Continuous-Time System Monitoring via a Continuous-Time Model Parameter Estimation

This research article primarily focuses on the estimation of parameters of a linear regression model by the method of ordinary least squares and depicts Gauss-Mark off theorem for linear estimation which is useful to find the BLUE of a linear parametric function of the classical linear regression model. A proof of generalized Gauss-Mark off theorem for linear estimation has been presented in this memoir. Ordinary Least Squares (OLS) regression is one of the major techniques applied to analyse data and forms the basics of many other techniques, e.g. ANOVA and generalized linear models [1]. The use of this method can be extended with the use of dummy variable coding to include grouped explanatory variables [2] and data transformation models [3]. OLS regression is particularly powerful as it relatively easy to check the model assumption such as linearity, constant, variance and the effect of outliers using simple graphical methods [4]. J.T. Kilmer et.al [5] applied OLS method to evolutionary and studies of algometry.

Based on a multilevel nonlinear mixed-effect model approach, a stand basal area model was developed for Cunninghamia lanceolata (Lamb.) Hook. plantations belonging to the National Forest Inventory in China. The database consists of 583 plots embracing 18 different blocks within three seed source sites in this study. Of the plots, 80% were randomly selected for model fitting and 20% were carried out for model validation. The modified Chapman–Richards and Schumacher models were evaluated to find a basic model. The explanatory variables included stand dominant height, stand age and total number of stems per hectare. After selection of the basic model, the fitting and predictive ability of a multilevel nonlinear mixed-effect model was analysed. Site-, block- and plot-level random-effects terms were assessed for their contributions to improve model prediction over the ordinary least squares (OLS) method widely used in forest management. In addition, within-plot variance–covariance structure was taken into account owing to the repeated measurements and hierarchical structure of the data set. When evaluating the predictive accuracy of the final model, the first measurement was used for estimation of random parameters. The Chapman–Richards model was finally selected for the basic model based on model-fitting statistics, and both the fitting model and validation data with site-, block- and plot-level random effects showed a substantial improvement compared with the OLS method. After taking into account a reasonable variance–covariance structure, the model performed better than the model developed using only random effects.

Investigation on drought characteristics such as severity, duration, and frequency is crucial for water resources planning and management in a river basin. While the methodology for multivariate drought frequency analysis is well established by applying the copulas, the estimation on the associated parameters by various parameter estimation methods and the effects on the obtained results have not yet been investigated. This research aims at conducting a comparative analysis between the maximum likelihood parametric and non-parametric method of the Kendall \(\tau \) estimation method for copulas parameter estimation. The methods were employed to study joint severity–duration probability and recurrence intervals in Karkheh River basin (southwest Iran) which is facing severe water-deficit problems. Daily streamflow data at three hydrological gauging stations (Tang Sazbon, Huleilan and Polchehr) near the Karkheh dam were used to draw flow duration curves (FDC) of these three stations. The \(Q_{75}\) index extracted from the FDC were set as threshold level to abstract drought characteristics such as drought duration and severity on the basis of the run theory. Drought duration and severity were separately modeled using the univariate probabilistic distributions and gamma–GEV, LN2–exponential, and LN2–gamma were selected as the best paired drought severity–duration inputs for copulas according to the Akaike Information Criteria (AIC), Kolmogorov–Smirnov and chi-square tests. Archimedean Clayton, Frank, and extreme value Gumbel copulas were employed to construct joint cumulative distribution functions (JCDF) of droughts for each station. Frank copula at Tang Sazbon and Gumbel at Huleilan and Polchehr stations were identified as the best copulas based on the performance evaluation criteria including AIC, BIC, log-likelihood and root mean square error (RMSE) values. Based on the RMSE values, nonparametric Kendall-\(\tau \) is preferred to the parametric maximum likelihood estimation method. The results showed greater drought return periods by the parametric ML method in comparison to the nonparametric Kendall \(\tau \) estimation method. The results also showed that stations located in tributaries (Huleilan and Polchehr) have close return periods, while the station along the main river (Tang Sazbon) has the smaller return periods for the drought events with identical drought duration and severity.

