms of its associated copula and marginals. Since marginal distributions can easily be determined by making use of a variety of techniques, we address the problem of securing the distribution of the copula. This will be done by using several approaches. For example, we will obtain bivariate least-squares approximations of the empirical copulas, modify the kernel density estimation technique and propose a criterion for selecting appropriate bandwidths, differentiate linearized empirical copulas, secure Bernstein polynomial approximations of suitable degrees, and apply a corollary to Sklar's result. Illustrative examples involving actual observations will be presented. The proposed methodologies will as well be applied to a sample generated from a known copula distribution in order to validate their effectiveness.

Keywords: copulas, Bernstein polynomial approximation, least-squares polynomial approximation, kernel density estimation, density approximation

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Authors: Jiemiao Chen, Shuoxun Xu

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The main goal of our research is to apply the Vine copula measuring dependency between price, temperature, and precipitation indices to calculate a fair crop insurance premium. This research is focused on Worth, Iowa, United States, over the period from 2000 to 2020, where the farmers are dependent on precipitation and average temperature during the growth period of corn. Our proposed insurance considers both the natural risk and the price risk in agricultural production. We first estimate the distributions of crops using parametric methods based on Goodness of Fit tests, and then Vine Copula is applied to model dependence between yield price, crop yield, and weather indices. Once the vine structure and its parameters are determined based on AIC/BIC criteria and forecasting price and yield are obtained from the ARIMA model, we calculate this crop insurance premium using the simulation data generated from the vine copula by the Monte Carlo Simulation method. It is shown that, compared with traditional crop insurance, our proposed insurance is more fair and thus less costly for the farmers and government.

Keywords: vine copula, weather index, crop insurance premium, insurance risk management, Monte Carlo simulation

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Authors: Gabriel I. Loaiza Ossa, Carlos A. Cadavid Moreno, Juan C. Arango Parra

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It is known that the Riemannian manifold determined by the family of inverse Gaussian distributions endowed with the Fisher metric has negative constant curvature κ= -1/2, as does the family of usual Gaussian distributions. In the present paper, firstly, we arrive at this result by following a different path, much simpler than the previous ones. We first put the family in exponential form, thus endowing the family with a new set of parameters, or coordinates, θ₁, θ₂; then we determine the matrix of the Fisher metric in terms of these parameters; and finally we compute this matrix in the original parameters. Secondly, we define the inverse q-Gaussian distribution family (q < 3) as the family obtained by replacing the usual exponential function with the Tsallis q-exponential function in the expression for the inverse Gaussian distribution and observe that it supports two possible geometries, the Fisher and the q-Fisher geometry. And finally, we apply our strategy to obtain results about the Fisher and q-Fisher geometry of the inverse q-Gaussian distribution family, similar to the ones obtained in the case of the inverse Gaussian distribution family.

Keywords: base of changes, information geometry, inverse Gaussian distribution, inverse q-Gaussian distribution, statistical manifolds

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Authors: Hina Munir Abbasi

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The current paper examines the relationship and diversification ability of Islamic stock indices /conventional stocks indices and Real Estate Investment Trust (REITs).To represent conditional dependency between stocks and REITs in a more realistic way, new modeling technique, time-varying copula with switching dependence is used. It represents reliance structure more accurately and realistically than a single copula regime as dependence may alter between positive and negative correlation regimes with time. The fluctuating behavior of markets has significant impact on economic variables; especially the downward trend during crisis. Overall addition of Real Estate Investment Trust in stocks portfolio reduces risks and provide better diversification benefit. Results varied depending upon the circumstances of the country. REITs provides better diversification benefits for Islamic Stocks, when both markets are bearish and can provide hedging benefit for conventional stocks portfolio.

