Inequality Constrained Least Squares Research Articles

Negative VTEC with spherical harmonic expansion model under different solar activity conditions

The Earth’s ionosphere can be described by a spherical harmonic (SH) expansion up to a specific degree. However, there exist negative vertical total electron content (VTEC) values in the global ionosphere map (GIM) with the SH expansion model. In this contribution, we specifically investigated the negative VTEC values that are induced by the SH expansion model and validated the performance of the inequality-constrained least squares (ICLS) method in eliminating the negative VTEC values. The GPS data from 2004 to 2017 was selected to cover one solar cycle and the experiments under different solar activity conditions were analyzed. The results in our work show that the occurrence of the negative VTEC values is attributed to the deficiency of the SH expansion model when the VTEC itself is small instead of the unevenly distribution of the GNSS stations. The negative VTEC values appear periodically in the temporal domain, showing apparently one year and half year periods. During one year, two peaks in June and December can be observed in the time series of the negative VTEC values. The number of negative VTEC values in June is obvious larger than that in December. During one solar cycle, the number of negative VTEC values under quiet solar activity condition is obvious larger than that under strong solar activity condition. In the spatial domain, the appearance of the negative VTEC values is strongly related with the movement of the subsolar point. In the latitude of the subsolar point has the largest magnitude, the negative values will appear on the opposite hemisphere and the further from the subsolar point the more negative values. The maximum number of the negative VTEC values in the southern hemisphere appears in June, while the peak value in the northern hemisphere appears in December. The maximum number of negative VTEC values in the southern hemisphere is generally larger than that in the northern hemisphere. In addition, the negative VTEC values are distributed both at middle latitude and high latitude in the southern hemisphere, while they are mainly distributed at high latitude in the northern hemisphere. When the ICLS method is used, the negative VTEC values can be eliminated efficiently and it has nearly no influence on the positive VTEC values. The ICLS method can also improve the receiver’s differential code bias (DCB) and significantly decrease the unreasonable negative slant TEC (STEC) values along the lines of sight. Using the final GIM product of the Jet Propulsion Laboratory (JPLG) as a reference, the root mean square (RMS) of the ICLS solution shows maximum 25%, 20% and 45% improvement relative to the least squares (LS) solution at northern high latitude, southern middle latitude and southern high latitude, respectively.

EM algorithm for one-shot device testing with competing risks under exponential distribution

This paper provides an extension of the work of Balakrishnan and Ling [1] by introducing a competing risks model into a one-shot device testing analysis under an accelerated life test setting. An Expectation Maximization (EM) algorithm is then developed for the estimation of the model parameters. An extensive Monte Carlo simulation study is carried out to assess the performance of the EM algorithm and then compare the obtained results with the initial estimates obtained by the Inequality Constrained Least Squares (ICLS) method of estimation. Finally, we apply the EM algorithm to a clinical data, ED01, to illustrate the method of inference developed here.

Ordinal Neural Networks Without Iterative Tuning

Ordinal regression (OR) is an important branch of supervised learning in between the multiclass classification and regression. In this paper, the traditional classification scheme of neural network is adapted to learn ordinal ranks. The model proposed imposes monotonicity constraints on the weights connecting the hidden layer with the output layer. To do so, the weights are transcribed using padding variables. This reformulation leads to the so-called inequality constrained least squares (ICLS) problem. Its numerical solution can be obtained by several iterative methods, for example, trust region or line search algorithms. In this proposal, the optimum is determined analytically according to the closed-form solution of the ICLS problem estimated from the Karush-Kuhn-Tucker conditions. Furthermore, following the guidelines of the extreme learning machine framework, the weights connecting the input and the hidden layers are randomly generated, so the final model estimates all its parameters without iterative tuning. The model proposed achieves competitive performance compared with the state-of-the-art neural networks methods for OR.

Inequality constrained ridge regression estimator

We carry out the idea of inequality constrained least squares (ICLS) estimation of Liew (1976) to the inequality constrained ridge regression (ICRR) estimation. We propose ICRR estimator by reducing the primal–dual relation to the fundamental problem of Dantzig–Cottle (1967,1974) with Lemke (1962) algorithm. Furthermore, we conduct a Monte Carlo experiment.