In lossless image coding, it has been demonstrated that using the method of ordinary least squares (OLS) to design a linear predictor for each pixel results in better compression performance than that of the state-of-the-art. In previous studies, the order of the predictor is chosen empirically and fixed for the whole image. Since images are nonstationary signals, the order should be adapted to the local characteristics of the image. In this paper, we tackle this problem by using a model averaging approach. We show that by averaging over a group of OLS predictors, the effective number of parameter of the resultant predictor is adjusted adaptively. We show that the proposed method is robust to changes in the size of the training block. It also leads to better performance than the OLS predictor.

The paper presents the estimation of lateral-directional aerodynamic derivatives (parameters) using conventional and neural based methods from real flight data of Hansa-3 aircraft. The conventional methods such as least squares (LS) and maximum likelihood (ML) require an a priori postulation of mathematical model to estimate the parameters. Whereas the neural-based method such as Neural-Gauss-Newton (NGN) is an algorithm that utilises feed forward neural network and Gauss-Newton optimisation to estimate the parameters and does not require any a priori postulation of mathematical model or solution of equations of motion. In the paper, the LS, ML and NGN methods are applied to lateral-directional flight data in order to estimate parameters. The results obtained in terms of lateral-directional aerodynamic derivatives are reasonably accurate to establish LS, ML and NGN as parameter estimation methods along with NGN method having an additional advantage of non-requirement of a priori mathematical model. The paper also highlights the effect of different types of control inputs on parameter estimation. For this, three types of control inputs were used to generate real flight data. The ailerons and rudder were deflected in the first, the ailerons were deflected while keeping rudder at trim condition in the second and the rudder was deflected while keeping ailerons at trim condition in the third type of control input to generate the real flight data. The paper presents the effect of three different types of control inputs in terms of aerodynamic parameters estimated through conventional and neural based methods using flight data generated through these inputs.

There is a widespread assumption that lower corporate taxation will lead to higher rates of job creation, but this belief may be false, as challenged by Anderson and Pizzigati (2017). This article studies the relationship between job creation and 3 tax aggressiveness proxies in Brazilian non-financial firms listed in B3 in the period between 2011 and 2016. The estimates were created using the methods of ordinary least squares (OLS) and quantile regression. The results obtained from the estimates using OLS did not present significant evidence of an association between the rate of job creation and effective tax rates. In the quantile regressions, it was possible to find a significant and positive relationship between job creation and effective tax rates, exclusively in quantile 25 of the effective tax rate on value added. Quantiles 50 and 75 presented negative relationships between job creation and effective tax rates in different tax aggressiveness metrics. The results suggest that the firm’s tax aggressiveness profile influences the relationship. While in the less tax-aggressive companies a reduction of the tax burden would potentially stimulate job creation, in those companies that are more tax-aggressive, the tax break may lead to the undesirable effect of a fall in job creation.

This study analyzes the factors influencing Indonesia’s import value during the period 2021–2025 using the Ordinary Least Squares (OLS) method. To ensure the validity of the model, a series of classical assumption tests were conducted in accordance with the Best Linear Unbiased Estimator (BLUE) criteria, including tests for normality, multicollinearity, heteroscedasticity, and autocorrelation. The data were obtained from official publications of the Central Statistics Agency (BPS) and other relevant sources. The estimation results demonstrate that the independent variables, namely the exchange rate (X₁), national income (X₂), foreign exchange reserves (X₃), inflation rate (X₄), and interest rate (X₅), exert varying effects on Indonesia’s import value, with certain variables exhibiting significant influence while others remain insignificant. The model is free from violations of the classical assumptions, thereby meeting the criteria of the Best Linear Unbiased Estimator (BLUE). Keywords: Import Value, OLS, Classical Assumption Tests, Macroeconomics