Keywords: conventional stocks, real estate investment trust, copula, diversification, risk spillover, safe heaven

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Authors: Imen Dhaou

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This study examines conditional Value at Risk by applying the GJR-EVT-Copula model, and finds the optimal portfolio for eight Dow Jones Islamic-conventional pairs. Our methodology consists of modeling the data by a bivariate GJR-GARCH model in which we extract the filtered residuals and then apply the Peak over threshold model (POT) to fit the residual tails in order to model marginal distributions. After that, we use pair-copula to find the optimal portfolio risk dependence structure. Finally, with Monte Carlo simulations, we estimate the Value at Risk (VaR) and the conditional Value at Risk (CVaR). The empirical results show the VaR and CVaR values for an equally weighted portfolio of Dow Jones Islamic-conventional pairs. In sum, we found that the optimal investment focuses on Islamic-conventional US Market index pairs because of high investment proportion; however, all other index pairs have low investment proportion. These results deliver some real repercussions for portfolio managers and policymakers concerning to optimal asset allocations, portfolio risk management and the diversification advantages of these markets.

Keywords: CVaR, Dow Jones Islamic index, GJR-GARCH-EVT-pair copula, portfolio optimization

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Authors: A. Keshavarz

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A physical model for guiding the wave in photorefractive media is studied. Propagation of cos-Gaussian beam as the special cases of sinusoidal-Gaussian beams in photorefractive crystal is simulated numerically by the Crank-Nicolson method in one dimension. Results show that the beam profile deforms as the energy transfers from the center to the tails under propagation. This simulation approach is of significant interest for application in optical telecommunication. The results are presented graphically and discussed.

Keywords: beam propagation, cos-Gaussian beam, numerical simulation, photorefractive crystal

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Authors: Keunhong Chae, Seokho Yoon

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In this paper, frequency offset (FO) estimation schemes robust to the non-Gaussian noise environments are proposed for orthogonal frequency division multiplexing (OFDM) systems. First, a maximum-likelihood (ML) estimation scheme in non-Gaussian noise environments is proposed, and then, the complexity of the ML estimation scheme is reduced by employing a reduced set of candidate values. In numerical results, it is demonstrated that the proposed schemes provide a significant performance improvement over the conventional estimation scheme in non-Gaussian noise environments while maintaining the performance similar to the estimation performance in Gaussian noise environments.

Keywords: frequency offset estimation, maximum-likelihood, non-Gaussian noise environment, OFDM, training symbol

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Authors: Zhichao Jiao, Craig Farnham, Jihui Yuan, Kazuo Emura

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It is common to use the outdoor design weather data to select the air-conditioning capacity in the building design stage. The outdoor design weather data are usually comprised of multiple meteorological elements for a 24-hour period separately, but the dependency between the elements is not well considered, which may cause an overestimation of selecting air-conditioning capacity. Considering the dependency between the air temperature and global solar radiation, we used the copula approach to model the joint distributions of those two weather elements and suggest a new method of selecting more credible outdoor design conditions based on the specific simultaneous occurrence probability of air temperature and global solar radiation. In this paper, the 10-year period hourly weather data from 2001 to 2010 in Osaka, Japan, was used to analyze the dependency structure and joint distribution, the result shows that the Joe-Frank copula fit for almost all hourly data. According to calculating the simultaneous occurrence probability and the common exceeding probability of air temperature and global solar radiation, the results have shown that the maximum difference in design air temperature and global solar radiation of the day is about 2 degrees Celsius and 30W/m2, respectively.

Keywords: energy conservation, design weather database, HVAC, copula approach

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Authors: Bruna G. M. Araújo, Pedro M. M. Q. Cruz

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In this letter, we review the literature of the major concepts that govern Gaussian quantum information. As we work with quantum information and computation with continuous variables, Gaussian states are needed to better describe these systems. Analyzing a single ion locked in a Paul trap we use the interaction picture to obtain a toolbox of Gaussian operations with the ion-laser interaction Hamiltionian. This is achieved exciting the ion through the combination of two lasers of distinct frequencies corresponding to different sidebands of the external degrees of freedom. First we study the case of a trap with 1 mode and then the case with 2 modes. In this way, we achieve different continuous variables gates just by changing the external degrees of freedom of the trap and combining the Hamiltonians of blue and red sidebands.