Non-linear panel estimation of import quotas: The evolution of quota premiums under the ATC

We develop a panel-based inequality constrained least-squares (ICLS) framework for estimating the export tax equivalent (ETE) of quotas where the ETEs vary over time. Working with a panel of bilateral data on textile and clothing trade,underlying bilateral tariffs, and the country-pair coverage of quotas under the WTO's Agreement on Textiles and Clothing (ATC), we use this framework to examine the evolution of market access conditions in the textile and clothing sectors. Our estimating framework takes advantage of the panel nature of trade data when calculating export tax equivalents while allowing for inequality constraints on the quota premium estimates. We also implement quadrature methods for calculating confidence intervals for our regression-based NTB measures.

An Aggregate Constraint Method for Inequality-constrained Least Squares Problems

The inequality-constrained least squares (ICLS) problem can be solved by the simplex algorithm of quadratic programming. The ICLS problem may also be reformulated as a Bayesian problem and solved by using the Bayesian principle. This paper proposes using the aggregate constraint method of non-linear programming to solve the ICLS problem by converting many inequality constraints into one equality constraint, which is a basic augmented Lagrangean algorithm for deriving the solution to equality-constrained non-linear programming problems. Since the new approach finds the active constraints, we can derive the approximate algorithm-dependent statistical properties of the solution. As a result, some conclusions about the superiority of the estimator can be approximately made. Two simulated examples are given to show how to compute the approximate statistical properties and to show that the reasonable inequality constraints can improve the results of geodetic network with an ill-conditioned normal matrix.

A closed form solution for tre least squares regression problem with linear inequality constraints

In this paper we consider a linear model Y = Xβ+e with linear inequality constraints R'β≥r, where X and R are known and full column rank matrices. The closed form of the inequality constrained least squares (ICLS) estimator is given. We provide two examples which illustrate the use of this closed form in the computation of estimates.

Inequality Constraints, Multicollinearity and Models of Police Expenditure

Inequality constraints have been used very infrequently in the analysis of linear statistical models. This is not due to an inability to construct estimators. Rather, it is a result of an inability to say very much about the properties of the inequality constrained estimator. Hanson [15] proved the existence of inequality constrained maximum likelihood estimators. A year later Judge and Takayama [17] demonstrated the value of quadratic programming for obtaining estimates of the parameters in a model constrained by inequalities. More recently, Liew [20; 21] and Klemm and Sposito [19] have suggested closed form estimates. The work by Liew makes some reference to the properties of the inequality constrained least squares (ICLS) estimator. A more formal effort to determine the properties of the ICLS estimator was made by Lovell and Prescott [23]. Their paper considered only the case of one location parameter being restricted to either the positive or negative half of the real line. In a more recent paper Wardle [32] has considered a Bayesian approach to the problem of determining the small sample properties of the ICLS estimator. Our more classical results are a confirmation of his work.

A Computer Program for Inequality Constrained Least-Squares Estimation

A Computer Program for Inequality Constrained Least-Squares Estimation

Inequality Constrained Least-Squares Estimation

Abstract There are growing demands to use prior and sample information for parameter estimation of a regression model in order to maintain consistency with underlying theory. To meet such demands, this paper provides an inequality constrained least-squares (ICLS) estimation, specifies an untruncated variance-covariance matrix of the ICLS estimates, and discusses their statistical properties in large-and small-sample cases. Finally, the ICLS and the ordinary least-squares OLS estimates are compared in terms of sample bias, sample mean-square error MSE and sample variance of the estimates by a Monte Carlo study.

Inequality Constrained Least-Squares Estimation

Abstract There are growing demands to use prior and sample information for parameter estimation of a regression model in order to maintain consistency with underlying theory. To meet such demands, this paper provides an inequality constrained least-squares (ICLS) estimation, specifies an untruncated variance-covariance matrix of the ICLS estimates, and discusses their statistical properties in large-and small-sample cases. Finally, the ICLS and the ordinary least-squares OLS estimates are compared in terms of sample bias, sample mean-square error MSE and sample variance of the estimates by a Monte Carlo study.