Models for biological systems derived from the generalized mass action law are typically a group of nonlinear ordinary differential equations. However, parameters in such models can be separated into two groups: one group of parameters linear in the model and another group of parameters nonlinear in the model. This paper introduces a separable parameter estimation method to estimate the parameters in such models. The separable parameter estimation method has three steps: in the first step, parameters linear in a model are estimated by optimizing the objective function using linear least squares method, assuming all parameters nonlinear in the model are known. In the second step, substituting the estimated parameters in the first step into the objective function yields a new objective function which is only of parameters nonlinear in the model. Then parameters nonlinear in the model are estimated by proper nonlinear estimation methods. In the last step, the estimates of parameters linear in the model are calculated using the estimates of parameters in the second step. To investigate its performance, the separable estimation method is applied to a biological system and is compared with the conventional parameter estimation methods. Simulation results show the improvement of the separable estimation method.

The Extreme Learning Machine (ELM), recently proposed by Huang et al. [6], is a single-hidden-layered neural network architecture which has been successfully applied to nonlinear regression and classification tasks [5]. A crucial step in the design of the ELM is the computation of the output weight matrix, a step usually performed by means of the ordinary least-squares (OLS) method - a.k.a. Moore-Penrose generalized inverse technique. However, it is well-known that the OLS method produces predictive models which are highly sensitive to outliers in the data. In this paper, we develop an extension of ELM which is robust to outliers caused by labelling errors. To deal with this problem, we suggest the use of M-estimators, a parameter estimation framework widely used in robust regression, to compute the output weight matrix, instead of using the standard OLS solution. The proposed model is robust to label noise not only near the class boundaries, but also far from the class boundaries which can result from mistakes in labelling or gross errors in measuring the input features. We show the usefulness of the proposed classification approach through simulation results using synthetic and real-world data.

The Pareto distribution has been introduced in Chapter 19. Also discussed there is a brief review of literature and methods of estimating its parameters. Further elaboration of the distribution is given in Chapter 20. Methods of parameter estimation were reviewed by Hosking and Wallis (1987). The methods of moments (MOM), maximum likelihood estimation (MLE) and probability weighted moments (PWM) were included in the review. Guo and Singh (1992) and Singh and Guo (1995) employed the principle of maximum entropy (POME) to develop a new competitive method of parameter estimation (Singh and Rajagopal, 1986) for the 3-parameter generalized Pareto (GP3) distribution and compared it with MOM, MLE and PWM using Monte Carlo simulated data The parameter estimates yielded by POME were either superior or comparable for high skewness.KeywordsRoot Mean Square ErrorMaximum Likelihood EstimationPareto DistributionGeneralize Pareto DistributionQuantile EstimationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Many existing software reliability models are based on the assumption of statistical independence among successive software failures. In reality, this assumption could be easily violated. In recent years, efforts have been made to relax this unrealistic assumption and a software reliability modeling framework considering failure correlation was developed by Goseva-Popstojanova and Trivedi. However, some important issues that are crucial to the proposed modeling framework to be used in practice remain unstudied, such as the method of estimation of model parameters. In this paper, we study the parameter estimation problem for the software reliability modeling framework developed in. We propose a relationship function among model parameters which could be essential to the reduction of the number of parameters to be estimated as well as to the reliability prediction using the proposed modeling framework. Two parameter estimation methods are developed based on deferent types of data available, using Maximum Likelihood Estimation (MLE) method. Simulation results preliminarily show that the accuracy of both proposed estimation methods seem to be satisfactory.