Keywords: Paul trap, ion-laser interaction, Gaussian operations

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Authors: Roengchai Tansuchat, Woraphon Yamaka, Paravee Maneejuk

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From the past recent, the global financial crisis, economic instability, and large fluctuation in agricultural commodity price have led to increased concerns about the volatility transmission among them. The problem is further exacerbated by commodities volatility caused by other commodity price fluctuations, hence the decision on hedging strategy has become both costly and useless. Thus, this paper is conducted to analysis the volatility spillover effect among major agriculture including corn, soybeans, wheat and rice, to help the commodity suppliers hedge their portfolios, and manage the risk and co-volatility of them. We provide a switching regime approach to analyzing the issue of volatility spillovers in different economic conditions, namely upturn and downturn economic. In particular, we investigate relationships and volatility transmissions between these commodities in different economic conditions. We purposed a Copula-based multivariate Markov Switching GARCH model with two regimes that depend on an economic conditions and perform simulation study to check the accuracy of our proposed model. In this study, the correlation term in the cross-hedge ratio is obtained from six copula families – two elliptical copulas (Gaussian and Student-t) and four Archimedean copulas (Clayton, Gumbel, Frank, and Joe). We use one-step maximum likelihood estimation techniques to estimate our models and compare the performance of these copula using Akaike information criterion (AIC) and Bayesian information criteria (BIC). In the application study of agriculture commodities, the weekly data used are conducted from 4 January 2005 to 1 September 2016, covering 612 observations. The empirical results indicate that the volatility spillover effects among cereal futures are different, as response of different economic condition. In addition, the results of hedge effectiveness will also suggest the optimal cross hedge strategies in different economic condition especially upturn and downturn economic.

Keywords: agricultural commodity futures, cereal, cross-hedge, spillover effect, switching regime approach

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Authors: A. Keshavarz, Z. Roosta

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In this paper, propagation of cos-Gaussian beam in strongly nonlocal nonlinear media has been stimulated by using paraxial group transformation. At first, cos-Gaussian beam, nonlocal nonlinear media, critical power, transfer matrix, and paraxial group transformation are introduced. Then, the propagation of the cos-Gaussian beam in strongly nonlocal nonlinear media is simulated. Results show that beam propagation has periodic structure during self-focusing effect in this case. However, this simple method can be used for investigation of propagation of kinds of beams in ABCD optical media.

Keywords: paraxial group transformation, nonlocal nonlinear media, cos-Gaussian beam, ABCD law

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Authors: Cuneyt Yucelbas, Seral Ozsen, Sule Yucelbas, Gulay Tezel

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Due to the fact that there exist only a small number of complex systems in artificial immune system (AIS) that work out nonlinear problems, nonlinear AIS approaches, among the well-known solution techniques, need to be developed. Gaussian function is usually used as similarity estimation in classification problems and pattern recognition. In this study, diagnosis of breast cancer, the second type of the most widespread cancer in women, was performed with different distance calculation functions that euclidean, gaussian and gaussian-euclidean hybrid function in the clonal selection model of classical AIS on Wisconsin Breast Cancer Dataset (WBCD), which was taken from the University of California, Irvine Machine-Learning Repository. We used 3-fold cross validation method to train and test the dataset. According to the results, the maximum test classification accuracy was reported as 97.35% by using of gaussian-euclidean hybrid function for fold-3. Also, mean of test classification accuracies for all of functions were obtained as 94.78%, 94.45% and 95.31% with use of euclidean, gaussian and gaussian-euclidean, respectively. With these results, gaussian-euclidean hybrid function seems to be a potential distance calculation method, and it may be considered as an alternative distance calculation method for hard nonlinear classification problems.