BackgroundParameter estimation is often the bottlenecking step in biological system modeling. For ordinary differential equation (ODE) models, the challenge in this estimation has been attributed to not only the lack of parameter identifiability, but also computational issues such as finding globally optimal parameter estimates over highly multidimensional search space. Recent methods using incremental estimation approach could alleviate the computational difficulty by performing the parameter estimation one-reaction-at-a-time. However, incremental estimation strategies usually require data smoothing and are known to produce biased parameter estimates.ResultsIn this article, we presented a new parameter estimation method called integrated flux parameter estimation (IFPE). We employed the integral form of the ODE such that we could compute the integral of reaction fluxes from time-series concentration data without data smoothing. Here, we formulated the parameter estimation as a nested optimization problem. In the outer optimization, we performed a minimization of model prediction errors over parameters associated with a subset of reactions labeled as independent. The dimension of the independent reaction subset was equal to the degrees of freedom in the calculation of integrated fluxes (IF) from concentration data. We selected the independent reactions such that given their IF values, the IFs of the remaining (dependent) reactions could be uniquely determined. Meanwhile, in the inner optimization, we estimated the model parameters associated with the dependent reactions, one-reaction-at-a-time, by minimizing the dependent IF prediction errors. We demonstrated the performance of the IFPE method using two case studies: a generalized mass action model of a branched pathway and a lin-log ODE model of Lactococcus lactis glycolytic pathway.ConclusionsThe IFPE significantly outperformed standard simultaneous parameter estimation in terms of computational efficiency and scaling. In comparison to incremental parameter estimation (IPE) method, the IFPE produced parameter estimates with significantly lower bias and did not require time-series data smoothing. The advantages of IFPE over the IPE however came at the cost of a small increase in the computational time.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-014-0127-x) contains supplementary material, which is available to authorized users.

Ordinary Least Squares (OLS) is one method of parameter estimation in regression analysis. However, the presence of outliers can cause estimation of regression coefficients obtained are not exact. Act of throwing away an outlier is not a wise move, because sometimes outliers provide significant information. Therefore, robust regression methods are needed to data contain outliers. This paper will use robust regression estimation method by M-estimation. This estimation use Iteratively Reweighted Least Squares (IRLS) method with weighting function by Huber and Tukey Bisquare. IRLS is applied to the case of food security in Central Java in 2007 that is influenced by the stock of rice, harvested area, average production, price of rice and the amount of consumption. The purpose of this writing is to compare goodness of M-estimation IRLS using Huber and Tukey Bisquare function in estimating the model parameters of food security in Central Java in 2007. Based on the research results can be concluded that the M-estimation by the Tukey Bisquare is better recommended than Huber function. This can be seen by value results of Mean Square Error and determination coefficient

In this paper we presented the new algorithm for astrometric reduction of the images received from modern large telescopes with very wide field of view. This algorithm is based on the iterative using of the method of ordinary least squares (OLS) and statistical Student t-criterion. The paper contains information about selecting the appropriate reduction model and system of conditional equations for determination of the reduction model parameters with the aim to solve it using the OLS method. The proposed algorithm also provides the automatic selection of the most probabilistic reduction model. At the first iteration the fifth degree polynomial is used, which provides 21 constant plates. The reduction error from the each iteration is used in the next iteration to eliminate reference objects whose residuals more than three sigma. In the developed algorithm the statistical Student t-criterion of reliability is applied after several iterations of getting rid of the noisy objects. This approach allows us to eliminate almost all systematic errors that are caused by imperfections of the optical system of modern large telescopes. The developed software based on the proposed algorithm was applied to perform the astrometric reduction of all measured positions of objects on the digitized photographic plates of SuperCOSMOS data. The research results showed that the new proposed algorithm allows performing the reduction into the system of reference catalogue with the highest accuracy level.

The method of ordinary least squares (OLS) and generalizations of it have been the mainstay of most forecasting methodologies for many years. It is well‐known, however, that outliers or unusual values can have a large influence on least‐squares estimators. Users of automatic forecasting packages, in particular, need to be aware of the influence that outlying data values can have on statistical analyses and forecasting results. Robust methods are available to modify least‐squares procedures so that outliers have much less influence on the final estimates; yet these formal methods have not found their way into general forecasting procedures. This paper provides a case study in which classical least‐square‐estimation procedures are complemented with a robust alternative to enhance statistical fit criteria and improve forecasting performance. The study suggests that much can be gained in understanding the nature of outliers and their influence on forecasting performance by performing a robust regression in addition to OLS.