Keywords: artificial immune system, breast cancer diagnosis, Euclidean function, Gaussian function

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Authors: Rabia Khan, Nargis Munir, Suriya Gharib, Syeda Roshana Ali

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In this paper linear equations are discussed in detail along with elimination method. Gaussian elimination and Gauss Jordan schemes are carried out to solve the linear system of equation. This paper comprises of matrix introduction, and the direct methods for linear equations. The goal of this research was to analyze different elimination techniques of linear equations and measure the performance of Gaussian elimination and Gauss Jordan method, in order to find their relative importance and advantage in the field of symbolic and numeric computation. The purpose of this research is to revise an introductory concept of linear equations, matrix theory and forms of Gaussian elimination through which the performance of Gauss Jordan and Gaussian elimination can be measured.

Keywords: direct, indirect, backward stage, forward stage

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Authors: Sandeep Kumar, Naveen Gupta

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The propagation of the Non-Gaussian laser beam results in strong self-focusing as compare to the Gaussian laser beam, which helps to achieve a prerequisite of the plasma-based electron, Terahertz generation, and higher harmonic generations. The theoretical investigation on the evolution of non-Gaussian laser beam through the collisional plasma with ramped density has been presented. The non-uniform irradiance over the cross-section of the laser beam results in redistribution of the carriers that modifies the optical response of the plasma in such a way that the plasma behaves like a converging lens to the laser beam. The formulation is based on finding a semi-analytical solution of the nonlinear Schrodinger wave equation (NLSE) with the help of variational theory. It has been observed that the decentred parameter ‘q’ of laser and wavenumber of ripples of medium contribute to providing the required conditions for the improvement of self-focusing.

Keywords: non-Gaussian beam, collisional plasma, variational theory, self-focusing

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Authors: Lina Pan

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In non-Gaussian clutter of a spherically invariant random vector, in the cases that a certain estimated covariance matrix could become singular, the adaptive target detection of high-range-resolution radar is addressed. Firstly, the restricted maximum likelihood (RML) estimates of unknown covariance matrix and scatterer amplitudes are derived for non-Gaussian clutter. And then the RML estimate of texture is obtained. Finally, a novel detector is devised. It is showed that, without secondary data, the proposed detector outperforms the existing Kelly binary integrator.

Keywords: non-Gaussian clutter, covariance matrix estimation, target detection, maximum likelihood

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Authors: Sie Long Kek, Wah June Leong, Kok Lay Teo

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Linear quadratic Gaussian model is a standard mathematical model for the stochastic optimal control problem. The combination of the linear quadratic estimation and the linear quadratic regulator allows the state estimation and the optimal control policy to be designed separately. This is known as the separation principle. In this paper, an efficient computational method is proposed to solve the linear quadratic Gaussian problem. In our approach, the Hamiltonian function is defined, and the necessary conditions are derived. In addition to this, the output error is defined and the least-square optimization problem is introduced. By determining the first-order necessary condition, the gradient of the sum squares of output error is established. On this point of view, the stochastic approximation approach is employed such that the optimal control policy is updated. Within a given tolerance, the iteration procedure would be stopped and the optimal solution of the linear-quadratic Gaussian problem is obtained. For illustration, an example of the linear-quadratic Gaussian problem is studied. The result shows the efficiency of the approach proposed. In conclusion, the applicability of the approach proposed for solving the linear quadratic Gaussian problem is highly demonstrated.

Keywords: iteration procedure, least squares solution, linear quadratic Gaussian, output error, stochastic approximation

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Authors: M. T. Alodat

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In this paper, we propose a generalization to the Gaussian process regression(GPR) model called the extended skew Gaussian process for regression(ESGPr) model. The ESGPR model works better than the GPR model when the errors are skewed. We derive the predictive distribution for the ESGPR model at a new input. Also we apply the ESGPR model to FOREX data and we find that it fits the Forex data better than the GPR model.

Keywords: extended skew normal distribution, Gaussian process for regression, predictive distribution, ESGPr model

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